Introduction to Slab Dielectric Waveguides

Transcription

1 Notes on Integrated Optics Introduction to Slab Dielectric Waveguides Prof. Elias N. Glytsis Dec. 6, 26 School of Electrical & Computer Engineering National Technical University of Athens

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3 Contents Ray Approach for Guided Modes. Alternate Approach Electromagnetic Approach for Guided Modes 5 2. TE Guided Modes TM Guided Modes Substrate Modes 3 4 Radiation Modes 5 5 Unphysical Modes 6 6 Evanescent Modes 9 7 Normalized Slab-Waveguide Variables 22 8 Cutoff Conditions 24 9 Power Considerations TE Modes TM Modes Multi-Layered Slab Waveguides 3. TE Guided Modes TM Guided Modes Finite-Difference Frequency-Domain Method 37. FDFD Method Based on Yee s Cell Average-Scheme Modification for Arbitrary Boundary Location Graded-Index Slab Waveguides 54 References 65 3

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5 INTRODUCTION TO SLAB DIELECTRIC WAVEGUIDES. Ray Approach for Guided Modes The basic geometry of a dielectric slab waveguide is shown in Fig.. The light is confined along the x-axis and propagates along the z-axis in the coordinate system shown. The waveguide is assumed to be uniform along the y-axis and therefore this comprises a two-dimensional electromagnetic problem. The basic principle behind the concept of light guiding in dielectric waveguide is the phenomenon of total internal reflection. If light can be launched into the film layer of the waveguide it can remain in the film, provided that the angle of incidence θ between the film-cover and the film-substrate regions is greater than both of the corresponding critical angles. The refractive indices of the three regions are n c, n f, and n s for cover, film, and substrate regions, respectively while the film thickness is h and the free-space wavelength of the light that can travel guided within this structure is λ. Of course it is necessary that n f > max{n c, n s } in order to guarantee that total internal reflection can occur in both filmsubstrate and film-cover boundaries. The angle θ, known also as the zig-zag angle, for a specific mode should satisfy the condition max{θ cr,fs, θ cr,fc } < θ < π/2, where θ cr,fs = sin (n s /n f ) and θ cr,fc = sin (n c /n f ), are the critical angles for the film-substrate and film-cover boundaries, respectively. Without loss of generality, in the following discussion, it is assumed that n s n c (which is usually the most practical case). In the special case that n s = n c the waveguide is characterized as a symmetric slab waveguide. It is well known that there is an induced phase shift upon total internal reflection at both boundaries. Therefore there should be a self-consistency condition that must be satisfied in order for the plane waves shown in the ray picture of Fig. to comprise a valid solution for a guided mode. In order to quantify the self-consistency condition Fig. 2 will be used []. In this figure two possible rays are shown along with two wavefronts the one specified by AB and the other by DC. The wavefronts AB and DC are selected to be at an infinitesimal distance away from the film-cover and film-substrate boundaries, respectively. This implies that the ray from point c 26 Prof. Elias N. Glytsis, Last Update: Dec. 6, 26

6 Figure : The geometric configuration of a three-region slab dielectric waveguide. B to point D has not suffered any total internal reflections. In contrast, the ray from point A to point C has suffered two total internal reflections at the two boundaries. Following the phases accumulated by points A and B of the initial wavefront as the rays move towards to their positions C and D of the second wavefront shown, it is necessary to require that the accumulated phase from A C and from B D should differ at most by 2πν where ν is an integer. This condition will guarantee that the second wavefront will remain a valid wavefront of the guided mode. Mathematically, the previous conditions can be written as follows { k n f (AC) + 2φ p fs + 2φp fc} } {{ } phase from A C { k n f (BD)} }{{} = 2πν, ν =, ±, ±2, () phase from B D where k = 2π/λ, and φ p fs and φp fc (p = TE or TM) are the phase shifts that occur upon total internal reflection at the film-substrate and film-cover boundaries respectively. These phase shifts are functions of the angle θ and are given by the equations (as it was presented in the Review of Electromagnetic Principles notes) n 2 φ TE f sin2 θ n 2 w fw (θ) = tan, for TE Polarization, (2) n f cosθ φ TM fw where w = c or s. (θ) = tan n 2 f n 2 w n 2 f sin2 θ n 2 w, for TM Polarization, (3) n f cos θ 2

7 6 7 5 : ' ( ) * +! " # $, -., / + / * % & Figure 2: The slab waveguide geometric configuration showing two rays and and two selective wavefronts (AB and DC). For self-consistency it is necessary that the accumulated phase from A C and from B D should differ by 2πν where ν is an integer. The points A and C are assumed to be infinitesimaly before/after the film-cover/film-substrate boundaries respectively. The distances (AC) and (BD) can be easily determined from the geometry shown in Fig. 2. Specifically, (AC) = h/cos θ and (BD) = (AD)sin θ = [(OA) (OD)] sin θ = [htan θ h/tan θ] sinθ = h[sin 2 θ cos 2 θ]/cos θ. Replacing the previous expressions in Eq. () and changing the signs assigning only positive (or zero) values for ν the following equation is derived 2k n f hcos θ 2φ p fs (θ) 2φp fc (θ) = 2πν, ν =,, 2, (4) where p = TE or TM. Of course solution for θ of the previous equation has meaning for a guided mode only when max{θ cr,fs, θ cr,fc } < θ < π/2. This equation can be solved only numerically (using for example the bisection method). For every value of ν solutions for θ p ν (p = TE or TM) can determined. There can be none, one, or multiple solutions depending on the parameters of the waveguide and the free-space wavelength. A graphical representation of the solution is shown in Fig. 3 for the example case where λ =.µm, h =.2µm, n c =., n f = 2.2, and n s =.5. The solutions for the zig-zag angles θ p ν can be visualized as the intersection of two curves representing the functions f (θ) = 2k n f hcos θ and f 2 (θ) = 2φ p fs (θ) + 2φp fc (θ) + 2πν (p = TE or TM). In this particular example there are 4 TE modes (TE, TE, TE 2, and TE 3 ) and 4 TM modes (TM, TM, TM 2, and TM 3 ) that can be supported. It can be observed that the solutions θ TE ν shift [φ TE fw and θ TM ν satisfy the inequality θ TE ν > θ TM ν (for any ν) due to the larger phase (θ) < φtm fw (θ), w = c or s] upon total internal reflection for the TM polarization. 3

8 4 λ = µm, n c =, n f = 2.2, n s =.5, h =.2µm Function f (θ) 2 2 Function f 2 (θ) TE TM Angle θ,(deg) Figure 3: The graphical representation of the solution of the dispersion equation 2k hn f cos θ 2φ p fc 2φ p fs = 2νπ (p = TE or TM, and ν =,, ). The solution for the zig-zag angle θ is shown as the intersection of two simple curves, the f (θ) = 2k hn f cosθ and f 2 (θ) = 2φ p fc (θ) + 2φp fs (θ) + 2νπ. For the example case shown λ =.µm, h =.2µm, n c =., n f = 2.2, and n s =.5. The solutions for the zigzag angles are θ TE = 8.537, θ TE = 7.854, θ2 TE = , and θ3 TE = for the TE modes, and θ TM = , θ TM = , and θ2 TM = , and θ3 TM = for the TM modes. The corresponding critical angles are θ cr,fc = and θ cr,fs = In general, the solution for the zig-zag angle θ = θν p (where p = TE or TM) is characteristic of the guided mode. The associated to θν p electromagnetic field represents the guided mode TE ν or TM ν and has a characteristic profile. The complete electromagnetic field will be determined in the next section when the full electromagnetic approach will be presented. From the angle θν, p the effective refractive index of the mode can be defined as Nν p = n f sinθν. p Furthermore, the effective propagation constant is related to the effective index as βν p = k Nν p = k n f sin θν p (p = TE or TM).. Alternate Approach A similar approach based on a self-consistency condition is presented in this section and is based on the geometry shown in Fig.. In this ray diagram of a guided mode in the waveguide a plane wave is associated with each ray. Specifically (neglecting polarization) the three plane 4

9 waves, S, S, and S 2, can be described by the following expressions S = S (x, z) = E e jk n f (x cosθ+z sin θ), (5) S = S (x, z) = E e jk n f ( x cosθ+z sinθ), (6) S 2 = S 2 (x, z) = E 2 e jk n f (x cosθ+z sin θ). (7) The plane wave S is produced by a reflection of S at the film-cover boundary. Similarly, plane wave S 2 is produced by reflection of S at the film-substrate boundary. These arguments can be quantified as follows S (x = h, z) = r fc S (x = h, z) E e +jk n f hcosθ = e j2φp fc E e jk n f hcosθ, (8) S 2 (x =, z) = r fs S (x = h, z) E 2 = e j2φp fs E, (9) where r fc and r fs are the reflection coefficients upon total internal reflection at the film-cover and film-substrate interfaces, respectively, and p = T E or T M. From the last two equations it can be deduced that E 2 = E e j( 2k n f hcosθ+2φ p fs +2φp fc ). () The last equation requires that in order for S 2 to be equivalent to S their amplitudes must differ by at most by 2πν, i.e. E 2 = E e j2πν = E e j( 2k n f hcosθ+2φ p fs +2φp fc ), () and from the last expression Eq. (4) is derived again. 2. Electromagnetic Approach for Guided Modes The electromagnetic fields representing a mode for the slab waveguide problem (using the coordinate axes system shown in Fig. ) can be written in the form E = [E x (x)ˆx + E y (x)ŷ + E z (x)ẑ] exp( jβz), (2) H = [H x (x)ˆx + H y (x)ŷ + H z (x)ẑ] exp( jβz), (3) where E w and H w (w = x, y, z) represent the electric and magnetic field components and β is a propagation constant. Using Maxwell equations in their differential time-harmonic form, for a homogeneous, linear, lossless, and isotropic material, of permittivity ɛ and permeability µ 5

10 (non-magnetic) with the above field expressions results in the following sets of equations (ω is the radial frequency) d dx E y H z H y E z jωµ = jωɛ + j β2 E y ωµ H z jωɛ H y, (4) jωµ j β2 E z ωɛ [ ] Hx β [ ] ωµ Ey =. (5) E x β H y ωɛ From the above equations it is straightforward to distinguish two set of field components, the {E y, H x, H z } and {H y, E x, E z }, which are independent from each other due to the zeros appearing in the matrices of Eqs. (4) and (5). These two sets can be used to define the TE-modes solutions for {E y, H x, H z }, and the TM-modes solutions for {H y, E x, E z }. It is mentioned that these two triplets of field components are independent only in the isotropic case. If the material becomes anisotropic the zero elements are replaced by non-zero ones and all six field components become coupled. Then a waveguide mode, usually called hybrid mode, can be characterized by all six field components as {E x, E y, E z, H x, H y, H z }. In a homogeneous, isotropic, lossless, non-magnetic, and linear material the solutions to the Helmholtz s equation are in the form of plane waves. For example, if solutions of the form of Eqs. (2), (3) are sought the Helmholtz s equation becomes d 2 U dx 2 + (k2 n2 β 2 ) U =, (6) where U = E or H for electric of magnetic field, respectively, and n = ε = ɛ/ɛ is the refractive index of the material (ε is the relative permittivity of the material). The solutions of the above equation have the form U = U + exp( j k + r) + U exp( j k r), (7) where U + and U are the vector amplitudes of the two plane wave solutions and k + and k are their corresponding wave-vectors. The k + /k terms represent the waves propagating towards 6

11 the positive/negative x direction. By applying the above solutions to each of the three separate regions of the slab waveguide of Fig., one can write the following equations U c e j(kcxx+kczz) + U c2 e j( kcxx+kczz), x > h, U = U f e j(k fxx+k fz z) + U f2 e j( k fxx+k fz z), < x < h, (8) U s e j(ksxx+kszz) + U s2 e j( ksxx+kszz), x <, where the subscripts c, f, and s denote fields as well as wavevector components in cover, film, and substrate, respectively. From the phase matching condition along the boundaries (x = for film-substrate and x = h for film-cover) the following relation is necessary k cz = k fz = k sz = β. (9) At the same time the wave-vector components should satisfy the plane wave dispersion equation of the form kwx 2 + β2 = k 2n2 w (for w = c, f, and s). In order to warrant total internal reflection at the film-cover and film-substrate boundaries β must satisfy the inequality k n c k n s < β < k n f. Then the x components of the wavevectors become kcx 2 = k2 n2 c β2 < k cx = ±j β 2 k 2n2 c = ±jγ c, (2) kfx 2 = kn 2 2 f β 2 > k fx = ± kn 2 2 f β2, (2) ksx 2 = kn 2 2 s β 2 < k sx = ±j β 2 kn 2 2 s = ±jγ s, (22) where the signs of the imaginary wave-vector components must be selected in such a way in order to warrant exponentially decaying solutions in the cover and the substrate regions. Thus, from the two possible solutions in the cover and the substrate regions only the exponentially decaying one are retained. Therefore the field solutions can be summarized as U c e γc(x h) e jβz, x > h, [ U = U f e jkfxx + U f2 e fxx] +jk e jβz, < x < h, (23) U s e γsx e jβz, x <, The unknown constants of the above equations (field amplitudes and β) should be determined by suitable application of the boundary conditions. The boundary conditions are the continuity of the tangential electric field and tangential magnetic field components at the film-cover and 7

12 film-substrate boundaries. In order to proceed with the boundary conditions and the determination of the unknowns the two distinguished families of modes (TE and TM) will be treated separately since they are decoupled (in isotropic regions). 2. TE Guided Modes As it was explained in the previous section the TE mode solutions include the field triplet {E y, H x, H z }. Then the electric field described in Eq. (23) can written in the form E c e γc(x h) e jβz, x > h, [ E = ŷ Ef e jkfxx + E f2 e ] +jk fxx e jβz, < x < h, E s e γsx e jβz, x <, The magnetic field components can be determined from Maxwell s equations, H x = (/jωµ )(de y /dz) and H z = (/jωµ )(de y /dx), and are given by and by H x = β ωµ H z = ωµ E c e γc(x h) e jβz, x > h, [ Ef e jk fxx + E f2 e +jk fxx ] e jβz, < x < h, E s e γsx e jβz, x <, jγ c E c e γc(x h) e jβz, x > h, [ kfx E f e jk fxx k fx E f2 e +jk fxx ] e jβz, < x < h, jγ s E s e γsx e jβz, x <. Using the continuity of the tangential electric and magnetic field components across the film-cover [E y (x = h + ) = E y (x = h ) and H z (x = h + ) = H z (x = h )] and film-substrate boundaries [E y (x = + ) = E y (x = ) and H z (x = + ) = H z (x = )] the following system of equations is formed e jk fxh e +jk fxh jγ c k fx e jk fxh k fx e +jk fxh k fx k fx jγ s } {{ } Ã TE (β 2 ) E c E f E f2 E s = (24) (25) (26). (27) 8

13 In order to have nontrivial solutions of the previous equation it is necessary that the determinant of à TE be set to zero. After some manipulations the following dispersion equation is derived det{ãte(β 2 )} = = tan(k fx h) = γ s + γ c k fx k fx γ s γ c. (28) k fx k fx A graphical representation of Eq. (28) is shown in Fig. 4 (for p = TE). There can be none, one, or multiple solutions depending on the waveguide parameters and the free-space wavelength. For this particular example (the waveguide parameters are given in the figure caption) there 4 TE and 4 TM solutions. This example case is the same with the one used for the graphical representation of Eq. (4). It is straightforward to show that the last equation is equivalent to Eq. (4). It is worth mentioning that the dispersion equation is actually a function of β 2 (or N 2 ) and not of just β (or N). This implies that if β ν is a solution of Eq. (28) then β ν is also a solution that corresponds to the same mode propagating backwards (along the z axis). If β ν satisfies Eq. (28) then the matrix ÃTE becomes singular and correspondingly the boundary conditions contained in Eq. (27) become dependent. This fact provides the flexibility of selecting a free parameter and then solve for the electric field amplitudes E c, E f, E f2, and E s as functions of this parameter. It is a simple task to perform this procedure and find that E c = E cos(k fx h φ TE fs ), E f = (E /2)exp(+jφ TE fs ), E f2 = (E /2)exp( jφ TE fs ), and E s = E cos φ TE fs where E is the free parameter and φ TE fs = tan (γ s /k fx ). The amplitude E is the free parameter and can also be determined if information is known about the power that the mode carries. It is mentioned that all amplitudes are calculated for β = β ν and therefore are characteristic of the TE ν mode. Then the final form of the electric field of the TE ν mode is given by cos(k fx h φ TE fs ) e γc(x h) e jβνz, x > h, E ν = ŷe cos(k fx x φ TE fs ) e jβνz, < x < h, cos φ TE fs e γsx e jβνz, x <, where for simplicity the subscript ν, which is characteristic of the TE ν mode, has been omitted from E, k fx, γ c, γ s and φ TE fs. Example TE ν mode electric field patterns are shown in Fig. 5 where the effective index solutions are also included. It can be observed that the number of zero-crossings of the electric field profile for TE ν mode is ν. 9 (29)

14 n c =, n f = 2.2, n s =.5, h =.2µm, λ = µm Functions f (N) or f 2 (N) f f 2 for TE f 2 for TM Effective Index, N Figure 4: A graphical representation of the solution of Eq. (28) for TE modes. The functions f and f 2 are given by f (N) = tan(k fx h) and f 2 (N) = [(γ s /k fx ) + (γ c /k fx )]/[ (γ c γ s /k 2 fx )], where k fx = k (n 2 f N2 ) /2, γ c = k (N 2 n 2 c) /2, γ s = k (N 2 n 2 s) /2, and N is the effective index. The case parameters are n c =, n f = 2.2, n s =.5, h =.2µm, and λ =.µm. The points along the effective index axis where f is discontinuous and infinite are given by N(m) = [n 2 f ((2m + )/4)2 (λ /h) 2 ] /2 (m =,, ). Similarly f 2 becomes infinite and discontinuous at N = [(n 4 f n2 sn 2 c)/(2n 2 f n2 c n 2 s)] /2. In the case of TM modes the solution of Eq. (34) is also shown. In the latter case f remains the same but f 2 becomes f 2 (N) = [(n 2 f /n2 s)(γ s /k fx ) + (n 2 f /n2 c)(γ c /k fx )]/[ (n 4 f /n2 cn 2 s)(γ c γ s /k 2 fx )]. In the TM case f 2 becomes discontinuous at N being the solution of ( a 2 )N 4 + (2n 2 f a2 n 2 s n2 c )N2 + (n 2 c n2 s n4 f a2 ) =, that lies in the interval n s < N < n f (where a = n 2 sn 2 c/n 4 f ). 2.2 TM Guided Modes The TM mode solutions include the field triplet {H y, E x, E z }. Then the magnetic field represented in Eqs. (23) can written in the form H c e γc(x h) e jβz, x > h, [ H = ŷ Hf e jkfxx + H f2 e ] +jk fxx e jβz, < x < h, (3) H s e γsx e jβz, x <, The electric field components can be easily determined from Maxwell s equations, E x = (/jωɛ)(dh y /dz) and E z = (/jωɛ)(dh y /dx), and are given by

15 TE (N eff = 2.7) TE (N eff = 2.783) Normalized E y Normalized E y TE 2 (N eff =.99) TE 3 (N eff =.683) Normalized E y Normalized E y Figure 5: The normalized electric field profiles for TE ν modes (ν =,, 2,and 3). The slab waveguide parameters are n c =, n f = 2.2, n s =.5, h =.2µm, and λ =.µm. The vertical lines indicate the film-substrate and film-cover boundaries. The effective indices are also shown on the top of each plot. E x = β ωɛ H c e γc(x h) e jβz, x > h, n 2 c [ Hf e jkfxx + H f2 e ] +jk fxx e jβz, < x < h, n 2 f (3) and by E z = ωɛ H s e γsx e jβz, x <, n 2 s +j γ c H n 2 c e γc(x h) e jβz, x > h, c [ ] k fx n 2 f H f e jkfxx k fx H n 2 f2 e +jk fxx f e jβz, < x < h, (32) j γ s H n 2 s e γsx e jβz, x <. s Using the continuity of the tangential electric and magnetic field components across the

16 film-cover [H y (x = h + ) = H y (x = h ) and E z (x = h + ) = E z (x = h )] and film-substrate boundaries [H y (x = + ) = H y (x = ) and E z (x = + ) = E z (x = )] the following system of equations is formed e jk fxh e +jk fxh j γ c k fx e jk k fxh fx e +jk fxh n 2 c n 2 f n 2 f k fx k fx j γ s n 2 f n 2 f } {{ } à TM (β 2 ) In order to have nontrivial solutions of the previous equation it is necessary that the determinant n 2 s H c H f H f2 H s =. (33) of à TM be set to zero. After some manipulations, similar to the TE polarization case, the following dispersion equation is derived det{ãtm(β 2 )} = = tan(k fx h) = n 2 f n 2 s γ s k fx + n2 f n 2 c γ c k fx n4 f γ c γ s n 2 sn 2 c kfx 2. (34) It is straightforward to show again that the last equation is equivalent to Eq. (4) (for p = TM). A graphical representation of Eq. (34) is also shown in Fig. 4. As for the TE polarization case the solution ±β ν of Eq. (34) correspond to the ±z-axis propagating mode (assuming that k n s < β ν < k n f ). If β ν satisfies Eq. (34) then the matrix à TM becomes singular and correspondingly the boundary conditions contained in Eq. (33) become dependent. This provide the flexibility of selecting a free parameter and then solve for the magnetic field amplitudes H c, H f, H f2, and H s as functions of this parameter. It is a simple task to perform this procedure and find that H c = H cos(k fx h φ TM fs ), H f = (H /2)exp(+jφ TM fs ), H f2 = (H /2)exp( jφ TM fs ), and H s = H cosφ TM fs where H is the free parameter and φ TM fs = tan [n 2 f γ s/(n 2 sk fx )]. The amplitude H is the free parameter and can also be determined if information about the power that the mode carries is known. It is mentioned that all amplitudes are calculated for β = β ν and therefore are characteristic of the TM ν mode. Then the final form of the magnetic field of the TM ν mode is given by H ν = ŷh cos(k fx h φ TM fs ) e γc(x h) e jβνz, x > h, cos(k fx x φ TM fs ) e jβνz, < x < h, cos φ TM fs e γsx e jβνz, x <, 2 (35)

17 where for simplicity the subscript ν, which is characteristic of the TM ν mode, has been omitted from H, k fx, γ c, γ s and φ TM fs. Example TM ν mode electric field patterns are shown in Fig. 6 where the effective index solutions are also included. Again it is observed that the number of zero-crossings of the magnetic field profile for TM ν mode is ν. TM (N eff = 2.642) TM (N eff = 2.542) Normalized H y Normalized H y TM 2 (N eff =.8636) TM 3 (N eff =.5968) Normalized H y Normalized H y Figure 6: The normalized electric field profiles for TM ν modes (ν =,, 2, and 3). The slab waveguide parameters are n c =, n f = 2.2, n s =.5, h =.2µm, and λ =.µm. The vertical lines indicate the film-substrate and film-cover boundaries. The effective indices are also shown on the top of each plot. 3. Substrate Modes In the previous sections solutions for guided modes were determined. This was based on the assumption that k n s < β < k n f (or n s < N < n f ) in order to guarantee that total internal reflection occurs across the film-cover and film-substrate boundaries. When k n c < β < k n s (or n c < N < n s ) then the electromagnetic field in the substrate region is propagating instead of evanescent. The modes that have this property are called substrate modes and radiate power into the substrate as they propagate. For guided modes the solutions for β (or N) were discretized. In the case of the substrate modes the solutions for β form a continuum. Thus, any 3

18 β in the interval k n c < β < k n s can be a solution for a substrate mode. It is straightforward to determine the electromagnetic field that correspond to a substrate mode. The electric or magnetic field of a TE or TM substrate mode can be written as U c e γc(x h) e jβz, x > h, [ U = U(x, z)ŷ = ŷ Uf e jkfxx + U f2 e ] +jk fxx e jβz, < x < h, (36) [ Us e jksxx + U s2 e +jksxx] e jβz, x <, where U = E y for TE substrate modes and U = H y for TM substrate modes, and k sx = (kn 2 2 s β 2 ) /2 > for the range of β that is valid for substrate modes. Using the continuity of the tangential electric and magnetic field components 4 equations can be specified. However, the number of unknowns is six, i.e. U c, U f, U f2, U s, U s2, and β. Therefore, there is some flexibility in satisfying the boundary conditions. For example, any β in the interval k n c < β < k n s can satisfy the boundary conditions. After some manipulations of the resulting boundary conditions it can be shown that the electric field of a TE substrate mode is given by E β = ŷe cos φ TE fc e γc(x h) e jβz, x > h, cos[k fx (x h) + φ TE fc ] e jβz, < x < h, [ cos(k fx h φ TE fc )cos(k sxx) + k ] fx sin(k fx h φ TE fc k )sin(k sxx) sx e jβz, x <, where E is a free parameter and φ TE fc = tan (γ c /k fx ). Similarly the magnetic field for a TM substrate mode is given by cosφ TM fc e γc(x h) e jβz, x > h, H β = ŷh cos[k fx (x h) + φ TM fc ] e jβz, < x < h, [ cos(k fx h φ TM fc )cos(k sxx) + k fx/n 2 f k sx /n 2 s where H is a free parameter and φ TM fc ] sin(k fx h φ TM fc )sin(k sxx) (37) e jβz, x <, (38) = tan (γ c /n 2 c )/(k fx/n 2 f ). Sample normalized electric/magnetic field profiles for TE/TM substrate modes are shown in Fig. 7. 4

19 TE Substrate Mode, (N eff =.45) TM Substrate Mode, (N eff =.45).5.5 Normalized E y Normalized H y TE Substrate Mode, (N eff =.3) TM Substrate Mode, (N eff =.3).5.5 Normalized E y Normalized H y Figure 7: The normalized electric/magnetic field profiles for T E/T M substrate modes. The slab waveguide parameters are n c =, n f = 2.2, n s =.5, h =.2µm, and λ =.µm. The vertical lines indicate the film-substrate and film-cover boundaries. The effective indices are also shown on the top of each plot. 4. Radiation Modes When < β < k n c (or < N < n c ) then the field in all three regions (cover, film, and substrate) is propagating. The modes that have this property are called r adiation modes and radiate power into both the cover and the substrate regions as they propagate. As in the case of the substrate modes, the solutions for β form a continuum and for the radiation modes. Thus, any β in the interval < β < k n c can be a solution for a radiation mode. It is straightforward to determine the electromagnetic field of a radiation mode. The electric or magnetic field of a TE or TM radiation mode can be written as [ Uc e jkcx(x h) + U c2 e +jkcx(x h)] e jβz, x > h, U = U(x, z)ŷ = ŷ [ Uf e jk fxx + U f2 e +jk fxx ] e jβz, < x < h, [ Us e jksxx + U s2 e +jksxx] e jβz, x <, (39) 5

20 where U = E y for TE radiation modes and U = H y for TM radiation modes, and k cx = (k 2 n2 c β2 ) /2 > for the range of β that is valid for radiation modes. Using the continuity of the tangential electric and magnetic field components 4 equations can be specified. However, the number of unknowns is seven, i.e. U c, U c2, U f, U f2, U s, U s2, and β. Therefore, there is even more flexibility in satisfying the boundary conditions. For example, any β in the interval < β < k n c can satisfy the boundary conditions. After some manipulations of the resulting boundary conditions it can be shown that the electric field of a T E radiation mode is given by [ ( + k fx )cos[k cx (x h) + k fx h φ] + 2 k cx ( k ] fx )cos[k cx (x h) k fx h + φ] e jβz, x > h, k cx E β = ŷe (4) cos(k fx x φ) e jβz, < x < h, [ ( + k fx )cos[k sx x φ] + ( k ] fx )cos[k sx x + φ] e jβz, x <, 2 k sx k sx where E and φ are two free parameters. Similarly the magnetic field for a TM radiation mode is given by H β = ŷe [ ( + k fx/n 2 f )cos[k 2 k cx /n 2 cx (x h) + k fx h φ] + c ( k fx/n 2 f )cos[k k cx /n 2 cx (x h) k fx h + φ] c ] e jβz, x > h, cos(k fx x φ) e jβz, < x < h, [ ( + k fx/n 2 f )cos[k 2 k sx /n 2 sx x φ] + ( k fx/n 2 ] f )cos[k s k sx /n 2 sx x + φ] e jβz, x <, s where H and φ are two free parameters. Sample normalized electric/magnetic field profiles for TE/TM radiation modes are shown in Fig Unphysical Modes When k n f < β < (or n f < N < ) then the field in all three regions (cover, film, and substrate) is comprised of evanescent terms, usually including both exponentially increasing and exponentially decreasing terms. The modes that have this property are called unphysical modes since there are valid solutions of the Maxwell s equations but they cannot be excited since they (4) 6

21 TE Radiation Mode,, N eff =.95, φ = 35 TM Radiation Mode,, N eff =.95, φ = Normalized E y Normalized H y TE Radiation Mode,, N eff =.95, φ = 3 TM Radiation Mode,, N eff =.95, φ = Normalized E y Normalized H y Figure 8: The normalized electric/magnetic field profiles for T E/T M radiation modes. The slab waveguide parameters are n c =, n f = 2.2, n s =.5, h =.2µm, and λ =.µm. The vertical lines indicate the film-substrate and film-cover boundaries. The effective indices are also shown on the top of each plot along with the choice of the free parameter φ. have infinite power. For this reason are called unphysical modes and are not usually referred to the textbooks. The analysis of these modes is similar with the case of radiation modes with the exception that all field components are comprised of both increasing and decreasing exponential terms. As in the case of the substrate/radiation modes, their solutions for β form a continuum. Thus, any β in the interval k n f < β < can be a solution for an unphysical mode. It is straightforward to determine the electromagnetic field of an unphysical mode. The electric or magnetic field of a TE or TM unphysical mode can be written as [ Uc e γc(x h) + U c2 e +γc(x h)] e jβz, x > h, U = U(x, z)ŷ = ŷ [U f e γ fx + U f2 e +γ fx ] e jβz, < x < h, [U s e γsx + U s2 e +γsx ] e jβz, x <, (42) where U = E y for TE unphysical modes and U = H y for TM unphysical modes, and γ f = (β 2 k 2 n 2 f )/2 > for the range of β that is valid for unphysical modes. If one tries to find 7

22 a solution without the increasing exponentials in the cover and the substrate regions (i.e., for U c2 = U s = ), one ends with a dispersion equation of the form (for TE polarization) tanh(γ f h) = [(γ s /γ f ) + (γ c /γ f )]/[ + (γ c γ s /γf 2 )] that does not have a real solution for β in the range of the unphysical modes (a similar equation does not have a solution for the TM polarization too). Therefore, it is required to retain all the terms appearing in the above equation similarly to the radiation modes case. Using the continuity of the tangential electric and magnetic field components 4 equations can be specified. However, the number of unknowns is seven, i.e. U c, U c2, U f, U f2, U s, U s2, and β (similar to th radiation modes case). Therefore, there is a lot of flexibility in satisfying the boundary conditions. For example, any β in the interval k n f < β < can satisfy the boundary conditions. After some manipulations of the resulting boundary conditions it can be shown that the electric field of a TE unphysical mode is given by E β = ŷe [ ( + γ f )cosh[γ c (x h) + γ f h φ] + 2 γ c ( γ ] f )cosh[γ c (x h) γ f h + φ] e jβz, x > h, γ c cosh(γ f x φ) e jβz, < x < h, [ ( + γ f )cosh[γ s x φ] + ( γ ] f )cosh[γ s x + φ] e jβz, x <, 2 γ s γ s where E and φ are two free parameters. Similarly the magnetic field for a TM unphysical mode is given by H β = ŷe [ ( + γ f/n 2 f )cosh[γ 2 γ c /n 2 c (x h) + γ f h φ] + c ( γ f/n 2 f )cosh[γ γ c /n 2 c (x h) γ f h + φ] c ] e jβz, x > h, cosh(γ f x φ) e jβz, < x < h, [ ( + γ f/n 2 f )cosh[γ 2 γ s /n 2 s x φ] + ( γ f/n 2 ] f )cosh[γ s γ s /n 2 s x + φ] e jβz, x <, s where H and φ are two free parameters. Sample normalized electric/magnetic field profiles for TE/TM radiation modes are shown in Fig. 9. (43) (44) 8

23 TE Unphysical Mode,, N eff = 2.25, φ = TM Unphysical Mode,, N eff = 2.25, φ =.8.8 Normalized E y.6.4 Normalized H y TE Unphysical Mode,, N eff = 2.25, φ s =.7783 TM Unphysical Mode,, N eff = 2.25, φ s = Normalized E y.6.4 Normalized H y Figure 9: The normalized electric/magnetic field profiles for T E/T M unphysical modes. The slab waveguide parameters are n c =, n f = 2.2, n s =.5, h =.2µm, and λ =.µm. The vertical lines indicate the filmsubstrate and film-cover boundaries. The effective indices are also shown on the top of each plot along with the choice of the free parameter φ. For symmetry of the fields in the film layer the φ parameter was chosen as φ = φ s = γ f h/2 in the bottom two plots. 6. Evanescent Modes In some cases there might be a need to use waveguide modes that have a purely imaginary propagation constant β, such that β = ±jβ (where β > ). In this case the waveguide field could be written as U = U(x)exp( j βz) = U(x)exp( βz), where for forward evanescent modes the β = jβ was selected in order for the evanescent mode to decay along the propagation direction z. Strictly speaking, the evanescent modes are not propagating since they decay along the propagation direction. However, they are proper modal solutions of the Helmholtz wave equation as it is applied to the slab waveguide problem. The acceptable imaginary propagation constants are of the form β = jβ (for the forward evanescent modes) and β = +jβ (for the backward evanescent modes), i.e. they all lie along the imaginary β axis. These evanescent modes form a continuum. For example any value of β > gives a propagation constant β = jβ 9

24 which is an acceptable solution to Maxwell equations in the slab waveguide geometry. It is straightforward to determine the electromagnetic field of an evanescent mode. The electric or magnetic field of a T E or T M forward propagating evanescent mode can be written as [ Uc e jk cx(x h) + U c2 e ] +jk cx(x h) e βz, x > h, U = U(x, z)ŷ = ŷ [U f e jk fx x + U f2 e +jk fx x ] e βz, < x < h, [ Us e jk sxx + U s2 e +jk sxx ] e βz, x <, (45) where U = E y for TE forward evanescent modes and U = H y for TM forward evanescent modes, and k cx = (k 2 n 2 c + β 2 ) /2 >, k fx = (k2 n 2 f + β2 ) /2 >, and k sx = (k 2 n 2 s + β 2 ) /2 >, for the range of β that is valid for forward evanescent modes (i.e. the negative imaginary axis of the propagation constant region). Using the continuity of the tangential electric and magnetic field components 4 equations can be specified. However, the number of unknowns is seven, i.e. U c, U c2, U f, U f2, U s, U s2, and β. Therefore, there is even more flexibility in satisfying the boundary conditions. For example, any β in the interval < β < k n c can satisfy the boundary conditions. After some manipulations of the resulting boundary conditions it can be shown that the electric field of a TE radiation mode is given by [ ( + k fx )cos[k 2 k cx(x h) + k fxh φ] + cx E β = ŷe ( k fx )cos[k k cx(x h) k fxh + φ] cx ] e βz, x > h, cos(k fx x φ) e βz, < x < h, (46) [ 2 ( + k fx k sx ] )cos[k sx x φ] + ( k fx )cos[k k sx sx x + φ] e βz, x <, where E and φ are two free parameters (similar to the radiation modes). In an analogous 2

25 manner, the magnetic field for a TM forward evanescent mode is given by H β = ŷe [ ( + k fx/n 2 f )cos[k 2 k cx/n 2 cx (x h) + k fx h φ] + c ( k fx /n2 ] f )cos[k k cx /n2 cx (x h) k fx h + φ] e βz, x > h, c cos(k fx x φ) e βz, < x < h, [ ( + k fx /n2 f )cos[k 2 k sxx φ] + ( k fx /n2 ] f )cos[k sx /n2 s k sxx + φ] e βz, x <, sx /n2 s where H and φ are again the two free parameters. Sample normalized electric/magnetic field profiles for TE/TM forward evanescent modes are shown in Fig.. (47) TE Evanescent Mode,, N eff = - j.5, φ = 3 TM Evanescent Mode,, N eff = - j.5, φ = Normalized E y Normalized H y TE Evanescent Mode,, N eff = - j.5, φ = 35 TM Evanescent Mode,, N eff = - j.5, φ = Normalized E y Normalized H y Figure : The normalized electric/magnetic field profiles for T E/T M forward evanescent modes. The slab waveguide parameters are n c =, n f = 2.2, n s =.5, h =.2µm, and λ =.µm. The vertical lines indicate the film-substrate and film-cover boundaries. The effective indices are also shown on the top of each plot along with the choice of the free parameter φ. The various modes of the slab waveguide problem (the example is for the single-film layer slab waveguide but it can be easily generalized to multilayer-film slab waveguides) are shown 2

26 < < < V V V V in the complex propagation constant ( β = β r + jβ i ) diagram in Fig.. The only discrete spectrum is of the guided modes as it can be seen. The substrate, radiation, unphysical, and evanescent modes from a continuum. In the same diagram the leaky modes are also shown. The leaky modes is an approximation of the radiation field by a series of discrete modes of complex propagation constant. These leaky modes will be discussed at a future section of these notes. u n v w m n l o h [ d i Z [ Y \ ] ^ _ ` a b c d [ e f X \ g b p q r s k t T L U P K L J M N O L P Q R I M O S E C G E C F E C D h [ d i Z [ Y \ } ~ b z Y [ z g f X \ g b h [ d i Z [ Y \ { c \ g \ f X \ g b = A [ \ c [ z c X ^ f X \ g b X ^ z c ^ } _ g d z Y ƒ ˆ Š Œ Š ˆ Ž h [ d i Z [ Y \ x y [ ^ g b d g ^ z f X \ g b W X Y Z [ Y \ } ~ b z Y [ z g f X \ g b C D W X Y Z [ Y \ x y [ ^ g b d g ^ z f X \ g b j k l m n l o W X Y Z [ Y \ { c \ g \ f X \ g b C F ; ; ; C G p q r s k t H I J K L J M N O L P Q R I M O S W X Y Z [ Y \ ] ^ _ ` a b c d [ e f X \ g b = A Figure : A diagram of the various modes of a slab waveguide. The guided modes (forward or backward) from a discrete spectrum while the radiation, substrate, and evanescent modes (forward or backward form a continuum spectrum. The unphysical mode solutions have also a continuum spectrum (forward or backward). The leaky modes are not exactly formal modes but they represent some form of normalization of the radiation spectrum (forward or backward). 7. Normalized Slab-Waveguide Variables The dispersion Eq. (4) can also be written in terms of normalized variables. Such a representation is useful for any three-region slab waveguide. The normalized parameters are the normalized frequency, V, the normalized effective propagation constant (or normalized effective 22

27 index), b, and are defined as follows V = 2π h n 2 f λ n2 s, (48) b = N2 n 2 s n 2 f n2 s = n2 f sin2 θ n 2 s n 2 f, (49) n2 s a TE a TM = n2 s n2 c n 2 f, (5) n2 s = n4 f n 4 c n 2 s n 2 c n 2 f, (5) n2 s where the a TE and a TM are defined as the asymmetry factors for TE and TM modes respectively. Using these normalized variables in Eq. (4) it is straightforward to derive the following dispersion equation for the case of TE polarization V { } { b b + b tan tan ate b b } = νπ, ν =,,. (52) A similar normalized dispersion equation can not be done for the TM modes. It can be shown that by defining the normalized variables, b TM = (n 2 f /qn2 s)[(n 2 n 2 s)/(n 2 f n2 s)] and q = (N 2 /n 2 f ) + (N2 /n 2 s), the resulting normalized dispersion equation for the TM modes is given by V q n f n s btm tan } tan b TM + a TM ( b TM d) = νπ, (53) b TM b TM { btm where ν =,, and d = [ (n s /n f ) 2 ][ (n c /n f ) 2 ]. It is obvious that the above normalized equation is not completely normalized since the ratios n s /n f, n c /n f, and n s, n f, are necessary for the evaluation of b TM. However, if n f n s then the normalized dispersion equation for the T M modes becomes similar to the one for the T E modes if the corresponding to the T M modes asymmetry factor, a TM, is used. Solving Eq. (52) for various values of the normalized frequency, V, of the asymmetry factor, a (TE or TM), and of ν, normalized diagrams such the one shown in Fig. 2 can be generated. These are exact for TE modes and only approximately correct for TM modes as it was explained above. Measuring from the diagram the value(s) of b ν, for given V and a, the effective index of the TE ν mode(s) (and approximately for TM ν mode(s) with a = a TM ) can be determined as N ν = [n 2 s + b ν(n 2 f n2 s )]/2. 23

28 ..9 Normalized Propagation Constant, b a = Normalized Frequency, V Figure 2: The normalized propagation constant b versus normalized frequency V for four selected asymmetry factors (a =,,, and 6 ) for the first 5 modes (ν =,, 2, 3, and 4). A small segment for ν = 5 and a = is shown on the right bottom corner of the diagram. 8. Cutoff Conditions When the effective index of a guided mode becomes smaller than the substrate refractive index (N ν < n s ) then the mode starts radiating into the substrate and becomes a substrate mode as it was discussed in the previous sections. The condition N ν = n s (or equivalently β ν = k n s ) represents a critical condition below which the guided field is cut-off and from guided mode becomes a substrate (or in general radiation) mode. Equivalently, this occurs when the zig-zag angle θ becomes equal to the critical angle at the film-substrate interface, θ cr,fs (it is reminded that n c < n s < n f ). Then, Eq. (4) becomes k h n 2 f n2 s tan ( a w ) = νπ, w = TE, TM. (54) The last equation is a cutoff condition and can be solved for one parameter retaining the others constant. For example, if it solved with respect to the waveguide thickness, h, the cutoff 24

29 thicknesses can be determined as follows h TE cut,ν = νπ + tan ( a TE ), (55) k n 2 f n2 s cut,ν = νπ + tan ( a TM ), (56) k n 2 f n2 s h TM where a TE and a TM are the asymmetry factors for TE and TM polarization respectively and are defined in Eqs. (5) and (5). It is interesting to notice that for symmetric waveguides, a TE = a TM = the cutoff thicknesses for the lowest order mode (ν = ) are zero. I.e., for symmetric waveguides, any film thickness, independently of how small it is, can support the lowest order modes TE and TM. In Fig. 3 the effective index of various modes is shown as a function of the film layer thickness. From these plots the cutoff thicknesses are the points from which each of the β ν curves starts. If Eq. (54) is solved with respect to the free-space wavelength or frequency the following cutoff wavelengths and frequencies can be defined for each mode and polarization 2πh n 2 λ TE f n2 s,cut,ν = νπ + tan ( a TE ), (57) 2πh n 2 λ TM f n2 s,cut,ν = νπ + tan ( a TM ), (58) cut,ν = c νπ + tan ( a TE ), (59) h n 2 f n2 s ω TE cut,ν = c νπ + tan ( a TM ). (6) h n 2 f n2 s ω TM It is evident again that for symmetric waveguides the cutoff wavelength is infinite and the corresponding cutoff frequencies are zero for the zero order modes. An example ω β diagram is shown in Fig. 4 where the cutoff frequencies can be shown for the first five TE and TM modes. The waveguide parameters are included in the figure caption. 9. Power Considerations In order to determine the power that each mode can carry it is necessary to evaluate the Poynting vector and integrate it along the x-axis. The power carried by the ν-th mode can be 25

30 Effective Index, N TE TM n c =, n f =.56, n s =.45, λ = µm Film Thickness, h (µm) 2.3 n c =, n f = 2.2, n s =.5, λ = µm Effective Index, N TE TM Film Thickness, h (µm) Figure 3: The effective index variation as a function of the waveguide film thickness. The slab waveguide parameters are shown on the top of each plot. It is mentioned that the effective indices of up to the tenth TE or TM are included. express as P = Re{ E H } ẑdxdy, (6) where the integration is over the entire xy plane. However, in slab waveguides, there is no y-dependence, and therefore the integration over the y-axis should give infinite power! As a result in slab waveguides the power per unit y length is of interest and the previous equation is written as P = = + { } 2 Re E H ẑdx = h 2 Re{ E H } ẑdx + 2 Re{ E H } ẑdx }{{}}{{} P s P f Re{ E H } ẑdx,(62) h } {{ } P c

31 x 5 n c =, n f = 2.2, n s =.5, h = µm 3 c/n c c/n s c/n f 2.5 Radial Frequency, ω (rad/sec) Propagation Constant, β (m ) TE TM x 7 Figure 4: A sample ω β diagram for a three layer slab waveguide with n c =., n f = 2.2, n s =.5, and h = µm for both TE and TM polarizations. where it is implied that P is expressed in Watts/per unit y length and the integral has been separated in three parts P s, P f, and P c each representing the power per unit y length in each of the three waveguide regions, substrate, film, and cover, respectively. 27

32 9. TE Modes Using the field Eqs. (29) and (25) in the case of TE modes the Poynting vector can be written as (/2)Re{ E y H x} and the resulting power components become P s P f = NTE ν ɛ (φ TE E 2cos2 fs ), (63) 4 µ γ s [ = NTE ν ɛ E 2 h + sin(2k fxh 2φ TE fs ) + sin(2φte 4 µ 2k fx fs ) ], (64) P c = NTE ν ɛ (k E 2cos2 fx h φ TE fs ), (65) 4 µ γ c P = NTE ν ɛ E 2 4 µ ] [h + γs + γc = NTE ν ɛ E 2 h TE eff,ν 4 µ, (66) where h TE eff,ν is the effective thickness of the TE ν mode and P = P s + P f + P c. It is reminded that φ TE fs = tan (γ s /k fx ). It is interesting to specify the percentage of power being carried by the TE ν mode in each of the three regions. Specifically, from the above equations the following ratios can be defined P s P = h TE eff,ν cos 2 (φ TE fs ) γ s, (67) [ P f P = h h + sin(2kfxh 2φTE fs ) + sin(2φte TE eff,ν 2k fx fs ) ], (68) P c P = h TE eff,ν cos 2 (k fx h φ TE fs ) γ c. (69) For example, the dependence of h TE eff,ν, and of P s/p, P f /P, and P c /P of the film thickness h is shown in Figs. 5 and 6 for the first modes of a slab waveguide. 28

33 Effective Thickness, h eff TE TM n c =, n f =.56, n s =.45, λ = µm Film Thickness, h (µm) Figure 5: A sample h eff,ν versus film thickness diagram for a three layer slab waveguide with n c =., n f =.56, n s =.45, and h = µm for both TE and TM polarizations. The effective thicknesses of the first TE and TM modes are shown. TE, n c =, n f =.56, n s =.45, λ = µm TM, n c =, n f =.56, n s =.45, λ = µm Normalized Power, P w /P, (w=c,f,s) P c /P P f /P P s /P Normalized Power, P w /P, (w=c,f,s) P c /P P f /P P s /P Film Thickness, h (µm) Film Thickness, h (µm) Figure 6: The power percentages in the cover, film and substrate regions as a function of the waveguide film thickness. The slab waveguide parameters are shown on the top of each plot. It is mentioned that the first ten TE and TM modes are included. 9.2 TM Modes Using the field Eqs. (35) and (3) in the case of TM modes the Poynting vector can be written as (/2)Re{E x H y} and the resulting power components become P s P f = NTM ν 4 µ = NTM ν µ H 2 4 ɛ n 2 f fs ) (φ TM H 2cos2, (7) ɛ n 2 sγ s [ h + sin(2k fxh 2φ TM fs ) + sin(2φtm 2k fx fs ) ], (7) P c 29 = NTM ν µ (k H 2cos2 fx h φ TM fs ), (72) 4 ɛ n 2 cγ c

34 where h TM eff,ν is the effective thickness of the TM ν mode and P = P s + P f + P c, and q c = (N TM ν case φ TM fs ) 2 /n 2 c + (NTM ν ) 2 /n 2 f, q s = (N TM ν ) 2 /n 2 s + (NTM ν ) 2 /n 2 f. It is reminded that in this = tan [(γ s /n 2 s )/(k fx/n 2 f )]. It is interesting to specify the percentage of power being carried by the TM ν mode in each of the three regions. Specifically, from the above equations the following ratios can be defined P s P = n 2 f h TM eff,ν P f P = n 2 f h TM eff,ν cos 2 (φ TM n 2 f n 2 s γ s [ fs ), (74) h + sin(2k fxh 2φ TM fs ) + sin(2φtm 2k fx fs ) ], (75) P c P = n 2 f h TM eff,ν cos 2 (k fx h φ TM n 2 c γ c fs ). (76) For example, the dependence of h TM eff,ν, and of P s/p, P f /P, and P c /P of the film thickness h is shown in Figs. 5 and 6 for the first modes of a slab waveguide. 3

35 Ÿ. Multi-Layered Slab Waveguides Multi-layered slab waveguides are of practical interest since they represent realizable semiconductor waveguides that are used in semiconductor lasers. Furthermore, even dielectric waveguides of more than a single film region are used. An example is the case that a buffer layer may be needed to be inserted between the film region and metallic electrodes that are used in electro-optic applications. Also, many times from a fabrication point of view, the resulting waveguides end up with a refractive index profile that is of graded (smooth) variation instead of step (abrupt) variation. Examples of such waveguides are the ones fabricated via ion-exchange, ion-bombardment, or metal-in-diffusion techniques. In latter cases the graded-index waveguides can be approximated by an arbitrary number of layers of constant index thus approximating the smoothly varying refractive index. This technique can also be used to characterized fabricated waveguides. For example, if the mode propagation constants can be measured (which is feasible through a prism-coupling method) the parameters of the fabricated waveguide can be determined by an inverse problem to the one that determines the propagation constants. For all these reasons, the extension of the three region slab waveguide to the general multi-layered case is presented here. ª «ž œ š «ª Figure 7: A general multilayered slab waveguide. The waveguide is comprised of N layers, with refractive indices, n i, and layer thicknesses, h i (i =, 2,, N). The cover (x < ) has a refractive index of n c, whereas the substrate (x > N h i) has a refractive index of n s. 3

36 A general multi-layered slab waveguide is shown in Fig. 7. The waveguide is comprised of N film regions (of refractive index n i and thickness h i ), in addition to the cover and substrate regions, of refractive indices, n c and n s, respectively. The x = is now selected at the cover-film layer # boundary and its direction is towards the substrate. This direction is opposite of the one used in Fig.. This is chosen since it matches the coordinate system that was utilized in the computer program that was pre-written to these notes in order to implement the procedure for the multi-layered slab waveguide. All the basic equations are the same with the only care needed in the selection of the signs of the exponential terms of the evanescent waves in cover and substrate regions. The approach to determine the modes (TE or TM) in the case of a multi-layered slab waveguide is conceptually identical with the one that was followed in Secs. 2. and 2.2. For example, if the T E modes should be determined, then the unknown electric field amplitude coefficients are E c, E, E 2,, E i, E i2,, E N, E N2, E s, as well as the propagation constant β ν. Therefore the number of unknowns is 2(N + ) excluding β ν. The number of equations that can be derived from the boundary conditions are 2(N + ). The boundary conditions result in a matrix equation similar to Eq. (27) but of dimension 2(N + ) 2(N + ). The determinant of the coefficient matrix is the dispersion equation and solutions of this in the interval k n s < β ν < k max i {n i } determine the propagation constants β ν. However, in this section the Transfer Matrix technique as described in Ref. [2] will be presented.. TE Guided Modes As in the case of the three-layer slab waveguide, the T E modes are characterized by the electromagnetic fields {E y, H x, H z }. Using the first two of Eqs. (4) and the first of Eqs. (5) the following equations can be written for a region of refractive index n [ ] jωµ d [ ] Ey = dx H z jωɛ + j β2 Ey, (77) H z ωµ H x = β ωµ E y. (78) 32