Asymptotic evaluation of the normalisedcut-off frequencies of an optical waveguide

with quadratic index variationJ.M. Arnold

Indexing term: Optical waveguide theory

Abstract: Asymptotic expressions for normalised cut-off frequencies of modes having large meridional ordern and small azimuthal order m are obtained for an inhomogeneous waveguide having quadratic index variation.A fractional integral technique is used to construct solutions directly from the differential equation in theform of an integral representation, which is expanded by the steepest-descent procedure.

1 Introduction

In the theoretical description of propagation in opticalwaveguides, an important set of parameters is formed by thecut-off frequencies of each mode. A number of authors1""3

have considered the problem of computation of the cut-offfrequencies by numerical means, but comparison of theresults in the case of a quadratic index dependence revealswide discrepancies between several sources. For example,the results of Someda and Zoboli,1 who used an obscurevariant of coupled mode theory, are not consistent with cal-culations made by Bianciardi and Rizzolo2 using the strati-fication technique.4

Analytical investigations have tended to ignore this prob-lem. Expressions for the cut-off frequencies can be deducedfrom the work of Gloge5 (using the WKB method) andMarcuse6 (using modal theory) but neither takes properaccount of the true effect of the core-cladding interface.Essentially, these expressions are obtained by computingthe axial phase coefficient j3 for an unbounded medium,and setting /3 equal to the value for a plane wave propagatingin the cladding medium (|3 = n2k, where n2 is the refractiveindex of the cladding, k is the free-space wavenumber). Thisgives the normalised cut-off frequencies for a quadraticradial dependence of index as

Vc = 4\nm + l ne{0,1,2,...}

me{0,1,2. . .}

where n is the meridional mode number and m is theazimuthal mode number. In this paper, it will be shownthat the correct value is

m + 1

me{0, l ,2 . .}

where 5 is a small quantity such that 5 -*• 0 as n -*• °°.The method we use will be to allow the normalised fre-

quency V to be represented by

V = A\vm + l

Paper T94M, first received 16th May and in revised form 22nd July1977Dr. Arnold is with the Department of Electrical & ElectronicEngineering, Queen Mary College, University of London, Mile EndRoad, London El 4NS, England

and we then find 5 such that

„ = n + 8-%

under the condition that 6 is small enough to be regarded asa perturbation. This method is well suited to calculationson quadratic index media, and has been usefully exploitedalso by Hashimoto et a/.13 These authors, however, wereonly concerned in obtaining propagation characteristics forstrongly confined modes, and their formulas do not applywhen V is close to its cut-off value, as the asymptotic rep-resentations fail for these values of V. Here, we constructexplicitly solutions whose representations are valid in thisregion.

The result of our labour is a short table of cut-off fre-quencies, which can be compared with existing numericalcalculations.1""3 Such a comparison leads to the conclusionthat some of these previous calculations1'3 do not give thecorrect values of cut-off frequency. It is essential that anytheoretical model of optical waveguide propagation becapable of predicting propagation characteristics to highorders of accuracy, since what is ultimately required fromthe model is the accurate prediction of differential groupdelays: therefore, a standard test by which conflictingnumerical predictions may be evaluated is of some interestin allowing discrimination to be made between good andbad methods. Comparison of predicted values of cut-off fre-quencies is such a standard test if the true (asymptotic)values are available.

2 Formulation of the problem

The problem to be considered is as follows: given the differ-ential equation

(1)

(2a)

(2b)

=5-}* = 0P

find solutions satisfying the conditions

(0 -f (Pm"1/20) = o p = idp

(ii) W = 0The differential equation is the scalar wave equation forthe quadratic index circular waveguide. The parameters aredefined as follows: if nx and n2 are values of the refractiveindex at p = 0 and p > 1, respectively, and the core radiusis a

MICROWAVES, OPTICS AND ACOUSTICS, NOVEMBER 1977, Vol. l,No. 6 203

V2 = (n\ -n\)k2a2

U2 = (n\k2 -$2)a2

W2 = (f -n\k2)a2

(3a)

(3b)

(3c)

(3d)

where m is any integer, and |3 is the axial phase constant.Cut-off conditions are defined by j3 = n2k, which gives eqn.2b; eqn. 2a is obtained by considering the limiting formof the usual boundary condition

Km(W)P = 1 (4)

as W-+ 0, where Km(w) is the modified Bessel function.Supposing, initially, W^O, eqn. 1 can be brought to a

more familiar form by making the transformations x =Vl/2p, which then gives

d2<t>V

(5)

This equation is customarily referred to as Laguerre's7 dif-ferential equation, and its solutions have been completelyclassified in terms of the Whitaker or confluent hypergeo-

metric functions;8*9 in particular, when — = 2(2n + m +

1), n and m being integers, one solution is well known interms of the Laguerre polynomial

where

(6)

(7)

(known as Rodrigues formula). In general, however,(U2lV)^2(2n +m + 1): in particular, here, we haveU= V at cut-off, so we set (U2 IV) = V, and represent

V = 2Qv + m + 1)

where v is a number to be found, subject to the boundaryconditions eqn. 2.

3 Solutions of Laugerre's differential equation

Although solutions to eqn. 5 can be found in terms ofWhitaker functions, here we choose to construct solutionsdirectly from the differential equation in a novel mannerusing fractional calculus. This is done for two reasons; first,this method leads directly and rapidly to integral rep-resentations for 0; secondly, a notation consistent with thefamiliar L^(%) in the theory of the unbounded medium isperfectly natural. All the results can be interpreted in termsof Whitaker functions if so desired.

Accordingly, we define the fractional derivative10 asfollows:

0Dvz{f(z)} =

1

T(6 - 1 ) dzn+1( (z-ZoT8f(zo)dzo v = n

n>0

(8)

This operator is identical to the differential operator ofordinary calculus when v is an integer. Using this operatorfor v = n, Rodrigues formula is

z~me*n K) T(n + l)° zX '

We define the generalised Laguerre function by extension as

4m)00 = ^ A T + v 2 } (io)

It is then easy to show that the function

) ( i i )

is a solution of Laguerre's differential equation (eqn. 5)when U2/V= 2(2y + m + 1) for any value of v. By lettingv = n + 5 we obtain, immediately, from eqn. 8,

(12)

By expanding the integrand in powers of z0 it. is readilyproved that L^"\z) is regular near z = 0.

The second (logarithmic) solution £/J,m)(z) is found byreplacing the operator QDV

Z with its adjoint10 (or Weyl)operator J^L.

4 Contour integrals

Having established that

, z (13)

it follows immediately from the definition given in eqn. 8that

\-v-l

(14)

which is valid as long as the integral exists (i.e. — (m + 1) <v < 0). Substitution for the variable of integration gives

where

m + 1X = v+—7— T =

(15)

(16)

which is a familiar integral for the confluent hypergeometricfunction. By converting to contour integrals in the mannershown by Buchholz8 or Slater 9 one shows that

204

0<v<\ " v ' 27rT~ * ~ " " ( 1 7 )

MICROWAVES, OPTICS AND ACOUSTICS, NOVEMBER 1977, Vol. l.No. 6

(18a)

where

Wi =

W, =

and C! and c2 are the contours in Fig. la. These are validfor -{m +\)<v.

Fig. 1a Contours in s-plane

t-plane

Fig. 1b Contours in t-plane

5 Steepest-descent evaluation

Steepest-descent evaluations of the type given in eqns. 18aand b have been performed by Buchholz8 and Tricomi.7

The results appear in a rather cumbersome form, however,and are not conducted to high degrees of accuracy. We shalloutline briefly the procedure, which, although tedious, isquite familiar.

We have to compute the integrals

[1-sJ

we also require

dW,

dz

with

L = f G(s)e-Z8nds

(20)

(21)

(22)

In this paper we consider m fixed and v -» °°. If v is fixedand m -*• °°, or y and m are both small, different methodsare required. Using the superscript 1 to indicate that thefunctions are to be evaluated at p = 1 (i.e. z = x2 = Vp2 =V) we have

Wj f ( l -Jcy

= - i f s2 JCJ

(23)

(24)

(25)

and we have used the fact that, for the cut-off problem,V

where

X~v+ + \z— = ~ from eqn. 8. The leading term in the

expansion of \p(s) is

3

By the steepest-descent12 method we obtain

W} = I Ar(T) f t2re2«3l*dtr = 0 Jcj

^T = t Br(T) f t2r+le2^'3dtdz r=0 JC{

(26)

(27)

(28)

where the {cj} are shown in Fig. Ib. {4r(r)}and {5r(r)}aredetermined in Appendix 10. Evaluating these integrals gives

(29)

where

'2X

-1/3

-1/3

and

dW\ l-i

W< = F(s)e-*s/2ds (19)

where 7 = 1 , 2 and ~c

MICROWAVES, OPTICS AND ACOUSTICS, NOVEMBER 1977, Vol. l,No. 6

hi

(30)

(31)

(32)

(33)

(34)

205

where

(35)

(36)

It now remains to use these series to calculate the cut-offfrequencies.

6 Boundary conditions

The boundary condition to be satisfied is, as we have seen,

— {pm"1/20} = 0 p = 1 (37)dp

and the function 0 is

/h — n"l + l/2--Vp2/2/(m)/'^ 2\ /oo\

Thus, we require

dp= 0, p = 1

or-{zme-2/2LiJn)(z)} = 0 , z = V = 4Xdz

Denoting e~znL<^n)(z )> this becomes

Using the integrals found in Section 5,

y

= —{e-^Wl +emiWl2}

(39)

(40)

(41)

(42)

2TT/ I dz dz

Using the expansions 33-36 we find that, to satisfy eqn.41,

(43)

Now we observe that, since\-4/3> ,-2/3

and K = 4x = 6{— j , roots of the above equation must be

-5 /3

IHSOl

— lyn = 0 (44)

as x (and v)~*°°, since the right-hand side dominatesasymptotically. Accordingly, we write

v=n-\+5 (45)

where n is an integer, and 5 is to be found; we expect 5 -*• 0as «-><». In this way, we deduce that for a root of eqn. 46

tan87T =R i cos TT/6

R2 —RI sin7r/6

where

m

= ie.-£

(Ala)

(476)

Asymptotic expansions for Rt andR2 follow directly fromthose for Qx, Q2, S2 and S2

\-4/3 „ L \~2Q

(48a)

«=o(486)

where {Q} and {Da} are given in Appendix 10.For large values of x (and hence V) we have

and so, to first order

(49)

(50)

Inserting the expressions for Co and Do from Appendix 10,we obtain the final result

2/31 /2XoV2/3r(4/3)/5m-3\ nb~^\T) F(27i)\-T-jC0S6

and, since V = 4x, the cut-off frequency is

2 6

(51)

(52)

As yet, n is any integer.

7 Numerical calculations

While it is possible to obtain higher-order terms in eqn. 53,such calculations are very tedious, and for the most partunnecessary; eqn. 53 suffices for most practical purposeswhen n > 3, and m < 3.

First, we compare our result with a formula due toOkamoto and Okoshi,3 which gives (in a notation compar-able to ours), for a quadratic index profile,

) (53)2 4

whereas we have shown that

(54)

It is immediately apparent that the two do not agree, dif-fering in several respects; this can be traced to an error inthe variational analysis used to obtain eqn. 53*.

Secondly, we turn to the actual evaluation of the normal-ised cut-off frequencies. Table I shows a sample of such cal-culations directly from eqn. 51. Later, we show how resultscan be obtained to complete the table for other values

*OKOSHI, T.: Private communication

206 MICROWAVES, OPTICS AND ACOUSTICS, NOVEMBER 1977, Vol. 1, No. 6

of m and n. Eqn. 51 may be used to extend the table in-definitely to the right, but not to the left or downwards.The numerical results are not in agreement with those ofSomeda and Zoboli,1 suggesting that the coupled modetheory is not suitable for this type of calculation. Verygood agreement is found, however, with calculations per-formed by Bianciardi and Rizzoli,2 who used a purelynumerical technique.

Table 1 : Calculated values of Vr for various values of m and n

m0123

313-20215-40817-58119-730

417-22119-39721-54923-683

n

521-23723-38925-52527-648

625-24327-38329-50731 -590

8 Conclusions

This paper has shown how asymptotic formulas may beobtained for the cut-off frequencies of large meridionalorder modes in a quadratic index optical waveguide. Resultsof typical calculations are given.

9 References

1 SOMEDA, C.G., and ZOBOLI, M.: 'Cut-off wavelengths ofguided modes in optical fibres with square-law core profile',Electron. Lett., 1975,11, pp. 602-603

2 BIANCIARDI, E., and RIZZOLI, V.: 'Propagation in graded-core fibres; a unified numerical description', Opt. & Quant.Electron., 1977, 9, p. 121

3 OKAMOTO, K., and OKOSHI, T.: 'Analysis of wave propagationin optical fibres having core with a-power refractive index distri-bution and uniform cladding', IEEE Trans., 1976, MTT-24, p.416

4 CLARRICOATS, P.J.B., and CHAN,. K.B.: 'Electromagneticwave propagation along radially inhomogeneous dielectric cylin-ders', Electron. Lett, 1970, 6, pp. 694-695

5 GLOGE, D., and MARCATILI, E.A.J.: 'Multimode theory ofgraded index fibres', Bell Syst. Tech. J., 1973, 52, p. 1563

6 MARCUSE, D.: 'Cut-off condition of optical fibres',/. Opt. Soc.Am., 1973, 63, p. 1369

7 TRICOMI, F.G.: 'Differential equations' (Blackie, Glasgow,1961)

8 BUCHHOLZ, H.: 'The confluent hypergeometric function'(Springer, Berlin, 1961)

9 SLATER, L.J.: 'Confuent hypergeometric functions' (CambridgeUniversity Press, 1960)

10 ROSS, B. (Ed.): 'Fractional calculus and its applications. Lecturenotes in mathematics (Springer Verlag, New York, 1975)

11 ERDELY, A., MAGNUS' W., OBERHETTINGER, F., andTRICOMI, F.G.: 'Higher transcendental functions: Batemanproject - Vol. 1', (McGraw-Hill, 1953)

12 FELSEN' L.B., and MARCUVITZ, N.: 'Radiation and scatteringof waves' (Prentice Hall Inc., 1973)

13 HASHIMOTO, M., NEMOTO, S., and MAKIMOTO, T.: 'Analsyisof guided waves along the cladded optical fibre; parabolic indexcore and homogeneous cladding', IEEE Trans., 1977, MTT-25, p.11

10 Appendix

Evaluation of the coefficients {Ar}and {Br}

(I _ e 2 \ — = V A pr- — - n — s2}— = Y B t2r+1

\ J £mU 1 j * ' 9 /y V J ^^ -Oft

When |s| < 1 we have, referring to eqn. 25 (55)

(56)

we substitute

i.e.

t = s{l+fs2 + ̂ 4 + !s6

= s{\ +h2 + * 4 +By reverting this series we find

s = '{l-^+T^-d

and

s6. • •}

(57)

(58)

(59)

(60)

Expanding (1 — s2)7 in powers of s2 gives

where

Z4

Pi = r,p2 = T ( T - 1 ) p3 = T ( T - 1 X T - 2 )

and

p4= T ( T - 1 ) ( T - 2 ) ( T - 3 )

By calculating powers of s from eqn. 59 we obtain

(1 —s2)7 = a0 +axt2 + a2f

4 +a3t6 + a4f8. . . (62)

where

a0 = 1 (63a)

fli = ~Pi (63b)

03 =13

(64c)

(6*0

- E± +" 24

27 34Pi

* 175"* 7875

From the definitions of {Ar} and {Br} in eqn. 55, we find

(64e)

Ao = 1

- T% Pi -

and

-2B0 = 1

-2BX = - f - p ,

-2B2 = t% + fPi + ip2

-2£3 = - # ^ - f^Pi ~ f P2 -

—ZD4 — —;

MICROWAVES, OPTICS AND ACOUSTICS, NOVEMBER 1977, Vol. l,No. 6

(65a)

(65b)

(65c)

(6&0

(65e)

(66a)

(66b)

(66c)

(66d)

(66e)

207

The coefficients {Cq} and {DQ} may now be determined, and using the previous values for A0,B0 and/?! we obtainbut it is not practicable to proceed beyond Co and Do.T h e s e a r e Co _ I r(4/3) (Sm-3

U /\ " l c , (68)m . _j l \ ^ 0 ^ 1 ^ / J ; \ 5

(67a)as required in eqn. 50.

Do = Bor(l) (61b)

ErratumSITCH, J.E., and JOHNS, P.B.: 'Transmission-line matrixanalysis of continuous waveguiding Structures usingStepped-impedance cavities', IEE Journal on Microwaves,Optics and Acoustics, 1977,1,(5), pp. 181-184

The authors wish to make the following correction:After eqns. 8 the next line should read

where

\\-R+ R '

208 MICROWA VES, OPTICS AND ACOUSTICS, NOVEMBER 1977, Vol. 1, No. 6