Vol. 8, No. 8/August 1991/J. Opt. Soc. Am. A 1245

Solving the problem of a radially inhomogeneouscircular waveguide with a polynomial profile

Radhey Shyam Kaushal*

Department of Physics and Astrophysics, University of Delhi, Delhi-110007, India

Received June 26, 1990; revised manuscript received February 12, 1991; accepted March 14, 1991

Using an ansatz for the eigenfunction, I obtain an exact analytic solution of the radially inhomogeneous circu-lar waveguide problem for a polynomial profile of the form f(r) = ar2

- br4 + cr6, where the expansion coeffi-cients a, b, and c, are positive and satisfy the constraint b2

= 8c3 12 (m + 2) + 4ac, with m = 0,1, 2....

It is well known' 3 that, for electromagnetic vector wavesassociated with a radially inhomogeneous circular wave-guide, the angular components of electric and magneticfields satisfy a pair of coupled-wave equations. Theseequations, after the introduction of a real and scalar am-plitude function +(r) in the spirit of Kurtz and Streifer,4

reduce to a single, uncoupled second-order differentialequation of the form3 5

+"(r) + - d'(r) + -ko'n(r) _ 2 - 2(r) = 0, (1)rr2

where m = 0,1,2,...,ko(=2r/Ao) is the wave number.Semivectorial approaches have also been used67 for ana-lyzing these problems. For a multimode graded-indexfiber, n(r) describes the refractive index variation with ra-dial coordinate r and is normally expressed as3

n2(r) = n2[1 - f(r)], (2)

where the profile function f(r) is assumed to be symmetricand analytic at r = 0 and also satisfies f(O) = 0.

In view of the fact that the dispersion property of thefiber is sensitive to the nature of f(r), several forms of f(r)have been suggested in the literature.'~' 2 The most com-monly used form of f(r), also consistent with the experi-ments, is

f(r) = k ,2 [a(r ) - b(r) +(r ) (3)

where the coefficients a,, bi, and c, are positive and ro isthe radius of the fiber. Theoretically, there have beenseveral attempts2 5"3 to obtain the solution of Eq. (1) forthe form of Eq. (3) approximately. While Hashimoto2

suggested the corrections to the WKB results, Ikuno13applied Maslov's perturbation formula'4 to obtain the solu-tion of Eq. (1). In all these papers, however, the eigenval-ues are obtained only approximately, and no attempts aremade to derive eigenfunctions explicitly.

In this paper an exact analytic solution of Eq. (1) for theform of Eq. (3) is presented. In particular, we shall ob-tain eigenvalues and eigenfunctions in a closed form for alimited choice of the expansion coefficients a,, b1, and c1

of Eq. (3). We shall first recast Eq. (1) in the form

d2+ 1 d + (A - ap2 + bp4 - Cp6

- 02,)(P)

by redefining the independent variable as p = (r/ro) andby setting

a =ar 0 b =br02, c=2cr0 ,A = ko2no2rO2 -

2ro2(5)

To solve Eq. (4), we eliminate the (do/dp) term by usingthe transformation +(p) = qf(p)/p"1

2 and rewrite Eq. (4) inthe form

l/(p) + (A-ap2 + bP4 -p 6-_ 2 _ )(p) 0 (6)

where primes now represent the differentiation with re-spect to p. Further, for the function ij(p), we make anansatz:

(7)ip(p) = exp[g(p)],

with

g(p) = /2cp 2 _ 1/4ap4 + 3 ln p,

which implies that

0,, = (g,, + g 2)q,

or

4,"(p) + -(25 + 1)o- + (3a + 2aB - 2)p2

+ 2acrp4 - a2p6 - 5(3- 1)14(p) = 0-

Comparing Eq. (8) with Eq. (6), we obtain

A = -(23 + 1)o,

o.2 - 3a + 2aS = a,

2aa = b,

a 2 = C,

5(8- 1) = M2 _ 1/4.

(8)

(9a)

(9b)

(9c)

(9d)

(9e)

0740-3232/91/081245-03$05.00 © 1991 Optical Society of America

Radhey Shyam. Kaushal

1246 J. Opt. Soc. Am. A/Vol. 8, No. 8/August 1991

f 1.43 -

1.42

1.41

1.40

1.39

1.38,, I I I I , , I

0 5 10 15 20 25 30 35 40 45r ( pm)

Fig. 1. Variation of n(r) with r. Curves 1 and 2 correspond totwo sets of coefficients a and c, namely, a = 125,000, c = 500,000and a = 125,000, c = 375,000, respectively.

These equations yield

a = V, = b/2V , = 1/2 m .

In what follows we shall use only the positive sign in 8 as itcorresponds to a well-behaved solution at the origin. Theeigenvalues A are obtained from Eq. (9a):

Am = -(m + 1)b/V (m = 0,1,2,...,). (10)

Here b is fixed from the constraint [Eq. (9b)]

b = (sgn b)2Vc (3Vc + 2 8 + a)" 2. (11)

The normalized eigenfunctions are given by [cf. Eq. (7)]

'km(P) = Nmpm exp[(bp 2- Cp4)/4VC],

with the normalization constant Nm obtained from

I0m(r)12 rdr = 1, (13)

as' 5

Nm = N/2C(m+l)/8 exp(-b2 /32c31 2 )

X [F(m + 1)D_(m+l)(_b/\/¶C3/2)i1l 2 . (14)

the coefficient b involves m dependence through andhence limits the choice of the profile function [Eq. (3)].

From Eqs. (3), (5), and (11), it is clear that the profilefunction f(r) develops an m dependence in the presentmethod of solving the problem. We look for a domain ofparameters a,, b, and cl of Eq. (3) in which such anm dependence of f(r) becomes rather mild. Interestingly,f(r), when computed in this domain, exhibits almost allthe features suggested by the experiments."",2 In fact,corresponding to the wavelength A = 633 nm and theaxial refractive index" no = 1.463, we can compute n(r)from Eq. (2) for a wide range of values of a and c. It isfound that the variation of n(r) with m (for fixed a, c, andro) is rather negligibly small as we go from m = 0 to m =1; the variation occurs only at the fifth decimal place. InFig. 1 the plot of n(r) as a function of r is shown for twotypical values of the parameters a and c, namely, fora = 125,000, c = 500,000 (curve 1) and a = 125,000, c =375,000 (curve 2) with the use of ro = 50 Am for the ra-dius of the fiber. Depending on the fabrication method,some fibers have an index dip near the axis. 2 These fea-tures can also be reproduced by the form of Eq. (3) with asuitable adjustment of the parameters, particularly forlarger values of c; such a trend can be seen in Fig. 1. Re-garding the m dependence of the eigenfunction ckm(p), itis clear that the exponential factor in Eq. (12) dominatesover the power factor. Therefore again the variation of'km(p) with m is not that pronounced for small values of m.

Regarding the use of ansatz [Eq. (7)] a few remarks arein order. First, this ansatz can easily be applied to thechoices of higher-degree polynomial profiles, and exacteigenvalues can be obtained as above. For example, forthe choice of

f(r) = k02fl0 [ ( r+)2 ( r 4 (r )6

- di + elf(-) ,

one can use fr(p) = exp[g(p)], with

g(p) = 1/2Up2 _ 1/4ap4 + '/6Tp6 + 8 In p.

The expression for the eigenvalues now becomes

Am = -(m + 1)oa [o- = (4ce - d2 )/8e 3 2, 8 = 1/2

(15)

Here D(x) is the parabolic cylindrical function. Notethat this exact normalization is meaningful only when theradius of the waveguide is large as compared with thewavelength or for an infinitely extended profile; other-wise one must solve the integral of Eq. (13) numericallyfrom r = 0 to r = r.

Thus the eigenvalues and eigenfunctions correspondingto different modes of excitation inside the circular wave-guide are given by Eqs. (10) and (12), respectively. Notethat the eigenvalue corresponding to the fundamentalmode (ground state) m = 0, is not zero but is given byIAol = b/Vs. In earlier papers2" 3 a parameter was intro-duced in Eq. (4), and the same is used as an expansion pa-rameter in obtaining the approximate solutions of theproblem. Present results, for the form of Eq. (3), are exactto the extent that the coefficients a,, bl, and cl of Eq. (3)redefined in Eqs. (5) satisfy condition Eq. (11). Further,

+ m],

(16)

where various parameters are again redefined in thespirit of Eqs. (5) and satisfy the constraints

a2 - 2a8 -3a = a, 2ac-2T-5T = b, (17)

with

= e-, a = d/2Ve,

and the eigenfunctions (not normalized) are given by

(0.(p) - pm exp(/2up2- 1¼p4

+ /6srp6 ). (18)

Second, the extended'6 version of an ansatz [Eq. (7)] canalso be applied without any difficulty to the case of non-radial polynomial profiles. In that case, however, it maybe mentioned that the corresponding eigenfunctions will

Radhey Shyani Kaushal

Vol. 8, No. 8/August 1991/J. Opt. Soc. Am. A 1247

not be normalizable unless some inverse harmonic termsof the form al/x 2 + a 2 /y

2 and (or) the cross terms of thetype (a3x/y + a 4 y/x),..., are added16 at least to quartic-type polynomial profiles.

To summarize, I have solved the radially inhomoge-neous circular waveguide problem rather exactly for theprofile of Eq. (3) satisfying the constraint of Eq. (11) andobtained closed-form expressions for the eigenvalues andeigenfunctions corresponding to different modes of vibra-tion. While the results for the profile of Eq. (15) are out-lined here, the scope of the applicability of an ansatz[Eq. (7)] is brought out with reference to noncentral poly-nomial profiles. The results are expected to be goodenough for single-mode fibers.

ACKNOWLEDGMENTS

The author is grateful to H. J. W Muller-Kirsten for mak-ing available the research of Hashimoto on this subject.Thanks are also due A. K. Ghatak, K. Thiagrajan, andR. P. Saxena for several helpful discussions.

*R. H. Kaushal is also a UGC research scientist.

REFERENCES1. M. Hashimoto, "On the asymptotic eigenvalues of an inhomo-

geneous circular waveguide," Opt. Commun. 32, 383-384(1980); see also, Int. J. Electron. 46, 125 (1979).

2. M. Hashimoto, 'Asymptotic eigenvalues of vector waves in aninhomogeneous circular waveguide," Opt. Commun. 34, 43-45 (1980).

3. A. Ghatak and K. Thiagrajan, "Graded index optical wave-guides: Review," in Progress in Optics XVIII, E. Wolf, ed.(North-Holland, Amsterdam, 1980), preprint pp. 1-126.

4. C. N. Kurtz and W Streifer, "Guided wave in inhomogeneousfocusing media, part I: formulation, solution for quadraticinhomogeniety," IEEE Trans. Microwave Theory Tech. 17,11-15 (1969); "part II: asymptotic solutions for generalweak inhomogeneity," IEEE Trans. Microwave Theory Tech.17, 250 (1969).

5. For the earlier suggestions about the distributed index wave-guides see, Y. Suematsu and K. Iga, "Mode conversion in lightbeam waveguides," J. IECE Jpn. 49, 1645-1652 (1966); S. E.Miller, Bell Syst. Tech. J. 46, 2017 (1965).

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8. E. G. Rawson and R. G. Murray, "Interferometric mea-surements of Selfoc dielectric constant coefficients to sixthorder," IEEE J. Quantum Electron. QE-9, 1114 (1973).

9. M. Ikeda, M. Takeda, and H. Yoshikiyo, "Refractive indexprofile of a graded index fiber: measurement by a reflectionmethod," Appl. Opt. 14, 814-815 (1975).

10. J. S. Cook, "Minimum impulse response in graded indexfibers," Bell Syst. Tech. J. 56, 719 (1977).

11. K. Iga and Y Kokubun, "Formulas for calculating the refrac-tive index profile of optical fibers from their transverse inter-ference patterns," Appl. Opt. 17, 1972-1974 (1978).

12. W A. Gambling, D. N. Payne, C. R. Hammond, and S. R.Norman, "Optical fibers based on phosphosilicate glass,"Proc. Inst. Electr. Eng. 123, 570-576 (1976).

13. H. Ikuno, Electron. Lett. 15, 762 (1979).14. V P. Maslov, cited in Ref. 2.15. I. S. Gradshteyn and I. Ryzhik, Table of Integrals, Series and

Products (Academic, New York, 1969), p. 337.16. R. S. Kaushal, "Quantum mechanics of noncentral harmonic

and anharmonic potentials in two dimensions," Ann. Phys.(NY) 206, 90-105 (1991).

Radhey Shyam Kaushal