Analysis of optical fibers by the effective-index method
K. S. Chiang
The effective-index method for determining waveguide dispersion is derived from the scalar wave equationand applied to optical fibers of arbitrary cross-sectional shapes. In the simplest use of the method, the opticalfiber is replaced by an equivalent slab waveguide with an index profile derived from the geometrical shape ofthe fiber. Results from analyzing circular, elliptical, and cusp-shaped fibers are used to illustrate the generalfeatures of the method. A procedure is also given for improving the accuracy of the method applied to a classof single-mode fiber.
1. Introduction
Optical fibers of general cross-sectional shapes havebeen successfully analyzed by various methods, amongwhich the finite element method,1-3 the finite differ-ence method,4 the point-matching method,5 and themethod of Eyges et al. 6 are typical ones. The effec-tive-index method, since its discovery, 7 has been wide-ly applied to waveguides of rectangular shapes.- 12
Apart from geometrically perturbed slabs 3 and trape-zoidal waveguides,14 however, the application of theeffective-index method to waveguides of more generalcross sections has not been investigated. In this paperthe effective-index method is derived from the scalarwave equation and, for the first time, applied to opticalfibers of general shapes. The central idea of the meth-od is to replace the optical fiber by an equivalent slabwaveguide whose refractive-index profile is deter-mined from the geometrical shape of the fiber. Thesuccess of converting the 2-D problem into a 1-D prob-lem makes the method significantly more efficient andsimpler than other methods.
Two effective-index methods are first discussed:the simple effective-index method and the compositeeffective-index method. In the former, a single effec-tive-index profile is used to characterize the fiber,while in the latter two such profiles are required. Nu-merical results show that both methods are only mod-erately accurate and their accuracy depends very muchon the shape of the fiber. A highly accurate method,
The author is with University of New South Wales, School of Electri-cal Engineering & Computer Science, P.O. Box 1, Kensington, NSW2033, Australia.
Received 31 May 1985.0003-6935/86/030348-07$02.00/0.
Equation (7) is just the scalar wave equation (or TEmode equation) for an inhomogeneous slab waveguidewith index profile neff. Once neff is obtained, the prop-agation constant of the fiber can be solved from Eq. (7).The above derivation can be regarded as a process usedto single out x(x) from O(x,y), which mostly accountsfor the variation of the mode field with respect to x.
B. Determination of the Effective-index Profile
The essence of the method is to determine neff fromEq. (6). At a particular x = xi, Eq. (6) can be written as
dkY (iy) + [n2(x-,y)k2 l neff (xi)kkxy(xi'y) = 0. (8)
Obviously, Eq. (8) is the scalar wave equation for a slabwaveguide with index profile n(xi,y) and propagationconstant neff(xi)k, which can be easily calculated. Inprinciple, neff(x) can be determined from Eq. (8), atleast pointwise.
The method can be significantly simplified for step-index fibers with n(x,y) = n, in the core, n(x,y) = n 2 inthe cladding, and n1 > n2. We assume that n(xi,y)represents a symmetrical three-layered slab of half-thickness t = t(xi) for all xi in the core region of thefiber, as shown in Fig. 1. The problem of findingneff(xi) is then replaced by the problem of determiningthe propagation constant of a symmetrical three-lay-ered slab of thickness depending on x. Miyagi andNishida1 5 have given an accurate explicit formula forthe propagation constant of such a slab. In our con-text, their result is expressed as
u=gv- gv - v sin(g) (9)1 + vt cos(g)
where
u = kn,2- eff(x)]1/2,
v = k(n, 2- n2
2)
1/2,
(m + 1)7rg 2(vt + 1)
m = mode order = 0, 1, 2 ....
The effective-index profile neff(X) is then given as
neff(X) [n- U_ (n- n2)] (10)
It is obvious that t(x) is actually the function thatdescribes the cross-sectional geometry of the fiber, forexample, t = (a2
- x2)11 2 for a circular fiber of radius a.With the help of Eqs. (9) and (10), neff(x) can be explic-itly related to the shape of the fiber and hence analyti-cally determined. The propagation constant of theoptical fiber can now be obtained from a single slabwaveguide, which can be analyzed by a number of well-known methods (for example, see Refs. 16-20).
For graded-index fibers, where Eq. (9) is no longerapplicable, the effective-index profile must bepointwise determined from Eq. (8). In practice, dis-
n2
I
2t (. )
In 2 "2 1
4 1.ef eff X) I 2
ne'~ f f (X ) = L
I *
(X),
Fig. 1. Effective index at x = xi, nff(xi), is obtained from thepropagation constant of the symmetrical three-layered slab of width
equal to the thickness of the fiber core at x = xi.
cretization of fiber cross section into narrow stripesmay be necessary. The number of 1-D problems thathave to be solved thus depends on the number ofstripes and the accuracy of the method may be degrad-ed due to the discretization.
The method described in this section is named thesimple effective-index method (SEIM).
Ill. Composite Effective-index Method (CEIM)
In the method discussed in the previous section, wedefine an x-dependent effective-index profile, which isnow denoted as n.(x) [equivalent to the neff(x) usedearlier], based on the assumption given by Eqs. (2) and(4). In fact, following the same principle, we can de-fine a new y-dependent effective-index profile, ny(y),by expressing the field in the fiber as
qO(x,Y) = 0()Y(') (11)
where kyx(y,x) is assumed to be a slowly varying func-tion of y, i.e., O90yx/9y and 20yx/Oy2 are negligible.With ny(y) as the effective-index profile, another 1-Dequation similar to Eq. (7) can be derived. (Equiva-lently, we obtain the new effective-index profile byrotating the fiber by 900. To avoid confusion, weprefer mathematically rotating the simple effective-index method to physically rotating the fiber. In fact,different fiber orientations will yield different effec-tive-index profiles. Among those effective index pro-files, there must be one that gives the best approxima-tion. We shall come to this point in the next section.)The propagation constants associated with nx(x) andny(y) are A., and by, respectively. The correspondingfields are O., and Oy, respectively. The idea is shown inFig. 2. Since O. and oy represent the extracted parts ofthe complete mode field that chiefly account for thevariations of the field with respect to x and y, respec-tively, we then propose that the complete 2-D field inthe fiber can be approximated by
Fig. 2. Two effective-index profiles, n(x) and ny(y), are construct-ed by following the procedures illustrated in Fig. 1.
validity must be checked. Nevertheless, Eq. (12) isconsistent with the Gaussian beam approximation ap-plied to single-mode fibers.21 22 Substituting Eq. (12)into the function of Eq. (1) and simplifying the resultwith the help of the functional of Eq. (7), we obtain anapproximation for the propagation constant of thefiber:
To use Eq. (13) we need to apply the simple effective-index method to the fiber twice and perform someintegrations. The procedure described in this sectionis called the composite effective-index method(CEIM) to be distinguished from the simple methodgiven in the previous section. It should be understoodthat the calculated from Eq. (13) is not necessarilymore accurate than either ox or f3y although both ox andOy are required in evaluating . However, the accuracyof Eq. (13), which can only be checked numerically,may be used to justify the validity of Eq. (12).
The formulation of the composite effective-indexmethod actually suggests another approach for analyz-ing graded-index fibers. The graded-index fiber canbe iteratively replaced by equivalent step-index fi-bers23 to which the composite effective-index methodcan be easily applied. In each iteration, the field of thestep-index fiber is approximately by Eq. (12) and thecorresponding propagation constant is calculated fromEq. (13).
IV. Discussions
A. Fiber Geometry and Modes
Since the effective-index methods are derived in theCartesian coordinate system, it is convenient to desig-nate the modes as Eijx and EijY.24 In a weakly guidingfiber, Eijx and EijY modes are approximately degener-ate so we denote E/xY as the fiber modes. The sub-
scripts i and j are used to specify the number of fieldmaxima in the x and y directions, respectively. Ac-cording to the simple effective-index method, it is clearthat the E- "xY mode can be analyzed by using the TEj_mode to construct the effective-index profile [i.e., m =j - 1 in Eq. (9)]. Then the propagation constant of theTEi_1 mode obtained by solving the resultant 1-Dequation [i.e., Eq. (7)] is the required propagation con-stant of the E-x^Y mode. It is then obvious that theeffective-index methods can only be applied to fibermodes which can be designated asE xy modes. Wecan summarize the discussions as follows:
(i) For fibers of symmetry classes C4, (e.g., square)and C2,, (e.g., ellipse), all modes can be analyzed.
(ii) For fibers of other symmetry classes, only partof the modes can be analyzed. For example, for circu-lar fibers (symmetry class C,, the LP01 (i.e., EixY)mode and the LP11 (i.e., E 2 1 X'Y or E1 2 X'Y) mode can beanalyzed but the LP02 mode cannot.
(iii) For all fibers, regardless of their symmetryclasses, the fundamental mode (E11 xY) can always beanalyzed.
B. Fiber Shape and Aspect Ratio
As we have mentioned, construction of the effective-index profile depends on the orientation of the fiber.How should the fiber be oriented to generate the effec-tive-index profile that gives the most accurate result?How does the fiber shape affect the accuracy of themethod? It is very difficult to answer these questionswith rigorous mathematical arguments. However, it ispossible, by qualitatively investigating the assump-tions involved in the derivation, to predict the perfor-mance of the method.
The derivation of the simple effective-index methodis based on the assumption that, in the average sense,the variation of kxy(X,y) with respect to x is muchslower than its variation with respect to y and alsoslower than the variation of ox (x) with respect to x [Eq.(4)]. The better this assumption is satisfied, the moreaccurate is the result. It is then required, according toEq. (6), that the effective-index profile nff(x) be aslowly varying function of x. To make neff vary slowly,the thickness of the fiber, i.e., t(x), should vary slowlywith respect to x.
From the above discussions, it is easy to explain thefollowing features of the method applied to fibers withat least two preferred axes of symmetry.
(i) In the simple effective-index method, if the x-dependent effective-index profile is wanted, the longaxis (or major axis) of the fiber should be aligned withthe x axis to obtaia the best result.
(ii) When the fiber is properly oriented as describedin (i), the simple effective-index method is more accu-rate for fibers with larger aspect ratio R, defined as theratio of the long axis to the short axis of the fiber (henceR 1), that is, the accuracy of the method increaseswith R. In the limit of a slab waveguide, i.e., R = -, themethod becomes exact.
(iii) Both the simple and the composite effective-index methods are more accurate when the fiber shape
Fig. 3. Dispersion curves of the LP01 and LP1 1 modes of the circularstep-index fiber
is closer to a rectangle with the aspect ratio remainingunchanged. In the special case of rectangular wave-guides with perfectly conducting walls, where separa-tion of variables applies, both methods become exact.
(iv) At low frequencies, the mode field spreads overthe entire cross section of the fiber so oxky,/8x and82 xy9/X2 become significant in comparison witha20x/aX2 and a20.y/ay2 , i.e., Eq. (4) is no longer valid.We should then expect that the accuracy of the simpleeffective-index method decreases with frequency.
An evaluation of the simple effective-index methodapplied to rectangular structures has been de-tailed.10 "11 However, it may be difficult to extend theapproach given by Refs. 10 and 11 to nonrectangularwaveguides. Little can be said about the behavior ofthe composite effective-index method, which is devel-oped on the basis of Eq. (12). which has a lot of intu-itive appeal but little mathematical backup.
V. Results
In the first example, the effective-index methods aretested with a circular step-index fiber. The dispersioncurves of the first two modes obtained from the presentmethods are compared with the exact ones as shown inFig. 3, where P2 = [(3/k)2 - n2
2]/(n,2 - n22), the nor-malized propagation constant, is plotted against v, thenormalized frequency. The percentage error of P2 as afunction of v is plotted in Fig. 4. It is interesting tonote that the simple effective-index method gives anupper bound while the composite effective-index,method gives a lower bound for p2 . The accuracy ofthe simple effective-index method increases with fre-quency, as expected. The accuracy of the compositeeffective-index method first increases rapidly with fre-quency and then decreases very slowly. Both methodsare just as accurate as many other simple methodsapplied to single-mode circular step-index fibers.25
The 1-D field O. obtained from the effective-indexmethods for the LP0 1 mode is shown in Fig. 5, fromwhich it is clear that Ax agrees with the exact field verywell at high frequency. Figure 5 justifies the use of the
+10.0
+8.0
+6.0
+4.0
+2.01 error 0.0
- 2 0
-4.0
-6.0
-8.0
-10.0
0.0 2.0 4.0 6.0 8.0 10.0
v-a k(n 2 n 2)i2
Fig. 4. Accuracy of the simple effective-index method (SEIM) andthe composite effective-index method (CEIM) for the circular fiber.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0 1.0 2.0 3 .0
,/a
Fig. 5. Comparison between the exact field and the approximatefield of the LP01 mode of the circular fiber at various frequencies.
composite effective-index method for obtaining fiberfields.
The second example is an elliptical step-index fiberof aspect ratio R = 2. The first five modes are ana-lyzed by the simple effective-index method and thefirst two modes by the composite effective-indexmethod. Results from Eyges et al.,6 which are be-lieved to be very accurate, are used as reference. Thedispersion curves are shown in Fig. 6. Most of thecomments for the circular fiber apply here. As expect-ed, the simple effective-index method is more accuratefor the elliptical fiber than for the circular fiber sincethe aspect ratio of the elliptical fiber is larger. Thecomposite effective-index method is also very accuratefor the fundamental mode but its accuracy is signifi-cantly degraded for the second-order mode.
In the last example, we analyze a cusp-shaped fiberand compare the results with those in Ref. 6. Thedispersion curves of the first two modes are shown inFig. 7. According to our discussions in Sec. IV, thisfiber, with the smallest aspect ratio R = 1 and relative-
cording to Eq. (13) does not improve the accuracy ofthe analysis. To obtain a more accurate analysis, weuse the following procedures. To be specific, let usconsider a class of step-index fiber whose core-clad-ding boundary is described by
x2N + (Ry)2N = a2, (14)
2 ~ / i nEN' I \ where R = a/b 1 is the aspect ratio with a the, Al/ SA //' 22 a-2b 2a semimajor axis and b the semiminor axis, and N is a
I .ai /Y,,", .r , I positive real number describing the shape of the0.0 2.0 4.0 6.0 8.o 10.0 12.0 114.0 boundary. For example, the previously analyzed el-
ak(n1 2 2 liptical fiber and cusp-shaped fiber correspond to N =2) 1.0 and 0.3, respectively. Similar to the composite
Dispersion curves of the elliptical step-index fiber. effective-index method, two approximates for thepropagation constants of the fiber, namely, O., and y,can be obtained by performing the simple effective-
1.0 so..,......, index method twice according to Fig. 2. We then write
p"2 = 02 + e(R,N,3 2),
f 2 = 2 + G(RN, #2)ex(RN, 32 ),
(15)
(16)
where /3 is the true propagation constant, and Ex andGe. represent the absolute errors of /3 2and #Y2 , respec-tively. Both ex and G are functions of fiber geometryand /32. With ex eliminated from Eqs. (14) and (15), weobtain
0.0 2.0 4.0 6.o 8.0 10.0 12.0
v - ak(n 2 - 2
Fig. 7. Dispersion curves of the cusp-shaped fiber.
ly rapidly varying boundary, represents a rather ex-treme test for the validity of the effective-index meth-ods. Indeed, the results for this fiber are not asaccurate as those for the circular fiber but we mustadmit that the results are still reasonably good even inthis fault-finding example. One may note that, if thecusp-shaped fiber is rotated by 450, it will look morelike a square which should favor the applicability ofthe effectice-index methods. However, with such arotation, Eq. (9) will no longer be applicable for all x inthe core region of the fiber because five-layered (in-stead of three-layered) slabs must be used to obtain therequired effective-index profile for some x. An accu-rate explicit formula similar to Eq. (9) for five-layeredslabs is unfortunately not available yet.
In all examples, the effective-index methods are notaccurate at frequency near cutoff. This fact is mainlydue to the approximate nature of the method as dis-cussed in the previous section, and secondarily due tothe errors incurred by Eq. (9) which is derived on thebasis of far-from-cutoff approximation.' 5
Through the illustrations given by the examples, weshould be confident of the effective-index methodsbeing applied to other fiber shapes.
VI. Dual Effective-Index Method (DEIM)
As demonstrated by numerical examples, the com-posite effective-index method which evaluates two so-lutions from the simple effective-index method ac-
2 =G# 2 - V G -I (17)
Clearly, /32 is simply the weighted sum of/f32and fy2 If
G is accurately determined, /3 can be accurately calcu-lated from Eq. (17). According to earlier discussions,Ox is always more accurate than fly since the semimajoraxis of the fiber is aligned with the x axis. We thushave
G(R,N,3 2 ) _ 1, (18)
where the equality holds for R = 1. So Eq. (17) isuseful only for R > 1, i.e., for the case /x 3# fly. (R = 1gives O3x = fly because of symmetry.) Alternatively, wecan write
I 2 = 02 + ef(R,N,# 2 ), (19)
where ey is the absolute error of /lY2. Because of theexchanged roles of a and b due to symmetry, we musthave
A = 2 + G (1, NI32) e (RNP 2).
It is easy to see, from Eqs. (15), (16), (19), and (20), thatG must satisfy the following functional equation:
Fig. 8. Empirical determination of the a function for the funda-mental mode of the class of step-index fiber described by Eq. (14).
To determine a, an empirical procedure is used. If theexact / is known for a fiber of specified R and N, then acan be solved from Eq. (23). An accurate method isthus required to evaluate the whole class of fiber de-scribed by Eq. (14). The accurate finite elementmethod recently developed by the author3 is employedfor the present purpose. The a values calculated forthe fundamental mode are shown in Fig. 3 as a functionof normalized propagation constant P2 for various Rand N. Indeed, a is independent of R. The solidcurves as shown in Fig. 8 are obtained from the follow-ing empirical formula fitting the data points:
a(N,P 2 ) = A exp[-D(p 2- B)2 ] + C, (24)
where
A = 1.75 exp(-0.42N 2 ) + 0.75,
B = 1.0 - 0.56 exp(-1.65N 2 ),
C = 0.25 exp(-2.4N) + 0.25,
even at low frequencies and therefore not shown in thefigures. We should bear in mind that the a given byEq. (24) is only good for the fundamental mode of thespecial class of fiber described by Eq. (14). For otherfiber classes or higher-order modes, different a func-tions (or in general, different G functions) are expect-ed. The present method is named the dual effective-index method (DEIM) to be distinguished from theprevious ones.
VIl. Efficiency of the Effective-index Methods
Of the three effective-index methods described inthe previous sections, the simple effective-index meth-od, being the building block of the other two, is themost efficient one. For step-index fibers, where Eq.(9) is applicable, the simple effective-index method isreduced to solving a single 1-D scalar wave equationwhile the composite and the dual effective-indexmethods require the solutions of two such equations.Therefore, the effective-index methods are nearly asfast as any method used for solving the 1-D waveequation (for example, see Refs. 16-20), and henceincomparably more efficient than those methodsl-6
that directly solve the 2-D problem to give the samedegree of accuracy. However, those direct methodsl-6
have the advantage that accuracy can always be im-proved at the expense of efficiency, for example, byrefining the discretization in the finite element meth-od,1-3 whereas there is no convergence scheme for theeffective-index methods. For graded-index fibers, theefficiency of the effective-index methods may be se-verely reduced if a very large number of inhomogen-eous slabs have to be analyzed to maintain the accura-cy. Nevertheless, the effective-index methods areparticularly suitable for small computers as the re-quired memory storage and programming effort arevery small.
D 1.52 + 048N 2 + 0.14
Equation (23) has been extensively tested using a giv-en by Eq. (24). As a is a function of 32 (i.e., P2 ), it isnecessary to use Eq. (23) iteratively by putting /32 = /3,2into Eq. (24) as the first trial. It is found that for N _0.3, in most cases, the calculated P2 is accurate to thefourth decimal place that corresponds to <0.5% errorfor medium and large P2 and a few percent error atP2
_
0.1. It is noted that the accuracy of /32 calculated fromEq. (23) is not sensitive to a at high frequencies, espe-cially for N _ 1, as in these cases, both /3,2 and /Y2 aregood approximations and any linear combination ofthem is also a good approximation. In principle, Eq.(23) is not useful for R = 1. In practice, however, Rslightly larger than 1, say 1.01, can be used to approxi-mate the R = 1 fiber. The previously analyzed circu-lar, elliptical, and cusp-shaped fibers have also beenanalyzed by using the present approach. The calcu-lated dispersion curves (for the fundamental modesonly) are nearly indistinguishable from the exact ones
Vil. Conclusion
Three effective-index methods, i.e, the simple, thecomposite, and the dual effective-index methods, havebeen presented for efficient analysis of optical fibers.As already illustrated by the numerical examples, thesimple effective-index method overestimates thepropagation constant of the fiber while the compositeeffective-index method underestimates it. The accu-racy of the latter is comparable with that of the formerfor the fundamental mode but gets poorer for higher-order modes. Both methods are useful in applicationswhere moderately accurate results are required. Thedual effective-index method is very accurate but thepresent development is only good for a specific class ofsingle-mode fiber.
The author would like to thank A. E. Karbowiak andP. L. Chu for their guidance and many useful discus-sions. This work is supported by the Australian Com-monwealth Research Scholarship.
References1. C. Yeh, K. Ha, S. B. Dong, and W. P. Brown, Jr., "Single-Mode
Optical Waveguides," Appl. Opt. 18, 1490 (1979).2. B. M. A. Rahman and J. B. Davies, "Finite-Element Analysis of
Optical and Microwave Waveguide Problems," IEEE Trans.Microwave Theory Tech. MTT-32, 20 (1984).
3. K. S. Chiang, "Finite Element Analysis of Optical Fibres withIterative Treatment of the Infinite 2-D Space," Opt. QuantumElectron. 17 (1985), to be published.
4. E. Schweig and W. B. Bridges, "Computer Analysis of DielectricWaveguides: A Finite-Difference Method," IEEE Trans. Mi-crowave Theory Tech. MTT-32, 531 (1984).
5. J. R. James and I. N. L. Gallett, "Point-Matched Solutions forPropagating Modes on Arbitrarily-Shaped Dielectric Rods,"Radio Electron. Eng. 42, 103 (1972).
6. L. Eyges, P. Wintersteiner, and P. D. Gianino, "Modes of Dielec-tric Waveguides of Arbitrary Cross Sectional Shape," J. Opt.Soc. Am. 69, 1226 (1979).
7. R. M. Knox and P. P. Toulios, "Integrated Circuits for theMillimeter through Optical Frequency Range," in Proceedings,Symposium on Submillimeter Waves, J. Fox, Ed. (PolytyechnicPress, Brooklyn, 1970), p. 497.
8. W. V. McLevige, T. Itoh, and R. Mittra, "New Waveguide Struc-tures for Millimeter-Wave and Optical Integrated Circuits,"IEEE Trans. Microwave Theory Tech. MTT-23, 788 (1975).
9. G. B. Hocker and W. K. Burns, "Mode Dispersion in DiffusedChannel Waveguides by the Effective Index Method," Appl.Opt. 16, 113 (1977).
10. S. T. Peng and A. A. Oliner, "Guidance and Leakage Propertiesof a Class of Open Dielectric Waveguides: Part I-Mathemati-cal Formulations," IEEE Trans. Microwave Theory Tech.MTT-29, 845 (1981).
11. A. A. Oliner, S. T. Peng, T. I. Hsu, and A. Sanchez, "Guidanceand Leakage Properties of a Class of Open Dielectric Wave-guides: Part II-New Physical Effects," IEEE Trans. Micro-wave Theory Tech. MTT-29, 855 (1981).
12. M. N. Armenise and M. De Sario, "Optical Rectangular Wave-
guide in Titanium-Diffused Lithium Niobate Having Its OpticalAxis in the Transverse Plane," J. Opt. Soc. Am. 72,1514 (1982).
13. J. Buus, "Application of the Effective Index Method to Non-planar Structures," IEEE J. Quantum Electron. QE-20, 1106(1984).
14. J. G. Gallagher, "Mode Dispersion of Trapezoidal Cross-SectionDielectric Optical Waveguides by the Effective-Index Method,"Electron. Lett. 15, 734 (1979).
15. M. Miyagi and S. Nishida, "Approximate Formula for Describ-ing Dispersion Properties of Optical Dielectric Slab and FiberWaveguides," J. Opt. Soc. Am. 69, 291 (1979).
16. A. Gedeon, "Comparison between Rigorous Theory and WKB-Analysis of Modes in Graded-Index Waveguides," Opt. Com-mun. 12, 329 (1974).
17. C. Pichot, "The Inhomogeneous Slab, a Rigorous Solution to thePropagation Problem," Opt. Commun. 23, 285 (1977).
18. J. Janta and J. Ctyroky, "On the Accuracy of WKB Analysis ofTE and TM Modes in Planar Graded-Index Waveguides," Opt.Commun. 25, 49 (1978).
19. M. Hashimoto, "A Numerical Method of Determining Propaga-tion Characteristics of Guided Waves along InhomogeneousPlanar Waveguides," J. Appl. Phys. 50, 2512 (1979).
20. J. P. Meunier, J. Pigeon, and J. N. Massot, "A Numerical Tech-nique for the Determination of Propagation Characteristics ofInhomogeneous Planar Optical Waveguides," Opt. QuantumElectron. 15, 77 (1983).
21. D. Marcuse, "Gaussian Approximation of the FundamentalModes of Graded-Index Fibers," J. Opt. Soc. Am. 68,103 (1978).
22. A. W. Snyder and J. D. Love, Optical Waveguide Theory,(Chapman & Hall, London, 1983), Chap. 17.
23. A. W. Snyder and R. A. Sammut, "Fundamental (HE,,) Modesof Graded Optical Fibers," J. Opt. Soc. Am. 69, 1663 (1979).
24. J. E. Goell, "A Circular-Harmonic Computer Analysis of Rect-angular Dielectric Waveguides," Bell Syst. Tech. J. 48, 2133(1969).
25. R. A. Sammut, "Analysis of Approximations for the Mode Dis-persion in Monomode Fibres," Electron. Lett. 15, 590 (1979).
Meetings Calendar continued front page 339
1986March
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