L.M.Walpita Vol.2,No.4/April 1985/J.Opt.Soc.Am.A 597 condition of the TE and TM fields at each boundary,similar For TM waves to an isotropic medium;this results in a relationship between the four forward and backward waves in two unconfined re- Hjy Aj1 exp[-pjz1(z-2j-2)] gions through a transfer matrix. +Bj1exp[pz1(2-3-2小， (9a) iwnjo2eoEjx =-Pjz1Aj1 exp[-pjz1(z-zj-2)] 2.MULTILAYER-WAVEGUIDE EQUATIONS piz1Bi1 exp piz1(z-zj-2), (9b) The general form of a section of a multilayer dielectric wave- iwnje2coEjz pjx1Aj1 exp[-pjz1(z-zj-2)] guide is shown in Fig.3.Assuming that there is no free charge +pjx1Bj1 exp(pjz1(z-2j-2)], (9c) in any layer,we write Maxwell's equations11 for each jth an- where isotropic layer as j represents the layer number, x,y,z represents the vector direction. X Ejm =>-iwujmiuoHjt, 0,1 indicates TE or TM,respectively, E0 is the free-space permittivity, 40 is the free-space permeability, 7XHjm=∑iwejmteoEj1: (5) A is the forward wave,and Bi is the backward wave. The losses in the structure may be taken into consideration by making ejmi complex.As we shall confine our interest to Equations(8)and(9)now are rearranged in a matrix format magnetically isotropic media,the above equations may be considering only the field components in the plane of the reduced further by writing the relative permeability tensor film: as Ejy expl-pjzy(2-zj-2)] explpjzy(z-2j-2)]Ajy LiwuuoHjz] TE(y=0),(10a) -pjay expl-pjzy(z-2j-2)1pjay exp[pjzy(z-zj-2)]]Bjy] Hjy exp[-pjay(z-zj-2)] exp[pjzy(2-zj-2)] Lde0Ejx」 expl-piay( explpjzy(z-zj-2) %2 B TM(y=1).(10b) The constants Ajy and Biy are the amplitudes of the for- ward (positive direction)and backward (negative direction) 0 waves,respectively(Fig.3).The fields may now be matched (6) 0 2p2-k2 Pxi明 j*1.n where uj is same for all the layers and u=j=1. The relative permittivity tensor for the dielectric layers is considered to be of the form discussed in Subsection 1.B and w。n1m is given by [ej11 0 07 Ejml= 0 6j22 0 (7) 0 j33 where ejmt njml2,ej11=cj22=njo2,and ej33=nje2. "wt5!” Solving Eqs.(5),we obtain the magnetic and electric fields =-2 of the guided modes and,as mentioned earlier,the wave equation will give rise to two types of field distributions(TE and TM).It is assumed that the waveguide is infinitely long, and hence there is no reflection in the direction of propagation (x).The field components in each layer for TE and TM waves are therefore W3 nglm x21 For TE waves Ejy Ajo exp[-pjzo(z-zj-2)] 2=20 Bjo exp[pjzo(z-2j-2)], (8a) iwuuoHjx =-pjzoAjo expl-pjzo(z-zj-2)] +pjzoBjo explpjz0(z-zj-2)], (8b) Fig.3.General form of an anistropic multilayer structure.Each layer (is of uniform index njim.Piar and Pizy are propagation iωμuoHjz=pjx0Aj0exp[-Pjzo(2-zj-2】 constants in the z and x directions,respectively.Ajy and Biy are forward-and backward-propagation wave amplitudes.Wi is the +pjxoBjo exp[pjzo(z-2j-2)]; (8c) layer thickness.Vol. 2, No. 4/April 1985/J. Opt. Soc. Am. A 597 condition of the TE and TM fields at each boundary, similar to an isotropic medium; this results in a relationship between the four forward and backward waves in two unconfined re￾gions through a transfer matrix. 2. MULTILAYER-WAVEGUIDE EQUATIONS The general form of a section of a multilayer dielectric wave￾guide is shown in Fig. 3. Assuming that there is no free charge in any layer, we write Maxwell's equations" for each jth an￾isotropic layer as v X Ejm = -iwSjIjmloHjl, V X Hjm = iCOEjmiEOEjj. (5) The losses in the structure may be taken into consideration by making cjml complex. As we shall confine our interest to magnetically isotropic media, the above equations may be reduced further by writing the relative permeability tensor as For TM waves Hjy = Aji exp[-pjz,(z -Z-2)] + Bj1 exp[pjz1 (z - Zj-2)], iconj,2eoEjr =-pj1 ,Aj, exp[-pjzi(z - Z-2)] + pj1,Bj1 exp[pj~,(z -Z-2)], iwnje2eoEjz = pj.,Aji exp[-pjz,(z -Z-2)] + pjlBjl exp[pjz,(z - Zj-2)], (9a) (9b) (9c) where j represents the layer number, x, y, z represents the vector direction, 0, 1 indicates TE or TM, respectively, EO is the free-space permittivity, AuO is the free-space permeability, Aj is the forward wave, and Bj is the backward wave. Equations (8) and (9) now are rearranged in a matrix format considering only the field components in the plane of the film: Ejy 1 = exp1-pj.(z - Z-2)] licvoHjI . LPjz.- exp[-pjzy(z - Zj-2)Pjz7 exp[-pjzy(z - Z-2)] [i H ~I = - Pj2 exp[-pjzy(z - j-2)] iGO~oE,~ njo exppiz, (Z - zj- 2 )] 1 1Aj, exp[pizy(z - z-2)]J [Bjj exp[p,.-(z -Z-2)] A exp[Pj(Z - Z-2)] 1Bj.z1 I TE ( = 0), (a) TM (y = 1). (lOb) The constants Aj, and Bj, are the amplitudes of the for￾ward (positive direction) and backward (negative direction) waves, respectively (Fig. 3). The fields may now be matched p. 2 =p2 -k 2 n2 P. Yp J-ZY 3 i Y = l,n where uj is same for all the layers and ,u = j = 1. The relative permittivity tensor for the dielectric layers is considered to be of the form discussed in Subsection 1.B and is given by 0 01 fj22 0° 0 ej33] B BnA Wfl ~nOM tPnzy Pn- Ant Z -Zn-2 (7) where Elmi = njm 2 , Ejll = ej2 2 = njo2 , and ej33 = nje2. Solving Eqs. (5), we obtain the magnetic and electric fields of the guided modes and, as mentioned earlier, the wave equation will give rise to two types of field distributions (TE and TM). It is assumed that the waveguide is infinitely long, and hence there is no reflection in the direction of propagation (x). The field components in each layer for TE and TM waves are therefore For TE waves z j 2 z =Z . Z-Z 1 Ej = Ajo exp[-pjzo(z - Z-2)] + BjO exp[pjpo(z - Zj-2)b i&4lIIoHjx = -pjzOAjO exp[-pjo(z - Zj-2)] + pjzOBjo exp[pjzo(z - Zj-2)] iCOIL/loHjz = pjxoAjo exp[-pzo(z - Z-2)] + pjxoBjo exp[pjzo(z - Zj-2)]; z=z z:0 (8a) 1ZY 1 l1Y Fig. 3. General form of an anistropic multilayer structure. Each (8b) layer (j) is of uniform index njlm. Pj,, and P are propagation constants in the z and x directions, respectively. A and Bj, are forward- and backward-propagation wave amplitudes. W is the (8c) . layer thickness. Ajmi = 0 0 0 0 A (6) Emill 'Ej. = -0 B W. n P. A. I Ij j y W3 n31m t P3ZY P3UY At | 3Y W2 n2lm t PzY P2xyAt I 1 Y . , B. L. M. Walpita