Y.-F.Li and J.W.Y.Lit Vol.4,No.4/April 1987/J.Opt.Soc.Am.A 671 General formulas for the guiding properties of a multilayer slab waveguide Yi-Fan Li Guelph-Waterloo Program for Graduate Work in Physics,University of Waterloo,Waterloo,Ontario N2L 3G1. Canada John W.Y.Lit Department of Physics and Computing.Wilfrid Laurier University,Waterloo,Ontario N2L3C5,Canada Received May 26,1986:accepted November 24,1986 General formulas describing field distributions and eigenvalue equations are obtained for both transverse-electric and transverse magnetic modes in a multilayer slab waveguide.New results show that additional multilayers can produce useful effects,such as increasing the cutoff values and the confinement factors of guided modes. INTRODUCTION THEORY Multilayer structured waveguides have been widely used The general structure of an L-layer medium is shown in Fig. recently in many optical devices,such as modulators,switch- 1,where L=l+m 1.ni and di are,respectively,the es,directional couplers,Bragg deflectors,spectrum analyz- refractive index and the thickness of the layer i.We have ers,and semiconductor lasers. chosen no to be the highest refractive index for convenience, A three-layer slab waveguide is the simplest optical wave- but this is not a restriction: guide that has been well studied and documented.1-5 Wave- (1) guides with more than three layers have been studied by no>ni (i=-m,,-1,+1,…,+0. many authors.6-19 The eigenvalue equations for the four- With the choice of the coordinate system in Fig.1,we have layer structure have been derived by the wave theory and the t0=±do, (2a) ray theory.7-11 The five-layer symmetrical guide with aniso- tropic dielectric permittivity has been considered by Nelson and McKenna.12 A special structure of a five-layer wave- d guide,the so-called W waveguide,has interesting properties with respect to mode cutoffs and confinement factors.13 (t:i=1,,..,1-1;-:i=1,..,m-1),(2b) Ruschin and Marom14 have obtained the explicit eigenvalue equations of the symmetrical seven-layer waveguide for both where t+is the x coordinate of the interface between the even and odd modes by using matrix treatment.Multilayer layers +i and +(i+1)above the interfacex =0,and t-iis the waveguides with periodic index distributions have also been x coordinate of the interface between the layers -i and-(i+ studied.15-17 An explicit eigenvalue equation of a periodic 1)below the interface x=0. stratified waveguide has been obtained by Yeh et al.17 By In order to obtain a complete description of the modes of using the matrix method,Walpita and Revelli have studied multilayer waveguides,we begin with Maxwell's equations: the general N-layer waveguide,but their results involved complex matrices.18,19 X E=-udH/ot, (3) In this paper we derive the explicit formulas for the field 7×H=en;2aE/at (i=-m,,-1,0,+1,,+0. distributions and the eigenvalue equations for both trans- (4) verse-electric (TE)and transverse-magnetic (TM)modes in a general multilayer slab waveguide,starting with Maxwell's e and u are,respectively,the dielectric permittivity and the equations.The results are compared with those obtained magnetic permeability of vacuum.We do not consider mag- by some other authors,and some applications of the formu- netic materials in this paper,so the use of the vacuum value las are also considered. u is sufficient. A one-dimensional analysis is presented here.However, We simplify the description of the waveguide by assuming it may be applied to the more general case of two-dimension- that there is no variation in the y direction,which means al guides by using the effective-index approximation20.21 to chat d/dy =0.The time dependence of the field is harmonic, separate a two-dimensional problem into two one-dimen- expressed as exp(jwt).Since we are interested in obtaining sional cases. the normal modes of the waveguide,we assume also that the 0740-3232/87/040671-07$02.00 @1987 Optical Society of AmericaVol. 4, No. 4/April 1987/J. Opt. Soc. Am. A 671 General formulas for the guiding properties of a multilayer slab waveguide Yi-Fan Li Guelph-Waterloo Program for Graduate Work in Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada John W. Y. Lit Department of Physics and Computing, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada Received May 26, 1986; accepted November 24, 1986 General formulas describing field distributions and eigenvalue equations are obtained for both transverse-electric and transverse magnetic modes in a multilayer slab waveguide. New results show that additional multilayers can produce useful effects, such as increasing the cutoff values and the confinement factors of guided modes. INTRODUCTION Multilayer structured waveguides have been widely used recently in many optical devices, such as modulators, switch￾es, directional couplers, Bragg deflectors, spectrum analyz￾ers, and semiconductor lasers. A three-layer slab waveguide is the simplest optical wave￾guide that has been well studied and documented.'- 5 Wave￾guides with more than three layers have been studied by many authors. 6-19 The eigenvalue equations for the four￾layer structure have been derived by the wave theory and the ray theory. 7 -"1 The five-layer symmetrical guide with aniso￾tropic dielectric permittivity has been considered by Nelson and McKenna.12 A special structure of a five-layer wave￾guide, the so-called W waveguide, has interesting properties with respect to mode cutoffs and confinement factors.'3 Ruschin and Marom14 have obtained the explicit eigenvalue equations of the symmetrical seven-layer waveguide for both even and odd modes by using matrix treatment. Multilayer waveguides with periodic index distributions have also been studied.' 5 - 17 An explicit eigenvalue equation of a periodic stratified waveguide has been obtained by Yeh et al.17 By using the matrix method, Walpita and Revelli have studied the general N-layer waveguide, but their results involved complex matrices. 18,19 In this paper we derive the explicit formulas for the field distributions and the eigenvalue equations for both trans￾verse-electric (TE) and transverse-magnetic (TM) modes in a general multilayer slab waveguide, starting with Maxwell's equations. The results are compared with those obtained by some other authors, and some applications of the formu￾las are also considered. A one-dimensional analysis is presented here. However, it may be applied to the more general case of two-dimension￾al guides by using the effective-index approximation 2 0,21 to separate a two-dimensional problem into two one-dimen￾sional cases. THEORY The general structure of an L-layer medium is shown in Fig. 1, where L = 1 + m + 1. ni and di are, respectively, the refractive index and the thickness of the layer i. We have chosen no to be the highest refractive index for convenience, but this is not a restriction: (1) With the choice of the coordinate system in Fig. 1, we have tlo = +do, (2a) t~i = (do + E d ) k=l (+ =1 . .., I - 1; -: i = J, . ,m - J), (2b) where t+j is the x coordinate of the interface between the layers +i and +(i + 1) above the interface x = 0, and t-i is the x coordinate of the interface between the layers -i and -(i + 1) below the interface x = 0. In order to obtain a complete description of the modes of multilayer waveguides, we begin with Maxwell's equations: V X E = -yH/at, (3) V X H =,Eni 2Ma/t (i =-,...-1, 0, +1, .. ., +1). (4) E and At are, respectively, the dielectric permittivity and the magnetic permeability of vacuum. We do not consider mag￾netic materials in this paper, so the use of the vacuum value ut is sufficient. We simplify the description of the waveguide by assuming that there is no variation in the y direction, which means that alay = 0. The time dependence of the field is harmonic, expressed as exp(jcot). Since we are interested in obtaining the normal modes of the waveguide, we assume also that the 0740-3232/87/040671-07$02.00 (© 1987 Optical Society of America Y.-F. Li and J. W. Y. Lit no > ni (i = -M,..., -1, + 1, ... , + 1)