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). 0<ξ<a, (4) index-method (EIM)procedure in closed form for diffused A exp(). <0 channel waveguides.In this paper we demonstrate the application of the closed-form expressions to obtain the where A is the normalization constant;K,a,and a are the characteristics of diffused channel waveguides by a com- variational parameters to be optimized;and,by the con bined variational and effective-index method(VEIM). tinuity condition. For a comparison of results obtained by our semiana- lytical calculations,we have also implemented the con- ka y=2a2a- ventional EIM and a complete two-dimensional scalar fi- 1+ka nite difference method(FDM).Our calculations over a wide range of parameters show that our results compare In the single-mode region we were able to fix empirically better with the FDM results than with those obtained by the value of the third parameter a for different profiles, EIM calculations and also give us a good analytical esti- and by empirical curve fitting to the curves obtained for mate of the field. the optimal values we obtained simple closed-form ex- pressions for k and a in terms of the waveguide param- eters V=kohv2n,An and the asymmetry parameter p =(n2-n2)/2nAn.For all profiles,k can be written 2.CLOSED-FORM FIELD EXPRESSIONS as The refractive-index profiles of waveguides generally ob- K=ao ai pi2+a2V+a3 VpR; (5) tained in ion-exchange processes are asymmetric and can be written asFig.1(a)] the expression for a is profile dependent and is given by Gaussian and error functions (n2 2nAng(). E>0 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 AmericaClosed-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 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 ana￾lytical solutions only for a few specific planar refractive￾index profiles, and one has to use either approximate methods or numerically intensive direct methods. Varia￾tional 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 re￾quired for each calculation, and hence the procedures are not readily usable in repetitive design problems. Re￾cently 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 vari￾ous refractive-index profiles that best model the actual profiles.5 The closed-form results of the planar analysis are readily applicable in implementing the effective￾index-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 com￾bined variational and effective-index method (VEIM). For a comparison of results obtained by our semiana￾lytical calculations, we have also implemented the con￾ventional 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 esti￾mate of the field. 2. CLOSED-FORM FIELD EXPRESSIONS The refractive-index profiles of waveguides generally ob￾tained 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 sym￾metric Gaussian described by n2~ y! 5 ns 2 1 2nsDn exp~2j2!. (3) Suitable three-parameter variational fields for the asym￾metric 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 con￾tinuity 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 ex￾pressions for k and a in terms of the waveguide param￾eters 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