Slab Waveguide Optical Sensors Using Negative-Index Materials

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2 Islamic University of Gaza Deanery of Higher Studies Faculty of Science Department of Physics الجاهعة االسالهية بغزة عوادة الدراسات العليا كلية العلوم قسن الفيزياء Slab Waveguide Optical Sensors Using Negative-Inde Materials مجسات الموجهات الضوئية باستخذام مواد رات معامل إنكسار سالب By Anwar Atiya Jarada Supervisor Dr. Sofyan A. Taya Associate Professor of Physics Submitted to the Faculty of Science as a Partial Fulfillment of the Master of Science (M. Sc.) in Physics Palestine, Gaza

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4 بسى هللا انشح انشح ى إهذاء إن سوذ أي انطاهشة وع ش أب ان ذ ذ إن صوخخ أب ائ إخىح وأخىاح I

5 ACKNOWLEDGMENT I am etremely indebted to my advisor and supervisor Dr. Sofyan A. Taya for his constant support, help, guidance, stimulating suggestions and encouragement in all time of research, for arranging and writing of this thesis and for the inspiration I eperienced with him. II

6 ABSTRACT In this thesis, two waveguide structures were assumed as sensors for detection variation in the inde of refraction of an analyte. The first structure comprises a Left-Handed Material (LHM) in the core layer whereas the second includes a LHM in the substrate. In each waveguide configuration, different mode orders were investigated. For a given mode order, the sensitivity of the proposed waveguide sensor was studied with the thickness of the core layer, frequency of the guided wave, and the parameters of the LHM. Moreover, with each waveguide structure, two analytes were presented: water and air. The results of the two waveguide configurations were compared to find out the optimal structure with the highest sensitivity. It was found that using LHMs in slab waveguide sensors can enhance the sensitivity and the LHM substrate structure has an improved sensitivity over the LHM core layer. III

7 ARABIC ABSTRACT ف هزة انشسانت حى دساست ىرخا ي يىخهاث ان ىخت ك دساث نهكشف ع انخغ ش ف يعايم ا كساس انىسظ يىضىع انذساست. ان ىرج األول حخى يادة حخ خع بس اح ت كهشبائ ت سانبت و فار ت يغ اط س ت سانبت ف طبقت انف هى ب ا ف انثا حكى ان ادة راث انس اح ت انكهشبائ ت انسانبت وان فار ت ان غ اط س ت انسانبت ف طبقت انشك ضة وحى ي اقشت عذة ا اط ف كم ىرج و حى دساست انخغ ش ف حساس ت يدساث يىخهاث ان ىخت نكم ظ يع انخغ ش ف كم ي س ك انف هى و حشدد يىخهاث ان ىخت وانعىايم انخ حعخ ذ عه ها ان ادة راث انس اح ت انكهشبائ ت انسانبت وان فار ت ان غ اط س ت انسانبت ك ا واسخخذو ف كم ىرج يشة ان اء ويشة اخشي انهىاء ك ادة يشاد دساسخها. ثى ح ج يقاس ت ب خائح انحانخ وان ىرخ ح ث وخذ أ انحساس ت ف ىرج انشك ضة -ان ص ىعت ي يىاد راث س اح ت كهشب ت سانبت و فار ت يغ اط س ت سانبت- أفضم ي ها ف ان ىرج األول ووخذ ا ان ظ األساس ف ان ىرج األول غ ش يىخىد ب ا ىخذ ف ان ىرج انثا. IV

8 List of Figure Captions CHAPTER ONE Figure 1.1 Refraction of light into medium 6 Figure 1. An electromagnetic wave incident at a plane interface. 7 Figure 1.3 Refraction of light at different angles including TIR. 10 Figure 1.4 Figure 1.5 Figure 1.6 Figure 1.7 Evanescent wave in medium after TIR. Goos-Hänchen shift effect. Basic structure of a slab waveguide. Light confinement in a planar waveguide by TIR CHAPTER TWO Figure.1 Figure. Wave propagation in a) RHM b) LHM Schematic illustrating of an evanescent field penetrating the cover medium 0 3 CHAPTER THREE Figure 3.1 Three-layers slab waveguide structure. 30 Figure 3. Figure 3.3 Sensitivity versus the guiding film thickness when the film is LHM and the cladding is water for different values of the frequency of the guided wave with F=0.56, ω p =10GHz, ω 0 =4GHz, γ=0.01ωp, ε 1 =.161, µ 1 =1, ε 3 =1.77, and µ 3 =1. Sensitivity versus the guiding film thickness when the film is LHM and the cladding is air for different values of the wave frequency with F=0.56, ω p =10GHz, ω 0 =4GHz, γ=0.01ω p, ε 1 =.161, µ 1 =1, ε 3 =1, and µ 3 = V

9 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Sensitivity versus frequency of the guided wave when the film is LHM and the cladding is water for different values of the guiding film thickness for F = 0.56, ω p = 10 GHz, ω 0 = 4 GHz, γ = 0.01ω p, ε 1 =.161, µ 1 = 1, ε 3 = 1.77, and µ 3 = 1 Sensitivity versus frequency of the guided wave when the film is LHM and the cladding is air for different values of the guiding film thickness for F = 0.56, ω p = 10 GHz, ω 0 = 4 GHz, γ = 0.01ω p, ε 1 =.161, µ 1 = 1, ε 3 = 1, and µ 3 = 1. Sensitivity versus the damping coefficient ( γ ) when the film is LHM and the cladding is water for different values of frequency of the guided wave for F = 0.56, ω p = 10 GHz, ω 0 = 4 GHz, d =7.0 mm, ε 1 =.161, µ 1 = 1, ε 3 = 1.77, and µ 3 =1. Sensitivity versus F when the film is LHM and the cladding is water for different values of frequency of the guided wave for γ = 0.01ω p, ω p = 10 GHz, ω 0 = 4 GHz, d =7.0 mm, ε 1 =.161, µ 1 = 1, ε 3 = 1.77, and µ 3 = 1 Sensitivity versus the guiding film thickness when the film is LHM and the cladding is water for different values of the frequency of the guided wave for F = 0.56, ω p = 10 GHz, ω 0 = 4 GHz, γ =0.01ω p, ε 1 =.161, µ 1 =1, ε 3 =1.77, and µ 3 =1. Sensitivity versus frequency of the guided wave when the film is LHM and the cladding is water for different values of the guiding film thickness for F = 0.56, ω p = 10 GHz, ω 0 = 4 GHz, γ = 0.01ω p, ε 1 =.161, µ 1 = 1, ε 3 = 1.77, and µ 3 = 1. Sensitivity versus the ratio (γ/ω p ) when the film is LHM and the cladding is water for different values of frequency of the guided wave for F = 0.56, ω p = 10 GHz, ω 0 = 4 GHz, d =7.0 mm, ε 1 =.161, µ 1 = 1, ε 3 = 1.77, and µ 3 = 1. Sensitivity versus F when the film is LHM and the cladding is water for different values of frequency of the guided wave for γ = 0.01ω p, ω p = 10 GHz, ω 0 = 4 GHz, d =7.0 mm, ε 1 =.161, µ 1 = 1, ε 3 = 1.77, and µ 3 = VI

10 CHAPTER FOURE Figure 4.1 Figure 4. Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Sensitivity versus the guiding film thickness when the substrate is LHM and the cladding is water for different values of the frequency of the guided wave for F=0.56, ω p =10GHz, ω 0 =4GHz, γ=0.01ω p, ε =3.06, µ =1, ε 3 =1.77, and µ 3 =1. Sensitivity versus the guiding film thickness when the substrate is LHM and the cladding is air for different values of the frequency of the guided wave for F=0.56, ω p =10GHz, ω 0 = 4 GHz, γ = 0.01ω p, ε = 3.06, µ = 1, ε 3 =1, and µ 3 =1. Sensitivity versus frequency of the guided wave when the cladding is water for different values of d, F= 0.56, ω p =10 GHz, ω 0 =4GHz, γ=0.01ω p, ε 1 =.161, µ 1 =1, ε 3 =1.77, and µ 3 =1. Sensitivity versus frequency of the guided wave when the cladding is air for different values of d, F=0.56, ω p =10GHz, ω 0 =4GHz, γ=0.01ω p, ε =3.06, µ =1, ε 3 =1, and µ 3 =1. Sensitivity versus the ratio (γ/ω p ) when the substrate is LHM and the cladding is water for different values of frequency of the guided wave for F=0.56, ω p =10GHz, ω 0 =4GHz, d=7.0mm, ε =3.06, µ =1, ε 3 =1.77, and µ 3 =1. Sensitivity versus F when the substrate is LHM and the cladding is water for different values of frequency of the guided wave for γ=0.01ω p, ω p =10GHz, ω 0 =4GHz, d=7.0mm, ε =3.06, µ =1, ε 3 =1.77, and µ 3 =1. Sensitivity versus the guiding film thickness when the substrate is LHM and the cladding is water for different values of the frequency of the guided wave for F=0.56, ω p =10GHz, ω 0 =4GHz, γ=0.01ω p, ε =3.06, µ =1, ε 3 =1.77, µ 3 = VII

11 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.1 Sensitivity versus frequency of the guided wave for different values of the guiding film thickness when the substrate is LHM and the cladding is water for F = 0.56, ω p = 10 GHz, ω 0 = 4 GHz, γ = 0.01ω p, ε = 3.06, µ = 1, ε 3 = 1.77, µ 3 = 1. Sensitivity versus the ratio (γ/ω p ) when the substrate is LHM and the cladding is water for different values of frequency of the guided wave for F = 0.56, ω p = 10 GHz, ω 0 = 4 GHz, d =7.0 mm, ε = 3.06, µ = 1, ε 3 = 1.77, µ 3 = 1. Sensitivity versus F when the substrate is LHM and the cladding is water for different values of frequency of the guided wave for γ = 0.01ω p, ω p = 10 GHz, ω 0 = 4 GHz, d =1.50 mm, ε = 3.06, µ = 1, ε 3 = 1.77, µ 3 = 1. Sensitivity versus the guiding film thickness when the substrate is LHM and the cladding is water for different values of the frequency of the guided wave for F=0.56, ω p =10GHz, ω 0 =4GHz, γ=0.01 ω p, ε =3.06, µ =1, ε 3 =1.77, µ 3 =1. Sensitivity versus the guiding film thickness when the substrate is LHM and the cladding is water for different values of the frequency of the guided wave for F=0.56, ω p =10GHz, ω 0 =4GHz, γ=0.01ω p, ε =3.06, µ =1, ε 3 =1.77, µ 3 = VIII

12 List of abbreviations: LHM NIM DNG TE TM TIR Left- Handed Material Negative- Inde Material Double Negative Transverse Electric Transverse Magnetic Total internal angle List of symbols: ε electric permittivity of the medium µ magnetic permeability of the medium surface conductivity. 0 permittivity of vacuum r relative permittivity of the medium 0 permeability of vacuum r relative permeability of the medium. k wave vector k wave number γ damping coefficient ω angular frequency F the fractional area of the unit cell occupied by the split ring ω p the plasma frequency ω 0 the resonance frequency E electric field H magnetic field D electric flu density B magnetic flu density electric charge density J current density n refractive inde of a medium N effective inde of the guided mode IX

13 CONTENTS Dedication I Acknowledgment II Abstract.III Arabic Abstract IV List of Figure Captions..V List of abbreviations IX List of symbols IX Chapter one Basics of Electromagnetic and Waveguide Theories Electromagnetism Mawell's equation Constitutive relation Wave equation Boundary conditions Medium and wave parameters Reflection and refraction Total internal reflection and evanescent fields 9 1. Waveguide theory Introduction Waveguide structure Formulation of waveguide equations Transverse electric (TE) modes 16 X

14 1..3. Transverse magnetic (TM) modes Power considerations 18 Chapter two Left-Handed Materials and Optical Waveguide Sensors Left-handed materials Concept of left-handed materials 0.1. A brief historical review Uses and applications 1. Slab waveguide optical sensors..1 Definition and principle of this work..... Advantages of optical sensors Classification of optical sensors Sensor sensitivity Previous studies.5 Chapter Three Three-Layers LHM Waveguide Structure as an Optical Sensor Dispersion relation Sensitivity of the structure Power flow Results and discussion The first mode (m=1) The second mode (m=) Chapter four Slab Waveguide Sensor with LHM Substrate Layer.46 XI

15 4.1 Basic equation Results and Discussion The fundamental mode (m=0) The first mode (m=1) The second mode (m=) Comparison between LHM core and LHM substrate structures.58 Conclusion.. 59 References..61 XII

16 Chapter One Basics of Electromagnetic and Waveguide Theories This chapter is intended to present electromagnetic field theory and basics of slab waveguides. A review of electromagnetism and physics of waveguide theory are presented. Mawell's equations, constitutive relations, and boundary conditions for the fields at an interface between two media are studied. Solution of the wave equation, applying the boundary conditions, and finally derivation of the dispersion relations for both transverse electric field (TE) and transverse magnetic field (TM) modes are investigated. 1.1 Electromagnetism Electromagnetic waves consist of perpendicular electric and magnetic fields which have been described by Mawell's equations. J. Mawell unified the basic laws of electric and magnetic fields into four fundamental equations, which state the relations between electromagnetic fields and their sources (current and charge) Mawell's equations A group of four wonderful equations that describes the theory of electromagnetism. These equations describe how electromagnetic waves propagate, interact, and how they are influenced by objects. Mawell was the first to determine the speed of propagation of electromagnetic waves which was the same as the speed of light, and hence to conclude that electromagnetic waves and visible light were really the same thing. Mawell's equations are written in terms of time-varying quantities as B E, (1.1) t 1

17 D H J, t (1.). D, (1.3). B 0, (1.4) where E (Volt/meter) and H (Ampere/meter) are the electric and magnetic fields, respectively, D (Coulomb/meter ) and B (Tesla) are the electric and magnetic flu densities, respectively, and (Coulomb/meter 3 ) and J (Ampere/meter ) are the electric charge and the current densities, respectively Constitutive relations The fields and flu densities in Eqs. (1.1)- (1.4) are related to each other by the constitutive relations, sometimes called material equations [1]. For a linear, isotropic, homogeneous medium D E, (1.5) B H, (1.6) J E, (1.7) where (Coulomb/Volt.meter) and (Tesla.meter/Ampere) are the electric permittivity and the magnetic permeability of the medium respectively, and (Volt.meter/Ampere) is the surface conductivity. The permittivity and permeability of a medium can be written as = 0, (1.8) r = 0 r, (1.9) where 0 is the permittivity of vacuum and equals F / m and r is the relative permittivity of the medium. 0 is the permeability of vacuum and equals to H / m and r is the relative permeability of the medium.

18 1.1.3 Wave equation Taking the curl of Eq. (1.1) to derive the electric field wave equation E H. (1.10) t Substituting Eq. (1.) into Eq. (1.10) D E E. (1.11) t t Using the vector identity E (. E) E. (1.1) Substituting Eq. (1.1) into Eq. (1.11) E E. E. (1.13) t For a source free isotropic medium. E =0 from Eq. (1.3), so that Eq. (1.13) becomes E E 0. (1.14) t Similarly, taking the curl of Eq. (1.) and following the same procedure we get the magnetic field wave equation as H H 0. (1.15) t Equations (1.14) and (1.15) are called Helmholtz equations for electric and magnetic fields, respectively. They have a sinusoidal solution if the coefficient of the second term is positive and an eponentially increasing or decreasing solution if it is negative. For time harmonic fields, the wave function takes the form i( k. r t A r, t A e, (1.16) ) 0 3

19 where A is either E or H, A 0 is the amplitude of the wave, k is the wave vector and is the angular frequency. Mawell's equations for time harmonic fields can be written as E ih, (1.17) H ie. (1.18) Helmholtz equation for either E or H becomes A A A A 0, (1.19) v where v is the speed of the wave Boundary conditions When electromagnetic waves cross an interface between two different media, they satisfy a set of equations called boundary conditions at the interface. These equations can be summarized as: * The continuity of normal component of the magnetic flu i.e. n B B, (1.0) where n is a unit vector normal to the interface. * The discontinuity of the normal component of the electric displacement when there is a surface charge of density, i.e. n 1. D D 1. (1.1) Otherwise there is a continuity of both normal and tangential components of the electric field, i.e. n E E (1.) * The discontinuity of the tangential component of the magnetic field when there is a surface current of density J, i.e. n H H. (1.3) 1 J 1 4

20 1.1.5 Medium and wave parameters Permittivity ( ) and permeability ( ) are the basic physical parameters for describing the medium-electromagnetic wave interaction. All physical properties of media are characterized by these parameters. Media, in optics, are characterized by the refractive inde which is the ratio of the speed of light ϲ in vacuum 8 ( 3 10 m/ s ) to its speed in the medium. where c n, 1 c, and 0 0 v 1 (1.4) The refractive inde of a medium is given in terms of relative permittivity and permeability as n. (1.5) r r There is another parameter called the etinction coefficient ( ) that should be mentioned when dealing with lossy media []. The intensity of the wave decreases eponentially when propagating in lossy media. In this case the comple refractive inde n ~ of such media is given by n ~ n i, (1.6) where n is given by Eqs. (1.4) and (1.5), while is the etinction coefficient. As a result, both the electric permittivity and magnetic permeability are also comple, and given by ~ r i and ~ i r i. i It is clear that the parameters of a medium are closely related to the wave parameters. The main wave parameters are the frequency and wavelength, from which one can get the angular frequency and the wave number k, where and k. The free space wave number is given by 5

21 k 0, (1.7) 0 Where 0 is the free space wavelength. Consequently, one can get the wave parameters in a medium if the wave free space parameters and the medium refractive inde are known 0 and k nk0. (1.8) n Reflection and refraction A ray changes its direction as it travels from a medium to another that has a different refractive inde. The change in speed that occurs is responsible for the bending of light, or refraction, that takes place at the interface between the two media. In Fig. 1.1, light is travelling from medium 1 into medium, and angles are measured from the normal to the interface. The angle of transmission of light into the second medium is related to the angle of incidence by Snell's law (n 1 sinᶿ1=n sinᶿ) Fig Refraction of light into medium 6

22 During the propagation of light in a medium, if the wave impinges an interface with another medium, it might be completely reflected back in the same medium, completely transmitted into the second medium or partially reflected and partially transmitted. The situation depends on the parameters of the two media, specifically the refractive indices and the angle of incidence. Consider two adjacent dielectric media with the interface separating them in the y- z plane. An electromagnetic wave with the electric field perpendicular to the plane of incidence, transverse electric (TE) polarized wave, is incident from region I with refractive inde n 1 to medium II with refractive inde n as shown in Fig. 1.. Fig. 1.. An electromagnetic wave incident at a plane interface. In this case, part of the wave is reflected back into the first medium and the other part is transmitted into the second medium. Reflection obeys the following laws: 7

23 1- The incident ray, the reflected ray and the normal to the reflection surface at the point of incidence lie in the same plane. - The angle which the incident ray makes with the normal ( i ) is equal to that which the reflected ray makes to the same normal ( r ). 3- The reflected and incident rays are on the opposite sides of the normal. While refraction laws are: 1- The incident ray, the refracted ray and the normal to the reflection surface at the point of incidence lie in the same plane. - The angle of the refracted ray ( t ) is governed by Snell's law. The refraction occurs because the phase velocity of the wave changes, which causes the change in direction. As a result, the wavelength is altered while the frequency remains constant. The reflection (r) and transmission (t) coefficients are given by Fresnel equations r n cos n cos i i t t TE, (1.9) ni cos i nt cos t t TE i i, (1.30) n cos n cos i n i cos t t r TM n cos n cos n cos n cos t i i t, (1.31) i t t i t n cos i i TM, (1.3) ni cos t nt cos i 8

24 where TE and TM denote transverse electric and transverse magnetic polarizations, respectively, ni and n t are the refractive indices of the incident and refraction media. When the wave is totally transmitted i.e. there's no reflection. Then the angle of incidence is called Brewster's angle. For an interface between two right-handed materials of positive refractive indices, Brewster angle only eists for TM polarized waves which is given by [1]. t i i t cos. (1.33) B TM t i i t Total internal reflection and evanescent fields The phenomenon of total internal reflection (TIR), where a wave striking an interface between two different media is being totally reflected back into the first medium and never transmitted to the other, takes place when the second medium is less denser than the first, i.e. n1 n, and the angle of incidence is greater than a particular angle called the critical angle The critical angle is given by c, as shown in Fig n c sin. (1.34) n1 9

25 Fig Refraction of light at different angles including TIR. TIR is a very important phenomenon as it is the basic principle of operation of slab waveguides and optical fibers. An important result of TIR is the propagation of an evanescent wave across the interface. Essentially, even though the entire incident wave is reflected back into the first medium, there is some penetration into the second medium at the boundary, as shown in Fig.1.4. For, from Snell's law it can be shown that i c n1 cos t i sin 1 i. (1.35) n The transmitted electric field is given by: So that, t t0 t t E E ep ik ( z cos ysin ) (1.36) E t n 1 n1 E t0 ep ik y sin i ep kz sin 1 i. (1.37) n n 11

26 As seen from Fig. 1.4 and Eq. (1.37), the refracted wave propagates only parallel to the surface and is attenuated eponentially in the second medium. Fig Evanescent wave in medium after TIR. The wave makes a lateral shift along the interface and does not reflect back at the same point it strikes the interface as shown in Fig This shift is called Goos- Hänchen shift [3]. The magnitude of the shift depends on the light polarization. 11

27 Fig. 1.5 Goos-Hänchen shift effect. The eplanation of this shift is that the incident light first penetrates the low-inde medium as an evanescent wave before being totally reflected back into the highinde medium [3]. 1. Waveguide theory 1..1 Introduction A waveguide is a structure that is used to confine and guide waves. There are different types of waveguides for each type of waves. The original and most common meaning of a waveguide is a hollow conductive metal pipe used to carry high frequency radio waves, particularly microwaves. 1

28 The geometry of a waveguide reflects its function. Slab waveguides confine energy to travel only in one dimension, fiber or channel waveguides for two dimensions. A waveguide is usually a dispersive structure that confines and directs a wave. It conveys a wave between its ends. An optical waveguide is used to guide waves in the optical spectrum. The first waveguide was proposed by J.J. Thomson in 1893, and was eperimentally verified by O.J. Lodge in There are many aspects to classify optical waveguides, they can be classified according to geometrical shape (planar, strip or fiber), mode structure (single or multimode), refractive inde distribution (step-inde or graded-inde), material (glass, polymer or semiconductor). Optical waveguides utilize the TIR phenomena to confine and convey light. 1.. Waveguide Structure Figure 1.6. shows the structure of the basic dielectric step inde waveguide, which consists a guiding film sandwiched between a cover and substrate. The guiding film has the highest refractive inde n. The film thickness is usually comparable f to the operating wavelength of guided light. The substrate and cover are relatively thick and have refractive indices n s and n c respectively such that n f > n s > n c. The thickness of both cover and substrate is much larger than the wavelength so that they are considered as semi-infinite media. 13

29 n c Cover/cladding n f film k f k h z n s substrate y Fig Basic structure of a slab waveguide. Light is confined in the guiding film by TIR at both film/cover and film/substrate interfaces as shown in Fig Fig Light confinement in a planar waveguide by TIR Formulation of waveguide equations Electromagnetic waveguides are analyzed by solving Mawell's equations, and applying the boundary conditions which are determined by the properties of the materials and their interfaces. These equations have multiple solutions, or modes, which are eigen- functions of the equation system. Consider again the waveguide shown in Fig The wave is supposed to propagate in the z-direction. The waveguide is assumed to be infinitely etended 14

30 15 in the y-direction, so that there's symmetry in the field distribution in this direction and this imposes that 0 y. Epanding Eq. (1.17) in three dimensions using the phasor notation for the harmonic waves, we get ˆ) ˆ ˆ ( ˆ ˆ ˆ k H j H i H i E E E z y k j i z y z y. (1.38) Let be the longitudinal propagation constant, so i z, and using the fact 0 y, ) ( 0 ), ( z t i e y E E,we get y H E, (1.39) y z H i E E i, (1.40) z y H i E. (1.41) In a similar manner, epanding Eq. (1.18) gives ˆ) ˆ ˆ ( ˆ ˆ ˆ k E j E i E i H H H z y k j i z y z y. (1.4) We get y E H, (1.43) y z E i H H i, (1.44) z y E i H. (1.45)

31 Guided modes depend on the light wavelength and polarization and the shape and size of the guide. The longitudinal mode of a waveguide is a particular standing wave pattern formed by waves confined in the cavity. The transverse modes are classified into different types. 1- TE modes (transverse electric) have no electric field in the direction of propagation. - TM modes (transverse magnetic) have no magnetic field in the direction of propagation. 3- TEM modes (transverse electromagnetic) have neither electric nor magnetic field in the direction of propagation. 4- Hybrid modes have both electric and magnetic field components in the direction of propagation Transverse electric (TE) modes The electric field vector of the incident wave is parallel to the interface between the layers in TE polarization. The boundary conditions state that the electric fields and magnetic fields components parallel to an interface are continuous at the boundary. There's a non-zero y-component of the electric field, while the magnetic field has two nonzero components. Then, E E H 0. From Eqs. (1.39), (1.41) and (1.44) we get z y where E y k k 0 0 ( N ) E 0, (1.46) y N is the effective refractive inde of the guided mode, and k 0 is the free space wave number. The solutions of Eq. (1.46) in the three layers are given by 16

32 B Ae c ( d ), d E y ( ) 1 cos( f ) B sin( f ),0 d, (1.47) s Ce, 0 where A, B 1, B and C represent the wave amplitude in the three media and the parameters given by and c c, f and s where c and s are called decay constants and k 0 N cc, (1.48) k, (1.49) s f 0 f f N k 0 N ss. (1.50) The dispersion relation can be obtained by applying the boundary conditions, we set 1 d tan 1 q tan q m, (1.51) f c where m=0, 1,, is the mode order and c q c and q s f f s s. (1.5) As can be seen from Eq. (1.51) there are discrete values of the longitudinal wave vector that satisfy the transverse resonance condition. These allowed solutions are called modes Transverse magnetic (TM) modes The magnetic field vector of the incident wave is parallel to the interface between the layers in TM modes. The magnetic field has only one nonzero component 17

33 while the electric field has two nonzero components. In other words, H H E 0. z y From Eqs. (1.40), (1.43) and (1.45) we get H y k 0 ( N ) H 0. (1.53) y In a similar manner to the TE case, the TM dispersion relation is given by where, 1 d tan 1 Q tan Q m, (1.54) f c s Q c c f and f c Q s s f. (1.55) f s 1..4 Power considerations Power vector density P describes the flow of energy carried by electromagnetic waves at any point in space. It specifies both the power density and the direction of flow. In terms of E and H it is defined as [4] P E H. (1.56) Since P represents a physical quantity it should be real, so only real parts of the three vectors in Eq. (1.56) should be considered. Moreover, it is more convenient to deal with the average of the Poynting vector for time harmonic fields as in Eq. (1.16). P av 1 T dtre E Re H. (1.57) T 0 Using phasor notation and Mawell's equations, Eq. (1.57) yields [4] 1 k 1 k P Re[ E ] Re[ H av ]. (1.58) 18

34 Eq. (1.58) is valid in a homogeneous medium. For an inhomogeneous medium such as a waveguide structure as in figure 1.6. the power is distributed in the three layers. The average Poynting vector is given by P total 1 Re[ E y ( ) ] d, (1.59) ( ) for TE polarization and P total 1 Re[ H y ( ) ] d ( ) for TM polarization [5]. i i (1.60) 19

35 Chapter Two Left-Handed Materials (Negative-Inde materials) and Optical Waveguide Sensors In this chapter, left-handed materials (LHMs) of simultaneously negative magnetic permeability and electric permittivity are presented. The possible applications of LHMs are also investigated. The use of slab waveguides as optical sensors is discussed in details. The previous work in the field of optical sensing is reviewed..1 Left-handed materials.1.1 Concept of left-handed materials Materials that have both negative electric permittivity ε and negative magnetic permeability μ in some frequency ranges are called left handed materials (LHMs) or Negative-Inde materials (NIMs). V. Veselago predicted a number of remarkable properties of waves in a LHM [6]. He showed some new features like reversal Doppler shift and backward Vavilov-Cerenkov radiation in metamaterials [6]. LHMs have wave vector and hence direction of propagation in opposite direction of energy flow as shown in Fig..1. E E H K P K P H a) RHM b) LHM Fig..1. Wave propagation in a) RHM b) LHM 1

36 .1. A brief historical review Pendry et al. [7,8] eperimentally demonstrated that a composite medium of periodically placed thin metallic wires can behave as an effective plasma medium for radiation with wavelength much larger than the spatial periodicity of the structure. For frequencies lower than a particular (plasma) frequency, the thin wire structure therefore ehibits a negative permittivity ε [9]. Smith et al. [10] constructed a LHM using the combination of periodic rods and split rings and they performed many eperiments in the microwave range to point out that the nature of this material is unlike any eisting material. The first eperimental investigation of negative inde of refraction was achieved by Shelby et al. in 001 [11]. The interaction of electromagnetic waves with stratified isotropic LHMs was investigated by Kong [1]. He investigated the reflection and transmission beams, field solution of guided waves, and linear and dipole antennas in stratified structure of LHMs. The theory of LHMs and their electromagnetic properties, possible future applications, physical remarks, and intuitive justifications are provided by Engheta in 003 [13]. Chew [14] analyzed the energy conservation property of a LHM and the realistic Sommerfeld problem of a point source over a LHM half space and a LHM slab. In 006, Sabah et al. [15] presented the reflected and transmitted powers due to the interaction of electromagnetic waves with a LHM. They studied the effects of the structure parameters, incidence angle, and the frequency on the reflected and transmitted powers for lossless LHM. The electromagnetic wave propagation through frequency-dispersive and lossy double-negative slab embedded between two different semi-infinite media was presented by Sabah et al. [16]. 1

37 .1.3 Uses and applications Due to the fabrication technologies, the LHMs are widely used in filters, imaging, optical imaging, and nondestructive detections. In 004, A. Grbic et al. realized the first superlens in the microwave regime [17], which demonstrated resolution three times better than the diffraction limit. Fang et al. proposed the first optical superlens using thin silver film in 005 [18], which breaks the diffraction limit and produces super-resolution images. For LHMs, the cloaking devices have gained more attention [19-1]. The successful demonstrations of invisible cloaks in the microwave regime [0-1] make it possible to realize cloaking devices in the future such as with the vast development of metamaterials, more applications will be found in the future. absorbers, lens, microwave components, and antennas, etc. The most eciting potential application is the perfect lens. The LHM lens looks quite eotic in that it does not have any ais or curvature, nor does it focus parallel rays or magnify small objects. All of these features were recognized in the seminal paper by Veselago [6]. The other outstanding feature of the metamaterial is superlens [-3]. LHM is to be used as a radar absorber and in the design of a light weight directive antenna substrate [1].. Slab waveguide optical sensors Slab waveguides were first proposed for telecommunication applications. In 1983 Teifenthelar and Lukosz, proposed that slab waveguides can be used as optical sensors for humidity application. While studying a slab waveguide with grating coupler, they discovered changes in coupling angles due to humidity variation. Since then, slab waveguide sensors have been the subject to a large number of investigations which led to a considerable advance in the field and many techniques have been developed. In this dissertation, evanescent field optical sensors using LHMs will be presented.

38 ..1 Definition and principle of work An optical biosensor is simply a device in which light interacts with the material to be detected (analyte) and then converts the affected light signal into a readable electric signal, which carries the necessary information about the analyte. As illustrated before the guided wave inside the guiding film etends as an evanescent wave into the surrounding layers. This decaying field penetrates and interacts with the cladding as shown in Fig..1. Changing the refractive inde of the cladding will change the effective inde of the guided mode. As a result the coupled mode will change which can be realized by a change in intensity, phase, polarization, frequency, emission or reflection. Fig..1. Schematic illustrating of an evanescent field penetrating the cover medium [4]. 3

39 .. Advantages of optical sensors Slab waveguide sensors have a wide range of applications 1- Detecting the eistence and/or concentrations of dangerous gases and liquids. - Sensing bacteria and viruses. 3- Monitoring biochemical reactions. 4- Monitoring humidity. 5- Estimating blood glucose levels. 6- Industrial applications. Optical slab waveguide sensors have very important features which make them more competitive than other sensing techniques [17-0]. Among these advantages, one can mention the following: 1- Rapid response. - Immunity to electromagnetic interference. 3- Small size/weight. 4- Resistance to chemically aggressive and ionizing environments. 5- Easy to interface with optical data communication systems and secure data transmission. 6- High sensitivity. 7- Low cost...3 Classification of optical sensors Optical sensors can be classified according to: -The role of the waveguide into etrinsic or intrinsic [19]. In etrinsic sensors the waveguide is used to transport light to and from the sensing region, i.e. sensing is done outside the waveguide. While in intrinsic sensors, both transportation of light and sensing are performed by the waveguide. 4

40 -The number of coupled modes used in the sensing process into single mode and multimode sensors. The waveguide in single mode sensors is very thin film with thickness comparable to the operating wavelength. While in multimode sensors, the waveguide thickness is much greater than the operating wavelength. H. Mukundan et al. [0] presented a comprehensive review on both types...4. Sensor sensitivity The effective inde of a guided mode depends on the structure parameters, i.e. the refractive indices of the three layers and the thickness of the guiding layer. The evanescent field of the guided mode penetrates both the substrate and cover layers. Any change in the cover inde (n c ) will change the effective inde (N) of the guided mode, this is the principle of operation of optical sensing. Mathematically the sensitivity S of a homogeneous sensor is given by Sn c N. (.1) n c For surface sensors in which an additional layer called adlayer is formed at the film clad interface, the sensitivity is defined as the change of the effective inde due to the change in the adlayer s thickness d A or refractive inde n A as [1], Sd A N or d A Sn A N. (.) n A Increasing the interaction of the evanescent field with the cladding medium increases the sensitivity. In other words, maimizing the field strength at the film/clad interface and the penetration depth is the key to maimize the sensitivity [0]. For a given analyte, the sensitivity of slab waveguide sensor can be enhanced by 1- Proper choice of other media (film and substrate). - Proper choice of the coupled mode. 5

41 3- Adding etra layers...5 Previous studies In 1989, grating couplers on planar waveguide were used as integrated-optical sensors responding to changes in the refractive inde of a liquid sample covering the waveguide. A comprehensive theory of the sensor sensitivities was developed and conditions for the waveguide parameters in order to obtain high sensitivities were derived. In this analysis both nonporous and microporous waveguiding films were considered [5]. A new type of optical sensors for the detection of gases was proposed. The working principle was as follows: the gas absorbed by a sensitive film, which changes its refractive inde with gas absorption, was measured. The sensitive film was deposited onto an integrated optical interferometer. As the intensity distribution of waveguide modes was not totally confined to the waveguide, the interferometer phase changes with the refractive inde of the film. Interferometers of Fabry-Perot and Mach-Zehnder type and films sensitive towards, e.g., CO and SO, were tested under different gas atmospheres. The temperature dependence of the sensor signal and sensitivities towards other gases and humidity were investigated. A sensor element fabricated that is suitable for the detection of the threshold limit value of CO was fabricated [6]. In 1998, closed-form analytical epressions and normalized charts were presented to provid the conditions for the maimum sensitivity of transverse electric (TE) and transverse magnetic (TM) evanescent-wave step-inde waveguide sensors. The analysis coverd both cases where the measurand is homogeneously distributed in the semi-infinite waveguide cover, and where it is an ultrathin film at the waveguide-cover interface [4]. 6

42 A recent study provided the conditions for maimum geometric wavelength dispersion in three-layer slab waveguides. The resulting optimization procedure was illustrated by the design of a dispersive channel waveguide for use in a novel SiON-technology based integrated optical wavelength sensor [7]. O. Parriaua et al., eamined in 000 a ray optic description of evanescent wave sensing which allowed a homogeneous treatment of the sensitivity of free space and guided wave slab sensors. The sensitivity of evanescent wave sensors was shown to be smaller or much smaller than that of a free space wave traversing the probed medium ecept in the remarkable case of large contrast slab waveguides propagating a restricted number of TM modes where the sensitivity eceeded that of a free space wave [8]. Two types of operation for metal-clad waveguide sensors, peak-type and dip-type operation, were described. A newly discovered peak-type operation could be achieved by use of a few-nanometers-thick cladding of a metal with a large imaginary permittivity, whereas conventional dip-type operation was obtained with a metal cladding with small imaginary permittivity and some tens of nanometers thick. Both types of operation were described, and the main differences were illustrated [9]. In 008, Mehjez et al. investigated nonlinear waveguide sensors with a LHM guiding layer for both transverse electric (TE) and transverse magnetic (TM) waves. The proposed structure consisted of a left-handed material (LHM) as a guiding layer sandwiched between a linear substrate and a nonlinear cover with an intensity dependent refractive inde [30]. A metal-clad planar optical waveguide biosensor with five layer structure was studied theoretically for the detection of Pseudomonas and Pseudomonas-like bacteria. Using a very simple boundary matching technique, the mode equation and other necessary formulae for the proposed biosensor were derived and 7

43 analyzed its performance under different conditions related to its constituents. Significant optimization of the important design parameters to sense micro-scale biological objects were shown numerically and compared with the results given for a biosensor with four layer structure. The importance and need of the inclusion of the thickness of an affinity layer as fifth layer in the four layer structure of the metal clad planar waveguide was discussed [31]. In 011, a four layer slab waveguide structure with one of these layers made of LHM was studied as a sensor. Using Fresnel reflection coefficients, the reflectance of the structure was studied in details. The sensitivity of the effective refractive inde to variations in the refractive inde of a measurand homogeneously distributed in the cladding layer was also presented [3]. In 01, two optical sensor structures based on Fabry-Perot resonators and frings of equal thickness structure were proposed. The first proposed sensor comprised a piezoelectric material as a substrate with the driving potential difference was the sensing probe for refractive inde changes of the sample. The second sensor comprised fringes of equal thickness structure with the number of fringes per unit length was the probe for changes in the inde of the sample. The calculations revealed that the proposed sensors have high sensitivity to changes in the refractive inde of the sample [33]. Kullab et al, investigated a four-layer waveguide structure comprising a dielectric substrate, a metal layer, a left-handed material (LHM) as a guiding layer, and a cladding as a metal-clad waveguide sensor. Fresnel reflection coefficients were used to study the resonance dips at which the reflectance minimizes. The effects of the LHM permittivity, permeability, and thickness on the reflectance curves were studied [34]. A symmetric three-layer slab waveguide with a left-handed material as a guiding layer was eamined analytically for cover refractive inde detection. The TM 8

44 mode dispersion relation of the proposed waveguide was derived. The sensitivity of the proposed sensor to changes in the cover refractive inde and the power flowing within each layer were presented. Some unusual features were founded; the sensitivity improvement compared with the conventional three-layer waveguide sensor is approimately a factor of 6 [35]. A four-layer metal-clad waveguide structure was considered as an optical sensor for refractometry applications. The structure had a LHM as a core layer. The structure parameters were chosen so that the reflectance profile of the proposed structure shows a sharp peak, which was appropriate for sensing applications. The sensor was found to ehibit a considerable angular shift of the reflectance peak for small changes in the refractive inde of the analyte, due to the LHM layer [36]. Transverse magnetic (TM) waves in a four-layer slab waveguide structure were studied for optical sensing applications. The structure consisted of a semi-infinite substrate, a thin metal layer, a LHM guiding layer, and a semi-infinite layer as a cover. The proposed sensor was operated in reflection mode in which the angular position of the reflectance peak was used to detect small changes in the refractive inde of the cover medium. The optimal structure parameters that correspond to the sharpest and highest peak were presented. The results revealed that for aluminum metal layer, a thickness of about 9 nm represents the optimum metal thickness. Moreover, the thickness, negative permittivity, and negative permeability of the guiding layer were found to have great impacts on the performance of the proposed optical waveguide sensor [37]. 9

45 Chapter Three Slab Waveguide Sensor With LHM Core Layer In this chapter, three-layers slab waveguide structure with a left-handed material (LHM) is treated as an optical sensor. The LHM is assumed to present in the core layer. Three different guided modes are studied. The sensitivity of the effective refractive inde to any variation in an analyte inde is presented in details for each mode. 3.1 Dispersion relation Figure 3.1. shows a schematic diagram of the slab waveguide structure under consideration. It consists of a thin film of high refractive inde having parameters (, ) and thickness (d), deposited on a semi-infinite substrate with parameters ( 1, 1 ) and a semi-infinite cladding at the top of the film with parameters ( 3, 3 ) which represents the sample to be tested. Cladding 3, 3 Film, d z Substrate 1, 1 y Fig Three-layers slab waveguide structure. 31

46 31 In TE mode the only nonzero components are y H E, and z H. Rewriting Mawell's equations in the epanded form, ˆ) ˆ ˆ ( 0 0 ˆ ˆ ˆ k H j H i H i E z y k j i z y y, (3.1) and ˆ) ˆ ˆ ( 0 ˆ ˆ ˆ k E j E i E i H H z y k j i z y z. (3.) Eq. (3.1) gives H E y, (3.3) E i H y z 1, (3.4) whereas Eq. (3.) gives z H H E i z y. (3.5) Substituting Eqs. (3.3) and (3.4) into Eq. (3.5) yields, y y y E i E i E i 1. (3.6) From which the TE wave equation can be obtained as 0 ) ( y y E E. (3.7) The above equation is called Helmholtz equation for TE polarized light. The solution of Eq. (3.7) in the three layers is given by

47 3 0, ),0 sin( ) cos(, ) ( ) ( Ce d B B d Ae E d y, (3.8) where A, B 1, B and C are coefficients representing the wave amplitudes in the three layers and the parameters 1, and 3 have the forms 1/ ) ( k, (3.9) 1/ 0 ) ( k, (3.10) and 1/ ) ( k. (3.11) Consequently, the nonzero magnetic field components can be written as d e C B B d e A H d 0 : 0 : )) sin( ) cos( ( 1 : ) ( ) ( 3, (3.1) 0 : 0 )) : cos( ) sin( ( : 1 ) ( ) ( 3 3 Ce d B B d Ae i H d z. (3.13) Applying the boundary conditions at = 0 and = d, the dispersion relation can be obtained as m d tan tan, (3.14) where 3 3 3, (3.15) and

48 1 1. (3.16) 1 3. Sensitivity of the structure For homogeneous sensing in which the analyte is homogeneously distributed in the cover layer, the sensitivity of the effective refractive inde to the variation of the cover inde is given by N S, (3.17) n 3 where n 3 is the cover refractive inde. To derive S, the dispersion relation given by Eq. (3.14) is differentiated with respect to N. We get, n S d. (3.18) 3 3N Power flow The time-average power flowing in the three layers of the waveguide is given by Eq. (1.65). Making use of the three equations giving (3.8)), we get E y in the three layers ( Eq. 1 P1 Ce d C, (3.19) 4 and ( ) 3 d 3 Ae d A, (3.0) 3 4 d 3 3 P d P B1 cos B sin 0 d, 33

49 1 B 1B B B cos dsin d B B d d 1 1 sin 4 P (3.1) The coefficients A, B 1, B and C can be connected to each other through the boundary conditions. It is found that B C 1, (3.) and B 1, (3.3) 1 C A 1 cos d sin d C. (3.4) 1 The total power flowing in the proposed waveguide structure can be obtained by summing the power in all layers P total P. (3.5) P3 P1 The sensitivity of slab waveguides to any change in the inde of the analyte is essentially dependent on the fraction of total power flowing in the cladding layer Results and discussion In the following computations, a three-layer waveguide structure is assumed in which the core layer is assumed to be LHM. Different mode orders are investigated. For a given mode order, the sensitivity of the proposed waveguide sensor is studied with the thickness of the core layer, frequency of the guided wave, and the parameters of the LHM. Moreover, with each waveguide structure, two analytes are presented, namely, water and air. In the following analysis, the core layer is assumed to be a lossy dispersive LHM in which ε and µ are given by[7,8] p 1, (3.6) i 34

50 F 1, (3.7) i 0 where F is the fractional area of the unit cell occupied by the split ring, p is the plasma frequency, ω 0 is the resonance frequency and γ is the damping coefficient. The substrate is considered to be a lossless dielectric with ε 1 =.161, µ 1 = 1. As mentioned above, the cladding layer is assumed to be either water (ε 3 = 1.77, µ 3 = 1) or air (ε 3 = 1, µ 3 = 1). The frequency range was taken from 4.0 GHz to 6.0 GHz in which ε and µ are simultaneously negative according to Eqs. (3.6) and (3.7). When the core layer is LHM, the fundamental mode (m = 0) does not eist as the numerical calculations showed. This is in agreement with the published work on the propagation of electromagnetic waves in LHM waveguide structures [38]. As a result, the following calculations represent slab waveguide sensors operated in higher modes The first mode (m=1) Figures 3. and 3.3 show the sensitivity of the proposed waveguide sensor with a LHM core layer versus the thickness of the LHM layer for different values of the wave frequency. Figures 3. and 3.3 are plotted for water and air analytes respectively. It is obvious from the figures that the sensitivity increases with increasing the thickness of the core layer. In conventional slab waveguide sensors comprising a dielectric core layer, the sensitivity increases with increasing the core layer thickness and peaks at an optimal value of the thickness. For thicknesses beyond the optimal thickness, the sensitivity decays to zero due to the high confinement of the wave in the core layer and the reduction of the evanescent tail in the surrounding media. The sensitivity of the proposed sensor behaves in a similar manner to that of the conventional waveguide sensor for thicknesses below 35

51 optimal value. As can be seen from the two figures, the sensitivity can reach up to 5% and 18% for water and air analytes, respectively. These values are comparable to the sensitivities obtained in conventional waveguide sensors consisting of lossless dielectric materials for the fundamental guided mode [4, 5]. When these values are compared to the sensitivity of the first mode of conventional waveguide sensors, it is found that a dramatic sensitivity enhancement is obtained in the proposed structure due to the presence of the LHM core layer. This enhancement can be attributed to the enlargement of the evanescent field caused by the LHM as reported in the literature [39]. The figures also illustrate the dependence of the sensitivity on the guided wave frequency. The sensitivity can be slightly enhanced with decreasing the guided wave frequency. As the frequency increases, the wavelength decreases and the wave confinement in the core layer increases. Consequently the evanescent field in the surrounding media decreases and so does the sensitivity as shown in Figs. 3.. and 3.3. The behavior of the sensitivity is eactly the same for water and air analytes with larger values in case of water. This can be attributed to the ratio which when increases, for constant n 1, enhances the evanescent tail in the cladding layer. n n 3 36