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 AmericaVol. 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 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 graded￾index 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 propa￾gation 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 deter￾minant 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 en￾ergy 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 ampli￾tudes 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