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, 18<1 waveguide shown in Fig.1(b),the first-order pertur- =Bexp[-(V2-a2)1/%钔l8>1, 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< RI > 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 tran￾scendental 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 nor￾malized propagation constant P2 with the parameter B = (2a) V2 for rectangular waveguides corresponding to a/b = 1 and a/b = 2, respectively. The results ob￾tained by Marcatili2 are the same as those obtained by using Eq. (7). As can be seen from the figures, the re￾sults obtained by using first-order perturbation theory [namely, Eq. (8)] agree well with the numerical calcu￾lations 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 ,