598 Opt.Soc.Am.A/Vol.2,No.4/April 1985 L.M.Walpita ment method needed much less computer time than did N=1.4= Vassell's technique.12 1.67 M-1.B -A value obtained for the same structure b=1.5124 3.TWO-LAYER AND SINGLE-LAYER 1.63 by Vessell's nethod. 飞。 ANISOTROPIC WAVEGUIDES AS SPECIAL CASES OF MULTILAYER WAVEGUIDES 1.59 As the first example in the application of our theory,we con- sider a waveguide structure (see Fig.5)consisting of two 1.55 step-index films bonded together and sandwiched between semi-infinite substrate and superstrate,so that the guide is 1.5 in fact a four-layer structure in which the two outside layers 60.0 a0.0 100.0 120.0140.0 are homogeneous and infinitely thick.We obtain 4(3/k)for Fig.4.Characteristic TEo curve for a multilayer waveguide. The this structure by multiplying the characteristic matrix for each isotropic multilayer structure has a substrate index 1.5124 and a su- perstrate index 1.000.Each layer in the structure consists of a layer and obtaining the resultant matrix(Appendix A).The high-index body (refractive index,1.80000)20 A thick and a low-index @4 element in the resultant matrix is termination(refractive index,1.400)7 A thick.Number of layers indicated on the abscissa. a4=1/(T2T4)(T4y+T3Y ×tanh[p3zy(22-zJl{T2zy+Titanh(p2zyz】 at each interface,and the constants An and By in the su- +1/(T3yT4)(T+T4y perstrate(nth semi-infinite layer)may be written in terms of X tanh[p7(z2-z1)][T+T2zy tanh(p2yz1)], a transfer matrix and the constants Aiy and B1y of the sub- (12) strate (first semi-infinite layer).These constants for the different layers are related in Appendix A. where Now the condition for wave propagation in the waveguide may be applied.If the energy is to be trapped within the re- Tjat=pjay/njo2Y. gion z=0and2=2n-2,ie.,within the outermost boundaries of the guide,any outside electromagnetic field must be eva- The characteristic equation for the single-slab anisotropic nescent,and,in addition,there should be no forward-propa- waveguide(see Fig.6)is derived by further simplification of gating wave in the substrate and no backward-propagating the above equation by substituting z2 =z1.In that case we plane wave in the superstrate.In order to satisfy this latter obtain condition,Bny(the backward-wave amplitude in the super- strate j=n)and Ai(the forward-wave amplitude in the substrate j=1)obviously should be zero: Superetrate []-[ (11) Qulding layer 2 Equation (11)can be satisifed only if the element a of the matrix is equal to zero. @4 is a function of B/k,the guide-normalized propagation constant in the x direction.The waveguide may be charac- ,020 Qulding layer 1 terized in terms of the normalized propagation constant (3/k) as a func ion of layer thickness as well as of the refractive in- 20✉0 e dices of the layers.The B/k values,which are sometimes re- aubetrate ferred to as the mode indices,are always larger than the sub- 10 strate and superstrate indices.The propagation constants Fig.5.Two-layer waveguide.In this case both layers are guiding, in the z direction in both the superstrate and the substrate are i.e.,the electric fields are sinusoidal in both the layers.Sometimes therefore always imaginary.This implies that the fields are it could also be the case that the field in one layer is evanescent. evanescent in both the superstrate and the substrate. It may now be shown that this method is a useful tool for Superstrate economically analyzing multilayer waveguides.If the index profile of a waveguide is known,the dispersion characteristics of the guide may be determined by equating a4(8/k)=0 and then solving for B/k.The dispersion characteristics of the Qulding layer zero-order mode of a multilayer waveguide,as obtained by this technique,are illustrated in Fig.4.In this case,the guide consists of a stack of twin layers in which each twin layer has a thin low-index region(7 A)and a thick high-index region(20 Subetrate A).The model dispersion characteristics (i.e.,the change of B/k with film thickness)has been compared with the model Fig.6.Step-index waveguide.The simplest form of the optical dispersion as given by Vassell's3 technique,and exact agree- waveguide and also considered a special case of the two-layer or ment was obtained.However,the zero-transfer matrix ele- multilayer waveguide.598 Opt. Soc. Am. A/Vol. 2, No. 4/April 1985 I I 1.67 1.63 1.59 1 .55 1 .51 N = 1._ 2 N = 1 .81 Nsu 1.5124 0- A value obtained for the same structure sub by Vessell's method. /i.. 60.0 80.0 100.0 120.0 140.0 Fig. 4. Characteristic TEO curve for a multilayer waveguide. The isotropic multilayer structure has a substrate index 1.5124 and a su￾perstrate index 1.000. Each layer in the structure consists of a high-index body (refractive index, 1.80000) 20 A thick and a low-index termination (refractive index, 1.400) 7 A thick. Number of layers indicated on the abscissa. at each interface, and the constants An,. and Bn7 in the su￾perstrate (nth semi-infinite layer) may be written in terms of a transfer matrix and the constants Al. and B17 of the sub￾strate (first semi-infinite layer). These constants for the different layers are related in Appendix A. Now the condition for wave propagation in the waveguide may be applied. If the energy is to be trapped within the re￾gion z = 0 and z = Zn-2, i.e., within the outermost boundaries of the guide, any outside electromagnetic field must be eva￾nescent, and, in addition, there should be no forward-propa￾gating wave in the substrate and no backward-propagating plane wave in the superstrate. In order to satisfy this latter condition, Bn, (the backward-wave amplitude in the super￾strate j = n) and Al 7 (the forward-wave amplitude in the substrate i = 1) obviously should be zero: 0A~nt =a3 a2] [1 a4 ] LB (11) Equation (11) can be satisifed only if the element a 4 of the matrix is equal to zero. a 4 is a function of 3/k, the guide-normalized propagation constant in the x direction. The waveguide may be charac￾terized in terms of the normalized propagation constant (13/k) as a func ion of layer thickness as well as of the refractive in￾dices of the layers. The /3/k values, which are sometimes re￾ferred to as the mode indices, are always larger than the sub￾strate and superstrate indices. The propagation constants in the z direction in both the superstrate and the substrate are therefore always imaginary. This implies that the fields are evanescent in both the superstrate and the substrate. It may now be shown that this method is a useful tool for economically analyzing multilayer waveguides. If the index profile of a waveguide is known, the dispersion characteristics of the guide may be determined by equating a4 (/3/k) = 0 and then solving for 1/k. The dispersion characteristics of the zero-order mode of a multilayer waveguide, as obtained by this technique, are illustrated in Fig. 4. In this case, the guide consists of a stack of twin layers in which each twin layer has a thin low-index region (7 A) and a thick high-index region (20' A). The model dispersion characteristics (i.e., the change of //k with film thickness) has been compared with the model dispersion as given by Vassell's 3 technique, and exact agree￾ment was obtained. However, the zero-transfer matrix ele￾ment method needed much less computer time than did Vassell's technique.'2 3. TWO-LAYER AND SINGLE-LAYER ANISOTROPIC WAVEGUIDES AS SPECIAL CASES OF MULTILAYER WAVEGUIDES As the first example in the application of our theory, we con￾sider a waveguide structure (see Fig. 5) consisting of two step-index films bonded together and sandwiched between semi-infinite substrate and superstrate, so that the guide is in fact a four-layer structure in which the two outside layers are homogeneous and infinitely thick. We obtain a4(3/k) for this structure by multiplying the characteristic matrix for each layer and obtaining the resultant matrix (Appendix A). The a4 element in the resultant matrix is a4 = i/(r2zr4z7)r4z + r3z7 X tanh[p 3 z7 (z 2 - zm)IIFnz 2 + rFz, tanh(p 2 zDZ1] + 1/(r3zyr4z)r3z + r4z7 X tanh[p 3,7(z2 - zi)I1[rz, + r2zy tanh(p 2 z7 zz)], (12) where r,.Ze pj._/nj.2,,. The characteristic equation for the single-slab anisotropic waveguide (see Fig. 6) is derived by further simplification of the above equation by substituting Z2 = zl. In that case we obtain n4e Ln40 Z2 ZI z0 =0 Supertrate n3e L n Guidig layer 2 -0 n 'e2L Guiding layer nle t 0 1o &lbstrate Fig. 5. Two-layer waveguide. In this case both layers are guiding, i.e., the electric fields are sinusoidal in both the layers. Sometimes it could also be the case that the field in one layer is evanescent. n4e Z2 1 z0 = 0 L nIeL Superatrate n40 In2 2-¢ l'- W 2 \ Guiding layer 2o 2 Substrate nio Fig. 6. Step-index waveguide. The simplest form of the optical waveguide and also considered a special case of the two-layer or multilayer waveguide. L. M. Walpita --n L-