The Pennsylvania State University. The Graduate School. College of Engineering PROPAGATION AND EXCITATION OF MULTIPLE SURFACE WAVES

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1 The Pennsylvania State University The Graduate School College of Engineering PROPAGATION AND EXCITATION OF MULTIPLE SURFACE WAVES A Dissertation in Engineering Science and Mechanics by Muhammad Faryad c 2012 Muhammad Faryad Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2012

2 The dissertation of Muhammad Faryad was reviewed and approved by the following: Akhlesh Lakhtakia Charles Godfrey Binder Professor of Engineering Science and Mechanics Dissertation Adviser Chair of Committee Michael T. Lanagan Professor of Engineering Science and Mechanics Associate Director Materials Research Institute Osama O. Awadelkarim Professor of Engineering Science and Mechanics Jainendra K. Jain Erwin W. Mueller Professor of Physics Judith A. Todd P. B. Breneman Professor of Engineering Science and Mechanics Head of the Department of Engineering Science and Mechanics Signatures are on file in the Graduate School. ii

3 Abstract Surface waves are the solutions of the frequency-domain Maxwell equations at the planar interface of two dissimilar materials. The time-averaged Poynting vector of a surface wave (i) has a significant component parallel to the interface and (ii) decays at sufficiently large distances normal to the interface. If one of the partnering materials is a metal and the other a dielectric, the surface waves are called surface plasmon-polariton (SPP) waves. If both partnering materials are dielectric, with at least one being periodically nonhomogeneous normal to the interface, the surface waves are called Tamm waves; and if that dielectric material is also anisotropic, the surface waves are called Dyakonov Tamm waves. SPP waves also decays along the direction of propagation, whereas Tamm and Dyakonov Tamm waves propagate with negligible losses. The propagation and excitation of multiple SPP waves guided by the interface of a metal with a periodically nonhomogeneous sculptured nematic thin film (SNTF), and the interface of a metal with a rugate filter were theoretically investigated. The SNTF is an anisotropic material with a permittivity dyadic that is periodically nonhomogeneous in the thickness direction. A rugate filter is also a periodically nonhomogeneous dielectric material; however, it is an isotropic material. Multiple SPP waves of the same frequency but with different polarization states, phase speeds, attenuation rates, and spatial field profiles were found to be guided by a metal/sntf interface, a metal/rugate-filter interface, and a metal slab in the SNTF. Multiple Dyakonov Tamm waves of the same frequency but different polarization states, phase speeds, and spatial field profiles were found to be guided by a structural defect in an SNTF, and by a dielectric slab in an SNTF. The characteristics of multiple SPP and Dyakonov Tamm waves were established by the investigations on canonical boundary-value problems. The Turbadar-Kretschmann-Raether (TKR) and the grating-coupled configurations were used to study the excitation of multiple SPP waves. In the TKR configuration, which is easy to implement in a laboratory, a plane wave of either of the two linear polarization states was made incident on the metal-capped rugate filter of finite thickness and the absorptances were calculated using a numerically stable algorithm. In the grating-coupled configuration, which is required for solar cell applications, a plane wave of either polarization state was made incident on iii

4 a rugate filter or an SNTF backed by a finitely thick metallic surface-relief grating and the total absorptance of the structure was calculated using the rigorous coupled-wave approach. In both the configurations, the excitation of SPP waves was inferred by the presence of those peaks in the absorptance curves that were independent of the thickness of the dielectric material. It was found that (i) it is the periodic nonhomogeneity (not the anisotropy) of a partnering dielectric material normal to the interface that is responsible for the multiplicity of surface waves; (ii) multiple SPP, Tamm, Dyakonov Tamm, and Fano waves of the same frequency and different phase speeds and spatial profiles can be guided by an interface of two different materials provided that at least one of them is periodically nonhomogeneous normal to the interface; (iii) the morphology of the partnering dielectric material affects the number, the phase speeds, the spatial profiles, and the degrees of localization of the surface waves; (iv) the number of surface waves can be increased further by the coupling of two interfaces separated by a sufficiently thin layer; and (v) multiple surface waves can be excited in the TKR and the grating-coupled configurations both with the isotropic and anisotropic but periodically nonhomogeneous dielectric materials. iv

5 Nontechnical Abstract The type of electromagnetic waves that propagate along the interface of two dissimilar materials are called surface waves. These waves are important for many applications because most of their energy is localized close to the interface, and the characteristics of these waves depend heavily on the properties of the partnering materials close to the interface. The localization of the energy of surface waves close to the interface can be used in sensing applications and harvesting solar energy. In sensing applications, an unknown chemical infiltrates one of the partnering materials, thereby changing the characteristics of propagating surface waves. This change in the characteristics of surface waves can be used to sense the properties of infiltrating chemical. In thin-film solar cells, the excitation of surface waves can increase the absorption of solar energy because a part of the energy of the light that is incident on the solar cell can be be used to launch surface waves instead of being wasted by reflection. If one of the two dissimilar materials is a metal and the other a dielectric material the surface waves are called surface plasmon-polariton (SPP) waves. Among various types of surface waves, SPP waves are the most extensively studied because it is easy to excite and use them in many optical applications, especially sensing. However, SPP waves can propagate only for short distances along the interface before their energy is converted to thermal energy. Moreover, for a given color (wavelength) of the incident light, only one SPP wave can be excited if both the partnering materials are homogeneous. Other types of surface waves include Fano waves, Tamm waves and Dyakonov Tamm waves that propagate guided by the interface of two dielectric materials. These waves can propagate much longer distances along the interface than SPP waves; however, it is much more difficult to excite them than SPP waves. The purpose of this thesis was to theoretically investigate the propagation and excitation of multiple surface waves of the same color but different polarization states, phase speeds, and spatial profiles. It was found that more than one surface wave of the same color can be guided by the interface of two materials if the partnering dielectric material is made to have periodically changing dielectric properties along a direction normal to the interface. This holds true for SPP waves, Tamm waves, Dyakonov Tamm waves, and Fano waves. Multiple SPP waves are guided by the interface of a metal and a periodically nonhomogeneous sculptured v

6 nematic thin film (SNTF) and the interface of the metal and a rugate filter. Both the SNTF and the rugate filter are periodically nonhomogeneous dielectric materials; however, an SNTF is an anisotropic and porous material, whereas a rugate filter is an isotropic material. The porosity of the SNTF can be used in sensing applications and the isotropy of the rugate filter makes possible the application of multiple SPP waves in solar cells because rugate filters can be fabricated with semiconductor materials. The number of multiple surface waves can be increased further if two interfaces are placed in close proximity. The coupling of the two interfaces leads to the emergence of new surface waves that are not guided by either interface independently. Since surface waves propagate with a different phase speed than the waves in the bulk of either of the partnering materials, the excitation of the surface waves requires a technique to match the phase speed of the incident light to that of the possible surface wave. For this purpose, attenuated total reflection (ATR) and scattering by a surface-relief grating, among others, are used. In this thesis, it is shown that multiple SPP waves can also be excited using both the ATR and the surface-relief gratings for metal/sntf and metal/rugate-filter interfaces. Availability of multiple surface waves, all of the same color, offers exciting possibilities in applications. For sensing applications, more than one surface waves can be used to detect more than one chemicals at the same time. For solar energy harvesting, the availability of multiple SPP waves can increase the absorption of light as compared to the case when only one SPP wave can be excited. vi

7 Contents List of Acronyms List of Symbols List of Figures List of Tables Acknowledgments xi xii xv xxvii xxx 1 Introduction Surface Plasmon-Polariton Waves Dyakonov Tamm Waves Sculptured Nematic Thin Films Rugate Filters Excitation of Surface Waves Prism-Coupled Configuration Grating-Coupled Configuration Waveguide-Coupled Configuration Objectives of the Thesis Organization of the Thesis SPP Waves Guided by Metal/SNTF Interface Introduction Theory Numerical Results and Discussion ψ = ψ = Concluding Remarks SPP Waves Guided by Metal/Rugate-Filter Interface Introduction Theory vii

8 3.3 Numerical Results and Discussion Concluding Remarks Propagation of Multiple Fano Waves Introduction Numerical Results and Discussion Concluding Remarks Dyakonov Tamm Waves Guided by a Phase-Twist Defect in an SNTF Introduction Theory Numerical Results and Discussion Multiple solutions of dispersion equation Decay constants Spatial profiles Concluding Remarks SPP Waves Guided by a Metal Slab in an SNTF Introduction Canonical Boundary-Value Problem Numerical Results and Discussion Bulk aluminum defect layer Electron-beam evaporated aluminum thin film Concluding Remarks Guided-Wave Propagation by a Dielectric Slab in an SNTF Introduction Numerical Results and Discussion Dyakonov Tamm waves guided by a single dielectric/sntf interface SNTF/dielectric/SNTF system Comparison with SNTF/metal/SNTF system of Ch Concluding Remarks Prism-Coupled Excitation of Multiple SPP Waves Introduction Theoretical Formulations TKR configuration Canonical-boundary value problem for coupled-spp-wave propagation Numerical Results and Discussion p-polarization state viii

9 8.3.2 s-polarization state Concluding Remarks Grating-Coupled Excitation of Multiple SPP Waves Guided by Metal/Rugate-Filter Interface Introduction Boundary-Value Problem Description Coupled ordinary differential equations Solution algorithm Numerical Results and Discussion Homogeneous dielectric partnering material Periodically nonhomogeneous dielectric partnering material Concluding Remarks Enhanced Absorption of Light Due to Multiple SPP Waves Introduction Numerical Results and Discussion Homogeneous semiconductor partnering material Periodically nonhomogeneous semiconductor partnering material Concluding Remarks Grating-Coupled Excitation of Multiple SPP Waves Guided by Metal/SNTF Interface Introduction Boundary-Value Problem Description Coupled ordinary differential equations Solution algorithm Absorptance Numerical Results and Discussion γ = γ = Comparison with the TKR configuration Concluding Remarks Conclusions and Suggestions for Future Work Conclusions Suggestions for Future Work Excitation of multiple surface waves with a finite source Simultaneous excitation of all possible SPP waves using quasiperiodic surface-relief grating ix

10 Excitation of Tamm and Dyakonov Tamm waves Thin-film solar cell with actual configuration A Propagation of Multiple Tamm Waves 171 A.1 Introduction A.2 Theory A.2.1 s-polarized surface waves A.2.2 p-polarized surface waves A.3 Numerical Results and Discussion A.3.1 Homogeneous-dielectric/rugate-filter interface A.3.2 Rugate filter with a phase defect A.3.3 Rugate filter with sudden change of mean refractive index. 182 A.3.4 Rugate filter with sudden change of amplitude A.3.5 Interface of two distinct rugate filters A.4 Concluding Remarks B Mathematica TM Codes 188 B.1 Newton-Raphson Method to Find κ in the Canonical Boundary- Value Problem of Ch B.2 Plotting the Components of P of a p-polarized SPP Wave in the Canonical Boundary-Value Problem of Ch B.3 Newton-Raphson Method to Find κ in the Canonical Boundary- Value Problem of Ch B.4 Newton-Raphson Method to Find κ in the Canonical Boundary- Value Problem of Chs. 6 and B.5 A p vs. θ in the TKR Configuration of Ch B.6 A p vs. θ in the Grating-Coupled Configuration of Chs. 9 and B.7 A p and A s vs. θ in the Grating-Coupled Configuration of Ch Bibliography 223 x

11 List of Acronyms ATR CTF CSTF deg Im PV PVD RCWA Re SNTF SPP STF TKR TO attenuated total reflection columnar thin film chiral sculptured thin film degrees imaginary part photovoltaic physical vapor deposition rigorous coupled-wave approach real part sculptured nematic thin film surface plasmon-polariton sculptured thin film Turbadar-Kretschmann-Raether Turbadar-Otto xi

12 List of Symbols A a p, a s A p, A s α met α n α ± n d 1 d 2 d 3 met planewave absorptance scalar amplitudes representing p- and s-polarized waves absorptances for p- and s-polarized incidence wavenumber of SPP wave in the metal normal to the direction of propagation nth eigenvalue corresponding to nth eigenvector [t] (n) of 4 4 matrix [ Q] nth eigenvalue corresponding to nth eigenvector [t ± ] (n) of 4 4 matrix [ Q ± ] thickness of the dielectric layer in the grating-coupled configuration combined thickness of the dielectric layer and the grating depth (d 2 = d 1 + L g ) total thickness of the structure in the grating-coupled configuration (d 3 = d 2 + L m ) e-folding distance into the dielectric material skin depth of the metal ± n amplitude of refractive-index modulation of rugate filter for z 0 e E exp( u 1,2 ) exp( v 1,2 ) ϵ 0 ϵ a, ϵ b, ϵ c ϵ l ϵ r, ϵ d auxiliary electric field phasor electric field phasor decay constants of Dyakonov Tamm wave when z decay constants of Dyakonov Tamm wave when z permittivity of free space relative permittivity scalars relative permittivity of the prism material in the TKR configuration relative permittivity of dielectric material xii

13 ϵ m ϵ (n) ϵ SNT F η 0 [f] γ γ ± h H k met k (n) x κ k 0 L L 1 L g L m L met L s λ 0 n a n b relative permittivity of metal nth coefficient of Fourier series of permittivity ϵ(x) permittivity dyadic of the SNTF intrinsic impedance of free space column vector containing x- and y-components of e and h fraction of the amplitude of sinusoidal variation in refractive index of a rugate filter the angle between the morphologically significant plane of an SNTF and the x-axis in the region z 0 auxiliary magnetic field phasor magnetic field phasor wavevector of the SPP wave in the metal x-component of nth Floquet harmonic wavenumber of a surface wave in the canonical problem along the direction of propagation wavenumber in free space period of surface-relief grating width of the bump in the surface-relief grating depth of the surface-relief grating thickness of the metal film in the TKR and grating-coupled configurations thickness of the metal slab in the canonical problem thickness of dielectric slab wavelength in free space lowest value of refractive index in a rugate filter highest value of refractive index in a rugate filter n ± avg mean refractive index of rugate filter for z 0 n l N d N g N p refractive index of the prism material in the TKR configuration number of slices in the dielectric material number of slices in the grating region number of period of the rugate filter in the TKR configuration ±N t ending and starting indexes in summations in RCWA xiii

14 [P ] coefficient matrix of matrix ordinary differential equation [P ± ] matrix [P ] in the region z 0 P P 1 P 2 P x,y,z [Q] time-averaged Poynting vector the component of P along the direction of propagation of an SPP wave the component of P in the interface plane and normal to the direction of propagation of an SPP wave x-, y- and z-components of P optical response of one period of an SNTF [Q ± ] matrix [Q] in the region z 0 [ Q] auxiliary matrix defined by [Q] = exp{i2ω[ Q]} r p, r s ψ t p, t s û x, û y, û y reflection amplitudes of p- and s-polarized waves angle between the direction of propagation of a surface wave and the morphologically significant plane of an SNTF transmission amplitudes of p- and s-polarized waves unit vectors along x-, y- and z- axis µ 0 permeability of free space θ ϕ incidence angle with the z-axis incidence angle with the x-axis in the xy plane ϕ ± phase shift in the SNTF or rugate filter in the region z 0 ω Ω angular frequency half-period of an SNTF or a rugate filter Ω ± half-period of an SNTF or a rugate filter in the region z 0 χ v δ v χ v vapor incidence angle amplitude of periodic variation of incident vapor the tilt of columns in an STF xiv

15 List of Figures 1.1 Schematic for the TKR configuration Schematic for the TO configuration Schematic for the grating-coupled configuration Schematic for the waveguide-coupled configuration A flow diagram showing the interconnections among different chapters of this thesis. The boxes with blue light background represent the chapters containing the canonical boundary-value problems, and the boxes with purple dark background represent the chapters that contain the boundary-value problems for the excitation of multiple surface waves. The boxes with white background do not contain any of the boundary-value problems (left) Real and (right) imaginary parts of κ as functions of ψ, for SPP-wave propagation guided by the planar interface of aluminum and a titanium-oxide SNTF. Either two or three modes are possible, depending on ψ Variations of components of e (in V m 1 ), h (in A m 1 ), and P (in W m 2 ) with z along the line {x = 0, y = 0}, for κ = ( i )k 0 and ψ = 0. The components parallel to û 1, û 2, and û z, are represented by black solid, red dashed, and blue chaindashed lines, respectively. The data were computed by setting a p = 1 V m 1, with a s = 0, b 1 = 0, and b 2 = i then obtained using (2.15) Same as Fig. 2.2 except for κ = ( i )k 0. The data were computed by setting a s = 1 V m 1, with a p = 0, b 1 = 1, and b 2 = 0 then obtained using (2.15). Theoretical analysis confirms that û 1 P > 0 for z < 0 for this case Same as Fig. 2.2 except for κ = ( i )k 0. The data were computed by setting a p = 1 V m 1, with a s = 0, b 1 = 0, and b 2 = i then obtained using (2.15) xv

16 2.5 Same as Fig. 2.2 except for κ = ( i )k 0 and ψ = 75. The data were computed by setting a p = 1 V m 1, with a s = i V m 1, b 1 = i5.3299, and b 2 = i then obtained using (2.15) Same as Fig. 2.2 except for κ = ( i )k 0 and ψ = 75. The data were computed by setting a p = 1 V m 1, with a s = i V m 1, b 1 = i , and b 2 = i then obtained using (2.15) (left) Real and (right) imaginary parts of κ/k 0 as functions of Ω/λ 0 for SPP-wave propagation guided by the planar interface of aluminum and a rugate filter described by Eq. (3.1) with n a = 1.45 and n b = Variations with z of the Cartesian components of e (in V m 1 ), h (in A m 1 ), and P (in W m 2 ) along the line {x = 0, y = 0}. The components parallel to û x, û y, and û z, are represented by red solid, blue dashed, and black chain-dashed lines, respectively. The data were computed by setting a p = 1 V m 1. (left) Ω/λ 0 = 0.1, κ/k 0 = i, and (right) Ω/λ 0 = 1, κ/k 0 = i. Both solutions lie on the branch labeled p8 in Fig Same as Fig. 3.2 except for (left) Ω/λ 0 = 1, κ/k 0 = i, and (right) Ω/λ 0 = 1.5, κ/k 0 = i, and the data were computed by setting a s = 1 V m 1. Both solutions lie on the branch labeled s2 in Fig (left) Real and (right) imaginary parts of κ/k 0 as functions γ [1, 0.001] with Ω = 2λ 0 for SPP-wave propagation guided by the planar interface of aluminum and a rugate filter described by Eq. (3.17) with n a = 1.45, n b = 2.32, and Ω = 2λ Same as Fig. 3.2 except for (left) γ = 0.5 and κ/k 0 = i on the branch labeled p3 in Fig. 3.4, and (right) γ = 0.1 and κ/k 0 = i on on the branch labeled p10 in Fig Variation of relative permittivity along the z axis for n a = 1.45, n b = 2.32, and ϵ m = 2. Although the semi-infinite rugate filter depicted here is a continuously nonhomogeneous medium, it can also be piecewise homogeneous Relative wavenuber κ/k 0 versus ϵ m [ 6, 0] for Fano-wave propagation when Ω = λ 0 = 633 nm, n a = 1.45, and n b = The red circles represent s-polarized, while the black triangles represent p-polarized, Fano waves. The gap in one of the solution branches appears to be a numerical artifact xvi

17 4.3 Variations of the magnitudes of the Cartesian components of electric and magnetic field phasors (in V m 1 and A m 1, respectively) with z. The x-, y-, and z-directed components are represented by solid red, blue dashed, and black chain-dashed lines, respectively for ϵ m = 6. Left: κ/k 0 = and p-polarization state. Right: κ/k 0 = and s-polarization state Same as Fig. 4.3 except for ϵ m = 0. Left: κ/k 0 = and p-polarization state. Right: κ/k 0 = and s-polarization state Same as Fig. 4.2, except that ϵ m [0, 2]. The waves represented by these solutions have to be classified as Tamm waves [16] Schematic illustration of the geometry of the problem, when γ + = γ The solutions κ/k 0 of the dispersion equation (5.18) as functions of γ + for certain specific values of γ. (a) First, (b) second, (c) third, and (d) fourth sets of solutions The decay constants exp( u 1 ), exp( u 2 ), exp( v 1 ), and exp( v 2 ) for the (a) first, (b) second, (c) third, and (d) fourth set of solutions in Fig Variations with z of the magnitudes of the Cartesian components of E (in V m 1 ), H (in A m 1 ), and P (in W m 2 ), when γ = 60, γ + = 30, and κ/k 0 = The components parallel to û x, û y, and û z, are represented by red solid, blue dashed, and black chain-dashed lines, respectively Same as Fig. 5.4 except that κ/k 0 = Same as Fig. 5.4 except that κ/k 0 = Same as Fig. 5.4 except that κ/k 0 = Schematic illustration of the geometry of the canonical boundaryvalue problem for γ + = γ Variation of real and imaginary parts of κ/k 0 with γ +, when γ = γ +. (a) L ± = ±7.5 nm, (b) L ± = ±12.5 nm, (c) L ± = ±25 nm, and (d) L ± = ±45 nm Variation of the Cartesian components of the time-averaged Poynting vector P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±7.5 nm, and γ = γ +. (a-c) γ + = 0, and (d-f) γ + = 25. (a) κ/k 0 = i0.1839, (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i0.1848, (e) κ/k 0 = i , and (f) κ/k 0 = i The x-, y- and z-directed components of P(x, z) are represented by solid red, dashed blue, and chain-dashed black lines, respectively xvii

18 6.4 Same as Fig. 6.3 except for L ± = ±45 nm. (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i Same as Fig. 6.2 except that γ = γ Variation of the Cartesian components of the time-averaged Poynting vector P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±7.5 nm, and γ = γ (a-c) γ + = 0, and (d-f) γ + = 25. The following values of κ were chosen for rough correspondence with those in Fig. 6.3: (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i Variation of the Cartesian components of the time-averaged Poynting vector P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±45 nm, and γ = γ The following values of κ and γ + were chosen to highlight the uncoupling of the two metal/s- NTF interfaces, when the metal slab is sufficiently thick. (a-e) γ + = 25 and (f) γ + = 65. (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i Same as Fig. 6.2 except γ = γ Variation of the Cartesian components of P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±7.5 nm, and γ = γ (a-c) γ + = 25, and (d-f) γ + = 150. (a) κ/k 0 = i0.1865, (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i0.1847, (e) κ/k 0 = i , and (f) κ/k 0 = i Same as Fig. 6.4 except for γ = γ Real and imaginary parts of κ/k 0, which represent SPP-wave propagation guided by the single interface of the chosen SNTF and electron-beam-evaporated aluminum: ϵ m = ( i3.9) Variations of real and imaginary parts of κ/k 0 with γ +. (a) L ± = ±7.5 nm, (b) L ± = ±25 nm, (c) L ± = ±45 nm, and (d) L ± = ±75 nm. The other parameters are provided at the beginning of Sec xviii

19 6.13 Variation of the Cartesian components of the time-averaged Poynting vector P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±7.5 nm. (a-c) γ + = 0, and (d-f) γ + = 25. The following values of κ were to highlight the coupled SPP-waves propagation: (a) κ/k 0 = i0.015, (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i The components parallel to û x, û y, and û z, are represented by red solid, blue dashed, and black chain-dashed lines, respectively Same as Fig except for L ± = ±75 nm, and γ + = 25. The following values of κ were chosen to highlight the decoupling of the two metal/sntf interfaces: (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , and (d) κ/k 0 = i Relative wavenumber κ/k 0, relative phase speed v r, the e-folding distance, and the decay constants exp (u 1,2 ) = exp ( 2Ω Im [ ]) α 1,2 + as functions of γ + for Dyakonov Tamm waves guided by the single interface of the chosen dielectric material and the SNTF. The black symbols (square) identify the solutions also found by Agarwal et al. [64], but the solutions identified by the red symbols (circular) were missed in that work Variation of relative wavenumber κ/k 0 with γ +, when γ = γ +. (a) L ± = ±1 Ω, (b) L ± = ±1.5 Ω, (c) L ± = ±3 Ω, and (d) L ± = ±4 Ω. Solutions in the shaded regions represent Dyakonov Tamm waves, but those in the unshaded regions represent waveguide modes, the boundary between the two regions being delineated for the chosen parameters by κ/k 0 = n s Variation of the Cartesian components of P(z) (in W m 2 ) with z for γ = γ + and L ± = ±Ω. The x-, y-, and z-directed components are represented by solid red, dashed blue and chain-dashed black lines. The orange-shaded region represents the dielectric slab. γ + = (a-d) 10 and (e-f) 80. κ/k 0 = (a) , (b) , (c) , (d) , (e) , and (f) Same as Fig. 7.3 except for L ± = ±4Ω and κ/k 0 = (a) , (b) , (c) , (d) , (e) , and (f) Same as Fig. 7.2 except for γ = γ Same as Fig. 7.3 except for γ = γ γ + = (a-c) 10, and (d-f) 40. κ/k 0 = (a) , (b) , (c) , (d) , (e) , and (f) xix

20 7.7 Same as Fig. 7.3 except for γ = γ , and L ± = ±4Ω. γ + = (a-c) 6, (d-e) 30, and (f) 20. κ/k 0 = (a) , (b) , (c) , (d) , (e) , and (f) Schematic of the TKR configuration Schematic of the canonical boundary-value problem for coupled- SPP-wave propagation due to the metal film Absorptance A p as function of the incidence angle θ in the TKR configuration, when λ 0 = 633 nm, n l = 2.58, L m = 30 nm, and Ω = 1.5λ 0. Solid red line is for N p = 3 and dashed blue line is for N p = 4. Others parameters are given at the beginning of Sec Variation of A p with L m at the θ-values of the A p -peaks for N p = 4 in the TKR configuration. (a) Green solid line is for θ = 33.23, black dashed line for θ = 37.20, red chain-dashed line for θ = 42.41, and blue dotted line is for θ = 48.01; (b) red solid line is for θ = 53.86, black dashed line for θ = 59.66, and blue chain-dashed line is for θ = Variations of the Cartesian components P x and P z (in W m 2 ) of the time-averaged Poynting vector along the z axis in (left) the metal film and (right) the rugate filter for L m = 30 nm in the TKR configuration for a p-polarized incident plane wave (a p = 1 V m 1, a s = 0). (top) θ = 33.23, (middle) θ = 42.41, and (bottom) θ = Red solid line represents P x, blue dashed line represents P z, and P y is identically zero Variations of the Cartesian components of the time-averaged Poynting vector P(x = 0, z) (in W m 2 ) along the z axis in (top) the prism material, (middle) the metal film with L m = 30 nm, and (bottom) the rugate filter for the canonical boundary-value problem formulated in Sec for a p-polarized SPP wave with (left) κ/k 0 = i, and (right) κ/k 0 = i. Red solid line represents P x, blue dashed line represents P z, and P y is identically zero. The computations were made with b p = 1 V m Same as Fig. 8.3 except that A s is plotted instead of A p Variation of A s vs. the thickness of the metal film L m at the θ- position of the A s -peaks for N p = 3 in the TKR configuration. Solid red line is for θ = 38.97, black dashed line for θ = 44.01, blue chain-dashed for θ = 49.22, green dotted line for θ = 54.63, and orange dashed line (with larger dashes) is for θ = xx

21 8.9 Variations of the Cartesian components P x and P z (in W m 2 ) of the time-averaged Poynting vector along the z axis in (left) the metal film and (right) the rugate filter for L m = 30 nm in the TKR configuration for an s-polarized incident plane wave (a p = 0, a s = 1 V m 1 ). (top) θ = 49.22, and (bottom) θ = Red solid line represents P x, blue dashed line represents P z, and P y is identically zero Variations of the Cartesian components of the time-averaged Poynting vector P(x = 0, z) (in W m 2 ) along the z axis in (top) the prism material, (middle) the metal film with L m = 30 nm, and (bottom) the rugate filter for two s-polarized SPP waves obtained from the solution of the canonical boundary-value problem shown in Fig (left) κ/k 0 = i, and (right) κ/k 0 = i. Red solid line represents P x, blue dashed line represents P z, and P y is identically zero. The computations were made with b s = 1 Vm Schematic of the boundary-value problem solved using the RCWA Absorptance A p as a function of the incidence angle θ when the surface-relief grating is defined by either (a) Eq. (9.53) or (b) Eq. (9.2). Black squares represent d 1 = 1500 nm, red circles d 1 = 1000 nm, and blue triangles d 1 = 800 nm. The grating depth (d 2 d 1 = 50 nm) and the thickness of the metallic layer (d 3 d 2 = 30 nm) are the same for all cases. The vertical arrows identify SPP waves Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L for θ = 12.5, when the surface-relief grating is defined by either (a) Eq. (9.53) or (b) Eq. (9.2) and the incident plane wave is p polarized. Other parameters are the same as for Fig Absorptances (a) A p and (b) A s as functions of the incidence angle θ, when the surface-relief grating is defined by Eq. (9.2) with L 1 = 0.5L, λ 0 = 633 nm, Ω = λ 0, and L = λ 0. Black squares are for d 1 = 6Ω, red circles for d 1 = 5Ω, and blue triangles for d 1 = 4Ω. The grating depth (d 2 d 1 = 50 nm) and the thickness of the metallic layer (d 3 d 2 = 30 nm) are the same for all plots. Each vertical arrow identifies an SPP wave Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L, when the surface-relief grating is defined by Eq. (9.2). The grating period L = λ 0 and the incident plane wave is p polarized. Other parameters are the same as for Fig xxi

22 9.6 Same as Fig. 9.5 except that the incident plane wave is s polarized Same as Fig. 9.4 except for L = 0.75λ Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L, when the surface-relief grating is defined by Eq. (9.2) and the incident plane wave is p polarized. The grating period L = 0.75λ 0, Ω = λ 0, and d 1 = 6Ω Same as Fig. 9.8 except that d 1 = 4Ω Same as Fig. 9.6 except for L = 0.75λ Absorptance A p as a function of the incidence angle θ, when the surface-relief grating is defined by Eq. (9.2) with L 1 = 0.5L, λ 0 = 633 nm, Ω = 1.5λ 0, and L = 0.8λ 0. Black squares are for d 1 = 6Ω, red circles for d 1 = 5Ω, and blue triangles for d 1 = 4Ω. The grating depth (d 2 d 1 = 50 nm) and the width of the metallic layer (d 3 d 2 = 30 nm) are the same for all the plots. Each vertical arrow indicates an SPP wave Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L, when the surface-relief grating is defined by Eq. (9.2) and the incident plane wave is p polarized. The grating period L = 0.8λ 0 and d 1 = 6Ω Same as Fig except that d 1 = 4Ω Same as Fig except that A s is plotted instead of A p, and L = 0.6λ Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L for two s-polarized incident plane waves, when the surface-relief grating is defined by Eq. (9.2). The grating period L = 0.6λ 0, d 1 = 4Ω, and Ω = 1.5λ Same as Fig except that d 1 = 6Ω Absorptances (a) A p and (b) A s vs. the angle of incidence θ, when L = 186 nm and d 3 d 2 = 30 nm. The vertical arrow identifies the excitation of an SPP wave Variation of the x-component of the time-averaged Poynting vector P x along the z axis at x = 0.75L for (a) a p-polarized incident plane wave when θ = 17, and (b) an s-polarized incident plane wave when θ = The period of the surface-relief grating L = 186 nm and the free-space wavelength λ 0 = 620 nm. All other parameters are the same as for Fig The horizontal scale for z (d 1, d 3 ) is exaggerated with respect to that for z (0, d 1 ) xxii

23 10.3 Absorptances (a) A p and (b) A s vs. the angle of incidence θ, when Ω = 200 nm, γ = 0.1, d 3 d 2 = 30 nm, and λ 0 = 620 nm. Also, (a) L = 170 nm, and (b) L = 200 nm. Each vertical arrow indicates the excitation of an SPP wave Variation of the x-component of the time-averaged Poynting vector P x along the z axis at x = 0.75L for (a) two p-polarized incident plane waves and (b) an s-polarized incident plane wave, at the θ- values of the absorptance peaks identified in Fig by vertical arrows. The horizontal scale for z (d 1, d 3 ) is exaggerated with respect to that for z (0, d 1 ) Same as Fig except for Ω = 300 nm, and (a) L = 195 nm and (b) L = 210 nm Variation of the x-component of the time-averaged Poynting vector P x along the z axis at x = 0.75L for (a) three p-polarized incident plane waves and (b) two s-polarized incident plane waves, at the θ-values of the absorptance peaks identified in Fig by vertical arrows. The horizontal scale for z (d 1, d 3 ) is exaggerated with respect to that for z (0, d 1 ) Same as Fig except for λ 0 = 827 nm, ϵ r = i, ϵ m = i, and (a) L = nm and (b) L = 282 nm Schematic of the boundary-value problem solved using the RCWA Absorptance A p vs. the angle of incidence θ when L = 380 nm, ϕ = γ = 0, and d 3 d 2 = 30 nm. The absorptance peak represents the excitation of a p-polarized SPP wave Variation of the x-component P x (x, z) of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3, when L = 380 nm and ϕ = γ = 0. The incident plane wave is p polarized and the angle of incidence θ = Same as Fig except that L = 280 nm Same as Fig except that θ = 13.6 and L = 280 nm Absorptance A s vs. the angle of incidence θ when L = 340 nm, ϕ = γ = 0, and d 3 d 2 = 30 nm. A vertical arrow identifies the peak that represents the excitation of an s-polarized SPP wave Variation of the x-component P x (x, z) of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3. The incident plane wave is s polarized and the angle of incidence θ = Absorptances A p and A s vs. θ when L = 286 nm, ϕ = 0, γ = 75, and d 3 d 2 = 30 nm. The vertical arrows identify the peaks that represent the excitation of SPP waves xxiii

24 11.9 Variation of the x- and y-components of the time-averaged Poynting vector P(0.75L, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 for p- and s-polarized incident plane waves when θ = 9.2, L = 286 nm, ϕ = 0, γ = 75, d 1 = 6Ω, L g = 20 nm, and d 3 d 2 = 30 nm Same as Fig except for θ = Absorptance A vs. α when L = 286 nm, ϕ = 0, γ = 75, L g = 20 nm, and d 3 d 2 = 30 nm. The electric field phasor of the incident plane wave is defined by Eq. (11.80) A flow diagram showing the interconnections among different chapters of this thesis. The boxes with blue light background represent the chapters containing the canonical boundary-value problems, and the boxes with purple dark background represent the chapters that contain the boundary-value problems for the excitation of multiple surface waves. The boxes with white background do not contain any of the boundary-value problems (reproduced from Ch. 1) A.1 κ/k 0 versus n avg for Tamm waves localized to the interface of a homogeneous dielectric material ( n = 0) and a rugate filter (n + avg = 1.885, + n = 0.87, Ω + = λ 0, and ϕ + = 0), when the free-space wavelength λ 0 = 633 nm. The red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves. I failed to find solutions to bridge the gaps in two branches of solutions; these gaps are likely to be numerical artefacts, as there is no physical reason for them to exist A.2 Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of Tamm waves along the z axis, when λ 0 = 633 nm, n avg = 2, n = 0, n + avg = 1.885, + n = 0.87, Ω + = λ 0, and ϕ + = 0. The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively. All calculations were made after setting a + = a = 1 V m 1. (top) p-polarization state and κ/k 0 = , (middle) s-polarization state and κ/k 0 = , and (bottom) s- polarization state and κ/k 0 = A.3 κ/k 0 versus + n for Tamm waves localized to the interface of a homogeneous dielectric material (n avg = 2.5, n = 0) and a rugate filter (n + avg = 1.885, Ω + = λ 0, and ϕ + = 0), when the free-space wavelength λ 0 = 633 nm. The red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves xxiv

25 A.4 κ/k 0 versus Ω + /λ 0 for Tamm waves localized to the interface of a homogeneous dielectric material (n avg = 2.5, n = 0) and a rugate filter (n + avg = 1.885, + n = 0.87, and ϕ + = 0), when the free-space wavelength λ 0 = 633 nm. The red circles indicate s- polarized Tamm waves and the black triangles are for p-polarized Tamm waves A.5 κ/k 0 versus ϕ for Tamm waves localized to the phase-defect plane z = 0 in a rugate filter, with n + avg = n avg = 1.885, + n = n = 0.87, Ω + = Ω = λ 0, and ϕ + = 0, when the free-space wavelength λ 0 = 633 nm. The red circles indicate s polarized Tamm waves and the black triangles are for p-polarized Tamm waves. No solutions exist for ϕ {0, π} A.6 Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of Tamm waves along the z axis. All parameters are same as for Fig. A.5 except ϕ = 8 for the top and middle rows, and ϕ = 174 for the bottom row. The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively. All calculations were made after setting a + = a = 1 V m 1. (top) p- polarization state and κ/k 0 = , (middle) s-polarization state and κ/k 0 = , and (bottom) p-polarization state and κ/k 0 = A.7 Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of an optical Tamm state along the z axis. All parameters are same as for Fig. A.5 except ϕ = 30 and κ/k 0 = The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively. All calculations were made after setting a + = a = 1 V m 1. The optical Tamm state is p-polarized A.8 κ/k 0 versus n avg for Tamm waves localized to the plane z = 0 in a rugate filter, with n + avg = 1.885, n = + n = 0.87, Ω = Ω + = λ 0, and ϕ = ϕ + = 0, when the free-space wavelength λ 0 = 633 nm. The red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves. No solutions exist when n avg = n + avg, because the physical discontinuity across the interface z = 0 then disappears. The gaps including n avg = n + avg are physical because the discontinuity across the interface z = 0 then is too weak to support surface waves; however, other gaps in the solutions are more likely to be numerical artefacts as there is no physical reasons for them to exist xxv

26 A.9 Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of p-polarized Tamm waves along the z axis. All parameters are same as for Fig. A.8 except (top) n avg = 1.5 and κ/k 0 = , and (bottom) n avg = 2.3 and κ/k 0 = The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively A.10 κ/k 0 versus n for Tamm waves localized to the plane z = 0 in a rugate filter, with n + avg = n avg = 1.885, + n = 0.87, Ω = Ω + = λ 0, ϕ = ϕ + = 0, and n [0, 0.87], when the free-space wavelength λ 0 = 633 nm.the red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves. No solutions can exist when n = A.11 Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of Tamm waves along the z axis. Distinct rugate filters having parameters n + avg = 1.885, + n = 0.87, ϕ + = 0 and Ω + = λ 0, and n avg = 1.6, n = 0.6, ϕ = 90 and Ω = 0.5λ 0, respectively, were chosen with the freespace wavelength fixed at λ 0 = 633 nm. The field distributions were calculated for κ/k 0 = The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively xxvi

27 List of Tables 6.1 Penetration depths + z = z for L ± = ±7.5 nm and γ = γ +. The solutions are numbered in descending values of Re [κ/k 0 ] Penetration depths + z = z for L ± = ±45 nm and γ = γ +. The solutions are numbered in descending values of Re [κ/k 0 ] Values of the incidence angle θ and the relative wavenumber k x /k 0, where a peak is present in Fig. 8.3 independent of the value of N p. Each peak represents a p-polarized SPP wave, not a waveguide mode Relative wavenumbers κ/k 0 of p-polarized SPP waves obtained by the solution of the canonical boundary-value problem formulated in Sec for L m = 30 nm. Other parameters are given at the beginning of Sec Relative wavenumbers κ/k 0 of p-polarized SPP waves guided by the interface between semi-infinite metal and semi-infinite rugate filter (Ch. 3). All the parameters are the same as for Table 8.2 except that L m Values of the incidence angles θ, and the relative wavenumbers k x /k 0, where a peak is present in Fig. 8.7 independent of the value of N p Same as Table 8.2 except that the relative wavenumbers of s-polarized SPP waves are given instead of p-polarized SPP waves Same as Table 8.3 except that the relative wavenumbers of s-polarized SPP waves are given instead of p-polarized SPP waves Relative wavenumbers k (n) x /k 0 of Floquet harmonics at the θ-value of the peak identified in Fig. 9.2 by a vertical arrow. A boldface entry signifies an SPP waves Relative wavenumbers κ/k 0 of possible SPP waves obtained by the solution of the canonical boundary-value problem (Ch. 3) for Ω = λ 0. Other parameters are provided in the beginning of Sec If κ represents an SPP wave propagating in the û x direction, κ represents an SPP wave propagating in the û x direction xxvii

28 9.3 Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig. 9.4 by vertical arrows when Ω = λ 0 and L = λ 0. Boldface entries signify SPP waves Relative wavenumbers k (n) x /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig. 9.7 by vertical arrows when Ω = λ 0 and L = 0.75λ 0. Boldface entries signify SPP waves Same as Table 9.2 except for Ω = 1.5λ Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig by vertical arrows when Ω = 1.5λ 0 and L = 0.8λ 0. Boldface entries signify SPP waves Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig by vertical arrows when Ω = 1.5λ 0 and L = 0.6λ 0. Boldface entries signify SPP waves Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-position of the A p -peak identified in Fig. 10.1(a). A boldface entry signifies an SPP wave Relative wavenumbers κ/k 0 of p-polarized and s-polarized SPP waves supported by the planar interface of bulk aluminum and the semiconductor characterized by Eq. (10.3), when Ω = 200 nm, γ = 0.1, and λ 0 = 620 nm Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the absorptance peaks in Fig Boldface entries signify SPP waves Same as Table 10.2 except for Ω = 300 nm /k 0 of Floquet harmonics at the θ-values of the absorptance peaks in Fig Boldface entries signify SPP waves Same as Table 10.4 except for λ 0 = 827 nm, ϵ r = i, and ϵ m = i Relative wavenumbers k (n) x 10.7 Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the absorptance peaks in Fig Boldface entries signify SPP waves Relative wavenumbers κ/k 0 of SPP waves obtained by the solution of the canonical boundary-value problem (Ch. 2) when γ = ϕ = 0. The constitutive parameters of the periodically nonhomogeneous SNTF and the metal are provided at the beginning of Sec If κ represents an SPP wave propagating in the û x direction, κ represents an SPP wave propagating in the û x direction Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-value of the absorptance peak in Fig when L = 380 nm and ϕ = γ = 0. A boldface entry signifies an SPP wave xxviii

29 11.3 Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-value of the absorptance peak in Fig when L = 280 nm. A boldface entry signifies an SPP wave Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-value of the peak identified by a vertical arrow in Fig when L = 340 nm. A boldface entry signifies an SPP wave Relative wavenumbers κ/k 0 of SPP waves obtained by the solution of the canonical boundary-value problem (Ch. 2) for propagation at an angle of 75 to the morphologically significant plane of the SNTF. The constitutive parameters of the SNTF and the metal are provided at the beginning of Sec The SPP waves are neither p nor s polarized. If κ represents an SPP wave propagating in the û x direction, κ represents an SPP wave propagating in the û x direction Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig by vertical arrows when L = 286 nm. Boldface entries signify SPP waves Relative wavenumbers n l sin θ l of SPP waves in the TKR configuration excited by s- and p-polarized incident plane waves propagating in the morphologically significant plane of the SNTF [70]. The constitutive parameters of the periodically nonhomogeneous SNTF and the metal are provided at the beginning of Sec. 11.3, whereas n l = Same as Table 11.7 except that the morphologically significant plane of the SNTF makes an angle of 75 with the incidence plane [71] A.1 Relative wavenumber κ/k 0 of s-polarized and p-polarized Tamm waves supported by the interface of two distinct rugate filters, whose parameters are provided in Section A.3.5. The free-space wavelength λ 0 = 633 nm xxix

30 Acknowledgments I would like to take this opportunity to thank my dissertation adviser, Prof. Akhlesh Lakhtakia, whose relentless supervision helped steer my research through thick and thin. If it were not for his constant advice and guidance, this thesis would never have come to exist in this form. It was not only his guidance in my academic matters that made my journey through the Ph. D. smooth, but also his personal advice in almost all aspects of my life. He was not only the thesis advisor, but also a mentor who always gave priority to my professional and personal developments. I am also grateful to the members of my thesis committee, Prof. Michael T. Lanagan, Prof. Osama O. Awadelkarim, and Prof. Jainendra K. Jain, for taking time out of their busy schedules to evaluate this thesis and provide valuable feedback. Thanks are also due to more than a dozen anonymous referees for their thankless job of painfully reviewing the papers submitted for publication in various journals and making suggestions to improve the quality of the research reported in this thesis. I thank Dr. John A. Polo Jr. for his guidance and valuable inputs on the initial work on SPP-wave propagation, and Dr. Husnul Maab for working with me on Fano and Tamm waves. I am greatly indebted to my wife, Hina Akhtar, for her love and support throughout my Ph. D. studies. She made many sacrifices in order for me to finish my dissertation in a timely manner. Finally, the following funding sources are gratefully acknowledged: (i) University Graduate Fellowship from the Graduate School ( ), (ii) The Charles Godfrey Binder Endowment at the Department of Engineering Science and Mechanics (Summers 2010, 2011), (iii) Teaching assistantship from the Department of Engineering Science and Mechanics ( , Fall 2011), and (iv) US National Science Foundation research grant: DMR (Spring 2012). xxx

31 Chapter 1 Introduction The objective of the research conducted for this thesis was to theoretically investigate the propagation and excitation of multiple surface waves all at the same frequency but with different polarization states, phase speeds, and spatial characteristics guided by single or double interfaces present in a periodically nonhomogeneous dielectric material. If one of the partnering materials is a metal, the surface waves are called surface plasmon-polariton (SPP) waves. If both partnering materials are dielectric, with at least one being periodically nonhomogeneous normal to the interface, the surface waves are called Tamm waves; and if that dielectric material is also anisotropic, the surface waves are called Dyakonov Tamm waves. SPP waves also decays along the direction of propagation, whereas Tamm and Dyakonov Tamm waves propagate with negligible losses. The surface waves chiefly studied for this thesis are SPP waves and Dyakonov Tamm waves. The former are guided by an interface of a metal and a dielectric material, and the latter by an interface of two dielectric materials with at least one being anisotropic and periodically nonhomogeneous normal to the interface. Two types of periodically nonhomogeneous dielectric materials have been considered in this thesis: sculptured nematic thin film (SNTF) [1] and rugate filter [2]. An SNTF is an anisotropic and optically continuous medium with a relative permittivity dyadic that is periodically nonhomogeneous in the thickness direction. A rugate filter is an isotropic dielectric material with a refractive index that varies periodically, usually in a sinusoidal fashion, in one direction, which is also taken to be the thickness direction in the present context. Surface waves guided by a planar interface between a periodically nonhomogeneous SNTF and an isotropic homogeneous medium possess remarkable properties and offer many possibilities for their use in chemical sensors, subwavelength optics and on-chip communication. Surface-wave propagation guided by four types of interfaces with the SNTF was studied for this thesis: (i) an interface of a metal and a periodically nonhomogeneous SNTF, (ii) an interface between two different SNTFs, (iii) a metal slab inserted in a periodically nonhomogeneous SNTF, and 1

32 (iv) a dielectric slab inserted in a periodically nonhomogeneous SNTF. Moreover, the excitation of multiple surface waves guided by a metal/sntf interface was studied in the grating-coupled configuration. The SNTF was chosen as a periodically nonhomogeneous material to provide two main characteristics: periodic nonhomogeneity and porosity. The former property made possible the propagation of multiple surface waves while the latter can be used for sensing applications. A rugate filter, being isotropic, is attractive for light-harvesting applications in thin-film solar cells due to the coupling of a part of the incident light with the surface waves [3], thereby reducing the reflectance and transmittance of light. Therefore, the propagation of multiple surface waves guided by a metal/rugatefilter interface is an attractive subject for practical applications of immense technological value. The propagation of multiple surface waves by a metal/rugate-filter interface and the interface of two rugate filters was studied. The excitation of multiple SPP waves was studied in the Turbadar Kretschmann Raether (TKR) and grating-coupled configurations for the metal/rugate-filter interface. The rugate filter was chosen because it is also periodically nonhomogeneous like an SNTF; however, it is an isotropic dielectric material unlike an SNTF. The investigations on surface-wave propagation by the interfaces of isotropic material and a rugate filter revealed that the multiplicity of surface waves is due to the periodic nonhomogeneity of the partnering dielectric material and not because of its anisotropy. This turned out to be a cornerstone for subsequent research on multiple surface waves, as a large variety of the materials used in practice are isotropic. The work presented in this thesis was chiefly motivated by the desire to be able to launch multiple surface waves of the same frequency but different polarization states, phase speeds, and spatial profiles. The possibility of exciting multiple SPP waves provides exciting prospects for enhancing the scope of the applications of SPP waves. For sensing applications, the use of more than one distinct SPP waves would increase confidence in a reported measurement; also, more than one analyte could be sensed at the same time, thereby increasing the capabilities of multi-analyte sensors. For imaging applications, the simultaneous creation of two images may become possible. For plasmonic communications, the availability of multiple channels would make information transmission more reliable as well as enhance capacity. Moreover, light absorption can be enhanced in thin-film solar cells by the use of multiple SPP waves. In this chapter, basic concepts needed for the rest of the thesis are provided: SPP waves in Sec. 1.1, Dyakonov Tamm waves in Sec. 1.2, SNTFs in Sec. 1.3, rugate filters in Sec. 1.4, and common methods for excitation of surface waves are presented in Sec Finally, the objectives of the research conducted and the organization of the thesis are presented in Secs. 1.6 and 1.7, respectively. In this thesis, an exp( iωt) time-dependence is implicit, with ω denoting the angular frequency, t the time, and i = 1. The free-space wavenumber, the free-space wavelength, and the intrinsic impedance of free space are denoted by 2

33 k 0 = ω ϵ 0 µ 0, λ 0 = 2π/k 0, and η 0 = µ 0 /ϵ 0, respectively, with µ 0 and ϵ 0 being the permeability and permittivity of free space. Vectors are in boldface, dyadics are underlined twice, column 4-vectors are in boldface and enclosed within square brackets, and 4 4 matrixes are underlined twice and square-bracketed. Dyadics have been treated as 3 3 matrixes in this thesis [4]. The asterisk denotes the complex conjugate, the superscript T denotes the transpose, and the Cartesian unit vectors are identified as û x, û y, and û z. 1.1 Surface Plasmon-Polariton Waves Among the various forms of electromagnetic surface waves, the SPP wave has the longest history of theoretical development and application [5 7]. More than a century ago, Zenneck [8] proposed that an electromagnetic wave in the microwave regime could travel along the planar interface of air and ground. Sommerfeld [9] provided rigorous mathematical analysis of what has since become known as the Zenneck wave [10, 11]. The underlying concept emerged again, about 60 years ago [12], in the form of SPP wave which is guided by the planar interface of two homogeneous, isotropic, dielectric materials, the real parts of whose relative permittivity scalars have opposite signs [13, 14]. Commonly, the partnering material with negative real permittivity is a metal [15], but other materials can also be appropriate [16, 17]. The theory has evolved to encompass interfaces between a metal and various dielectric materials of greater complexity. The inclusion of anisotropic, homogeneous, dielectric materials [18 25] in the study of electromagnetic surface waves has been considered for some time now. SPP waves guided by the interface of a metal and a periodically nonhomogeneous dielectric material exhibit remarkable characteristics [16]. The nonhomogeneous dielectric materials investigated include continuously varying materials [26 29] such as cholestric liquid crystals, as well as layered structures [30 33]. This technoscientific ferment is due to a resonance phenomenon that arises when the energy carried by photons in the partnering dielectric material is transferred to free electrons in the metal partner at that interface, and vice versa. Different dielectric materials will become differently polarized on interrogation by an electromagnetic field, thereby enabling a widely used technique for sensing chemicals and biochemicals [34, 35]. Furthermore, SPP imaging systems are used for high-throughput analysis of biomolecular interactions for proteomics, drug discovery, and pathway elucidation [36, 37]. SPP-based imaging techniques are also going to be useful for lithography [13, 39]. SPP-based sensing technology has been successfully applied to the screening of bioaffinity interactions with DNA, carbohydrates, peptides, phage display libraries, and proteins [38]. Finally, as SPP waves can be excited in the terahertz and optical regimes, they may be useful for high-speed communication of information on computer chips [40]. Whereas conventional wires are very attenuative at frequencies beyond a few tens of GHz, 3

34 ohmic losses are minimal for plasmonic transmission [14] which enables long-range communications [41]. At a specific frequency, the solution of a canonical boundary-value problem [13,14,35] shows that only one SPP wave can propagate along the interface, if the partnering dielectric material is isotropic and homogeneous. The same conclusion holds true even if that material is anisotropic [20, 42]. However, if the partnering dielectric material is both anisotropic and periodically nonhomogeneous in the direction normal to the interface, the solutions of the canonical boundary-value problem [26] show that more than one SPP waves with different phase speeds, attenuation rates, and field distributions, but of the same frequency [27] can propagate guided by the interface. Experimental verification of this theoretical prediction has been found [43,44]. Moreover, some researchers [45 48] have shown experimentally and theoretically that s-polarized SPP waves can also be guided by an interface of a metal and a periodic multi-layered dielectric material. 1.2 Dyakonov Tamm Waves Although there are two earlier reports [49, 50], the research on surface waves guided by the interface of two dielectric materials started in earnest in 1988, when Dyakonov [51] studied the surface waves guided by an interface of an isotropic dielectric material and a uniaxial dielectric material. These surface waves are called Dyakonov waves. Dyakonov waves are found to be guided by the interface of two homogeneous dielectric materials, of which at least one material must be anisotropic [16, 52 55]. Since very restrictive conditions need to be satisfied in order for Dyakonov waves to exist [54, 56], it took two decades for experimental evidence of these new surface waves to emerge [57]. Dyakonov waves have potential applications in integrated optics, optical sensing, and waveguiding [53, 58, 59]. Lakhtakia and Polo investigated the effects of periodic nonhomogeneity of one of the two partnering dielectric materials on surface-wave propagation, when the direction of nonhomogeneity is normal to the interface [60]. They used a methodology traceable to Tamm for a realistic Kronig Penney model (that is, assuming the solid to occupy only a half-space instead of the entire space [61]), leading to the emergence of electronic states localized to the interface called Tamm states, observed experimentally in 1990 [62]. The new type of surface waves are called Dyakonov Tamm waves [60]. A significant difference between the Dyakonov waves and the Dyakonov Tamm waves is the puny range of propagation directions in the interface plane of the former type of waves [54, 56] in comparison to the wide range for the latter type of waves. The extension of the range of directions must be due to the periodic nonhomogeneity of either one or both partnering dielectric materials [16]. This periodic nonhomogeneity can be introduced by using a periodic sculptured thin film (STF) [1, 63] as a partnering dielectric material. Lakhtakia and Polo 4

35 chose a chiral STF as one of the two partnering dielectric materials, the other being isotropic and homogeneous. Agarwal et al. studied the propagation of the Dyakonov Tamm waves guided by the interface of an isotropic dielectric material and a periodically nonhomogeneous SNTF [64]. More recently, Gao et al. theoretically examined the propagation of Dyakonov Tamm waves guided by a twist-defect interface in a chiral STF. Most significantly, they found that multiple Dyakonov Tamm waves of same frequency, but different phase speed, field distribution and the degree of localization to the interface can be guided by that interface [65 67]. 1 The same conclusion was found to hold for the propagation of the Dyakonov Tamm waves guided by an interface between two chiral STFs that differ only in handedness [68]. In both instances, the most strongly localized Dyakonov Tamm waves are essentially confined to within two or three structural periods normal to the interface. 1.3 Sculptured Nematic Thin Films A sculptured thin film (STF) is an assembly of parallel columns of nanoscale crosssectional diameter, microscopically, where each column is of the same shape [1,69]. An STF is grown commonly using physical vapor deposition (PVD), where a directional vapor flux is incident on a substrate, which could be fixed, rotating and/or rocking. Under the right temperature and pressure, the film grows with columns whose shape is determined by the motion of the substrate. Macroscopically, for optical purposes, an STF is a material continuum that is periodically nonhomogeneous in a particular direction. Depending on the shape of the columns in an STF, it can be classified into three categories: (i) Columnar thin film (CTF), where all columns are parallel to a straight line; (ii) Sculptured nematic thin film (SNTF), where the shape of each column is described by a two dimensional curve in space; and (iii) Chiral sculptured thin film (CSTF), where each column is a helix. For the work undertaken for this thesis, only periodically nonhomogeneous SNTFs were considered. While an SNTF is grown, the substrate is only rocked about a tangential axis [1, Chap. 8]. The nature of the rocking defines the shape of the columns of the film and hence its macroscopic electromagnetic properties. The plane in which the columns lie is the morphologically significant plane of the SNTF. Let the z-axis be parallel to the thickness direction of an SNTF. Then, a periodically nonhomogeneous SNTF s permittivity dyadic is of the form [44,70,71] ϵ SNT F (z) = ϵ 0 S y (z) ϵ ref (z) S 1(z), (1.1) y 1 Agarwal et al. [64] found only one Dyakonov Tamm wave but later more than one solutions were found for the same interface (Ch. 7). 5

36 where the dyadics S y (z) = (û x û x + û z û z ) cos [χ(z)] + (û z û x û x û z ) sin [χ(z)] + û y û y ϵ ref (z) = ϵ a(z) û z û z + ϵ b (z) û x û x + ϵ c (z) û y û y } (1.2) depend on the vapor incidence angle χ v (z) = χ v (z ± 2Ω) with respect to the substrate (xy) plane, where 2Ω is the period. For this thesis, χ v (z) = χ v + δ v sin(πz/ω) that varies sinusoidally about the mean value χ v with period 2Ω. The dyadic S y (z) is a rotation matrix that describes the tilt of the columns of the SNTF at a given value of z. The tilt angle χ(z) with respect to the substrate plane depends upon the vapor incidence angle χ v (z). The quantities ϵ a (z ), ϵ b (z ), and ϵ c (z ) are the eigenvalues of ϵ ref (z ) and hence of ϵ SNT F (z) and should be interpreted as the principal relative permittivity scalars in the plane z = z [72]. These three quantities and χ(z ) depend on χ v (z ), the conditions for the fabrication of the SNTF, and the material(s) evaporated to fabricate the SNTF, and therefore need to be found experimentally. For all the numerical results presented in this thesis, an SNTF made of titanium oxide was considered for numerical results. The parameters of a CTF of titanium oxide were found experimentally by Hodgkinson et al. [73] which have been used for an SNTF in this work. These parameters are [70] ϵ a (z) = [ v(z) v 2 (z)] 2 ϵ b (z) = [ v(z) v 2 (z)] 2, (1.3) ϵ c (z) = [ v(z) v 2 (z)] 2 χ(z) = tan 1 [ tan χ v (z)] where v(z) = 2χ v (z)/π. 1.4 Rugate Filters Rugate filters are isotropic dielectric thin films that have continuously varying and periodic refractive index along the thickness direction [2]. A multi-layered structure with a large number of sufficiently thin layers made of isotropic dielectric materials of different refractive indexes may also be classified as a rugate filter [2]. Rugate filters find applications as optical filters [74] and light-trapping layers in thin-film solar cells [75]. Rugate filters can be fabricated with various physical and chemical deposition techniques. Physical deposition techniques include cluster beam deposition [76], magnetron sputtering [77], and ion-beam sputtering [78], whereas chemical deposition techniques include molecular assembly [79] and electrolysis [80]. 6

37 The relative permittivity of a rugate filter with sinusoidally changing permittivity along the z axis is given by [( ) ( ) nb + n a nb n ( a ϵ r (z) = + sin π z ) ] 2, (1.4) 2 2 Ω where n a and n b are the lowest and the highest indexes of refraction, respectively, and 2Ω is the period. 1.5 Excitation of Surface Waves The propagation of multiple surface waves is studied in this thesis using canonical boundary-value problems. The canonical problems provide an understanding of the underlying phenomenon and different factors affecting the propagation of surface waves, but they cannot be implemented experimentally because they involve at least one half-space occupied by a periodically nonhomogeneous dielectric material. However, the solution of a canonical problem guides experimental implementation. For instance, the solution to a canonical problem of SPP-wave propagation guided by a metal/sntf interface gives the information on the spatial extent of surface waves into the partnering materials in a practical configuration. This information is crucial in deciding the thickness of the metal layer and that of the SNTF in an experimental setup. The excitation of the surface waves is a very challenging objective because of the different value of the phase speed of the surface waves than the phase speed in the bulk partnering materials. Many schemes have been developed to overcome this difficulty, and more common among them are explained in the following subsections Prism-Coupled Configuration The most common method for excitation of SPP wave is prism-coupled configuration using the attenuated total reflection (ATR). There are two configurations to implement the prism coupling: TKR and Turbadar-Otto (TO) 2 [34]. In the TKR configuration, as shown in Fig. 1.1, a high-refractive-index prism is interfaced with a metal and a dielectric thin film. An oil, with refractive index the same as that of the prism is used between the prism and the metal to remove air pockets. When light propagating in the prism is made incident on the metal film, a part of it is reflected back into the prism and a part is refracted into the metal. The wave refracting into the metal decays exponentially in the direction perpendicular to 2 Turbadar in 1959 [81] had anticipated the 1968 papers of both Otto [82] and Kretschmann and Raether [83], but had not used the word plasmon. 7

38 the prism/metal interface. If the metal film is sufficiently thin, the wave penetrates through the metal and couples with the surface plasmons at the boundary of the metal at metal/dielectric interface. At a particular angle of incidence at the prism/metal interface, the electromagnetic boundary conditions are satisfied to launch the SPP wave at the metal/dielectric interface. The TKR configuration is used to excite multiple SPP waves at the metal/rugate-filter interface in Ch. 8. Incident light Reflected light Prism Metal Dielectric Index matching oil SPP wave Figure 1.1: Schematic for the TKR configuration. Incident light Reflected light Prism Dielectric Index matching oil SPP wave Metal Figure 1.2: Schematic for the TO configuration. If the metal and dielectric films are interchanged in the TKR configuration, the new configuration is called the TO configuration. In this configuration, as shown in Fig. 1.2, the light incident on the prism/dielectric interface at an angle larger than the critical angle of incidence for these two materials produces an evanescent wave in the partnering dielectric material. If the dielectric film is sufficiently thin and the phase speed of the evanescent wave parallel to the interface and the SPP wave are the same, an SPP wave is launched guided by the metal/dielectric interface. The TKR configuration can only be used to excite SPP waves, whereas the TO configuration may be used to launch Tamm or Dyakonov-Tamm waves in addition to SPP waves if the metal film is replaced, respectively, by an isotropic or anisotropic but periodically nonhomogeneous dielectric material. 8

39 The prism-coupled configuration is particularly useful in exciting an SPP wave with the desired polarization state and is easy to implement. However, a lot of care is needed to distinguish between the SPP waves guided by the metal/dielectric interface and the waveguide modes propagating in the bulk of the partnering dielectric material of finite thickness Grating-Coupled Configuration An alternative to the TKR configuration is the grating-coupled configuration, schematically shown in Fig This configuration is typically used to excite SPP waves; however, it can be used to excite Tamm and Dyakonov Tamm waves as well. SPP waves can be excited in the grating-coupled configuration by the illumination of the periodic corrugations of a metallic surface-relief grating coated with the dielectric partnering material. Fields in the two partnering materials must be represented as linear superpositions of Floquet harmonics. If the component of the wavevector of a Floquet harmonic in the plane of the grating is the same as that of the SPP wave, the Floquet harmonic can couple with the SPP wave. The grating-coupled configuration also allows the reverse process: the efficient coupling of SPP waves, which are otherwise nonradiative, with light [84, 85]. This is an important advantage over the TKR configuration because it allows for better incorporation of chemical sensors based on SPP waves [86] in integrated optical circuits [87]. Incident Dielectric Metal Metal SPP waves Figure 1.3: Schematic for the grating-coupled configuration. 9

40 Waveguide mode Dielectric superstrate Metal Dielectric layer SPP wave Dielectric substrate Figure 1.4: Schematic for the waveguide-coupled configuration. This configuration is particularly useful if one wishes to excite multiple SPP waves simultaneously by using a finite light source and/or a quasi-periodic surfacerelief grating. This technique is used to excite SPP waves at the metal/rugate-filter and metal/sntf interfaces in Chs. 9, 10, and Waveguide-Coupled Configuration A less common technique for exciting SPP waves is through an optical dielectric waveguide. Typically, a dielectric waveguide is integrated with a metal/dielectric interface, as shown schematically in Fig When a waveguide mode propagating in the dielectric waveguide has the same phase speed as that of an SPP wave guided by the metal/dielectric interface, the electromagnetic energy from the waveguide mode in the dielectric waveguide couples with the SPP wave guided by the metal/dielectric interface. This configuration has the advantage of exciting SPP waves directly into the metal/dielectric interface. 1.6 Objectives of the Thesis The objectives of the research conducted for this thesis were to (a) find the basic property of the partnering dielectric materials that is responsible for the multiplicity of surface waves; (b) elucidate the effects of the morphology of the partnering dielectric materials on the characteristics of surface waves; (c) find the minimum spatial dimensions of the partnering materials in order to implement the structures for experimental research; (d) find other ways to increase the number of possible surface waves; 10

41 (e) study the excitation of multiple surface waves in prism- and grating-coupled configurations with periodically nonhomogeneous partnering dielectric materials; and (f) see if multiple SPP waves can lead to enhanced absorption of light in thinfilm solar cells. 1.7 Organization of the Thesis To achieve objectives (b) and (c) of the forgoing section, the canonical boundaryvalue problem of SPP-wave propagation guided by a single planar interface of a metal and an SNTF is formulated and solved in Ch. 2. A dispersion equation is obtained and solved to find the complex wavenumber for SPP waves which can be guided by the metal/sntf interface. The spatial field and power profiles are also given. In Ch. 3, an interface of a metal and a rugate filter with a sinusoidal refractive index profile is analyzed for objective (a) to elucidate the effect of periodic nonhomogeneity of the partnering dielectric material. The wavenumbers of the possible SPP waves that can be supported by the metal/rugate-filter interface are found as functions of the period of the rugate filter. In Ch. 4, the propagation of multiple Fano waves by an interface of a rugate filter and a material that has a negative real permittivity is studied. This problem is an extension of the one solved in Ch. 3. To achieve objectives (b) and (c), Dyakonov Tamm waves guided by a phasetwist combination defect in a periodically nonhomogeneous SNTF are studied in Ch. 5. The phase defect is fixed at 180, whereas the twist defect is kept variable as also the direction of propagation, for numerical results. Multiple Dyakonov Tamm waves that differ in spatial profile, degree of localization, and phase speed are found to propagate guided by the phase-twist combination defect, depending on the angle between the morphologically significant planes of the SNTF on either side of the defect as well as on the direction of propagation. SPP-wave propagation guided by a metallic slab inserted in a periodically nonhomogeneous SNTF is considered in Ch. 6. The morphologically significant planes of the SNTF on both sides of the metal slab could either be aligned or twisted with respect to each other. The effect of slab thickness on the multiplicity and the spatial profiles of SPP waves is analyzed. In Ch. 7, the metal slab in the SNTF is replaced by a dielectric slab to study the effect of the coupling of the two interface on the propagation of Dyakonov Tamm waves. The investigations presented in Chs. 6 and 7 were conducted to achieve objective (d). The excitation of multiple SPP waves at the metal/rugate-filter interface in the TKR and the grating-coupled configurations is studied, respectively, in Chs. 8 and 9 to achieve objective (e). The results of the TKR and the grating-coupled con- 11

42 Ch. 2 Single metal/sntf interface Ch. 3 Single metal/rugatefilter interface Ch. 4 Multiple Fano waves Ch. 6 SNTF/metal/ SNTF interface Ch. 5 SNTF/SNTF interface Ch. 8 TKR configuration (metal/rugate-filter interface) Ch. 7 SNTF/dielectric/SNTF interface Ch. 11 Grating-coupled configuration (metal/sntf interface) Ch. 9 Grating-coupled configuration (metal/rugate-filter interface) Ch. 10 Application of multiple SPP waves in solar cells Ch. 12 Conclusions Future work Appendix A Rugate-filter/rugate-filter interface Appendix B Mathematica TM Codes Figure 1.5: A flow diagram showing the interconnections among different chapters of this thesis. The boxes with blue light background represent the chapters containing the canonical boundary-value problems, and the boxes with purple dark background represent the chapters that contain the boundary-value problems for the excitation of multiple surface waves. The boxes with white background do not contain any of the boundary-value problems. figurations are successfully correlated with the canonical boundary-value problem solved in Ch. 3. For objective (f), the excitation of multiple SPP waves has been shown to increase the absorptance of light in thin-film solar cells in Ch. 10. The excitation of multiple SPP waves in the grating-coupled configuration for the metal/sntf interface is studied in Ch. 11 to achieve objective (e), and the results are correlated successfully with those obtained in Ch. 2. Finally, the overall conclusions and the suggestions for future work are presented in Ch. 12. A flow diagram showing the interconnections among various chapters of this thesis is presented in Fig The boxes with light blue background represent the chapters that contain the formulations and solutions of the canonical boundaryvalue problems, and the boxes with dark purple background represent the chapters dealing with the excitation of multiple SPP waves in the TKR or grating-coupled configurations. A patient reader can read the thesis from the beginning till the end in the order the chapters are arranged in the thesis. However, if the reader is interested only in a particular part of the thesis, it is advised that she follows the 12

43 flow diagram. For instance, if the reader wishes to read Ch. 11, she should begin with Ch. 2 and then jump to to Ch. 11 or if the reader wishes to read Ch. 10, she should begin with Ch. 3 and then jump to Ch. 10 preferably going through Ch. 9 as well. 13

44 Chapter 2 SPP Waves Guided by Metal/SNTF Interface 2.1 Introduction This chapter is devoted to the investigation on the propagation of a surface wave guided by the planar interface of a metal and an SNTF, the latter being periodically nonhomogeneous in the direction normal to the interface. In Refs. 70 and 71, the absorptance, reflectance and transmittance of an incident linearly polarized plane wave were calculated when this planar interface is implemented via a metal-sntf bilayer and a prism in the TKR configuration. The wavevector of the incident plane wave was supposed to lie wholly in the morphologically significant plane of the SNTF in Ref. 70, but that restriction was lifted in Ref. 71. The computed results showed that multiple SPP waves, all of the same frequency or color, can be excited at the metal/sntf interface. The guided SPP waves possess different field structures as well as different phase speeds. In Ref. 44, experimentally obtained data was presented in support of the theoretical predictions of Ref. 70. The absorptances of two different metal-sntf bilayers were evaluated as the difference between unity and the measured reflectance, the transmittance being assumed to be null-valued at angles of incidence exceeding the critical angle for the interface of the SNTF with the prism. Analysis of the collected data confirmed the possibility of exciting multiple SPP waves with different phase speeds and field structures. The method adopted for theoretical predictions and experimental verification in Refs. 70, 71, and 44 was an indirect one, because the existence of SPP waves was deduced from the absorptance, reflectance and transmittance characteristics of a metal-sntf bilayer. Each layer in the bilayer is of finite thickness. In the This chapter is based on: M. Faryad, J. A. Polo Jr., and A. Lakhtakia, Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. Part IV: Canonical problem, J. Nanophoton. 4, (2010). 14

45 canonical problem of SPP-wave propagation, the metal and the SNTF occupy halfspaces. The solution of the canonical problem provides incontrovertible proof of the existence of multiple SPP-wave modes, and also eliminates possible confusion with waveguide modes spread over the entirety of the SNTF. Accordingly, in this chapter, the goal was set to prove the existence of multiple SPP waves directly. The approach adopted is similar to that of Agarwal et al. [64], being independent of any incident plane wave. The formulated boundary-value problem was solved numerically for the SPP wavenumbers. The canonical problem is formulated in Sec. 2.2, and numerical results are presented and discussed in Sec The concluding remarks are given in Sec Theory Let the half-space z 0 be occupied by an isotropic and homogeneous metal with complex-valued relative permittivity scalar ϵ m. The region z 0 is occupied by the chosen SNTF as described in Sec Let the SPP wave propagate parallel to the unit vector û x cos ψ + û y sin ψ along the interface z = 0, and attenuate as z ±. Therefore, in the region z 0, the wave vector may be written as k met = κ û 1 α met û z, (2.1) where κ 2 + αmet 2 = k 2 0 ϵ m, κ is complex-valued, and Im(α met ) > 0 for attenuation as z ; here and hereafter, the unit vectors û 1 = û x cos ψ + û y sin ψ and û 2 = û x sin ψ + û y cos ψ. Accordingly, the field phasors in the metal may be written as [ ( αmet E(r) = a p û 1 + κ ) ] û z + a s û 2 exp(ik met r), z 0, (2.2) k 0 k 0 and H(r) = η 1 0 [ ( αmet a p ϵ met û 2 + a s û 1 + κ )] û z exp(ik met r), z 0. (2.3) k 0 k 0 Here a p and a s are unknown scalars with the same units as the electric field, with the subscripts p and s, respectively, denoting the parallel and perpendicular polarization states with respect to the plane formed by û 1 and û z. For field representation in the SNTF, E(r) = e(z) exp (iκû 1 r) and H(r) = h(z) exp (iκû 1 r). The components e z (z) and h z (z) of the field phasors can be found in terms of the other components as follows: e z (z) = ϵ d(z) [ϵ a (z) ϵ b (z)] sin [χ(z)] cos [χ(z)] e x (z) ϵ a (z)ϵ b (z) ϵ d (z) +κ ωϵ 0 ϵ a (z)ϵ b (z) [h x(z) sin ψ h y (z) cos ψ], z > 0, (2.4) h z (z) = κ ωµ 0 [e x (z) sin ψ e y (z) cos ψ], z > 0, (2.5) 15

46 where ϵ d (z) = ϵ a (z) ϵ b (z) ϵ a (z) cos 2 [χ(z)] + ϵ b (z) sin 2 [χ(z)]. (2.6) The other components of the electric and magnetic field phasors are used in the column vector [f(z)] = [e x (z) e y (z) h x (z) h y (z)] T (2.7) which satisfies the matrix differential equation [64] where the 4 4 matrix [P (z)] = ω d dz [f(z)] = i [ P (z) ] [f(z)], z > 0, (2.8) µ µ ϵ 0 ϵ c (z) 0 0 ϵ 0 ϵ d (z) κ ϵ d(z) [ϵ a (z) ϵ b (z)] ϵ a (z) ϵ b (z) + κ2 ωϵ 0 + κ2 ωµ 0 ϵ d (z) ϵ a (z) ϵ b (z) sin [χ(z)] cos [χ(z)] 0 0 cos ψ sin ψ cos 2 ψ 0 0 sin 2 ψ cos ψ sin ψ cos ψ sin ψ cos 2 ψ 0 0 sin 2 ψ cos ψ sin ψ 0 0 cos ψ sin ψ sin ψ cos ψ. (2.9) The piecewise uniform approximation technique [1] is used to determine the matrix [Q] that appears in the relation [f(2ω)] = [Q] [f(0+)] (2.10) to characterize the optical response of one period of the chosen SNTF for specific values of κ and ψ. By virtue of the Floquet Lyapunov theorem [88], a matrix [ Q] can be defined such that { [Q] = exp i2ω[ Q] }. (2.11) 16

47 Both [Q] and [ Q] share the same eigenvectors, and their eigenvalues are also related. Let [t] (n), n [1, 4], be the eigenvector corresponding to the nth eigenvalue σ n of [Q]; then, the corresponding eigenvalue α n of [ Q] is given by α n = i ln σ n 2Ω After ensuring that Im(α 1,2 ) > 0, [f(0+)] = [ [t] (1) [t] (2) ], n [1, 4]. (2.12) [ b1 b 2 ] (2.13) for SPP-wave propagation, where b 1 and b 2 are unknown dimensionless scalars; the other two eigenvalues of [ Q] pertain to waves that amplify as z and cannot therefore contribute to the SPP wave. At the same time, [f(0 )] = α met k 0 cos ψ sin ψ α met k 0 sin ψ cos ψ ϵ met η 0 sin ψ α met k 0 η 0 cos ψ ϵ met η 0 cos ψ α met k 0 η 0 sin ψ [ ap a s ], (2.14) by virtue of (2.2) and (2.3). Continuity of the tangential components of the electric and magnetic field phasors across the plane z = 0 requires that [f(0 )] = [f(0+)], which may be rearranged as the matrix equation [Y ] a p a s b 1 b 2 = (2.15) For a nontrivial solution, the 4 4 matrix [Y ] must be singular, so that det [Y ] = 0 (2.16) is the dispersion equation for the SPP wave. This equation has to be solved in order to determine the SPP wavenumber κ. 2.3 Numerical Results and Discussion A Mathematica TM program was written and implemented to solve (2.16) using the Newton Raphson method [89] to obtain κ for a specific value of ψ. The program is provided in Appendix B.1. The solutions of the dispersion equation were searched when Re [κ/k 0 ] > 1. The free-space wavelength was fixed at λ 0 = 633 nm. The 17

48 Re Κ k Ψ deg Im Κ k Ψ deg Figure 2.1: (left) Real and (right) imaginary parts of κ as functions of ψ, for SPPwave propagation guided by the planar interface of aluminum and a titanium-oxide SNTF. Either two or three modes are possible, depending on ψ e 1, 2, e 1, 2, h 1, 2, h 1, 2, P 1, 2, z P 1, 2, z Figure 2.2: Variations of components of e (in V m 1 ), h (in A m 1 ), and P (in W m 2 ) with z along the line {x = 0, y = 0}, for κ = ( i )k 0 and ψ = 0. The components parallel to û 1, û 2, and û z, are represented by black solid, red dashed, and blue chain-dashed lines, respectively. The data were computed by setting a p = 1 V m 1, with a s = 0, b 1 = 0, and b 2 = i then obtained using (2.15). metal was taken to be aluminum: ϵ m = 56 + i21. As in Ref. 71, the angles χ v and δ v were taken to be 45 and 30, respectively, and Ω = 200 nm for all results 18

49 e 1, 2, e 1, 2, 4 2 h 1, 2, P 1, 2, z h 1, 2, P 1, 2, z Figure 2.3: Same as Fig. 2.2 except for κ = ( i )k 0. The data were computed by setting a s = 1 V m 1, with a p = 0, b 1 = 1, and b 2 = 0 then obtained using (2.15). Theoretical analysis confirms that û 1 P > 0 for z < 0 for this case. presented in this chapter. Computed values of the real and imaginary parts of κ for the canonical problem are shown in Fig These solutions are organized into three branches. For 0 ψ 36, three values of κ are found which satisfy (2.16) and therefore represent SPP waves. For 36 ψ 90, there are two values of κ which satisfy (2.16). This trend is fully consistent with the conclusions drawn in Ref. 71 for the TKR configuration. The different solutions of (2.16) for any specific value of ψ indicate that the SPP waves have different phase speeds ω/re(κ) as theoretically predicted in Refs. 70 and 71, and experimentally confirmed in Ref. 44 and different e-folding distances 1/Im(κ) along the direction of propagation. Although the wavenumber κ must be real-valued for the TKR configuration [70, 71], the solution branch represented by the blue chain-dashed lines in Fig. 2.1 suggests that κ has to be complex-valued for the canonical problem. As ψ increases from 0, Im(κ) decreases monotonically on this solution branch. For ψ = , κ/k 0 = i and no solutions emerged on this branch for slightly higher values of ψ. 19

50 e 1, 2, 4 2 e 1, 2, h 1, 2, h 1, 2, P 1, 2, z P 1, 2, z Figure 2.4: Same as Fig. 2.2 except for κ = ( i )k 0. The data were computed by setting a p = 1 V m 1, with a s = 0, b 1 = 0, and b 2 = i then obtained using (2.15). In the following subsections, the results obtained for two values of ψ are examined in some detail ψ = 0 Specifically, for ψ = 0 the values of κ which satisfy (2.16) are κ 1 = ( i )k 0, κ 2 = ( i )k 0, and κ 3 = ( i )k 0. These solutions represent SPP waves with wave vectors lying wholly in the morphologically significant plane of the SNTF, as addressed theoretically in Ref. 70 and experimentally in Ref. 44. The Cartesian components of the electric and magnetic field phasors and the time-averaged Poynting vector P = 1Re (E 2 H ) as functions of z along the line (x = 0, y = 0) are shown for κ 1 in Fig. 2.2, and for κ 3 in Fig The Mathematica TM program is provided in Appendix B.2. These SPP waves are p- polarized, and were respectively labeled as p2 and p1 in Ref. 44. Figure 2.3 shows the variations of e, h, and P along the z axis for κ = κ 2. This SPP wave is s-polarized and was labeled as s3 in Ref. 44. The localization of all three SPP 20

51 8 30 e 1, 2, e 1, 2, h 1, 2, h 1, 2, P 1, 2, z P 1, 2, z Figure 2.5: Same as Fig. 2.2 except for κ = (2.459+i )k 0 and ψ = 75. The data were computed by setting a p = 1 V m 1, with a s = i V m 1, b 1 = i5.3299, and b 2 = i then obtained using (2.15). waves around the interface z = 0 is evident from Figs Also, the SPP wave p2 is more localized inside the SNTF than either p1 or s3. Furthermore, the phase speed of p1 is higher than that of s3, which exceeds the phase speed of p2. However, s3 will travel a longer distance along the interface than either p1 or p2, which could not have been deduced from the theoretical analysis for the TKR configuration in Ref. 70. Examination of Figs shows that, after spatial averaging over an appropriate z-range, the component of the time-averaged Poynting vector along the direction of propagation is higher in magnitude in the SNTF than in the metal, regardless of the polarization state of the SPP wave. This means that the energy of the SPP wave primarily resides in the SNTF ψ = 75 Only two values of κ were found to satisfy (2.16) for ψ = 75 : κ 1 = ( i )k 0 and κ 2 = ( i )k 0. The phase speed of the SPP wave with κ = κ 2 exceeds the phase speed of the other SPP wave (κ = κ 1 ), and the 21

52 e 1, 2, 10 5 e 1, 2, h 1, 2, h 1, 2, P 1, 2, z P 1, 2, z Figure 2.6: Same as Fig. 2.2 except for κ = (2.066+i )k 0 and ψ = 75. The data were computed by setting a p = 1 V m 1, with a s = i V m 1, b 1 = i , and b 2 = i then obtained using (2.15). former SPP wave will propagate a longer distance along the interface than the latter SPP wave. The variations of e, h, and P along the z axis are shown in Fig. 2.5 for κ 1, and in Fig. 2.6 for κ 2. These two SPP waves cannot be rigorously classified as either p- or s-polarized, which is consistent with the deductions in Ref. 71 for the TKR configuration. However, the electric field in Fig. 2.5 is predominantly oriented in the plane formed by û 1 and û z, thereby suggesting that the SPP wave for κ 1 could be classified as quasi-p-polarized on both sides of the interface. In contrast, the electric field in Fig. 2.6 is predominantly oriented parallel to û 2, which implies that the SPP wave for κ 2 is quasi-s-polarized on both sides of the interface. Finally, it can be deduced from Figs. 2.5 and 2.6 that the energy content of either SPP wave for ψ = 75 resides primarily in the SNTF, just as of any of the three SPP waves for ψ = 0. 22

53 2.4 Concluding Remarks The solution of the canonical boundary-value problem (i) proved directly that multiple SPP waves of the same frequency but different phase speeds, attenuation rates, spatial profiles, and polarization states can be guided by the planar interface of semi-infinite expanses of the metal and the periodically nonhomogeneous SNTF; and (ii) not only upheld the conclusions obtained in Refs. 44, 70 and 71, but also gave additional information on the e-folding distance along the direction of propagation and the spatial extent of the SPP waves into the metal and the SNTF. The excitation of multiple SPP waves at the metal/sntf interface is studied in Ch. 11 and the results of that chapter are successfully correlated with those obtained in this chapter. 23

54 Chapter 3 SPP Waves Guided by Metal/Rugate-Filter Interface 3.1 Introduction At a specific frequency, the solution of a canonical boundary-value problem [13, 14, 35] shows that only one SPP wave can propagate along the interface, if the partnering dielectric material is isotropic and homogeneous. The same conclusion holds true even if that material is anisotropic [20] [42]. However, if the partnering dielectric material is both anisotropic and periodically nonhomogeneous in the direction normal to the interface, the solutions of the canonical boundary-value problem [26], as is also shown in Ch. 2, shows that more than one SPP waves with different phase speeds, attenuation rates, and field distributions, but of the same frequency [27] can propagate guided by the interface. Experimental verification of this theoretical prediction has been found [43, 44]. On the basis of foregoing discussion, it can be hypothesized that the phenomenon of the propagation of multiple SPP waves guided by a single interface could simply be due to the periodic nonhomogeneity of the dielectric partnering material. In order to test this hypothesis, a canonical problem involving two halfspaces is set up, one filled homogeneously with a metal, the other occupied by an isotropic dielectric material whose refractive index varies sinusoidally along the normal direction (as in a rugate filter [2,74]). The canonical boundary-value problem is formulated in Sec. 3.2, and numerical results are presented and discussed in Sec Concluding remarks are presented in Sec This chapter is based on: M. Faryad and A. Lakhtakia, On surface plasmon-polariton waves guided by the interface of a metal and a rugate filter with sinusoidal refractive-index profile, J. Opt. Soc. Am. B 27, (2010). 24

55 3.2 Theory Let the half-space z 0 be occupied by an isotropic and homogeneous metal with complex-valued relative permittivity scalar ϵ m. The region z 0 is occupied by a rugate filter with relative permittivity [( ) ( ) nb + n a nb n ( a ϵ d (z) = + sin π z ) ] 2, (3.1) 2 2 Ω where n a and n b are lowest and highest indexes of refraction, respectively, and 2Ω is the period. Without loss of generality, let the SPP wave propagate parallel to the unit vector û x guided by the interface z = 0, and attenuate as z ±. Therefore, in the region z 0, the wave vector may be written as k met = κ û x α met û z, (3.2) where κ 2 +αmet 2 = k 2 0 ϵ m, κ is complex-valued, and Im(α met ) > 0 for attenuation as z. Accordingly, the field phasors in the metallic half-space may be written as [ ( αmet E(r) = a p û x + κ ) ] û z + a s û y exp(ik met r), z 0, (3.3) k 0 k 0 and H(r) = η 1 0 [ ( αmet a p ϵ met û y + a s û x + κ )] û z exp(ik met r), z 0. (3.4) k 0 k 0 Here a p and a s are unknown scalars with the same units as the electric field, and the subscripts p and s, respectively, denote the p- (parallel-) and s- (perpendicular-) polarization states with respect to the xz plane. For field representation in the rugate filter, let me write E(r) = e(z) exp (iκx) and H(r) = h(z) exp (iκx). The components e z (z) and h z (z) of the field phasors can be found in terms of the other components as follows: e z (z) = κ ωϵ 0 ϵ r (z) h y(z), z > 0, (3.5) h z (z) = κ ωµ 0 e y (z), z > 0. (3.6) The other components of the electric and magnetic field phasors are used in the column vector [f(z)] = [e x (z) e y (z) h x (z) h y (z)] T (3.7) which satisfies the matrix ordinary differential equation d dz [f(z)] = i [ P (z) ] [f(z)], z > 0, (3.8) 25

56 where the 4 4 matrix [P (z)] = ω + ωϵ 0 ϵ d (z)µ µ µ ϵ 0 ϵ d (z) 0 0 ϵ 0 ϵ d (z) κ µ ϵ 0 ϵ d (z) (3.9) The 4 4 matrix ordinary differential equation (3.8) can be partitioned into two autonomous 2 2 matrix ordinary differential equations, one for p-polarized light involving {e x, h y } and the other for s-polarized light involving {e y, h x }, but it would be better to treat Eq. (3.8) as a whole for both linear polarization states simultaneously, to have a holistic view of the problem. The piecewise uniform approximation technique [1] to determine the matrix [Q] that appears in the relation [f(2ω)] = [Q] [f(0+)] (3.10) to characterize the optical response of one period of the rugate filter for specific values of κ. By virtue of the Floquet Lyapunov theorem [88], a matrix [ Q] can be defined such that { [Q] = exp i2ω[ Q] }. (3.11) Both [Q] and [ Q] share the same (linearly independent) eigenvectors, and their eigenvalues are also related. Let [t] (n), n [1, 4], be the eigenvector corresponding to the nth eigenvalue σ n of [Q]; then, the corresponding eigenvalue α n of [ Q] is given by α n = i ln σ n, n [1, 4]. (3.12) 2Ω After labeling the eigenvalues of [Q] such that Im(α 1,2 ) > 0,it is set [94] [f(0+)] = [ [t] (1) [t] ] [ ] (2) b1 (3.13) for SPP-wave propagation, where b 1 and b 2 are unknown dimensionless scalars; the other two eigenvalues of [ Q] pertain to waves that amplify as z and cannot therefore contribute to the SPP wave. At the same time, [f(0 )] = α met k α met k 0 η 0 ϵ met η b 2 [ ap a s ], (3.14)

57 by virtue of Eqs. (3.3) and (3.4). Continuity of the tangential components of the electric and magnetic field phasors across the plane z = 0 requires that [f(0 )] = [f(0+)], which may be rearranged as the matrix equation [Y ] a p a s b 1 b 2 = (3.15) For a nontrivial solution, the 4 4 matrix [Y ] must be singular, so that det [Y ] = 0 (3.16) is the dispersion equation for the SPP wave. This equation has to be solved in order to determine the SPP wavenumber κ. 3.3 Numerical Results and Discussion A Mathematica TM program was written and implemented to solve Eq. (3.16) using the Newton Raphson method [89], and the solutions were searched when Re [κ/k 0 ] > 1. The free-space wavelength was fixed at λ 0 = 633 nm, and the metal was taken to be bulk aluminum: ϵ m = 56 + i21. While Ω was kept variable, the minimum and maximum indexes of refraction of the rugate filter were fixed from an example provided by Baumeister [74]: n a = 1.45 and n b = The solutions of the dispersion equation, calculated for Ω/λ 0 2, are shown in Fig These solutions are organized in 16 branches: eight branches labeled as s1 s8 represent SPP waves with the s-polarization state, and eight branches labeled as p1 p8 represent SPP waves with the p-polarization state. At any value of Ω 0.145λ 0, there are more than one possible SPP waves. Evidence for the excitement of s-polarized SPP waves guided by the interface of a metal and an isotropic dielectric material does exist, although it is very rare [33, 47]. Clearly from Fig. 3.1, more than one SPP waves can propagate for Ω 0.145λ 0. These SPP waves have different phase speeds ω/re(κ) and attenuation rates 1/Im(κ). As both partnering materials are isotropic, this multiplicity must be surely due to the periodic nonhomogeneity of the dielectric partnering material (i.e., the rugate filter) which is the chief result presented in this chapter. Although the observability of two SPP waves one of either linear polarization state when the partnering dielectric material is periodically nonhomogeneous in a piecewise uniform fashion had recently been established numerically for the 1 Figures 3.1 and 3.4 are updated versions of Figs. 1 and 4, respectively, in: M. Faryad and A. Lakhtakia, On surface plasmon-polariton waves guided by the interface of a metal and a rugate filter with sinusoidal refractive-index profile, J. Opt. Soc. Am. B 27, (2010). 27

58 Sarid and the TKR configurations [33], the solution of the canonical boundaryvalue problem not only provides necessary mathematical rigor but also shows that the number of simultaneously excitable SPP waves can exceed two. Re k p1 p2 p3 s1 p4 p5 p6 p7 p8 s2 s3 s4 s5 s6 s7 s Im k Figure 3.1: (left) Real and (right) imaginary parts of κ/k 0 as functions of Ω/λ 0 for SPP-wave propagation guided by the planar interface of aluminum and a rugate filter described by Eq. (3.1) with n a = 1.45 and n b = Representative plots of the phasors e and h, and of the time-averaged Poynting vector P = 1 2 Re(E H ) along the line {x = 0, y = 0}, are given in Figs. 3.2 and 3.3 as functions of z. For Fig. 3.2, two solutions one at Ω = 0.1λ 0 and the other at Ω = λ 0 from the branch p8 were selected and e, h, and P were computed by setting a p = 1 V m 1. These plots represent p-polarized SPP waves. For both solutions, the maximums of the fields and the power density in the metal lie at z = 0. But the distributions inside the rugate filter are different for the two solutions: the maximums lie at z = 0+ when Ω = 0.1λ 0, but at z 0.5Ω when Ω = λ 0. Profiles of the fields and the power density are shown in Fig. 3.3 for two solutions on the branch s2: Ω = λ 0 and Ω = 1.5λ 0. The data were computed by setting a s = 1 V m 1. The plots show the localization of SPP waves to the interface. The power density resides in the rugate filter almost wholly within a distance equal to the period 2Ω for both cases. The profiles in the metal are of the same type as in Fig

59 e x,y, h x,y, P x,y,z e x,y, h x,y, P x,y,z Figure 3.2: Variations with z of the Cartesian components of e (in V m 1 ), h (in A m 1 ), and P (in W m 2 ) along the line {x = 0, y = 0}. The components parallel to û x, û y, and û z, are represented by red solid, blue dashed, and black chaindashed lines, respectively. The data were computed by setting a p = 1 V m 1. (left) Ω/λ 0 = 0.1, κ/k 0 = i, and (right) Ω/λ 0 = 1, κ/k 0 = i. Both solutions lie on the branch labeled p8 in Fig As the period 2Ω of rugate filter increases in relation to λ 0, the branches in Fig. 3.1 come closer to each other. At some very high value of Ω/λ 0, only one solution indicating a p-polarized SPP wave should survive, because the rugate filter would be virtually homogeneous over 1.5λ 0 closest to the interface. Although this possibility could be tested by setting a very large value of Ω/λ 0, the correct identification of α 1,2 became problematic for Ω/λ 0 > 2, owing to numerical errors in computing [Q]. In order to overcome this difficulty, I modified the ϵ d (z) of the rugate filter to [( ) ( ) nb + n a nb n ( a ϵ d (z) = + γ sin π z ) ] 2, γ [0, 1], (3.17) 2 2 Ω with Ω = 2λ 0 fixed, and decreased the parameter γ from 1 to Close to the interface, a decrease in γ tantamounts to an increase in Ω. 29

60 e x,y, h x,y, P x,y,z e x,y, h x,y, P x,y,z Figure 3.3: Same as Fig. 3.2 except for (left) Ω/λ 0 = 1, κ/k 0 = i, and (right) Ω/λ 0 = 1.5, κ/k 0 = i, and the data were computed by setting a s = 1 V m 1. Both solutions lie on the branch labeled s2 in Fig In Fig. 3.4, the calculated solutions of the dispersion equation are organized as functions of γ [0.001, 1] in nine branches. The branches labeled as p1 p5, p7, s1, s4, and s8 are continuations of the similarly labeled branches in Fig. 3.1, while two new branches p9 and p10 emerge. All solution branches in Figs. 3.1 and 3.4 die out when the imaginary part of κ decreases to zero, the sole exception being the branch p10. This branch extends to γ = in Fig As γ decreases even further, the branch p10 approaches the solution for a metal/dielectric interface with the relative permittivity of the dielectric partnering material uniform at ϵ d = (1/4) (n a + n b ) 2. For instance, at γ = 0.001, the solution of the dispersion equation gives a p-polarized SPP wave with wavenumber κ = ( i)k 0, while the p-polarized wave at the interface of aluminum and a homogeneous isotropic dielectric with relative permittivity ϵ d = has wavenumber κ = ( i)k 0 [34, 35]. Figure 3.5 shows the profiles of the fields and the time-averaged Poynting vector for (i) γ = 0.5 and κ/k 0 = i on the branch labeled p3 in Fig. 3.4 and (ii) γ = 0.1 and κ/k 0 = i on the branch labeled p10 in the same figure. Strong localization to the interface of the metal and the rugate filter 30

61 Re k p1 p2 p3 s1 p4 p5 p7 p9 p10 s3 s4 s Im k Figure 3.4: (left) Real and (right) imaginary parts of κ/k 0 as functions γ [1, 0.001] with Ω = 2λ 0 for SPP-wave propagation guided by the planar interface of aluminum and a rugate filter described by Eq. (3.17) with n a = 1.45, n b = 2.32, and Ω = 2λ 0. is indicated, with the field maximum inside the rugate filter being found within half a period of the interface. Neither in Fig. 3.1 nor in Fig. 3.4 does a solution branch for the p-polarization state intersect a solution branch for the s-polarization state. Accordingly, SPP waves guided by the interface under study cannot have a polarization state other than linear. The spatial profiles provided in Figs. 3.2, 3.3 and 3.5 are useful for experimentally implementing the TKR configuration [13, 14] in order to observe and exploit SPP-wave propagation guided by the planar interface of a metal and a rugate filter. Clearly then, the rugate filter cannot be semi-infinitely thick, but it must be sufficiently thick so that reflection from its back surface has negligible significance [43, 90]. The presented spatial profiles show that, in the rugate filter, the SPP waves are confined within three structural periods of the interface with the metal. So a four-period-thick rugate filter deposited upon a metallic thin film, which is thicker than the penetration depth of the metal, can be used in the TKR configuration to excite SPP waves. 31

62 e x,y, h x,y, e x,y, h x,y, P x,y,z P x,y,z Figure 3.5: Same as Fig. 3.2 except for (left) γ = 0.5 and κ/k 0 = i on the branch labeled p3 in Fig. 3.4, and (right) γ = 0.1 and κ/k 0 = i on on the branch labeled p10 in Fig Concluding Remarks SPP-wave propagation guided by the interface of a metal and a rugate filter with a sinusoidal refractive-index profile was studied to validate the hypothesis that the periodic nonhomogeneity of the dielectric partnering material is responsible for the existence of multiple SPP waves, all of the same frequency but different phase speeds, attenuation rates, linear polarization states, and field distributions. Moreover, it was found that (i) the period of the nonhomogeneity must exceed a minimum value for more than one SPP waves to exist, and (ii) only a single p-polarized SPP wave can propagate when the period becomes very large such that the rugate filter is essentially uniform close to the interface. The results obtained in this chapter are used as a guide to excite multiple SPP waves in the TKR and the grating-coupled configurations in Ch. 8 and 9, respectively. 32

63 Chapter 4 Propagation of Multiple Fano Waves 4.1 Introduction Fano waves are the surface waves that are guided by the planar interface of two isotropic, lossless, homogeneous mediums with relative permittivities of opposite signs [11,91,92]. Similar to SPP waves which arise when the negative-permittivity partnering medium is not lossless [92,93], Fano waves have the following properties: (i) unattenuated propagation along the interface with a phase speed smaller than the phase speed of light in the positive-permittivity partnering medium, (ii) exponentially decaying field amplitudes on both sides of the interface, and (iii) a p-polarization state. The magnitude of the permittivity of the negative-permittivity partnering medium must exceed that of the positive-permittivity partnering medium. At a given frequency, at most one Fano wave can propagate in a specified direction along the interface. As has been shown in Ch. 3, the planar interface of a metal and an isotropic, lossless, periodically nonhomogeneous, dielectric medium (a rugate filter) with positive permittivity can guide multiple SPP waves at a specific frequency in the optical regime. For this chapter, it was investigated if multiple Fano waves could also be supported by the interface of a positive-permittivity rugate filter and a negative-permittivity dielectric medium, both of which are isotropic and lossless. The theoretical formulation to tackle the underlying canonical boundary-value problem is exactly the same as in Ch. 3, for which reason it is not repeated in this This chapter is based on: M. Faryad, H. Maab, and A. Lakhtakia, Rugate-filter-guided propagation of multiple Fano waves, J. Opt. (United Kingdom) 13, (2011). 33

64 chapter. Let me proceed directly to the presentation and discussion of illustrative numerical results in Sec Concluding remarks are presented in Sec Numerical Results and Discussion Let the half-space z < 0 be occupied by an isotropic and homogeneous medium with relative permittivity ϵ m < 0. The region z > 0 is occupied by a semi-infinite rugate filter with relative permittivity [( ) ( ) nb + n a nb n ( a ϵ d (z) = + sin π z ) ] 2, z > 0, (4.1) 2 2 Ω where n b > n a > 0 and 2Ω is the period. The variation of relative permittivity along the z axis is shown in Fig An exp( iωt) time-dependence is implicit, with ω denoting the angular frequency. Without loss of generality, surface-wave propagation is taken to occur along the x axis with an exp(iκx) dependence. Field amplitudes must decay as z ±. The free-space wavenumber and the freespace wavelength are denoted by k 0 = ω ϵ 0 µ 0 and λ 0 = 2π/k 0, respectively, with µ 0 and ϵ 0 being the permeability and permittivity of free space. Before proceeding to the results for the current boundary-value problem, let me consider the case when n b = n a so that the rugate filter is replaced by a homogeneous medium. A p-polarized surface wave can then propagate with wavenumber κ = k 0 n a ϵm / (n 2 a + ϵ m ), (4.2) nb =n a but only if ϵ m < n 2 a. (4.3) The situation changes dramatically when n b and n a are dissimilar with ϵ d (z) > 0 z > 0, of course. This becomes evident from Fig. 4.2, which contains solutions of the dispersion equation for surface-wave propagation when ϵ m [ 6, 0], Ω = λ 0 = 633 nm, n a = 1.45, and n b = The minimum and maximum indexes of refraction of the rugate filter were fixed from an example provided by Baumeister [74], and the search for the solutions of the dispersion equation was restricted to κ/k 0 > n a. Nine solutions of the dispersion equation were found for ϵ m n 2 b and up to eight solutions were found for ϵ m ( n 2 b, 0]; indeed, solutions exist even when ϵ m > n 2 a. Since κ/k 0 is real-valued for all the solutions found, the wave propagation is lossless. Some solutions possess the p-polarization state, the others being s-polarized. Clearly, the presence of periodic nonhomogeneity in the positivepermittivity dielectric partnering medium has (i) engendered multiple Fano waves of two different linear polarization states, and 34

65 Relative Permittivity Figure 4.1: Variation of relative permittivity along the z axis for n a = 1.45, n b = 2.32, and ϵ m = 2. Although the semi-infinite rugate filter depicted here is a continuously nonhomogeneous medium, it can also be piecewise homogeneous. / k m Figure 4.2: Relative wavenuber κ/k 0 versus ϵ m [ 6, 0] for Fano-wave propagation when Ω = λ 0 = 633 nm, n a = 1.45, and n b = The red circles represent s- polarized, while the black triangles represent p-polarized, Fano waves. The gap in one of the solution branches appears to be a numerical artifact. 35

66 (ii) permitted Fano-wave propagation at low values of ϵ m, even as low as 0. Without that periodicity, only one Fano wave is possible, that too if ϵ m is sufficiently large. The exponential decay rate( normal to the interface in the half-space z < 0 is ϵm ) proportional to Im (α m ) = Im k 2 0 κ 2 > 0. Since ϵ m < 0, α m is purely imaginary for real-valued κ, signifying a very high attenuation in the half-space z < 0. Moreover, for fixed ϵ m, the attenuation rate in the half space z < 0 is higher for a Fano wave with a higher κ. For SPP-wave propagation, α m is generally a complex number because both ϵ m and κ are complex-valued. 6 6 E x,y, 4 2 E x,y, H x,y, H x,y, Figure 4.3: Variations of the magnitudes of the Cartesian components of electric and magnetic field phasors (in V m 1 and A m 1, respectively) with z. The x-, y-, and z-directed components are represented by solid red, blue dashed, and black chain-dashed lines, respectively for ϵ m = 6. Left: κ/k 0 = and p-polarization state. Right: κ/k 0 = and s-polarization state. Spatial profiles of the magnitudes of the Cartesian components of the electric and magnetic field phasors along a line normal to the interface are given in Fig. 4.3 for two Fano waves, one p-polarized and the other s-polarized, when ϵ m = 6. The figure shows relatively strong localization of the p-polarized wave to the plane z = 0 than of the s-polarized wave due to the higher value of κ for the former wave. Seven other Fano waves are also possible, per Fig. 4.2, and their spatial profiles are qualitatively similar to the ones presented. The spatial profiles in Fig. 4.4 are for two of the eight Fano waves possible when ϵ m = 0. One of the two spatial profiles is for a p-polarized Fano wave, the 36

67 4 2 E x,y, E x,y, H x,y, H x,y, Figure 4.4: Same as Fig. 4.3 except for ϵ m = 0. Left: κ/k 0 = and p- polarization state. Right: κ/k 0 = and s-polarization state. / k m Figure 4.5: Same as Fig. 4.2, except that ϵ m [0, 2]. The waves represented by these solutions have to be classified as Tamm waves [16]. other for an s-polarized Fano wave. Since ϵ m = 0, all components of the magnetic 37

68 field phasor vanish in the half space z < 0 for the p-polarized Fano wave, which means that energy transport occurs only in the rugate filter. As noted previously, when n b = n a > 0, Fano-wave propagation can occur only if ϵ m is sufficiently negative; surface-wave propagation is impossible if ϵ m > 0. However, when n b > n a, surface-wave propagation can occur even if ϵ m is positive. Solutions of the dispersion equation are presented in Fig. 4.5 for the same parameters as for Fig. 4.2, except that ϵ m [0, 2]. Such surface waves have to be classified as Tamm waves [16]. Just a change in the sign of the relative permittivity of the homogeneous medium occupying the half space z < 0 leads to the propagation of Tamm/Fano waves instead of Fano/Tamm waves, the periodically nonhomogeneous medium in the other half space remaining unchanged. Thus, the hitherto different concepts of Fano waves and Tamm waves can be conceptually unified. Parenthetically, let me also note that Fano waves are replaced by surface plasmon-polariton waves, if the homogeneous medium in the region z < 0 with Re(ϵ m ) < 0 is also dissipative. 4.3 Concluding Remarks A surface wave, called a Fano wave, can be guided by the interface of two isotropic, homogeneous, lossless, dielectric mediums only (i) if the product of their relative permittivities is negative, and (ii) the magnitude of the permittivity of the negative-permittivity partnering medium is sufficiently high. It has been shown in this chapter that multiple Fano waves with different phase speeds and polarization states can propagate if the positive-permittivity partnering medium is periodically nonhomogeneous normal to the interface. No restriction exists on the magnitude of the permittivity of the negative-permittivity partnering medium. Furthermore, (i) the additional Fano waves, whose creation can be attributed to the periodic nonhomogeneity of the medium occupying the half space z > 0, are not waveguide modes; and (ii) Fano waves transmute into Tamm waves when both partnering mediums have positive permittivities. These findings buttress the hypothesis that periodic nonhomogeneity normal to the interface results in the possibility of multiple surface waves as was seen in the last chapter. 38

69 Chapter 5 Dyakonov Tamm Waves Guided by a Phase-Twist Defect in an SNTF 5.1 Introduction Gao et al. [65 68] theoretically examined the propagation of Dyakonov Tamm waves guided by a twist defect in a chiral STF. Most significantly, they found that multiple Dyakonov Tamm waves of the same frequency, but different phase speeds, field distributions, and the degrees of localization to the interface can be guided by the interface. The same conclusion was found to hold true for the propagation of the Dyakonov Tamm waves guided by an interface between two chiral STFs that differ only in handedness [68]. In both instances, the most strongly localized Dyakonov Tamm waves are essentially confined within two or three structural periods normal to the interface. The propagation of Dyakonov Tamm waves guided by the interface formed by a phase-twist combination defect in a periodically nonhomogeneous SNTF is studied in this chapter in order to shed light on the effect of the morphology of the periodically nonhomogeneous STF and the degree of localization of the Dyakonov Tamm waves to the interface. The relevant canonical boundary-value problem is formulated in Sec Numerical results are discussed in Sec. 5.3, and concluding remarks are presented in Sec This chapter is based on: M. Faryad and A. Lakhtakia, Dyakonov Tamm waves guided by a phase-twist combination defect in a sculptured nematic thin film, Opt. Commun. 284, (2011). 39

70 5.2 Theory The canonical boundary-value problem to be formulated is described as follows. Let the half-spaces z < 0 and z > 0 be occupied by the chosen SNTF with periodically nonhomogeneous permittivity dyadic [64] ε SNT F (z) = ϵ 0 S z (γ ± ) S y (z) ε (z) S 1(z) S 1 ref y z (γ± ), z 0. (5.1) The expression for ε SNT F (z) given here is more general than the one given in Sec. 1.3 because it involves a new rotation matrix S z (γ ± ) which describes the angle γ ± between the x axis and the morphologically significant plane of the SNTF. This dyadic is S z (γ ± ) = (û x û x + û y û y ) cos γ ± + (û y û x û x û y ) sin γ ± + û z û z. (5.2) Thus, the morphologically significant plane is formed by the unit vectors û z and û x cos γ ± +û y sin γ ± for z 0, and γ + γ is the twist between the morphologically significant planes on the two sides of the interface. The remaining dyadics in Eq. (6.1) are explained in Sec. 1.3 whereas the vapor incidence angle χ v (z) is taken to vary sinusoidally [44]: ( πz ) χ v (z) = χ v + δ v sin Ω ± ϕ±, z 0, (5.3) where χ v is the mean value and δ v the amplitude of sinusoidal variation of the vapor incidence angle, and 2Ω is the period of the SNTF normal to the plane z = 0. The phase defect is introduced through the structural phase constants ϕ + and ϕ in the half-spaces z > 0 and z < 0, respectively. The geometry of the problem for γ + = γ is shown schematically in Fig The phase-twist combination defect can be introduced during the vapor deposition process as follows. Suppose that an SNTF is being deposited by rocking the substrate sinusoidally about an axis passing through the plane of the substrate, in accordance with Eq. (5.3) [44]. First, the rocking is stopped and the vapor flux is shut off. Then, the substrate is rotated by an angle γ + γ about an axis that is normal to the substrate plane [95, 96], and the vapor deposition angle is changed by ϕ + ϕ. Finally, the vapor flux is turned on and the sinusoidal rocking is resumed. Since γ + and γ are independent of each other, let me fix the direction of wave propagation in the xy plane to be parallel to the x axis; the direction of propagation is thus variable relative to the material coordinate system intrinsic to the anisotropy of the SNTF. Accordingly, E(r) = e(z) exp (iκx), H(r) = h(z) exp (iκx), (5.4) where the wavenumber κ is a complex-valued scalar, and create the column 4- vector [f(z)] = [e x (z) e y (z) h x (z) h y (z)] T. (5.5) 40

71 z SNTF 2Ω x, y Figure 5.1: Schematic illustration of the geometry of the problem, when γ + = γ. Whereas the axial field components e z (z) and h z (z) can be expressed as e z (z) κ 0 h z (z) = [ A ± (z) ] [f(z)], z 0, (5.6) 0 through the 4 4 matrix [ A ± (z) ] = κ2 ωϵ κ 2 ωµ κ ϵ d(z) [ϵ a (z) ϵ b (z)] ϵ a (z) ϵ b (z) containing ϵ d (z) ϵ a(z) ϵ b (z) sin [χ(z)] cos [χ(z)] cos γ ± sin γ ± sin γ ± cos γ ± (5.7) ϵ d (z) = ϵ a (z) ϵ b (z)/ { ϵ a (z) cos 2 [χ(z)] + ϵ b (z) sin 2 [χ(z)] }. (5.8) The column 4-vector [f(z)] satisfies the matrix ordinary differential equation d dz [f(z)] = i [ P ± (z) ] [f(z)], z 0, (5.9) 41

72 where the 4 4 matrix [P ± (z)] = [A ± (z)] + ω µ µ 0 0 ϵ 0 [ϵ c (z) ϵ d (z)] cos γ ± sin γ ± ϵ 0 [ ϵc (z) cos 2 γ ± + ϵ d (z) sin 2 γ ±] 0 0 ϵ 0 [ ϵc (z) sin 2 γ ± + ϵ d (z) cos 2 γ ±] ϵ 0 [ϵ c (z) ϵ d (z)] cos γ ± sin γ ± 0 0 Equation (5.9) requires numerical solution by the piecewise uniform approximation technique [64]. The ultimate aim is to determine the matrixes ] [Q + and ] [Q that characterize the optical response of one period of the SNTF on either side of the defect as follows: ] [f(±2ω)] = [Q ± [f(0±)]. (5.11) By virtue of the Floquet Lyapunov theorem [88], the matrixes [ Q + ] and [ Q ] can be defined such that { [Q ± ] = exp ±i2ω[ Q } ± ]. (5.12) Both [Q ± ] and [ Q ± ] share the same eigenvectors, and their eigenvalues are also related as follows. Let [t ± ]] (n), n [1, 4], be the eigenvector corresponding to the nth eigenvalue σ n ± of [Q ± ; then, the corresponding eigenvalue α n ± of [ Q ± ] is given by α ± n = i ln σ± n 2Ω. (5.13) The electromagnetic fields of the Dyakonov Tamm wave must diminish in magnitude as z ±. Therefore, in the half-space z > 0, the eigenvalues of [ Q + ] are labeled such that Im [ α 1,2] + > 0 and then set [94] [ [ ] A + [f (0+)] = [t + ] (1) [t + ] (2) 1 A + 2 ], (5.14) where A + 1,2 are unknown scalars; the other two eigenvalues of [ Q + ] describe fields that amplify as z + and cannot therefore contribute to the Dyakonov Tamm wave. A similar argument for the half-space z < 0 requires us to ensure that Im [ α1,2] < 0 and then set [94] [ ] [ ] A [f (0 )] = [t ] (1) [t ] (2) 1, (5.15) 42 A 2. (5.10)

73 where A 1,2 are unknown scalars. Using Eqs. (5.14) and (5.15) to satisfy the continuity condition [f (0 )] = [f (0+)], a matrix equation is obtained that may be rearranged as A A [M] + 2 A 1 = 0 0, (5.16) 0 A 2 where [ [M] = ] [t + ] (1) [t + ] (2) [t ] (1) [t ] (2). (5.17) For a nontrivial solution, the 4 4 matrix [ M(κ) ] must be singular, so that det [ M(κ) ] = 0 (5.18) is the dispersion equation for the Dyakonov Tamm wave. 5.3 Numerical Results and Discussion The Newton-Raphson technique [89] was implemented on Mathematica TM to solve the dispersion equation (5.18) for κ. The code for the program is provided in Appendix B.3. The free-space wavelength was fixed at λ 0 = 633 nm and the halfperiod Ω = 200 nm for all calculations reported here. The angles χ v and δ v were taken to be 45 and 30, respectively, for all results presented in this chapter. The phase constants ϕ + and ϕ were taken to be 0 and 180, respectively, to generate a 180 -phase defect. The angles γ + and γ were left as variables in the computer program. Moreover, the solutions of the dispersion equation were searched when κ/k 0 > 1. The matrixes [Q + ] and [Q ] were calculated using the piecewise uniform approximation technique. This technique consists of subdividing each period of the SNTF into a cascade of electrically thin sublayers parallel to the plane z = 0, and assuming the dielectric properties to be spatially uniform in each sublayer. A sufficiently large number N + 1 points z n ± = ±2Ω (n/n), n [0, N], are defined on each side of the phase-twist combination defect and the matrixes [ ( z ± )] } [W {±i ± ] = exp P ± n 1 + z n ± 2Ω, n [1, N], (5.19) n 2 N are calculated for a specific value of κ; then [Q ± ] = [W ± N ] [W ± N 1 ]... [W ± 2 ] [W ± 1 ]. (5.20) 43

74 A sublayer thickness of 2Ω/N = 2 nm gave reasonable results. Due to the symmetry of the problem, the solutions for γ + [180, 360 ] are the same as for γ + [0, 180 ], at a specific value of γ. Also, the solutions for {γ, γ + } are the same as for {180 γ, 180 γ + } = 0 o 2.05 = 0 o = 30 o 2.04 = 30 o k o = 60 o = 90 o (a) (deg) k o = 60 o = 90 o (b) (deg) k o = 0 o = 30 o = 60 o = 90 o (c) (deg) k o = 0 o = 30 o = 60 o = 90 o (d) (deg) Figure 5.2: The solutions κ/k 0 of the dispersion equation (5.18) as functions of γ + for certain specific values of γ. (a) First, (b) second, (c) third, and (d) fourth sets of solutions. 44

75 5.3.1 Multiple solutions of dispersion equation Four sets of solutions of the dispersion equation (5.18) are found for γ + [0, 180 ] when γ was fixed in the interval [0, 90 ]. Representative results for γ = 0, 30, 60, and 90 are presented in Fig. 5.2 as functions of γ +. The values of κ/k 0 lie between 1.3 and 1.4 in the first set of solutions, presented in Fig. 5.2(a). Solutions in this set exist (i) for γ + [0, 65 ] [115, 180 ] when γ = 0, 30 and 60 ; and (ii) for γ + [0, 55 ] [125, 180 ] when γ = 90. It can be noted that solutions do not exist at and in some neighborhood of γ + = 90. The second set of solutions is given in Fig. 5.2(b). These solutions span the whole range of γ + for all values of γ, and the values of the relative wavenumber κ/k 0 lies between 2.0 and The third set of solutions is given in Fig. 5.2(c). The values of the relative wavenumber κ/k 0 lie between 2.23 and A solution in the third set exists (i) for γ + [0, 180 ] when γ = 0, (ii) for γ + [0, 65 ] [115, 180 ] when γ = 30, (iii) for γ + [0, 45 ] [135, 180 ] when γ = 60, and (iv) for γ + [0, 15 ] [165, 180 ] when γ = 90. The absence of solutions at and in some neighborhood of γ + = 90, for γ > 0, is reminiscent of the first set. The γ + -range in which no solution exists widens as γ increases towards 90. The solutions in the fourth set are shown in Fig. 5.2(d). Just like in the second set, the solutions in the fourth set spans the whole range of γ + for all values of γ. The values of the relative wavenumber κ/k 0 lie between 2.9 and Decay constants As mentioned in Sec. 5.2, Dyakonov Tamm waves must decay away from the interface. To ensure this, the eigenvalues of [ Q + ] are labeled such that Im [ α + 1,2] > 0, and the eigenvalues of [ Q ] such that Im [ α 1,2] < 0. To study the decay of Dyakonov Tamm waves at distances sufficiently far away from the interface, the constants that represent the decay after one period (2Ω) of the SNTF are calculated. Since there are two eigenvalues each of [ Q + ] and [ Q ] that appear in the representation of a Dyakonov Tamm wave, there are two decay constants 45

76 exp(-v 2 ) exp(-u 2 ) exp(-v 1 ) exp(-u 1 ) = 0 o = 30 o = 60 o = 90 o (a) (deg) exp(-u 1 ) exp(-v 1 ) exp(-u 2 ) exp(-v 2 ) 0.4 = 0 o = 30 o = 60 o = 90 o (b) (deg) exp(-u 1 ) exp(-v 1 ) exp(-u 2 ) exp(-v 2 ) = 0 o = 30 o = 60 o = 90 o (c) (deg) exp(-v 1 ) exp(-u 2 ) exp(-v 2 ) exp(-u 1 ) = 0 o = 30 o = 60 o = 90 o (d) (deg) Figure 5.3: The decay constants exp( u 1 ), exp( u 2 ), exp( v 1 ), and exp( v 2 ) for the (a) first, (b) second, (c) third, and (d) fourth set of solutions in Fig on either side of the interface. Four decay constants are defined [65, 67, 68] as exp( u 1 ), exp( u 2 ), exp( v 1 ), and exp( v 2 ), where u 1,2 = 2Ω Im [ α 1,2] + and v 1,2 = 2Ω Im [ α1,2]. The decay constants for all solutions of the dispersion equation provided in Fig. 5.2 are presented in Fig The degree of localization of the Dyakonov Tamm wave to the defect interface depends on the value of the decay constants. The Dyakonov Tamm wave is strongly localized if the decay 46

77 1.5 E x,y, H x,y, P x,y,z Figure 5.4: Variations with z of the magnitudes of the Cartesian components of E (in V m 1 ), H (in A m 1 ), and P (in W m 2 ), when γ = 60, γ + = 30, and κ/k 0 = The components parallel to û x, û y, and û z, are represented by red solid, blue dashed, and black chain-dashed lines, respectively. constants are closer to zero. If the decay constants are close to unity, the wave is very loosely bound to the defect interface. The decay constants corresponding to the first set of solutions are given in Fig. 5.3(a). These decay constants vary between 0.7 and Particularly, the decay constants exp( u 1 ) and exp( v 1 ) vary between 0.9 and 0.99 for all values of γ + and γ, for which a solution of the dispersion equation exists. The very low decay rates implies that the solutions given in Fig. 5.2(a) represent those Dyakonov Tamm waves that are not tightly bound to the interface z = 0. Figure 5.3(b) shows that the decay constants for the second set of solutions vary between 0.1 and Hence, the solutions given in Fig. 5.2(b) represent Dyakonov Tamm waves that are strongly localized to the interface. The decay constants corresponding to the solutions in the third set vary widely between 0.0 and 0.9, as can be seen in Fig. 5.3(c). It can be concluded from the figure that as γ increases, the range of γ + for the propagation of Dyakonov Tamm waves reduces, because the 47

78 E x,y, H x,y, P x,y,z Figure 5.5: Same as Fig. 5.4 except that κ/k 0 = decay constant exp( v 1 ) grows rapidly as γ + either increases from 0 or decreases from 180. The growth rate of exp( v 1 ) is higher at a higher value of γ. Finally, the decay constants for the solutions in Fig. 5.2(d) are given in Fig. 5.3(d). These decay constants lie between 0.0 and Evidently, the Dyakonov Tamm waves corresponding to the solutions in Fig. 5.2(d) are the most tightly bound to the phase-twist combination defect Spatial profiles In order to further illustrate the anatomy of Dyakonov Tamm waves, the spatial profiles of the electric field phasor E, the magnetic field phasor H and the timeaveraged Poynting vector P = (1/2) Re (E H ) are computed for all four solutions of the dispersion equation (5.18) when γ = 60 and γ + = 30. The coefficients A + 2, A 1, and A 2 were computed using Eq. (5.16) after setting A + 1 = 1 V m 1 therein. Spatial profiles of the four solutions are presented in Figs The spatial profiles for the first solution (κ/k 0 = ) are presented in Fig The decay is faster in the half-space z > 0 relative to the half-space z < 0, 48

79 E x,y, H x,y, P x,y,z Figure 5.6: Same as Fig. 5.4 except that κ/k 0 = as also signified by the following values of the four decay constants: exp( u 1 ) = 0.94 and exp( u 2 ) = 0.74 for z > 0, exp( v 1 ) = 0.99 and exp( v 2 ) = 0.84 for z < 0. The spatial profiles (not shown) and decay constants for other solutions in Fig. 5.2(a) show qualitatively similar disparities, thereby permitting the conclusion that the Dyakonov Tamm waves from the first set of solutions are quite loosely bound to the phase-twist combination defect. In Fig. 5.5, the spatial profiles of the magnitudes of the components of E, H, and P are given for the second solution (κ/k 0 = ). The Dyakonov Tamm wave is confined within one period of the SNTF on either side of the defect. The asymmetry in the spatial profiles is due to the asymmetry of the arrangement of the morphologically significant planes of the SNTF on either side of the interface. The same observations and conclusions hold for the spatial profiles depicted in Fig. 5.6 for the third solution (κ/k 0 = ) as well as for the spatial profiles depicted in Fig. 5.7 for the fourth solution (κ/k 0 = ). Thus, it can be concluded that all of these three Dyakonov Tamm waves are tightly bound to the phase-twist combination defect. 49

80 E x,y, H x,y, P x,y,z Figure 5.7: Same as Fig. 5.4 except that κ/k 0 = Concluding Remarks The canonical boundary-value problem of surface-wave propagation guided by a phase-twist combination defect in a periodically nonhomogeneous SNTF is formulated and numerically solved. A structural phase difference ϕ + ϕ was fixed at 180 while the twist γ + γ between the morphologically significant planes of the SNTF on either side of the combination defect was kept variable. The direction of propagation (in the plane of the defect interface) was varied by choosing various values of γ + and γ. It was found that (i) multiple Dyakonov Tamm waves can propagate, guided by the combination defect, depending upon the twist and the direction of propagation, with different phase speeds and degrees of localization to the interface; and (ii) the most strongly localized Dyakonov Tamm waves are confined within one structural period of the SNTF on either side of the defect interface. Conclusion (ii) is in contrast to the results obtained for Dyakonov Tamm waves guided by a twist defect in a chiral STF, where Dyakonov Tamm waves are at 50

81 best confined within two structural periods. As the chiral STFs of Gao et al. [65, 67] and the SNTFs in this chapter are taken to be made of the same material (titanium oxide), and the periods differ by just 1.5%, the difference in the highest degrees of localization must be attributed to the differences in the morphological dimensionality of the two types of STFs: (i) the morphology of a chiral STF requires a three-dimensional curve for its description, whereas (ii) that of an SNTF only a two-dimensional curve. Let me note that only Tamm waves will be guided by the structural defect if the partnering dielectric materials of this chapter were to be made isotropic, as has been shown in Appendix A. 51

82 Chapter 6 SPP Waves Guided by a Metal Slab in an SNTF 6.1 Introduction The possibility of multiple SPP waves localized to the single interface of a metal and a periodically nonhomogeneous SNTF has been demonstrated both theoretically in Ch. 2 and elsewhere [70, 71], and experimentally [44]. Also, SPP-wave propagation along the interface of a metal and a chiral STF has been shown both theoretically [26] and experimentally [43] to admit more than one type of SPP waves. A further increase in the number of SPP waves would require the use of multiple parallel metal/dielectric interfaces, which is already established well with isotropic dielectric materials [ ]. Pursuing this line of thinking, the canonical problem of wave propagation guided by a sufficiently thin metallic slab inserted in a periodically nonhomogeneous SNTF that completely occupies the half-spaces on both sides of the slab, is undertaken in this chapter. In Sec. 6.2, the theoretical formulation of the canonical boundary-value problem is presented, where a dispersion equation is obtained. Representative numerical results are discussed in Sec. 6.3 for a metallic slab made of bulk aluminum and for an electron-beam evaporated thin film. Finally, concluding remarks are given in Sec It must be noted that the canonical problem treated here underlies pragmatic configurations to actually excite and exploit SPP waves [35, 116, 117].. This chapter is based on: (i) M. Faryad and A. Lakhtakia, Surface plasmon-polariton wave propagation guided by a metal slab in a sculptured nematic thin film, J. Opt. (United Kingdom) 12, (2010); (ii) M. Faryad and A. Lakhtakia, Multiple surface-plasmon-polariton waves localized to a metallic defect layer in a sculptured nematic thin film, Phys. Status Solidi Rapid Res. Lett. 4, (2010); and (iii) M. Faryad and A. Lakhtakia, Coupled surface-plasmonpolariton waves in a sculptured nematic thin film with a metallic defect layer, Proc. SPIE 7766, 77660L (2010). 52

83 6.2 Canonical Boundary-Value Problem Suppose that the region L z L + is occupied by an isotropic and homogeneous metal with complex-valued relative permittivity scalar ϵ m. The thickness of the metal slab is denoted by L met = L + L. The regions z L ± are occupied by the chosen SNTF with periodically nonhomogeneous permittivity dyadic [70, 71] ϵ SNT F (z) = ϵ 0 S z (γ ± ) S y (z) ϵ o ref (z) S 1 y (z) S 1 z (γ± ), z L ±, (6.1) where the locally orthorhombic symmetry is expressed through the diagonal dyadic and the local tilt dyadic ϵ o ref (z) = ϵ a(z) û z û z + ϵ b (z) û x û x + ϵ c (z) û y û y (6.2) S y (z) = (û x û x + û z û z ) cos [χ(z)] + (û z û x û x û z ) sin [χ(z)] + û y û y (6.3) expresses nematicity. Both the relative permittivity scalars ϵ a,b,c (z) and the tilt angle χ(z) are supposed to have been nano-engineered by a periodic variation of the direction of the vapor flux during fabrication by physical vapor deposition [1, 44]. This periodic variation is captured by the vapor incidence angle [44] [ ] π(z L± ) χ v (z) = χ v ± δ v sin, z L ±, (6.4) Ω that varies sinusoidally with z, where χ v is the mean value and δ v the amplitude of the sinusoidal variation of the vapor incidence angle, and 2Ω is the period of the SNTF normal to the interface. The third dyadic in Eq. (6.1) was chosen as S z (γ ± ) = (û x û x + û y û y ) cos γ ± + (û y û x û x û y ) sin γ ± + û z û z, (6.5) so that plane formed by the unit vectors û z and û x cos γ ± + û y sin γ ± is the morphologically significant plane for z L ±. Thus, there is sufficient flexibility in the formulation with respect to the twist γ + γ of the two morphologically significant planes. The geometry of the canonical problem is shown schematically in Fig Without loss of generality, let me choose the direction of SPP-wave propagation in the xy plane to be parallel to the x axis. Accordingly, where κ is a complex-valued scalar. E(r) = e(z) exp (iκx) H(r) = h(z) exp (iκx) }, (6.6) 53

84 z 2Ω SNTF L met x, y metal slab Figure 6.1: Schematic illustration of the geometry of the canonical boundary-value problem for γ + = γ. The axial field components e z (z) and h z (z) can be expressed in terms of the column 4-vector [f(z)] = [e x (z) e y (z) h x (z) h y (z)] T (6.7) via κ e z (z) 0 h z (z) 0 where one 4 4 matrix [ A ± (z) ] = + κ ϵ d(z) [ϵ a (z) ϵ b (z)] ϵ a (z) ϵ b (z) = involves the auxiliary quantity { [ A ± (z) ] [f(z)], z L ±, [ A met (z) ] [f(z)], z (L, L + ), κ2 ωϵ κ 2 ωµ sin [χ(z)] cos [χ(z)] ϵ d (z) ϵ a (z) ϵ b (z) cos γ ± sin γ ± sin γ ± cos γ ± ϵ d (z) = ϵ a (z) ϵ b (z)/ { ϵ a (z) cos 2 [χ(z)] + ϵ b (z) sin 2 [χ(z)] }, and the other 4 4 matrix [ A met (z) ] = κ2 ωϵ 0 ϵ m κ 2 ωµ (6.8)

85 ω The column 4-vector [f(z)] satisfies the matrix differential equations d dz [f(z)] = i [ P ± (z) ] [f(z)], z L ±, (6.9) and d dz [f(z)] = i [ P met (z) ] [f(z)], z (L, L + ), (6.10) where the 4 4 matrixes [ P ± (z) ] = [ A ± (z) ] µ µ 0 0 [ ϵ 0 [ϵ c (z) ϵ d (z)] cos γ ± sin γ ± ϵ 0 ϵc (z) cos 2 γ ± + ϵ d (z) sin 2 γ ±] [ 0 0 ϵ 0 ϵc (z) sin 2 γ ± + ϵ d (z) cos 2 γ ±] ϵ 0 [ϵ c (z) ϵ d (z)] cos γ ± sin γ ± 0 0 and [ P met (z) ] = [ A met (z) ] + ω µ µ ϵ 0 ϵ m 0 0 ϵ 0 ϵ m Equation (6.10) can be solved straightforwardly to yield. [f(l + )] = exp { i [ P met] (L + L ) } [f(l ]. (6.11) Equation (6.9) requires numerical solution by the piecewise uniform approximation ] technique [64]. The ultimate aim is to determine the matrixes [Q ± that appear in the relations ] [f(l ± ± 2Ω)] = [Q ± [f(l ± )] (6.12) to characterize the optical response of one period of the SNTF on either side of the metal slab. Basically, this technique consists of subdividing each period of the SNTF into a cascade of electrically thin sublayers parallel to the plane z = 0, and assuming the dielectric properties to be spatially uniform in each sublayer. A sufficiently large number N + 1 points z n ± = L ± ± 2Ω (n/n), n [0, N], are defined on each side of the metal slab and the matrixes [ ( z ± )] } [W {±i ± ] = exp P ± n 1 + z n ± 2Ω, n 2 N n [1, N], (6.13) are calculated for a specific value of κ; then [Q ± ] = [W ± N ] [W ± N 1 ]... [W ± 2 ] [W ± 1 ]. (6.14) 55

86 A sublayer thickness 2Ω/N = 2 nm was adequate for the results reported in Sec By virtue of the Floquet Lyapunov theorem [88], the matrixes [ Q ± ] can be defined such that { [Q ± ] = exp ±i2ω[ Q } ± ]. (6.15) Both [Q ± ] and [ Q ± ] share the same eigenvectors, and their eigenvalues are also related as follows. Let [t ± ]] (n), n [1, 4], be the eigenvector corresponding to the nth eigenvalue σ n ± of [Q ± ; then, the corresponding eigenvalue α n ± of [ Q ± ] is given by α ± n = i ln σ± n 2Ω. (6.16) The electromagnetic fields of the SPP wave must diminish in magnitude as z ±. Therefore, in the half-space z > L +, the eigenvalues of [ Q + ] are labeled such that Im [ α + 1,2] > 0 and then set [ [ ] A + [f (L + )] = [t + ] (1) [t + ] (2) 1 A + 2 ], (6.17) where A + 1,2 are unknown scalars; the other two eigenvalues of [ Q + ] describe fields that amplify as z + and cannot therefore contribute to the SPP wave. A similar argument for the half-space z < L requires us to ensure that Im [ α1,2] < 0 and then to set [ ] [ ] A [f (L )] = [t ] (1) [t ] (2) 1, (6.18) where A 1,2 are unknown scalars. Combining Eqs. (6.11), (6.17), and (6.18) to ensure the continuity of the tangential components of the electric and magnetic fields across each of the two metal/sntf interfaces, a matrix equation can be obtained: [ [t + ] (1) [t + ] (2) ] [ A + 1 A + 2 which may be rearranged as ] A 2 = exp { i [ P s] (L + L ) } [ [M(κ)] A + 1 A + 2 A 1 A 2 56 [t ] (1) [t ] (2) ] [ A 1 A 2 ], (6.19) 0 = 0 0. (6.20) 0

87 For a nontrivial solution, the 4 4 matrix [ M(κ) ] must be singular, so that is the dispersion equation for SPP-wave propagation. det [ M(κ) ] = 0 (6.21) 6.3 Numerical Results and Discussion A Mathematica TM program was written and implemented to solve (6.21) to obtain κ for specific values of γ + and γ. The code for the program is provided in Appendix B.4. The free-space wavelength was fixed at λ 0 = 633 nm and the halfperiod Ω = 200 nm for all calculations. The angles χ v and δ v were taken to be 45 and 30, respectively, for all results presented in this chapter. The dispersion equation (6.21) was solved using the Newton-Raphson technique [89] for a defect layer made of bulk aluminum and electron-beam evaporated aluminum thin film, and the solutions were searched when Re [κ/k 0 ] > Bulk aluminum defect layer The dispersion equation for bulk aluminum defect layer was solved for three different values of the twist between the morphologically significant planes on either side of the metal slab; (i) γ = γ +, (ii) γ = γ , and (iii) γ = γ , while γ + was kept as a variable. For each choice, the boundaries of the metal slab were taken to be at L ± = ±7.5, ±12.5, ±25, or ±45 nm. These selections adequately represent the results of my investigation. The metal was taken to be bulk aluminum (ϵ m = i). The skin depth of aluminum at the chosen wavelength is met = { Im [ k 0 ϵm ]} 1 = nm, a quantity of interest in relation to the thickness of the metal slab. γ = γ + Let me begin with the case when the morphologically significant planes of the SNTF on both sides of the metal slab are aligned with each other and make an angle γ = γ + with respect to the direction of SPP-wave propagation in the xy plane. The real and imaginary parts of κ/k 0 which satisfies Eq. (6.21) are presented in Fig. 6.2 as functions of γ + [0, 90 ]; by virtue of symmetry, the solutions 57

88 Re [ k o ] Im [ k o ] Re [ /k o ] Im [ /k o ] (deg) (a) (deg) (deg) (c) (deg) Re [ /k o ] Im [ /k o ] Re [ /k o ] Im [ /k o ] (deg) (b) (deg) + (deg) (d) (deg) Figure 6.2: Variation of real and imaginary parts of κ/k 0 with γ +, when γ = γ +. (a) L ± = ±7.5 nm, (b) L ± = ±12.5 nm, (c) L ± = ±25 nm, and (d) L ± = ±45 nm. for γ + and 360 γ + are the same as for γ +. For the thinnest metal slab (L ± = ±7.5 nm), the solutions are organized in five branches which span the entire range of γ +. As the metal slab thickens to 25 nm (L ± = ±12.5 nm), the number of branches does not change, but only four of those branches span the entire range of γ + and one is confined to γ + [0, 49 ]. Further thickening of the metal slab to 50 nm (L ± = ±25 nm) leads to five values of κ satisfying the dispersion equation only in the range γ + [0, 35 ], four in the range γ + (35, 37 ], and three in the range γ + (37, 90 ]. Finally, for a 90-nm thick metal slab (L ± = ±45 nm) only three solutions exist for γ [0, 36 ] and two for γ (36, 90 ], these solutions being the same as for a single metal/sntf interface as given in Ch. 2. It can be concluded that, as the thickness of the metal slab is increased, the coupling between two metal/sntf interfaces z = L and z = L + decreases; ultimately, the two interfaces decouple from each other when the thickness L met significantly exceeds twice the skin depth met in the metal. The solutions in Fig. 6.2 can be categorized into three sets. The first set comprises those solutions for which Re[κ/k 0 ] lies between 2.3 and 2.7. This set has two branches when the metal slab is 15-nm thick, both branches spanning 58

89 P x,y,z (a) P x,y,z (d) P x,y,z P x,y,z (b) (c) P x,y,z P x,y,z (e) (f) Figure 6.3: Variation of the Cartesian components of the time-averaged Poynting vector P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±7.5 nm, and γ = γ +. (a-c) γ + = 0, and (d-f) γ + = 25. (a) κ/k 0 = i0.1839, (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i0.1848, (e) κ/k 0 = i , and (f) κ/k 0 = i The x-, y- and z-directed components of P(x, z) are represented by solid red, dashed blue, and chain-dashed black lines, respectively. the entire range of γ +. As the thickness L met increases, the two branches come closer to each other and eventually merge completely. The second set comprises solutions for which Re [κ/k 0 ] lies between 2.05 and 2.1. It also has two branches when L met = 15 nm, both branches coalescing into one branch as the thickness increases. Complete merger of the two branches of the first set occurs at a value of L met smaller than that for the two branches of the first set. Regardless of the value of L met, solutions in the second set can be found over the entire range of γ +. The third set consists of solutions lying on just one branch (1.85 < Re [κ/k 0 ] < 1.95), but the maximum value of γ + on this branch decreases rapidly from 90 as L met increases from 15 nm. The foregoing categorization is also meaningful as the spatial profiles of the fields are very similar for both solutions (for a specific value of γ + ) in the first two sets. The same remark can be made for the spatial profiles of the time-averaged 59

90 P x,y,z P x,y,z P x,y,z (a) (b) (c) P x,y,z P x,y,z P x,y,z (d) (e) (f) Figure 6.4: Same as Fig. 6.3 except for L ± = ±45 nm. (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i Poynting vector P(x, z) = 1 2 Re [E(x, z) H (x, z)]. (6.22) Representative plots of P(0, z) against z are presented in Figs. 6.3 and 6.4 for combinations of (i) three values of κ, one from each set, and (ii) two values of γ +. Whereas γ + = 0 was chosen because it represents the propagation of SPP waves in the morphologically significant plane of the SNTF on either side of the metal slab, γ + = 25 was chosen for the direction of SPP-wave propagation oblique to that plane. Figure 6.3 shows the spatial profiles of P(0, z) for L ± = ±7.5 nm and six selected values of κ. In order to determine these profiles, first A + 1 was set equal to unity and then the remaining coefficients were found using Eq. (6.20); the exception is Figure 6.3(b), for which A + 2 = 1 was set and the remaining coefficients were found using Eq. (6.20). The spatial profiles shown allow us to conc4lude that (i) the SPP waves are bound strongly to both metal/sntf interfaces, and (ii) the power density mostly resides in the SNTF. 60

91 Representative calculated values of the penetration depth ± z of the SPP wave into the metal, defined as the distance along the z axis (in the metal slab) at which the amplitude of the electric or magnetic field decays to e 1 of its value at the nearest metal/sntf interface z = L ±, are tabulated in Table 1 for L ± = ±7.5 nm. The penetration depths for all SPP waves are 12.7 nm, which confirms strong coupling of the two metal/sntf interfaces. This is not surprising since z = + z is very close to met. Table 6.1: Penetration depths + z = z for L ± = ±7.5 nm and γ = γ +. The solutions are numbered in descending values of Re [κ/k 0 ]. + z = z (nm) Solution γ + = γ 1st 2nd 3rd 4th 5th The spatial profiles of P(0, z) for L ± = ±45 nm are presented in Fig. 6.4 for six selected values of κ/k 0 identified in the figure caption. These profiles were obtained after A + 1 was set equal to unity, except that A + 2 = 1 was used for κ/k 0 = i The spatial profiles of P(0, z) for L ± = ±12.5 nm and ±25 nm have not been shown here because the spatial profiles for these cases are similar to those shown in Figs. 6.3 and 6.4. Representative values of the penetration depths ± z for L ± = ±45 nm are given in Table 2. The penetration depths for this case are of the same order as for L ± = ±7.5 nm. As the thickness of the metal slab is 90 nm, it can be safely conc4luded that the SPP waves propagating on the two metal/sntf interfaces are not coupled to each other but are propagating independently. γ = γ The real and imaginary parts of κ/k that satisfies the dispersion equation (6.21) for γ = γ are given in Fig. 6.5 as functions of γ + [0, 90 ]. Due to the symmetry of the problem, the solutions for 90 ± γ +, 180 ± γ +, 270 ± γ + and 360 γ + are the same as for γ +. Five solutions are found which span the whole range of γ + for L ± = ±7.5 and ± 12.5 nm. However, when L ± = 61

92 Table 6.2: Penetration depths + z = z for L ± = ±45 nm and γ = γ +. The solutions are numbered in descending values of Re [κ/k 0 ]. + z = z (nm) Solution γ + = γ 1st 2nd 3rd Re [ k o ] Im [ k o ] Re [ k o ] Im [ k o ] (deg) (a) (deg) (deg) (c) (deg) Re [ k o ] Im [ k o ] Re [ k o ] Im [ k o ] (deg) (b) (deg) (deg) (d) (deg) Figure 6.5: Same as Fig. 6.2 except that γ = γ ±25 and ± 45 nm, only four solutions exist for γ (37, 53 ), but five for other values of γ + [0, 37 ] [53, 90 ]. 62

93 P x,y,z P x,y,z P x,y,z (a) (b) (c) P x,y,z P x,y,z P x,y,z (d) (e) (f) Figure 6.6: Variation of the Cartesian components of the time-averaged Poynting vector P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±7.5 nm, and γ = γ (a-c) γ + = 0, and (d-f) γ + = 25. The following values of κ were chosen for rough correspondence with those in Fig. 6.3: (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i For the thickest slab considered, one can see from comparing the results presented in Fig. 6.5(d) with those in Fig. 6.2(d) which are same as that for a single metal/sntf interface that the two metal/sntf interfaces are actually uncoupled from each other. The solutions in Fig. 6.5(d) represent SPP-wave propagation guided by a metal/sntf interface with the direction of propagation in the xy plane either making an angle γ + or γ with the morphologically significant plane of the SNTF. As in Sec , the solutions of the dispersion equation can be categorized into three sets with the same criterions as given in Sec However, in this case, the two branches in either of the first two sets do not merge into one branch as L met is increased. Instead, the two branches remain distinct, each holding for SPP-wave propagation guided by one of the two metal/sntf interfaces uncoupled from the other metal/sntf interface. The single branch in the third set actually vanishes for mid-range values of γ +, with the two parts of that branch signifying 63

94 P x,y,z (a) P x,y,z (d) P x,y,z P x,y,z (b) (c) P x,y,z P x,y,z (e) (f) Figure 6.7: Variation of the Cartesian components of the time-averaged Poynting vector P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±45 nm, and γ = γ The following values of κ and γ + were chosen to highlight the uncoupling of the two metal/sntf interfaces, when the metal slab is sufficiently thick. (a-e) γ + = 25 and (f) γ + = 65. (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i independent propagation guided by the two metal/sntf interfaces. Examination of the field profiles confirms the conclusions made in Sec regarding the effect of the thickness of the metal slab. The asymmetry in the alignment of the morphologically significant planes for z < L and z > L + is reflected in the spatial profiles of the time-averaged Poynting vector presented in Figs. 6.6 and 6.7 for L met = 15 and 90 nm, respectively. Fig. 6.6 also suggests that the coupling of the SPP-wave is directly proportional to the real part of wavenumber. So slower SPP waves are more strongly coupled to both interfaces. This observation is also confirmed by the spatial profiles of SPP waves when L met = 25 and 50 nm (not shown). Fig. 6.7 shows the uncoupling of the two metal/sntf interfaces when the metal slab is sufficiently thick. 64

95 Re [ k o ] Im [ k o ] Re [ k o ] Im [ k o ] (deg) (a) (deg) (deg) (c) (deg) Re [ k o ] Im [ k o ] Re [ k o ] Im [ k o ] (deg) (b) (deg) (deg) (d) (deg) Figure 6.8: Same as Fig. 6.2 except γ = γ γ = γ The solutions of the dispersion equation (6.21) for γ = γ are given in Fig. 6.8 for γ [0, 180 ]. By virtue of symmetry, the solutions for γ are the same as for γ +. Five solutions exist for the entire range of γ + for L ± = ±7.5 nm. When L ± = ±12.5 nm, five solutions exist for γ [0, 37 ] [98, 180 ] but only four for γ (37, 98 ). Further thickening of the metal to 50 nm (L ± = ±25 nm) yields five solutions for γ [0, 36 ] [99, 180 ] but four for γ (36, 99 ). Decoupling of the two metal/sntf interfaces becomes very pronounced for L ± = ±45 nm, when all solutions found are the same for either (i) a metal/sntf interface for which the direction of propagation in the xy plane makes an angel γ + with the morphologically significant plane, or (ii) a metal/sntf interface for which the direction of propagation in the xy plane is inclined at γ to the morphologically significant plane. Similar to Secs and 6.3.1, the solutions can be grouped into three sets, with the same criterions given in Sec Representative field profiles for 15- nm- and 90-nm-thick metal slabs are given in Figs. 6.9 and 6.10, respectively. One value of κ is selected from each set at γ + = 25 and 150. Whereas γ + = 25 65

96 P x,y,z (a) P x,y,z (d) P x,y,z (b) P x,y,z (e) P x,y,z (c) P x,y,z (f) Figure 6.9: Variation of the Cartesian components of P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±7.5 nm, and γ = γ (a-c) γ + = 25, and (d-f) γ + = 150. (a) κ/k 0 = i0.1865, (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i0.1847, (e) κ/k 0 = i , and (f) κ/k 0 = i corresponds to the propagation in the xy plane at an angle of 25 with respect to the morphologically significant plane in the region z > L + and at 70 in the region z < L, the analogous angles are 30 and 15, respectively, when γ + = 150. Fig. 6.9 shows that SPP waves are guided by both interfaces of thin metal slab, but the power density profile is asymmetric due to the fact that morphologically significant planes are not parallel to each other in the two regions occupied by the SNTF. Figure 6.10 shows that when L met = 90 nm, any SPP wave propagates predominantly guided by one of the two metal/sntf interfaces.it can be deduced from these two figures that the conclusions drawn in Secs and hold true for this case as well, and therefore are general enough Electron-beam evaporated aluminum thin film Let me now consider a more realistic metal film which is deposited by electronbeam evaporation method. The electron-beam-evaporated thin film of aluminum has ϵ m = ( i3.9) 2 [44]. Before the discussion on the SPP-wave propagation 66

97 P x,y,z (a) P x,y,z (d) (b) 150 (e) P x,y,z 0.10 P x,y,z P x,y,z (c) P x,y,z (f) Figure 6.10: Same as Fig. 6.9 except that L ± = ±45 nm. (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i Re [ k o ] Im [ k o ] (deg) (deg) Figure 6.11: Real and imaginary parts of κ/k 0, which represent SPP-wave propagation guided by the single interface of the chosen SNTF and electron-beamevaporated aluminum: ϵ m = ( i3.9) 2. 67

98 Re [ k o ] (deg) (a) (deg) Im [ k o ] Re [ k o ] (deg) (c) (deg) Im [ k o ] Re [ k o ] (deg) (b) (deg) Im [ k o ] Re [ k o ] (deg) (d) (deg) Im [ k o ] Figure 6.12: Variations of real and imaginary parts of κ/k 0 with γ +. (a) L ± = ±7.5 nm, (b) L ± = ±25 nm, (c) L ± = ±45 nm, and (d) L ± = ±75 nm. The other parameters are provided at the beginning of Sec guided by the metallic defect layer in the SNTF, let me go back to the problem discussed in Ch. 2. By setting L = and L + = 0, the problem reduces to the SPP-wave propagation guided by a single interface at z = 0 between an SNTF and a metal. The solutions of the dispersion equation for this single interface are given in Fig For this case, four SPP waves can propagate localized to the interface when γ + [0, 65 ] and three SPP waves when γ + (65, 90 ]. Next let me move on to the e-beam evaporated aluminum defect layer. The solutions of the dispersion equation for four values of thickness of the metallic defect layer are presented in Fig for γ + [0, 90 ]. Five solutions span the whole range of γ + for L ± = ±7.5 nm. However, when L ± = ±25, ± 45, and ± 75 nm, Fig shows that, respectively, (i) five, six, and seven solutions exist for γ + [0, 25 ] [65, 90 ]; and (ii) six, seven, and eight solutions for γ (25, 65 ). For the metallic defect layer with L met = 150 nm, it can be seen from comparing the results presented in Fig. 6.12(d) with those in Fig that the two 68

99 P x,y,z P x,y,z P x,y,z (a) (b) (c) P x,y,z P x,y,z P x,y,z (d) (e) (f) Figure 6.13: Variation of the Cartesian components of the time-averaged Poynting vector P(x, z) (in W m 2 ) along the z axis when x = 0, L ± = ±7.5 nm. (ac) γ + = 0, and (d-f) γ + = 25. The following values of κ were to highlight the coupled SPP-waves propagation: (a) κ/k 0 = i0.015, (b) κ/k 0 = i , (c) κ/k 0 = i , (d) κ/k 0 = i , (e) κ/k 0 = i , and (f) κ/k 0 = i The components parallel to û x, û y, and û z, are represented by red solid, blue dashed, and black chain-dashed lines, respectively. metal/sntf interfaces are actually uncoupled from each other. The solutions in Fig. 6.12(d) represent SPP-wave propagation guided independently by the two metal/sntf interfaces (as in Fig. 6.11). It can be concluded that, as the thickness of the metallic defect layer is increased, the coupling between two metal/sntf interfaces located at z = L and z = L + decreases; ultimately, the two interfaces decouple from each other when the thickness L met significantly exceeds twice the skin depth met = [ Im ( )] 1 k 0 ϵm in the metal. The skin depth in the electronbeam-evaporated aluminum thin film at the chosen wavelength is nm. The solutions in Fig can be divided into four sets. The first set comprises those solutions for which Re(κ/k 0 ) lies between 2.3 and 2.8. This set has only one branch representing the coupled SPP-wave propagation when L met = 15 nm, but it 69

100 P x,y,z (a) P x,y,z (c) P x,y,z (b) P x,y,z (d) Figure 6.14: Same as Fig except for L ± = ±75 nm, and γ + = 25. The following values of κ were chosen to highlight the decoupling of the two metal/s- NTF interfaces: (a) κ/k 0 = i , (b) κ/k 0 = i , (c) κ/k 0 = i , and (d) κ/k 0 = i splits into two branches one branch representing the uncoupled SPP-wave propagation guided by one interface and the other branch representing propagation guided by the other interface when L met = 150 nm. The second set comprises those solutions for which Re(κ/k 0 ) lies between 2.08 and This set also consists of one branch representing coupled SPP-wave propagation when the defect layer thickness is very small. As the defect layer thickness increases, this single branch splits into two branches eventually representing uncoupled SPP waves guided by the two metal/sntf interfaces independently. The third set consists of the solutions for which Re(κ/k 0 ) lies between 1.9 and This set comprises two branches when L met = 15 nm, both representing coupled SPP waves; as the defect layer thickness increases, this set reorganizes into two branches which represent decoupling of the two metal/sntf interfaces. The fourth set comprises those solutions for which Re(κ/k 0 ) lies between 1.3 and 1.4. This set has only a single branch of solutions representing the coupled SPP-wave propagation spanning the whole range of γ + for L met = 15 nm, but two new branches appear as the thickness of the metallic defect layer increases. These branches eventually converge to their common analog in Fig The spatial profiles of the fields are similar for the solutions in each set at a given value of the defect layer s thickness. The same remark can be made for the spatial profiles of the time-averaged Poynting vector. Examination of the spatial profiles shows that SPP waves propagate guided by the two metal/sntf interfaces independently when the defect layer s thickness is much greater than 70

101 twice the penetration depth in the metal. As that thickness is reduced, coupling of the two interfaces results in new SPP waves. The smaller is the thickness of the metallic defect layer, the stronger is the coupling effect. The asymmetry in the alignment of the morphologically significant planes for z < L and z > L + is reflected in the spatial profiles of the time-averaged Poynting vector presented in Figs and 6.14 for L met = 15 and 150 nm, respectively. In Fig. 6.13, the variations of the Cartesian components of P for γ + = 0 and 25 and for three values of κ selected from the first, third and fourth sets are given to show the coupled SPP-wave propagation. Figure 6.13 also suggests that the coupling is directly proportional to the real part of the wavenumber κ. Hence, slower SPP waves are more strongly coupled to both interfaces. This observation is also confirmed by the spatial profiles of SPP waves when L met = 50 and 90 nm (not shown). To highlight the uncoupled SPP-wave propagation guided by the two interfaces independently when L met = 150 nm, the spatial distribution of power density is shown in Fig For these plots one solution was selected from each set at γ + = Concluding Remarks The canonical boundary-value problem to examine the characteristics of SPP waves guided by a thin metal slab inserted in a periodically nonhomogeneous SNTF is formulated and numerically solved. The morphologically significant planes on the two sides of the metal slab could be either parallel to or twisted with respect to each other. It was found that (i) when the metal slab is very thin, its two interfaces with the SNTF couple to each other, thereby generating more modes of SPP-wave propagation; (ii) as the metal slab thickness increases, the coupling between the two interfaces decreases; (iii) both the phase speed and the attenuation of an SPP wave depend on the twist between the morphologically significant planes of the SNTF on the two sides of the metal slab; (iv) smaller phase speeds are obtained with a thinner metal slab; and (v) the two interfaces of a sufficiently thick metal slab with the SNTF independently guide SPP waves. The results for bulk aluminum and an electron-beam evaporated thin film are similar except for the different penetration depths. 71

102 Chapter 7 Guided-Wave Propagation by a Dielectric Slab in an SNTF 7.1 Introduction The last two decades have opened new technoscientific avenues, including the capability to design materials with nanoscale morphology [97 99]. Since solid-state technology is mostly planar, the need for two distinct modalities of guided-wave propagation, among others, has arisen. First, efficient planar optical waveguides in the cladding/core/cladding configuration are needed to transport optical signals [100]; second, surface-plasmon-polariton (SPP) waves guided by planar metal/dielectric interfaces are attractive as they are faster than purely electronic waves traveling in good conductors [13, 14]. A major challenge to the adoption of SPPbased communication is the high attenuation of SPP waves [101], as becomes evident from the solutions of a relevant canonical boundary-value problem [102], primarily due to ohmic losses in the metal. Though several mitigation strategies are being tried out [41], an obvious one is to replace the metal by a dielectric material. That is the attraction of Dyakonov Tamm waves. Because of continued interest in conventional dielectric waveguides and optical surface waves, I decided to find a structure which can transport electromagnetic energy via waveguide modes and Dyakonov Tamm waves. The interface of a homogeneous, isotropic dielectric material and a periodically nonhomogeneous SNTF [70] can support Dyakonov Tamm waves [64]. Therefore, I set out to investigate wave propagation guided by a homogeneous, isotropic dielectric slab inserted in a periodically nonhomogeneous SNTF, the dielectric slab being the core and the SNTF being the cladding. The structure thus formed can be considered to be a canonical structure that allows (i) interaction between waves localized to differ- This chapter is based on: M. Faryad and A. Lakhtakia, Propagation of surface waves and waveguide modes guided by a dielectric slab inserted in a sculptured nematic thin film, Phys. Rev. A. 83, (2011). 72

103 ent parallel surfaces, as well as (ii) waveguide modes with energy largely confined to the region between two parallel surfaces. Moreover, this structure allows the study the study of effect of the coupling of the two interfaces on the multiplicity of Dyakonov Tamm waves. Since this problem is the same as discussed in Ch. 6, except that the metallic slab is replaced by a dielectric slab, the theretical formulation for this problem is the same as given in Ch. 6. However, the results for the dielectric slab are quite different than that of a metallic slab. The numerical results for the dielectric slab are discussed in Sec Concluding remarks are given in Sec Numerical Results and Discussion A Mathematica TM program was written and implemented to solve (6.21) to obtain κ for specific values of γ + and γ. The code for the program is the same as in Appendix B.4 except that the relative permittivity of the metal slab be replaced by that of the dielectric slab. Following Agarwal et al. [64], δ v = 16.2, χ v = 19.1, and Ω = 197 nm was fixed. Moreover, the search of the solutions of the dispersion equation was restricted to κ/k 0 > Dyakonov Tamm waves guided by a single dielectric/sntf interface Before I discuss the solutions of the dispersion equation for guided-wave propagation by the SNTF/dielectric/SNTF system described Sec. 7.1, let me discuss the solutions of the dispersion equation for a single dielectric/sntf interface. The waves localized to this interface are Dyakonov Tamm waves [64]. Effectively, I set L + = 0 and revised the formulation in the limit L. The relative wavenumber κ/k 0, the relative phase speed v r = n s k 0 /κ, the e- folding distance = 1/Im [+ ] (k 0 n s ) 2 κ 2 into the homogeneous dielectric material perpendicular to the interface, and the decay constants exp (u 1,2 ) = exp ( 2Ω Im [ α + 1,2]) are presented as functions of γ + in Fig Either none, one or two Dyakonov Tamm waves can be guided by this interface, depending on the angle γ + between the direction of propagation and the morphologically significant plane of the SNTF. This situation has been analyzed in detail by Agarwal et al. [64] for various values of ϵ s, χ v, and δ v. However, for the specific case chosen, the shorter branch of the solutions given in Fig. 7.1 was missed in Ref. 64. The e-folding distance varies between 0.8 Ω and 2 Ω. The decay constants exp (u 1,2 ) represent the decay of the Dyakonov Tamm waves after one structural period (i.e., 2Ω) into the SNTF and perpendicular to the interface. If both of its decay constants are close to zero, a Dyakonov Tamm wave is strongly localized. If either 73

104 one or both of its decay constants are close to unity, a Dyakonov Tamm wave is loosely bound to the interface on the SNTF side. k o Decay Const exp(-u ) exp(-u 2 ) 0.3 exp(-u 1 ) 0.0 exp(-u 2 ) (deg) Figure 7.1: Relative wavenumber κ/k 0, relative phase speed v r, the e-folding distance, and the decay constants exp (u 1,2 ) = exp ( 2Ω Im [ α + 1,2]) as functions of γ + for Dyakonov Tamm waves guided by the single interface of the chosen dielectric material and the SNTF. The black symbols (square) identify the solutions also found by Agarwal et al. [64], but the solutions identified by the red symbols (circular) were missed in that work SNTF/dielectric/SNTF system Returning to the boundary-value problem (with finite L ± ) actually tackled for this chapter, I solved the dispersion equation (6.21) for two different values of the twist between the morphologically significant planes on either side of the dielectric slab. I chose (i) γ = γ +, and (ii) γ = γ , 74

105 k o k o (a) (deg) (b) (deg) k o k o (c) (deg) (d) (deg) Figure 7.2: Variation of relative wavenumber κ/k 0 with γ +, when γ = γ +. (a) L ± = ±1 Ω, (b) L ± = ±1.5 Ω, (c) L ± = ±3 Ω, and (d) L ± = ±4 Ω. Solutions in the shaded regions represent Dyakonov Tamm waves, but those in the unshaded regions represent waveguide modes, the boundary between the two regions being delineated for the chosen parameters by κ/k 0 = n s. 75

106 while γ + was kept as a variable. For each choice, the boundaries of the dielectric slab were taken to be L ± = ±Ω, ±1.5 Ω, ±3 Ω, or ±4 Ω nm. These selections were made so as to have L s to be less than and greater than twice the e-folding distance within the dielectric slab along the z axis, in order to study the effect of coupling of the two interfaces. γ = γ + Let me begin with the case when the morphologically significant planes of the SNTF on both sides of the dielectric slab are aligned with each other and make an angle γ = γ + with respect to the direction of wave propagation in the xy plane. By virtue of symmetry, the solutions for γ + and 360 γ + are the same as for γ +. The solutions of the dispersion equation for various values of slab thickness given in Fig. 7.2 as functions of γ + [0, 90 ] can be grouped into two categories, based on the value of κ. Within the dielectric slab, k z = + (k 0 n s ) 2 κ 2 is the wavenumber along the z axis. If κ/k 0 > n s, k z /k 0 must be a complex number, signifying that these solutions do not represent waveguide modes, but Dyakonov Tamm waves localized to the interfaces between the dielectric slab and the SNTF. However, the solution of the dispersion equation with κ/k 0 < n s represent waveguide modes as k z would be a real number. This categorization is supported by a comparative analysis of the solutions presented in Figs. 7.1 and 7.2 as follows. The solutions given in Fig. 7.2(a) for L ± = ±Ω may be organized in three branches with κ/k 0 > n s and several branches with κ/k 0 < n s. As the slab thickness L s increases to 8 Ω in Fig. 7.2(d), the three branches with κ/k 0 > n s merge into two branches, but the number of branches for κ/k 0 < n s goes on increasing. The decrease in the number of solutions representing Dyakonov Tamm waves is due to the uncoupling of the two interfaces z = L and z = L + as the thickness L + L increases. For L ± = ±Ω, the slab thickness L s is smaller than twice the e-folding distance = 1/Im (k z ). Hence, due to the coupling of the two interfaces I have more solutions representing Dyakonov Tamm waves. But L s > 2 when L ± = ±4 Ω (because /Ω [0.8, 2]), and the two dielectric/sntf interfaces are uncoupled from each other so that I have exactly the same solutions with κ/k 0 > n s, as given in Fig. 7.1 for a single dielectric/sntf interface. The increase in the number of waveguide modes with the increase in the thickness L s of the dielectric slab is in accordance with the general behavior of waveguiding structures [104, 105]: the number of waveguide modes increases with the increase in the cross-sectional dimensions of a waveguide. The foregoing analysis is buttressed by the spatial profiles of the time-averaged Poynting vector P(x, z) = 1 2 Re [E(x, z) H (x, z)] ; (7.1) 76

107 P x,y,z (a) P x,y,z (b) P x,y,z (c) P x,y,z (d) P x,y,z (e) P x,y,z (f) Figure 7.3: Variation of the Cartesian components of P(z) (in W m 2 ) with z for γ = γ + and L ± = ±Ω. The x-, y-, and z-directed components are represented by solid red, dashed blue and chain-dashed black lines. The orange-shaded region represents the dielectric slab. γ + = (a-d) 10 and (e-f) 80. κ/k 0 = (a) , (b) , (c) , (d) , (e) , and (f) as κ is real, P(x, z) P(z). Representative plots of P(z) against z are presented in Figs. 7.3 and 7.4 for four values of κ at γ + = 10 and two values of κ at γ + = 80. These spatial profiles were calculated by setting A + 1 = 1 Vm 1 and determining the remaining coefficients using Eq. (6.20). The shaded region in each plot represents the region occupied by the dielectric slab. Each of Figs. 7.3(a), (b), and (c) represents the spatial profile of the power density of a Dyakonov Tamm wave, as the relative wavenumber κ/k 0 > n s and the power density is strong at or close to the interfaces. In contrast, Figs. 7.3(d), (e), and (f) contain waveguide modes because κ/k 0 < n s. The distinction between Dyakonov Tamm waves and the waveguide modes is more evident in the spatial profiles presented in Fig. 7.4 for L ± = ±4 Ω. The spatial profiles given in Figs. 7.4(a) and (b) represent Dyakonov Tamm waves, whereas Figs. 7.4(c), (d), (e), and (f) contain the spatial profiles of waveguide modes. Some of the waveguide modes do have maximums in their profiles at or near the two dielectric SNTF/interfaces, but some other waveguide modes contain most of the power near the central axis. 77

108 P x,y,z (a) P x,y,z (b) P x,y,z (c) P x,y,z (d) P x,y,z (e) P x,y,z (f) Figure 7.4: Same as Fig. 7.3 except for L ± = ±4Ω and κ/k , (c) , (d) , (e) , and (f) = (a) , (b) γ = γ The solutions of the dispersion equation for γ = γ the morphologically significant planes of the SNTF on both sides are perpendicular to each other are given in Fig. 7.5 as functions of γ + [0, 90 ] for various values of slab thickness. Due to the symmetry of the problem, the solutions for 90 ±γ +, 180 ±γ +, 270 ±γ + and 360 γ + are the same as for γ +. The solutions for this case can also be divided into two categories based on the value of κ/k 0. The solutions with κ/k 0 > n s represent Dyakonov Tamm waves that strongly couple to either one or both of the dielectric/sntf interfaces, while the solutions with κ/k 0 < n s represent waveguide modes. For the thinnest dielectric slab, there are three branches representing Dyakonov Tamm waves that are coupled to both interfaces. As the slab thickness increases up to 8 Ω, the coupling diminishes and the solutions representing Dyakonov Tamm waves eventually become identical to the solutions of two independent dielectric/sntf interfaces with the morphologically significant plane located either at γ + = 0 or at γ + = 90. The Dyakonov Tamm waves propagate independently guided by individual interfaces for L s = 8 Ω because the slab thickness is then much greater than the e-folding distance for all γ +. The number of solution branches representing waveguide modes increases as the thickness of the dielectric slab increases from 2 Ω to 8 Ω. Representative plots of P(z) vs. z for L ± = ±1 Ω are given in Fig. 7.6 for three 78

109 k o k o (a) (deg) (b) (deg) k o k o (c) (deg) (d) (deg) Figure 7.5: Same as Fig. 7.2 except for γ = γ values of κ each at γ + = 10 and 40. The spatial profiles given in Figs. 7.6(a), (b), and (d) represent Dyakonov Tamm waves while those in Figs. 7.6(c), (e), and (f) represent waveguide modes. Figure 7.7 contains representative spatial profiles of the time-averaged Poynting vector for L ± = ±4 Ω for three values of κ at γ + = 6, two values of κ at γ + = 30, and one value of κ at γ + = 20. The spatial profiles given in Figs. 7.7(a) and (d) represent Dyakonov Tamm waves 79

110 P x,y,z P x,y,z (a) (d) P x,y,z P x,y,z (b) (e) P x,y,z P x,y,z (c) (f) Figure 7.6: Same as Fig. 7.3 except for γ = γ γ + = (a-c) 10, and (d-f) 40. κ/k 0 = (a) , (b) , (c) , (d) , (e) , and (f) P x,y,z (a) P x,y,z 0.05 (b) P x,y,z (c) P x,y,z (d) P x,y,z (e) (f) Figure 7.7: Same as Fig. 7.3 except for γ = γ , and L ± = ±4Ω. γ + = (a-c) 6, (d-e) 30, and (f) 20. κ/k 0 = (a) , (b) , (c) , (d) , (e) , and (f) P x,y,z 80

111 localized to one interface while the profiles given in Figs. 7.7(b), (c), (e), and (f) represent waveguide modes. However, in Figs. 7.7(e) and (f) the decay rate of the Dyakonov Tamm wave in the SNTF z > L + is very low. The asymmetry in the power profiles in Figs. 7.6 and 7.7 is due to the asymmetry in the arrangement of the morphologically significant planes of the SNTF on either side of the dielectric slab. Before continuing, let me note that Dyakonov Tamm waves alone would propagate if the dielectric slab were to be absent (i.e. L s = 0), as has been shown in Ch. 5. Analysis of Figs. 7.1, 7.2 and 7.5 reveals a major advantage of using the SNT- F/dielectric/SNTF system for the propagation of Dyakonov Tamm waves as compared to the single SNTF/dielectric interface: the angular range of propagation and the possible number of Dyakonov Tamm waves can be increased if the thickness of the dielectric slab is sufficiently small. Guided by a single SNTF/dielectric interface, two Dyakonov Tamm waves can propagate for γ + [0, 10 ], one for γ + (10, 66 ], and none for γ + (66, 90 ]. However, the presented data indicate that an SNTF/dielectric/SNTF system with a dielectric slab of thickness 2Ω or less can support the propagation of three Dyakonov Tamm waves for γ + [0, 10 ], two for γ + (10, 48 ], and one for γ + (48, 90 ], when γ = γ +, two Dyakonov Tamm waves for γ + [0, 10 ] [22, 68 ] [80, 90 ] and one for γ + (10, 22 ) (68, 80 ), when γ = γ Comparison with SNTF/metal/SNTF system of Ch. 6 Wave propagation guided by an aluminum slab inserted in a periodically nonhomogeneous SNTF was studied in detail in Ch. 6 for Ω = 200 nm, χ v = 45 and δ v = 30. However, similar calculations with Ω = 197 nm, χ v = 19.1 and δ v = 16.2 were performed, for direct comparison with the data presented in Figs , and the results showed same characteristics as in Ch. 6. The waves propagating in the SNTF/metal/SNTF system are classified as SPP waves which may be coupled to either one or both of the metal/sntf interfaces, depending on the thickness of metal slab. Some attributes of wave propagation in the SNTF/metal/SNTF system are similar to those of guided-wave propagation in the SNTF/dielectric/SNTF system, but others are not. The following differences were noted: The wavenumber κ is a complex number for the SNTF/metal/SNTF system P(x, z) = P(z) exp [ 2Im(κ)x], but real for the SNTF/dielectric/SNTF system so that P(x, z) = P(z). Consequently, whereas attenuation must occur in the direction of propagation in the SNTF/metal/SNTF system, but the propagation is lossless in the SNTF/dielectric/SNTF system. 81

112 The dielectric core can support the propagation of waveguide modes whereas the metallic core cannot. Therefore, both surface waves (Dyakonov Tamm waves) and waveguide modes can propagate in the SNTF/dielectric/SNTF system, but only surface waves (SPP waves) can propagate in the SNT- F/metal/SNTF system. Coupling between the two interfaces exists only for very thin metal slabs (L s 60 nm), but for much thicker dielectric slabs (L s 1000 nm). This is because the imaginary parts of the relative permittivities of metals are very high in the optical regime, but those of commonplace dielectric materials are vanishingly small. When both the morphologically significant planes of the SNTF on either side of the dielectric slab are aligned parallel to each other, the allowable range of the direction of propagation for the Dyakonov Tamm waves is restricted, as can be seen from Fig However, SPP waves can propagate in any direction in the interface plane as is shown in Ch. 6, when the dielectric slab is replaced by a metal slab. If one compares only surface-wave propagation in the two systems, three similarities must be noted: The wavenumber κ for Dyakonov Tamm waves guided by the dielectric slab and Re (κ) for SPP waves guided by the metallic slab are greater than k 0, so both SPP and Dyakonov Tamm waves have smaller phase speeds than the speed of light in free space. The surface waves are strongly coupled to both interfaces when the slab thickness is less than twice the e-folding distance into the slab material. As the slab gets thicker, the coupling decreases. The coupling of two interfaces increases the number of possible surface waves that can be guided by the dielectric or metallic slab, although the effect is more pronounced with a metallic slab. 7.3 Concluding Remarks The canonical boundary-value problem of a dielectric slab inserted in an SNTF was solved numerically to analyze guided-wave propagation in the SNTF/dielectric/S- NTF system. The problem was solved for two cases: (i) when the morphologically significant planes of the SNTF on either sides are aligned with each other and (ii) when they are perpendicular to each other. For both the cases, (i) multiple surface (Dyakonov Tamm) waves and waveguide modes with different phase speeds and spatial profiles propagate, and 82

113 (ii) as the thickness of the dielectric slab increases, the number of waveguide modes increases; however, (iii) with the increase of the slab thickness, the two dielectric/sntf interfaces begin to uncouple and ultimately, each interface guides multiple Dyakonov Tamm waves all by itself. If the dielectric slab is replaced by a metal slab, coupling of the two metal/s- NTF interfaces will also occur when the slab is very thin as was seen in Ch. 6. Multiple surface (SPP) waves guided by one or both of the metal/sntf interfaces can propagate, but the SNTF/metal/SNTF system does not exhibit waveguide modes. 83

114 Chapter 8 Prism-Coupled Excitation of Multiple SPP Waves 8.1 Introduction As shown in Ch. 3, multiple SPP waves can still be guided by metal/dielectric interface even when the periodically nonhomogeneous dielectric material is isotropic. The possibility of exciting multiple SPP waves of a specific frequency using isotropic materials could lead to new applications if a simple way can be found to excite them. For this purpose, I decided to theoretically investigate the most commonly used method of excitation of SPP waves: TKR configuration [34]. The theoretical problem undertaken involves a metal-capped rugate filter with a prism on top of the metal film. Since the periodically nonhomogeneous and isotropic dielectric material can be fabricated as a rugate filter [2, 76, 78, 80], the problem undertaken in this chapter can be experimentally implemented. In the TKR configuration shown in Fig. 8.1, waves can be guided by (i) the metal/rugate-filter interface (SPP waves), as seen in Ch. 3; (ii) the prism/metal interface (SPP waves); (iii) the prism/metal and the metal/rugate-filter interfaces jointly (coupled SPP waves), as seen in Ch. 6; and (iv) the rugate filter due to its finite thickness (waveguide modes [104, 105]). The propagation of coupled SPP waves critically depends on the thickness of the metal film in the TKR configuration, for which purpose the canonical boundaryvalue problem of coupled-spp-wave propagation guided by a thin metal film, with This chapter is based on: M. Faryad and A. Lakhtakia, On multiple surface-plasmonpolariton waves guided by the interface of a metal film and a rugate filter in the Kretschmann configuration, Opt. Commun. 284, (2011). 84

115 a semi-infinite prism on one side and a semi-infinite rugate filter on the other side, has to be solved. This canonical problem, shown schematically in Fig. 8.2, was also taken up for this chapter. The propagation of waveguide modes depends on the thickness of the rugate filter. The theoretical formulations of the TKR configuration and the aforementioned canonical boundary-value problem are provided in Sec Numerical results are presented and discussed in Sec Finally, concluding remarks are presented in Sec Incident light x Reflected light z = 0 z= L m z Metal film Homogeneous dielectric SPP waves Rugate filter z= L Σ Homogeneous dielectric Transmitted light Figure 8.1: Schematic of the TKR configuration. 8.2 Theoretical Formulations TKR configuration Let the half-spaces z 0 and z L be occupied by a dielectric material with relative permittivity ϵ l = n 2 l, the region 0 < z < L m by a metal with relative permittivity ϵ m, and the region L m < z < L m + L d = L Σ by a rugate filter with relative permittivity [( ) ( ) ( nb + n a nb n a ϵ d (z) = + sin π z L )] 2 m, z (L m, L Σ ), (8.1) 2 2 Ω where n a and n b are the minimum and maximum refractive indexes, respectively, and 2Ω is the period of the rugate filter. The boundary-value problem is shown schematically in Fig The rugate filter was chosen to contain an integral number of periods, that is, L d = 2ΩN p, N p {1, 2, 3,...}. (8.2) 85

116 Furthermore, n l > n b is required for the TKR configuration. Let a linearly polarized plane wave, propagating in the half-space z 0 and with its wave vector making an angle θ with the z axis, be incident on the metalcapped rugate filter. The incident, reflected and transmitted field phasors may be represented as [70] E inc (r) = ( a p p + + a s s ) exp (ik x x + ik 0 n l z cos θ), z 0, (8.3) H inc (r) = n l η 0 ( as p + a p s ) exp (ik x x + ik 0 n l z cos θ), z 0, (8.4) E ref (r) = ( r p p + r s s ) exp (ik x x ik 0 n l z cos θ), z 0, (8.5) H ref (r) = n l η 0 ( rs p r p s ) exp (ik x x ik 0 n l z cos θ), z 0, (8.6) E tr (r) = ( t p p + + t s s ) exp [ik x x + ik 0 n l (z L Σ ) cos θ], z L Σ, (8.7) H tr (r) = n l η 0 ( ts p + t p s ) exp [ik x x + ik 0 n l (z L Σ ) cos θ], z L Σ,(8.8) where the unit vectors s = û y, and p ± = û x cos θ + û z sin θ represent s- and p-polarization states, respectively, and k x = k 0 n l sin θ R. (8.9) The reflection amplitudes r p,s and the transmission amplitudes t p,s are to be determined in terms of known amplitudes a p,s of the incident plane wave. Let me define the electric and magnetic field phasors for z (0, L Σ ) as E(r) = e(z) exp (ik x x) H(r) = h(z) exp (ik x x) }. (8.10) Using them in the frequency-domain Maxwell curl postulates, a 2 2 matrix ordinary differential equation can be obtained for each linear polarization state. p-polarization state Substitution of Eqs. (8.10) in the Maxwell curl postulates yields the 2 2 matrix ordinary differential equation where d [ f (p) (z) ] [ ] = i P (p) (z) [f (p) (z) ], z (0, L Σ ), (8.11) dz [ f (p) (z) ] = [ ex (z) h y (z) ], (8.12) 86

117 and [ ] [ P (p) (z) = [ ] [ P (p) (z) = P (p) d P (p) m [ ] = [ ] (z) = 0 ωµ 0 k2 x ωϵ 0 ϵ m ωϵ 0 ϵ m 0 ] 0 ωµ 0 k2 x ωϵ 0 ϵ d (z) ωϵ 0 ϵ d (z) 0, z (0, L m ], (8.13) ], z (L m, L Σ ). (8.14) The column vectors [ f (p) (0) ] and [ f (p) (L Σ ) ] can be written using Eqs. (8.3) (8.8) as [ f (p) (0) ] [ ] [ ] cos θ cos θ ap = (8.15) and n l η 0 n l η 0 [ f (p) (L) ] = t p [ cos θ n l η 0 r p ]. (8.16) To find [ f (p) (z) ] for z (0, L Σ ), let me divide the region L m < z < L Σ into N d slices, and consider the region 0 z L m as one slice. So, in the region 0 < z < L Σ, I have N d + 1 slices and N d + 2 interfaces. In the jth slice, j [1, N d + 1], I approximate [ ] ] [ ( )] P (z) (p) = [P (p) = P (p) zj + z j 1, z (z j, z j 1 ), (8.17) j 2 so that Eq. (8.11) yields [ f (p) (z j 1 ) ] ] = [G (p) j ] } ] 1 exp { i j [D (p) [G (p) [f (p) (z j ) ], (8.18) j j ] where j = z j z j 1, [G (p) is a square matrix comprising the eigenvectors of ] j ] [P (p) as its columns, and the diagonal matrix [D (p) contains the eigenvalues j ] j of [P (p) in the same order. j After the solution of Eq. (8.11) using Eq. (8.18), the straightforward method to find the unknown reflection amplitude r p and transmission amplitude t p requires the implementation of standard boundary conditions at the interface planes z = 0, z = L m, and z = L Σ. This method was successfully used in a similar problem [70], where an SNTF [1] was used instead of a rugate filter. However, when I implemented this method for the TKR configuration described in this chapter, the solution did not converge for all values of the incidence angle θ. Usually, the solution converged only for θ [0, θ c ), where θ c < 90 decreases as n l increases. So I had to reformulate the problem using a stable algorithm [ ], which is 87

118 popular in finding reflected and transmitted fields for various gratings using the rigorous coupled-wave approach. This method is described now. [ Let ] me define auxiliary transmission coefficients t (j) p and transmission matrixes Z (p) by the relation [108] j [ f (p) (z j ) ] [ = t ] (j) p Z (p), j [0, N j d + 1], (8.19) where z 0 = 0, t (N d+1) p To find t (j) p and [Z (p) ] results in the relation ] t (j 1) p [Z (p) j 1 = t p, and j [ Z (p) ] N d +1 = [ cos θ n l η 0 ]. (8.20) for j [0, N d ], substitute Eq. (8.19) in (8.18), which ] = t (j) p [G (p) j (1) e i jdj 0 0 e i jd (2) j ] 1 ] [G (p) [Z (p) j [1, N d + 1], (8.21) ] where d (1) j and d (2) j are the eigenvalues of the square matrix [P (p) such that the imaginary part of d (1) j is greater than that of d (2) j. ] Since t (j) p and [Z (p) cannot be determined simultaneously from Eq. (8.21), let me define [108] j t (j 1) p = t (j) p w (1) j ( exp where the scalars w (1) j and its counterpart w (2) j [ ] w (1) ] 1 j w (2) = [G (p) j j i j d (1) j j j ), (8.22) are defined via [Z (p) ] j. (8.23) Substitution of Eq. (8.22) in (8.21) results in the relation ] ] 1 [Z (p) = [G (p) ( ) w (2) j j 1 j exp i w (1) j d (2) j + i j d (1), j [1, N d + 1].(8.24) j j ] ] From Eqs. (8.23) and (8.24), I find [Z (p) in terms of [Z (p). After partitioning [Z (p) ] 0 = [ 88 0 z (1) 0 z (2) 0 ] N d +1, (8.25) j,

119 and using Eqs. (8.32) and (8.19), r p and t (0) p are found as follows: [ t (0) p r p ] = a p [ ] 1 [ ] 0 cos θ cos θ. (8.26) z (1) z (2) 0 n l η 0 Equation (8.26) is obtained by enforcing the usual boundary conditions across the plane z = 0. After t (0) p is known, t p = t (N d+1) p is found by reversing the sense of iterations in Eq. (8.22). s-polarization state For the s-polarization state, the 2 2 matrix ordinary differential equation can be obtained in the metal and the rugate filter as where and [ ] [ P (s) (z) = n l η 0 d [ f (s) (z) ] [ ] = i P (s) (z) [f (s) (z) ], z (0, L Σ ), (8.27) dz P (s) m [ ] = [ ] [ ] P (s) (z) = P (s) (z) = d [ [ f (s) (z) ] = [ ey (z) h x (z) 0 ωµ 0 k 2 x ωµ 0 ωϵ 0 ϵ m 0 ], (8.28) ] 0 ωµ 0 k 2 x ωµ 0 ωϵ 0 ϵ d (z) 0, z (0, L m ], (8.29) ], z (L m, L Σ ). (8.30) The column vectors [ f (s) (0) ] and [ f (s) (L Σ ) ] can be written using Eqs. (8.3) (8.8) as [ f (s) (0) ] [ ] [ ] 1 1 as = n l n η 0 cos θ l (8.31) η 0 cos θ r s and [ f (s) (L) ] = t s [ 1 n l η 0 cos θ ]. (8.32) The rest of the procedure to find the unknown reflection amplitude r s and transmission amplitude t s is the same as in Sec for the p-polarization state. 89

120 8.2.2 Canonical-boundary value problem for coupled-sppwave propagation Let me next consider the canonical boundary-value problem shown in Fig The geometry of the problem is the same as for the TKR configuration except that the rugate filter is semi-infinite in thickness, i.e., N p. Let the coupled SPP wave propagate parallel to x axis and attenuate as z ±. Therefore, in the region z 0, the electric and magnetic field phasors may be written as and E(r) = H(r) = η 1 0 [ b p ( αl k 0 û x + κ k 0 û z ) + b s û y ] exp(iκx iα l z), z 0, (8.33) [ ( αl b p ϵ l û y + b s û x + κ )] û z exp(iκx iα l z), z 0. (8.34) k 0 k 0 where κ 2 + αl 2 = k2 0 ϵ l, κ is complex-valued, and Im(α l ) > 0 for attenuation as z. Here b p and b s are unknown scalars with the same units as the electric field, with the subscripts p and s, respectively, denoting the s- and p-polarization states. The field phasors in the metal film and the rugate filter are taken to be } E(r) = e(z) exp (iκx), z > 0. (8.35) H(r) = h(z) exp (iκx) x z = 0 z= L m z Homogeneous dielectric Metal film SPP waves Rugate filter Figure 8.2: Schematic of the canonical boundary-value problem for coupled-sppwave propagation due to the metal film. p-polarization state For p-polarized coupled SPP waves, the substitution of Eq. (8.35) in the Maxwell curl postulates results in a 2 2 matrix ordinary differential equation. The solution 90

121 of this equation in the metal film is [ f (p) (L m ) ] { [ = exp i with [ P (p) mc ] [ = P (p) mc ] } L m [f (p) (0+) ]. (8.36) 0 ωµ 0 κ2 ωϵ 0 ϵ m ωϵ 0 ϵ m 0 ]. (8.37) The optical response of one period [1] of the rugate filter is given by a matrix [Q (p) ], defined via [ f (p) (L m + 2Ω) ] ] = [Q (p) [f (p) (L m +) ], (8.38) that has to be found using a piecewise uniform approximation. By virtue of the Floquet Lyapunov theorem [88], a matrix [ Q (p) ] can be defined such that { [Q (p) ] = exp i2ω[ Q } (p) ]. (8.39) Both [Q (p) ] and [ Q (p) ] share the same eigenvectors, and their eigenvalues are also related. Let [t] (1) and [t] (2), be the eigenvectors corresponding to the eigenvalues σ 1 and σ 2, respectively, of [Q (p) ]; then, the corresponding eigenvalue α n of [ Q (p) ] is given by α n = i ln σ n, 2Ω n {1, 2}. (8.40) After ensuring that Im(α 1 ) > 0, I set [f (p) (L m +)] = c p [t] (1), (8.41) where c p is an unknown scalar; the other eigenvalue of [ Q (p) ] pertains to a wave that amplifies as z. The field at z = 0 can be written using Eqs. (8.33) and (8.34) as [ αl ] [f (p) k (0 )] = b 0 p ϵ. (8.42) l η 0 Implementing the standard boundary conditions across the planes z = 0 and z = L m, I arrive at the matrix equation [ { [ ] } αl ] c p [t] (1) = b p exp i P (p) k L m,c m 0 ϵ (8.43) l η 0 91

122 by making the use of Eqs. (8.36), (8.41) and (8.42). Equation (8.43) can be rearranged as ] [ ] [M (p) cp = [0]. (8.44) For a nontrivial solution, the 2 2 matrix [M (p) ] must be singular, so that ] det [M (p) = 0 (8.45) is the required dispersion equation. This equation was solved for κ C using the Newton-Raphson method [89]. s-polarization state The dispersion equation for s-polarized coupled SPP waves can be obtained in the same way as in the foregoing subsection. 8.3 Numerical Results and Discussion After finding the transmission and reflection amplitudes in the TKR configuration, two reflection coefficients and two transmission coefficients b p r pp = r p /a p, r ss = r s /a s, (8.46) t pp = t p /a p, t ss = t s /a s, (8.47) may be obtained. Now, the absorptances for p- and s-polarization states can be defined as A p = 1 ( r pp 2 + t pp 2), A s = 1 ( r ss 2 + t ss 2), (8.48) whereas A p,s [0, 1] due to the principle of conservation of energy [70]. For all the calculations reported in the remainder of this section, n a = 1.45, n b = 2.32, and λ 0 = 633 nm was fixed as in Ch. 3. The half-spaces z < 0 and z > L Σ were supposed to be occupied by zinc selenide: n l = The metal film was taken be an aluminum thin film: ϵ m = ( i) 2 [44]. The half-period of the rugate filter was taken to be Ω = 1.5λ 0 because the interface of a semi-infinite metal and rugate filter can guide multiple p- and multiple s-polarized SPP waves at this value of the half-period as was seen in Ch. 3. For the computation of absorptances A p and A s, the number of slices N d were selected so as to have 2-nm-thick slices in L m < z < L Σ. The results for p-polarized SPP waves and s-polarized SPP waves guided by the metal/rugate-filter in the TKR configuration of Sec are now presented and related to the results obtained from the solution of the canonical boundary-value problem of Sec

123 8.3.1 p-polarization state The variation of absorptance A p vs. the angle of incidence θ is given in Fig. 8.3 for N p {3, 4} and L m = 30 nm. The absorptances were calculated for two values of N p in order to identify waveguide modes [104, 105], which must depend on L d, as has been shown elsewhere [70,120]. A Mathematica TM program for calculating A p versus θ is provided in Appendix B.5. At seven values of the incidence angle θ, given in Table 8.1, a peak is present independent of the value of N p. The relative wavenumbers k x /k 0 at these values of θ are also given in Table 8.1. The seven A p -peaks in Fig. 8.3 represent the excitation of p-polarized SPP waves because of the independence of their θ-value from the value of N p. I arrived at the same conclusion from calculations made with N p = 10 (not presented) Ap Θ deg Figure 8.3: Absorptance A p as function of the incidence angle θ in the TKR configuration, when λ 0 = 633 nm, n l = 2.58, L m = 30 nm, and Ω = 1.5λ 0. Solid red line is for N p = 3 and dashed blue line is for N p = 4. Others parameters are given at the beginning of Sec Now, the relative wavenumber of a p-polarized SPP wave that can be guided by the prism/metal interface in the canonical configuration [34] is κ/k 0 = ϵ l ϵ m /(ϵ l + ϵ m ) = i, (8.49) which is certainly not close to any of the relative wavenumbers given in Table 8.1. Therefore, any peak identified in Table 8.1 represents either an SPP wave guided by the metal/rugate-filter interface or a coupled SPP wave guided by the metal film. The variation of A p with the thickness L m of the metal film is illustrated in Fig. 8.4 at the θ-values of the A p -peaks for N p = 4. The value of A p at each peak first increases and then decreases with the increase in the thickness of the metal film. This behavior is typical of the absorptance peak that represents the 93

124 Table 8.1: Values of the incidence angle θ and the relative wavenumber k x /k 0, where a peak is present in Fig. 8.3 independent of the value of N p. Each peak represents a p-polarized SPP wave, not a waveguide mode. θ k x /k 0 = n l sin θ Ap (a) L m nm Ap (b) L m nm Figure 8.4: Variation of A p with L m at the θ-values of the A p -peaks for N p = 4 in the TKR configuration. (a) Green solid line is for θ = 33.23, black dashed line for θ = 37.20, red chain-dashed line for θ = 42.41, and blue dotted line is for θ = 48.01; (b) red solid line is for θ = 53.86, black dashed line for θ = 59.66, and blue chain-dashed line is for θ = excitation of an SPP wave. The peak absorptance is maximum when L m is close to the penetration depth met of the aluminum thin film at the chosen wavelength: met = [ Im ( k 0 ϵm )] 1 = nm. All the peaks have Ap 1 at L m 27 nm except the peak at θ = that has A p 0.8 at L m 24 nm. This indicates that the excitation of the SPP wave at θ = is not as efficient as that of the remaining SPP waves. Also, Fig. 8.4 indicates that the optimal value of L m is close to met for a given metal. Further evidence in support of the conclusion that the peaks identified in Table 8.1 represent the excitation of p-polarized SPP waves was provided by the solution of the canonical boundary-value problem formulated in Sec The solutions of the dispersion Eq. (8.45) for p-polarized SPP waves are given in Table 8.2 for L m = 30 nm. Let me repeat that rugate filter occupies the half-space 94

125 Table 8.2: Relative wavenumbers κ/k 0 of p-polarized SPP waves obtained by the solution of the canonical boundary-value problem formulated in Sec for L m = 30 nm. Other parameters are given at the beginning of Sec p-pol i i i i i i i z > L m in that canonical boundary-value problem instead of having a finite thickness L d = 2ΩN p as for the TKR configuration. All solutions of the canonical boundary-value problem represent SPP waves. Depending on the thickness of the metal film, these SPP waves can either couple the two interfaces of the metal film or not. The relative wavenumbers κ/k 0 of the seven possible p-polarized solutions are given in Table 8.2. Comparison of Tables 8.1 and 8.2 shows that Re [κ/k 0 ] of each item in Table 8.2 is close to the relative wavenumber k x /k 0 at one of the A p -peaks, signifying thereby that the seven peaks identified in Table 8.1 represent the excitation of p-polarized SPP waves. To analyze the spatial power-density profiles of the SPP waves excited by an incident p-polarized plane wave in the TKR configuration, I have shown the variation of the Cartesian components of time-averaged Poynting vector P(x, z) = 1 2 Re [E(x, z) H (x, z)] (8.50) in Fig. 8.5 for θ = 33.21, 42.41, and along a line normal to the plane z = L m ; P(x, z) P(z) as k x R. The figure shows that P z decays in the metal film as z L m and is negligible in the rugate filter (z > L m ). This should be expected because the incident plane wave decays as it penetrates into the metal. However, P x is localized to the metal/rugate-filter interface and decays away from that interface on both sides. The localization of the P x to the interface plane z = L m is a clear indicator of an SPP wave. Le me note that, while the variation of P x is similar in the metal for the three SPP waves, the spatial profiles of P x in the rugate filter are different for different SPP waves, indicating the different degree of localization of the SPP wave excited with a different value of θ. In all the plots presented in Fig. 8.5, P x (z = 0) 0, whereas P x (z = L Σ ) 0. For all other SPP waves excited by p-polarized incident plane waves, the same behavior of P x and P z was noted (profiles not shown). Because the thickness of the metal film is almost equal to the penetration depth of the metal, P x does not decay to zero in the thin metal film. For a sufficiently thick metal film, all the power density may reside completely in the rugate filter and the metal film; however, no SPP waves may be excited in the TKR configuration if the metal film is made too thick. 95

126 P x,z P x,z P x,z z L m z L m z L m P x,z P x,z P x,z z L m z L m z L m Figure 8.5: Variations of the Cartesian components P x and P z (in W m 2 ) of the time-averaged Poynting vector along the z axis in (left) the metal film and (right) the rugate filter for L m = 30 nm in the TKR configuration for a p-polarized incident plane wave (a p = 1 V m 1, a s = 0). (top) θ = 33.23, (middle) θ = 42.41, and (bottom) θ = Red solid line represents P x, blue dashed line represents P z, and P y is identically zero. The thin metal film in the TKR configuration, however, may lead to the coupling of the two interface, that is, an SPP wave may be coupled to both prism/metal and metal/rugate-filter interfaces, or an SPP wave may be excited on the metal/rugate-filter interface. To distinguish between coupled and uncoupled p-polarized SPP waves, the spatial power-density profiles of the SPP waves guided by the thin metal film were examined. Representative spatial profiles of power density for two different p-polarized SPP waves are given in Fig. 8.6 for the canonical boundary-value problem formulated in Sec The SPP wave with κ/k 0 = i is loosely localized to prism/metal interface in the prism. However, the SPP wave with κ/k 0 = i is strongly localized to both the prism/metal and the metal/rugate-filter interfaces. The spatial power density profiles (not shown) of the remaining p-polarized SPP waves showed that all the 96

127 P x,z P x,z P x,z P x,z P x,z z L m P x,z z L m z L m z L m Figure 8.6: Variations of the Cartesian components of the time-averaged Poynting vector P(x = 0, z) (in W m 2 ) along the z axis in (top) the prism material, (middle) the metal film with L m = 30 nm, and (bottom) the rugate filter for the canonical boundary-value problem formulated in Sec for a p-polarized SPP wave with (left) κ/k 0 = i, and (right) κ/k 0 = i. Red solid line represents P x, blue dashed line represents P z, and P y is identically zero. The computations were made with b p = 1 V m 1. SPP waves are strongly localized to both the interfaces except the two SPP waves with κ/k 0 = i and κ/k 0 = i; however, most of the power density of each p-polarized SPP waves resides in the rugate filter. So, five out of the seven p-polarized SPP waves are coupled SPP waves. To see if the coupling due to thin metal film results in new SPP waves in relation to SPP waves guided by the metal/rugate-filter interface, the canonical boundary-value problem of SPP-wave propagation by that interface was solved. The relative wavenumbers κ/k 0 of seven possible p-polarized SPP waves that are guided by that interface are presented in Table 8.3. All the p-polarized SPP waves that are excited in the TKR configuration are also guided by the interface of semiinfinite metal and rugate filter. This shows that the coupling of the two interfaces 97

128 Table 8.3: Relative wavenumbers κ/k 0 of p-polarized SPP waves guided by the interface between semi-infinite metal and semi-infinite rugate filter (Ch. 3). All the parameters are the same as for Table 8.2 except that L m. p-pol i i i i i i i in the TKR configuration is not strong enough to be able to guide new p-polarized SPP waves, provided that L m is comparable to met s-polarization state The absorptance A s for an s-polarized incident plane wave is given in Fig. 8.7 as a function of the incidence angle θ for the TKR configuration with a 30-nm-thick metal film for N p {3, 4}. For five values of the incidence angle θ, an absorptance peak is present, independent of the value of N p. The θ-values of these peaks along with the relative wavenumbers k x /k 0 = n l sin θ are given in Table 8.4. All of the A s -peaks represent the excitation of s-polarized SPP waves because the peaks are independent of the value of N p. From the figure, one may suspect that there is also a peak at θ 35 that appears to be independent of N p. However, closer scrutiny revealed that the θ- value of the peak changes significantly with a change in N p ; therefore, it does not represent the excitation of an s-polarized SPP wave As Θ deg Figure 8.7: Same as Fig. 8.3 except that A s is plotted instead of A p. In Fig. 8.7, the value of A s at all the peaks, that represent the excitation of s-polarized SPP waves, is greater than 0.8 except for the peak at θ = So the excitation of the s-polarized SPP wave at θ = is not that efficient 98

129 Table 8.4: Values of the incidence angles θ, and the relative wavenumbers k x /k 0, where a peak is present in Fig. 8.7 independent of the value of N p. θ k x /k 0 = n l sin θ as those of the rest of the s-polarized SPP waves. Moreover, the comparison of Figs. 8.3 and 8.7 shows that the peaks representing the excitation of s-polarized SPP waves are narrower than those representing the excitation of p-polarized SPP waves As L m nm Figure 8.8: Variation of A s vs. the thickness of the metal film L m at the θ- position of the A s -peaks for N p = 3 in the TKR configuration. Solid red line is for θ = 38.97, black dashed line for θ = 44.01, blue chain-dashed for θ = 49.22, green dotted line for θ = 54.63, and orange dashed line (with larger dashes) is for θ = To analyze the effect of the metal film s thickness on the coupling efficiency, the variation of A s vs. L m is given in Fig. 8.8 for θ-values of all five A s -peaks of interest. The graphs show that A s reaches a maximum value close to L m = 27 nm (the same value as for p-polarized incident plane waves) for all θ-values except for θ = The maximum of the A s -peak at θ = occurs at L m 12 nm; however, at L m = 12 nm the value of A s for other four A s -peaks is less than 0.4. Therefore, to excite maximum number of s-polarized SPP waves efficiently, a value of L m close to 27 nm should be used. This is in contrast to the excitation 99

130 of p-polarized SPP waves, where all possible SPP waves can be excited efficiently using a metal film with L m = 27 nm. The solution of the dispersion equation for s-polarized SPP waves guided by the metal film, with the semi-infinite prism on one side and the semi-infinite rugate filter on the other, are presented in Table 8.4. The value of Re [κ/k 0 ] of each item in Table 8.5 is close to the relative wavenumber k x /k 0 at one of the A s -peaks given in Table 8.4, reinforcing the conclusion that the peaks identified in Table 8.4 represent the excitation of s-polarized SPP waves in the TKR configuration. Table 8.5: Same as Table 8.2 except that the relative wavenumbers of s-polarized SPP waves are given instead of p-polarized SPP waves. s-pol i i i i i P x,z P x,z z L m z L m P x,z P x,z z L m z L m Figure 8.9: Variations of the Cartesian components P x and P z (in W m 2 ) of the time-averaged Poynting vector along the z axis in (left) the metal film and (right) the rugate filter for L m = 30 nm in the TKR configuration for an s-polarized incident plane wave (a p = 0, a s = 1 V m 1 ). (top) θ = 49.22, and (bottom) θ = Red solid line represents P x, blue dashed line represents P z, and P y is identically zero. Representative spatial profiles of the power density for two s-polarized SPP waves are provided in Fig. 8.9 in the TKR configuration for θ = and For both the cases, P z decays in the metal film as z L m and is negligible in the rugate filter, as was the case for p-polarized incident plane wave. In the rugate 100

131 P x,z P x,z P x,z P x,z P x,z z L m P x,z z L m z L m z L m Figure 8.10: Variations of the Cartesian components of the time-averaged Poynting vector P(x = 0, z) (in W m 2 ) along the z axis in (top) the prism material, (middle) the metal film with L m = 30 nm, and (bottom) the rugate filter for two s-polarized SPP waves obtained from the solution of the canonical boundaryvalue problem shown in Fig (left) κ/k 0 = i, and (right) κ/k 0 = i. Red solid line represents P x, blue dashed line represents P z, and P y is identically zero. The computations were made with b s = 1 Vm 1. Table 8.6: Same as Table 8.3 except that the relative wavenumbers of s-polarized SPP waves are given instead of p-polarized SPP waves. s-pol i i i i i filter, P x is confined to within one period 2Ω of the rugate filter for both s-polarized SPP waves. For θ = 49.22, P x decays away from the interface in the metal film. However, for θ = 60.66, P x decays away from the interface at z = L m but again starts to amplify as z 0, indicating that the SPP wave is not only guided by the interface at z = L m but may also be coupled to the interface at z = 0. The spatial 101

132 profiles of the power density of s-polarized SPP waves at θ = 38.97, 44.01, and were similar to that of the s-polarized SPP wave at θ = (profiles not shown). The spatial profiles of the power density of s-polarized SPP waves were examined to see if the s-polarized SPP waves are coupled for the canonical boundaryvalue problem formulated in Sec Representative spatial profiles for two s-polarized SPP waves guided by the metal thin film with a semi-infinite rugate filter on one side and a semi-infinite prism on the other side are given in Fig for κ/k 0 = i and i. The parameters for the prism, the metal film, and the rugate filter are the same as for the TKR configuration. These two SPP waves correspond to the two SPP waves in Fig The spatial profiles (not shown) for the rest of s-polarized SPP waves of the canonical boundary-value problem, are also similar to those shown in Fig Both profiles show that the s-polarized SPP waves are strongly localized to the metal/rugate-filter interface and very loosely localized to the prism/metal interface, indicating that the s-polarized SPP waves are not coupled. This is in contrast to the p-polarization state, where five of the seven SPP waves are coupled to both interfaces. The canonical boundary-value problem of SPP-wave propagation by the interface of a semi-infinite metal and the semi-infinite rugate filter (Ch. 3) was solved for the s-polarization state, and the relative wavenumbers κ/k 0 of five s-polarized SPP waves are given in Table 8.6. The wavenumbers in Tables 8.5 and 8.6 are almost the same, so the coupling due to the thin metal film has not resulted in new s-polarized SPP waves in TKR configuration, as was also the case with p-polarization state. 8.4 Concluding Remarks The plane waves of either p- or s-polarization state, propagating in an optically denser dielectric material, were made incident on a metal-capped rugate filter. The absorptances were calculated using a stable algorithm as functions of the incidence angle. The excitation of SPP waves was inferred from the presence of those peaks in the absorptance spectrum that were independent of the thickness of the rugate filter. It was concluded that (i) multiple p- and s-polarized SPP waves can be excited using the TKR configuration, (ii) most of the possible SPP waves can be excited efficiently using a metal film with a thickness that is close to the penetration depth, and (iii) the absorptance peaks representing the excitation of s-polarized SPP waves are narrower than those representing p-polarized SPP waves. 102

133 A canonical boundary-value problem to study the propagation of coupled SPP waves by a metal film, with a semi-infinite rugate filter on one side and a semiinfinite homogeneous dielectric material on the other side, was also formulated to obtain a dispersion equation for each of the two linear polarization states. The solution of the dispersion equations and the spatial profiles of the SPP waves (i) reinforced the results of the TKR configuration, and (ii) showed that p-polarized SPP waves are more likely to be coupled to both the prism/metal and metal/rugate-filter interfaces than s-polarized SPP waves. However, the solution of another canonical boundary-value problem (Ch. 3) of SPP-wave propagation by the interface of a semi-infinite metal and a semi-infinite rugate filter revealed that the coupling due to the thin metal film in the TKR configuration does not result in new SPP waves. 103

134 Chapter 9 Grating-Coupled Excitation of Multiple SPP Waves Guided by Metal/Rugate-Filter Interface 9.1 Introduction The canonical boundary-value problem solved in Ch. 3 showed that multiple SPP waves can be guided even if the dielectric partnering material is isotropic provided that the material is also periodically nonhomogeneous normal to the interface. This is a very attractive result, because both partnering materials are isotropic and because the dielectric partnering material can be fabricated as a rugate filter [2, 76, 78, 80, 110]. In Ch. 8, I presented the investigations on the prism-coupled (TKR configuration) excitation of multiple SPP waves. It was seen in that chapter that the absorptance peaks independent of the thickness of the dielectric layer represent the excitation of SPP waves. For this chapter, I set out to investigate the excitation of multiple SPP waves by the periodically corrugated interface of a metal and a rugate filter, which is an alternative to the TKR configuration. Though the TKR configuration is easier to implement in a laboratory than the grating-coupled configuration, it does not allow the application of multiple SPP waves in the solar cells. However, as is shown in Ch. 10, the grating-coupled configuration can be used to enhance the absorption of light in thin-film solar cells. In the gratingcoupled configuration, fields in the two partnering materials are represented as a linear superposition of Floquet harmonics. If the component of the wavevector of a Floquet harmonic in the plane of the grating is the same as that of the SPP wave, the Floquet harmonic can couple with the SPP wave. The interplay of the periodic nonhomogeneity of the dielectric partnering material and a periodically This chapter is based on: M. Faryad and A. Lakhtakia, Grating-coupled excitation of multiple surface-plasmon-polariton waves, Phys. Rev. A 84, (2011). 104

135 corrugated interface is phenomenologically rich [111, 112], and should lead to the excitation of multiple SPP waves as different Floquet harmonics. The relevant boundary-value problem was formulated using the rigorous coupledwave approach (RCWA) [106, 118]. In this numerical technique, the constitutive parameters are expanded in terms of Fourier series with known expansion coefficients, and the electromagnetic field phasors are expanded in terms of Floquet harmonics whose coefficients are determined by substitution in the frequencydomain Maxwell curl postulates. The accuracy of solution is conventionally held to depend only on the number of Floquet harmonics actually used in the computations [107]. The RCWA has been used to solve for scattering by a variety of surface-relief gratings [ , 119], generally with both partnering materials being homogeneous. The theoretical formulation of the boundary-value problem is provided in Sec. 9.2 and the numerical results are discussed in Sec Concluding remarks are presented in Sec Boundary-Value Problem Description Let me consider the boundary-value problem shown schematically in Fig The regions z < 0 and z > d 3 are vacuous, the region 0 z d 1 is occupied by the dielectric partnering material with relative permittivity ϵ d (z), and the region d 2 z d 3 by the metallic partnering material with spatially uniform relative permittivity ϵ m. The region d 1 < z < d 2 contains a surface-relief grating of period L along the x axis. The relative permittivity ϵ g (x, z) = ϵ g (x ± L, z) in this region is taken to be as { ϵ m [ϵ m ϵ d (z)]u [d 2 z g(x)], x (0, L 1 ), ϵ g (x, z) = (9.1) ϵ d (z), x (L 1, L), for z (d 1, d 2 ), with ( ) πx g(x) = (d 2 d 1 ) sin, L 1 (0, L), (9.2) and U(ζ) = { L 1 1, ζ 0, 0, ζ < 0. (9.3) The depth of the surface-relief grating defined by Eq. (9.2) is d 2 d 1. This particular grating shape is chosen for the ease of fabrication; however, the theoretical formulation given in the remainder of this section is independent of the shape of the surface-relief grating. 105

136 In the vacuous half-space z 0, let a plane wave propagating in the xz plane at an angle θ to the z axis, be incident on the structure. Hence, the incident, reflected, and transmitted field phasors may be written in terms of Floquet harmonics as follows: E inc (r) = n Z ( sn a (n) s + p + n a (n) p ) [ ( exp i k (n) x x + k z (n) z )], z 0, (9.4) H inc (r) = η 0 1 n Z ( p + n a (n) s s n a (n) p ) [ ( exp i k (n) x x + k z (n) z )], z 0, (9.5) E ref (r) = n Z ( sn r (n) s + p n r (n) p ) exp [ i ( k (n) x x k (n) z z )], z 0, (9.6) H ref (r) = η 0 1 n Z ( p n r (n) s s n r (n) p ) [ ( exp i k (n) x k z (n) z )], z 0, (9.7) x where k (n) x E tr (r) = n Z ( sn t (n) s H tr (r) = η 0 1 n Z The unit vectors and + p + n t (n) p ( p + n t (n) s ) { [ exp i k (n) x x + k z (n) (z d 3 ) ]}, s n t (n) p z d 3, (9.8) ) { [ exp i k (n) x x + k z (n) (z d 3 ) ]}, z d 3, (9.9) = k 0 sin θ + nκ x, κ x = 2π/L, and + k 2 k z (n) 0 (k x (n) ) 2, k 2 0 > (k x (n) ) 2 =. (9.10) +i (k x (n) ) 2 k 2 0, k 2 0 < (k x (n) ) 2 s n = û y (9.11) p ± n = k(n) z û x + k(n) x û z (9.12) k 0 k 0 represent the s- and p-polarization states, respectively Coupled ordinary differential equations The relative permittivity in the region 0 z d 3 can be expanded as a Fourier series with respect to x, viz., ϵ(x, z) = n Z ϵ (n) (z) exp(inκ x x), z [0, d 3 ], (9.13) 106

137 Incident -1 0 z = 0 z = d 1 z = d 2 z = d Rugate filter Metal Metal -1 z +1 SPP waves x +1 0 Figure 9.1: Schematic of the boundary-value problem solved using the RCWA. where and ϵ (n) (z) = ϵ d (z), z [0, d 1 ], ϵ (0) 1 L (z) = L 0 ϵ g(x, z)dx, z (d 1, d 2 ), ϵ m, z [d 2, d 3 ], { 1 L L 0 g(x, z) exp( inκ x x)dx, z [d 1, d 2 ] 0, otherwise H(r) = n Z H (n) (z) exp(ik (n) x x) (9.14) ; n 0. (9.15) The field phasors may be written in the region 0 z d 3 in terms of Floquet harmonics as E(r) = E (n) (z) exp(ik x (n) x) n Z, z [0, d 3 ], (9.16) with unknown functions E (n) (z) = E x (n) (z)û x +E y (n) (z)û y +E z (n) (z)û z and H (n) (z) = H x (n) (z)û x + H y (n) (z)û y + H z (n) (z)û z. 107

138 Substitution of Eqs. (9.13) and (9.16) in the frequency-domain Maxwell curl postulates results in a system of four ordinary differential equations and two algebraic equations as follows: d dz E(n) x d dz E(n) y k x (n) E y (n) d dz H(n) x (z) ik (n) x E z (n) (z) = ik 0 η 0 H y (n) (z), (9.17) (z) = ik 0 η 0 H (n) x (z), (9.18) (z) = k 0 η 0 H z (n) (z), (9.19) (z) ik x (n) H z (n) (z) = ik 0 ϵ (n m) (z)e y (m) (z), (9.20) d dz H(n) y (z) = ik 0 η 0 m Z η 0 m Z ϵ (n m) (z)e (m) x (z), (9.21) k x (n) H y (n) (z) = k 0 ϵ (n m) (z)e z (m) (z). (9.22) η 0 m Z Equations (9.17) (9.22) hold z (0, d 3 ) and n Z. These equations can be recast into an infinite system of coupled first-order ordinary differential equations. This system can not be implemented on a digital computer. Therefore, n N t was restricted and then define the column (2N t + 1)-vectors [X σ (z)] = [X ( N t) σ (z), X ( N t) σ (z),..., X σ (0) (z),..., X (N t 1) σ (z), X (N t) σ (z)] T, (9.23) for X {E, H} and σ {x, y, z}. Similarly, let me define (2N t + 1) (2N t + 1)- matrixes [K x ] = diag[k (n) x ], [ϵ(z)] = [ ϵ (n m) (z) ], (9.24) where diag[k x (n) ] is a diagonal matrix. Equations (9.19) and (9.22) yield and [E z (z)] = η 0 k 0 [ ϵ(z) ] 1 ] [K x [H y (z)] (9.25) [H z (z)] = 1 η 0 k 0 [ K x ] [E y (z)], (9.26) the use of which in Eqs. (9.17), (9.18), (9.20) and (9.21) eliminates E z (n) n Z, and gives the matrix ordinary differential equation H (n) z and d dz [f(z)] = i [ P (z) ] [f(z)], z (0, d 3 ), (9.27) 108

139 where the column vector [f(z)] with 4(2N t + 1) components is defined as [f(z)] = [[E ] x (z)] T, [E y (z)] T, η 0 [H x (z)] T, η 0 [H y (z)] T T (9.28) and the 4(2N t + 1) 4(2N t + 1)-matrix [ P (z) ] is given by [ ] P (z) = [ ] [ ] [ ] [ ] P 14 (z) [ ] [ ] [ ] [ ] 0 0 k 0 I 0 [ ] [ ] [ ] [ ] 0 P 32 (z) 0 0 [ ] [ ] [ ] [ ] P 41 (z) 0 0 0, (9.29) where [ 0 ] is the (2N t +1) (2N t +1) null matrix and [ I ] is the (2N t +1) (2N t +1) identity matrix, the three nonnull submatrixes on the right side of Eq. (9.29) are as follows: [ ] [ ] 1 [ ] P 14 (z) = k 0 I K k x [ϵ(z) ] ] 1 [K x, (9.30) 0 [ ] P 32 (z) = 1 ] 2 [ ] [K k x k0 ϵ(z), (9.31) [ ] 0 [ ] P 41 (z) = k 0 ϵ(z). (9.32) Solution algorithm The column vectors [f(0)] and [f(d 3 )] can be written using Eqs. (9.4) (9.9) as [ ] [ ] [ ] Y + Y [ ] [f(0)] = [ e ] [ e ] [A] Y +, [f(d Y + Y [R] 3 )] = [ e ] [T], (9.33) Y + h h h where [A] = [ a ( N t) s, a ( N t+1) s,..., a (0) s,..., a (N t 1) s, a (N t) s, ] a ( Nt) p, a ( Nt+1) p,..., a (0) p,..., a (Nt 1) p, a (Nt) T p, (9.34) [R] = [ r ( N t) s, r ( N t+1) s,..., r s (0),..., r (N t 1) s, r (N t) s, ] r ( N t) p, r ( N t+1) p,..., r p (0),..., r (N t 1) p, r (N t) T p, (9.35) [T] = [ t ( Nt) s, t ( Nt+1) s,..., t (0) s,..., t (Nt 1) s, t (Nt) s, ] t ( N t) p, t ( N t+1) p,..., t (0) p,..., t (N t 1) p, t (N t) T p, (9.36) 109

140 [ ] and the nonzero entries of (4N t + 2) (4N t + 2)-matrixes Y ± are as follows: e,h ( Y ± e )nm = 1, n = m + 2N t + 1, (9.37) ( ) Y ± = k(n) z e nm k 0, n = m 2N t 1, (9.38) ( Y ± h )nm = k(n) z k 0, n = m [1, 2N t + 1], (9.39) ( ) Y ± h = 1, n = m [2N nm t + 2, 4N t + 2]. (9.40) In order to devise a stable algorithm [ ], the region 0 z d 1 is divided into N d slices and the region d 1 < z < d 2 into N g slices, but the region d 2 z d 3 is kept as just one slice. So, there are N d +N g +1 slices and N d +N g +2 interfaces. In the jth slice, j [1, N d + N g + 1], bounded by the planes z = z j 1 and z = z j, I approximate [ ( )] [ ] [ ] P (z) = P = zj + z j 1 P, z (z j j, z j 1 ), (9.41) 2 so that Eq. (9.27) yields [f(z j 1 )] = [ G ] j exp { i j [ D ] j } [G ] 1 j [f(z j )], (9.42) where j = z j z j 1, [ G ] is a square matrix comprising the eigenvectors of [ P ] j j as its columns, and the diagonal matrix [ D ] contains the eigenvalues of [ P ] in j j the same order. Let me define auxiliary column vectors [T] j and auxiliary transmission matrixes [ ] Z by the relation [108] j [f(z j )] = [ Z ] j [T] j, j [0, N d + N g + 1], (9.43) where z 0 = 0, [T] Nd +N g +1 = [T], and [ Z ]N = d +N g +1 [ Y + e [ Y + h ] ]. (9.44) To find [T] j and [ Z ] for j [0, N j d + N g ], I substitute Eq. (9.43) in (9.42), which results in the relation [ [ Z ]j 1 [T] j 1 = [ ] e i G j ] [D] u j 0 j 0 e i [G ] 1 [Z ] j[d] [T] l j j j j, j [1, N d + N g + 1], (9.45) 110

141 where [ D ] u and [ D ] l are the upper and lower diagonal submatrixes of [ D ], respectively, when the eigenvalues are arranged in decreasing order of the imaginary j j j part. Since [T] j and [ Z ] cannot be determined simultaneously from Eq. (9.45), let j me define [108] { [ ] } u [T] j 1 = exp i j D [W ] u [T] j j j, (9.46) where the square matrix [ W ] u and its counterpart [ W ] l are defined via j j [ [ ] u ] W j [ ] l = [ G ] 1 [Z ]. (9.47) W j j j Substitution of Eq. (9.46) in (9.45) results in the relation [ ] Z = [ G ] [ [ ] ] I j 1 j exp { [ ] { i j [D] j} l l [W W ] } u 1 { }, j exp i j [D] u j j [1, N d + N g + 1]. (9.48) From Eqs. (9.47) and (9.48), I find [ Z ] in terms of [ Z ]. After partitioning 0 N d +N g+1 [ Z ] 0 = [ [ Z ] u [ ] 0 l Z 0 ] j, (9.49) and using Eqs. (9.33) and (9.43), [R] and [T] 0 are found as follows: [ [T]0 [R] ] [ ] u Z = 0 [ ] l Z 0 [ ] Y [ e ] Y h 1 [ Y + e [ Y + h ] ] [A]. (9.50) Equation (9.50) is obtained by enforcing the usual boundary conditions across the plane z = 0. After [T] 0 is known, [T] = [T] Nd +N g+1 is found by reversing the sense of iterations in Eq. (9.46). 9.3 Numerical Results and Discussion Homogeneous dielectric partnering material Let me begin with the dielectric partnering material being homogeneous, i.e., ϵ d (z) is independent of z. This case has been numerically illustrated by Homola [34, p. 38] and I adopted the same parameters: λ 0 = 800 nm, ϵ d = (water), 111

142 ϵ m = i (gold), and L = 672 nm. The incident plane wave is p polarized (a (n) p = δ n0 V m 1 and a s (n) 0 n Z) and the quantity of importance is the absorptance A p = 1 N t ( r (n) 2 s + r (n) 2 p + t (n) 2 s + ( ) t (n) 2) k z (n) p Re, k z (0) (9.51) which simplifies to n= N t A p = 1 N t n= N t ( r (n) 2 p + ( ) t (n) 2) k z (n) p Re, (9.52) k z (0) because all materials are isotropic. Figure 9.2(a) shows the variation of A p versus the incidence angle θ for a sinusoidal surface-relief grating defined by [34] g(x) = 1 [ ( )] 2πx 2 (d 2 d 1 ) 1 + sin (9.53) L instead of Eq. (9.2), and Fig. 9.2(b) shows the same for the surface-relief grating defined by Eq. (9.2) with L 1 = 0.5L. For computational purposes, I set N d = 1, N g = 50, and N t = 10, after ascertaining that all nonzero reflectances r p (n) 2 Re ( k z (n) /k z (0) ) and transmittances t (n) p 2 Re ( k z (n) /k z (0) ) converged within ±0.5% for all n [ N t, N t ]. Each figure shows plots of A p vs. θ for three different values of the thickness d 1, in order to distinguish [70] between (i) surface waves, which must be independent of d 1 for sufficiently large values of that parameter, and (ii) waveguide modes [104, 105], which must depend on d 1, as has been shown in Ref. 120 and in Ch. 7. In both figures, an absorptance peak at θ 12.5 for all three values of d 1 indicates the excitation of an SPP wave. Parenthetically, I note here that an SPP wave is a solution of a canonical boundary-value problem involving the planar interface of two semi-infinite half spaces, one of which is occupied by a metal and the other by a dielectric material; but, as the canonical boundary-value problem cannot be implemented practically, both materials must be present as sufficiently thick layers in a real situation so that the SPP wave decays appreciably through the thickness of each layer. The relative wavenumbers k x (n) /k 0 of a few Floquet harmonics at θ = 12.5 are given in Table 9.1. The solution of the canonical boundary-value problem [34] 112

143 A p A p (a) (b) (deg) Figure 9.2: Absorptance A p as a function of the incidence angle θ when the surfacerelief grating is defined by either (a) Eq. (9.53) or (b) Eq. (9.2). Black squares represent d 1 = 1500 nm, red circles d 1 = 1000 nm, and blue triangles d 1 = 800 nm. The grating depth (d 2 d 1 = 50 nm) and the thickness of the metallic layer (d 3 d 2 = 30 nm) are the same for all cases. The vertical arrows identify SPP waves. Table 9.1: Relative wavenumbers k (n) x /k 0 of Floquet harmonics at the θ-value of the peak identified in Fig. 9.2 by a vertical arrow. A boldface entry signifies an SPP waves. n = 2 n = 1 n = 0 n = 1 n = 2 θ = (when both partnering materials are semi-infinite along the z axis and their interface is planar) shows that the relative wavenumber κ/k 0 of the SPP wave that can be guided by the planar gold-water interface is κ/k 0 = ϵ d ϵ m /(ϵ d + ϵ m ) = i. (9.54) 113

144 P x (Wm -2 ) (a) (b) d 1 = 1000 nm z (nm) P x (Wm -2 ) (a) (b) d 1 = 1000 nm z-d 1 (nm) Figure 9.3: Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L for θ = 12.5, when the surface-relief grating is defined by either (a) Eq. (9.53) or (b) Eq. (9.2) and the incident plane wave is p polarized. Other parameters are the same as for Fig A comparison of Table 9.1 and Eq. (9.54) confirms that an SPP wave is excited at θ = 12.5 as the Floquet harmonic of order n = 1. The spatial profiles of the x-component of the time-averaged Poynting vector P(x, z) = 1 2 Re [E(x, z) H (x, z)] (9.55) along the z axis for x = 0.75L for the p-polarized incident plane wave at θ = 12.5 (the θ-value of the peak identified in Fig. 9.2 by a vertical arrow) also indicate that p-polarized SPP waves are indeed excited for both types of the surface-relief grating because P x decays quickly away from the interface z = d 1. Let me note that the absorptance peak in Fig. 9.2(b) is not only wider than in Fig. 9.2(a), but also of lower magnitude, which points out the critical importance of the shape function g(x) of the surface-relief grating. The incidence angle θ determined by Homola [34, p. 38] is approximately 11, the small difference between his and my results being (i) due to the different methods of computation and (ii) the fact that, while Homola had semi-infinite dielectric and metallic partnering materials, I have the two of finite thickness. 114

145 9.3.2 Periodically nonhomogeneous dielectric partnering material Now let me move on to the excitation of multiple SPP waves by a surface-relief grating where the dielectric partnering material has a periodic nonhomogeneity normal to the mean metal/dielectric interface: [( ) ( ) ( nb + n a nb n a ϵ d (z) = + sin π d )] 2 2 z, z > 0, (9.56) 2 2 Ω where 2Ω is the period. I chose n a = 1.45 and n b = 2.32, the same parameters as were used in Ch. 3. For all calculations reported in the remainder of this paper, I chose the metal to be bulk aluminum (ϵ m = i) and the free-space wavelength λ 0 = 633 nm. The surface-relief grating is defined by Eq. (9.2) with L 1 = 0.5L. I fixed N t = 8 after ascertaining that the absorptances for N t = 8 converged to within ±1% of the absorptances calculated with N t = 9. The grating depth d 2 d 1 = 50 nm and the thickness d 3 d 2 = 30 nm were also fixed, as their variations would not qualitatively affect the excitation of multiple SPP waves. Numerical results for Ω = λ 0 and Ω = 1.5λ 0 are now presented. Ω = λ 0 Let me commence with Ω = λ 0. The solution of the corresponding canonical boundary-value problem (when both the rugate filter and the metal are semiinfinite in thickness and their interface is planar) results in five p-polarized and two s-polarized SPP waves, the relative wavenmbers κ/k 0 being provided in Table 9.2. I used the solution of the canonical boundary-value problem as a guide to choose the grating period L and as a reference for the relative wavenumbers of SPP waves. To analyze the excitation of s-polarized SPP waves in the grating-coupled configuration, I calculated the absorptance n=n t ( r (n) A s = 1 2 s + ( ) t (n) 2) k z (n) s Re (9.57) k z (0) n= N t for a (n) s = δ n0 V m 1 and a (n) p 0 n Z. Both A p and A s were calculated as functions of θ for d 1 {4Ω, 5Ω, 6Ω}, with N g and N d selected to have slices of thickness 2 nm in the region 0 z d 1 but 1 nm in the region d 1 < z < d 2. The Mathematica TM program to calculate A p versus θ is provided in Appendix B.6. The code for A s is the same except that the matrix [P ] is different for s-polarized incidence. For all three values of d 1, a peak is present at θ = 37.7 in the plots of A p vs. θ in Fig. 9.4(a). The relative wavenumbers k x (n) /k 0 of several Floquet harmonics at this incidence angle are given in Table 9.3. At θ = 37.7, k x (1) /k 0 = is close 115

146 Table 9.2: Relative wavenumbers κ/k 0 of possible SPP waves obtained by the solution of the canonical boundary-value problem (Ch. 3) for Ω = λ 0. Other parameters are provided in the beginning of Sec If κ represents an SPP wave propagating in the û x direction, κ represents an SPP wave propagating in the û x direction. s-pol i i i i p-pol i i i i i These solutions were missed in: M. Faryad and A. Lakhtakia, Grating-coupled excitation of multiple surface-plasmon-polariton waves, Phys. Rev. A 84, (2011) (a) A p 0.4 A s (b) (deg) Figure 9.4: Absorptances (a) A p and (b) A s as functions of the incidence angle θ, when the surface-relief grating is defined by Eq. (9.2) with L 1 = 0.5L, λ 0 = 633 nm, Ω = λ 0, and L = λ 0. Black squares are for d 1 = 6Ω, red circles for d 1 = 5Ω, and blue triangles for d 1 = 4Ω. The grating depth (d 2 d 1 = 50 nm) and the thickness of the metallic layer (d 3 d 2 = 30 nm) are the same for all plots. Each vertical arrow identifies an SPP wave. 116

147 Table 9.3: Relative wavenumbers k (n) x /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig. 9.4 by vertical arrows when Ω = λ 0 and L = λ 0. Boldface entries signify SPP waves. n = 2 n = 1 n = 0 n = 1 n = 2 θ = θ = θ = θ = θ = to Re[κ/k 0 ] = , where κ/k 0 is the relative wavenumber of a p-polarized SPP wave in the canonical boundary-value problem as provided in Table 9.2. Thus, this A p -peak represents the excitation of a p-polarized SPP wave as a Floquet harmonic of order n = 1. In order to confirm this conclusion, I plotted the spatial profile of P x (0.75L, z) in Fig. 9.5 for θ = Indeed, P x decays quickly away from the plane z = d 1 in the region containing metal, and it also decays periodically, according to the Floquet Lyapunov theorem [88] inside the rugate filter away from the same interface, thereby providing confirmation. For the A p -peak at θ 21, the angular location changes slightly with the change in the value of d 1. However, this peak also represents the excitation of a p- polarized SPP wave because (i) k x (1) /k 0 = (Table 9.3) is close to Re[κ/k 0 ] = (Table 9.2), and (ii) the spatial profile of P x (0.75L, z) provided in Fig. 9.5 is also indicative of a surface wave guided by the metal/rugate-filter interface. The reason for the change in the θ-value of the A p -peak is the weak localization of this SPP wave in the region z < d 1 (see the left panel in Fig. 9.5 for θ = 21 ). However, for a sufficiently large value of d 1, the peak should be independent of the value of d 1. Three A s -peaks are present at θ 16.3, 28.4, and 31.6 in the plots of A s vs. θ, for all three values of d 1 in Fig. 9.4(b). The relative wavenumbers of Floquet harmonics at these values of the incidence angle are also provided in Table 9.3. At θ = 16.3, an s-polarized SPP wave is excited as a Floquet harmonic of order n = 2 because k x ( 2) /k 0 = (Table 9.3) is close to Re[κ/k 0 ] = (Table 9.2). The spatial profile of P x (0.75L, z) given in Fig. 9.6 for θ = 16.3 also confirms this conclusion. I also note that the s-polarized SPP wave is propagating in the û x direction because it is excited as a Floquet harmonic of a negative order. The A s -peak at θ = 28.4 represents the excitation of an s-polarized SPP wave, as a Floquet harmonic of order n = 1, because (i) k x (1) /k 0 = (Table 9.3) is close to Re[κ/k 0 ] = (Table 9.2), and (ii) the spatial profile of P x (0.75L, z) 117

148 P x (Wm -2 ) d 1 = 6 = 21 o = 37.7 o P x (Wm -2 ) d 1 = 6 = 21 o = 37.7 o z / z-d 1 (nm) Figure 9.5: Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L, when the surface-relief grating is defined by Eq. (9.2). The grating period L = λ 0 and the incident plane wave is p polarized. Other parameters are the same as for Fig P x (Wm -2 ) d 1 = 6 = 16.3 o = 28.4 o = 31.6 o P x (Wm -2 ) d 1 = 6 = 16.3 o = 28.4 o = 31.6 o z / z-d 1 (nm) Figure 9.6: Same as Fig. 9.5 except that the incident plane wave is s polarized. provided in Fig. 9.6 shows that an s-polarized SPP wave is guided by the interface z = d 1 in the +û x direction. Coincidently, the A s -peak at θ = 31.6 represents the excitation of the same s-polarized SPP wave but as a Floquet harmonic of order n = 2 because k x ( 2) /k 0 = (Table 9.3) is close to Re[κ/k 0 ] = (Table 9.2). This is also evident from the comparison of the spatial profiles given in Fig. 9.6 for θ = 28.4 and θ = Although the two spatial profiles are mirror images of each other, the excitation of the s-polarized SPP wave at θ = 31.6 is 118

149 not very efficient because it is excited as a Floquet harmonic of a higher order ( n = 2) (a) A s A p (b) (deg) Figure 9.7: Same as Fig. 9.4 except for L = 0.75λ 0. Table 9.4: Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig. 9.7 by vertical arrows when Ω = λ 0 and L = 0.75λ 0. Boldface entries signify SPP waves. n = 2 n = 1 n = 0 n = 1 n = 2 θ = θ = θ = θ = θ = θ = Since not all possible SPP waves (predicted from the solution of the canonical boundary-value problem) can be excited with period L = λ 0 of the surface-relief grating, the grating period needs to be changed in order to excite the remaining SPP waves. The plots of A p and A s vs. θ for L = 0.75λ 0 are presented in 119

150 P x (Wm -2 ) = 32.5 o d 1 = 6 P x (Wm -2 ) = 32.5 o d 1 = z / z-d 1 (nm) Figure 9.8: Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L, when the surface-relief grating is defined by Eq. (9.2) and the incident plane wave is p polarized. The grating period L = 0.75λ 0, Ω = λ 0, and d 1 = 6Ω. P x (Wm -2 ) d 1 = 4 = 50.9 o = 64.2 o P x (Wm -2 ) d 1 = 4 = 50.9 o = 64.2 o z / z-d 1 (nm) Figure 9.9: Same as Fig. 9.8 except that d 1 = 4Ω. Fig , again for d 1 {4Ω, 5Ω, 6Ω}. Figure 9.7(a) shows three A p -peaks at θ 32.5, 50.9, and 64.2 that are present for all three chosen values of d 1. The relative wavenumbers of several Floquet harmonics at these values of θ are given in Table 9.4. The A p -peak at θ = 32.5 represents the excitation of a p-polarized SPP wave as a Floquet harmonic of order n = 1 because k x (1) /k 0 = is close 1 Figures 9.7(b) and 9.10 are not present in: M. Faryad and A. Lakhtakia, Grating-coupled excitation of multiple surface-plasmon-polariton waves, Phys. Rev. A 84, (2011). 120

151 P x (Wm -2 ) = 22.8 o = 40.1 o = 22.8 o = 40.1 o = 61.4 o = 61.4 o d 1 = 6 d 1 = z / P x (Wm -2 ) z-d 1 (nm) Figure 9.10: Same as Fig. 9.6 except for L = 0.75λ 0. to Re ( i) in Table 9.2. The spatial profile of P x (0.75L, z) given in Fig. 9.8 also supports this conclusion. Similarly, the A p -peaks at θ = 50.9 and 64.2 represent the excitation of two other p-polarized SPP waves as a Floquet harmonic of the same order (n = 1), as is evident from the comparison of Tables 9.3 and 9.4, and from the spatial profiles of P x (0.75L, z) provided in Fig In the plots of A s vs. θ in Fig. 9.7(b), three peaks at θ 22.8, 40.1, and 61.4 are present independent of the value of d 1. The relative wavenumbers of several Floquet harmonics at these values of θ are also given in Table 9.4. All of these A s -peaks represent the excitation of s-polarized SPP waves as Floquet harmonics of order n = 1 as can be seen by the comparison of Tables 9.3 and 9.4, and from the spatial profiles of P x (0.75L, z) provided in Fig It may be noted that the same s-polarized SPP wave is excited as a Floquet harmonic of order n = 1 at θ = 22.8 in Fig. 9.7(b) and as a Floquet harmonic of order n = 1 in Fig. 9.4(b) at θ = Ω = 1.5λ 0 The relative wavenumbers of possible SPP waves that can be guided by the planar interface of the chosen rugate filter and the metal are given in Table 9.5 for Ω = 1.5λ 0. In this case, the solution of the canonical boundary-value problem indicated that four s-polarized and six p-polarized SPP waves can be guided by the metal/rugate-filter interface. For computations, the region d 1 < z < d 2 was again divided into 1-nm-thick slices; however, the region 0 z d 1 was divided into 3-nm-thick slices to reduce the computation time, after ascertaining that the accuracy of the computed reflectances and transmittances had not been adversely affected. In the plots of A p vs. θ for L = 0.8λ 0, provided in Fig. 9.11, the excitation of 121

152 Table 9.5: Same as Table 9.2 except for Ω = 1.5λ 0. s-pol i i i i p-pol i i i i i i A p (deg) Figure 9.11: Absorptance A p as a function of the incidence angle θ, when the surface-relief grating is defined by Eq. (9.2) with L 1 = 0.5L, λ 0 = 633 nm, Ω = 1.5λ 0, and L = 0.8λ 0. Black squares are for d 1 = 6Ω, red circles for d 1 = 5Ω, and blue triangles for d 1 = 4Ω. The grating depth (d 2 d 1 = 50 nm) and the width of the metallic layer (d 3 d 2 = 30 nm) are the same for all the plots. Each vertical arrow indicates an SPP wave. Table 9.6: Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig by vertical arrows when Ω = 1.5λ 0 and L = 0.8λ 0. Boldface entries signify SPP waves. n = 2 n = 1 n = 0 n = 1 n = 2 θ = θ = θ = θ = θ = θ = θ = p-polarized SPP waves is indicated at seven values of the incidence angle: θ 8.8, 16.3, 20.8, 27.5, 37.3, 40, and The relative wavenumbers k x (n) /k 0 of a few 122

153 Floquet harmonics at these values of the incidence angle are given in Table 9.6. The A p -peak at θ = 8.8 represents the excitation of a p-polarized SPP wave because k x (1) /k 0 = is close to Re [κ/k 0 ] = (Table 9.5). The spatial profile of P x (0.75L, z) given in Fig for θ = 8.8 confirms the excitation of a p- polarized SPP wave; however, the SPP wave is very loosely bound to the interface z = d 1 in the region 0 < z < d 1. P x (Wm -2 ) d 1 = 6 = 8.8 o P x (Wm -2 ) d 1 = 6 = 8.8 o z / z-d 1 (nm) Figure 9.12: Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L, when the surface-relief grating is defined by Eq. (9.2) and the incident plane wave is p polarized. The grating period L = 0.8λ 0 and d 1 = 6Ω. The A p -peak at θ = 16.3 represents the excitation of another p-polarized SPP wave, because k x (1) /k 0 = is close to Re [κ/k 0 ] = (Table 9.5). The A p -peak at θ = 20.8 also represents a p-polarized SPP wave because k x ( 2) /k 0 = is close to Re [κ/k 0 ] = (Table 9.5). Similarly, the A p -peak at θ = 27.5 is due to the excitation of another p-polarized SPP wave as a Floquet harmonic of order n = 1. The spatial profiles of P x (0.75L, z) given in Fig. 9.13(a) for three different p-polarized plane waves incident at θ = 16.3, 20.8, and 27.5 also confirm that SPP waves are excited as Floquet harmonic of order n = 1, n = 2, and n = 1, respectively. A comparison of Tables 9.5 and 9.6 shows that the A p -peaks at θ = 37.3 and 40 represent the excitation of the same p-polarized SPP wave; however, the SPP wave is excited as a Floquet harmonic of order n = 2 at θ = 37.3 but of order n = 1 at θ = 40. Similarly, a p-polarized SPP wave is excited as a Floquet harmonic of order n = 1 at θ = The spatial profiles of P x (0.75L, z) given in Fig. 9.13(b) also support these conclusions. In the plots of A s vs. θ for L = 0.6λ 0, provided in Fig. 9.14, four peaks at 123

154 P x (Wm -2 ) d 1 = 4 = 16.3 o = 20.8 o = 27.5 o P x (Wm -2 ) d 1 = 4 = 16.3 o = 20.8 o = 27.5 o z / (a) z-d 1 (nm) P x (Wm -2 ) d 1 = 4 = 37.3 o = 40 o = 51.8 o P x (Wm -2 ) d 1 = 4 = 37.3 o = 40 o = 51.8 o z / (b) z-d 1 (nm) Figure 9.13: Same as Fig except that d 1 = 4Ω. A s (deg) Figure 9.14: Same as Fig except that A s is plotted instead of A p, and L = 0.6λ

155 Table 9.7: Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig by vertical arrows when Ω = 1.5λ 0 and L = 0.6λ 0. Boldface entries signify SPP waves. n = 2 n = 1 n = 0 n = 1 n = 2 θ = θ = θ = θ = P x (Wm -2 ) d 1 = 4 = 3.7 o = 6.4 o z / P x (Wm -2 ) d 1 = 4 = 3.7 o = 6.4 o z-d 1 (nm) Figure 9.15: Variation of the x-component of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 at x = 0.75L for two s-polarized incident plane waves, when the surface-relief grating is defined by Eq. (9.2). The grating period L = 0.6λ 0, d 1 = 4Ω, and Ω = 1.5λ 0. θ 3.7, 6.4, 13.5, and 16.2 are present for all values of d 1. The A s -peak at θ = 3.7 represents the excitation of an s-polarized SPP wave as a Floquet harmonic of order n = 1 because k x (1) /k 0 = is close to Re [κ/k 0 ] = (Table 9.5), which is a solution of the canonical boundary-value problem for an s-polarized SPP wave, whereas the A s -peak at θ = 6.4 represents the excitation of another s-polarized SPP wave because k x (1) /k 0 = is close to Re [κ/k 0 ] = (Table 9.5). Similarly, two s-polarized SPP waves are excited as a Floquet harmonic of order n = 1 at θ = 13.5 and of order n = 1 at θ = 16.2, respectively, as is evident from the comparison of Tables 9.6 and 9.7. The spatial profiles of P x (0.75L, z) given in Figs and 9.16 confirm these conclusions. 125

156 P x (Wm -2 ) d 1 = 6 = 13.5 o = 16.2 o P x (Wm -2 ) d 1 = 6 = 13.5 o = 16.2 o z / z-d 1 (nm) Figure 9.16: Same as Fig except that d 1 = 6Ω. 9.4 Concluding Remarks In the last two subsections, I have deciphered a host of numerical results and identified those absorptance peaks that indicate the excitation of SPP waves in the grating-coupled configuration, when the dielectric partnering material is periodically nonhomogeneous normal to the mean plane of the surface-relief grating. It was found that (i) the periodic nonhomogeneity of the dielectric partnering material enables the excitation of multiple SPP waves of both p- and s-polarization states; (ii) fewer s-polarized SPP waves are excited than p-polarized SPP waves; (iii) for a given period of the surface-relief grating, it is possible for two plane waves with different angles of incidence to excite the same SPP wave (Figs. 9.4 (b) and 9.11); (iv) not all SPP waves predicted by the solution of the canonical problem may be excited in the grating-coupled configuration for a given period of the surface-relief grating; (v) the absorptance peaks representing the excitation of p-polarized SPP waves are generally wider than those representing s-polarized SPP waves; (vi) the absorptance peak is narrower for an SPP wave of higher phase speed (i.e. smaller Re(κ)); and (vii) an SPP wave that is excited as a Floquet harmonic of order n = +1 for θ [0, π/2) or n = 1 for θ ( π/2, 0], by virtue of symmetry is the most efficient (Fig. 9.6). 126

157 It may be noted that some other combination of the periodic functions ϵ d (z) and g(x) may allow all solutions of the canonical boundary-value problem to be excited in the grating-coupled configuration with a specific {d 1, d 2, d 3 }. 127

158 Chapter 10 Enhanced Absorption of Light Due to Multiple SPP Waves 10.1 Introduction For the last three decades, research to bring down the cost of photovoltaic (PV) solar cells has gained a huge momentum and many techniques to increase the efficiency of light harvesting by solar cells have been investigated. Among other methods [121], the use of plasmonic structures to enhance the absorption of light by PV solar cells has been studied [3,122,123]. The basic idea is to have periodic texturing of the metallic backing layer of a thin-film solar cell to help excite surfaceplasmon-polariton (SPP) waves. As the partnering semiconductor in these studies has been homogeneous, only one SPP wave (of p-polarization state) at a given frequency can be excited [13, 16] leading to modest gains in the absorption of light. In the last chapters, it has been shown that multiple SPP waves of different polarization states, phase speeds, and attenuation rates can be guided by the interface of a metal and a dielectric material that is periodically nonhomogeneous in the direction normal to the metallic/dielectric interface. Also, it was shown in Ch. 9 that multiple SPP waves can be excited using a grating-coupled configuration. Therefore, I set out to investigate if light absorption can be enhanced due to the excitation of multiple SPP waves by a surface-relief grating on the metallic backing layer in a PV solar cell with a periodically nonhomogeneous semiconductor. For this purpose, a boundary-value problem of reflection by a surface-relief grating coated with a periodically nonhomogeneous semiconductor was investigated. The theoretical formulation of the boundary-value problem using the rigorous coupledwave approach (RCWA) is explained in Ch. 9, and is not repeated in this chapter. Numerical results are presented and discussed in Sec. 10.2, and concluding remarks This chapter is based on: M. Faryad and A. Lakhtakia, Enhanced absorption of light due multiple surface-plasmon-polariton waves, Proc. SPIE 8110, 81100F (2011). 128

159 are presented in Sec Numerical Results and Discussion The absorptance A p = 1 n=n t n= N t ( r (n) 2 p + ( ) t (n) 2) k z (n) p Re k z (0) (10.1) for a p-polarized incident plane wave (a p (n) absorptance A s = 1 n=n t n= N t = δ n0 and a (n) s ( r (n) 2 + ( t (n) 2) Re s s k (n) z k (0) z 0 n Z) and the ), (10.2) for an s-polarized incident plane wave (a (n) s = δ n0 and a (n) p 0 n Z) were calculated as functions of the incidence angle θ. The value of N d was chosen so as to have 2 nm-thick slices in the region 0 < z < d 1, and fixed N t = 8 and N g = 50 after ascertaining that the absorptances converged. I also fixed d 3 d 2 = 30 nm and set ( ϵ d (z) = ϵ r [1 + γ sin π d )] 2 2 z, (10.3) Ω where 2Ω is the period and γ [0, 1]. Numerical results for homogeneous and periodically nonhomogeneous semiconductor partnering materials are now presented Homogeneous semiconductor partnering material Suppose that the semiconductor is homogeneous: γ = 0. For illustrative purposes, I chose this material to be a-si 1 x C x : H with ϵ r = i [124, Fig. 1(c)] at λ 0 = 620 nm, and the metal as bulk aluminum: ϵ m = i [125]. The interface of the chosen homogeneous semiconductor and metal can guide only one p-polarized SPP wave with the relative wavenumber κ/k 0 = ϵ r ϵ m /(ϵ r + ϵ m ) = i (10.4) but no s-polarized SPP waves [13, 16]. The absorptances A p and A s vs. θ are given in Fig for two values of d 1 with a surface-relief grating of period L = 186 nm, and for one value of d 1 without a surface-relief grating (i.e., L g = 0). The absorptances were calculated for two values of d 1 in order to distinguish between [70] (i) waveguide modes that depend on the thickness of the semiconductor layer [104] and (ii) surface waves that should be independent of that thickness when the semiconductor layer is sufficiently thick. In Fig. 10.1(a), a peak at θ 17 is present regardless of the value of 129

160 0.8 d 1 =800 nm, L g = 25 nm d 1 =1000 nm, 0.8 L g = 25 nm d 1 =1000 nm, L g = A p 0.4 A s (a) (deg) (b) (deg) Figure 10.1: Absorptances (a) A p and (b) A s vs. the angle of incidence θ, when L = 186 nm and d 3 d 2 = 30 nm. The vertical arrow identifies the excitation of an SPP wave. d 1. This A p -peak represents the excitation of a p-polarized SPP wave as a Floquet harmonic of order n = 1 because k x (1) /k 0 = (see Table 10.1) is very close to Re[κ/k 0 ] = , where κ/k 0 given by Eq. (10.4) is the relative wavenumber of the p-polarized SPP wave. In Fig. 10.1(b), all absorptance peaks are dependent on the value of d 1 and, therefore, represent the excitation of waveguide modes but not of s-polarized SPP waves. Table 10.1: Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-position of the A p -peak identified in Fig. 10.1(a). A boldface entry signifies an SPP wave. L (nm) θ (deg) n = 2 n = 1 n = 0 n = 1 n = Further evidence in support of the claim that the absorptance peaks independent of the value of d 1 (for sufficiently large d 1 ) represent SPP waves is provided by the spatial profiles of the x-component of the time-averaged Poynting vector P(x, z) = 1 2 Re [E(x, z) H (x, z)]. (10.5) The variations of P x on the z axis for x = 0.75L for (i) a p-polarized incident plane 130

161 P x d 1 = 800 nm d 2 = d nm d 3 = d nm P x d 1 = 1000 nm d 2 = d nm d 3 = d nm (a) (b) 0 z d 1 d 2 d 3 0 d 1 d 2 z d 3 Figure 10.2: Variation of the x-component of the time-averaged Poynting vector P x along the z axis at x = 0.75L for (a) a p-polarized incident plane wave when θ = 17, and (b) an s-polarized incident plane wave when θ = The period of the surface-relief grating L = 186 nm and the free-space wavelength λ 0 = 620 nm. All other parameters are the same as for Fig The horizontal scale for z (d 1, d 3 ) is exaggerated with respect to that for z (0, d 1 ). wave at θ = 17 (the θ-value of the peak identified in Fig 10.1(a) by an arrow) and (ii) an s-polarized incident plane wave at θ = 13.3 (the θ-value of a randomly chosen peak in Fig. 10.1(b) for d 1 = 1000 nm) are shown in Figs. 10.2(a) and (b), respectively. In Fig. 10.2(a), the energy is tightly bound to the plane z = d 1 suggesting that an SPP wave is excited as was claimed in the foregoing paragraph. However, Fig. 10.2(b) shows that the wave is being guided by the homogeneous semiconductor layer, indicating a waveguide mode. Figure 10.1 shows that the absorptance in the presence of a surface-relief grating is generally higher than the absorptance without a surface-relief grating for p- polarized incident plane waves, while the difference is not significant for s-polarized incident plane waves. Since both SPP waves and waveguide modes are excited by p-polarized incident plane waves, while only waveguide modes can be excited by s-polaarized incident plane waves, it can be inferred that the excitation of an SPP wave enhances the absorption Periodically nonhomogeneous semiconductor partnering material Let me now introduce periodic nonhomogeneity (γ 0) in the semiconductor partnering material normal to the mean interface of the surface-relief grating. The solution of the relevant canonical boundary-value problem (Ch. 3) shows that 131

162 the planar interface of a metal and a periodically nonhomogeneous semiconductor can guide multiple SPP waves of both p- and s-polarization states. The results for Ω = 200 nm and 300 nm are now presented and discussed. Ω = 200 nm Suppose that Ω = 200 nm, γ = 0.1, λ 0 = 620 nm, ϵ r = i [124, Fig. 1(c)], and ϵ m = i [125]. The relative wavenumbers of SPP waves that can be guided by the planar interface of the metal and the chosen periodically nonhomogeneous semiconductor were obtained by the solution of the canonical boundaryvalue problem and are given in Table This table shows that two p- and one s-polarized SPP waves can be guided by the planar interface of the two chosen partnering materials. Table 10.2: Relative wavenumbers κ/k 0 of p-polarized and s-polarized SPP waves supported by the planar interface of bulk aluminum and the semiconductor characterized by Eq. (10.3), when Ω = 200 nm, γ = 0.1, and λ 0 = 620 nm. pol. state κ/k 0 p i i s i Plots of the absorptances A p and A s vs. θ are provided in Fig for L g = 0 and L g > 0. Two A p -peaks, at θ 12 and 25.1, in Fig. 10.3(a) are independent of the thickness of the semiconductor layer. The relative wavenumbers of Floquet /k 0 = at θ = 12 is close to Re[κ/k 0 ] = (Table 10.2), the first A p -peak represents the excitation of a p-polarized SPP wave as a Floquet harmonic of order n = 1. Similarly, the A p -peak at θ = 25.1 in Fig. 10.3(a) represents the excitation of harmonics at these values of θ are given in Table Since k (1) x another p-polarized SPP wave as a Floquet harmonic of order n = 1. The spatial profiles of the x-component of the time-averaged Poynting vector for these two p-polarized SPP waves, provided in Fig. 10.4(a), support the conclusion that p-polarized SPP waves are excited. Let me note that the p-polarized SPP wave that is excited as a Floquet harmonic of order n = 1 propagates in the +û x direction while the SPP wave that is excited as a Floquet harmonic of order n = 1 propagates in the û x propagation. One A s -peak at θ 16.2 is present in Fig. 10.3(b) independent of the value of d 1 when L g 0. The relative wavenumbers of Floquet harmonics at this value of θ are also given in Table A comparison of Tables 10.2 and 10.3 shows that an s-polarized SPP wave is excited as a Floquet harmonic of order n = 1 at θ = The variation of P x along the z axis given in Fig. 10.4(b) also indicates the excitation of an s-polarized SPP wave. 132

163 d 1 =8, L g = 0 d 1 =8, L g = 25 nm d 1 =7, L g = 25 nm 0.6 A p 0.4 d 1 =7, L g = 20 nm d 1 =8, L g = 20 nm d 1 =8, L g = 0 A s (a) (deg) (b) (deg) Figure 10.3: Absorptances (a) A p and (b) A s vs. the angle of incidence θ, when Ω = 200 nm, γ = 0.1, d 3 d 2 = 30 nm, and λ 0 = 620 nm. Also, (a) L = 170 nm, and (b) L = 200 nm. Each vertical arrow indicates the excitation of an SPP wave. Table 10.3: Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the absorptance peaks in Fig Boldface entries signify SPP waves. L (nm) θ (deg) n = 2 n = 1 n = 0 n = 1 n = The excitation of two p-polarized SPP waves is accompanied by a very significant increase in the absorptance for a p-polarized incident plane wave, as is evident from the comparison of the absorptances for L g = 0 and L g > 0 in Fig. 10.3(a). Similarly, the excitation of an s-polarized SPP wave in the grating-coupled configuration is correlated with a significant increase in the absorptance in Fig. 10.3(b). Ω = 300 nm Let me now present the numerical results for Ω = 300 nm and γ = 0.1. The solution of the canonical boundary-value problem shows that three p- and two s- polarized SPP waves withe relative wavenumbers given in Table 10.4 are guided 133

164 P x = 12 o = 25.1 o d 1 = 8 d 2 = d nm d 3 = d nm (a) z P x = 16.2 o d 1 = 8 d 2 = d nm d 3 = d nm 0 d 1 d 2 d 3 0 d 1 d 2 d 3 Figure 10.4: Variation of the x-component of the time-averaged Poynting vector P x along the z axis at x = 0.75L for (a) two p-polarized incident plane waves and (b) an s-polarized incident plane wave, at the θ-values of the absorptance peaks identified in Fig by vertical arrows. The horizontal scale for z (d 1, d 3 ) is exaggerated with respect to that for z (0, d 1 ). (b) z by the planar interface of the chosen metal and the periodically nonhomogeneous semiconductor at λ 0 = 620 nm. Table 10.4: Same as Table 10.2 except for Ω = 300 nm. pol. state κ/k 0 p i i i s i i The absorptances A p and A s as functions of θ are given in Fig for L g = 0 and L g > 0. In Fig. 10.5(a), three peaks at θ 7.7, 13.4, and 39.4 are independent of the value of d 1 when L g > 0. The relative wavenumbers of Floquet harmonics at these values of θ are provided in Table 10.5, and a comparison of Tables 10.4 and 10.5 shows that each A p -peak represents the excitation of a p- polarized SPP wave. The spatial profiles of P x along the z axis at x = 0.75L given in Fig. 10.6(a) also suggests the same conclusion. Two peaks are present in Fig. 10.5(b) at θ 10.3 and 31.8, independent of the value of d 1 in the absorptance curves for the grating-coupled configuration (i.e., L g > 0). The relative wavenumbers of Floquet harmonics at these values of θ are also given in Table A comparison of Tables 10.4 and 10.5 shows that an s-polarized SPP wave is excited corresponding to each A s -peak as a Floquet 134

165 d 1 =8, L g = 20 nm d 1 =7, L g = 20 nm d 1 =8, L g = d 1 =8, L g = 25 nm d 1 =7, L g = 25 nm d 1 =8, L g = A p A s (a) (deg) 0.2 (b) (deg) Figure 10.5: Same as Fig except for Ω = 300 nm, and (a) L = 195 nm and (b) L = 210 nm. Table 10.5: Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the absorptance peaks in Fig Boldface entries signify SPP waves. L (nm) θ (deg) n = 2 n = 1 n = 0 n = 1 n = harmonic of order n = 1. The spatial profiles of P x (x = 0.75L, z), given in Fig. 10.6(b), at the θ-values of the A s -peaks also show that s-polarized SPP waves are indeed excited. A comparison of plots for L g = 0 and L g > 0 in Fig. 10.5(a) for p-polarized incident plane waves shows that the absorptance increases wherever an SPP wave is excited, but this conclusion appears unwarranted from Fig. 10.5(b) for s-polarized incident plane waves. However, the possibility of increasing A s by changing the grating period L still remains and requires further exploration. The periodically nonhomogeneous semiconductor with Ω = 300 nm and γ = 0.1 was also investigated at λ 0 = 827 nm. The relative permittivity of a-si 1 x C x : H at 135

166 P x d 1 = 8 d 2 = d nm d 3 = d nm (a) = 7.7 o = 13.4 o = 39.4 o 0.00 (b) z P x = 10.3 o = 31.8 o d 1 = 8 d 2 = d nm d 3 = d nm 0 d 1 d 2 d 3 0 d 1 d 2 d 3 Figure 10.6: Variation of the x-component of the time-averaged Poynting vector P x along the z axis at x = 0.75L for (a) three p-polarized incident plane waves and (b) two s-polarized incident plane waves, at the θ-values of the absorptance peaks identified in Fig by vertical arrows. The horizontal scale for z (d 1, d 3 ) is exaggerated with respect to that for z (0, d 1 ). z λ 0 = 827 nm was taken to be ϵ r = i [124, Fig. 1(c)] and of bulk aluminum as ϵ m = i [125]. The solution of the canonical boundary-value problem for these parameters shows that two p- and two s-polarized SPP waves can be guided by the planar interface of the metal and the periodically nonhomogeneous semiconductor, with relative wavenumbers available in Table Table 10.6: Same as Table 10.4 except for λ 0 ϵ m = i. = 827 nm, ϵ r = i, and pol. state κ/k 0 p i i s i i For λ 0 = 827 nm, the absorptances A p and A s are plotted as functions of θ in Fig. 10.7, for L g = 0 and L g > 0. Two A p -peaks at θ 2.6 and 16.4 in Fig. 10.7(a), along with two A s -peaks at θ 8.2 and 18.1 in Fig. 10.7(b), are independent of the value of d 1. The relative wavenumbers of Floquet harmonics at the θ-values of these peaks are given in Table Comparison of Tables 10.6 and 10.7 show that each A p - (A s -) peak represents the excitation of a p- (s-) polarized SPP wave. Moreover, Fig shows that, for both linear polarization states of the incident plane wave, the excitation of multiple SPP waves is accompanied by a significant enhancement of absorption. 136

167 d 1 =7, L g = 20 nm d 1 =8, L g = 20 nm d 1 =8, L g = d 1 =7, L g = 20 nm d 1 =8, L g = 20 nm d 1 =8, L g = 0 A p A s (a) (deg) (b) (deg) Figure 10.7: Same as Fig except for λ 0 = 827 nm, ϵ r = i, ϵ m = i, and (a) L = nm and (b) L = 282 nm. Table 10.7: Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the absorptance peaks in Fig Boldface entries signify SPP waves. L (nm) θ (deg) n = 2 n = 1 n = 0 n = 1 n = Concluding Remarks The absorptances for p- and s-polarized incident plane waves were calculated as functions of the incidence angle θ, using RCWA, for a finitely thick and periodically nonhomogeneous semiconductor layer on top of a metallic surface-relief grating. The excitation of surface plasmon-polariton waves was inferred from (i) the absorptance peaks whose θ-locations were independent of the thickness of the semiconductor layer when that layer was made sufficiently thick, and was confirmed by the (ii) analysis of the spatial profiles of the time-averaged Poynting vector. 137

168 Moreover, it was seen that (iii) the use of a metallic surface-relief grating, as opposed to that of a metallic slab with planar faces, can significantly enhance absorption for both p- and s-polarized incidence when one or more SPP waves are excited. 138

169 Chapter 11 Grating-Coupled Excitation of Multiple SPP Waves Guided by Metal/SNTF Interface 11.1 Introduction The excitation of SPP waves using a grating-coupled configuration which is an alternative to the TKR configuration was theoretically investigated for this chapter. These waves can be excited in the grating-coupled configuration by the illumination of the periodic corrugations of a metallic surface-relief grating coated with the dielectric partnering material. The grating-coupled configuration also allows the reverse process: the efficient coupling of SPP waves, which are otherwise nonradiative, with light [84, 85]. This is an important advantage over the TKR configuration because it allows for better incorporation of chemical sensors based on SPP waves [86] in integrated optical circuits [87]. The theoretical formulation of the boundary-value problem provided in Sec is such that the grating plane and the morphologically significant plane of the SNTF can be rotated about a common axis with respect to each other. The RCWA [106, 118] is employed to numerically solve the boundary-value problem. Numerical results are presented in Sec for the case when the wave vector of the illuminating plane wave lies wholly in the grating plane. Peaks in the plots of absorptance vs. the angle of incidence are studied carefully to elicit evidence of the excitation of multiple SPP waves and these peaks are correlated with those in the TKR configuration and with the results of canonical boundary-value problem of Ch. 2. Concluding remarks are presented in Sec This chapter is based on: M. Faryad and A. Lakhtakia, Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. Part V: Grating-coupled excitation, J. Nanophoton. 5, (2011). 139

170 11.2 Boundary-Value Problem Description Let me consider the boundary-value problem shown schematically in Fig The regions z < 0 and z > d 3 are vacuous. The region 0 z d 1 is occupied by an SNTF with a periodically nonhomogeneous permittivity dyadic [44, 70, 71] where the dyadics ϵ SNT F (z) = ϵ 0 S z (γ ) S y (z) ϵ (z) S 1(z) S 1 ref y z (γ ), (11.1) S y (z) = (û x û x + û z û z ) cos [χ(z)] + (û z û x û x û z ) sin [χ(z)] + û y û y ϵ ref (z) = ϵ a(z) û z û z + ϵ b (z) û x û x + ϵ c (z) û y û y } (11.2) depend on the vapor incidence angle χ v (z) = χ v + δ v sin [π(d 2 z)/ω] that is supposedly made to vary sinusoidally with period 2Ω during the deposition of the SNTF by a physical evaporation process in a vacuum chamber. The third dyadic in Eq. (11.1) was chosen as S z (γ ) = (û x û x + û y û y ) cos γ + (û y û x û x û y ) sin γ + û z û z, (11.3) so that the plane formed by the unit vectors û z and û x cos γ + û y sin γ is the morphologically significant plane of the SNTF. Thus, there is sufficient flexibility in the formulation with respect to the orientation of the morphologically significant plane. The region d 2 z d 3 is occupied by the partnering metallic material with spatially uniform relative permittivity ϵ m. The region d 1 < z < d 2 contains a metallic surface-relief grating of period L along the x axis, with the SNTF present in the troughs. The xz plane is the grating plane. The relative permittivity dyadic ϵ g (x, z) = ϵ g (x±l, z) in this region is taken to be as { [ ] ϵm I ϵ m I + ϵ ϵ g (x, z) = SNT F (z) U [d 2 z g(x)], x (0, L 1 ), (11.4) ϵ SNT F (z), x (L 1, L), for z (d 1, d 2 ), with and I = û x û x + û y û y + û z û z, (11.5) g(x) = (d 2 d 1 ) sin (πx/l 1 ), L 1 (0, L), (11.6) U(ζ) = { 1, ζ 0, 0, ζ < 0. (11.7) The depth of the surface-relief grating defined by Eq. (11.6) is L g = d 2 d 1. Let me note that the permittivity dyadic defined by Eq. (11.4) is a simplistic description 140

171 Incident -1 0 z = 0 z = d 1 z = d 2 z = d 3-2 SNTF Metal Metal -2-1 z +1 SPP waves x +1 0 Figure 11.1: Schematic of the boundary-value problem solved using the RCWA. of the region d 1 < z < d 2 and the actual morphology of the region is very hard to model for the fabricated device [126]. In the vacuous half-space z 0, let a plane wave propagating at an angle θ to the z axis and an angle ϕ to the x axis in the xy plane, illuminate the structure. Hence, the incident, reflected, and transmitted field phasors may be written in terms of Floquet harmonics as follows [ ]: E inc (r) = n Z η 0 H inc (r) = n Z ( sn a (n) s ( p + n a (n) s ) [ ( + p + n a p (n) exp i k (n) ) [ ( s n a (n) p exp i k (n) x x + k y (0) x x + k y (0) y + k (n) z z )], z 0, (11.8) y + k (n) z z )], z 0, (11.9) 141

172 E ref (r) = n Z η 0 H ref (r) = n Z E tr (r) = n Z η 0 H tr (r) = n Z ( sn t (n) s ( p + n t (n) s ( sn r (n) s ( p n r (n) s ) [ ( + p n r p (n) exp i k (n) ) [ ( s n r p (n) exp i k (n) ) { [ + p + n t (n) p exp i k (n) x x + k y (0) x x + k y (0) x x + k y (0) ) { [ s n t p (n) exp i k (n) x x + k y (0) where k (n) x = k 0 cos ϕ sin θ + nκ x, κ x = 2π/L, k (0) and The unit vectors and k (n) z = k (n) xy = p ± n = k(n) z k 0 y k (n) z z )], z 0, (11.10) y k (n) z z )], z 0, (11.11) y + k (n) z (z d 3 ) ]}, z d 3, (11.12) y + k z (n) (z d 3 ) ]}, z d 3, (11.13) y = k 0 sin ϕ sin θ, (k x (n) ) 2 + (k y (0) ) 2, (11.14) + k 2 0 (k xy (n) ) 2, k 2 0 > (k (n) +i xy ) 2 (k (n) xy ) 2 k 2 0, k 2 0 < (k (n) xy ) 2. (11.15) s n = k(0) y û x + k x (n) û y k xy (n) ( k x (n) û x + k y (0) k (n) xy û y ) (11.16) + k(n) xy û z (11.17) k 0 represent the s- and p-polarization states, respectively. Whereas { a (n) s, a (s) p }, n Z, are the { known} amplitudes of the incident electric field { phasor, } the reflection amplitudes r s (n), r p (s) and the transmission amplitudes t (n) s, t (s) p, n Z, have to be determined Coupled ordinary differential equations The relative permittivity dyadic in the region 0 z d 3 can be expanded as a Fourier series with respect to x, viz., ϵ(x, z) = n Z ϵ (n) (z) exp(inκ x x), z [0, d 3 ], (11.18) 142

173 where and ϵ (n) (z) = { 1 L ϵ SNT F (z), z [0, d 1 ], ϵ (0) 1 L (z) = L 0 ϵ (x, z)dx, z (d g 1, d 2 ), ϵ m I, z [d 2, d 3 ], L 0 g xx)dx, z [d 1, d 2 ] 0, otherwise (11.19) ; n 0, (11.20) where 0 is the null dyadic. The coefficient dyadic ϵ (n) (z) on the right side of Eq. (11.18) can be expanded as ϵ (n) (z) = ϵ (n) xx (z)û x û x + ϵ (n) xy (z)û x û y + ϵ (n) xz (z)û x û z + ϵ (n) yx (z)û y û x + ϵ (n) yy (z)û y û y + ϵ (n) yz (z)û y û z + ϵ (n) zx (z)û z û x + ϵ (n) zy (z)û z û y + ϵ (n) zz (z)û z û z. (11.21) The field phasors may be written in the region 0 z d 3 in terms of Floquet harmonics as E(r) = [ ] [ ( E (n) x (z)û x + E y (n) (z)û y + E z (n) (z)û z exp i k (n) x x + k y (0) y )] n Z H(r) = [ ] [ ( H (n) x (z)û x + H y (n) (z)û y + H z (n) (z)û z exp i k (n) x x + k y (0) y )], n Z z [0, d 3 ], (11.22) with unknown functions E x,y,z(z) (n) and H x,y,z(z). (n) Substitution of Eqs. (11.18) and (11.22) in the frequency-domain Maxwell curl postulates results in a system of four ordinary differential equations d dz E(n) x d dz E(n) d dz H(n) x (z) ik (n) x y (z) ik y (0) m Z d dz H(n) (z) ik (n) [ ϵ (n m) yx x y (z) ik y (0) [ ϵ (n m) xx m Z E z (n) E z (n) H (n) (z)e (m) x (z) = ik 0 η 0 H y (n) (z), (11.23) (z) = ik 0 η 0 H x (n) (z), (11.24) z (z) = ik 0 η 0 (z) + ϵ (n m) yy H (n) z (z) = ik 0 η 0 (z)e (m) x (z) + ϵ (n m) xy 143 (z)e (m) y (z)e (m) y (z) + ϵ (n m) (z)e z (m) (z) ], yz (z) + ϵ (n m) (z)e z (m) (z) ], xz (11.25) (11.26)

174 and two algebraic equations k x (n) E y (n) (z) k y (0) k x (n) H (n) y (z) k y (0) [ ϵ (n m) zx m Z E x (n) (z) = k 0 η 0 H z (n) (z), (11.27) H (n) x (z) = k 0 η 0 (z)e (m) x (z) + ϵ zy (n m) (z)e y (m) (z) + ϵ (n m) zz (z)e z (m) (z) ]. (11.28) Equations (11.23) (11.28) hold z (0, d 3 ) and n Z. These equations can be recast into an infinite system of coupled first-order ordinary differential equations, but that system can not be implemented on a digital computer. Therefore, I restrict n N t and then define the column (2N t + 1)-vectors [X σ (z)] = [X σ ( Nt) (z), X σ ( Nt) (z),..., X σ (0) (z),..., X σ (Nt 1) (z), X σ (Nt) (z)] T, (11.29) for X {E, H} and σ {x, y, z}. Similarly, I define (2N t +1) (2N t +1)-matrixes [K x ] = diag[k (n) x ], [ϵ αβ (z)] = [ ϵ (n m) αβ ] (z), (11.30) where diag[k x (n) ] is a diagonal matrix, α {x, y, z}, and β {x, y, z}. Substitution of Eqs. (11.27) and (11.28) into Eqs. (11.23) (11.26), to eliminate E (n) z and H (n) z n Z, gives the matrix ordinary differential equation d ] [ ] ] [ f(z) = i P (z) [ f(z), z (0, d 3 ), (11.31) dz where the column vector [ f(z)] with 4(2N t + 1) components is defined as ] [ f(z) = [ [E x (z)] T, [E y (z)] T, η 0 [H x (z)] T, η 0 [H y (z)] T ] T ; (11.32) [ ] and the 4(2N t + 1) 4(2N t + 1)-matrix P (z) is given by [ P 11 (z) ] [ P 12 (z) ] [ P 13 (z) ] [ P 14 (z) ] [ ] [ ] [ ] [ ] [ ] P (z) 21 P (z) 22 P (z) 23 P (z) 24 P (z) = [ ] [ ] [ ] [ ] P (z) 31 P (z) 32 P (z) 33 P [ ] [ (z), (11.33) ] [ 34 ] [ ] P (z) 41 P (z) 42 P (z) 43 P (z)

175 where [ ] P (z) 11 [ ] P (z) 12 [ ] P (z) 13 [ ] P (z) 14 [ ] P (z) 21 [ ] P (z) 22 [ ] P (z) 23 [ ] P (z) 24 ] [ 1 [ ] = [K x ϵ zz (z)] ϵ zx (z), (11.34) ] [ 1 [ ] = [K x ϵ zz (z)] ϵ zy (z), (11.35) [ ] [ = k(0) y 1 K k x ϵ zz (z)], (11.36) 0 [ ] 1 [ ] [ 1 ] = k 0 I K k x ϵ zz (z)] [K x, (11.37) 0 = k (0) y = k (0) y [ 1 [ ] ϵ zz (z)] ϵ zx (z), (11.38) [ 1 [ ] ϵ zz (z)] ϵ zy (z), (11.39) [ ] k (0) 2 [ y 1 = k 0 I + ϵ k zz (z)], 0 (11.40) [ = k(0) y 1 ] ϵ k zz (z)] [K x, 0 (11.41) [ ] P (z) 31 [ ] P (z) 32 [ ] P (z) 33 [ ] P (z) 34 [ ] P (z) 41 [ ] P (z) 42 [ ] P (z) 43 [ ] P (z) 44 ] [ ] [ 1 [ ] [ ] = k 0 [ϵ yx (z) + k 0 ϵ yz (z) ϵ zz (z)] ϵ zx (z) k(0) y K k x 0 = 1 [ ] 2 ] [ ] [ 1 [ K k x k0 [ϵ yy (z) + k 0 ϵ yz (z) ϵ zz (z)] ϵ zy (z) 0 (11.42) ], (11.43) [ ] [ 1 = k y (0) ϵ yz (z) ϵ zz (z)], (11.44) [ ] [ 1 ] = ϵ yz (z) ϵ zz (z)] [K x, (11.45) ] [ ] [ 1 [ ] 2 = k 0 [ϵ xx (z) k 0 ϵ xz (z) ϵ zz (z)] ϵ zx (z) k(0) y [ ] I, k 0 (11.46) ] [ ] [ 1 [ ] [ ] = k 0 [ϵ xy (z) k 0 ϵ xz (z) ϵ zz (z)] ϵ zy (z) + k(0) y K k x, 0 (11.47) [ ] [ 1 = k y (0) ϵ xz (z) ϵ zz (z)], (11.48) [ ] [ 1 ] = ϵ xz (z) ϵ zz (z)] [K x, (11.49) and [ I ] is the (2N t + 1) (2N t + 1) identity matrix. 145

176 When ϕ = 0, [ ] [ ] [ ] [ ] [ ] [ ] P (z) 13 = P (z) 21 = P (z) 22 = P (z) 24 = P (z) 33 = P (z) 43 = [ 0 ], (11.50) where [ 0 ] is the (2N t + 1) (2N t + 1) null matrix; and [ ] [ ] [ ] [ ] P (z) 12 = P (z) 31 = P (z) 34 = P (z) 42 = [ 0 ], (11.51) and ϵ SNT F (z) = ϵ 0 S y (z) ϵ ref (z) S 1(z) (11.52) y when γ = 0 as well. Thus, when γ = ϕ = 0, k y (0) = 0, [ ] [ ] ϵ xz (z) = ϵ zx (z), (11.53) [ ] [ ] [ ] [ ] ϵ xy (z) = ϵ yx (z) = ϵ yz (z) = ϵ zy (z) = [ 0 ], (11.54) and [ P 11 (z) ] [ 0 ] [ 0) ] [ P 14 (z) ] [ ] P (z) = [ ] [ ] [ ] [ ] 0 0 k 0 I 0 [ ] [ [ ] [ ] 0 P 32 (z)] 0 0 [ ] P (z) [ ] [ ] [ ] P (z) 44. (11.55) Solution algorithm [ f(0)] The column vectors and [ f(d 3 )] can be written using Eqs. (11.8) (11.13) as [ ] [ ] ] Y [ f(0) + Y [ ] [ ] = e e [A] Y [ ] [ ], [ f(d3 )] + = e [ ] [T], (11.56) Y + Y [R] Y + h h h where [A] = [ a ( N t) s, a ( N t+1) s,..., a (0) s,..., a (N t 1) s, a (N t) s, ] a ( N t) p, a ( N t+1) p,..., a (0) p,..., a (N t 1) p, a (N t) T p, (11.57) [R] = [ r ( N t) s, r ( N t+1) s,..., r s (0),..., r (N t 1) s, r (N t) s, ] r ( N t) p, r ( N t+1) p,..., r p (0),..., r (N t 1) p, r (N t) T p, (11.58) [T] = [ t ( Nt) s, t ( Nt+1) s,..., t (0) s,..., t (Nt 1) s, t (Nt) s, ] t ( Nt) p, t ( Nt+1) p,..., t (0) p,..., t (Nt 1) p, t (Nt) T p, (11.59) 146

177 [ ] and the nonzero entries of (4N t + 2) (4N t + 2)-matrixes Y ± are as follows: e,h s n û x, n = m [1, 2N t + 1], ( ) Y ± h = p ± n û y, n = m [2N t + 2, 4N t + 2], (11.60) nm p ± n û x, n = m + 2N t + 1, s n û y, n = m 2N t + 1, p ± n û x, n = m [1, 2N t + 1], ( Y ± e )nm = s n û y, n = m [2N t + 2, 4N t + 2], (11.61) s n û x, n = m + 2N t + 1, p ± n û y, n = m 2N t + 1. In order to devise a stable algorithm to determine the unknown [R] and [T] for known [A] [ ], the region 0 z d 1 is divided into N d slices and the region d 1 z d 2 into N g slices, but the region d 2 z d 3 is kept as just one slice. So, there are N d + N g + 1 slices and N d + N g + 2 interfaces. In the jth slice, j [1, N d + N g + 1], bounded by the planes z = z j 1 and z = z j, I approximate [ ] [ ] P (z) = P j [ = P ( zj + z j 1 2 )], z (z j, z j 1 ), (11.62) with z 0 = 0 and z Nd +N g+1 = d 3. Equation (11.31) yields [130] [ f(zj 1 )] = [ G ] exp [ ] } { i j j D [G ] ] 1 [ f(zj ), (11.63) j j where j = z j z j 1, [ G ] [ ] is a square matrix comprising the eigenvectors of j P j as its columns, and the diagonal matrix [ D ] [ ] contains the eigenvalues of j P in the same order. Let me define auxiliary column vectors [T] j and auxiliary transmission matrixes [ ] Z by the relation [108] j [ f(zj )] = [ Z ] [T] j j, j [0, N d + N g + 1] ; (11.64) j hence, [ ] [T] Nd +N = [T], and [ g+1 Z ]N = d +N g +1 Y + e [ Y + h ]. (11.65) To find [T] j and [ Z ] for j [0, N j d + N g ], I substitute Eq. (11.64) in (11.63), which results in the relation [ [ Z ]j 1 [T] j 1 = [ ] e i G j ] [D] u j 0 j 0 e i [G ] 1 [Z ] j[d] [T] l j j j j, j [1, N d + N g + 1], (11.66) 147

178 where [ D ] u and [ D ] l are the upper and lower diagonal submatrixes of [ D ], respectively, when the eigenvalues are arranged in decreasing order of the imaginary j j j part. Since both [T] j and [ Z ] cannot be determined simultaneously from Eq. (11.66), j let me formulate [108] { [ ] } u [T] j 1 = exp i j D [W ] u [T] j j j, (11.67) where the square matrix [ W ] u and its counterpart [ W ] l are defined via j j [ [ ] u ] W j [ ] l = [ G ] 1 [Z ]. (11.68) W j j j Substitution of Eq. (11.67) in (11.66) results in the relation [ ] Z = [ G ] [ [ ] ] I j 1 j exp { [ ] { i j [D] j} l l [W W ] } u 1 { }, j exp i j [D] u j j [1, N d + N g + 1]. (11.69) From Eqs. (11.68) and (11.69), I find [ Z ] in terms of [ Z ]. After partitioning [ [ ] u 0 N d +N g +1 ] Z [ Z ] 0 = [ ] 0 l Z 0 j, (11.70) and using Eqs. (11.56) and (11.64), [R] and [T] 0 are found as follows: [ [T]0 [R] ] [ ] u Z = 0 [ ] l Z 0 [ ] Y [ e ] Y h 1 [ Y + e [ Y + h ] ] [A]. (11.71) Equation (11.71) was obtained by enforcing the usual boundary conditions across the plane z = 0. After [T] 0 is known, [T] = [T] Nd +N g +1 is found by reversing the sense of iterations in Eq. (11.67) Absorptance For planewave illumination, a (n) s = a (n) p of the 2 2 matrixes in the relations [ r (n) s r (n) p ] = [ r (n) ss r (n) ps r (n) sp r (n) pp = 0 n [ N t, 1] [1, N t ]. The elements ] 148 [ a (0) s a (0) p ], n Z, (11.72)

179 and [ t (n) s t (n) p ] = [ t (n) ss t (n) ps t (n) sp t (n) pp ] [ a (0) s a (0) p ], n Z, (11.73) are the reflection and transmission coefficients. Co-polarized coefficients have both subscripts identical, but cross-polarized coefficients do not. Reflectances [ and transmittances of order n are defined, for example, as R sp (n) = r sp (n) 2 Re k z (n) /k z (0) and ] [ ] T sp (n) = t (n) sp 2 Re k z (n) /k z (0). The planewave absorptance of the structure is given by ( ) N t (n) rs 2 + r p (n) 2 + t (n) s 2 + t (n) p 2 k z (n) A = 1 Re. (11.74) n= N t a (0) s 2 + a (0) p 2 k z (0) This expression can be written in terms of the reflection and transmission coefficients as a (0) s 2 A = A s a (0) s 2 + a (0) p + + N t n= N t N t n= N t ( r (n) ( ss r sp (n) r (n) ss r (n) sp 2 + A p + r (n) ps r pp (n) + r ps (n) r (n) pp a (0) p 2 a (0) s 2 + a (0) p 2 + t (n) ss t sp (n) + t (n) ss t (n) sp + t (n) + t (n) ps ps t (n) pp t (n) pp s a (0) p a (0) p ) a (0) ) a (0) ( s 2 + a (0) Re 2 a (0) ( (0) s a p s 2 + a (0) p Re 2 k z (n) k z (0) k (n) z k (0) z ) ), (11.75) where A s = 1 n Z R (n) ss + R (n) ps + T (n) ss + T (n) ps (11.76) is the absorptance for s-polarized illumination and A p = 1 n Z R (n) sp + R (n) pp + T (n) sp + T (n) pp (11.77) is the absorptance for p-polarized illumination Numerical Results and Discussion For the illustrative numerical results presented in this section, I chose the same parameters for the periodically nonhomogeneous SNTF and the metal as in Ch. 2. The free-space wavelength was fixed at λ 0 = 633 nm, and the metal was taken 149

180 to be bulk aluminum: ϵ m = i. The SNTF was chosen to be made of titanium oxide [73], with ϵ a (z) = [ v(z) v 2 (z)] 2 ϵ b (z) = [ v(z) v 2 (z)] 2 (11.78) ϵ c (z) = [ v(z) v 2 (z)] 2 χ(z) = tan 1 [ tan χ v (z)] where v(z) = 2χ v (z)/π. The angles χ v and δ v were taken to be 45 and 30, respectively, and Ω = 200 nm was fixed. I also restricted the propagation of the incident plane wave to be in the grating plane, i.e., ϕ = 0, for all numerical results presented in this section. For computations, I set N t = 8, N g = 50, and N d = ( )d 1, after ascertaining that the absorptance converged within ±1% for all n [ N t, N t ]. The width of the metallic layer was kept the same in all calculations: d 3 d 2 = 30 nm; and L 1 = 0.5L was also fixed γ = 0 Since the plane of the incidence, the morphologically significant plane of the SNTF, and the grating plane are all the same when γ = ϕ = 0, any SPP wave that is excited can only propagate in the xz plane. The corresponding relative wavenumbers κ/k 0 of the SPP waves guided by the planar metal/sntf interface in the canonical boundary-value problem formulated in Ch. 2 are provided in Table This table shows that two p-polarized and one s-polarized SPP waves can be guided by the metal/sntf interface. Table 11.1: Relative wavenumbers κ/k 0 of SPP waves obtained by the solution of the canonical boundary-value problem (Ch. 2) when γ = ϕ = 0. The constitutive parameters of the periodically nonhomogeneous SNTF and the metal are provided at the beginning of Sec If κ represents an SPP wave propagating in the û x direction, κ represents an SPP wave propagating in the û x direction. p-pol i i s-pol i Numerical results for grating-coupled excitation of multiple SPP waves are now presented in the foregoing context. Because γ = ϕ = 0, depolarization does not occur; hence, if the incident plane wave is p (resp. s) polarized, the reflected and the transmitted waves are also p (resp. s) polarized and so is any SPP wave excited. 150

181 Excitation by a p-polarized incident plane wave d 1 = 7, L g = 25nm d 1 = 7, L g = 20nm d 1 = 8, L g = 20nm A p (deg) Figure 11.2: Absorptance A p vs. the angle of incidence θ when L = 380 nm, ϕ = γ = 0, and d 3 d 2 = 30 nm. The absorptance peak represents the excitation of a p-polarized SPP wave. The absorptance A p is plotted in Fig as a function of the angle of incidence θ, when L = 380 nm and ϕ = γ = 0, for three combinations of d 1 and L g. The Mathematica TM code is provided in Appendix B.7 to calculate A p and A s. The absorptance peak at θ 11.1 is present in Fig independent of thickness d 1 of the SNTF and the grating depth L g. The presence of the peak for both values of d 1 indicates that this peak represents the excitation of a p-polarized SPP wave [70, 71] and not of a waveguide mode [104, 105], which must depend on d 1. The increase in the height of the A p -peak with the increase in the value of L g is another indicator that the peak represents the excitation of an SPP wave because the excitation efficiency is commonly held to increase with increasing grating depth [34]. The relative wavenumbers k x (n) /k 0 of a few Floquet harmonics at θ = 11.1 when L = 380 nm are presented in Table A comparison of Tables 11.1 and 11.2 shows that, as k x (1) /k 0 = is very close to Re ( i), the p-polarized SPP wave is excited as a Floquet harmonic of order n = 1. To examine the spatial variation of the power density of the SPP wave in the 151

182 Table 11.2: Relative wavenumbers k (n) x /k 0 of Floquet harmonics at the θ-value of the absorptance peak in Fig when L = 380 nm and ϕ = γ = 0. A boldface entry signifies an SPP wave. n = 2 n = 1 n = 0 n = 1 n = 2 θ = structure, the time-averaged Poynting vector P(x, z) = 1 2 Re [E(x, z) H (x, z)], (11.79) was calculated. The x-component P x of P(x, z) is presented in Fig along the z axis when a p-polarized plane wave is incident on the structure at an angle θ = 11.1, L = 380 nm, and x = 0.25L or 0.75L. The spatial profiles show that the energy is bound to the plane z = d 1 and most of the power density resides in the SNTF. This shows that a surface wave a p-polarized SPP wave is being guided by the metal/sntf interface. However, the SPP wave is weakly localized to the plane z = d 1 in the SNTF. P x (W m -2 ) x = 0.25 L x = 0.75 L d 1 = 8 d 2 = d 1 +20nm d 3 = d 2 +30nm P x (W m -2 ) x = 0.25 L x = 0.75 L d 1 = 8 d 2 = d 1 +20nm d 3 = d 2 +30nm z / z-d 1 (nm) Figure 11.3: Variation of the x-component P x (x, z) of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3, when L = 380 nm and ϕ = γ = 0. The incident plane wave is p polarized and the angle of incidence θ = To excite the second p-polarized SPP wave predicted by the canonical boundaryvalue problem with κ/k 0 = i, the grating period L was reduced to 280 nm. The absorptance A p is presented as a function of θ in Fig. 11.4, when 152

183 d 1 = 6, L g = 20nm d 1 = 6, L g = 25nm d 1 = 7, L g = 20nm 0.4 A p (deg) Figure 11.4: Same as Fig except that L = 280 nm. L = 280 nm and γ = ϕ = 0, again for three combinations of d 1 and L g. An absorptance peak at θ 13.6, which is present in all three curves, indicates the excitation of a p-polarized SPP wave. The relative wavenumbers of a few Floquet harmonics at θ = 13.6 when L = 280 nm are provided in Table Since /k 0 = is very close to Re ( i), the p-polarized SPP wave is excited as a Floquet harmonic of order n = 1. k (1) x Table 11.3: Relative wavenumbers k (n) x /k 0 of Floquet harmonics at the θ-value of the absorptance peak in Fig when L = 280 nm. A boldface entry signifies an SPP wave. n = 2 n = 1 n = 0 n = 1 n = 2 θ = The variations of P x (0.25L, z) and P x (0.75L, z) along the z axis are shown in Fig when a p-polarized plane wave is incident on the structure at an angle θ = 13.6, and L = 280 nm. The plots show that the power density is strongly localized to the metal/sntf interface. A comparison of Figs and 11.4 show that the A p -peak representing the excitation of a p-polarized SPP wave with a smaller phase speed i.e., larger magnitude of Re(κ) is broader. Also, the SPP 153

184 P x (W m -2 ) x = 0.25 L x = 0.75 L d 1 = 7 d 2 = d 1 +20nm d 3 = d 2 +30nm z / P x (W m -2 ) x = 0.25 L x = 0.75 L d 1 = 7 d 2 = d 1 +20nm d 3 = d 2 +30nm z-d 1 (nm) Figure 11.5: Same as Fig except that θ = 13.6 and L = 280 nm. wave with the broader absorptance peak is more tightly localized in the SNTF to the plane z = d 1. Excitation by an s-polarized incident plane wave The absorptance A s was calculated as a function of the angle of incidence θ when L = 340 nm and γ = ϕ = 0. In Fig. 11.6, A s is presented with respect to θ for three combinations of d 1 and L g. The peaks present in the absorptance curves for d 1 = 8Ω at θ 8.5 and 15, and for d 1 = 7Ω at θ 12.1, are due to the excitation of waveguide modes because their θ-positions are dependent on the value of d 1. An A s -peak in Fig at θ 11.6 represents the excitation of an s-polarized SPP wave because that peak exists for all values of d 1. The change in the θ-position of the peak when L g is reduced from 25 to 20 nm is negligible. This absorptance peak, representing the excitation of an s-polarized SPP wave, is narrower than of those representing the two p-polarized SPP waves in Sec Finally, I note that the s-polarized SPP wave is also excited as a Floquet harmonic of order n = 1: k x (1) /k 0 = in Table 11.4 is very close to Re ( i) in Table Table 11.4: Relative wavenumbers k (n) x /k 0 of Floquet harmonics at the θ-value of the peak identified by a vertical arrow in Fig when L = 340 nm. A boldface entry signifies an SPP wave. n = 2 n = 1 n = 0 n = 1 n = 2 θ =

185 d 1 = 7, L g = 25nm d 1 = 8, L g = 20nm d 1 = 8, L g = 25nm 0.6 A s (deg) Figure 11.6: Absorptance A s vs. the angle of incidence θ when L = 340 nm, ϕ = γ = 0, and d 3 d 2 = 30 nm. A vertical arrow identifies the peak that represents the excitation of an s-polarized SPP wave. P x (W m -2 ) x = 0.25 L x = 0.75 L d 1 = 7 d 2 = d 1 +25nm d 3 = d 2 +30nm z / P x (W m -2 ) x = 0.25 L x = 0.75 L d 1 = 7 d 2 = d 1 +25nm d 3 = d 2 +30nm z-d 1 (nm) Figure 11.7: Variation of the x-component P x (x, z) of the time-averaged Poynting vector P(x, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3. The incident plane wave is s polarized and the angle of incidence θ = Two spatial profiles of the x-component of the time-averaged Poynting vector 155

186 P(x, z) when θ = 11.6 are given in Fig These spatial profiles show that the s-polarized SPP wave is guided by the metal/sntf interface because the power density is localized to the same plane. Before moving to the data obtained for γ = 75, let me note that the spatial profiles in the SNTF (0 < z < d 1 ) of the p- and s-polarized SPP waves presented in Figs. 11.3, 11.5, and 11.7 are similar to the profiles determined by the solution of the canonical boundary-value problem (Figs. 2.4, 2.2, and 2.3, respectively, in Ch γ = 75 When ϕ = 0 and γ = 75, the morphologically significant plane of the SNTF makes an angle of 75 with the xz plane the incidence plane and the grating plane. Relative wavenumbers κ/k 0, obtained from the solution of the canonical boundary-value problem for SPP waves propagating at an angle of 75 with respect to the morphologically significant plane, are presented in Table The two SPP waves, whose relative wavenumbers are provided in this table, are neither purely p nor s polarized because of the anisotropy of the SNTF. Table 11.5: Relative wavenumbers κ/k 0 of SPP waves obtained by the solution of the canonical boundary-value problem (Ch. 2) for propagation at an angle of 75 to the morphologically significant plane of the SNTF. The constitutive parameters of the SNTF and the metal are provided at the beginning of Sec The SPP waves are neither p nor s polarized. If κ represents an SPP wave propagating in the û x direction, κ represents an SPP wave propagating in the û x direction i i The absorptances A p and A s were calculated for L = 286 nm, ϕ = 0, and γ = 75, and are presented in Fig as functions of the angle of incidence θ. Independent of the value of d 1 and the polarization state of the incident plane wave, an absorptance peak is present at θ 9.2. The relative wavenumbers k x (n) /k 0 of Floquet harmonics at θ = 9.2 are provided in Table A comparison of Tables 11.5 and 11.6 show that the peak represents the excitation of an SPP wave as a Floquet harmonic of order n = 1 because k x ( 1) /k 0 = in Table 11.6 is very close to Re ( i) in Table The spatial profiles of the x- and y-components of P(0.75L, z) given in Fig for θ = 9.2 indicate that the SPP wave is localized to the metal/sntf interface. As the magnitude of P y is approximately ten times smaller than that of P x, the SPP wave transports energy mainly along the û x direction. This is reasonable because the SPP wave is excited in the grating-coupled configuration as a Floquet 156

187 0.4 A p, d 1 = 5, L g = 20nm A s, d 1 = 5, L g = 20nm A p, d 1 = 6, L g = 20nm A s, d 1 = 6, L g = 20nm Absorptance (deg) Figure 11.8: Absorptances A p and A s vs. θ when L = 286 nm, ϕ = 0, γ = 75, and d 3 d 2 = 30 nm. The vertical arrows identify the peaks that represent the excitation of SPP waves. harmonic of negative order. The notable characteristic of this SPP wave is that it can be excited by a plane wave of either polarization state; however, the excitation is more efficient if the incident plane wave is s polarized. Table 11.6: Relative wavenumbers k x (n) /k 0 of Floquet harmonics at the θ-values of the peaks identified in Fig by vertical arrows when L = 286 nm. Boldface entries signify SPP waves. n = 2 n = 1 n = 0 n = 1 n = 2 θ = θ = At θ 15.5 in Fig. 11.8, a peak is present independent of the value of d 1 in the plots of A p but not of A s. A comparison of Tables 11.5 and 11.6 shows that the A p -peak represents the excitation of an SPP wave as a Floquet harmonic of order n = 1 because k x (1) /k 0 = in Table 11.6 is very close to Re ( i) in Table Contrary to the SPP wave excited at θ = 9.2, the absence of the peak in the curves of A s shows that this SPP wave is excited only by a p-polarized 157

188 P x (W m -2 ) p pol. s pol. P x (W m -2 ) p pol. s pol z / z-d 1 (nm) p pol. s pol p pol. s pol. P y (W m -2 ) P y (W m -2 ) z / z-d 1 (nm) Figure 11.9: Variation of the x- and y-components of the time-averaged Poynting vector P(0.75L, z) along the z axis in the regions (left) 0 < z < d 1 and (right) d 1 < z < d 3 for p- and s-polarized incident plane waves when θ = 9.2, L = 286 nm, ϕ = 0, γ = 75, d 1 = 6Ω, L g = 20 nm, and d 3 d 2 = 30 nm. incident plane wave. The spatial profiles of P(0.75L, z) at θ = 15.5 for s- and p-polarized incident plane waves, given in Fig , also support this conclusion; furthermore, the SPP wave is strongly localized in the SNTF to the plane z = d 1. Let me note that the A p - and A s -peaks at θ = 9.2 are narrower than the A p - peak at θ = 15.5, thereby supporting the conclusion drawn in Sec that the absorptance peak representing the excitation of an SPP wave with smaller phase speed is broader. The spatial profiles in the SNTF (0 < z < d 1 ) of the SPP waves presented in Figs and are similar to the profiles for the corresponding SPP waves guided by the planar metal/sntf interface in the canonical boundaryvalue problem (Figs. 2.6 and 2.5, respectively, in Ch. 2). To analyze the effect of the direction of electric field phasor of the linearly 158

189 0.003 p pol. s pol P x (W m -2 ) P x (W m -2 ) p pol. s pol z / z-d 1 (nm) P y (W m -2 ) p pol. s pol. P y (W m -2 ) p pol. s pol z / z-d 1 (nm) Figure 11.10: Same as Fig except for θ = polarized incident plane wave on the efficiency of the excitation of SPP waves when ϕ = 0, I also calculated the absorptance A for an incident plane wave with electric field phasor E inc (r) = ( ) [ ( cos α s 0 + sin α p + 0 exp i k (0) x x + k z (0) z )], z 0. (11.80) This plane wave is s polarized when α {0, 180,...} and p polarized when α {90, 270,...}. By virtue of the symmetry of the problem, A(π + α) = A(α). The absorptance is plotted as a function of α in Fig when ϕ = 0, γ = 75, d 1 {5Ω, 6Ω}, and θ {9.2, 15.5 } the θ-values of the absorptance peaks that represent the excitation of SPP waves and identified in Fig by vertical arrows. As Figure shows that the absorptance is maximum at different values of α for different values of d 1, the efficient excitation of an SPP wave depends not only on the direction of the incident electric field but also on the thickness of the SNTF. 159

190 = 9.2 o, d 1 = 5 = 9.2 o, d 1 = 6 = 15.5 o, d 1 = 5 = 15.5 o, d 1 = A (deg) Figure 11.11: Absorptance A vs. α when L = 286 nm, ϕ = 0, γ = 75, L g = 20 nm, and d 3 d 2 = 30 nm. The electric field phasor of the incident plane wave is defined by Eq. (11.80). To explain this conclusion, let me use Eq. (11.80) in Eq. (11.75) to obtain A = A s cos 2 α + A p sin 2 α + 1 N t 2 sin 2α + t (n) ps t (n) pp + r (n) ss r (n) sp + r ps (n) r (n) pp n= N t + t (n) ss t (n) sp ( r (n) ss r sp (n) + t (n) ps + r (n) t (n) pp ps r pp (n) ( ) Re + t (n) k (n) z k (0) z ss t (n) sp ) (11.81) Equation (11.81) shows that A depends on α, and more generally, on the vibration ellipse of the incident plane wave. In addition, as depolarization affects the absorptance and depolarization must depend on d 1, the absorptance must also depend on the thickness of the SNTF. The higher the absorptance, the more efficient is the excitation of the SPP wave Comparison with the TKR configuration The excitation of multiple SPP waves guided by a planar metal/sntf interface in the Kretschmann configuration was studied theoretically earlier in this series [70, 71]. In this configuration, a plane wave propagating in a half space occupied 160

191 by a homogeneous dielectric material of real refractive index n l is taken to be incident on a metal-capped SNTF at an angle θ l with respect to the thickness direction of the SNTF. The SNTF is of finite but sufficiently large thickness and is terminated by a half space that can either be vacuous or be occupied by any homogeneous material, whereas the metal film is very thin. The theoretical aim is to compute the absorptance as a function of θ l and identify those values of the wavenumber k x = k 0 n l sin θ l for which the transmittance into the half space on the other side of the SNTF is nil while the absorptance is a maximum, regardless of the thickness of the SNTF beyond some threshold value. In Ref. 70, the incidence plane and the morphologically significant plane of the SNTF were taken to be the same. Implementing the formulation provided in Ref. 70 with n l = 2.58 and for the constitutive parameters of the SNTF and the metal the same as given in the beginning of Sec. 11.3, I determined the relative wavenumbers k x /k 0 of the SPP waves excited. A comparison of these values, provided in Table 11.7, with the boldface entries in Tables shows that the relative wavenumbers of the SPP waves excited in the grating-coupled configuration and in the TKR configuration are in an excellent agreement. Table 11.7: Relative wavenumbers n l sin θ l of SPP waves in the TKR configuration excited by s- and p-polarized incident plane waves propagating in the morphologically significant plane of the SNTF [70]. The constitutive parameters of the periodically nonhomogeneous SNTF and the metal are provided at the beginning of Sec. 11.3, whereas n l = θ l n l sin θ l p-pol p-pol s-pol Table 11.8: Same as Table 11.7 except that the morphologically significant plane of the SNTF makes an angle of 75 with the incidence plane [71]. θ l n l sin θ l p-pol s-pol In Ref. 71, the relative wavenumbers of SPP waves excited in the TKR configuration were provided for the case of the morphologically significant plane of the SNTF being rotated at angle of 75 about the z axis with respect to the 161

192 incidence plane, when n l = 2.58 and the constitutive parameters of the SNTF and the metal are the same as in the beginning of Sec These relative wavenumbers are given in Table 11.8, and are in excellent agreement with the boldface entries in Table 11.6 for the grating-coupled configuration. Besides the agreement of relative wavenumbers of SPP waves in the gratingcoupled and the TKR configurations, I noted that in both configurations (i) the absorptance peaks representing excitation by p-polarized plane waves are broader than those by s-polarized plane waves; and (ii) the higher the phase speed of an SPP wave, the narrower is the absorptance peak representing the excitation of that SPP wave. Moreover, (iii) the spatial profiles of SPP waves are similar in both configurations. This can be seen by comparing Figs. 11.3, 11.5, and 11.7 with Fig. 18 in Ref. 70, 1 and Figs and in this chapter with Figs. 6 and 3, respectively, in Ref Concluding Remarks The excitation of surface-plasmon-polariton waves by the illumination of a metallic surface-relief grating coated with a periodically nonhomogeneous SNTF by an obliquely incident plane wave was theoretically investigated. The absorptances were calculated using the rigorous coupled-wave approach as functions of the angle of incidence with respect to the thickness direction of the SNTF and the polarization state of the incident plane wave, when the incidence plane and the grating plane are identical. The excitation of SPP waves was inferred by those peaks in the absorptance curves (i) that were independent of the thickness of the SNTF when that thickness is sufficiently large; (ii) where the wavenumber of one of the Floquet harmonics matches that of an SPP wave predicted by the solution of the canonical boundary-value problem (Ch. 2); and (iii) the spatial profile of the components of time-averaged Poynting vector indicated the existence of a surface wave. 1 The difference in the values of χ v, 45 in this chapter and 30 in Ref. 70, does not create a significant difference in the spatial profiles. 162

193 If the morphologically significant plane of the SNTF is parallel to the incidence plane (and the grating plane), p- and s-polarized SPP waves can be excited, respectively, by p- and s-polarized incident plane waves. If the morphologically significant plane is not parallel to the incidence plane, SPP waves that are neither p nor s polarized can be excited. In the curves of absorptance as a function of the angle of incidence with respect to the thickness direction of the SNTF, (i) the peaks representing excitation by incident p-polarized plane waves are generally broader than those representing excitation by incident s-polarized plane waves, and (ii) the lower the phase speed of an SPP wave, the broader the peaks are in the absorptance curves. The efficient excitation of SPP waves depends on the vibration ellipse of the incident plane wave as well as on the thickness of the SNTF. 163

194 Chapter 12 Conclusions and Suggestions for Future Work 12.1 Conclusions As was discussed in Ch. 1, the objectives of the research conducted for this thesis were to (a) find the basic property of the partnering dielectric materials that is responsible for the multiplicity of surface waves; (b) elucidate the effects of the morphology of the partnering dielectric materials on the characteristics of surface waves; (c) find the minimum spatial dimensions of the partnering materials in order to implement the structures for experimental research; (d) find other ways to increase the number of possible surface waves; (e) study the excitation of multiple surface waves in prism- and grating-coupled configurations with periodically nonhomogeneous partnering dielectric materials; and (f) see if multiple SPP waves can lead to enhanced absorption of light in thinfilm solar cells. To achieve these objectives, the propagation and excitation of multiple surface waves guided by the interfaces of two dissimilar materials were theoretically studied. The various boundary-value problems investigated are shown in flow diagram in Fig (reproduced from Ch. 1). Two types of surface waves were extensively studied: surface plasmon-polariton (SPP) waves, and Dyakonov Tamm waves. Multiple SPP waves guided by the interface of a metal and a periodically nonhomogeneous sculptured nematic thin film (SNTF), the interface of a metal and a 164

195 rugate filter, and a metal slab inserted in a periodically nonhomogeneous SNTF were studied in the format of a canonical boundary-value problem. The propagation of multiple Dyakonov Tamm waves guided by a phase-twist combination defect in a periodically nonhomogeneous SNTF, and by a dielectric slab inserted in the SNTF. Moreover, multiple Tamm waves guided by an interface between two rugate filters were also studied. The excitation of multiple SPP waves in both the Turbadar-Kretschmann-Raether (TKR) and the grating-coupled configurations was also studied. It was also shown that the multiple SPP waves may enhance the absorption of light in thin-film solar cells. Ch. 2 Single metal/sntf interface Ch. 3 Single metal/rugatefilter interface Ch. 4 Multiple Fano waves Ch. 6 SNTF/metal/ SNTF interface Ch. 5 SNTF/SNTF interface Ch. 8 TKR configuration (metal/rugate-filter interface) Ch. 7 SNTF/dielectric/SNTF interface Ch. 11 Grating-coupled configuration (metal/sntf interface) Ch. 9 Grating-coupled configuration (metal/rugate-filter interface) Ch. 10 Application of multiple SPP waves in solar cells Ch. 12 Conclusions Future work Appendix A Rugate-filter/rugate-filter interface Appendix B Mathematica TM Codes Figure 12.1: A flow diagram showing the interconnections among different chapters of this thesis. The boxes with blue light background represent the chapters containing the canonical boundary-value problems, and the boxes with purple dark background represent the chapters that contain the boundary-value problems for the excitation of multiple surface waves. The boxes with white background do not contain any of the boundary-value problems (reproduced from Ch. 1). In Ch. 2, the solution of a canonical boundary-value problem showed that multiple SPP waves of both p- and s-polarization states can be guided by a metal/sntf interface. Furthermore, SPP waves that are neither p- nor s-polarized can also be guided. The solution of the canonical boundary-value problem unequivocally proved the existence of multiple SPP waves because the possibility of waveguide modes is not present in a canonical boundary-value problem as is the case in the TKR and the grating-coupled configuration. In Ch. 3, it was shown 165

196 that multiple SPP waves can be guided by an interface of a metal and a periodically nonhomogeneous dielectric material even if that material is isotropic. This was an important discovery as it showed that it is the periodic nonhomogeity of the partnering dielectric material that is responsible for the multiplicity of SPP waves provided that the period of the dielectric material is within a certain range of values. The results obtained in Ch. 3 proved to be a cornerstone for most of the research conducted by me. In Ch. 4, it was seen that SPP waves transmute into Fano waves when the imaginary part of the permittivity of the partnering metal is made zero. Moreover, the nonhomogeneity of the partnering dielectric material led to the excitation of not one but multiple Fano waves, supporting further the hypothesis that periodic nonhomogeneity of the dielectric material leads towards multiple surface waves. To further increase the number of possible surface waves, it was hypothesized that two interfaces in close proximity could lead to new surface waves. For this purpose, the canonical boundary-value problem of SPP wave-propagation by an SNTF/metal/SNTF system was solved in Ch. 6. The solution of the canonical problem showed that the coupling of the two metal/sntf interfaces results in new SPP waves provided the metallic layer between the two SNTFs is thinner than twice the penetration depth of the SPP waves into the metal. In Ch. 7, the effect of coupling on the Dyakonov Tamm waves was studied by solving a canonical problem of SNTF/dielectric/SNTF system. Again, it was seen that a sufficiently thin dielectric layer can lead to new Dyakonov Tamm waves that were otherwise absent from the single dielectric/sntf interface. Moreover, waveguide modes were also found to be guided by the dielectric slab in addition to the surface waves. Propagation guided by the SNTF/dielectric/SNTF system is lossless (or almost lossless with actual materials), but not by the SNTF/metal/SNTF system. However, one slight disadvantage of the SNTF/dielectric/SNTF system is the shorter angular sector when both the interfaces are uncoupled for the propagation directions of Dyakonov Tamm waves when compared to SPP waves. The capability for lossless propagation of surface waves in any direction could be useful for sensing and communication applications that are currently restricted by the attenuative propagation of SPP waves [14, 34, 35, 40, 41]. Multiple Dyakonov Tamm waves can lead to enhanced confidence in sensing measurements and may increase the capabilities of multi-analyte sensors. The coexistence of the Dyakonov Tamm waves and the waveguide modes could lead to enhanced sensing modalities. The presence of multiple Dyakonov Tamm waves and waveguide modes may also be useful for multi-channel communication at a specific frequency. The excitation of multiple SPP waves in the TKR configuration was studied in Ch. 8. A plane wave of either p- or s-polarization state, propagating in an optically denser dielectric material, was taken to be incident on a metal-capped rugate filter. The absorptances were calculated using a stable numerical algorithm as functions of the incidence angle. The excitation of SPP waves was inferred from 166

197 the presence of those peaks in the absorptance spectrum that were independent of the thickness of the rugate filter. A canonical boundary-value problem to study the propagation of coupled SPP waves by a metal film, with a semi-infinite rugate filter on one side and a semi-infinite homogeneous dielectric material on the other side, was also formulated to obtain a dispersion equation for each of the two linear polarization states. The solution of the dispersion equations and the spatial profiles of the SPP waves showed that p-polarized SPP waves are more likely to couple to the prism/metal interface than s-polarized SPP waves. However, the solution of another canonical boundary-value problem (Ch. 3) of SPP-wave propagation by the interface of a semi-infinite metal and a semi-infinite rugate filter revealed that the coupling due to the thin metal film in the TKR configuration does not result in new SPP waves. The excitation of multiple SPP waves by a surface-relief grating formed by a metal and a dielectric material, both of finite thickness, was studied theoretically in Ch. 9 using the rigorous coupled-wave approach (RCWA) for a practically implementable setup. The presence of an SPP wave was inferred by a peak in the plot of absorptance vs. the angle of incidence θ, provided that the θ-location of the peak turned out to be independent of the thickness of the dielectric partnering material. If that material is homogeneous, only one p-polarized SPP wave, that too of p-polarization state, is excited. In general, the absorptance peak is narrower for an s-polarized SPP wave than for of a p-polarized SPP wave, and the absorptance peak is narrower for an SPP wave of higher phase speed. The fact that only p-polarized SPP waves can be excited when the partnering dielectric material is homogeneous indicates the difficulty of exciting s-polarized SPP waves which is reflected both in the narrower peaks for A s than for A p in the results presented in Chs. 8, 9, 10, and 11, as well as in the greater number of solutions of a relevant canonical boundary-value problem in Ch. 3 for p- than for s-polarized SPP waves. Underlying all of these observations is the great mismatch in optical admittance [33] across metal/dielectric interfaces that usually prevails for s-polarized fields [48]. In Ch. 10, the excitation of multiple SPP waves was shown to increase the absorption in thin-film solar cells. For the chosen semiconductor material and the metal, it was seen that periodically corrugated metallic back surface may lead to excitation of both p- and s-polarized SPP waves, thereby, increasing the overall absorption in the structure. Though, the excitation of waveguide modes also plays a role in the enhancement of absorption in the grating-coupled configuration. However, the geometric parameters of the surface-relief grating and the periodically nonhomogeneous semiconductor layer shall have to be optimized carefully in order to obtain an overall enhanced absorption of the insolation flux over the nm wavelength range. I hope that this work will turn out to be a useful first step towards the exploitation of multiple SPP waves in enhancing the efficiency of thin-film solar cells. 167

198 Finally, the excitation of SPP waves guided by a metal/sntf interface was investigated in Ch. 11. The results obtained in this chapter not only reinforced the conclusions drawn by the investigations on the related canonical boundary-value problem, but also opened routes towards the excitation of multiple SPP waves guided by the interface of a metal and a periodically nonhomogeneous SNTF. It was seen that efficient excitation of SPP waves does not only depend on the polarization ellipse of the incident linearly polarized plane wave but also on the thickness of the SNTF. So, optimization of the grating profile and the SNTF thickness may be necessary for efficient simultaneous excitation of many SPP waves as well as for broadband performance. The excitation of different SPP waves in the grating-coupled configuration sometimes requires different periods of the surface-relief grating, as was seen in Chs. 9, 10, and 11. Therefore, the simultaneous excitation of all possible SPP waves may be achieved using a quasi-periodic grating [131]. In summary, it was found that (i) it is the periodic nonhomogeneity (not the anisotropy) of the partnering dielectric material normal to the interface that is responsible for the multiplicity of surface waves; (ii) multiple SPP, Tamm, Dyakonov Tamm, and Fano waves of the same frequency and different phase speeds and spatial profiles can be guided by an interface of two different materials provided that at least one of them is periodically nonhomogeneous normal to the interface; (iii) the morphology of the partnering dielectric material affects the number, the phase speeds, the spatial profiles, and the degrees of localization of the surface waves; (iv) the number of surface waves can be increased further by the coupling of two interfaces separated by a sufficiently thin layer; (v) multiple surface waves can be excited in the TKR and the grating-coupled configurations with periodically nonhomogeneous dielectric materials; and (vi) multiple SPP waves can be used to enhance the light absorption in a thin-film solar cell. These conclusions relate to objectives (a)-(f) in the same order. Since a variety of theoretical problems of practical significance have been formulated and solved in this thesis, it is hoped that it will provide a rigorous theoretical footing for later research on the exploitation of multiple surface waves in optical applications. Let me conclude by noting that all computations in this thesis were made at a single free-space wavelength λ 0 (except in Ch. 10, where two values of λ 0 168

199 were considered); however, multiple SPP waves can be expected at other values of λ 0 too. The numbers of p- and s-polarized SPP waves and their wavenumbers in any related canonical boundary-value problem must depend on λ 0. This is because all constitutive parameters of causal materials are frequency-dependent, not to mention that the effects of the spatial dimensions of the materials require interpretation in terms of λ Suggestions for Future Work The study of multiple surface waves is still in its infancy, and there is considerable room available to work both theoretically and experimentally, and both to gain understanding of the phenomenon of multiplicity and its exploitation for practical devices. I have a few suggestions for theoretical work on multiple surface waves based on the work reported in this thesis Excitation of multiple surface waves with a finite source The study of excitation of multiple surface waves with a finite source, such as a line source, in either the TKR or the grating-coupled configuration will not only increase the scope of applications of surface waves but can also provide a route towards exciting multiple surface waves without the inconvenience of changing the angle of incidence. This study can lead to the possibility of exciting multiple surface waves in nano-devices with embedded light sources. This problem can be analyzed easily by expressing the electromagnetic field of the finite source as a spectrum of planewaves [132, Sec. 2.2], and utilizing the same analytical and numerical tools explained in Chs. 8, 9, and Simultaneous excitation of all possible SPP waves using quasi-periodic surface-relief grating As was seen in Chs. 9, 10, and 11 that the excitation of different SPP waves requires that different choices may be needed for the period of the surface-relief grating. This is impossible to achieve with one fixed structure but can be overcome by having a new surface-relief grating such that each period of the new surface-relief grating contains, say, two or three periods of two (or as many as one likes) different surface-relief gratings. This new structure will allow the simultaneous excitation of all possible SPP waves without having to change anything. This problem can be tackled with the formulations provided in Chs. 9 and 11 by taking a new grating shape-function g(x). However, for computations, the number of terms in 169

200 the Fourier expansion of the grating shape-function g(x) needs to be much higher than what was used in Chs. 9 and 11 in order to get converged solutions Excitation of Tamm and Dyakonov Tamm waves As was described in the introduction to Ch. 7, the scope of application of SPP waves is limited due to the propagation losses along the direction of propagation. This limitation can be overcome by having the ability to excite other types of surface waves, that is, Tamm and Dyakonov Tamm waves. These waves propagate with negligible losses. So, the problem of excitation of multiple Tamm and Dyakonov Tamm waves is very important to enhance the scope of applications of multiple surface waves. The study of excitation of Tamm and Dyakonov Tamm waves can be done with the same formulation as provided in Chs. 8, 9, and 11: just the permittivity of the metal layers in those chapters needs to be replaced by the permittivity of the chosen dielectric materials Thin-film solar cell with actual configuration In Ch. 10, it was seen that the light absorption can be enhanced if the partnering semiconductor material in the grating-coupled configuration is made periodically nonhomogeneous normal to the mean metal/semiconductor interface. However, the boundary value problem in that chapter can not be used for actual solar cells because it does not have any p n junctions and no antireflection (AR) coatings or transparent conducting oxide (TCO) layer for electrical connection. Moreover, the problem solved in that chapter was limited only to the enhancement of light absorption and not the output power. I believe, a complete problem that include the geometry of actual solar cell with the aim to optimize the output power is worth investigating. For this purpose, the formulation given in Ch. 9 will have to be modified in order to include layered structure of the solar cell. 170

201 Appendix A Propagation of Multiple Tamm Waves A.1 Introduction The reports on the propagation of surface waves when both partnering dielectric materials are periodically nonhomogeneous normal to the interface are much scarcer and of very recent vintage [65 68, 133, 134]. In all of these reports, both partnering materials are also anisotropic. Numerical studies have shown that, depending on the direction of propagation in the interface plane, from none up to four surface waves with different phase speeds, spatial distributions of the electromagnetic field components, and degrees of localization can propagate at a particular frequency and for a chosen set of partnering materials. Would similar characteristics be exhibited if both partnering materials were isotropic, in contrast to those in previous works in Refs ,133, and 134, and in Ch. 5 of this thesis? The work reported in this appendix was undertaken to satisfy that curiosity. The isotropic but periodically nonhomogeneous material chosen for this purpose has a refractive index that varies sinusoidally about a mean value. Such continuously nonhomogeneous materials are routinely used as rugate filters for rejecting light in a specific frequency band [2, 74] and may have application in solar cells too [75]. Because of analogy with the surface Tamm states [61, 62] in solid-state electronics, the surface waves supported by rugate filters and other periodically nonhomogeneous materials such as the piecewise homogeneous photonic crystals [135, 136] are called Tamm waves [16, 137]. However, a subset of these can also be classified as optical Tamm states [138]. A synoptic view of the canonical problem of surface waves supported by the interface of two distinct, semi-infinite rugate filters with sinusoidal refractive-index This appendix is based on: H. Maab, M. Faryad, and A. Lakhtakia, Surface electromagnetic waves supported by the interface of two semi-infinite rugate filters with sinusoidal refractive index profiles, J. Opt. Soc. Am. A 28, (2011). 171

202 profiles is provided here. The two filters can differ in the mean refractive index, the amplitude of the refractive-index modulation, and the period of the same modulation. Alternatively, the two rugate filters can be identical except for a phase defect [65 67]. Since surface waves generally decay away from the interface, the canonical problem involving two semi-infinite rugate filters can be practically implemented using two rugate filters of sufficient thickness. Surface waves can be excited using an end-fire coupling technique [41] involving either a diffraction grating [14] or a fiber [35]. Section A.2 provides the necessary theoretical formulation, while numerical results are presented and discussed in Section A.3. Finally, the concluding remarks are presented in Sec. A.4. A.2 Theory Let the refractive index at the chosen angular frequency ω for all z (, ) be specified as n(z) = { n (z) = n avg + ( n /2) sin(ϕ + πz/ω ), z < 0, n + (z) = n + avg + ( + n /2) sin(ϕ + + πz/ω + ), z > 0, (A.1) so that n ± (z ± 2Ω ± ) = n ± (z). Here, n ± avg are the mean refractive indexes, whereas n ±, 2Ω ±, and ϕ ±, respectively, are the amplitudes, periods, and phases of the refractive-index modulation. Other periodic profiles of n ± (z) can also be accommodated by the formulation described in this section. Without any loss of generality, I take the propagation direction in the interface plane z = 0 to be parallel to the x axis. Since both dielectric materials are isotropic, the surface waves are either s-polarized or p-polarized. These two types of surface waves can be handled separately. A.2.1 s-polarized surface waves For an s-polarized surface wave, I have and E(r) = e y (z)û y exp (iκx) H(r) = [h x (z)û x + (κ/k 0 η 0 ) e y (z)û z ] exp (iκx), where ω/re(κ) is the phase speed of the surface wave along the x axis. Furthermore, the frequency-domain Maxwell equations yield the matrix ordinary differential equation [ d ey (z) dz h x (z) ] = iω 0 µ 0 ( ) ] 2 ϵ 0 [n 2 κ (z) k [ ey (z) h x (z) ], z 0. (A.2)

203 ] Equation (A.2) has to be solved in order to determine the matrixes [Q ± that appear in the relations [ ] [ ] ey (±2Ω ± ) ] ey (0±) = [Q ± h x (±2Ω ± ) h x (0±) (A.3) and characterize the optical responses of one period of the rugate filters on the two sides of the interface. These matrixes can be calculated using standard numerical techniques [65, 89]. Thereafter, the eigenvalues σ 1 ± and σ 2 ± = 1/σ 1 ±, and the corresponding eigenvectors [ ] 1, t ± T [ ] ] 1 and 1, t ± T 2, of [Q ± can be calculated, where the eigenvalues and the eigenvectors have been labeled such that Re ( ) ln σ 1 ± < 0 (Ch. 5). Consistent with the requirement that the electromagnetic field must decay as z ±, I get [136] [ ] [ ] ey (0±) 1 = a ±, (A.4) h x (0±) where a ± are unknown coefficients. As the usual boundary conditions must hold across the interface z = 0 i.e., e y (0+) = e y (0 ) and h x (0+) = h x (0 ) I obtain [ ] [ ] [ ] 1 1 a + 0 =, (A.5) t + 1 t 1 a 0 which yields the identity a + = a and the dispersion equation for the s-polarized surface wave. A.2.2 t + (κ) = t (κ) p-polarized surface waves For a p-polarized surface wave, I have and t ± 1 H(r) = h y (z)û y exp (iκx) E(r) = { e x (z)û x [ κη 0 /k 0 n 2 (z) ] h y (z)û z } exp (iκx). (A.6) Furthermore, the frequency-domain Maxwell equations yield the matrix ordinary differential equation [ [ ] ( ) ] 2 [ ] µ d ex (z) 0 0 n 2 κ (z) n = iω 2 (z) k 0 ex (z), z 0. dz h y (z) ϵ 0 n 2 (z) 0 h y (z) (A.7) The remainder of the procedure to determine the dispersion equation of the p- polarized surface wave is exactly the same as in Section A

204 A.3 Numerical Results and Discussion A computer program was prepared to find the solutions of the dispersion equations of s and p-polarized surface waves, using the Newton Raphson method [89]. All calculations were made for λ 0 = 633 nm. Several different cases were solved in order to understand the characteristics of surface waves supported by rugate filters with sinusoidal refractive-index profiles. The search for solutions was confined to real-valued κ, because all refractive indexes were taken to be real-valued. A.3.1 Homogeneous-dielectric/rugate-filter interface Let me begin with surface-wave propagation guided by the interface of a homogeneous dielectric material (z < 0) and a rugate filter (z > 0). For this purpose, let me set n = 0 and ϕ + = 0. Figure A.1 shows the solutions κ/k 0 of the dispersion equations for Tamm waves of both linear polarization states when n + avg = 1.885, + n = 0.87, Ω + = λ 0, and ϕ + = 0, while n avg [1.3, 2.2]; parenthetically, I note that these parameters for a rugate filter are realizable [74]. The immediately obvious conclusion from the figure is that more than one solutions are possible. The number of solutions decreases as n avg increases towards the maximum value n + avg + + n /2 of n + (z). Thus, in Fig. A.1, only two solutions exist for the highest value of n avg although eight solutions exist for the lowest value of n avg. The multiplicity of solutions in Fig. A.1 is just like that for surface-plasmonpolariton (SPP) waves guided by the interface of a metal and a rugate filter (Ch. 3). However, whereas those SPP waves are much more likely to be p-polarized than s-polarized as was seen in Ch. 3, the polarization states of Tamm waves here are almost equally but not always equally split between s and p. The presented data also show that the phase speed of a Tamm wave is smaller than the phase speed in the homogeneous partnering material, because κ > k 0 n avg. Furthermore, as k 0 (n + avg + n /2) < κ < k 0 (n + avg + + n /2), the phase speed of the Tamm wave is intermediate to both the minimum and the maximum phase speeds in the rugate filter. Exploration of the κ-range beyond k 0 (n + avg + + n /2) failed to turn up solutions. For a fixed value of n avg, the different Tamm waves have distinct phase speeds. They also have different spatial distributions of the components of the electromagnetic field despite the identity a + = a. Figure A.2 shows field distribution profiles for three of the eight Tamm waves for n avg = 2, the other parameters being the same as for Fig. A.1. Inside the homogeneous partnering material, the nonzero field components decay exponentially as z, the variation being of the type exp [ z κ 2 ( k 0 n avg ] ) 2 with κ > k 0 n avg. Thus, the higher the value of κ (i.e., the lower the phase speed) at fixed n avg, the more tightly bound is the 174

205 / k n - avg Figure A.1: κ/k 0 versus n avg for Tamm waves localized to the interface of a homogeneous dielectric material ( n = 0) and a rugate filter (n + avg = 1.885, + n = 0.87, Ω + = λ 0, and ϕ + = 0), when the free-space wavelength λ 0 = 633 nm. The red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves. I failed to find solutions to bridge the gaps in two branches of solutions; these gaps are likely to be numerical artefacts, as there is no physical reason for them to exist. Tamm wave to the plane z = 0. In contrast, the field distribution has to be far more complicated inside the rugate filter (z > 0). Following the dictates of the Floquet Lyapunov theorem [88], the z-dependence of each nonzero component of the electromagnetic field is the product of two factors. One of these factors decays exponentially as z, but the other factor is faithfully reproduced unit cell by unit cell. The undulations of the second factor in each unit cell can be cast in terms of a semiclassical coupled wave theory [139, 140]. Hence, in Fig. A.2, the average of a nonzero component over the q-th unit cell z [2(q 1)Ω +, 2qΩ + ] is higher than the average over the (q + 1)-th unit cell z [2qΩ +, 2(q + 1)Ω + ], q {1, 2,...}, but the variations inside two adjacent unit cells look quite similar in form. The maximum field strength of a Tamm wave does not necessarily lie at the interface plane. Thus, in Fig. A.2, the maximum magnitudes inside the rugate 175

206 e x,y,z h x,y,z e x,y,z h x,y,z e x,y,z h x,y,z Figure A.2: Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of Tamm waves along the z axis, when λ 0 = 633 nm, n avg = 2, n = 0, n + avg = 1.885, + n = 0.87, Ω + = λ 0, and ϕ + = 0. The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively. All calculations were made after setting a + = a = 1 V m 1. (top) p-polarization state and κ/k 0 = , (middle) s-polarization state and κ/k 0 = , and (bottom) s-polarization state and κ/k 0 = filter lie at z 1.4Ω + in the plots in the top and middle rows, and at z = 0.5Ω + in the bottom row. In order to quantify the degree of localization of the Tamm wave to the plane z = 0+, let me define the decay constant β + = exp [ Re ( )] ln σ 1 +. Since Re (ln σ + ) < 0, and thus 0 < β + < 1; the smaller the decay constant is, the stronger is the Tamm wave localized to the interface. The decay constant of the p-polarized Tamm wave (top row in Fig. A.2) is β + = , while the decay constants of the two s-polarized Tamm waves are β + = (middle row) and 176

207 β + = (bottom row), respectively. These values of β + illustrate the weaker localization of the p-polarized Tamm wave in comparison to the stronger localization of the two s-polarized Tamm waves, in Fig. A.2. The effect of the amplitude + n of the refractive-index modulation of the rugate filter was examined next. For this purpose, calculations were made with n avg = 2.5, Ω + = λ 0, and n + avg = fixed, while + n 2. No solutions can exist for + n = 0 because both the partnering materials then are homogeneous with relative permittivities that are positive and real-valued [14, 35], and none was found. The smallest value of + n for which a solutions was found is 0.01; the solution is for the s-polarization state, as shown in Fig. A.3. The smallest value of + n is a physical requirement in order for the discontinuity at the interface to be able to guide a surface wave. Fewer solutions are present for lower values than for the higher values of + n. For lower values of + n, the decay constant β + is close to unity signifying that the Tamm waves are weakly bound to the interface z = 0+; as + n increases, β + decreases signifying more tightly bound Tamm waves. Furthermore, solutions with higher phase speed ω/κ emerge and the phase-speed distribution generally widens as + n increases. In order to investigate the role of the period of the rugate filter, the same exercise as for Fig. A.3 was conducted, except that + n = 0.87 was fixed but Ω + /λ 0 was varied between 0 and 2.0. As Fig. A.4 shows, the number of solutions is generally small when the period of the rugate filter is small. I were not able to find any solutions of the dispersion equations when Ω + < 0.32λ 0. This limit suggests a physical restriction on the minimum value of the period in order for a surface wave to be guided by the interface of a homogeneous material and a periodically nonhomogeneous material. As the period 2Ω + increases, more solutions can emerge, but there is an upper limit to the number of solutions since all solutions in the figure are confined to the range k 0 (n + avg + n /2) < κ < k 0 (n + avg + + n /2). Further increase of the period 2Ω + should lead to a decrease in the number of solutions; indeed, when 2Ω + is very large, the periodically nonhomogeneous material becomes virtually homogeneous close to the interface and no surface wave can then be supported. A.3.2 Rugate filter with a phase defect If n avg = n + avg, n = + n, Ω = Ω +, but ϕ ϕ +, one can say that the entire space z (, ) is occupied by a single rugate filter which has a phase defect ϕ + ϕ at the plane z = 0. I chose to investigate the propagation of Tamm waves guided by the phase defect at λ 0 = 633 nm by fixing n avg = n + avg = 1.885, n = + n = 0.87, Ω = Ω + = λ 0, ϕ + = 0, but keeping ϕ [0, 2π) variable. Clearly, n(z) is continuous across the plane z = 0, and the phase defect vanishes, when ϕ = 0 or 2π. In contrast, a phase defect does exist despite the continuity of n(z) across the plane z = 0 when ϕ = π; however, I could not find any solutions 177

208 / k n Figure A.3: κ/k 0 versus + n for Tamm waves localized to the interface of a homogeneous dielectric material (n avg = 2.5, n = 0) and a rugate filter (n + avg = 1.885, Ω + = λ 0, and ϕ + = 0), when the free-space wavelength λ 0 = 633 nm. The red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves. of the dispersion equations that signifies Tamm waves. For other values of ϕ, n(z) is discontinuous across the plane z = 0. The solutions of the dispersion equations for Tamm waves bound to the phasedefect plane z = 0 are shown in Fig. A.5. Clearly, multiple Tamm waves can be guided by a phase defect in a rugate filter. For ϕ / {0, π}, up to 12 solutions are possible with κ > k 0 (n + avg + n /2), almost always equally divided between the s- and p-polarization states. As ϕ increases from 0, the number of solutions increases episodically. All branches of s-polarized solutions are paired with branches of p- polarized solutions, and all of these solutions satisfy the inequality κ > k 0 (n + avg + n /2). Four unpaired branches of p-polarized solutions also exist in Fig. A.5. One branch satisfies the inequality κ > k 0 (n + avg + n /2) and therefore indicates Tammwave propagation. For the other three unpaired branches, κ < k 0 (n + avg + n /2). Solutions on these three branches are special because κ/k 0 is less than the minimum value of the refractive index of the rugate filter on either side of the phase 178

209 / k / 0 Figure A.4: κ/k 0 versus Ω + /λ 0 for Tamm waves localized to the interface of a homogeneous dielectric material (n avg = 2.5, n = 0) and a rugate filter (n + avg = 1.885, + n = 0.87, and ϕ + = 0), when the free-space wavelength λ 0 = 633 nm. The red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves. defect: such solutions cannot exist if one of the region on either side of the phase defect is replaced by a homogeneous dielectric material. These solutions are to be called optical Tamm states [138], in stricter analogy to the electronic Tamm states [61, 62] than the other Tamm waves. Optical Tamm states can only be found at the interface of two photonic crystals having overlapping bandgaps, their wavenumber κ/k 0 must be lower than the smallest refractive index, and they can even possess infinite phase speed (i.e., κ = 0). Their use for polaritonic lasers is very promising [141]. Despite the fact that no solution exists for ϕ = 0 (or 2π), some solution branches, but not all, appear to wrap about ϕ = 2π in Fig. A.5. The branches with decay constants β ± = exp [ Re ( )] ln σ 1 ± much smaller than unity at ϕ 0 and ϕ 2π appear to wrap around have solutions. Branches of solutions with β ± close to unity for ϕ 0 do not show the wrapping characteristic. The gaps in several branches of solutions in Fig. A.5 were not categorized either numerical artefacts or physical gaps; alternatively, some gaps may simply be apparent but 179

210 / k (deg) Figure A.5: κ/k 0 versus ϕ for Tamm waves localized to the phase-defect plane z = 0 in a rugate filter, with n + avg = n avg = 1.885, + n = n = 0.87, Ω + = Ω = λ 0, and ϕ + = 0, when the free-space wavelength λ 0 = 633 nm. The red circles indicate s polarized Tamm waves and the black triangles are for p-polarized Tamm waves. No solutions exist for ϕ {0, π}. not real. ] The eigenvalues σ 1,2 ± depend on the matrixes [Q ±, which are responses of one period of the rugate filter on either side of the interface. ] As the] same rugate filter exists on both sides of a phase defect interface, [Q + and [Q are related such that σ + l = σ l, l {1, 2}; hence, β+ = β, and the decay constants on either side of the phase-defect interface are identical. Spatial profiles of the electromagnetic field are shown in Fig. A.6 for two p- polarized Tamm waves at ϕ = 8 and 174, and one s-polarized Tamm wave at ϕ = 8. The decay constants at ϕ = 8 for the p-polarized wave are β + = β = , and for the s-polarized wave are β + = β = The decay constants at ϕ = 174 for the p-polarized wave are β + = β = Of these three Tamm waves, the s-polarized wave is most strongly localized to the phase-defect 180

211 e x,y,z e x,y,z e x,y,z h x,y,z h x,y,z h x,y,z Figure A.6: Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of Tamm waves along the z axis. All parameters are same as for Fig. A.5 except ϕ = 8 for the top and middle rows, and ϕ = 174 for the bottom row. The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively. All calculations were made after setting a + = a = 1 V m 1. (top) p-polarization state and κ/k 0 = , (middle) s-polarization state and κ/k 0 = , and (bottom) p-polarization state and κ/k 0 = interface. Figure A.7 shows the spatial profile of the electromagnetic field at ϕ = 30 for a p-polarized optical Tamm state. The value of κ/k 0 is less than the minimum value of the refractive index on either side of the phase-defect interface. The decay constants are β + = β = , which indicate weak localization to the interface 181

212 e x,y,z h x,y,z Figure A.7: Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of an optical Tamm state along the z axis. All parameters are same as for Fig. A.5 except ϕ = 30 and κ/k 0 = The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively. All calculations were made after setting a + = a = 1 V m 1. The optical Tamm state is p-polarized. plane. A.3.3 Rugate filter with sudden change of mean refractive index Instead of a phase defect, suppose that the mean refractive index changes suddenly across the plane z = 0. The discontinuity thus caused could possibly guide Tamm waves. In order to explore this possibility, n + avg = 1.885, n = + n = 0.87, Ω = Ω + = λ 0, and ϕ = ϕ + = 0 was fixed but n avg [1.2, 2.3]. Figure A.8 shows that multiple solutions of the dispersion equations exist for k 0 (n avg n /2) < κ < k 0 (n + avg + + n /2). Both s- and p-polarized Tamm waves are represented by equal number of branches, except for a single branch of s-polarized Tamm waves with k 0 (n avg n /2) < κ < k 0 (n + avg + n /2). As the value of n avg approaches that of n + avg, the number of possible Tamm waves decreases, and at n avg = n + avg, no solutions of the dispersion equations were found as would be expected due to the disappearance of the discontinuity at z = 0. When n avg > n + avg, the relative wavenumbers of possible Tamm waves increase due to the increase in mean refractive index in the half space z < 0. The magnitudes of electric and magnetic field components for two p-polarized Tamm waves at n avg = 1.5 and n avg = 2.3 are shown in Fig. A.9. The decay constants for the solution κ = k 0 for n avg = 1.5 are β + = and β = ; hence, the decay is faster in the half space z > 0 than in the half space z < 0. The decay constants for the solution κ = k 0 for n avg = 2.3 are β + = and β = Thus, the decays in both half spaces are rapid, but the localization of the Tamm wave is stronger in the half space z > 0 as compared 182

213 / k n āvg Figure A.8: κ/k 0 versus n avg for Tamm waves localized to the plane z = 0 in a rugate filter, with n + avg = 1.885, n = + n = 0.87, Ω = Ω + = λ 0, and ϕ = ϕ + = 0, when the free-space wavelength λ 0 = 633 nm. The red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves. No solutions exist when n avg = n + avg, because the physical discontinuity across the interface z = 0 then disappears. The gaps including n avg = n + avg are physical because the discontinuity across the interface z = 0 then is too weak to support surface waves; however, other gaps in the solutions are more likely to be numerical artefacts as there is no physical reasons for them to exist. to the half space z < 0. As no solutions were found for κ < k 0 (n avg n /2), the chosen parameters did not admit solutions that could be classified as optical Tamm states [138]. A.3.4 Rugate filter with sudden change of amplitude Suppose that the discontinuity at plane z = 0 is caused by sudden change of the amplitude of the refractive index of the rugate filter. To find the relative wavenumbers of the Tamm waves that can be guided by this discontinuity, the dispersion equations were solved with n + avg = n avg = 1.885, Ω + = Ω = λ 0, ϕ + = ϕ = 0, and + n = 0.87, whereas n [0, 0.87] was kept variable. The solutions, presented in Fig. A.10, can be organized in four branches of p-polarized and three branches of s-polarized Tamm waves. None of the solutions are for 183

214 e x,y,z h x,y,z e x,y,z h x,y,z Figure A.9: Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of p-polarized Tamm waves along the z axis. All parameters are same as for Fig. A.8 except (top) n avg = 1.5 and κ/k 0 = , and (bottom) n avg = 2.3 and κ/k 0 = The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively. optical Tamm states [138]. As the discontinuity disappears at n = 0.87, no solutions can then exist. The Tamm waves represented by the upper four branches of the solutions given in Fig. A.10 are strongly localized to the plane z = 0 because their decay constants β ± are close to zero. However, the Tamm waves represented by the lower three branches are weakly localized to the plane z = 0; furthermore, the localization is weaker in the half space z < 0 than in the half space z > 0. A.3.5 Interface of two distinct rugate filters Finally, the propagation of Tamm waves guided by the interface of two distinct rugate filters was investigated. The parameters of the rugate filter occupying the half space z > 0 were chosen as n + avg = 1.885, + n = 0.87, ϕ + = 0, and Ω + = λ 0 ; those of the rugate filter in the half space z < 0 were set as n avg = 1.6, n = 0.6, ϕ = 90, and Ω = 0.5λ 0. The free-space wavelength λ 0 = 633 nm for the calculations. The solutions of the dispersion equations are tabulated in Table A.1. Four solutions are for s-polarized Tamm waves while three solutions are for p-polarized 184

215 / k n Figure A.10: κ/k 0 versus n for Tamm waves localized to the plane z = 0 in a rugate filter, with n + avg = n avg = 1.885, + n = 0.87, Ω = Ω + = λ 0, ϕ = ϕ + = 0, and n [0, 0.87], when the free-space wavelength λ 0 = 633 nm.the red circles indicate s-polarized Tamm waves and the black triangles are for p-polarized Tamm waves. No solutions can exist when n = Table A.1: Relative wavenumber κ/k 0 of s-polarized and p-polarized Tamm waves supported by the interface of two distinct rugate filters, whose parameters are provided in Section A.3.5. The free-space wavelength λ 0 = 633 nm. s-pol p-pol Tamm waves; no solutions representing optical Tamm states were found. The 185

216 e x,y,z h x,y,z Figure A.11: Variation of the magnitudes of the nonzero Cartesian components of (left) E (in V m 1 ) and (right) H (in A m 1 ) of Tamm waves along the z axis. Distinct rugate filters having parameters n + avg = 1.885, + n = 0.87, ϕ + = 0 and Ω + = λ 0, and n avg = 1.6, n = 0.6, ϕ = 90 and Ω = 0.5λ 0, respectively, were chosen with the free-space wavelength fixed at λ 0 = 633 nm. The field distributions were calculated for κ/k 0 = The components parallel to û x, û y, and û z are represented by red dotted, blue dashed, and black solid lines, respectively. spatial profiles of the electromagnetic field of a p-polarized Tamm wave are given in Fig. A.11. The decay constants for this Tamm wave are β + = and β = , which indicate stronger localization in the half space z < 0 than in the half space z > 0. A.4 Concluding Remarks The boundary-value problem for surface waves guided by the interface of two periodically nonhomogeneous and isotropic dielectric materials (semi-infinite rugate filters) was formulated and numerically solved for several cases. Both rugate filters were taken to possess a sinusoidally varying refractive index. Multiple Tamm waves and optical Tamm states with different phase speed, polarization state, and degree of localization to the interface were found to be guided by (i) an interface of a homogeneous material and a rugate filter; (ii) an interface created by a phase defect in a rugate filter; (iii) an interface created by a sudden change of either the mean refractive index or the amplitude of the sinusoidal variation of the refractive index; and (iv) an interface between two distinct rugate filters. The number of possible Tamm waves depend upon 186

217 (i) the refractive index of the homogeneous dielectric partner, (ii) the amplitude and the period of the sinusoidal nonhomogeneity, and (iii) the mean refractive index of the rugate filter. The period of the rugate filter has to be greater than a certain value in order for the Tamm waves to exist. While the relative wavenumbers of Tamm waves guided by the interface of a homogeneous dielectric material and a rugate filter are always greater than the refractive index of the homogeneous material, the phase-defect interface in a rugate filter was also found to guide surface waves with relative wavenumbers that are smaller than the minimum refractive index on either side of the phase defect. These surface waves can be classified as optical Tamm states. The Tamm waves can be either p- or s-polarized, whereas all optical Tamm states were found to be p-polarized. 187

218 Appendix B Mathematica TM Codes B.1 Newton-Raphson Method to Find κ in the Canonical Boundary-Value Problem of Ch. 2 munot = 4 Pi 10ˆ( 7); epsnot = ˆ( 12); lamnot = ˆ( 9); knot = (2 Pi )/ lamnot ; etanot = Sqrt [ munot/ epsnot ] ; omega = knot /Sqrt [ epsnot munot ] ; ns = I ; epsmet = 56 + I 21; Cap omega = ˆ( 9); d e l t a v = (30 Pi ) / ; Chit = (45 Pi ) / ; Cap delta = 2 10ˆ( 9); Ns = 2 Cap omega/ Cap delta ; d e l t a k = 10ˆ( 6); ( Module to c a l c u l a t e determinent o f M ) getmdet [ kappa, s t a t e ] := Module[ { kappa = kappa, f l a g 2 = s t a t e }, f l a g = 0 ; While [ f l a g == 0, CN = IdentityMatrix [ 4 ] ; For [ n = 1, n <= 2 Ns, n++, x1 = 2 n Cap omega/ns ; x2 = 2 (n 1) Cap omega/ns ; 188

219 z = 0. 5 ( x1 + x2 ) ; Chiv = Chit + d e l t a v Sin [ ( Pi z )/ Cap omega ] ; Chi = ArcTan[ Tan[ Chiv ] ] ; v = (2 Chiv )/ Pi ; epsa = ( v v ˆ 2 ) ˆ 2 ; epsb = ( v v ˆ 2 ) ˆ 2 ; epsc = ( v v ˆ 2 ) ˆ 2 ; epsd = ( epsa epsb ) / ( epsa (Cos [ Chi ] ) ˆ 2 +epsb ( Sin [ Chi ] ) ˆ 2 ) ; A = {{0, 0, 0, munot }, {0, 0, munot, 0}, {0, epsnot epsc, 0, 0}, { epsnot epsd, 0, 0, 0}}; B = {{Cos [ p s i ], 0, 0, 0}, {Sin [ p s i ], 0, 0, 0}, {0, 0, 0, 0}, {0, 0, Sin [ p s i ], Cos [ p s i ] } } ; F = {{0, 0, Cos [ p s i ] Sin [ p s i ], (Cos [ p s i ] ) ˆ 2 }, {0, 0, ( Sin [ p s i ] ) ˆ 2, Cos [ p s i ] Sin [ p s i ] }, {0, 0, 0, 0}, {0, 0, 0, 0}}; G = {{0, 0, 0, 0}, {0, 0, 0, 0}, { Cos [ p s i ] Sin [ p s i ], (Cos [ p s i ] ) ˆ 2, 0, 0}, { (Sin [ p s i ] ) ˆ 2, Cos [ p s i ] Sin [ p s i ], 0, 0}}; P = omega A + ( kappa epsd ( epsa epsb ) Sin [ Chi ] Cos [ Chi ] ) / ( epsa epsb ) B + ( kappa ˆ2 epsd ) / ( omega epsnot epsa epsb ) F + kappa ˆ2/( omega munot) G; CN = MatrixExp [ ( I P 2 Cap omega )/Ns ].CN ; ] ; alphas = Sqrt [ epsmet knot ˆ2 kappa ˆ 2 ] ; I f [Im[ alphas ] < 0, alphas = alphas ; ] ; Print [ "alpha s=", alphas ] ; M = {{ alphas Cos [ p s i ] / knot, Sin [ p s i ], 0, 0}, { alphas Sin [ p s i ] / knot, Cos [ p s i ], 0, 0}, {epsmet Sin [ p s i ] / ( etanot ), alphas Cos [ p s i ] / ( etanot knot ), 0, 0}, { epsmet Cos [ p s i ] / ( etanot ), alphas Sin [ p s i ] / ( etanot knot ), 0, 0} } ; 189

220 { eigvaluen, eigvectorn } = Eigensystem [CN] ; eigvectorn = Transpose [ eigvectorn ] ; eigvalueq = I Log [ eigvaluen ] / ( 2 Cap omega ) ; n = 3 ; For [m = 1, m < 5, m++, I f [Im[ eigvalueq [ [m] ] ] > 0, Print [Im[ eigvalueq [ [m ] ] ] ] ; M[ [ All, n ] ] = eigvectorn [ [ All, m] ] ; n++]; I f [ n == 5, m = 5 ; ] ; ] ; detm = Det [M] ; I f [ detm!= 0, f l a g = 1 ; Break [ ] ; ] ; I f [ detm == 0 && f l a g 2 == 1, kappa = knot + kappa ; f l a g = 0 ; ] ; I f [ detm == 0 && f l a g 2 == 2, kappa = knot + kappa ; f l a g = 0 ; ] ; ] ; detm ] ( Newton Raphson Method ) p s i = (0 Pi ) / ; y = 1 ; mold = 1 ; t o l = 10ˆ( 14); kappa = N[ ( ) knot ] While [Abs[Re[ mold ] ] > t o l Abs[Im[ mold ] ] > tol, Print [ S t y l e [ "Iteration No = ", 18, Red], y ] ; y = y + 1 ; kappaold = (1) kappa ; mold = getmdet [ kappaold, 1 ] ; kappanew = (1 + d e l t a k ) kappa ; mol = getmdet [ kappanew, 2 ] ; qold = ( mol mold ) / ( kappanew kappaold ) ; kappa = kappa mold/ qold ; Print [ "Det M old=", mold ] ; Print [ "kappa=", kappa/ knot ] ; I f [ y == 50, Quit [ ] ; ] ; ] 190

221 B.2 Plotting the Components of P of a p-polarized SPP Wave in the Canonical Boundary-Value Problem of Ch. 2 munot = 4 Pi 10ˆ( 7); epsnot = ˆ( 12); lamnot = ˆ( 9); knot = (2 Pi )/ lamnot ; etanot = Sqrt [ munot/ epsnot ] ; omega = knot /Sqrt [ epsnot munot ] ; ns = I ; epsmet = 56 + I 21; Cap omega = ˆ( 9); d e l t a v = (30 Pi ) / ; Chit = (45 Pi ) / ; p s i = (0 Pi ) / ; kappa = ( I ) knot ; Cap delta = 2 10ˆ( 9); Ns = 2 Cap omega/ Cap delta ; d e l t a k = 10ˆ( 6); ( Module to c a l c u l a t e determinent o f M ) CN = IdentityMatrix [ 4 ] ; For [ n = 1, n <= Ns, n++, x1 = 2 n Cap omega/ns ; x2 = 2 (n 1) Cap omega/ns ; z = 0. 5 ( x1 + x2 ) ; Chiv = Chit + d e l t a v Sin [ ( Pi z )/ Cap omega ] ; Chi = ArcTan[ Tan[ Chiv ] ] ; v = (2 Chiv )/ Pi ; epsa = ( v v ˆ 2 ) ˆ 2 ; epsb = ( v v ˆ 2 ) ˆ 2 ; epsc = ( v v ˆ 2 ) ˆ 2 ; epsd = ( epsa epsb ) / ( epsa (Cos [ Chi ] ) ˆ 2 + epsb ( Sin [ Chi ] ) ˆ 2 ) ; A = {{0, 0, 0, munot }, {0, 0, munot, 0}, {0, epsnot epsc, 0, 0}, 191

222 { epsnot epsd, 0, 0, 0} } ; B = {{Cos [ p s i ], 0, 0, 0}, {Sin [ p s i ], 0, 0, 0}, {0, 0, 0, 0}, {0, 0, Sin [ p s i ], Cos [ p s i ] } } ; F = {{0, 0, Cos [ p s i ] Sin [ p s i ], (Cos [ p s i ] ) ˆ 2 }, {0, 0, ( Sin [ p s i ] ) ˆ 2, Cos [ p s i ] Sin [ p s i ] }, {0, 0, 0, 0}, {0, 0, 0, 0}}; G = {{0, 0, 0, 0}, {0, 0, 0, 0}, { Cos [ p s i ] Sin [ p s i ], (Cos [ p s i ] ) ˆ 2, 0, 0}, { (Sin [ p s i ] ) ˆ 2, Cos [ p s i ] Sin [ p s i ], 0, 0}}; P = omega A + ( kappa epsd ( epsa epsb ) Sin [ Chi ] Cos [ Chi ] ) / ( epsa epsb ) B + ( kappa ˆ2 epsd ) / ( omega epsnot epsa epsb ) F + kappa ˆ2/( omega munot) G; CN = MatrixExp [ ( I P 2 Cap omega )/Ns ].CN ; ] ; alphas = Sqrt [ knot ˆ2 ns ˆ2 kappa ˆ 2 ] ; I f [Im[ alphas ] < 0, alphas = alphas ; ] ; M = {{ alphas Cos [ p s i ] / knot, Sin [ p s i ], 0, 0}, { alphas Sin [ p s i ] / knot, Cos [ p s i ], 0, 0}, {epsmet Sin [ p s i ] / ( etanot ), alphas Cos [ p s i ] / ( etanot knot ), 0, 0}, { epsmet Cos [ p s i ] / ( etanot ), alphas Sin [ p s i ] / ( etanot knot ), 0, 0}}; { eigvaluen, eigvectorn } = Eigensystem [CN] ; eigvectorn = Transpose [ eigvectorn ] ; eigvalueq = I Log [ eigvaluen ] / ( 2 Cap omega ) ; n = 3 ; For [m = 1, m < 5, m++, I f [Im[ eigvalueq [ [m] ] ] > 0, M[ [ All, n ] ] = eigvectorn [ [ All, m] ] ; n + + ] ; ] ; detm = Det [M] ; newm = {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}; newm = M[ [ 2 ; ; 4, 2 ; ; 4 ] ] ; bmat = M[ [ 2 ; ; 4, 1 ] ] ; 192

223 Mp = {{ alphas Cos [ p s i ] / knot, Sin [ p s i ] }, { alphas Sin [ p s i ] / knot, Cos [ p s i ] }, {epsmet Sin [ p s i ] / ( etanot ), alphas Cos [ p s i ] / ( etanot knot )}, { epsmet Cos [ p s i ] / ( etanot ), alphas Sin [ p s i ] / ( etanot knot ) } } ; ap = 1 ; as = 0 ; b1 = 0 ; b2 = M[ [ 1, 1 ] ] /M[ [ 1, 3 ] ] ; Print [ "b2 = ", b2 ] ; flp = Mp. { ap, as } e 1 p l i s t = {}; e 2 p l i s t = {}; e z p l i s t = {}; h 1 p l i s t = {}; h 2 p l i s t = {}; h z p l i s t = {}; p 1 p l i s t = {} p 2 p l i s t = {}; p z p l i s t = {}; CN = IdentityMatrix [ 4 ] ; For [ n = 1, n <= Ns, n++, x1 = 2 n Cap omega/ns ; x2 = 2 (n 1) Cap omega/ns ; z = 0. 5 ( x1 + x2 ) ; Chiv = Chit + d e l t a v Sin [ ( Pi z )/ Cap omega ] ; Chi = ArcTan[ Tan[ Chiv ] ] ; v = (2 Chiv )/ Pi ; epsa = ( v v ˆ 2 ) ˆ 2 ; epsb = ( v v ˆ 2 ) ˆ 2 ; epsc = ( v v ˆ 2 ) ˆ 2 ; epsd = ( epsa epsb ) / ( epsa (Cos [ Chi ] ) ˆ 2 + epsb ( Sin [ Chi ] ) ˆ 2 ) ; A = {{0, 0, 0, munot }, {0, 0, munot, 0}, 193

224 {0, epsnot epsc, 0, 0}, { epsnot epsd, 0, 0, 0}}; B = {{Cos [ p s i ], 0, 0, 0}, {Sin [ p s i ], 0, 0, 0}, {0, 0, 0, 0}, {0, 0, Sin [ p s i ], Cos [ p s i ] } } ; F = {{0, 0, Cos [ p s i ] Sin [ p s i ], (Cos [ p s i ] ) ˆ 2 }, {0, 0, ( Sin [ p s i ] ) ˆ 2, Cos [ p s i ] Sin [ p s i ] }, {0, 0, 0, 0}, {0, 0, 0, 0}}; G = {{0, 0, 0, 0}, {0, 0, 0, 0}, { Cos [ p s i ] Sin [ p s i ], (Cos [ p s i ] ) ˆ 2, 0, 0}, { (Sin [ p s i ] ) ˆ 2, Cos [ p s i ] Sin [ p s i ], 0, 0}}; P = omega A +B ( kappa epsd ( epsa epsb ) Sin [ Chi ] Cos [ Chi ] ) / ( epsa epsb ) + ( kappa ˆ2 epsd ) / ( omega epsnot epsa epsb ) F + kappa ˆ2/( omega munot ) G; {exp, eyp, hxp, hyp} = CN. flp ; ezp = ( epsd ( epsa epsb ) Sin [ Chi ] Cos [ Chi ] ) / ( epsa epsb ) exp + ( kappa epsd ) / ( omega epsnot epsa epsb ) ( hxp Sin [ p s i ] hyp Cos [ p s i ] ) ; hzp = (kappa /( omega munot ) ) ( exp Sin [ p s i ] eyp Cos [ p s i ] ) ; hxpc = Conjugate [ hxp ] ; hypc = Conjugate [ hyp ] ; hzpc = Conjugate [ hzp ] ; pxp = Re[ eyp hzpc ezp hypc ] / 2 ; pyp = Re[ ezp hxpc exp hzpc ] / 2 ; pzp = Re[ exp hypc eyp hxpc ] / 2 ; e1p = exp Cos [ p s i ] + eyp Sin [ p s i ] ; e2p = exp Sin [ p s i ] + eyp Cos [ p s i ] ; h1p = hxp Cos [ p s i ] + hyp Sin [ p s i ] ; h2p = hxp Sin [ p s i ] + hyp Cos [ p s i ] ; p1p = pxp Cos [ p s i ] + pyp Sin [ p s i ] ; p2p = pxp Sin [ p s i ] + pyp Cos [ p s i ] ; e 1 p l i s t = Append[ e 1 p l i s t, {( z )/ Cap omega, Abs[ exp Cos [ p s i ] + eyp Sin [ p s i ] ] } ] ; e 2 p l i s t = Append[ e 2 p l i s t, {( z )/ Cap omega, Abs[ exp Sin [ p s i ] + eyp Cos [ p s i ] ] } ] ; 194

225 e z p l i s t = Append[ e z p l i s t, {( z )/ Cap omega, Abs[ ezp ] } ] ; h 1 p l i s t = Append[ h 1 p l i s t, {( z )/ Cap omega, Abs[ hxp Cos [ p s i ] + hyp Sin [ p s i ] ] } ] ; h 2 p l i s t = Append[ h 2 p l i s t, {( z )/ Cap omega, Abs[ hxp Sin [ p s i ] + hyp Cos [ p s i ] ] } ] ; h z p l i s t = Append[ h z p l i s t, {( z )/ Cap omega, Abs[ hzp ] } ] ; p 1 p l i s t = Append[ p 1 p l i s t, {( z )/ Cap omega, pxp Cos [ p s i ] + pyp Sin [ p s i ] } ] ; p 2 p l i s t = Append[ p 2 p l i s t, {( z )/ Cap omega, pxp Sin [ p s i ] + pyp Cos [ p s i ] } ] ; p z p l i s t = Append[ p z p l i s t, {( z )/ Cap omega, pzp } ] ; CN = MatrixExp [ ( I P 2 Cap omega )/Ns ].CN ; ] ; e 1 m l i s t = {}; e 2 m l i s t = {}; e z m l i s t = {}; h1mlist = {}; h2mlist = {}; h z m l i s t = {}; p1mlist = {} p2mlist = {}; p z m l i s t = {}; For [ z = 0 Cap omega, z >= 0.41 Cap omega, z = z 0.01 Cap omega, exm = ( ap alphas Cos [ p s i ] / knot as Sin [ p s i ] ) Exp[ I alphas z ] ; eym = ( ap alphas Sin [ p s i ] / knot + as Cos [ p s i ] ) Exp[ I alphas z ] ; ezm = ap kappa Exp[ I alphas z ] / knot ; hxm = ( as alphas Cos [ p s i ] / knot + ap epsmet Sin [ p s i ] ) Exp[ I alphas z ] / ( etanot ) ; hym = ( as alphas Sin [ p s i ] / knot ap epsmet Cos [ p s i ] ) Exp[ I alphas z ] / ( etanot ) ; hzm = as kappa Exp[ I alphas z ] / ( knot etanot ) ; hxmc = Conjugate [ hxm ] ; hymc = Conjugate [ hym ] ; hzmc = Conjugate [ hzm ] ; pxm = Re[ eym hzmc ezm hymc ] / 2 ; pym = Re[ ezm hxmc exm hzmc ] / 2 ; pzm = Re[ exm hymc eym hxmc ] / 2 ; e1m = exm Cos [ p s i ] + eym Sin [ p s i ] ; 195

226 e2m = exm Sin [ p s i ] + eym Cos [ p s i ] ; h1m = hxm Cos [ p s i ] + hym Sin [ p s i ] ; h2m = hxm Sin [ p s i ] + hym Cos [ p s i ] ; p1m = pxm Cos [ p s i ] + pym Sin [ p s i ] ; p2m = pxm Sin [ p s i ] + pym Cos [ p s i ] ; e 1 m l i s t = Append[ e1mlist, {( z )/ Cap omega, Abs[ exm Cos [ p s i ] + eym Sin [ p s i ] ] } ] ; e 2 m l i s t = Append[ e2mlist, {( z )/ Cap omega, Abs[ exm Sin [ p s i ] + eym Cos [ p s i ] ] } ] ; e z m l i s t = Append[ ezmlist, {( z )/ Cap omega, Abs[ ezm ] } ] ; h1mlist = Append[ h1mlist, {( z )/ Cap omega, Abs[ hxm Cos [ p s i ] + hym Sin [ p s i ] ] } ] ; h2mlist = Append[ h2mlist, {( z )/ Cap omega, Abs[ hxm Sin [ p s i ] + hym Cos [ p s i ] ] } ] ; h z m list = Append[ hzmlist, {( z )/ Cap omega, Abs[ hzm ] } ] ; p1mlist = Append[ p1mlist, {( z )/ Cap omega, pxm Cos [ p s i ] + pym Sin [ p s i ] } ] ; p2mlist = Append[ p2mlist, {( z )/ Cap omega, pxm Sin [ p s i ] + pym Cos [ p s i ] } ] ; p z m list = Append[ pzmlist, {( z )/ Cap omega, pzm } ] ; ] ep = L i s t L i n e P l o t [ { e 1 p l i s t, e 2 p l i s t, e z p l i s t } ] ; hp = L i s t L i n e P l o t [ { h 1 p l i s t, h 2 p l i s t, h z p l i s t } ] ; pp = L i s t L i n e P l o t [ { p 1 p l i s t, p 2 p l i s t, p z p l i s t } ] ; em = L i s t L i n e P l o t [ { e1mlist, e2mlist, e z m l i s t } ] ; hm = L i s t L i n e P l o t [ { h1mlist, h2mlist, h zmlist } ] ; pm = L i s t L i n e P l o t [ { p1mlist, p2mlist, p zmlist } ] ; g = Grid [{{em, ep }, {hm, hp }, {pm, pp }}, Spacings > {0, 0 }] B.3 Newton-Raphson Method to Find κ in the Canonical Boundary-Value Problem of Ch. 5 munot = 4 Pi 10ˆ( 7); epsnot = ˆ( 12); lamnot = ˆ( 9); knot = (2 Pi )/ lamnot ; etanot = Sqrt [ munot/ epsnot ] ; omega = knot /Sqrt [ epsnot munot ] ; ns = I ; epsmet = 56 + I 21; 196

227 Cap omega = ˆ( 9); d e l t a v = (30 Pi ) / ; Chit = (45 Pi ) / ; Cap delta = 2 10ˆ( 9); Ns = 2 Cap omega/ Cap delta ; d e l t a k = 10ˆ( 6); ( Module to c a l c u l a t e determinent o f M ) getmdet [ kappa, s t a t e ] := Module[ { kappa = kappa, f l a g 2 = s t a t e }, f l a g = 0 ; While [ f l a g == 0, CNp = IdentityMatrix [ 4 ] ; CNn = IdentityMatrix [ 4 ] ; For [ n = 1, n <= Ns, n++, x1 = 2 n Cap omega/ns ; x2 = 2 (n 1) Cap omega/ns ; z = 0. 5 ( x1 + x2 ) ; Chivp = Chit + d e l t a v Sin [ ( Pi ( z ) ) / Cap omega ] ; Chivn = Chit d e l t a v Sin [ ( Pi ( z ) ) / Cap omega ] ; Chip = ArcTan[ Tan[ Chivp ] ] ; Chin = ArcTan[ Tan[ Chivn ] ] ; vp = (2 Chivp )/ Pi ; vn = (2 Chivn )/ Pi ; epsap = ( vp vp ˆ 2 ) ˆ 2 ; epsbp = ( vp vp ˆ 2 ) ˆ 2 ; epscp = ( vp vp ˆ 2 ) ˆ 2 ; epsdp = ( epsap epsbp ) / ( epsap (Cos [ Chip ] ) ˆ 2 + epsbp ( Sin [ Chip ] ) ˆ 2 ) ; epsan = ( vn vn ˆ 2 ) ˆ 2 ; epsbn = ( vn vn ˆ 2 ) ˆ 2 ; epscn = ( vn vn ˆ 2 ) ˆ 2 ; epsdn = ( epsan epsbn ) / ( epsan (Cos [ Chin ] ) ˆ 2 + epsbn ( Sin [ Chin ] ) ˆ 2 ) ; Ap = {{0, 0, 0, munot }, {0, 0, munot, 0}, { epsnot ( epscp epsdp ) Cos [ gammap] 197

228 Sin [ gammap ], epsnot ( epscp (Cos [ gammap ] ) ˆ 2 + epsdp ( Sin [ gammap ] ) ˆ 2 ), 0, 0}, { epsnot ( epscp ( Sin [ gammap] ) ˆ 2 + epsdp (Cos [ gammap ] ) ˆ 2 ), epsnot ( epscp epsdp ) Cos [ gammap ] Sin [ gammap ], 0, 0}}; Bp = {{Cos [ gammap ], Sin [ gammap ], 0, 0}, {0, 0, 0, 0}, {0, 0, 0, Sin [ gammap ] }, {0, 0, 0, Cos [ gammap ] } } ; Fp = {{0, 0, 0, ((kappa ˆ2 epsdp ) / ( omega epsnot epsap epsbp ) ) }, {0, 0, 0, 0}, {0, kappa ˆ2/( omega munot ), 0, 0}, {0, 0, 0, 0} } ; Pp = omega Ap + ( kappa epsdp ( epsap epsbp ) Sin [ Chip ] Cos [ Chip ] ) / ( epsap epsbp ) Bp + Fp ; An = {{0, 0, 0, munot },{0, 0, munot, 0}, { epsnot ( epscn epsdn ) Cos [ gamman] Sin [ gamman ], epsnot ( epscn (Cos [ gamman] ) ˆ 2 + epsdn ( Sin [ gamman ] ) ˆ 2 ), 0, 0}, { epsnot ( epscn ( Sin [ gamman] ) ˆ 2 + epsdn (Cos [ gamman ] ) ˆ 2 ), epsnot ( epscn epsdn ) Cos [ gamman ] Sin [ gamman ], 0, 0}}; Bn = {{Cos [ gamman ], Sin [ gamman ], 0, 0}, {0, 0, 0, 0}, {0, 0, 0, Sin [ gamman ] }, {0, 0, 0, Cos [ gamman ] } } ; Fn = {{0, 0, 0, ((kappa ˆ2 epsdn ) / ( omega epsnot epsan epsbn ) ) }, {0, 0, 0, 0}, {0, kappa ˆ2/( omega munot ), 0, 0}, {0, 0, 0, 0} } ; Pn = omega An + ( kappa epsdn ( epsan epsbn ) Sin [ Chin ] Cos [ Chin ] ) / ( epsan epsbn ) Bn + Fn ; CNp = MatrixExp [ ( I Pp 2 Cap omega )/Ns ]. CNp; CNn = MatrixExp[ (( I Pn 2 Cap omega )/Ns ) ]. CNn ; ] ; Mp = {{0, 0},{0, 0},{0, 0},{0, 0}}; Mn = {{0, 0},{0, 0},{0, 0},{0, 0}}; { eigvaluenp, eigvectornp } = Eigensystem [CNp ] ; eigvectornp = Transpose [ eigvectornp ] ; 198

229 { eigvaluenn, eigvectornn } = Eigensystem [CNn ] ; eigvectornn = Transpose [ eigvectornn ] ; eigvalueqp = I Log [ eigvaluenp ] / ( 2 Cap omega ) ; eigvalueqn = I Log [ eigvaluenn ] / ( 2 Cap omega ) ; n = 1 ; For [m = 1, m < 5, m++, I f [Im[ eigvalueqp [ [m] ] ] > 0, Mp[ [ All, n ] ] = eigvectornp [ [ All, m] ] ; n + + ] ; ] ; n = 1 ; For [m = 1, m < 5, m++, I f [Im[ eigvalueqn [ [m] ] ] < 0, Mn[ [ All, n ] ] = eigvectornn [ [ All, m] ] ; n + + ] ; ] ; M = Transpose [ Join [ Transpose [Mp], Transpose [Mn ] ] ] ; detm = Det [M] ; I f [ detm!= 0, f l a g = 1 ; Break [ ] ; ] ; I f [ detm == 0 && f l a g 2 == 1, kappa = knot + kappa ; f l a g = 0 ; ] ; I f [ detm == 0 && f l a g 2 == 2, kappa = knot + kappa ; f l a g = 0 ; ] ; ] ; detm ] ( Newton Raphson Method ) gammap = (0 Pi ) / ; gamman = (90 Pi ) / ; y = 1 ; mold = 1 ; t o l = 10ˆ( 14); kappa = N[ ( ) knot ] While [Abs[ mold ] > tol, Print [ S t y l e [ "Iteration No = ", 18, Red], y ] ; y = y + 1 ; kappaold = (1) kappa ; mold = getmdet [ kappaold, 1 ] ; kappanew = (1 + d e l t a k ) kappa ; mol = getmdet [ kappanew, 2 ] ; 199

230 qold = ( mol mold ) / ( kappanew kappaold ) ; kappa = Re[ kappa mold/ qold ] ; Print [ " Det M old=", mold ] ; Print [ "kappa=", kappa/ knot ] ; I f [ y == 50, Quit [ ] ; ] ; ] B.4 Newton-Raphson Method to Find κ in the Canonical Boundary-Value Problem of Chs. 6 and 7 munot = 4 Pi 10ˆ( 7); epsnot = ˆ( 12); lamnot = ˆ( 9); knot = (2 Pi )/ lamnot ; etanot = Sqrt [ munot/ epsnot ] ; omega = knot /Sqrt [ epsnot munot ] ; ns = I ; epsmet = 56 + I 21; Cap omega = ˆ( 9); d e l t a v = (30 Pi ) / ; Chit = (45 Pi ) / ; Cap delta = 2 10ˆ( 9); Ns = 2 Cap omega/ Cap delta ; d e l t a k = 10ˆ( 6); ( Module to c a l c u l a t e determinent o f M ) getmdet [ kappa, s t a t e ] := Module[ { kappa = kappa, f l a g 2 = s t a t e }, f l a g = 0 ; While [ f l a g == 0, CNp = IdentityMatrix [ 4 ] ; CNn = IdentityMatrix [ 4 ] ; For [ n = 1, n <= Ns, n++, x1 = 2 n Cap omega/ns ; x2 = 2 (n 1) Cap omega/ns ; z = 0. 5 ( x1 + x2 ) ; Chivp = Chit + d e l t a v Sin [ ( Pi (Lp + z Lp ) ) / Cap omega ] ; Chivn =Chit d e l t a v Sin [ ( Pi (Ln z Ln ) ) / Cap omega ] ; Chip = ArcTan[ Tan[ Chivp ] ] ; Chin = ArcTan[ Tan[ Chivn ] ] ; vp = (2 Chivp )/ Pi ; 200

231 vn = (2 Chivn )/ Pi ; epsap = ( vp vp ˆ 2 ) ˆ 2 ; epsbp = ( vp vp ˆ 2 ) ˆ 2 ; epscp = ( vp vp ˆ 2 ) ˆ 2 ; epsdp = ( epsap epsbp ) / ( epsap (Cos [ Chip ] ) ˆ 2 + epsbp ( Sin [ Chip ] ) ˆ 2 ) ; epsan = ( vn vn ˆ 2 ) ˆ 2 ; epsbn = ( vn vn ˆ 2 ) ˆ 2 ; epscn = ( vn vn ˆ 2 ) ˆ 2 ; epsdn = ( epsan epsbn ) / ( epsan (Cos [ Chin ] ) ˆ 2 + epsbn ( Sin [ Chin ] ) ˆ 2 ) ; Ap = {{0, 0, 0, munot }, {0, 0, munot, 0}, { epsnot ( epscp epsdp ) Cos [ gammap] Sin [ gammap ], epsnot ( epscp (Cos [ gammap] ) ˆ 2 + epsdp ( Sin [ gammap ] ) ˆ 2 ), 0, 0}, { epsnot ( epscp ( Sin [ gammap] ) ˆ 2 + epsdp (Cos [ gammap ] ) ˆ 2 ), epsnot ( epscp epsdp ) Cos [ gammap ] Sin [ gammap ], 0, 0}}; Bp = {{Cos [ gammap ], Sin [ gammap ], 0, 0}, {0, 0, 0, 0}, {0, 0, 0, Sin [ gammap ] }, {0, 0, 0, Cos [ gammap] } } ; Fp = { {0, 0, 0, ((kappa ˆ2 epsdp ) / ( omega epsnot epsap epsbp ) ) }, {0, 0, 0, 0}, {0, kappa ˆ2/( omega munot ), 0, 0}, {0, 0, 0, 0}}; Pp = omega Ap + ( kappa epsdp ( epsap epsbp ) Sin [ Chip ] Cos [ Chip ] ) / ( epsap epsbp ) Bp + Fp ; An = {{0, 0, 0, munot }, {0, 0, munot, 0}, { epsnot ( epscn epsdn ) Cos [ gamman] Sin [ gamman ], epsnot ( epscn (Cos [ gamman] ) ˆ 2 + epsdn ( Sin [ gamman ] ) ˆ 2 ), 0, 0 }, { epsnot ( epscn ( Sin [ gamman] ) ˆ 2 + epsdn (Cos [ gamman ] ) ˆ 2 ), epsnot ( epscn epsdn ) Cos [ gamman ] Sin [ gamman ], 0, 0 } } ; Bn = { {Cos [ gamman ], Sin [ gamman ], 0, 0}, {0, 0, 0, 0}, {0, 0, 0, Sin [ gamman ] }, {0, 0, 0, Cos [ gamman ] } } ; 201

232 Fn = {{0, 0, 0, ((kappa ˆ2 epsdn ) / ( omega epsnot epsan epsbn ) ) }, {0, 0, 0, 0}, {0, kappa ˆ2/( omega munot ), 0, 0}, {0, 0, 0, 0}}; Pn = omega An + ( kappa epsdn ( epsan epsbn ) Sin [ Chin ] Cos [ Chin ] ) / ( epsan epsbn ) Bn + Fn ; CNp = MatrixExp [ ( I Pp 2 Cap omega )/Ns ]. CNp; CNn = MatrixExp[ (( I Pn 2 Cap omega )/Ns ) ]. CNn ; ] ; Amet = {{0, 0, 0, munot }, {0, 0, munot, 0}, {0, epsnot epsmet, 0, 0}, { epsnot epsmet, 0, 0, 0}}; Bmet = {{0, 0, 0, (kappa ˆ2/( omega epsnot epsmet ) ) }, {0, 0, 0, 0},{0, kappa ˆ2/( omega munot ), 0, 0}, {0, 0, 0, 0}}; Pmet = omega Amet + Bmet ; Mp = {{0, 0},{0, 0},{0, 0},{0, 0}}; Mn ={{0, 0},{0, 0},{0, 0},{0, 0}}; { eigvaluenp, eigvectornp } = Eigensystem [CNp ] ; eigvectornp = Transpose [ eigvectornp ] ; { eigvaluenn, eigvectornn } = Eigensystem [CNn ] ; eigvectornn = Transpose [ eigvectornn ] ; eigvalueqp = I Log [ eigvaluenp ] / ( 2 Cap omega ) ; eigvalueqn = I Log [ eigvaluenn ] / ( 2 Cap omega ) ; n = 1 ; For [m = 1, m < 5, m++, I f [Im[ eigvalueqp [ [m] ] ] > 0, Mp[ [ All, n ] ] = eigvectornp [ [ All, m] ] ; n + + ] ; ] ; n = 1 ; For [m = 1, m < 5, m++, I f [Im[ eigvalueqn [ [m] ] ] < 0, Mn[ [ All, n ] ] = eigvectornn [ [ All, m] ] ; n + + ] ; ] ; M = Transpose [ Join [ Transpose [Mp], Transpose [ MatrixExp [ I Pmet (Lp Ln ) ].Mn ] ] ] ; 202

233 detm = Det [M] ; I f [ detm!= 0, f l a g = 1 ; Break [ ] ; ] ; I f [ detm == 0 && f l a g 2 == 1, kappa = knot + kappa ; f l a g = 0 ; ] ; I f [ detm == 0 && f l a g 2 == 2, kappa = knot + kappa ; f l a g = 0 ; ] ; ] ; detm ] ( Newton Raphson Method ) gammap = (0 Pi ) / ; gamman = (0 Pi ) / ; Lp = ˆ( 9); Ln = ˆ( 9); y = 1 ; mold = 1 ; t o l = 10ˆ( 14); kappa = N[ ( ) knot ] While [Abs[Re[ mold ] ] > t o l Abs[Im[ mold ] ] > tol, Print [ S t y l e [ "Iteration No = ", 18, Red], y ] ; y = y + 1 ; kappaold = (1) kappa ; mold = getmdet [ kappaold, 1 ] ; kappanew = (1 + d e l t a k ) kappa ; mol = getmdet [ kappanew, 2 ] ; qold = ( mol mold ) / ( kappanew kappaold ) ; kappa = kappa mold/ qold ; Print [ " Det M old=", mold ] ; Print [ "kappa=", kappa/ knot ] ; I f [ y == 50, Quit [ ] ; ] ; ] B.5 A p vs. θ in the TKR Configuration of Ch. 8 munot = 4 Pi 10ˆ( 7); epsnot = ˆ( 12); lamnot = ˆ( 9); knot = N[ ( 2 Pi )/ lamnot ] ; etanot = Sqrt [ munot/ epsnot ] ; omega = knot /Sqrt [ epsnot munot ] ; nss = I 3. 9 ; epm = nss ˆ 2 ; 203

234 nl = ; na = ; nb = ; NA = ( na + nb )/2 NB = ( nb na ) / 2 ; Np = 3 ; HP = 1.5 lamnot ; Ld = 2 HP Np; Ns = 2 HP/(2 10ˆ( 9)); Print [ Ns ] ; Ns = 950; tn = 3 0 ; tm = 6 5 ; Lm = 30 10ˆ( 9); a p l i s t = {}; For [NN = tn, NN < tm, NN = NN , theta = NN Pi /180; kappa = knot nl Sin [ theta ] ; Z [ Ns + 1 ] = {{ Cos [ theta ] }, { nl / etanot }}; For [ n = Ns + 1, n >= 2, n, x1 = Lm + (Ld ) ( n 1)/Ns ; x2 = Lm + (Ld ) ( n 2)/Ns ; z = 0. 5 ( x1 + x2 ) ; epd = (NA + NB Sin [ Pi ( z Lm)/HP] ) ˆ 2 ; Pd = {{0, omega munot kappa ˆ2/( omega epsnot epd )}, {omega epsnot epd, 0}}; {DN1, GN11} = Eigensystem [ Pd ] ; GN1 = Transpose [ GN11 ] ; I f [Im[DN1 [ [ 1 ] ] ] < Im[DN1 [ [ 2 ] ] ], a1 = DN1 [ [ 1 ] ] ; b1 = GN1 [ [ All, 1 ] ] ; DN1 [ [ 1 ] ] = DN1 [ [ 2 ] ] ; DN1 [ [ 2 ] ] = a1 ; GN1 [ [ All, 1 ] ] = GN1 [ [ All, 2 ] ] ; GN1 [ [ All, 2 ] ] = b1 ; ] ; W = Inverse [GN1 ]. Z [ n ] ; WU[ n ] = W[ [ 1, 1 ] ] ; WL[ n ] = W[ [ 2, 1 ] ] ; DM = DiagonalMatrix [DN1 ] ; 204

235 DU[ n ] = DM[ [ 1, 1 ] ] ; DL[ n ] = DM[ [ 2, 2 ] ] ; Lower1 = Exp[ I Ld DL[ n ] / Ns ] WL[ n ] Exp[ I Ld DU[ n ] / Ns ] /WU[ n ] ; Dummy1 = {{1}, {Lower1 }}; Z [ n 1 ] = GN1.Dummy1 ; ] ; Pm = {{0, omega munot kappa ˆ2/( omega epsnot epm)}, {omega epsnot epm, 0}}; {DN3, GN33} = Eigensystem [Pm] ; GN3 = Transpose [ GN33 ] ; I f [Im[DN3 [ [ 1 ] ] ] < Im[DN3 [ [ 2 ] ] ], a3 = DN3 [ [ 1 ] ] ; b3 = GN3 [ [ All, 1 ] ] ; DN3 [ [ 1 ] ] = DN3 [ [ 2 ] ] ; DN3 [ [ 2 ] ] = a3 ; GN3 [ [ All, 1 ] ] = GN3 [ [ All, 2 ] ] ; GN3 [ [ All, 2 ] ] = b3 ; ] ; W = Inverse [GN3 ]. Z [ 1 ] ; WU[ 1 ] = W[ [ 1, 1 ] ] ; WL[ 1 ] = W[ [ 2, 1 ] ] ; DM = DiagonalMatrix [DN3 ] ; DU[ 1 ] = DM[ [ 1, 1 ] ] ; DL[ 1 ] = DM[ [ 2, 2 ] ] ; Lower2 = Exp[ I Lm DL [ 1 ] ] WL[ 1 ] Exp[ I Lm DU[ 1 ] ] /WU[ 1 ] ; Dummy2 = {{1}, {Lower2 }}; Z [ 0 ] = GN3.Dummy2; dummy60 = {{Z [ 0 ] [ [ 1, 1 ] ], Cos [ theta ] }, {Z [ 0 ] [ [ 2, 1 ] ], nl / etanot }}; T0RP = Inverse [ dummy60].{{ Cos [ theta ] }, { nl / etanot }}; T0P = T0RP [ [ 1, 1 ] ] ; RP = T0RP [ [ 2, 1 ] ] ; TP[ 0 ] = T0P ; TP[ 1 ] = Exp[ I Lm DU[ 1 ] ] TP[ 0 ] /WU[ 1 ] ; Clear [ i ] ; For [ i = 2, i <= Ns + 1, i ++, TP[ i ] = Exp[ I Ld DU[ i ] / Ns ] TP[ i 1 ] /WU[ i ] ; ] ; rpp = Abs[RP] ˆ 2 ; PAB = 1 Abs[TP[ Ns + 1 ] ] ˆ 2 Abs[RP] ˆ 2 ; 205

236 a p l i s t = Append[ a p l i s t, {NN, PAB} ] ; ] Needs [ "PlotLegends " ] ; applot = L i s t L i n e P l o t [ { a p l i s t } ] a p l i s t // MatrixForm B.6 A p vs. θ in the Grating-Coupled Configuration of Chs. 9 and 10 munot = 4 Pi 10ˆ( 7); epsnot = ˆ( 12); lamnot = ˆ( 9); knot = N[ ( 2 Pi )/ lamnot ] ; etanot = Sqrt [ munot/ epsnot ] ; omega = knot /Sqrt [ epsnot munot ] ; nss = I ; epm = nss ˆ 2 ; L = lamnot ; d1 = 4 lamnot ; d2 = d ˆ( 9); d3 = d ˆ( 9); L1 = 0. 5 L ; na = ; nb = ; NA = ( na + nb )/2 NB = ( nb na ) / 2 ; HP = lamnot ; a p l i s t = {}; Nt = 8 ; Nd = d1 / 2 ; Ng = 5 0 ; Ns = Nd + Ng ; ( begin Module e p s i l o n ( z, n ) ) epzn [ zz, nn ] := Module[ { z = zz, n = nn, y, A, B, CD}, A = epm + epd ; B = epm epd ; CD = d2 d1 ; y = L1 ArcSin [ ( d2 z )/CD] / Pi ; epd = (NA + NB Sin [ Pi ( d2 z )/HP] ) ˆ 2 ; I f [ n == 0, I f [ z <= d1, 206

237 e p s i l o n = epd ; ] ; I f [ z > d1 && z < d2, e p s i l o n = ( epd (L L1 + 2 y ) + epm (L1 2 y ) ) /L ; ] ; I f [ z >= d2, e p s i l o n = epm ; ] ; ] ; I f [ n!= 0, I f [ z <= d1, e p s i l o n = 0 ; ] ; I f [ z > d1 && z < d2, e p s i l o n = (B Exp[ I n (L1 y ) 2 Pi/L ] B Exp[ I n y 2 Pi/L ] ) I /(n 2 Pi ) ; ] ; I f [ z >= d2, e p s i l o n = 0 ; ] ; ] ; e p s i l o n ] ( end Module e p s i l o n ( z, n) ) For [NN = 1, NN <= 70, NN = NN , theta = NN Pi /180; Print [ "NN = ", NN] ; ( begin wavenumber c a l c u l a t i o n ) Block [ { n, kxnn, kxn }, kxn = {}; For [ n = Nt, n <= Nt, n = n + 1, kxnn = knot Sin [ theta ] + n 2 Pi/L ; kxn = Append[ kxn, N[ kxnn ] ] ; ] ; KX = DiagonalMatrix [ kxn ] ; ] ; Block [ { n, y, kxnn }, kzn = {}; y = 0 ; For [ n = Nt, n <= Nt, n = n + 1, y = y + 1 ; kxnn = knot Sin [ theta ] + n 2 Pi/L ; I f [ knot ˆ2 >= kxnn ˆ2, kzn = Append[ kzn, {y, N[ Sqrt [ knot ˆ2 kxnn ˆ 2 ] ] } ] ; ] ; I f [ knot ˆ2 < kxnn ˆ2, kzn = Append[ kzn, {y, I N[ Sqrt[ knot ˆ2 + kxnn ˆ 2 ] ] } ] ; ] ; ] ; ] ; 207

238 ( end wavenumber c a l c u l a t i o n ) NMat = DiagonalMatrix [ ConstantArray [ 0, 2 Nt + 1 ] ] ; Block [ { i1, j1, i2, j2, i3, j3, i4, j 4 }, Ype = Table [Which[ i 1 == j1, kzn [ [ i1, 2 ] ] / knot,! ( i 1 == j 1 ), 0 ], { i1,2 Nt + 1}, { j1, 2 Nt + 1 } ] ; Yme = Table [Which[ i 2 == j2, kzn [ [ i2, 2 ] ] / knot,! ( i 2 == j 2 ), 0 ], { i2,2 Nt + 1}, { j2, 2 Nt + 1 } ] ; Yph = Table [Which[ i 3 == j3, 1,! i 3 == j3, 0 ], { i3, 2 Nt + 1}, { j3,2 Nt + 1 } ] ; Ymh = Table [Which[ i 4 == j4, 1,! i 4 == j4, 0 ], { i4, 2 Nt + 1}, { j4,2 Nt + 1 } ] ; YP = ArrayFlatten [{{ Ype}, {Yph } } ] ; YM = ArrayFlatten [{{Yme}, {Ymh} } ] ; ] ; AP = Normal [ SparseArray [{{ Nt + 1, 1} > 1, {2 Nt + 1, 1} > 0 } ] ] ; ( STABLE method begins ) Z [ Ns + 1 ] = YP; Block [ { P3, i, j, p, q, a3, b3, EPZ3, GN33}, EPZ3 = Table [ I f [ i == j, epm, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; P143 = knot IdentityMatrix [ 2 Nt + 1 ] (1/ knot ) KX. Inverse [ EPZ3 ].KX; P233 = knot IdentityMatrix [2 Nt + 1 ] ; P323 = (1/ knot ) KX.KX knot EPZ3 ; P413 = knot EPZ3 ; P3 = ArrayFlatten [{{NMat, P143 }, {P413, NMat } } ] ; {DN3, GN33} = Eigensystem [ P3 ] ; GN3 = Transpose [ GN33 ] ; For [ p = 1, p < 2 (2 Nt + 1), p = p + 1, For [ q = 1, q <= 2 (2 Nt + 1) p, q = q + 1, I f [Im[DN3 [ [ q ] ] ] < Im[DN3 [ [ q + 1 ] ] ], a3 = DN3 [ [ q ] ] ; b3 = GN3 [ [ All, q ] ] ; DN3 [ [ q ] ] = DN3 [ [ q + 1 ] ] ; DN3 [ [ q + 1 ] ] = a3 ; GN3 [ [ All, q ] ] = GN3 [ [ All, q + 1 ] ] ; GN3 [ [ All, q + 1 ] ] = b3 ; 208

239 ] ; ] ; ] ; ] ; W = Inverse [GN3 ]. Z [ Ns + 1 ] ; WU[ Ns + 1 ] = W[ [ 1 ; ; 2 Nt + 1, All ] ] ; WL[ Ns + 1 ] = W[ [ 2 Nt + 2 ; ; (4 Nt + 2), All ] ] ; DM = DiagonalMatrix [DN3 ] ; DU[ Ns + 1 ] = DM[ [ 1 ; ; 2 Nt + 1, 1 ; ; 2 Nt + 1 ] ] ; DL[ Ns + 1 ] = DM[ [ 2 Nt + 2 ; ; (4 Nt + 2), 2 Nt + 2 ; ; (4 Nt + 2 ) ] ] ; Lower2 = MatrixExp[ I ( d3 d2 ) DL[ Ns + 1 ] ].WL[ Ns + 1 ]. Inverse [WU[ Ns + 1 ] ]. MatrixExp [ I ( d3 d2 ) DU[ Ns + 1 ] ] ; Dummy2 = ArrayFlatten [{{ IdentityMatrix [2 Nt + 1 ] }, {Lower2 } } ] ; Z [ Ns ] = GN3.Dummy2; For [ n = Ns, n >= Nd + 1, n, x1 = d1 ( d1 d2 ) ( n Nd)/Ng ; x2 = d1 ( d1 d2 ) ( n 1 Nd)/Ng ; z = 0. 5 ( x1 + x2 ) ; e l i s t = {}; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t = Append[ e l i s t, epzn [ z, ind ] ] ; ] ; EPZ2 = Table [ e l i s t [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; P142 = knot IdentityMatrix [ 2 Nt + 1 ] (1/ knot ) KX. Inverse [ EPZ2 ].KX; P412 = knot EPZ2 ; P2 = ArrayFlatten [{{NMat, P142 }, {P412, NMat } } ] ; {DN2, GN22} = Eigensystem [ P2 ] ; GN2 = Transpose [ GN22 ] ; For [ p = 1, p < 2 (2 Nt + 1), p = p + 1, For [ q = 1, q <= 2 (2 Nt + 1) p, q = q + 1, I f [Im[DN2 [ [ q ] ] ] < Im[DN2 [ [ q + 1 ] ] ], a = DN2 [ [ q ] ] ; b = GN2 [ [ All, q ] ] ; DN2 [ [ q ] ] = DN2 [ [ q + 1 ] ] ; DN2 [ [ q + 1 ] ] = a ; GN2 [ [ All, q ] ] = GN2 [ [ All, q + 1 ] ] ; GN2 [ [ All, q + 1 ] ] = b ; ] ; ] ; ] ; W = Inverse [GN2 ]. Z [ n ] ; 209

240 WU[ n ] = W[ [ 1 ; ; 2 Nt + 1, All ] ] ; WL[ n ] = W[ [ 2 Nt + 2 ; ; (4 Nt + 2), All ] ] ; DM = DiagonalMatrix [DN2 ] ; DU[ n ] = DM[ [ 1 ; ; 2 Nt + 1, 1 ; ; 2 Nt + 1 ] ] ; DL[ n ] = DM[ [ 2 Nt + 2 ; ; (4 Nt + 2), 2 Nt + 2 ; ; (4 Nt + 2 ) ] ] ; Lower2 = MatrixExp[ I ( d2 d1 ) DL[ n ] /Ng ].WL[ n ]. Inverse [WU[ n ] ]. MatrixExp [ I ( d2 d1 ) DU[ n ] /Ng ] ; Dummy2 = ArrayFlatten [{{ IdentityMatrix [2 Nt + 1 ] }, {Lower2 } } ] ; Z [ n 1 ] = GN2.Dummy2 ; ] ; Clear [ n, x1, x2, z, a, b, p, q ] ; For [ n = Nd, n >= 1, n, x1 = ( d1 ) n/nd; x2 = ( d1 ) ( n 1)/Nd; z = 0. 5 ( x1 + x2 ) ; epd = (NA + NB Sin [ Pi ( d2 z )/HP] ) ˆ 2 ; EPZ1 = Table [ I f [ i == j, epd, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; P141 = knot IdentityMatrix [ 2 Nt + 1 ] (1/ knot ) KX. Inverse [ EPZ1 ].KX; P411 = knot EPZ1 ; P1 = ArrayFlatten [{{NMat, P141 }, {P411, NMat } } ] ; {DN1, GN11} = Eigensystem [ P1 ] ; GN1 = Transpose [ GN11 ] ; For [ p = 1, p < 2 (2 Nt + 1), p = p + 1, For [ q = 1, q <= 2 (2 Nt + 1) p, q = q + 1, I f [Im[DN1 [ [ q ] ] ] < Im[DN1 [ [ q + 1 ] ] ], a = DN1 [ [ q ] ] ; b = GN1 [ [ All, q ] ] ; DN1 [ [ q ] ] = DN1 [ [ q + 1 ] ] ; DN1 [ [ q + 1 ] ] = a ; GN1 [ [ All, q ] ] = GN1 [ [ All, q + 1 ] ] ; GN1 [ [ All, q + 1 ] ] = b ; ] ; ] ; ] ; W = Inverse [GN1 ]. Z [ n ] ; WU[ n ] = W[ [ 1 ; ; 2 Nt + 1, All ] ] ; WL[ n ] = W[ [ 2 Nt + 2 ; ; (4 Nt + 2), All ] ] ; DM = DiagonalMatrix [DN1 ] ; DU[ n ] = DM[ [ 1 ; ; 2 Nt + 1, 1 ; ; 2 Nt + 1 ] ] ; DL[ n ] = DM[ [ 2 Nt + 2 ; ; (4 Nt + 2), 2 Nt + 2 ; ; (4 Nt + 2 ) ] ] ; 210

241 Lower1 = MatrixExp[ I ( d1 ) DL[ n ] /Nd ].WL[ n ]. Inverse [WU[ n ] ]. MatrixExp [ I ( d1 ) DU[ n ] /Nd ] ; Dummy1 = ArrayFlatten [{{ IdentityMatrix [2 Nt + 1 ] }, {Lower1 } } ] ; Z [ n 1 ] = GN1.Dummy1 ; ] ; dummy60 = ArrayFlatten [{{Z [ 0 ], YM} } ] ; T0RP = Inverse [ dummy60 ]. YP.AP; RP = T0RP[ [ 2 Nt + 2 ; ; (4 Nt + 2), All ] ] ; T0P = T0RP [ [ 1 ; ; 2 Nt + 1, All ] ] ; TP[ 0 ] = T0P ; Clear [ i ] ; For [ i = 1, i <= Nd, i ++, TP[ i ] = Inverse [WU[ i ] ]. MatrixExp [ I ( d1 ) DU[ i ] /Nd ]. TP[ i 1 ] ; ] ; Clear [ i ] ; For [ i = Nd + 1, i <= Ns, i ++, TP[ i ] = Inverse [WU[ i ] ]. MatrixExp [ I ( d2 d1 ) DU[ i ] /Ng ].TP[ i 1 ] ; ] ; TP[ Ns + 1 ] = Inverse [WU[ Ns + 1 ] ]. MatrixExp [ I ( d3 d2 ) DU[ Ns + 1 ] ]. TP[ Ns ] ; ( STABLE method ends ) RP2 = Abs[RP] ˆ 2 ; TP2 = Abs[TP[ Ns + 1 ] ] ˆ 2 ; Block [ { n, kxnn, kzr }, kzr = {}; For [ n = Nt, n <= Nt, n = n + 1, kxnn = knot Sin [ theta ] + n 2 Pi/L ; I f [ knot ˆ2 >= kxnn ˆ2, kzr = Append[ kzr, (N[ Sqrt [ knot ˆ2 kxnn ˆ 2 ] ] ) / ( knot Cos [ theta ] ) ] ; ] ; I f [ knot ˆ2 < kxnn ˆ2, kzr = Append[ kzr, 0 ] ; ] ; ] ; RKZ = DiagonalMatrix [ kzr ] ; ] ; ( p i n c i d e n t, p r e f l e c t e d and transmitted ) RPp = Transpose [ Transpose [ RP2 ]. RKZ ] ; TPp = Transpose [ Transpose [ TP2 ]. RKZ ] ; PRT = RPp + TPp; PAB = 1 Total [PRT ] ; 211

242 a p l i s t = Append[ a p l i s t, {NN, PAB [ [ 1 ] ] } ] ; ] L i s t L i n e P l o t [ a p l i s t ] Print [ a p l i s t // MatrixForm ] ; B.7 A p and A s vs. θ in the Grating-Coupled Configuration of Ch. 11 munot = 4 Pi 10ˆ( 7); epsnot = ˆ( 12); lamnot = ˆ( 9); knot = N[ ( 2 Pi )/ lamnot ] ; etanot = Sqrt [ munot/ epsnot ] ; omega = knot /Sqrt [ epsnot munot ] ; nss = I ; epm = nss ˆ 2 ; Cap omega = ˆ( 9); d e l t a v = (30 Pi ) / ; Chit = (45 Pi ) / ; gamma = 40 Pi /180; L = 1.7 Cap omega ; d1 = 7 Cap omega ; d2 = d ˆ( 9); d3 = d ˆ( 9); L1 = 0.5 L ; a p l i s t = {}; a s l i s t = {}; Nt = 8 ; Nd = d1 10ˆ9/2; Ng = 5 0 ; Ns = Nd + Ng ; Sz = {{Cos [ gamma ], Sin [ gamma ], 0}, {Sin [ gamma ], Cos [ gamma ], 0}, {0, 0,1}}; ( begin Module e p s i l o n ( z, n ) ) epzn [ epdd, zz, nn ] := Module[ { epd = epdd, z = zz, n = nn, y, A, B, CD}, A = epm + epd ; B = epm epd ; CD = d2 d1 ; y = L1 ArcSin [ ( d2 z )/CD] / Pi ; 212

243 I f [ n == 0, I f [ z <= d1, e p s i l o n = epd ; ] ; I f [ z > d1 && z < d2, e p s i l o n = ( epd (L L1 + 2 y ) + epm (L1 2 y ) ) /L ; ] ; I f [ z >= d2, e p s i l o n = epm ; ] ; ] ; I f [ n!= 0, I f [ z <= d1, e p s i l o n = 0 ; ] ; I f [ z > d1 && z < d2, e p s i l o n = (B Exp[ I n (L1 y ) 2 Pi/L ] B Exp[ I n y 2 Pi/L ] ) I /(n 2 Pi ) ; ] ; I f [ z >= d2, e p s i l o n = 0 ; ] ; ] ; e p s i l o n ] epzncross [ epdd, zz, nn ] := Module[ { epd = epdd, z = zz, n = nn, y, A, B, CD}, A = epd ; B = epd ; CD = d2 d1 ; y = L1 ArcSin [ ( d2 z )/CD] / Pi ; I f [ n == 0, I f [ z <= d1, e p s i l o n = epd ; ] ; I f [ z > d1 && z < d2, e p s i l o n = epd (L L1 + 2 y )/L ; ] ; I f [ z >= d2, e p s i l o n = 0 ; ] ; ] ; I f [ n!= 0, I f [ z <= d1, e p s i l o n = 0 ; ] ; I f [ z > d1 && z < d2, e p s i l o n = (B Exp[ I n (L1 y ) 2 Pi/L ] B Exp[ I n y 2 Pi/L ] ) I /(n 2 Pi ) ; ] ; 213

244 I f [ z >= d2, e p s i l o n = 0 ; ] ; ] ; e p s i l o n ] ( end Module e p s i l o n ( z, n) ) For [NN = 5, NN <= 20, NN = NN , theta = NN Pi /180; Print [ "NN = ", NN] ; ( begin wavenumber c a l c u l a t i o n ) Block [ { n, kxnn, kxn }, kxn = {}; For [ n = Nt, n <= Nt, n = n + 1, kxnn = knot Sin [ theta ] + n 2 Pi/L ; kxn = Append[ kxn, N[ kxnn ] ] ; ] ; KX = DiagonalMatrix [ kxn ] ; ] ; Block [ { n, y, kxnn }, kzn = {}; y = 0 ; For [ n = Nt, n <= Nt, n = n + 1, y = y + 1 ; kxnn = knot Sin [ theta ] + n 2 Pi/L ; I f [ knot ˆ2 >= kxnn ˆ2, kzn = Append[ kzn, {y, N[ Sqrt [ knot ˆ2 kxnn ˆ 2 ] ] } ] ; ] ; I f [ knot ˆ2 < kxnn ˆ2, kzn = Append[ kzn, {y, I N[ Sqrt[ knot ˆ2 + kxnn ˆ 2 ] ] } ] ; ] ; ] ; ] ; ( end wavenumber c a l c u l a t i o n ) NMat = DiagonalMatrix [ ConstantArray [ 0, 2 Nt + 1 ] ] ; Block [ { i1, j1, i2, j2, i3, j3, i4, j 4 }, Ype = Table [Which [ ( i 1 == j Nt + 1), 1, i 1 == j 1 2 Nt 1, kzn [ [ i1, 2 ] ] / knot,! ( ( ( i 1 == j Nt + 1) i 1 == j 1 2 Nt 1 ) ), 0 ], { i1, 4 Nt + 2}, { j1, 4 Nt + 2 } ] ; Yme = Table [Which [ ( i 2 == j Nt + 1), 1, i 2 == j 2 2 Nt 1, kzn [ [ i2, 2 ] ] / knot,! ( ( ( i 2 == j Nt + 1) i 2 == j 2 2 Nt 1 ) ), 0 ], { i2, 4 Nt + 2}, { j2, 4 Nt + 2 } ] ; Yph = Table [Which [ ( i 3 == j 3 && i 3 <= 2 Nt + 1), 214

245 kzn [ [ i3, 2 ] ] / knot, i 3 == j 3 && i 3 > 2 Nt + 1, 1,! i 3 == j3, 0 ], { i3,4 Nt + 2}, { j3, 4 Nt + 2 } ] ; Ymh = Table [Which [ ( i 4 == j 4 && i 4 <= 2 Nt + 1), kzn [ [ i4, 2 ] ] / knot, i 4 == j 4 && i 4 > 2 Nt + 1, 1,! i 4 == j4, 0 ], { i4,4 Nt + 2}, { j4, 4 Nt + 2 } ] ; YP = ArrayFlatten [{{ Ype}, {Yph } } ] ; YM = ArrayFlatten [{{Yme}, {Ymh} } ] ; ] ; ( Y matrixes ends ) AS = Normal [ SparseArray [{{ Nt + 1, 1} > 1, {4 Nt + 2, 1} > 0 } ] ] ; AP = Normal [ SparseArray [{{3 Nt + 2, 1} > 1, {4 Nt + 2, 1} > 0 } ] ] ; Z [ Ns + 1 ] = YP; Block [ { P3, i, j, p, q, a3, b3, EPZ3, GN33}, EPZ3 = Table [ I f [ i == j, epm, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; P143 = knot IdentityMatrix [ 2 Nt + 1 ] (1/ knot ) KX. Inverse [ EPZ3 ].KX; P233 = knot IdentityMatrix [2 Nt + 1 ] ; P323 = (1/ knot ) KX.KX knot EPZ3 ; P413 = knot EPZ3 ; P3 = ArrayFlatten [{{NMat, NMat, NMat, P143 }, {NMat, NMat, P233, NMat}, {NMat, P323, NMat, NMat}, {P413, NMat, NMat, NMat } } ] ; {DN3, GN33} = Eigensystem [ P3 ] ; GN3 = Transpose [ GN33 ] ; For [ p = 1, p < 4 (2 Nt + 1), p = p + 1, For [ q = 1, q <= 4 (2 Nt + 1) p, q = q + 1, I f [Im[DN3 [ [ q ] ] ] < Im[DN3 [ [ q + 1 ] ] ], a3 = DN3 [ [ q ] ] ; b3 = GN3 [ [ All, q ] ] ; DN3 [ [ q ] ] = DN3 [ [ q + 1 ] ] ; DN3 [ [ q + 1 ] ] = a3 ; GN3 [ [ All, q ] ] = GN3 [ [ All, q + 1 ] ] ; GN3 [ [ All, q + 1 ] ] = b3 ; ] ; ] ; ] ; ] ; 215

246 W = Inverse [GN3 ]. Z [ Ns + 1 ] ; WU[ Ns + 1 ] = W[ [ 1 ; ; 4 Nt + 2, All ] ] ; WL[ Ns + 1 ] = W[ [ 4 Nt + 3 ; ; 2 (4 Nt + 2), All ] ] ; DM = DiagonalMatrix [DN3 ] ; DU[ Ns + 1 ] = DM[ [ 1 ; ; 4 Nt + 2, 1 ; ; 4 Nt + 2 ] ] ; DL[ Ns + 1 ] = DM[ [ 4 Nt + 3 ; ; 2 (4 Nt + 2), 4 Nt + 3 ; ; 2 (4 Nt + 2 ) ] ] ; Lower2 = MatrixExp[ I ( d3 d2 ) DL[ Ns + 1 ] ].WL[ Ns + 1 ]. Inverse [WU[ Ns + 1 ] ]. MatrixExp [ I ( d3 d2 ) DU[ Ns + 1 ] ] ; Dummy2 = ArrayFlatten [{{ IdentityMatrix [4 Nt + 2 ] }, {Lower2 } } ] ; Z [ Ns ] = GN3.Dummy2; For [ n = Ns, n >= Nd + 1, n, x1 = d1 ( d1 d2 ) ( n Nd)/Ng ; x2 = d1 ( d1 d2 ) ( n 1 Nd)/Ng ; z = 0. 5 ( x1 + x2 ) ; Chiv = Chit + d e l t a v Sin [ ( Pi ( d2 z ) ) / Cap omega ] ; Chi = ArcTan[ Tan[ Chiv ] ] ; v = (2 Chiv )/ Pi ; epa = ( v v ˆ 2 ) ˆ 2 ; epb = ( v v ˆ 2 ) ˆ 2 ; epc = ( v v ˆ 2 ) ˆ 2 ; Sy = {{Cos [ Chi ], 0, Sin [ Chi ] }, {0, 1, 0}, {Sin [ Chi ], 0, Cos [ Chi ] } } ; e p r e f = {{epb, 0, 0}, {0, epc, 0}, {0, 0, epa }}; e p s n t f = Sz. Sy. e p r e f. Inverse [ Sy ]. Inverse [ Sz ] ; epsxx = e p s n t f [ [ 1, 1 ] ] ; epsxy = e p s n t f [ [ 1, 2 ] ] ; epsxz = e p s n t f [ [ 1, 3 ] ] ; epsyx = e p s n t f [ [ 2, 1 ] ] ; epsyy = e p s n t f [ [ 2, 2 ] ] ; epsyz = e p s n t f [ [ 2, 3 ] ] ; epszx = e p s n t f [ [ 3, 1 ] ] ; epszy = e p s n t f [ [ 3, 2 ] ] ; epszz = e p s n t f [ [ 3, 3 ] ] ; e l i s t x x = {}; e l i s t x y = {}; e l i s t x z = {}; e l i s t y x = {}; e l i s t y y = {}; 216

247 e l i s t y z = {}; e l i s t z x = {}; e l i s t z y = {}; e l i s t z z = {}; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t x x = Append[ e l i s t x x, epzn [ epsxx, z, ind ] ] ; ] ; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t y y = Append[ e l i s t y y, epzn [ epsyy, z, ind ] ] ; ] ; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t z z = Append[ e l i s t z z, epzn [ epszz, z, ind ] ] ; ] ; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t x y = Append[ e l i s t x y, epzncross [ epsxy, z, ind ] ] ; ] ; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t x z = Append[ e l i s t x z, epzncross [ epsxz, z, ind ] ] ; ] ; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t y x = Append[ e l i s t y x, epzncross [ epsyx, z, ind ] ] ; ] ; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t y z = Append[ e l i s t y z, epzncross [ epsyz, z, ind ] ] ; ] ; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t z x = Append[ e l i s t z x, epzncross [ epszx, z, ind ] ] ; ] ; For [ ind = 2 Nt, ind <= 2 Nt, ind = ind + 1, e l i s t z y = Append[ e l i s t z y, epzncross [ epszy, z, ind ] ] ; ] ; EPZ2xx = Table [ e l i s t x x [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; EPZ2xy = Table [ e l i s t x y [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; EPZ2xz = Table [ e l i s t x z [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; EPZ2yx = Table [ e l i s t y x [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; EPZ2yy = Table [ e l i s t y y [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; EPZ2yz = Table [ e l i s t y z [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; EPZ2zx = Table [ e l i s t z x [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; EPZ2zy = Table [ e l i s t z y [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; EPZ2zz = Table [ e l i s t z z [ [ j i + 2 Nt + 1 ] ], { i, 1, 2 Nt + 1}, { j, 1, 2 Nt + 1 } ] ; P112 = KX. Inverse [ EPZ2zz ]. EPZ2zx ; 217

248 P122 = KX. Inverse [ EPZ2zz ]. EPZ2zy ; P142 = knot IdentityMatrix [ 2 Nt + 1 ] (1/ knot ) KX. Inverse [ EPZ2zz ].KX; P232 = knot IdentityMatrix [2 Nt + 1 ] ; P312 = knot EPZ2yx + knot EPZ2yz. Inverse [ EPZ2zz ]. EPZ2zx ; P322 = ( 1/ knot ) KX.KX knot EPZ2yy + knot EPZ2yz. Inverse [ EPZ2zz ]. EPZ2zy ; P342 = EPZ2yz. Inverse [ EPZ2zz ]. KX; P412 = knot EPZ2xx knot EPZ2xz. Inverse [ EPZ2zz ]. EPZ2zx ; P422 = knot EPZ2xy knot EPZ2xz. Inverse [ EPZ2zz ]. EPZ2zy ; P442 = EPZ2xz. Inverse [ EPZ2zz ].KX; P2 = ArrayFlatten [{{ P112, P122, NMat, P142 }, {NMat, NMat, P232, NMat}, {P312, P322, NMat, P342 }, {P412, P422, NMat, P442 } } ] ; {DN2, GN22} = Eigensystem [ P2 ] ; GN2 = Transpose [ GN22 ] ; For [ p = 1, p < 4 (2 Nt + 1), p = p + 1, For [ q = 1, q <= 4 (2 Nt + 1) p, q = q + 1, I f [Im[DN2 [ [ q ] ] ] < Im[DN2 [ [ q + 1 ] ] ], a = DN2 [ [ q ] ] ; b = GN2 [ [ All, q ] ] ; DN2 [ [ q ] ] = DN2 [ [ q + 1 ] ] ; DN2 [ [ q + 1 ] ] = a ; GN2 [ [ All, q ] ] = GN2 [ [ All, q + 1 ] ] ; GN2 [ [ All, q + 1 ] ] = b ; ] ; ] ; ] ; W = Inverse [GN2 ]. Z [ n ] ; WU[ n ] = W[ [ 1 ; ; 4 Nt + 2, All ] ] ; WL[ n ] = W[ [ 4 Nt + 3 ; ; 2 (4 Nt + 2), All ] ] ; DM = DiagonalMatrix [DN2 ] ; DU[ n ] = DM[ [ 1 ; ; 4 Nt + 2, 1 ; ; 4 Nt + 2 ] ] ; DL[ n ] = DM[ [ 4 Nt + 3 ; ; 2 (4 Nt + 2), 4 Nt + 3 ; ; 2 (4 Nt + 2 ) ] ] ; Lower2 = MatrixExp[ I ( d2 d1 ) DL[ n ] /Ng ].WL[ n ]. Inverse [WU[ n ] ]. MatrixExp [ I ( d2 d1 ) DU[ n ] /Ng ] ; Dummy2 = ArrayFlatten [{{ IdentityMatrix [4 Nt + 2 ] }, {Lower2 } } ] ; Z [ n 1 ] = GN2.Dummy2 ; ] ; 218

249 Clear [ n, x1, x2, z, a, b, p, q ] ; For [ n = Nd, n >= 1, n, x1 = ( d1 ) n/nd; x2 = ( d1 ) ( n 1)/Nd; z = 0. 5 ( x1 + x2 ) ; Chiv = Chit + d e l t a v Sin [ ( Pi ( d2 z ) ) / Cap omega ] ; Chi = ArcTan[ Tan[ Chiv ] ] ; v = (2 Chiv )/ Pi ; epa = ( v v ˆ 2 ) ˆ 2 ; epb = ( v v ˆ 2 ) ˆ 2 ; epc = ( v v ˆ 2 ) ˆ 2 ; Sy = {{Cos [ Chi ], 0, Sin [ Chi ] }, {0, 1, 0}, {Sin [ Chi ], 0, Cos [ Chi ] } } ; e p r e f = {{epb, 0, 0}, {0, epc, 0}, {0, 0, epa }}; e p s n t f = Sz. Sy. e p r e f. Inverse [ Sy ]. Inverse [ Sz ] ; epsxx = e p s n t f [ [ 1, 1 ] ] ; epsxy = e p s n t f [ [ 1, 2 ] ] ; epsxz = e p s n t f [ [ 1, 3 ] ] ; epsyx = e p s n t f [ [ 2, 1 ] ] ; epsyy = e p s n t f [ [ 2, 2 ] ] ; epsyz = e p s n t f [ [ 2, 3 ] ] ; epszx = e p s n t f [ [ 3, 1 ] ] ; epszy = e p s n t f [ [ 3, 2 ] ] ; epszz = e p s n t f [ [ 3, 3 ] ] ; EPZ1xx = Table [ I f [ i == j, epsxx, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; EPZ1xy = Table [ I f [ i == j, epsxy, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; EPZ1xz = Table [ I f [ i == j, epsxz, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; EPZ1yx = Table [ I f [ i == j, epsyx, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; EPZ1yy = Table [ I f [ i == j, epsyy, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; EPZ1yz = Table [ I f [ i == j, epsyz, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; EPZ1zx = Table [ I f [ i == j, epszx, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; EPZ1zy = Table [ I f [ i == j, epszy, 0 ], { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; EPZ1zz = Table [ I f [ i == j, epszz, 0 ], 219

250 { i, 2 Nt + 1}, { j, 2 Nt + 1 } ] ; P111 = KX. Inverse [ EPZ1zz ]. EPZ1zx ; P121 = KX. Inverse [ EPZ1zz ]. EPZ1zy ; P141 = knot IdentityMatrix [ 2 Nt + 1 ] (1/ knot ) KX. Inverse [ EPZ1zz ].KX; P231 = knot IdentityMatrix [2 Nt + 1 ] ; P311 = knot EPZ1yx + knot EPZ1yz. Inverse [ EPZ1zz ]. EPZ1zx ; P321 = ( 1/ knot ) KX.KX knot EPZ1yy + knot EPZ1yz. Inverse [ EPZ1zz ]. EPZ1zy ; P341 = EPZ1yz. Inverse [ EPZ1zz ]. KX; P411 = knot EPZ1xx knot EPZ1xz. Inverse [ EPZ1zz ]. EPZ1zx ; P421 = knot EPZ1xy knot EPZ1xz. Inverse [ EPZ1zz ]. EPZ1zy ; P441 = EPZ1xz. Inverse [ EPZ1zz ].KX; P1 = ArrayFlatten [{{ P111, P121, NMat, P141 }, {NMat, NMat, P231, NMat}, {P311, P321, NMat, P341 }, {P411, P421, NMat, P441 } } ] ; {DN1, GN11} = Eigensystem [ P1 ] ; GN1 = Transpose [ GN11 ] ; For [ p = 1, p < 4 (2 Nt + 1), p = p + 1, For [ q = 1, q <= 4 (2 Nt + 1) p, q = q + 1, I f [Im[DN1 [ [ q ] ] ] < Im[DN1 [ [ q + 1 ] ] ], a = DN1 [ [ q ] ] ; b = GN1 [ [ All, q ] ] ; DN1 [ [ q ] ] = DN1 [ [ q + 1 ] ] ; DN1 [ [ q + 1 ] ] = a ; GN1 [ [ All, q ] ] = GN1 [ [ All, q + 1 ] ] ; GN1 [ [ All, q + 1 ] ] = b ; ] ; ] ; ] ; W = Inverse [GN1 ]. Z [ n ] ; WU[ n ] = W[ [ 1 ; ; 4 Nt + 2, All ] ] ; WL[ n ] = W[ [ 4 Nt + 3 ; ; 2 (4 Nt + 2), All ] ] ; DM = DiagonalMatrix [DN1 ] ; DU[ n ] = DM[ [ 1 ; ; 4 Nt + 2, 1 ; ; 4 Nt + 2 ] ] ; DL[ n ] = DM[ [ 4 Nt + 3 ; ; 2 (4 Nt + 2), 4 Nt + 3 ; ; 2 (4 Nt + 2 ) ] ] ; Lower1 = MatrixExp[ I ( d1 ) DL[ n ] /Nd ].WL[ n ]. Inverse [WU[ n ] ]. MatrixExp [ I ( d1 ) DU[ n ] /Nd ] ; Dummy1 = ArrayFlatten [{{ IdentityMatrix [4 Nt + 2 ] }, {Lower1 } } ] ; Z [ n 1 ] = GN1.Dummy1 ; ] ; 220

251 dummy60 = ArrayFlatten [{{Z [ 0 ], YM} } ] ; T0RP = Inverse [ dummy60 ]. YP.AP; T0P = T0RP [ [ 1 ; ; 4 Nt + 2, All ] ] ; RP = T0RP[ [ 4 Nt + 3 ; ; 2 (4 Nt + 2), All ] ] ; TP[ 0 ] = T0P ; Clear [ i ] ; For [ i = 1, i <= Nd, i ++, TP[ i ]=Inverse [WU[ i ] ]. MatrixExp [ I ( d1 ) DU[ i ] /Nd ]. TP[ i 1 ] ; ] ; Clear [ i ] ; For [ i = Nd + 1, i <= Ns, i ++, TP[ i ]=Inverse [WU[ i ] ]. MatrixExp [ I ( d2 d1 ) DU[ i ] /Ng ].TP[ i 1 ] ; ] ; TP[ Ns + 1]= Inverse [WU[ Ns + 1 ] ]. MatrixExp [ I ( d3 d2 ) DU[ Ns + 1 ] ]. TP[ Ns ] ; T0RS = Inverse [ dummy60 ]. YP.AS; T0S = T0RS [ [ 1 ; ; 4 Nt + 2, All ] ] ; RS = T0RS [ [ 4 Nt + 3 ; ; 2 (4 Nt + 2), All ] ] ; TS [ 0 ] = T0S ; Clear [ i ] ; For [ i = 1, i <= Nd, i ++, TS [ i ]=Inverse [WU[ i ] ]. MatrixExp [ I ( d1 ) DU[ i ] /Nd ]. TS [ i 1 ] ; ] ; Clear [ i ] ; For [ i = Nd + 1, i <= Ns, i ++, TS [ i ]=Inverse [WU[ i ] ]. MatrixExp [ I ( d2 d1 ) DU[ i ] /Ng ]. TS [ i 1 ] ; ] ; TS [ Ns + 1]= Inverse [WU[ Ns + 1 ] ]. MatrixExp [ I ( d3 d2 ) DU[ Ns + 1 ] ]. TS [ Ns ] ; RP2 = Abs[RP] ˆ 2 ; TP2 = Abs[TP[ Ns + 1 ] ] ˆ 2 ; RS2 = Abs[RS ] ˆ 2 ; TS2 = Abs[ TS [ Ns + 1 ] ] ˆ 2 ; Block [ { n, kxnn, kzr }, kzr = {}; For [ n = Nt, n <= Nt, n = n + 1, kxnn = knot Sin [ theta ] + n 2 Pi/L ; I f [ knot ˆ2 >= kxnn ˆ2, kzr = Append[ kzr, (N[ Sqrt [ knot ˆ2 kxnn ˆ 2 ] ] ) / ( knot Cos [ theta ] ) ] ; ] ; 221

252 I f [ knot ˆ2 < kxnn ˆ2, kzr = Append[ kzr, 0 ] ; ] ; ] ; RKZ = DiagonalMatrix [ kzr ] ; ] ; ( p i n c i d e n t, p r e f l e c t e d and transmitted Pp ) RPp = RP2 [ [ 2 Nt + 2 ; ; 4 Nt + 2, All ] ] ; RPpt = Transpose [ Transpose [RPp ]. RKZ ] ; TPp = TP2 [ [ 2 Nt + 2 ; ; 4 Nt + 2, All ] ] ; TPpt = Transpose [ Transpose [TPp ]. RKZ ] ; ( p i n c i d e n t, s r e f l e c t e d transmitted Sp ) RSp = RP2 [ [ 1 ; ; 2 Nt + 1, All ] ] ; RSpt = Transpose [ Transpose [ RSp ]. RKZ ] ; TSp = TP2 [ [ 1 ; ; 2 Nt + 1, All ] ] ; TSpt = Transpose [ Transpose [ TSp ]. RKZ ] ; ( s i n c i d e n t, s r e f l e c t e d and transmitted Ss ) RSs = RS2 [ [ 1 ; ; 2 Nt + 1, All ] ] ; RSst = Transpose [ Transpose [ RSs ]. RKZ ] ; TSs = TS2 [ [ 1 ; ; 2 Nt + 1, All ] ] ; TSst = Transpose [ Transpose [ TSs ]. RKZ ] ; ( s i n c i d e n t, p r e f l e c t e d and transmitted Ps ) RPs = RS2 [ [ 2 Nt + 2 ; ; 4 Nt + 2, All ] ] ; RPst = Transpose [ Transpose [ RPs ]. RKZ ] ; TPs = TS2 [ [ 2 Nt + 2 ; ; 4 Nt + 2, All ] ] ; TPst = Transpose [ Transpose [ TPs ]. RKZ ] ; ( ) PRT = RPpt + TPpt + RSpt + TSpt ; PAB = 1 Total [PRT ] ; SRT = RSst + TSst + RPst + TPst ; SAB = 1 Total [SRT ] ; a p l i s t = Append[ a p l i s t, {NN, PAB [ [ 1 ] ] } ] ; a s l i s t = Append[ a s l i s t, {NN, SAB [ [ 1 ] ] } ] ; ] L i s t L i n e P l o t [ { a p l i s t, a s l i s t } ] Print [ a p l i s t // MatrixForm, a s l i s t // MatrixForm ] ; 222

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264 VITA Muhammad Faryad Muhammad Faryad was born on Marh 18, 1982 in Sialkot, Pakistan. He received his B.Sc. degree in Mathematics and Physics from University of Punjab, Lahore, Pakistan, in 2002, and his M.Sc. and M.Phil. degrees in Electronics from Quaidi-Azam University, Islamabad, Pakistan in 2006 and 2008, respectively. He was awarded Certificates of Merit in M. Sc. and M. Phil. in Quaid-i-Azam University for outstanding academic achievements. He coauthored ten journal articles and two conference proceeding papers during his M. Phil. Faryad joined the Department of Engineering Science and Mechanics at the Pennsylvania State University as a Ph. D. student in August He was advised by Akhlesh Lakhtakia, the Charles Godfrey Binder Endowment Professor. During his Ph. D., Faryad coauthored twelve journal articles and two conference proceeding papers, and received the following awards: (i) College of Engineering Fellowship from College of Engineering ( ), (ii) Sabih & Güler Hayek Graduate Scholarship in Engineering Science and Mechanics (2011), (iii) A Society of Photo-Optical Instrumentation Engineers (SPIE) scholarship (2011), and (iv) Alumni Association Dissertation Award (2012). Faryad is a student member of SPIE and OSA.