L.M.Walpita Vol.2,No.4/April 1985/J.Opt.Soc.Am.A 595 Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix L.M.Walpita Department of Electrical Engineering and Computer Sciences,C-014,University of California,San Diego,San Diego,California 92093 Received March 28,1984;accepted November 26,1984 The propagation properties of optical planar waveguides with multilayer index profiles are analyzed by the transfer matrix of transmitted and reflected beam amplitudes in multilayers.The propagation wave number for guided- wave modes is obtained from the condition that certain elements in the transfer matrix must be zero.This numeri- cal technique requires much shorter computer times compared with the usual method of solving the eigenvalue equations,obtained by setting the characteristic determinant to zero.The analysis is also applicable either to waveguides that have losses or to certain cases of uniaxial dielectric anisotropy.All waveguides are assumed to be magnetically isotropic.Some examples of the analysis of graded-index profiles and calculations of the effect of metal claddings and prism perturbations on guided modes are given. INTRODUCTION AND OVERVIEW the condition of evanescent fields outside the outer boundaries The theoretical work on the modeling of dielectric waveguides whereby some elements of the transfer matrix are equated to has been well documented.1-8 Using a ray approach,Tien! zero.The objective of this paper,therefore,is concerned with has obtained a characteristic equation for step-index isotropic deriving the condition under which some elements of the slab waveguides.This has been extended by Gia Russo and transfer matrix are zero.The theory is generalized to take Harris2 to characterize an anisotropic structure.The effect into consideration both the losses in optical waveguides and of metal claddings on such optical waveguides has also been some special cases of uniaxial anisotropy.The effects of metal studied.4 In addition,waveguide structures with graded- cladding and prism perturbations on optical waveguides are index profiles have been analyzed.5 In this paper,the theory analyzed as special cases of the general formalism. developed by Vassell3 for the anisotropic parallel boundary is modified to yield a much simpler numerical procedure to A. General Overview calculate the modes of a planar waveguide with a lossless We consider a medium consisting of stratified constant-index graded-index profile.A waveguide with a graded-index layers with parallel boundaries where a plane wave introduced profile,in which both the superstrate and the substrate are into the structure will undergo reflection and refraction at considered to be infinitely thick,can be approximated by each boundary.In order for the structure to behave as a layers of materials that have a constant index within each waveguide,the energy flow must be parallel to the layer layer.Vassell3 has shown that for solutions of waveguide boundaries.In the direction normal to the boundaries,the equations any guided wave must have decaying fields in both structure must behave as a resonator,and there is no net en- the substrate and the superstrate outside the two outermost ergy flow in this direction.The coordinate system for our guide boundaries in a direction transverse to the energy flow. structure is defined in Fig.1.The direction of propagation Basically,this is a transverse resonance condition,which is of the guided wave is considered to be the x direction,and the known in the field of microwaves,and the related equations direction of the guide thickness is the z direction.The could also be expressed in transverse impedance'terms.7,8 waveguide structures are planar,and therefore,as far as the Inhomogeneous dielectric slabs have been characterized by guided wave is concerned,there is no dependence of field impedance considerations when it is claimed that the propa- variations on the y coordinate.We now consider each layer gation constant could be obtained numerically with rapid to have forward-and backward-propagating plane waves in convergence to solution and with high accuracy.8 In essence, the direction (z)normal to the boundaries.The amplitude Vassell's technique matches four vectors corresponding to the and the phase of these plane waves are then related to those electric and magnetic fields at the boundary of each layer. in the neighboring layers by the continuity condition of the The field amplitudes of the forward and backward waves transverse electric and magnetic fields at the boundaries,i.e., propagating normal to the guide boundary in the substrate the amplitude and the phase of the forward and backward and superstrate regions outside the waveguide are then related waves in one layer are related by a matrix to those in the next by a 4X 4 transfer matrix.In his analysis,the eigenvalues are layer.When the boundary conditions are applied to the the solutions of the eigenequations obtained from the deter- waves in the subsequent layers and finally to the forward and minant of the transfer matrix.However,the procedures for backward waves of the unconfined superstrate and substrate numerically calculating the eigenvalues of the characteristic regions,we obtain a 2 X 2 transfer matrix relating the ampli- equations for waveguides with an arbitrary graded-index tudes and the phases of the forward and backward waves in profile are quite complex and lengthy.The same objective the superstrate region to those in the substrate region.In can be achieved by using a slightly different concept to satisfy order to satisfy the resonance condition in the z direction,i.e., 0740-3232/85/040595-08$02.00 @1985 Optical Society of America
Vol. 2, No. 4/April 1985/J. Opt. Soc. Am. A 595 Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix L. M. Walpita Department of Electrical Engineering and Computer Sciences, C-014, University of California, San Diego, San Diego, California 92093 Received March 28, 1984; accepted November 26, 1984 The propagation properties of optical planar waveguides with multilayer index profiles are analyzed by the transfer matrix of transmitted and reflected beam amplitudes in multilayers. The propagation wave number for guidedwave modes is obtained from the condition that certain elements in the transfer matrix must be zero. This numerical technique requires much shorter computer times compared with the usual method of solving the eigenvalue equations, obtained by setting the characteristic determinant to zero. The analysis is also applicable either to waveguides that have losses or to certain cases of uniaxial dielectric anisotropy. All waveguides are assumed to be magnetically isotropic. Some examples of the analysis of graded-index profiles and calculations of the effect of metal claddings and prism perturbations on guided modes are given. INTRODUCTION AND OVERVIEW The theoretical work on the modeling of dielectric waveguides has been well documented.1- 8 Using a ray approach, Tien' has obtained a characteristic equation for step-index isotropic slab waveguides. This has been extended by Gia Russo and Harris2 to characterize an anisotropic structure. The effect of metal claddings on such optical waveguides has also been studied.4 In addition, waveguide structures with gradedindex profiles have been analyzed.5 In this paper, the theory developed by Vassell3 for the anisotropic parallel boundary is modified to yield a much simpler numerical procedure to calculate the modes of a planar waveguide with a lossless graded-index profile. A waveguide with a graded-index profile, in which both the superstrate and the substrate are considered to be infinitely thick, can be approximated by layers of materials that have a constant index within each layer. Vassell3 has shown that for solutions of waveguide equations any guided wave must have decaying fields in both the substrate and the superstrate outside the two outermost guide boundaries in a direction transverse to the energy flow. Basically, this is a transverse resonance condition, which is known in the field of microwaves, and the related equations could also be expressed in transverse impedance'terms. 7 ' 8 Inhomogeneous dielectric slabs have been characterized by impedance considerations when it is claimed that the propagation constant could be obtained numerically with rapid convergence to solution and with high accuracy.8 In essence, Vassell's technique matches four vectors corresponding to the electric and magnetic fields at the boundary of each layer. The field amplitudes of the forward and backward waves propagating normal to the guide boundary in the substrate and superstrate regions outside the waveguide are then related by a 4 X 4 transfer matrix. In his analysis, the eigenvalues are the solutions of the eigenequations obtained from the determinant of the transfer matrix. However, the procedures for numerically calculating the eigenvalues of the characteristic equations for waveguides with an arbitrary graded-index profile are quite complex and lengthy. The same objective can be achieved by using a slightly different concept to satisfy the condition of evanescent fields outside the outer boundaries whereby some elements of the transfer matrix are equated to zero. The objective of this paper, therefore, is concerned with deriving the condition under which some elements of the transfer matrix are zero. The theory is generalized to take into consideration both the losses in optical waveguides and some special cases of uniaxial anisotropy. The effects of metal cladding and prism perturbations on optical waveguides are analyzed as special cases of the general formalism. A. General Overview We consider a medium consisting of stratified constant-index layers with parallel boundaries where a plane wave introduced into the structure will undergo reflection and refraction at each boundary. In order for the structure to behave as a waveguide, the energy flow must be parallel to the layer boundaries. In the direction normal to the boundaries, the structure must behave as a resonator, and there is no net energy flow in this direction. The coordinate system for our structure is defined in Fig. 1. The direction of propagation of the guided wave is considered to be the x direction, and the direction of the guide thickness is the z direction. The waveguide structures are planar, and therefore, as far as the guided wave is concerned, there is no dependence of field variations on the y coordinate. We now consider each layer to have forward- and backward-propagating plane waves in the direction (z) normal to the boundaries. The amplitude and the phase of these plane waves are then related to those in the neighboring layers by the continuity condition of the transverse electric and magnetic fields at the boundaries, i.e., the amplitude and the phase of the forward and backward waves in one layer are related by a matrix to those in the next layer. When the boundary conditions are applied to the waves in the subsequent layers and finally to the forward and backward waves of the unconfined superstrate and substrate regions, we obtain a 2 X 2 transfer matrix relating the amplitudes and the phases of the forward and backward waves in the superstrate region to those in the substrate region. In order to satisfy the resonance condition in the z direction, i.e., 0740-3232/85/040595-08$02.00 © 1985 Optical Society of America L. M. Walpita
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 continuity
596 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 extraordinary 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 propagation 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 elecx 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 superstrate regions are also imaginary. Such imaginary constants imply that the fields in these two regions are evanescent, 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 uniaxial-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 direction 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 ordinary (TE) and extraordinary (TM) forward- and backward-propagating plane waves will satisfy the continuity L. M. Walpita
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 regions through a transfer matrix. 2. MULTILAYER-WAVEGUIDE EQUATIONS The general form of a section of a multilayer dielectric waveguide is shown in Fig. 3. Assuming that there is no free charge in any layer, we write Maxwell's equations" for each jth anisotropic 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 forward (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
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 superstrate 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 superstrate (nth semi-infinite layer) may be written in terms of a transfer matrix and the constants Al. and B17 of the substrate (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 region z = 0 and z = Zn-2, i.e., within the outermost boundaries of the guide, any outside electromagnetic field must be evanescent, and, in addition, there should be no forward-propagating 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 superstrate 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 characterized in terms of the normalized propagation constant (13/k) as a func ion of layer thickness as well as of the refractive indices of the layers. The /3/k values, which are sometimes referred to as the mode indices, are always larger than the substrate 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 agreement was obtained. However, the zero-transfer matrix element 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 consider 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-
L.M.Walpita Vol.2,No.4/April 1985/J.Opt.Soc.Am.A 599 A.Determination of the Propagation Constant of a Graded-Index Guide 2.0 The propagation constant(B/k)of a waveguide with any given graded-index profile may be determined by using a piece- wise-linear approximation of the index-profile shape.For the calculation,let us assume the following index profiles: Gaussian: n(x)=ng+An exp(-x2/D2), Exponential: n(x)=ns An exp(-x/D), where ns is the substrate index,An is the maximum index change,and D is defined as the diffusion depth. We have divided the graded-index profile into 20 layers,as shown in Fig.8,and the normalized propagation constant is calculated for ns =2.2,An =0.02,and D=1 um.The nor- malized propagation constants,as obtained by solving for zero-element conditions,are 0.5 B/k Gaussian 2.2057, B/k Exponential 2.2047. ---TM TE Such an analysis may be extended further to obtain the nor- malized propagation constant of more-complex structures, 0.0 such as the effect of various claddings on graded-index 1.461.481.501.521.541.56 waveguides. Normalized propegation constant Fig.7.Characteristics curves of an anisotropic step-index asym- B.Effect of Metal Claddings on Guided Waves metrical waveguide.The normalized propagation constant versus As was mentioned earlier,we can take losses into consideration thickness curves for an anisotropic layer(no 1.525,ne =1.570)on by replacing the real dielectric constant with a complex di- a substrate(isotropic index,1.457).The superstrate is air(isotropic electric constant,i.e.,the refractive index N is replaced by N index,1.000). +ik for the metal situation.Now the propagation constant B/k of Eq.(11)is replaced by (B/k +iA).In order to show the tanh(p2zY21)=-T27(T4+r/(T2+T). (13) Previously it was stated that in order for the film to guide light, 2.22 the fields of the guided wave modes outside the guide Gausslan boundary must be evanascent.Therefore,in the case of this waveguide,B<kn2,B>kni,and B>kn4.Thus p2ay is 2.21 imaginary.We now may rewrite Eq.(13)as 2p2znlz1-221-224=2Mx, (14) 2.20 where tan 624=T4zy/T2y tan 21=Ti/T2 and M is an 1,0 .2.0 3.0 40 integer. Thus,in the case of a single-layer step-index waveguide,the Gulde depth-microns condition for 4 =0 is identical with the guiding condition obtained in the conventional theory by Gia Russo and Harris2 by extending the basic ray equation given by Tien.1 The calculated waveguide-propagation characteristics of anan- 2,22 isotropic step-index guide sandwiched between the isotropic Expon●ntlal superstrate and substrate is illustrated in Fig.7. 2.21 4.FURTHER EXAMPLES In order to demonstrate the flexibility of the zero-element 2.20 lip女 method,we have applied the theory to the following examples: 0 1.0 2.0 3,0 4.0 (1)the determination of the propagation properties of a guide with a given graded-index profile,(2)the effect of metal Gulde depth-microns cladding on guided modes,and(3)the effect of prism per- Fig.8.Typical index profiles for a diffused guide.Two types of turbation on a given guided-wave mode.All numerical ex- profiles are considered,Gaussian and exponential.The following amples were made at the He-Ne laser wavelength of 633 nm parameters are assumed for the evaluation of the propagation con- stant for the TEo mode:substrate index,2.20;maximum index,2.22: and for TE modes. diffusion depth,1.0 um
Vol. 2, No. 4/April 1985/J. Opt. Soc. Am. A 599 2.0 1.5 I C, 1.0 0.5 0.0 1.46 1.48 1.50 1.52 1.54 1.56 Nonalized propagation constant Fig. 7. Characteristics curves of an anisotropic step-index asymmetrical waveguide. The normalized propagation constant versus thickness curves for an anisotropic layer (nO = 1.525, ne = 1.570) on a substrate (isotropic index, 1.457). The superstrate is air (isotropic index, 1.000). tanh(p2 zz) 7 = -r2.y( r4, + r. 7)/(r2z 2 + iizzr4z. (13) Previously it was stated that in order for the film to guide light, the fields of the guided wave modes outside the guide boundary must be evanascent. Therefore, in the case of this waveguide, 13 < kn2 , > knj, and 1 > kn4 . Thus P2zY is imaginary. We now may rewrite Eq. (13) as 2Ip2z IZ1 - 2021 - 224 = 2M-7r, A. Determination of the Propagation Constant of a Graded-Index Guide The propagation constant (3/k) of a waveguide with any given graded-index profile may be determined by using a piecewise-linear approximation of the index-profile shape. For the calculation, let us assume the following index profiles: Gaussian: Exponential: n(x) = n, + An exp(-x 2 /D2 ), n(x) = n8 + An exp(-x/D), where n, is the substrate index, An is the maximum index change, and D is defined as the diffusion depth. We have divided the graded-index profile into 20 layers, as shown in Fig. 8, and the normalized propagation constant is calculated for n, = 2.2, An = 0.02, and D = 1 mm. The normalized propagation constants, as obtained by solving for zero-element conditions, are 13/k Gaussian = 2.2057, 13/k Exponential = 2.2047. Such an analysis may be extended further to obtain the normalized propagation constant of more-complex structures, such as the effect of various claddings on graded-index waveguides. B. Effect of Metal Claddings on Guided Waves As was mentioned earlier, we can take losses into consideration by replacing the real dielectric constant with a complex dielectric constant, i.e., the refractive index N is replaced by N + iK for the metal situation. Now the propagation constant 3/k of Eq. (11) is replaced by (1/k + iA). In order to show the x 0 0 V (14) 2.21 2.20 where tan 024 = r4z,/r2zy tan 021 = rPz,/r2 ,, and M is an integer. Thus, in the case of a single-layer step-index waveguide, the condition for a 4 = 0 is identical with the guiding condition obtained in the conventional theory by Gia Russo and Harris2 by extending the basic ray equation given by Tien.' The calculated waveguide-propagation characteristics of an anisotropic step-index guide sandwiched between the isotropic superstrate and substrate is illustrated in Fig. 7. 4. FURTHER EXAMPLES In order to demonstrate the flexibility of the zero-element method, we have applied the theory to the following examples: (1) the determination of the propagation properties of a guide with a given graded-index profile, (2) the effect of metal cladding on guided modes, and (3) the effect of prism perturbation on a given guided-wave mode. All numerical examples were made at the He-Ne laser Wavelength of 633 nm and for TE modes. 0. 1.0 .2.0 3.0 4 0 Guide depth -microns 2.22 x uD 2. 21 2.20 0 1.0 2.0 3.0 4.0 Guide depth - microns Fig. 8. Typical index profiles for a diffused guide. Two types of profiles are considered, Gaussian and exponential. The following parameters are assumed for the evaluation of the propagation con- stant for the TEo mode: substrate index, 2.20; maximum index, 2.22; diffusion depth, 1.0 ,um. 1G L. M. Walpita