A.K.Taneja and E.K.Sharma Vol.16,No.11/November 1999/J.Opt.Soc.Am.A 2781 Closed-form variational effective-index analysis for diffused optical channel waveguides Ashmeet Kaur Taneja and Enakshi Khular Sharma Department of Electronic Sciences,University of Delhi South Campus,New Delhi 110021,India Received February 11,1999;revised manuscript received June 4,1999;accepted July 1,1999 We present an application of simple closed-form expressions based on a variational approach for field param- eters and effective indices for a closed-form analysis of diffused planar single-mode optical waveguides to ob- tain the characteristics of diffused channel waveguides by a combination of the variational and effective-index methods.1999 Optical Society of AmericaS0740-3232(99)00511-6] OCIS code:230.7380. 1.INTRODUCTION where &y/h,h defines the diffusion depth and g()de- Analysis of diffused planar and channel waveguides is the fines the profile shape that can be any of the following functions': foundation of the design of integrated optical waveguide devices.The scalar wave equation has closed-form ana- exp(-2). Gaussian lytical solutions only for a few specific planar refractive )= erfc(). complementary error function. index profiles,and one has to use either approximate exp(-2), exponential function methods or numerically intensive direct methods.Varia- tional procedures!-3 that have been widely used with (2) one-,two-,or three-parameter trial fields are summarized A symmetric profile that is of interest and that models the in Ref.1.However,an optimization of parameters is re- lateral profile of diffused channel waveguides is the sym- quired for each calculation,and hence the procedures are metric Gaussian described by not readily usable in repetitive design problems.Re- cently we evolved accurate closed-form expressions4 for 2(y)=n,2+2n,△nexp(-2). (3) the fields and effective indices of diffused planar Suitable three-parameter variational fields for the asym- waveguides based on the variational approach,which are metric profiles of Eq.(2)can be written as valid in the useful single-mode range of operation for vari- ous refractive-index profiles that best model the actual A(1+ka)exp(-a2a2)exp[-y(-a)]. profiles.5 The closed-form results of the planar analysis t>a are readily applicable in implementing the effective- (y)= A(1 +KE)exp(-a2g2). 00 n2(y月= nc2, ξ<0 (1) a bo bi p-i2 b2 V+bap-2v 0740-3232/99/112781-05$15.00 1999 Optical Society of America
Closed-form variational effective-index analysis for diffused optical channel waveguides Ashmeet Kaur Taneja and Enakshi Khular Sharma Department of Electronic Sciences, University of Delhi South Campus, New Delhi 110021, India Received February 11, 1999; revised manuscript received June 4, 1999; accepted July 1, 1999 We present an application of simple closed-form expressions based on a variational approach for field parameters and effective indices for a closed-form analysis of diffused planar single-mode optical waveguides to obtain the characteristics of diffused channel waveguides by a combination of the variational and effective-index methods. © 1999 Optical Society of America [S0740-3232(99)00511-6] OCIS code: 230.7380. 1. INTRODUCTION Analysis of diffused planar and channel waveguides is the foundation of the design of integrated optical waveguide devices. The scalar wave equation has closed-form analytical solutions only for a few specific planar refractiveindex profiles, and one has to use either approximate methods or numerically intensive direct methods. Variational procedures1–3 that have been widely used with one-, two-, or three-parameter trial fields are summarized in Ref. 1. However, an optimization of parameters is required for each calculation, and hence the procedures are not readily usable in repetitive design problems. Recently we evolved accurate closed-form expressions4 for the fields and effective indices of diffused planar waveguides based on the variational approach, which are valid in the useful single-mode range of operation for various refractive-index profiles that best model the actual profiles.5 The closed-form results of the planar analysis are readily applicable in implementing the effectiveindex-method (EIM) procedure in closed form for diffused channel waveguides. In this paper we demonstrate the application of the closed-form expressions to obtain the characteristics of diffused channel waveguides by a combined variational and effective-index method (VEIM). For a comparison of results obtained by our semianalytical calculations, we have also implemented the conventional EIM and a complete two-dimensional scalar fi- nite difference method6 (FDM). Our calculations over a wide range of parameters show that our results compare better with the FDM results than with those obtained by EIM calculations and also give us a good analytical estimate of the field. 2. CLOSED-FORM FIELD EXPRESSIONS The refractive-index profiles of waveguides generally obtained in ion-exchange processes are asymmetric and can be written as [Fig. 1(a)] n2~ y! 5 H ns 2 1 2nsDng~j!, j . 0 nc 2, j , 0 , (1) where j 5 y/h, h defines the diffusion depth and g(j) de- fines the profile shape that can be any of the following functions5 : g~j! 5 H exp~2j2!, Gaussian erfc~j!, complementary error function. exp~22j!, exponential function (2) A symmetric profile that is of interest and that models the lateral profile of diffused channel waveguides is the symmetric Gaussian described by n2~ y! 5 ns 2 1 2nsDn exp~2j2!. (3) Suitable three-parameter variational fields for the asymmetric profiles of Eq. (2) can be written as4 c ~ y! 5 H A~1 1 ka!exp~2a2a2!exp@2g ~j 2 a!#, j . a A~1 1 kj!exp~2a2j2!, 0 , j , a, A exp~kj!, j , 0 (4) where A is the normalization constant; k, a, and a are the variational parameters to be optimized; and, by the continuity condition, g 5 2a2a 2 ka 1 1 ka . In the single-mode region we were able to fix empirically the value of the third parameter a for different profiles, and by empirical curve fitting to the curves obtained for the optimal values we obtained simple closed-form expressions for k and a in terms of the waveguide parameters V 5 k0hA2nsDn and the asymmetry parameter p 5 (ns 2 2 nc 2)/2nsDn. For all profiles, k can be written as k 5 a0 1 a1 p1/2 1 a2V 1 a3Vp1/2; (5) the expression for a is profile dependent and is given by Gaussian and error functions a 5 b0 1 b1 p21/2 1 b2V 1 b3 p21/2V A. K. Taneja and E. K. Sharma Vol. 16, No. 11/November 1999/J. Opt. Soc. Am. A 2781 0740-3232/99/112781-05$15.00 © 1999 Optical Society of America
2782 J.Opt.Soc.Am.A/Vol.16,No.11/November 1999 A.K.Taneja and E.K.Sharma exponential function 3.APPLICATION TO DIFFUSED CHANNEL a=b0+b1p2+b2V2+b3p22 WAVEGUIDES One of the common techniques for analysis of a typical step function channel waveguide with a two-dimensional refractive- a bo +bi p12 b2 V-i2 b3 p-v-V index distribution n(x,y)is the EIM,in which the struc- (6) ture is reduced to an effective planar structure by divid- where the eight constants a and b(i=1,2,3.4)are a ing the waveguide into planar waveguide segments along given set of numbers for a given profile function g(). x with only y confinement to obtain ner(x)(see,for ex- given in Table 1. ample,Ref.7).The closed-form analysis can be applied For the symmetric Gaussian profile the corresponding to achieve this by using the closed-form expression for symmetric field is given by each of the independent planar waveguides and to obtain the effective-index profile nef(x)analytically.Hence,for (y)=Bexp(a2a2)exp(-2a2a). >a, a typical diffused channel waveguide profile [Fig.1(b)]. =Bexp(-a2g2). -a0 ξ<0 and the closed-form expressions for a and a are (⑨) a=ao a vu2 for each segment in the lateral x direction,V and p are a=bo b1V. (8) given by The four constants are given in Table 1.The correspond- V=koh2n,An exp(-x2/w2)]12, ing closed-form expressions of the normalized effective index,b=(ne2-ns2)/2nAn,are given in Ap- (n2-n2) pendix A. exp(x2/w2). (10) 2n,△n The error in the values of k and a as obtained from the empirical formula and by direct maximization of the and the corresponding a,k,and a are obtained from Eqs. variational expression is less than 0.5%.The error in the (5)and(6)and Table 1.The corresponding expression normalized effective index b is less than 2%,and field forms compare well with FDM calculations. for b(given in Appendix A)then gives nefr(x).The so- obtained typical n()profile shown in Fig.2 resembles a Gaussian function and can be well fitted to the following y=0 ne function: n2(x)=n,2+2n,8nexp(-x21d), (11) h where n(y) 8n=[n2(0)-n,2]V2ns, (12) and d,which can in general be obtained by interpolation, is the value of x where [ne(x)-n2]falls to ith of its value at the center (x=0).One can obtain the varia- ns tional parameters a and a corresponding to V kodv2nsn from Eqs.(8)to obtain the final effective (a) index ne. ne A comparison of results obtained by our calculations with those obtained by the conventional EIM and a com- plete two-dimensional FDM calculation is given in Table 2.The accuracy of the closed-form VEIM calculation and n(x,0) h conventional numerically intensive EIM calculation is comparable.In fact,the VEIM results are usually closer to the FDM results,because the variational analysis al- ways estimates the effective indices to be lower than those obtained by the exact method,whereas the h(0,y) effective-index procedure always overestimates the effec- tive index of the structure,resulting in a fortutious can- ns cellation of errors.However,the closed-form calculation is efficient and fast,and the so-obtained field forms are (b) also analytical,with the y variation at each x defined by Fig.1.(a)Typical refractive-index profile of a diffused planar a(x),K(x),and a and the xvariation by the a and a of the optical waveguide.(b)Cross section view of the diffused chan- corresponding Gaussian profile.Figure 3(a)compares nel optical waveguide showing the coordinates used. the VEIM results with FDM results of the corresponding
exponential function a 5 b0 1 b1 p21/2 1 b2V3/2 1 b3 p21/2V3/2 step function a 5 b0 1 b1 p21/2 1 b2V21/2 1 b3 p21/2V21/2 (6) where the eight constants ai and bi (i 5 1, 2, 3, 4) are a given set of numbers for a given profile function g(j), given in Table 1. For the symmetric Gaussian profile the corresponding symmetric field is given by c ~ y! 5 B exp~a2a2!exp~22a2auju!, uju . a, 5 B exp~2a2j2!, 2a , j , a, (7) and the closed-form expressions for a and a are a 5 a0 1 a1V1/2, a 5 b0 1 b1V. (8) The four constants are given in Table 1. The corresponding closed-form expressions of the normalized effective index, b 5 (ne 2 2 ns 2)/2nsDn, are given in Appendix A. The error in the values of k and a as obtained from the empirical formula and by direct maximization of the variational expression is less than 0.5%. The error in the normalized effective index b is less than 2%, and field forms compare well with FDM calculations. 3. APPLICATION TO DIFFUSED CHANNEL WAVEGUIDES One of the common techniques for analysis of a typical channel waveguide with a two-dimensional refractiveindex distribution n(x, y) is the EIM, in which the structure is reduced to an effective planar structure by dividing the waveguide into planar waveguide segments along x with only y confinement to obtain neff (x) (see, for example, Ref. 7). The closed-form analysis can be applied to achieve this by using the closed-form expression for each of the independent planar waveguides and to obtain the effective-index profile neff (x) analytically. Hence, for a typical diffused channel waveguide profile [Fig. 1(b)], n2~x, y! 5 H ns 2 1 2nsDng~j!exp~2x2/w2!, j . 0 nc 2, j , 0 , (9) for each segment in the lateral x direction, V and p are given by V 5 k0h@2nsDn exp~2x2/w2!# 1/2, p 5 ~ns 2 2 nc 2! 2nsDn exp~x2/w2!, (10) and the corresponding a, k, and a are obtained from Eqs. (5) and (6) and Table 1. The corresponding expression for b (given in Appendix A) then gives neff (x). The soobtained typical neff 2 (x) profile shown in Fig. 2 resembles a Gaussian function and can be well fitted to the following function: n2~x! 5 ns 2 1 2nsdn exp~2x2/d2!, (11) where dn 5 @neff 2 ~0! 2 ns 2#/2ns , (12) and d, which can in general be obtained by interpolation, is the value of x where @neff 2 (x) 2 ns 2 # falls to 1 e th of its value at the center (x 5 0). One can obtain the variational parameters a and a corresponding to V 5 k0dA2nsdn from Eqs. (8) to obtain the final effective index ne . A comparison of results obtained by our calculations with those obtained by the conventional EIM and a complete two-dimensional FDM calculation is given in Table 2. The accuracy of the closed-form VEIM calculation and conventional numerically intensive EIM calculation is comparable. In fact, the VEIM results are usually closer to the FDM results, because the variational analysis always estimates the effective indices to be lower than those obtained by the exact method, whereas the effective-index procedure always overestimates the effective index of the structure, resulting in a fortutious cancellation of errors. However, the closed-form calculation is efficient and fast, and the so-obtained field forms are also analytical, with the y variation at each x defined by a(x), k(x), and a and the x variation by the a and a of the corresponding Gaussian profile. Figure 3(a) compares the VEIM results with FDM results of the corresponding Fig. 1. (a) Typical refractive-index profile of a diffused planar optical waveguide. (b) Cross section view of the diffused channel optical waveguide showing the coordinates used. 2782 J. Opt. Soc. Am. A/Vol. 16, No. 11/November 1999 A. K. Taneja and E. K. Sharma
A.K.Taneja and E.K.Sharma Vol.16.No.11/November 1999/J.Opt.Soc.Am.A 2783 Table 1.Empirical Constants for Various Profiles Symmetric Constants Gaussian Error Function Exponential Step Function Gaussian a 1.00 0.78 0.85 1.20 ao -0.42 -0.60 -0.52 -0.42 -0.26 a 0.15 0.38 0.53 -0.27 0.79 a2 0.07 0.03 -0.02 0.17 as 0.93 0.87 0.81 1.13 bo 0.44 0.50 0.63 2.22 1.11 bi 0.06 -0.03 -0.05 -0.17 -0.08 bz 0.19 0.18 0.06 -1.96 bs 0.02 0.04 0.02 0.39 APPENDIX A 1.515 The closed-form expressions obtained for the normalized 1.513 effective index b for the asymmetric profiles are nefs(z) 1.511 b=(r-s)/t. (A1) 1.509 where 1.507 -6 -4 0 r(4m) 2 4pa 1 Fig.2.ne(x)profile obtained for a Gauss error function chan- 个 nel waveguide with ns=1.50771,w=4.1 um,h=3.65 um. An 0.028,at yo 1.214 um.The thicker curve corresponds a exp(-2a2a2)(16a2a-6x2a to the Gaussian fitted profile. +16Ka2a2(1+Ka)-4k], (A2) Table 2.Comparison of n,for a Channel Waveguide,n(x,y)=ng+2n,Ang(g) ×exp(-x21w2),with ns=1.50771,W=4.1m, 4(a2+k2) T erf(V2aa) Vo kohv2n,An and po =(n32-ne2)/2n,An (4a2+k2) K +V2a2 g(的 Vo.Po VEIM EIM FDM 2K Gaussian V%=4.0.P%=151.514971.5150891.51486 -exp(-2a2a2)(ka-2) V%=2.8.po=591.508201.50821 1.50803 4a(1+Ka)2 + Error V%=5.5.pP0=151.513091.51322 1.51300 exp(-2a2a2). (A3) K Function V。=7.0,P0=591.509681.50972 1.50966 22a2a- 1+ka Exponential V。=4.0,Po=151.508611.508671.50825 V=10.0.P%=591.509941.509991.50995 and the profile-dependent term r is given by Gaussian normalized field as a function of yfor different values of x, and Fig.3(b)compares the field variation in the x direc- tion at two different values of y.The corresponding two- 2 a vn erf[(1+2a2)2a][2+k2(1+2a2)] dimensional field is shown in Fig.4. (1+2a2)32 Aak2a 4.CONCLUSIONS 1+2aexp[-(1+2a2a2】+8vra1+Ka2 We have developed an accurate analytical closed-form procedure for diffused channel waveguides based on a combination of the variational method and the effective- ×ex2a1+2a2a2a-1+Ka index method.The procedure is useful for synthesis pro- cedures that require repetitive calculations with changed k21/ parameters to obtain a satisfactory design. ka)erfc 2a2a+ak 1+ka (A4)
normalized field as a function of y for different values of x, and Fig. 3(b) compares the field variation in the x direction at two different values of y. The corresponding twodimensional field is shown in Fig. 4. 4. CONCLUSIONS We have developed an accurate analytical closed-form procedure for diffused channel waveguides based on a combination of the variational method and the effectiveindex method. The procedure is useful for synthesis procedures that require repetitive calculations with changed parameters to obtain a satisfactory design. APPENDIX A The closed-form expressions obtained for the normalized effective index b for the asymmetric profiles are b 5 ~r 2 s!/t, (A1) where s 5 4pa k 1 1 V2 F 4ka 1 Ap 2 erf~A2aa!~ 4a2 1 3k2! 2 a exp~ 22a2a2!~ 16a2a 2 6k2a 1 16ka2a2 ~ 1 1 ka! 2 4k#, (A2) t 5 4~a2 1 k2! ak 1 Ap 2 erf~A2aa! a2 ~4a2 1 k2! 2 2k a exp~22a2a2!~ka 2 2! 1 4a~1 1 ka! 2 2S 2a2a 2 k 1 1 kaD exp~22a2a2!, (A3) and the profile-dependent term r is given by Gaussian r 5 2aAp erf@~1 1 2a2! 1/2a#@2 1 k2~1 1 2a2!# ~1 1 2a2! 3/2 1 4ak2a 1 1 2a2 exp@2~1 1 2a2a2!# 1 8Apa~1 1 ka! 2 3 expF 2a~1 1 2a2!S a2a 2 k 1 1 kaD 1 k2 ~1 1 ka! 2G erfcS 2a2a 1 a 2 k 1 1 kaD , (A4) Fig. 2. neff (x) profile obtained for a Gauss error function channel waveguide with ns 5 1.50771, w 5 4.1 mm, h 5 3.65 mm, Dn 5 0.028, at g0 5 1.214 mm. The thicker curve corresponds to the Gaussian fitted profile. Table 1. Empirical Constants for Various Profiles Constants Gaussian Error Function Exponential Step Function Symmetric Gaussian a 1.00 0.78 0.85 1.20 a0 20.42 20.60 20.52 20.42 20.26 a1 0.15 0.38 0.53 20.27 0.79 a2 0.07 0.03 20.02 0.17 a3 0.93 0.87 0.81 1.13 b0 0.44 0.50 0.63 2.22 1.11 b1 0.06 20.03 20.05 20.17 20.08 b2 0.19 0.18 0.06 21.96 b3 0.02 0.04 0.02 0.39 Table 2. Comparison of ne for a Channel Waveguide, n2(x, y) 5 ns 2 1 2nsDng(j) 3 exp(2x2/w2), with ns 5 1.50771, w 5 4.1 mm, V0 5 k0hA2nsDn and p0 5 (ns 2 2 nc 2)/2nsDn g(j) V0 ,p0 VEIM EIM FDM Gaussian V0 5 4.0, p0 5 15 1.51497 1.515089 1.51486 V0 5 2.8, p0 5 59 1.50820 1.50821 1.50803 Error V0 5 5.5, p0 5 15 1.51309 1.51322 1.51300 Function V0 5 7.0, p0 5 59 1.50968 1.50972 1.50966 Exponential V0 5 4.0, p0 5 15 1.50861 1.50867 1.50825 V0 5 10.0, p0 5 59 1.50994 1.50999 1.50995 A. K. Taneja and E. K. Sharma Vol. 16, No. 11/November 1999/J. Opt. Soc. Am. A 2783
2784 J.Opt.Soc.Am.A/Vol.16,No.11/November 1999 A.K.Taneja and E.K.Sharma r=owaV目daa+ 0.2 x=2.1m m-剑 -x=4,27n 82a2a+42 -20 8101214.1618 y(um) (a) +-+4+- 4a(1+Ka)2 X exp[-2a(1+a2a)], K 02 +y=1.84m 1+2a2a-1+ka (A6) *9=0 step function 0 15 10 0 10 15 r(um) (b) 4K πerf(V2aa) Fig.3.(a)Comparison of normalized field forms obtained by the r=+V2 a2 (4a2+x3) VEIM and the FDM in the y direction at different values of x for a Gauss error function channel waveguide with parameters as in Fig.2.The thicker curves correspond to the VEIM results.(b) 2K -exp(-2a2a2)(Ka-2). A7) Comparison of normalized field forms obtained by the VEIM and the FDM in the x direction at different values of y for a Gauss error function channel waveguide with parameters as in Fig.2. The thicker curves correspond to the VEIM results. The normalization constant A is given by error function A =8a/th (A8) r=4a (1 +K)2 exp(-2a2g2)erfc()dgf 0 For the symmetric Gaussian profile,b is given by Ka 2a(1 ka)2 exp 2 a2a2- 1+ka 2a2a- 1+ka 0.27 exp 22a2a 1+ka 0 exp 2a'a-1+xa 1086 erfe 2a2a c(um) 1+ka (A5) 0。 (m】 Fig.4.Two-dimensional field obtained for a Gauss error func- exponential function tion channel waveguide with parameters as in Fig.2
error function r 5 4a E 0 a ~1 1 kj! 2 exp~22a2j2!erfc~j!djf 1 2a~1 1 ka! 2 expF 2S a2a2 2 ka 1 1 kaDG S 2a2a 2 k 1 1 kaD 3 H 0.27 expF 2S 2a2a 2 k 1 1 kaDG 2 expS 2a2a 2 k 1 1 kaD 2 erfcS 2a2a 2 k 1 1 ka 1 aD J , (A5) exponential function r 5 exp~1/2a2!HAp 2 F erf S A2aa 1 1 A2a D 2 erf S 1 A2a DGF 4S 1 2 k 2a2 D 2 1 k2 a2G 2 expS A2aa 1 1 A2a D 2 F 8A2a2a 1 4A2 1 4k a S 1 2 k 2a2 DGJ 1 4A2 1 4k a S 1 2 k 2a2 D 1 4a~1 1 ka! 2 1 1 2a2a 2 k 1 1 ka exp@22a~1 1 a2a!#, (A6) step function r 5 4k a 1 Ap 2 erf~A2aa! a2 ~4a2 1 k2! 2 2k a exp~22a2a2!~ka 2 2!. (A7) The normalization constant A is given by A 5 A8a/th . (A8) For the symmetric Gaussian profile, b is given by Fig. 3. (a) Comparison of normalized field forms obtained by the VEIM and the FDM in the y direction at different values of x for a Gauss error function channel waveguide with parameters as in Fig. 2. The thicker curves correspond to the VEIM results. (b) Comparison of normalized field forms obtained by the VEIM and the FDM in the x direction at different values of y for a Gauss error function channel waveguide with parameters as in Fig. 2. The thicker curves correspond to the VEIM results. Fig. 4. Two-dimensional field obtained for a Gauss error function channel waveguide with parameters as in Fig. 2. 2784 J. Opt. Soc. Am. A/Vol. 16, No. 11/November 1999 A. K. Taneja and E. K. Sharma
A.K.Taneja and E.K.Sharma Vol.16,No.11/November 1999/J.Opt.Soc.Am.A 2785 T Ver1a)+exp(2+1)erte[a(2+1]Vert(aa) b= .(A9) 1π a Vzerf(V2aa)+ 1 2a2aexp(-2a2a) and the normalization constant is given by 1/2 B= 1 (A10) a Vzerf(v2aa)+ h agexp(-2aa) Address correspondence to E.K.Sharma at waveguides,"J.Lightwave Technol.12.1543-1549(1994). enakshi@bol.net.in. 4.A.K.Taneja,S.Srivastava,and E.K.Sharma,"Closedform expressions for propagation characteristics of diffused pla- nar optical waveguides,"Microwave Opt.Technol.Lett.15. REFERENCES 305-310(1997). 5. S.I.Najafi,ed.,Introduction to Glass and Integrated Optics 1.E.K.Sharma,S.Sharma,and J.P.Meunier."Design of re- (Artech House,London,1992). fractive ion exchange integrated optical components,"IEEE 6.M.Stern."Finite-difference analysis of planar optical J.Sel.Top.Quantum Electron.2.163-175 (1996). waveguides,"in Methods for Modeling and Simulation of 2.W.-H.Tsai,S.-C.Chao,and M.-S.Wu,"Variational analysis Guided-Wave Optoelectronic Devices,W.P.Huang.ed., of single mode inhomogeneous planar optical waveguides," Progress in Electromagnetic Research,Vol.10(EMW,Cam- J.Lightwave Technol.10.747-751 (1992). bridge,Mass.,1995),pp.123-186. 3.S.-C.Chao,M.-S.Wu,and W.-H.Tsai,"Variational analysis 7.H.Nishihara,M.Haruna,and T.Suhara,Optical Inte- of modal coupling efficiency between graded-index grated Circuits (McGraw-Hill,New York)
Address correspondence to E. K. Sharma at enakshi@bol.net.in. REFERENCES 1. E. K. Sharma, S. Sharma, and J. P. Meunier, ‘‘Design of refractive ion exchange integrated optical components,’’ IEEE J. Sel. Top. Quantum Electron. 2, 163–175 (1996). 2. W.-H. Tsai, S.-C. Chao, and M.-S. Wu, ‘‘Variational analysis of single mode inhomogeneous planar optical waveguides,’’ J. Lightwave Technol. 10, 747–751 (1992). 3. S.-C. Chao, M.-S. Wu, and W.-H. Tsai, ‘‘Variational analysis of modal coupling efficiency between graded-index waveguides,’’ J. Lightwave Technol. 12, 1543–1549 (1994). 4. A. K. Taneja, S. Srivastava, and E. K. Sharma, ‘‘Closedform expressions for propagation characteristics of diffused planar optical waveguides,’’ Microwave Opt. Technol. Lett. 15, 305–310 (1997). 5. S. I. Najafi, ed., Introduction to Glass and Integrated Optics (Artech House, London, 1992). 6. M. Stern, ‘‘Finite-difference analysis of planar optical waveguides,’’ in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices, W. P. Huang, ed., Progress in Electromagnetic Research, Vol. 10 (EMW, Cambridge, Mass., 1995), pp. 123–186. 7. H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York). b 5 A p 2a2 1 1 erf~A2a2 1 1a! 1 Ap exp@2a2a2~2a2 1 1!#erfc @a~2a2 1 1!# 2 a V2 Ap 2 erf~A2aa! 1 a Ap 2 erf~A2aa! 1 1 2a2a exp~22a2a! , (A9) and the normalization constant is given by B 5 H 1 hF 1 a Ap 2 erf~A2aa! 1 1 2a2a exp~22a2a!GJ 1/2 . (A10) A. K. Taneja and E. K. Sharma Vol. 16, No. 11/November 1999/J. Opt. Soc. Am. A 2785