December 1988 Vol.13,No.12 OPTICS LETTERS 1129 Explanation of errors inherent in the effective-index method for analyzing rectangular-core waveguides A.Kumar,*D.F.Clark,and B.Culshaw Optoelectronics Division,Department of Electronic and Electrical Engineering,University of Strathclyde,204 George Street, Glasgow G1 1XW,UK 'Received March 19,1988;accepted September 13,1988 It is shown that use of the effective-index method for a rectangular-core waveguide is equivalent to analysis of a pseudorectangular-core waveguide,the dielectric constant of which is higher in some of the cladding regions than that of the actual waveguide.This explains why the effective-index method gives higher values for the propagation constants for the various guided modes than other methods and different results depending on whether one starts with the longer dimension or the shorter dimension to construct the effective-index waveguide. Most integrated-optical waveguides consist of either tive index of the mth mode of the waveguide shown in rectangular or near-rectangular cores,and therefore Fig.1(c)then corresponds to the approximate value of many analyticall-5 as well as numerical methods6.7 Bmn of the given waveguide [Fig.1(a)]. have been reported for analyzing such structures.Of The EIM inherently uses the method of separation these,the effective-index method(EIM)is one of the of variables to solve the wave equation,by approxi- most extensively used,as it is a relatively simple meth- mating the dielectric-constant profile no2(x,y)by od and can even be applied to nonrectangular wave- some other profile of the form guide geometries.It is also known that in the case of rectangular-core waveguides the EIM overestimates n2(x,y)=n2(x)+n"2(y). (2) the propagation constants of the various guided modes.3.4 Furthermore,different results are obtained depending on whether one starts with the longer di- If vmn(x,y)=Xm(x)Yn(y)is taken to represent the mension or the shorter dimension.However,to our modal-field distribution of the mode under consider- knowledge,no one has yet explained why these errors ation,then Xm(x)and Yn(y)will satisfy the scalar- should arise in the EIM. wave equation In this Letter we show that using the EIM for a rectangular-core waveguide is equivalent to applying the method of separation of variables (or Marcatili's method2)to a pseudorectangular-core waveguide,the dielectric constant of which is higher in some cladding regions and lower in the corner regions than that of the actual waveguide.We also explain why the errors 2a mentioned above are inherent in the EIM. We discuss the scalar-wave modes of a rectangular- n吃 core waveguide as shown in Fig.1(a)with core dimen- (a) sions 2a and 26 (a b),which can represent a channel waveguide (n2=n4),a rib waveguide (n3 n4),or a symmetric-core waveguide (n2 =n3 =n4).In the 20 scalar-wave approximation the modal-field patterns mn(x,y)and the corresponding propagation con- stants are the solutions of the equation 25 Vmn [hono(8mm0,(1) dy2 where ko is the free-space wave number and no2(x,y) represents the dielectric constant distribution of the (b) given waveguide [Fig.1(a)]. c According to the EIM,to obtain the propagation constant 8mn of the Emn mode of such a waveguide,one Fig.1.(a)Diagram of a rectangular-core waveguide.(b) first obtains the effective index Bv/ko of the nth-order and (c)Planar waveguides used to calculate propagation mode of the waveguide shown in Fig.1(b).The effec- constants through EIM. 0146-9592/88/121129-03$2.00/0 1988 Optical Society of America
December 1988 / Vol. 13, No. 12 / OPTICS LETTERS 1129 Explanation of errors inherent in the effective-index method for analyzing rectangular-core waveguides A. Kumar,* D. F. Clark, and B. Culshaw Optoelectronics Division, Department of Electronic and Electrical Engineering, University of Strathclyde, 204 George Street, Glasgow G1 lXW, UK Received March 19, 1988; accepted September 13, 1988 It is shown that use of the effective-index method for a rectangular-core waveguide is equivalent to analysis of a pseudorectangular-core waveguide, the dielectric constant of which is higher in some of the cladding regions than that of the actual waveguide. This explains why the effective-index method gives higher values for the propagation constants for the various guided modes than other methods and different results depending on whether one starts with the longer dimension or the shorter dimension to construct the effective-index waveguide. Most integrated-optical waveguides consist of either rectangular or near-rectangular cores, and therefore many analytical 1 - 5 as well as numerical methods 6 ' 7 have been reported for analyzing such structures. Of these, the effective-index method (EIM) is one of the most extensively used, as it is a relatively simple method and can even be applied to nonrectangular waveguide geometries. It is also known that in the case of rectangular-core waveguides the EIM overestimates the propagation constants of 'the various guided modes.3 ' 4 Furthermore, different results are obtained depending on whether one starts with the longer dimension or the shorter dimension. However, to our knowledge, no one has yet explained why these errors should arise in the EIM. In this Letter we show that using the EIM for a rectangular-core waveguide is equivalent to applying the method of separation of variables'(or Marcatili's method2) to a pseudorectangular-core waveguide, the dielectric constant of which is higher in some cladding regions and lower in the corner regions than that of the actual waveguide. We also explain why the errors mentioned above are inherent in the EIM. We discuss the scalar-wave modes of a rectangularcore waveguide as shown in Fig. 1(a) with core dimensions 2a and 2b (a > b), which can represent a channel waveguide (n2 = n4 ), a rib waveguide (n3 = n4 ), or a symmetric-core waveguide (n2 = n3= n4). In the scalar-wave approximation the modal-field patterns Vlmn(X, y) and the corresponding propagation constants are the solutions of the equation a2 tmna4m d2 + 2 + [ko2no2 (x Y) - Imn ]Imn = 0, (1) where ko is the free-space wave number and no2(x, y) represents the dielectric constant distribution of the given waveguide [Fig. 1(a)]. According to the EIM, to obtain the propagation constant lPmn of the Emn mode of such a waveguide, one first obtains the effective index ,3ny/ko of the nth-order mode of the waveguide shown in Fig. 1(b). The effective index of the mth mode of the waveguide shown in Fig. 1(c) then corresponds to the approximate value of tmn of the given waveguide [Fig. 1(a)]. The EIM inherently uses the method of separation of variables to solve the wave equation, by approximating the dielectric-constant profile no2(x, y) by some other profile of the form n2(x, y) = n'2 (x) + n" 2(y). (2) If iPmn(X, y) = Xm(x)Yn(y) is taken to represent the modal-field distribution of the mode under consideration, then Xm(x) and Yn(y) will satisfy the scalarwave equation A Y n23 n,2 2b _ _ _ ,_------- -+ x n2 2 4 ni 2Q -4 2 (a) 2n3 2b n2 1. n 2 n2 2a i h2 2 n4 (b) n2n4 (C) Fig. 1. (a) Diagram of a rectangular-core waveguide. (b) and (c) Planar waveguides used to calculate propagation constants through EIM. 0146-9592/88/121129-03$2.00/0 © 1988 Optical Society of America
1130 OPTICS LETTERS Vol.13,No.12 December 1988 1axm+是dya 9 号1 Xm dx2 Yn dy2 +2[n2(x)+n"26y]-Bmn23=0.(3) E11 According to the EIM,one first removes the y de- E12.E21 pendence from Eq.(3)by considering a planar wave- guide [shown in Fig.1(b)],thus giving n2(y)as 3 n"2y)=n12 lyl <b 2 =ng2 y>b 0.8 12 18 2.0 24 2832384.0 =n22 y<-b. (4) B Yn(y)are the modes of this waveguide and satisfy the 1.0 2 following equation: 0.8 1a2Y+kn20)-6nw9=0. 0 E21 Yn dy2 (5) 06 Subtracting Eq.(5)from Eq.(3),one gets the following 04 equation for Xm(x): E12 02 "n2(x)+8n/m0.(6) *: Xm dx2 0.4 0.8 1.21.62.02.428 323.64.0 This shows that Bmn is the propagation constant of the mth mode of a planar waveguide whose dielectric con- stant distribution is given by [n'2(x)+Bny2/ko2].Ac- Fig.3.Variation of the normalized propagation constant cording to the EIM,this is a waveguide shown in Fig. P2 as a function of B for a rectangular-core waveguide with 1(c),with n2=n3=n4 and (top)a/b=1 and (bottom)a/b=2.The dashed and dotted curves correspond to the EIM and to the n2(x)+Bny2/o2=Bny2/ko2 Ixl<a method of Goell,s respectively. =n42 xl>a, (7) which gives us n2(x)=0 xl <a .2 n+n- -晚x xl>a. (8) Ko to. Substituting n'2(x)and n"2(y)from Eqs.(8)and (4) 2b n 喉 into Eq.(2),one gets the separable dielectric-constant profile n2(x,y)(shown in Fig.2),which the EIM effec- tively analyzes,rather than the waveguide shown in 20 Fig.1(a). nn- 吃 好 Fig.4.Separable dielectric-constant distribution analyzed by EIM if one first starts from the x direction. !, 哈 - 畅 n *肠 It is clear from Fig.2 that n2(x,y)has a higher value in cladding region 4(lxl>a,lyl <b)than in the actual waveguide by an amount n2-Bny2/ko2 and a lower 2a value in the corner regions by an amount Bny2/ko2- n42.Since the fractional power in the cladding regions n吃 is much more than that in the corner regions,the net effect of the differences between the dielectric-con- stant distributions of the given waveguide Fig.1(a)] Fig.2.Separable dielectric constant distribution n2(x,y) and those of Fig.2 is that the propagation constants described by Eqs.(2),(4),and (8). for the different guided modes are always overestimat-
1130 OPTICS LETTERS / Vol. 13, No. 12 / December 1988 1 d 2Xm + 1 d2 Yn Xm dX2 Yn dy2 + {k0 2[n'2(x) + n"2(%)] - mn21 .. = 0. (3) According to the EIM, one first removes the y dependence from Eq. (3) by considering a planar waveguide [shown in Fig. 1(b)], thus giving n"2 (y) as n / 2 (y) = n 2 1.0 0.8 p2 0.8 0.4 0.2 lyl< b =n32 y>b = n22 y<-b. ,-I A.. E2~~~~ME2 1 0.4 0.8 1.2 1.8 2.0 2A 2.8 3.2 3.8 (4) Yn(y) are the modes of this waveguide and satisfy the following equation: 1 d 2 Yn 2 2(y) Yn~ dy2 + [k0 2 n"2 y- #p32] = 0. (5) Subtracting Eq. (5) from Eq. (3), one gets the following equation for Xm(x): 1d 2 X Xm d 2 M + {k 0 2 [n'2(x) + f3nY2/k0 2] -mn1 = 0. (6) This shows that O=n is the propagation constant of the mth mode of a planar waveguide whose dielectric constant distribution is given by [n'2 (x) + flnY2/ko2]. According to the EIM, this is a waveguide shown in Fig. 1(c), with n' 2 (x) + 3ny 2 /k 0 2 = Ony 2 /k 0 2 = n4 2 which gives us n'2 (x) = 0 - n4 2) lxi <a lxi > a, 1.0 a =2:2 081- p2 0.6 0.4 0.2 ,. * I 1 1.' ",S. 0.4 0.8 1.2 1.6 2.0 2.4 2.8 .3.2 3.6 4.0 Fig. 3. Variation of the normalized propagation constant P2 as a function of B for a rectangular-core waveguide with n2 = n3 = n4 and (top) a/b = 1 and (bottom) alb = 2. The dashed and dotted curves correspond to the EIM and to the method of Goell,6 respectively. (7) 1xI<a xl > a. (8) Substituting n'2 (x) and n"2 (y) from Eqs. (8) and (4) into Eq. (2), one gets the separable dielectric-constant profile n2 (x, y) (shown in Fig. 2), which the EIM effectively analyzes, rather than the waveguide shown in Fig. 1(a). n 2 4~ n2 -(p2- n4) _ _ 2n4 <__ 2 . 1 2 12 P2 1 ~~~~ko n2 _ n2) kox n32 2b 2a n 2 2 Fig. 2. Separable dielectric constant described by Eqs. (2), (4), and (8). I n,2 - 2 _- 2 , n- ( -n2 I_ __ ko__- n2 _ -,,22 k2 t n2r b t o n 2 ) distribution n2(X, y) Fig. 4. Separable dielectric-constant distribution analyzed by EIM if one first starts from the x direction. It is clear from Fig. 2 that n2(x, y) has a higher value in cladding region 4 (lxi > a, Iyl < b) than in the actual waveguide by an amount n1 2 - 6nY2/k0 2 and a lower value in the corner regions by an amount 3ny 2/ko2 - n4 2 . Since the fractional power in the cladding regions is much more than that in the corner regions, the net effect of the differences between the dielectric-constant distributions of the given waveguide [Fig. 1(a)] and those of Fig. 2 is that the propagation constants for the different guided modes are always overestimat- 4.0 a, 1. 1. , '.. . I1. I. '.. I . , . r.. I,._,,... .. Ony2 ko2
December 1988 Vol.13,No.12 OPTICS LETTERS 1131 ed by the EIM.If one decreases the v.value [=kob(n12 field penetration is greater than in cladding region 4. -n22)1/2]along the y direction,Bny/ko will move to- As a result the error introduced in the calculation of ward n2.As a result,the error in the EIM should the propagation constants will be.more than in the increase owing to the increase in the overestimation of previous case.This explains why the EIM gives dif- the dielectric constant in region 4.This is clear from ferent (less accurate)values for the propagation con- Fig.3,where we have plotted the values of the normal- stants of the various modes if one starts with the x-slab ized propagation constants p2 {=(8mn2 -ko2n22)/ waveguide instead of the y-slab waveguide for con- [ko2(n2-n22)]as a function of normalized frequency structing the effective-index waveguide. B [=(2/T)v]for the first few modes of two symmetric A.Kumar thanks the British Council for financial rectangular-core waveguides of aspect ratios 1 and 2. We have also obtained the propagation constants support. On leave from the Physics Department,Indian for the modes shown in Fig.3 by applying the method of separation of variables to the corresponding wave- Institute of Technology,Delhi,New Delhi,India. guides shown in Fig.2.It is found that the values obtained are the same as those obtained by the EIM. References This confirms the fact that applying the EIM to the waveguide shown in Fig.1(a)is equivalent to analyzing 1.R.M.Knox and P.P.Toulios,in Proceedings of MRI the corresponding waveguide shown in Fig.2. Symposium on Submillimeter Waves,J.Fox,ed.(Poly- Similarly,if one starts with the x waveguide,it can technic Press,New York,1970),p.497. be shown that the EIM is equivalent to analyzing a 2.E.A.J.Marcatili,Bell Syst.Tech.J.48,2071(1969). waveguide (shown in Fig.4)whose dielectric constant 3.G.B.Hocker and W.K.Burns,Appl.Opt.16,113(1977). is higher in the cladding regions 2(y <-b)and 3(y 4.A.Kumar,K.Thyagarajan,and A.K.Ghatak,Opt.Lett. 8.63(1983). b)by an amount n2-8mx2/ko2 and lower in the corner 5.R.K.Varshney and A.Kumar,IEEE J.Lightwave Tech- regions by an amount 8mx2/ko2-n42.It should be nol.LT-6,601(1988). noted that in this case the overestimation of the di- 6.J.E.Goell,Bell Syst.Tech.J.48,2133(1969). electric constant is made in the cladding regions along 7.C.Yeh,K.Ha,S.B.Dong,and W.P.Brown,Appl.Opt. the shorter dimension of the waveguide,where the 18,1490(1979)
December 1988 / Vol. 13, No. 12 / OPTICS LETTERS 1131 ed by the EIM. If one decreases the v.value [=kob(ni2 - n2 2)'/ 2] along the y direction, 3ny/ko will move toward n2. As a result, the error in the EIM should increase owing to the increase in the overestimation of the dielectric constant in region 4. This is clear from Fig. 3, where we have plotted the values of the normalized propagation constants p2 {= (13mn2 - ko2n2 2)/ [ko2(n1 2 - n2 2)]1 as a function of normalized frequency B [=(2hr)v] for the first few modes of two symmetric rectangular-core waveguides of aspect ratios 1 and 2. We have also obtained the propagation constants for the modes shown in Fig. 3 by applying the method of separation of variables to the corresponding waveguides shown in Fig. 2. It is found that the values obtained are the same as those obtained by the EIM. This confirms the fact that applying the EIM to the waveguide shown in Fig. 1(a) is equivalent to analyzing the corresponding waveguide shown in Fig. 2. Similarly, if one starts with the x waveguide, it can be shown that the EIM is equivalent to analyzing a waveguide (shown in Fig. 4) whose dielectric constant is higher in the cladding regions 2 (y < -b) and 3 (y > b) by an amount n1 2 - l.mx2/ko 2 and lower in the corner regions by an amount flmx2/ko2 -f4 2. It should be noted that in this case the overestimation of the dielectric constant is made in the cladding regions along the shorter dimension of the waveguide, where the field penetration is greater than in cladding region 4. As a result the error introduced in the calculation of the propagation constants will be. more than in the previous case. This explains why the EIM gives different (less accurate) values for the propagation constants of the various modes if one starts with the x-slab waveguide instead of the y-slab waveguide for constructing the effective-index waveguide. A. Kumar thanks the British Council for financial support. * On leave from the Physics Department, Indian Institute of Technology, Delhi, New Delhi, India. References 1. R. M. Knox and P. P. Toulios, in Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic Press, New York, 1970), p. 497. 2. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969), 3. G. B. Hocker and W. K. Burns, Appl. Opt. 16, 113 (1977). 4. A. Kumar, K. Thyagarajan, and A. K. Ghatak, Opt. Lett. 8, 63 (1983). 5. R. K. Varshney and A. Kumar, IEEE J. Lightwave Technol. LT-6,601 (1988). 6. J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969). 7. C. Yeh, K. Ha, S. B. Dong, and W. P. Brown, Appl. Opt. 18,1490 (1979)