January 1983 Vol.8,No.1 OPTICS LETTERS 63 Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach Arun Kumar,K.Thyagarajan,and A.K.Ghatak Department of Physics,Indian Institute of Tehnology,Delhi,New Delhi-110 016,India Received July 8,1982 We report the exact solution of the scalar-wave equation for a rectangular-core waveguide structure and develop a perturbation analysis for evaluating accurately the propagation characteristics of practical integrated-optical structures.We show that the present method gives results that are more accurate than other analytical methods reported earlier. The waveguides used in integrated optics consist of the effective-index method gives ambiguous results for rectangular or nearly rectangular dielectric cores sur- square-cross-section waveguides. rounded by a dielectric of slightly lower refractive index. We first discuss the rigorous solutions of the scalar- In order to be able to design efficient integrated-optical wave equation for the refractive-index distribution of devices,such as directional couplers and filters,it is the form [see Fig.1(a)] important to understand the modal properties of such rectangular-core waveguides,which have been a subject n2(x,y)=n'2(x)+n"2y), (1) of considerable interest in the recent past.1-6 Achieving where an accurate analytical solution of such waveguides is difficult because of the presence of corners.A consid- n2(x)=12 Ixa/2: (2) mate solutions corresponding to rectangular-core waveguides.The numerical method developed by n"26y)= 2， ly|<b/2 Goell is based on an expansion in circular harmonic functions and becomes cumbersome for nonunity aspect =n22-n12/2, ly 6/2. (3) ratios.Also,the method would become more difficult to use for asymmetric waveguides.The finite-element The propagation characteristics of the structure shown method involves considerable numerical calculation, in Fig.1(b)can then be obtained by perturbing the di- and the accuracy of the results depends on the number electric constant in the corners by an amount n12-n22, of elements chosen;for low V values (since the modal and,since for most practical waveguides n1 n2,the fields extend greatly beyond the waveguide core)a large effect of this perturbation is expected to be small.Thus number of elements are required for a given accuracy, we expect that the solutions corresponding to the re- and the method becomes quite cumbersome. fractive-index distributions given by Eqs.(1)-(3)will In this Letter,we give an exact solution of the sca- form an accurate basis for obtaining solutions for lar-wave equation for the waveguide shown in Fig.1(a); structures that are closely related to it. such a waveguide closely resembles the rectangular-core In the scalar-wave approximation,the transverse waveguide shown in Fig.1(b).We show that solutions corresponding to the waveguide in Fig.1(a)can be used as a basis for obtaining the propagation characteristics 2n-n of practical integrated-optical structures by using per- turbation techniques.We then make perturbation calculations to obtain the propagation characteristics of the waveguide shown in Fig.1(b).We have shown that the modal fields corresponding to the waveguide in Fig.1(a)are the same as those used by Marcatili,2 and n therefore the perturbation method developed gives b results that are much more accurate than the ones ob- tained by using Marcatili's method.It has also been Fig.1.(a)Rectangular core described by Egs.(1)-(3).The shown that the perturbation method gives results that modes of the waveguide shown in (b)can be obtained from the are more accurate than the ones obtained by using the modes of the waveguide shown in (a)by using the perturbation effective-index method.3 We should also mention that theory. 0146-9592/83/010063-03$1.00/0 ©1983，Optical Society of America
January 1983 / Vol. 8, No. 1 / OPTICS LETTERS 63 Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach Arun Kumar, K. Thyagarajan, and A. K. Ghatak Department of Physics, Indian Institute of Tehnology, Delhi, New Delhi-11o 016, India Received July 8,1982 We report the exact solution of the scalar-wave equation for a rectangular-core waveguide structure and develop a perturbation analysis for evaluating accurately the propagation characteristics of practical integrated-optical structures. We show that the present method gives results that are more accurate than other analytical methods reported earlier. The waveguides used in integrated optics consist of rectangular or nearly rectangular dielectric cores surrounded by a dielectric of slightly lower refractive index. In order to be able to design efficient integrated-optical devices, such as directional couplers and filters, it is important to understand the modal properties of such rectangular-core waveguides, which have been a subject of considerable interest in the recent past.1-6 Achieving an accurate analytical solution of such waveguides is difficult because of the presence of corners. A considerable amount of work has been done based on analytical2, 3, 6 and numerical' 4 ' 5 methods to obtain approximate solutions corresponding to rectangular-core waveguides. The numerical method developed by Goell' is based on an expansion in circular harmonic functions and becomes cumbersome for nonunity aspect ratios. Also, the method would become more difficult to use for asymmetric waveguides. The finite-element 4 method involves considerable numerical calculation, and the accuracy of the results depends on the number of elements chosen; for low V values (since the modal fields extend greatly beyond the waveguide core) a large number of elements are required for a given accuracy, and the method becomes quite cumbersome. In this Letter, we give an exact solution of the scalar-wave equation for the waveguide shown in Fig. 1(a); such a waveguide closely resembles the rectangular-core waveguide shown in Fig. 1 (b). We show that solutions corresponding to the waveguide in Fig. 1(a) can be used as a basis for obtaining the propagation characteristics of practical integrated-optical structures by using perturbation techniques. We then make perturbation calculations to obtain the propagation characteristics of the waveguide shown in Fig. 1(b). We have shown that the modal fields corresponding to the waveguide in Fig. 1 (a) are the same as those used by Marcatili, 2 and therefore the perturbation method developed gives results that are much more accurate than the ones obtained by using Marcatili's method. It has also been shown that the perturbation method gives results that are more accurate than the ones obtained by using the effective-index method. 3 We should also mention that the effective-index method gives ambiguous results for square-cross-section waveguides. We first discuss the rigorous solutions of the scalarwave equation for the refractive-index distribution of the form [see Fig. 1(a)] n2(x,y) = n'2 (x) + n" 2(y), where ,2(X) = 2 ' = n22 -n2/2, n"2(y) = n2 ' = n22 - ni2/2, (1) lxi a/2; (2) yjy b/2. (3) The propagation characteristics of the structure shown in Fig. 1(b) can then be obtained by perturbing the dielectric constant in the corners by an amount n1 2-n2 2 and, since for most practical waveguides ni n2, the effect of this perturbation is expected to be small. Thus we expect that the solutions corresponding to the refractive-index distributions given by Eqs. (1)-(3) will form an accurate basis for obtaining solutions for structures that are closely related to it. In the scalar-wave approximation, the transverse 2n.',fl A //I n2 V~' I2 -S 2n-nt ft 777 a2 . A na 4 - fl - 0 (b) Fig. 1. (a) Rectangular core described by Eqs. (1)-(3). The modes of the waveguide shown in (b) can be obtained from the modes of the waveguide shown in (a) by using the perturbation theory. 0146-9592/83/010063-03$1.00/0 ©0 1983, Optical Society of America W)
64 OPTICS LETTERS Vol.8,No.1 January 1983 component of the field satisfies the following equa- tion: n9产-1-片-诗 T) 2y+a2+o2n2x,y)-2]业=0， 8x2t0y21 (4) where u and v are obtained as solutions of the tran- where ko is the free-space wave number and 8 is the scendental equations given above.It should be pointed out that the field patterns obtained are similar to the propagation constant of the mode.For n2(x,y)given by Eqs.(1)-(3),we use the method of separation of one assumed by Marcatili2;however,we now have variables and writev(x,y)=X(x)Y(y),where X(x)and shown the configuration to which the fields correspond. Y(y)can be symmetric or antisymmetric functions of This then enables us to use perturbation theory for a x and y,respectively.For example,when X(x)is a structure that closely resembles the one given in Fig. 1(a). symmetric function,we have For example,if we now consider the rectangular X(x）=A cosua, 181, bation theory gives us the following expression for the (5) normalized propagation constant: where=2x/a,Vi ko(a/2)(n12-n22)1/2,and u is p2=P02+P2, determined from the following transcendental equation (which is obtained from the continuity conditions): where utan4=(V12-2)12. (6) Similarly,for the antisymmetric mode we would have P2= u cot u=-(V12-u2)1/2.In a similar manner we can n12-n22 consider the y-dependent solutions that would be either xdy symmetric or antisymmetric in the y coordinate Continuity conditions on Y(y)will give us v tany=(V22 -+图-2 -v2)1/2 for symmetric modes in y and y cot=-(V22 1+pcos24月 -)1/2 for antisymmetric modes in y;here V2= 1/2/2v g sin 2v]-1 (kob/2)(n12-n22)1/2. ×+鉴-1 1+g cos 2v (8) The normalized propagation constant is given by where p =1 (-1)for modes symmetric (antisymmetric) in x and g =1 (-1)for modes symmetric (antisymme- tric)in y;the perturbation technique is discussed in many papers;see,e.g.,Ref.7. 号 Figures 2(a)and 2(b)show the variation of the nor- malized propagation constant P2 with the parameter B=(2/)V2 for rectangular waveguides corresponding to a/b 1 and a/b =2,respectively.The results ob- E1 or En tained by Marcatili2 are the same as those obtained by using Eq.(7).As can be seen from the figures,the re- 0-2 sults obtained by using first-order perturbation theory namely,Eq.(8)]agree well with the numerical calcu- lations of Goell;indeed,the agreement is even better 12 than the one using the effective-index method3 in the entire practical range of B values except at small values of B;we may mention here that the labeling of the curves corresponding to Goell's calculations and the effective-index method is interchanged in Ref.3.In addition,the effective-index method for a square- cross-section waveguide does not give the same propa- gation constants for Epa and Eap modes with p q. For higher-order modes,our results are in much better agreement with those of Goell!than the ones obtained by using the effective-index method.It should also be pointed out that for higher-order modes it becomes 00 1024 36 cumbersome to get accurate results by using the circular harmonic analysis of Goell.For example,for the E22 Fig.2.Variation of the normalized propagation constant P2 mode the results reported by Goelll for a rectangular as a function of B for a rectangular waveguide with (a)a/b= waveguide with a/b=2 are certainly in error since even 1 and (b)a/b=2.- -Eq.(7)and Ref.2:- —,present the unperturbed value of P2 from our analysis is larger work,i.e.,Eq.(8);...,Goell'sl results;- than that of Goell,and the perturbation would further effective-index method. increase the value of p2
64 OPTICS LETTERS / Vol. 8, No. 1 / January 1983 component of the field satisfies the following equa- tion: a20 + dy2 + [h0 2n2 (x,y) _ 32 h1'= 0, (4) Ox2 1ay 2 where ko is the free-space wave number and j3 is the propagation constant of the mode. For n2(x, y) given by Eqs. (1)-(3), we use the method of separation of variables and write {'(x, y) = X(x) Y(y), where X(x) and Y(y) can be symmetric or antisymmetric functions of x and y, respectively. For example, when X(x) is a symmetric function, we have X(x) = A cosyu, = B exp[-( V12 - A2)1121 4j] W 1, where t = 2x/a, V1 = ko(a/2)(n,2 - n2 2 )1/2, and u is determined from the following transcendental equation (which is obtained from the continuity conditions): ,u tan g = (V12 -,U2)1/2 (6) Similarly, for the antisymmetric mode we would have g cot A =--(V2- g2 )l/ 2 . In a similar manner we can consider the y-dependent solutions that would be either symmetric or antisymmetric in the y coordinate. Continuity conditions on Y(y) will give us v tan v = (V22 - V2) 112 for symmetric modes in y and v cot v = -(V2 - v2 ) 1 / 2 for antisymmetric modes in y; here V2 = (kob/2)(n1 2 - n22)1/2 The normalized propagation constant is given by 1 B - 0I a' 2 0*4 0.2 00 04 0-8 1-2 1.6 0o 74 fl 32 34 4-0 B -. Fig. 2. Variation of the normalized propagation constant P2 as a function of B for a rectangular waveguide with (a) a/b = 1 and (b) a/b = 2. Ad -, Eq. (7) and Ref. 2;- , present work, i.e., Eq. (8); ...... , Goell'sl results; …----- effective-index method. , = 3O2 - = 1 la2 k2(n2 -n2 2) V12 V22 (7) where A' and v are obtained as solutions of the transcendental equations given above. It should be pointed out that the field patterns obtained are similar to the one assumed by Marcatili 2 ; however, we now have shown the configuration to which the fields correspond. This then enables us to use perturbation theory for a structure that closely resembles the one given in Fig. 1(a). For example, if we now consider the rectangular waveguide shown in Fig. 1(b), the first-order pertur- bation theory gives us the following expression for the normalized propagation constant: 1p2 = po2 + pJ2, where pL2= 1 n12- n2 X 1[+I I4 f rlfk 1t1i2(n2 - n22)dxdy 1 F S2 _ 1 /2 | 2 12 +p sin 2I ,-1 ) qpcos 2jcs ] ?I22 1)/2 2v + q sin 2Yvl X (8 where p = 1 (-1) for modes symmetric (antisymmetric) in x and q = 1 (-1) for modes symmetric (antisymme- tric) in y; the perturbation technique is discussed in many papers; see, e.g., Ref. 7. Figures 2(a) and 2(b) show the variation of the normalized propagation constant P2 with the parameter B = (2a) V2 for rectangular waveguides corresponding to a/b = 1 and a/b = 2, respectively. The results obtained by Marcatili2 are the same as those obtained by using Eq. (7). As can be seen from the figures, the results obtained by using first-order perturbation theory [namely, Eq. (8)] agree well with the numerical calculations of Goell1; indeed, the agreement is even better than the one using the effective-index method3 in the entire practical range of B values except at small values of B; we may mention here that the labeling of the curves corresponding to Goell's calculations and the effective-index method is interchanged in Ref. 3. In addition, the effective-index method for a square- cross-section waveguide does not give the same propa- gation constants for Epq and Eqp modes with p sd q. For higher-order modes, our results are in much better agreement with those of Goell' than the ones obtained by using the effective-index method. It should also be pointed out that for higher-order modes it becomes cumbersome to get accurate results by using the circular harmonic analysis of Goell.l For example, for theE 22 mode the results reported by Goell' for a rectangular waveguide with a/b = 2 are certainly in error since even the unperturbed value of P2 from our analysis is larger than that of Goell, and the perturbation would further increase the value of p2 . W, b - 21 .. Ezz I I I ... I I 1. I ,
January 1983 Vol.8,No.1 OPTICS LETTERS 65 n ntni- 生 n nn-n (E>pa).For example,the Epe*mode would approxi- ELE心 mately correspond to a TM mode in the x direction and to a TE mode in the y direction,and we assume that corresponds to Ex,which is the major electric-field n component.In order to get the eigenvalue equations, 772 we make Er and dEr/oy continuous at y =(b/2)and n 0 2n号-n 0 2n-n n2E:and dEx/ax continuous at x =+(a/2).The con- tinuity conditions aty=(b/2)will lead to the same a】 16) transcendental equation,namely,v tanv=(V22-v2)1/2; Fig.3.(a)An embedded channel waveguide.(b)The however,the transcendental equation corresponding structure that can be used as the basis for obtaining the to the continuity condition at x=(a/2)will be slightly propagation characteristics of the waveguide shown in (a). different. In summary,we have established specific rectangu- lar-core waveguide structures that can be used as a basis for evaluating the propagation characteristics of most The analysis presented above can easily be extended practical integrated-optical waveguides. to asymmetric rectangular waveguides.For example, This research was partially supported by the De- the embedded channel waveguide shown in Fig.3(a)can partment of Science and Technology,India.Thanks be approximated by the waveguide shown in Fig.3(b) ere due to Ajay Amar for help in numerical calcula- whose reiractive-index profile is given by Eq.(1),where tions. n'2(x)is the same as that given in Eq.(2)and n"2y）=ng2-n12f2,y>b/2 References =n122, 1y<b/2 1.J.E.Goell,Bell Syst.Tech.J.48,2133(1969). =n22-n122,y<-b/2. (9) 2.E.A.J.Marcatili,Bell.Syst.Tech.J.48,2071(1969). 3.G.B.Hocker and W.K.Burns,Appl.Opt.16,113 The above profile matches the exact profile everywhere (1979). except in the corner regions,where the approximate 4.C.Yeh,K.Ha,S.B.Dong,and W.P.Brown,Appl.Opt.18, profile differs only by n12-n22 from the actual profile; 1490(1979). this difference from the actual profile can then be in- 5.R.Mitra,Y.L.Hou,and V.Samejal,IEEE Trans.Micro- wave Theory Tech.MTT-28,36 (1980). corporated by using first-order perturbation theory. 6.V.V.Cherny,G.A.Juravley,and J.R.Whinnery,IEEE We may modify the above analysis to obtain the Trans.Microwave Theory Tech.MTT-28,401 (1980). propagation constants of predominantly x-polarized 7.M.S.Sodha and A.K.Ghatak,Inhomogeneous Optical modes (E*)and predominantly y-polarized modes Waveguides (Plenum,New York,1977)
January 1983 / Vol. 6, No. 1 / OPTICS LETTERS 65 n 3 n 2 +n'-n~'/ ... -,A Fl 2 Fl t n 2 k - a to-- 4 2nI - (a) n3 n3 ~'n' zn2-_n2 f f2 2 x . .2 ...... e22-n 'Is - 2 -1I I (b Fig. 3. (a) An embedded channel waveguide. (b) The structure that can be used as the basis for obtaining the propagation characteristics of the waveguide shown in (a). The analysis presented above can easily be extended to asymmetric rectangular waveguides. For example, the embedded channel waveguide shown in Fig. 3(a) can be approximated by the waveguide shown in Fig. 3(b whose refractivee-index profile is given by Eq. (1), where n'2 4x) is the same as that given in Eq. (2) and n1,() = 3-2 - nt12/2, y > b/2 = n1 2 12, Hyj < i/2 = n2 2 - n1 2/2, y < -b12. (9) The above profile matches the exact profile everywhere except in the corner regions, where the approximate profile differs only by nr2 - n22 from the actual profile; this difference from the actual profile can then be incorporated by using first-order perturbation theory. We may modify the above analysis to obtain the propagation constants of predominantly x-polarized modes (EXpq) and predominantly y-polarized modes (Eypq,). For example, the EpqX mode would approximately correspond to a TM mode in the x direction and to a TE mode in the y direction, and we assume that q4 corresponds to Ex, which is the major electric-field component. In order to get the eigenvalue equations, we make E, and dEX/ay continuous at y = d (b/2) and n 2 E, and dEX/8x continuous at x = ±(a12). The continuity conditions at y = (b/2) will lead to the same transcendental equation, namely, v tan v = (V22 - 2)1/2; however, the transcendental equation corresponding to the continuity condition at x = a/2) will be slightly different. In summary, we have established specific rectangular-core waveguide structures that can be used as a basis for evaluating the propagation characteristics of most practical integrated-optical waveguides. This research was partially supported by the Department of Science and Technology, India. Thanks are due to Ajay Amar for help in numerical calculations. References 1. J. E. Coell, Bell Syst. Tech. J. 48, 2133 (1969). 2. E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969). 3. G. B. Hocker and W. K. Burns, Appi. Opt. 16, 113 (1979). 4. C. Yeh, K. Ha, S. B. Dong, and W. P. Brown, Appl. Opt. 18, 1490 (1979). 5. R. Mitra, Y. L. Hou, and V. Samejal, IEEE Trans. Microwave Theory Tech. MTT-28, 36 (1980). 6. V. V. Cherny, G. A. Juravley, and J. R. Whinnery, IEEE Trans. Microwave Theory Tech. MTT-28, 401 (1980). 7. M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977). 1