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). -a<<a,(7) ∫n,2+2n,△ng)exp(-x21w2), n2(x,y)=n2 >0 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 correspondingexponential 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 correspond￾ing closed-form expressions of the normalized effective index, b 5 (ne 2 2 ns 2)/2nsDn, are given in Ap￾pendix 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 refractive￾index distribution n(x, y) is the EIM, in which the struc￾ture is reduced to an effective planar structure by divid￾ing the waveguide into planar waveguide segments along x with only y confinement to obtain neff (x) (see, for ex￾ample, 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 so￾obtained 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 varia￾tional 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 com￾plete 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 al￾ways estimates the effective indices to be lower than those obtained by the exact method, whereas the effective-index procedure always overestimates the effec￾tive index of the structure, resulting in a fortutious can￾cellation 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 chan￾nel 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