596 Opt.Soc.Am.A/Vol.2,No.4/April 1985 L.M.Walpita and the electric field in the plane xz(or magnetic field in the y direction),known as transverse magnetic (TM)waves. Wave propagstion Since the z axis of the index ellipsoid has been chosen to coincide with the z axis of the coordinate system,only the above two types of modes could exist in the waveguide. 二二二 2 Within each layer of constant index,the TE and TM modes Waveguide in the anisotropic waveguide correspond to ordinary and ex- Subatrate traordinary plane waves,respectively(Fig.2),traveling in a bulk anisotropic medium bouncing back and forth between -Z the boundaries.The ray direction R,the direction of the wave normal S,and the E and D vectors of these plane waves are Fig.1.Coordinate system on which the theory is based.The wave all solutions of the zero-element transfer-matrix condition, propagation is in the x direction,and the guide-thickness variation which permits only a certain direction of plane wave propa- is in the z direction.The waveguide structure is a planar slab,and therefore the y coordinate has no influence on the wave propagation, gation in the waveguide.Each ray direction corresponds to i.e.,the wave propagation is two dimensional. a mode order of the guided wave.The index ellipsoid9 of the anisotropic materials under investigation is of the form given by x2,y2,z2. n++n (1) With reference to Fig.2,the relationship between the elec- tric-field vector(E)and the displacement vector(D)is D& no2 0 07 「Ex7 D 0 no2 0 Ey (2) 0 0 ne2 LE:] In the case of TE waves,Dy =no2Ey,and hence the wave normal(S)and the ray direction(R)are identical,indicating that =0.The effective refractive index(n),therefore,is equal to the ordinary refractive index(no).For TM waves, D:=Ene2 cos 0 and Dx=Eno2 sin 0,in which case Dz/D:= tan 0'=no2/ne2 tan 0.The effective index for TM waves thus may be obtained as n'=neno/(ne2 sin20'+no2 cos2 0)1/2. (3) In the case of dielectric media without any boundary discontinuities,a wave will continue to propagate in the ray direction,i.e.,the wave propagation always will be in the di- Fig.2.Wave propagation in a uniaxial anistropic medium.Two rection R at an angle 0 to the x axis.Now let the wave be in cases of wave propagation are considered:(top)the electric field in a stratified layer structure in which the two outermost the y direction is influenced only by the ordinary refractive index (no), boundaries cause total internal reflection and the intermediate and (bottom)the electric field in the plane xz is influenced by both the ordinary (no)and the extraordinary (ne)refractive indices. boundaries cause both reflection and refraction.The relation between the wave vector (B)in the x direction,the wave vector no energy flow in that direction,the forward plane wave in the (pay)in the z direction,and the free-medium wave vector substrate and the backward plane wave in the superstrate (kn')for a given layer is1o must have zero amplitude.In terms of the transfer matrix, Pxy=iB=ikn'cos 0', this condition requires that an element of the transfer matrix be zero.When this zero transfer-matrix element condition Pay =kn'sin 0= (2-k2ne2)1/2 is satisfied,the propagation constants in the z direction of the ne two remaining nonzero waves in the substrate and the su- for TM(Y=1),(4a) perstrate regions are also imaginary.Such imaginary con- stants imply that the fields in these two regions are evanes- Px7=iB=ikno Cos 0, cent,matching the requirement of the guided-wave modes. Pay kno sin 0=(82-k2no2)1/2 B.Wave Propagation in Anisotropic Media for TE (Y=0),(4b) For guided-wave modes in an anisotropic medium,we shall where k is the free-space wave vector. consider only the case in which the optical axis of the uniax- It is clear from these relationships that the TM propagation ial-index ellipsoid is in the z direction.Electromagnetic constant is a function of both ne and no,whereas the TE modes propagating in planar dielectric media are divided into propagation constant is a function only of no.Both the or- two types according to their polarization:the electric field dinary (TE)and extraordinary (TM)forward-and back- in the y direction,known as transverse electric (TE)waves, ward-propagating plane waves will satisfy the continuity596 Opt. Soc. Am. A/Vol. 2, No. 4/April 1985 Fig. 1. Coordinate system on which the theory is based. The wave propagation is in the x direction, and the guide-thickness variation is in the z direction. The waveguide structure is a planar slab, and therefore the y coordinate has no influence on the wave propagation, i.e., the wave propagation is two dimensional. E(TM) -X z and the electric field in the plane xz (or magnetic field in the y direction), known as transverse magnetic (TM) waves. Since the z axis of the index ellipsoid has been chosen to coincide with the z axis of the coordinate system, only the above two types of modes could exist in the waveguide. Within each layer of constant index, the TE and TM modes in the anisotropic waveguide correspond to ordinary and ex￾traordinary plane waves, respectively (Fig. 2), traveling in a bulk anisotropic medium bouncing back and forth between the boundaries. The ray direction R, the direction of the wave normal S, and the E and D vectors of these plane waves are all solutions of the zero-element transfer-matrix condition, which permits only a certain direction of plane wave propa￾gation in the waveguide. Each ray direction corresponds to a mode order of the guided wave. The index ellipsoids of the anisotropic materials under investigation is of the form given by x2 y2 z2 * -+ + -= 1. no2 no2 ne2 (1) With reference to Fig. 2, the relationship between the elec￾x tric-field vector (E) and the displacement vector (D) is [D x- [no 2 0 IDy= 0 n L.;D L_O 0 R S 2 n 2 E x] D2 0 Ey . ne 2 E, (2) In the case of TE waves, D = n 2Ey, and hence the wave normal (S) and the ray direction (R) are identical, indicating that 0 = '. The effective refractive index (n'), therefore, is equal to the ordinary refractive index (no). For TM waves, Dz = En 2 cos 0 and D = En 2 sin 0, in which case DX/D = tan 6' = no 2 /n 2 tan 0. The effective index for TM waves thus may be obtained as n'= neno/(ne 2 sin2 6' + no 2 COS2 6/)1/2. lz Fig. 2. Wave propagation in a uniaxial anistropic medium. Two cases of wave propagation are considered: (top) the electric field in the y direction is influenced only by the ordinary refractive index (no), and (bottom) the electric field in the plane xz is influenced by both the ordinary (no) and the extraordinary (ne) refractive indices. no energy flow in that direction, the forward plane wave in the substrate and the backward plane wave in the superstrate must have zero amplitude. In terms of the transfer matrix, this condition requires that an element of the transfer matrix be zero. When this zero transfer-matrix element condition is satisfied, the propagation constants in the z direction of the two remaining nonzero waves in the substrate and the su￾perstrate regions are also imaginary. Such imaginary con￾stants imply that the fields in these two regions are evanes￾cent, matching the requirement of the guided-wave modes. B. Wave Propagation in Anisotropic Media For guided-wave modes in an anisotropic medium, we shall consider only the case in which the optical axis of the uniax￾ial-index ellipsoid is in the z direction. Electromagnetic modes propagating in planar dielectric media are divided into two types according to their polarization: the electric field in the y direction, known as transverse electric (TE) waves, (3) In the case of dielectric media without any boundary discontinuities, a wave will continue to propagate in the ray direction, i.e., the wave propagation always will be in the di￾rection R at an angle to the x axis. Now let the wave be in a stratified layer structure in which the two outermost boundaries cause total internal reflection and the intermediate boundaries cause both reflection and refraction. The relation between the wave vector () in the x direction, the wave vector (py) in the z direction, and the free-medium wave vector (kn') for a given layer is10 PxY Pz-y = = i kn' = sin ikn' ' cos =I = ', .2 (/32 - n )/ for TM ( = 1), (4a) PxY = i = ikno cos 6, PzY = kno sin 0 = (2 - k 2 no 2 )1/2 for TE ( = 0), (4b) where k is the free-space wave vector. It is clear from these relationships that the TM propagation constant is a function of both n and , whereas the TE propagation constant is a function only of n. Both the or￾dinary (TE) and extraordinary (TM) forward- and back￾ward-propagating plane waves will satisfy the continuity L. M. Walpita