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,Bkni,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 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
600 Opt.Soc.Am.A/Vol.2,No.4/April 1985 L.M.Walpita effect of lossy media on guided modes in dielectric waveguides, we used the examples shown in Fig.9(a)in which the metal Buffer layor cladding on a waveguide is separated from the waveguide by a buffer layer.Now it is necessary to solve the following equation: Subetrate a4N,N2,W2,N3,W3,(N4+ik4,(/h+iA)】=0, (a) where Ni is the substrate index,N2 the guide index,W2 the 105 guide thickness,N3 the buffer-layer index,Wa the buffer-layer thickness,and (N4 ik4)the complex index of the metal 2.20495 3.0 cladding. Since the guiding condition is satisfied only when both the imaginary and the real parts of are zero,we evaluated both 2.204851 2.0 the propagation constant and the loss constant of the guide. In Figure 9(b),we have plotted both of these constants as a _unperturbed norm:propagation sonstant-- function of the buffer-layer thickness.As expected,the loss 2.20475 1.0 constant is increased as the buffer-layer thickness is reduced, Attenuatlon constant and it has a maximum value for the zero buffer-layer thick- ness. It is also seen that the tendency of the metal cladding 2.20455 is to lower the propagation constant of the guide that is free 0.0 0 of cladding,a tendency also reported by Findaklay and 1000 2000 Chen.13 Buffer layer thickness (b) C. Effect of Prism Perturbation on the Optical Fig.9.Effect of metal claddings on buffered guided waves.The Waveguides normalized propagation constant and the attenuation constant are The prism has a much larger index than the effective index plotted for the TEomode as a function of the buffer-layer thickness (B/k)of the guided modes,and therefore the fields in the The structure consists of substrate index 2.20,guide index 2.210,guide thickness 1.5 um,buffer-layer index 1.47,and metal-cladding complex prism(now effective superstrate)are not evanescent.The index1.44+i3.70. result is that the light from the guide will leak into the prism. In reality,the prism is separated from the guide by a low-index gap (e.g.,air),and hence the prism perturbation of the guided Priam modes is considered weak.Even though the guiding condition still is not truly satisfied,we find approximate solutions for Gap the perturbed propagation constant by solving 4(8/k +iA). Gulde However,solutions for B/k cannot be found when the prism perturbation is large,i.e.,when the gap between the prism and Subatrate8丁+ the guide is small. We have analyzed the structure shown in Fig.10(a),and the effect of the gap between the prism and the guide was deter- llquld cladding Index 1.457 mined.It is seen from Fig.10(b)that B/k increases with the increase of the gap spacing before becoming constant.This demonstrates that beyond a certain gap thickness the modes 1.55 are little affected by the prism,i.e.,the guide may be consid- ered free of perturbation.We have compared two situations: (1)when the gap consists of free space and(2)when the gap 81.53 alr cladding Index 1.000 consists of an index liquid (e.g.,a liquid with a refractive index of 1.457).The analysis indicates that the liquid gap has to be much larger than the air gap before the guided modes be- 1.51 come unperturbed.Since the liquid has a larger index than air,it carries comparatively more optical energy in the evan- escent fields,resulting in long evanescent tails,thereby per- mitting larger prism perturbation. 1.49 5. CONCLUSIONS 1.47 We have analyzed isotropic and anisotropic waveguides using 0.0 0.2 0.4 0.6 the zero elements of the characteristic transfer matrix of the Coupling gap (um) guiding structure.This method has a practical advantage ) over other techniques (e.g.,that given by Vassell3),as the need Fig.10.Prism perturbation of the guided modes.The curves are to solve a determinate is eliminated and hence the computa- for a TEo mode in a guide (index,1.7)on a substrate (index,1.45) tion time is reduced.This is a general theory for anistropic The guide thickness is 0.2 um,and the prism index is 1.778. dielectric multilayer waveguides,provided that the index el-
600 Opt. Soc. Am. A/Vol. 2, No. 4/April 1985 Bufraye / / Buffer layer Wavegulde Substrate (a) 1000 Buffer layer thickness 2000 A effect of lossy media on guided modes in dielectric waveguides, we used the examples shown in Fig. 9(a) in which the metal cladding on a waveguide is separated from the waveguide by a buffer layer. Now it is necessary to solve the following equation: a4 [N,, N2 , W 2, N3 , W3, (N4 + ik4), (13/k + iA)] = 0, where N, is the substrate index, N2 the guide index, W2 the x10-5 guide thickness, N3 the buffer-layer index, W 3 the buffer-layer thickness, and (N4 + iK4) the complex index of the metal 3.0 cladding. Since the guiding condition is satisfied only when both the imaginary and the real parts of a 4 are zero, we evaluated both 2.0 o as the propagation constant and the loss constant of the guide. 8 In Figure 9(b), we have plotted both of these constants as a ° function of the buffer-layer thickness. As expected, the loss 1.0 constant is increased as the buffer-layer thickness is reduced, < and it has a maximum value for the zero buffer-layer thickness. It is also seen that the tendency of the metal cladding is to lower the propagation constant of the guide that is free °0 ° of cladding, a tendency also reported by Findaklay and Chen.13 (b) Fig. 9. Effect of metal claddings on buffered guided waves. The normalized propagation constant and the attenuation constant are plotted for the TEO mode as a function of the buffer-layer thickness. The structure consists of substrate index 2.20, guide index 2.210, guide thickness 1.5 gtm, buffer-layer index 1.47, and metal-cladding complex index 1.44 + i3.70. (a) 2 8 0 0, 0.0 0.2 0.4 Coupling gap (pm) (b) 0.6 Fig. 10. Prism perturbation of the guided modes. The curves are for a TEo mode in a guide (index, 1.7) on a substrate (index, 1.45). The guide thickness is 0.2 jum, and the prism index is 1.778. C. Effect of Prism Perturbation on the Optical Waveguides The prism has a much larger index than the effective index (13/k) of the guided modes, and therefore the fields in the prism (now effective superstrate) are not evanescent. The result is that the light from the guide will leak into the prism. In reality, the prism is separated from the guide by a low-index gap (e.g., air), and hence the prism perturbation of the guided modes is considered weak. Even though the guiding condition still is not truly satisfied, we find approximate solutions for the perturbed propagation constant by solving a 4(13/k + iA). However, solutions for 13/k cannot be found when the prism perturbation is large, i.e., when the gap between the prism and the guide is small. We have analyzed the structure shown in Fig. 10(a), and the effect of the gap between the prism and the guide was determined. It is seen from Fig. 10(b) that 13/k increases with the increase of the gap spacing before becoming constant. This demonstrates that beyond a certain gap thickness the modes are little affected by the prism, i.e., the guide may be considered free of perturbation. We have compared two situations: (1) when the gap consists of free space and (2) when the gap consists of an index liquid (e.g., a liquid with a refractive index of 1.457). The analysis indicates that the liquid gap has to be much larger than the air gap before the guided modes become unperturbed. Since the liquid has a larger index than air, it carries comparatively more optical energy in the evanescent fields, resulting in long evanescent tails, thereby permitting larger prism perturbation. 5. CONCLUSIONS We have analyzed isotropic and anisotropic waveguides using the zero elements of the characteristic transfer matrix of the guiding structure. This method has a practical advantage over other techniques (e.g., that given by Vassell3 ), as the need to solve a determinate is eliminated and hence the computation time is reduced. This is a general theory for anistropic dielectric multilayer waveguides, provided that the index el- 2.20495 .° 2.20485' i 2.20475. z 2.20465 Normallsed propagation constant / An ~~~~~~~~~~tan 0 L. M. Walpita
L.M.Walpita Vol.2,No.4/April 1985/J.Opt.Soc.Am.A 601 lipsoid axes coincide with the selected coordinate systems for the fourth semi-infinite layer The theory may be used to synthesize index profiles of any given shape.In addition,we have extended our analysis to E划 lossy situations when a guide contains a metal cladding. io44oH4x】 We have analyzed the guiding characteristics of various exp[-p4z7(z-22)] exp[p47(z-22)] selected structures of arbitrary index profiles,graded-index -T4y exp[-p4y (z-22)]T4 exp[p4=Y(z-z2)] profiles,anisotropic waveguides,and metal-cladded wave- guides.We also have investigated the effects of perturbation, × 百4 such as that of a prism,on guided modes.We have seen that y this is a convenient and easy way of solving various forms of where Tj=pjy/njo2.The above equations are for the TE planar optical waveguides. (Y =0)modes.In the case of TM(Y=1)modes,Ejy is re- placed by Hjy and iwuuoHjz is replaced by iweoEjz.The APPENDIX A boundary conditions imply that the fields in the plane of the guide are continuous at layer boundaries.We therefore relate From Eqs.(10):example,four-layer structure the fields as follows: Eix 2=0Ey(0刨1 [E2(0)1 liωoHjst Hix(0)] H2x(0) = exp[-pjzy(z-2j-2)] exp[pjzy(z-zj-2】 [E2y(211 _[E8y(21)1 2=21 L-Tjayexp[-pjs(z-2j-2)]T'jay exp[pjsy(2-2j-2)]] H2x(21)Har (21) i [E3(z2)1_[E4y(22)l 2=22 B Has(22)] HAz(22) For the substrate:first semi-infinite layer We may now write iwu4 oHx」 [exp[-P1zy(-2川 exp[p1zy(-z】1A1y in terms of a matrix and 【-1 exp[-p1zv(-z】I1 exp1zy(-z川lBJ Hence MiA Mg A4 1 1 -1 exp-p3zy(z2-2】expp3zy(22-z1】[11-1 -T3y exp[-pszy(22-21)]Iaay explps7(22-z1)]-Ts27 T3 Mp Mn X exp(-p2zy21) exp(p2zy21) 1[11 1-1111[A -T2:exp(-p22721)T2 exp(p2zY21-T27 T27] -(Mrs MrMr Mn [A1 B4】 This relationship is extended to a multilayer case as fol- lows: for the second layer2 E2y E=MaMa-1…MMsM2Mnl Bnyl iwupoH2z We may write = exp(-P2zxz) exp(p2zz) -T2z exp(-p2yz)T2y exp(p2zyz)][B2yl [a1a吗=MnMn-.M4M为MeMl 03 4 for the third layer3 For the guiding condition to be satisfactory,must be equal E3y to zero and the fields outside the guide boundaries normal to iwμ4oH3x the plane of the film must be evanescent. exp[-pazy(z-21)] exp[p3r(z-z】l = -3 expl-P3zy(2-z1】T3 zr exp[p3zY(2-z川 ACKNOWLEDGMENTS A37] The author is grateful to W.S.C.Chang of the University of [B37] California,San Diego,for his valuable suggestions and critical
Vol. 2, No. 4/April 1985/J. Opt. Soc. Am. A 601 lipsoid axes coincide with the selected coordinate systems. The theory may be used to synthesize index profiles of any given shape. In addition, we have extended our analysis to lossy situations when a guide contains a metal cladding. We have analyzed the guiding characteristics of various selected structures of arbitrary index profiles, graded-index profiles, anisotropic waveguides, and metal-cladded waveguides. We also have investigated the effects of perturbation, such as that of a prism, on guided modes. We have seen that this is a convenient and easy way of solving various forms of planar optical waveguides. APPENDIX A From Eqs. (10): example, four-layer structure Ejy [ exp [-pjzy(z - zi- 2)] exp [piz(z - zj-2)] 1 [-rjz 7exp[-pj.(z - Zj-2)] jzy exp[pji,,(z-zj-2)]] X -j For the substrate: first semi-infinite layer El liw~lpoHldj exp[-plzy(-z)] exp[pizy(-z)] 1 A1 l |-rlz-yexp[-Piz-r(z)] Plz exp[lz'y(-z)]] LB1 2' for the fourth semi-infinite layer E4y iAyoH4.1 [ exp[-p 4zY(Z - z 2)] exp[p 4z(z - Z2)] 1 [-I4zy exp[-p4zy(z - 2)] r 4 zy exp[p4zy(z -Z 2)] [B4 ,J where rjz = pjz/nj 0 2'y' The above equations are for the TE (,y = 0) modes. In the case of TM (y = 1) modes, Ejy is replaced by Hjy and i i oHj is replaced by ieoEjx. The boundary conditions imply that the fields in the plane of the guide are continuous at layer boundaries. We therefore relate the fields as follows: Z=o [Ey(O)] = [E2y(O)] IHlx(O)j M. (0)1 Z = zl [E2y(Zl)] = [E3y(Z ) tH2 ( 1)] [H3. (Z. 1)1 Z = Z2 [E3y(z2)] = [E4y (z2)1] tH3. (z 2)] j H4. (z 2) j We may now write [AJ7 LB- y in terms of a matrix and [A1 7 Hence Mf4 Mf 3 A4,] = exp[-P3z-,(z2 - 0] exp[p3zy(Z2 - Zl)]1 [B-4,y L-r4z- r4zy L-r 3zy exp[-P 3zy(Z2 -z)] Fr 3 z exp[p3,,(z 2 - z) -r3 r, Mf2 Mfl X I exp(-P2Zz-Z) exp(p 2zyZl) P 1y 1 ly -r2z exp(-P2zpZp) r2z exp(P2zZl)Pl-r2zy r2z[ -rz rz IB1 |4Bj =[Mf4 Mf 3 Mf 2 Mfji IByj. for the second layer 2 E2Y 1 [A oH2x] [ exp(-P2zzZ) exp(P2 zyz) 1[A2 1 [-P2 _ exp(-P2zzZ) 2zy exp(p2 z-)J 7 B2,y for the third layer3 E3y i(099oH3x. exp[-p3z,(z - )] exp[p3zy(z - l) 1 [-F3z, exp[-P3zy(Z- zi)] r3zy exp [Pzy(z- Z 1) LB3,J This relationship is extended to a multilayer case as follows: Ani] = [Mf. Mfn- ... Mf4 Mf3 M 2 Mf] Al]- We may write [ a21 = [Mf. n-.l . . Mf4 Mf3 M 2 Mfl]. For the guiding condition to be satisfactory, a 4 must be equal to zero and the fields outside the guide boundaries normal to the plane of the film must be evanescent. ACKNOWLEDGMENTS The author is grateful to W. S. C. Chang of the University of California, San Diego, for his valuable suggestions and critical L. M. Walpita
602 Opt.Soc.Am.A/Vol.2,No.4/April 1985 L.M.Walpita reading of the manuscript.The author also wishes to thank 6.T.Miyamoto and M.Momoda,"Propagation characteristics of J.B.Davis and C.W.Pitt of University College,London,for a multilayered thin film optical waveguide with buffer layer,"J. initial discussions.This work was initially carried out at 0pt.Soc.Am.72,1163-1166(1982). University College,London,and continued at the University 7.J.R.Wait,Electromagnetic Waves in Stratified Media (Perga- mon,London,1970),pp.8-21. of California,San Diego. 8.E.F.Kuester and D.C.Chang,"Propagation,attenuation,and dispersion characteristics of inhomogeneous dielectric slab waveguides,"IEEE Trans.Microwave Theory Tech.MTT-23, REFERENCES 98-106(1975). 9.M.Born andE.Wolf,Principles of Optics(Pergamon,London, 1.P.K.Tien,"Light waves in thin films and integrated optics," 1980),pp.665-718. Appl.0pt.10,2395-2413(1971). 10.V.Ramaswamy,"Ray model of energy and power flow in aniso- 2.D.P.Gia Russo and J.H.Harris,"Wave propagation in aniso- tropic film waveguides,"J.Opt.Soc.Am.64,1313-1320 tropic thin film optical waveguides,"J.Opt.Soc.Am.63,138-145 (1974). (1973). 11.E.C.Jordon and K.G.Balman,Electromagnetic Waves and 3.M.O.Vassell,"Structure of optical guided modes in planar Radiating Systems (Prentice-Hall,Englewood Cliffs,N.J.,1968), multilayers of optically anisotropic materials,"J.Opt.Soc.Am. pp.100-111. 64,166-173(1974). 12.The computer-time differential depends on the size of the prob- 4.Y.Yamato,T.Kamiya,and H.Yanai,"Characteristics of optical lem under investigation.As an example,in obtaining the nor guided modes in multilayer metal-clad planar optical guide with malized propagation constant of a 10-layer structure,the zero- low index dielectric buffer layer,"IEEE J.Quantum Electron element method takes less than 5%of the computation time that QE-11,729-736(1975). Vassell's method requires. 5.G.B.Hocker and W.K.Burns,"Modes in diffused optical 13.T.Findakly and C.L.Chen,"Diffused optical waveguides with waveguides of arbitrary index profile,"IEEE J.Quantum Elec- exponential profiles:effect of metal-clad and dielectric overlay," tron.QE-11,270-276(1975). Appl.0pt.17,469-474(1978)
602 Opt. Soc. Am. A/Vol. 2, No. 4/April 1985 reading of the manuscript. The author also wishes to thank J. B. Davis and C. W. Pitt of University College, London, for initial discussions. This work was initially carried out at University College, London, and continued at the University of California, San Diego. REFERENCES 1. P. K. Tien, "Light waves in thin films and integrated optics," Appl. Opt. 10, 2395-2413 (1971). 2. D. P. Gia Russo and J. H. Harris, "Wave propagation in anisotropic thin film optical waveguides," J. Opt. Soc. Am. 63, 138-145 (1973). 3. M. 0. Vassell, "Structure of optical guided modes in planar multilayers of optically anisotropic materials," J. Opt. Soc. Am. 64, 166-173 (1974). 4. Y. Yamato, T. Kamiya, and H. Yanai, "Characteristics of optical guided modes in multilayer metal-clad planar optical guide with low index dielectric buffer layer," IEEE J. Quantum Electron. QE-lI, 729-736 (1975). 5. G. B. Hocker and W. K. Burns, "Modes in diffused optical waveguides of arbitrary index profile," IEEE J. Quantum Electron. QE-lI, 270-276 (1975). 6. T. Miyamoto and M. Momoda, "Propagation characteristics of a multilayered thin film optical waveguide with buffer layer," J. Opt. Soc. Am. 72, 1163-1166 (1982). 7. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, London, 1970), pp. 8-21. 8. E. F. Kuester and D. C. Chang, "Propagation, attenuation, and dispersion characteristics of inhomogeneous dielectric slab waveguides," IEEE Trans. Microwave Theory Tech. MTT-23, 98-106 (1975). 9. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 665-718. 10. V. Ramaswamy, "Ray model of energy and power flow in anisotropic film waveguides," J. Opt. Soc. Am. 64, 1313-1320 (1974). 11. E. C. Jordon and K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), pp. 100-111. 12. The computer-time differential depends on the size of the problem under investigation. As an example, in obtaining the nor- malized propagation constant of a 10-layer structure, the zero- element method takes less than 5% of the computation time that Vassell's method requires. 13. T. Findakly and C. L. Chen, "Diffused optical waveguides with exponential profiles: effect of metal-clad and dielectric overlay," Appl. Opt. 17, 469-474 (1978). L. M. Walpita
