714 OPTICS LETTERS Vol.16,No.10 May 15,1991 Performance of the effective-index method for the analysis of dielectric waveguides Kin S.Chiang Commonwealth Scientific and Industrial Research Organization,Divi on of Applied Physics,Lindfield,2070,Australia Received June 11,1990 An asymptotic study of the effective-index method for the analysis of rectangular-core dielectric waveguides is given.Two ways of applying the effective-index method,depending on how the effective index is calculated,are considered,and expressions for the errors in the calculation of the propagation constant are derived.These expressions show explicitly how the accuracy of the method varies with the normalized frequency,the mode orders, the dimensions of the waveguide,and the relative refractive indices of the core and the surrounding media.Many novel properties of the method are revealed by these expressions.For example,it can be shown that the effective- index method can underestimate the propagation constant for a strip waveguide. The effective-index method!is probably the most Two different ways of applying the effective-index popular method for the analysis of rectangular dielec- method to this waveguide are possible.The effective- tric waveguides,which are the fundamental structures index method that results in an x-dependent profile is in many millimeter-wave and optical integrated cir- discussed first.The mode index (the propagation cuits (see the references in Ref.2).The basic idea of constant divided by the free-space wave number)of the method is to replace the rectangular structure by the TEn-1 mode in the slab of half-thickness b is used an equivalent slab with an effective refractive index as the effective refractive index nx of a second slab of obtained from another slab.Although the method is half-thickness a.The propagation constant of the simple,an assessment of its accuracy has mainly relied TEm-1 mode in the second slab is then regarded as the on comparisons with accurate numerical data.There approximate propagation constant of the Emn mode in has been little theoretical research on the performance the rectangular structure.It has been shown by Ku- of the method.How does the accuracy of the method mar et al.3 that this process is equivalent to solving a change with optical wavelength,waveguide dimen- profile as shown in Fig.2.This profile,which is exact sions,mode orders,and refractive-index profile?For for the effective-index method,differs from the origi- what waveguide structure is the method most accu- nal profile only in certain cladding regions.The di- rate?Does the method always give an upper bound electric constant of the n region of the original wave- for the propagation constant as widely believed?3 guide is increased by an amount n2-n2,while that of Questions such as these are yet to be answered. the corner regions is decreased by an amount n,2-n42. In this Letter,with the recent findings of Kumar et One can write B.2=82+ex,where B and Bx are the al.3 and an asymptotic analysis,the first attempt to exact and the calculated propagation constants,re- the author's knowledge is made to derive explicit ex- spectively,and ex is the error.According to a standard pressions for the errors of the effective-index method. perturbation analysis,the error ex is given by These expressions highlight the effects due to normal- ized frequency,mode orders,waveguide dimensions ex (n12-n2)k2Px1-(n32-n4)k2Px2 (1) and refractive indices and therefore provide a detailed description of the asymptotic behavior of the with method. ro Consider a rectangular-core waveguide as shown in Wdxdy Fig.1,where a and b are the half-width and the half- J0 P (2) thickness of the core,respectively,and n,n2,n3,and dxdy n4 are the refractive indices of the core and the sur- rounding claddings with ni>n22 n3,n4.The refrac- ro tive-index profile of this waveguide is characterized by 2 dxdy +2 wdxdy three relative index steps:A:=(n12-n2)/2n12 for i= 2,3,4.This structure is general enough to represent P2= (3) 「十网 several important classes of optical waveguide.Here dxdy we restrict ourselves to the solution of the scalar wave equation,which applies to many practical cases where where k is the free-space wave number and and are n2 is only slightly smaller than n,i.e.,A2<1.The the mode fields in the original and the approximating guided mode in the waveguide is denoted by the Emn waveguides,respectively.Only the situation where mode with m-1 and n-1 field zeros along the x andy the waveguide is operated at a large normalized fre- axes,respectively. quency V,which is defined by V=bkn(2A2)1/2,is 0146-9592/91/100714-03$5.00/0 1991 Optical Society of America714 OPTICS LETTERS / Vol. 16, No. 10 / May 15, 1991 Performance of the effective-index method for the analysis of dielectric waveguides Kin S. Chiang Commonwealth Scientific and Industrial Research Organization, Divi :.'n of Applied Physics, Lindfield, 2070, Australia Received June 11, 1990 An asymptotic study of the effective-index method for the analysis of rectangular-core dielectric waveguides is given. Two ways of applying the effective-index method, depending on how the effective index is calculated, are considered, and expressions for the errors in the calculation of the propagation constant are derived. These expressions show explicitly how the accuracy of the method varies with the normalized frequency, the mode orders, the dimensions of the waveguide, and the relative refractive indices of the core and the surrounding media. Many novel properties of the method are revealed by these expressions. For example, it can be shown that the effective￾index method can underestimate the propagation constant for a strip waveguide. The effective-index method' is probably the most popular method for the analysis of rectangular dielec￾tric waveguides, which are the fundamental structures in many millimeter-wave and optical integrated cir￾cuits (see the references in Ref. 2). The basic idea of the method is to replace the rectangular structure by an equivalent slab with an effective refractive index obtained from another slab. Although the method is simple, an assessment of its accuracy has mainly relied on comparisons with accurate numerical data. There has been little theoretical research on the performance of the method. How does the accuracy of the method change with optical wavelength, waveguide dimen￾sions, mode orders, and refractive-index profile? For what waveguide structure is the method most accu￾rate? Does the method always give an upper bound for the propagation constant as widely believed? 3 Questions such as these are yet to be answered. In this Letter, with the recent findings of Kumar et al. 3 and an asymptotic analysis, the first attempt to the author's knowledge is made to derive explicit ex￾pressions for the errors of the effective-index method. These expressions highlight the effects due to normal￾ized frequency, mode orders, waveguide dimensions, and refractive indices and therefore provide a detailed description of the asymptotic behavior of the method. Consider a rectangular-core waveguide as shown in Fig. 1, where a and b are the half-width and the half￾thickness of the core, respectively, and ni, n2, n3, and n4 are the refractive indices of the core and the sur￾rounding claddings with n1 > n2 > n3, n4. The refrac￾tive-index profile of this waveguide is characterized by three relative index steps: Ai = (n1 2 - n, 2)/2n, 2 for i = 2, 3, 4. This structure is general enough to represent several important classes of optical waveguide. Here we restrict ourselves to the solution of the scalar wave equation, which applies to many practical cases where n2 is only slightly smaller than nj, i.e., A2 << 1. The guided mode in the waveguide is denoted by the Emn mode with m - 1 and n - 1 field zeros along the x and y axes, respectively. Two different ways of applying the effective-index method to this waveguide are possible. The effective￾index method that results in an x-dependent profile is discussed first. The mode index (the propagation constant divided by the free-space wave number) of the TEn-1 mode in the slab of half-thickness b is used as the effective refractive index nx of a second slab of half-thickness a. The propagation constant of the TErn-i mode in the second slab is then regarded as the approximate propagation constant of the Emn mode in the rectangular structure. It has been shown by Ku￾mar et al. 3 that this process is equivalent to solving a profile as shown in Fig. 2. This profile, which is exact for the effective-index method, differs from the origi￾nal profile only in certain cladding regions. The di￾electric constant of the n4 2 region of the original wave￾guide is increased by an amount n1 2 - n. 2 , while that of the corner regions is decreased by an amount nx2 - n4 2. One can write #x 2 = f 2 + E, where f and f# are the exact and the calculated propagation constants, re￾spectively, and Ex is the error. According to a standard perturbation analysis, 4 the error Ex is given by ex = (n 2 -nx 2 )k2Px - (n2 - n4 2)k2P 2 , (1) with 2b fO 2 , I 9fdxdy fo -c Pxl r+0 ,+O II J Adxdy _C fO 2 J g dxdy + 2 | | &dxdy Px2 = r+<:o r+<nJSJ(3) If : Adxdy where k is the free-space wave number and iA and A are the mode fields in the original and the approximating waveguides, respectively. Only the situation where the waveguide is operated at a large normalized fre￾quency V, which is defined by V = bknl(2A2 ) / 2 , is 0146-9592/91/100714-03$5.00/0 © 1991 Optical Society of America