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 America
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 :.'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 effectiveindex 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 dielectric waveguides, which are the fundamental structures in many millimeter-wave and optical integrated circuits (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 dimensions, mode orders, and refractive-index profile? For what waveguide structure is the method most accurate? 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 expressions for the errors of the effective-index method. These expressions highlight the effects due to normalized 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 halfthickness of the core, respectively, and ni, n2, n3, and n4 are the refractive indices of the core and the surrounding claddings with n1 > n2 > n3, n4. The refractive-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 effectiveindex 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 Kumar 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 original profile only in certain cladding regions. The dielectric constant of the n4 2 region of the original waveguide 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, respectively, 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 frequency V, which is defined by V = bknl(2A2 ) / 2 , is 0146-9592/91/100714-03$5.00/0 © 1991 Optical Society of America
May 15,1991/Vol.16,No.10 OPTICS LETTERS 715 Using relation(4)to evaluate P:1 and P:2,and substi- n tuting the far-from-cutoff conditions,(n12-nz2)k2= n2m2/462 and (n22-n42)k2 A4V2/A262,into Eq.(1), we obtain n 2b xm2n2 16Rb28 4 51 2a n where R=a/b is the aspect ratio of the core.Equation Fig.1.Rectangular-core dielectric waveguide. (5)shows explicitly how the accuracy of the method improves with increasing normalized frequency V and aspect ratio R.It also shows that the Em and E modes have the same accuracy at a given V(provided ng-(n2-n2) ng-(n2-n2) that Vis large).For rectangular-core fibers and chan- nel waveguides,i.e.,△2=△4，the second and the third terms in Eq.(5)are negligible (since V>>1),and the effective-index method overestimates the propagation n2+n2-n3)2b n2x(n2-n2) constant.For a strip waveguide with△2《△g=△4， the error is small because of the factor(A2/A4)1/2.The third term in Eg.(4)is negligible,but the second term is larger than the first term when V is smaller than 2a n好-2-n) n2-2-n43) A4/2A2(which is a large number).This means that the n effective-index method is particularly accurate for a strip waveguide and underestimates the propagation Fig.2.Waveguide that is exactly analyzed by the effective- constant for this waveguide when v is not large. index method that results in an x-dependent index profile. These findings have been supported by existing nu- merical examples.2.6,7 The effective-index method can also result in a y- dependent refractive-index profile.The mode index n32-,2-n n子n2-n3) n2-(m2-n4) of the TEm-1 mode in the slab of half-thickness a can be used as the effective refractive index ny of a second slab,which has a half-thickness b.The propagation constant of the TEn-1 mode in the second slab is re- n2 2b n2 garded as the approximate constant of the Emn mode in the rectangular waveguide.This process is equiva- lent to analyzing a modified rectangular structure3 as shown in Fig.3,which differs from the original struc- n2-,2-n 2a n2-o,2-n ture in the upper and the lower cladding regions as nn子-n3 well as in the corner regions.Similarly,one can write Fig.3.Waveguide that is exactly analyzed by the effective- B,2=82 ey,with By and ey being the approximate propagation constant and the error,respectively. index method that results in a y-dependent index profile. Following the same procedures for obtaining er,we find that considered.When V is large (>>1),the mode fields and are well confined in the core area and are nearly identical to each other.According to an asymptotic 32R2b2V8 +会-( analysis,5 the mode fields are given by 6) sin max sin ny 0<x<2a,0<y<2b 2a 2b 2aV A -m<x≤0,0<y<2b, (4) 0(信（传[（台 -m<x≤0，-<y≤0， 4a2 (台)（会）（会）g-[(会)》 -m<x≤0,2b≤y≤+m. 4a2
May 15,1991 / Vol. 16, No. 10 / OPTICS LETTERS 715 Using relation (4) to evaluate PX1 and P. 2 , and substituting the far-from-cutoff conditions, (n1 2 - n.2)k2 = n2 ir2/4b2 and (n. 2 - n4 2 )k2 = A4 V2/A2 b2 , into Eq. (1), we obtain 7r 4 m2 n2 /A 2 \" 2 rA2 1 [ \3/211 EX 16R 3 b2W V ) AJ 4 2VL 4 1 )J (5) 2 Fig. 1. Rectangular-core dielectric waveguide. n2_(n 2_n 2) ~2 (X -4) I - 2a 2 n2_2(n 2_n Fig. 2. Waveguide that is exactly analyzed by the effectiveindex method that results in an x-dependent index profile. n3 -(n 2_n42) n2_(n 2_n 2) n n2+(n 2_n2) 3 (f 1 fY) 2a n ~2 2+(n 2_n2) +(l nY) n 2_(n 2_n n3 (y -f 4 214 n2 2(n2 2) ~2 yn -4) Fig. 3. Waveguide that is exactly analyzed by the effectiveindex method that results in a y-dependent index profile. where R = a/b is the aspect ratio of the core. Equation (5) shows explicitly how the accuracy of the method improves with increasing normalized frequency V and aspect ratio R. It also shows that the Emn and Enm modes have the same accuracy at a given V (provided that Vis large). For rectangular-core fibers and channel waveguides, i.e., A2 = A4 , the second and the third terms in Eq. (5) are negligible (since V >> 1), and the effective-index method overestimates the propagation constant. For a strip waveguide with A2 >1), the mode fields V, and ^6 are well confined in the core area and are nearly identical to each other. According to an asymptotic analysis,5 the mode fields are given by = r 4 m2n2F, Y 32R 2 b2VW L ( A \ [3/2 + A 3 JL (sin mrx sin n2by 0 < x < 2a, 0 < y < 2b m7rb ( 1Y2 sn ry epVI (A4 Y12 1 - 2aV 4 2b 2 Jb (-l)n+lmnir 2b (A2 1/2 / V \ ex V /A 4 \1/2 /4aV 2 k2 ) exPyb-Y~exP~jb tt x] 4 A3 A4 b V 2 2 mnir 2 b JA 2 1 / 2 A 2 1 . 2 ex V- A3 1 / 2 ( 0,0 <y < 2b, (4) -00<X x •O, -o <Y y•O, V (A 4 1/2 epb A^2 X - <x S 0,2b •y S +o. x 1 /A2 \1/21 RV rA4 ' (6)
716 OPTICS LETTERS Vol.16,No.10 May 15,1991 Table 1.Calculated Normalized Propagation index method.2.8 The improvement in accuracy with Constants for the First 13 Modes of a Eq.(7)is demonstrated by the results shown in Table Rectangular-Core Fiber with R=2 at V=2x 1 for the first 13 modes in a rectangular-core fiber with P2-P2 P2-P2 Pd2-p2 R=2 operating at V=2T,where P:2=[(8x/k)2-n22]/ Mode P2a (×10-4) (X10-4) (×10-4) (n12-n22),Py2=[(8y/h)2-n22]/n12-n22),and Pa2= 0.9402 0.3 0.7 (RP,2-P,2)/(R-1).Accurate numerical data from 0 0.9001 1.3 2.9 0 Eyges et al.9 are used as references.Although Eq.(7) 0.8334 3.6 7.5 0 is exact only at V=+c,it is accurate at a finite V,as 0.8023 1.5 3.1 0 shown by the results in the table.It is also worth 0.7622 7.3 14.2 1 noting from the results in Table 1 that the difference E 0.7405 62 14.4 -1 P,2-P:2 is nearly the same for the Emn and Enm E32 0.6957 1 33 1 modes.The fact that the errors are independent of E61 0.6219 12 -2 the order of the mode orders as implied by Eqs.(5)and E42 0.6030 32 63 0 (6)has been verified. 13 0.5776 5.4 91 E 0.5377 21 In the case of a channel waveguide with△2=△4《 3 s2 0.4848 53 109 -3 Aa,it is found that ey/exR/2.The relative accuracy 1 0.4786 19 42 -4 of the two methods depends on R.For R>2,the method resulting in an x-dependent profile is more Data from Ref.9. accurate,whereas for R1.The d method that results in an x-dependent index profile is in general more accurate. -0.001 In conclusion,the asymptotic expressions [Egs.(5) and(6)]have revealed many general properties of the effective-index method that were not known previous- ly.The present approach has also been applied to 0.00 more complicated structures,such as directional cou- 3.0 4.0 5.0 6.0 plers and waveguide arrays.10 To obtain error esti- mates for small normalized frequencies,much more Fig.4.Difference in the calculated normalized propaga- research needs to be done. tion constants P,2-P,2 as a function of V for the Eu mode in a channel waveguide with an aspect ratio R and indices n References =1.5,n2 n4 1.48492,and n3=1.0.Solid curves,numeri- cal results;dotted curves,asymptotic results. 1.R.M.Knox and P.P.Toulios,in Proceedings of MRI Symposium on Submillimeter Waves,J.Fox,ed.(Poly- technic,New York,1970),p.497. which is accurate when RV is large.We can now 2.K.S.Chiang,Appl.0pt.25,2169(1986) compare ex and ey to determine which method is more 3.A.Kumar,D.F.Clark,and B.Culshaw,Opt.Lett.13, accurate. 1129(1988). In the case of a rectangular-core fiber with A2=Aa= 4.A.W.Snyder and J.D.Love,Optical Wavegide Theory △4《l,we have ey/ex=R.This implies that the (Chapman Hall,London,1983),Chap.18. method starting with the thinner slab is more accurate 5.A.W.Snyder and X.-H.Zheng,J.Opt.Soc.Am.A3,600 than the one starting with the thicker slab.By cancel- (1986). ing the errors in the two methods,one can obtain a 6.B.M.A.Rahman and J.B.Davies,IEE Proc.Pt.J 132, 349(1985). more accurate solution from the two effective-index 7.M.J.Robertson,P.C.Kendall,S.Ritchie,P.W.A. solutions, Mcllory,and M.J.Adams,IEEE J.Lightwave Technol. 7,2105(1989). 2= R82-62 R-1 (7) 8.K.S.Chiang,Appl.0pt.25,348(1986). 9.L.Eyges,P.Gianino,and P.Wintersteiner,J.Opt.Soc. Am.69,1226(1979). This procedure has been named the dual effective- 10.K.S.Chiang,IEEE J.Lightwave Technol.9,62 (1991)
716 OPTICS LETTERS / Vol. 16, No. 10 / May 15, 1991 Table 1. Calculated Normalized Propagation Constants for the First 13 Modes of a Rectangular-Core Fiber with R = 2 at V = 27r PX 2 -P2 pY2. p2 Pd2 - p2 Mode p 2 a (X10- 4) (X10-4 ) (X10- 4 ) El, 0.9402 0.3 0.7 0 E21 0.9001 1.3 2.9 0 E31 0.8334 3.6 7.5 0 E1 2 0.8023 1.5 3.1 0 E2 2 0.7622 7.3 14.2 1 E 41 0.7405 6.7 14.4 -1 E3 2 0.6957 17 33 1 E 51 0.6219 12 25 -2 E4 2 0.6030 32 63 0 E 13 0.5776 5.4 9.1 2 E2 3 0.5377 21 36 5 E5 2 0.4848 53 109 -3 E61 0.4786 19 42 -4 a Data from Ref. 9. 0.002 0.001 I .4; 0.000 -0.001 -0.002 3.0 4.0 5.0 V Fig. 4. Difference in the calculated normalized propaga- tion constants PY2 - p,2 as a function of V for the El, mode in a channel waveguide with an aspect ratio R and indices n, = 1.5, n2 = n4 = 1.48492, and n3 = 1.0. Solid curves, numerical results; dotted curves, asymptotic results. which is accurate when RV is large. We can now compare ex and Ey to determine which method is more accurate. In the case of a rectangular-core fiber with A2 = A3 = A4 2, the method resulting in an x-dependent profile is more accurate, whereas for R > 1. The method that results in an x-dependent index profile is in general more accurate. In conclusion, the asymptotic expressions [Eqs. (5) and (6)] have revealed many general properties of the effective-index method that were not known previous- ly. The present approach has also been applied to more complicated structures, such as directional couplers and waveguide arrays.10 To obtain error estimates for small normalized frequencies, much more research needs to be done. References 1. R. M. Knox and P. P. Toulios, in Proceedings of MRI Symposium on Submillimeter Waves, J. Fox, ed. (Polytechnic, New York, 1970), p. 497. 2. K. S. Chiang, Appl. Opt. 25, 2169 (1986). 3. A. Kumar, D. F. Clark, and B. Culshaw, Opt. Lett. 13, 1129 (1988). 4. A. W. Snyder and J. D. Love, Optical Wavegide Theory (Chapman & Hall, London, 1983), Chap. 18. 5. A. W. Snyder and X.-H. Zheng, J. Opt. Soc. Am. A 3,600 (1986). 6. B. M. A. Rahman and J. B. Davies, IEE Proc. Pt. J 132, 349 (1985). 7. M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. McIlory, and M. J. Adams, IEEE J. Lightwave Technol. 7, 2105 (1989). 8. K. S. Chiang, Appl. Opt. 25, 348 (1986). 9. L. Eyges, P. Gianino, and P. Wintersteiner, J. Opt. Soc. Am. 69, 1226 (1979). 10. K. S. Chiang, IEEE J. Lightwave Technol. 9, 62 (1991). . R=1.5 1 5 ~~~~~~~~~............ . ............................ .... 4
