Propagation Characteristics of Single Mode Optical Fibers with

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-18, NO. 10, OCTOBER 1982

1484

Propagation Characteristics of Single Mode Optical Fibers with Arbitrary Index Profiles: A Simple Numerical Approach ENAKSHI KHULAR SHARMA, ANURAG SHARMA, AND I. C. GOYAL

Abstract-We present here a rapidly converging numerical procedure for the direct evaluation of the propagation constant and its first and second derivatives in single mode optical fibers with arbitrary refractive index profiles. To illustrate the procedure we have also used it to evaluate the propagation constant and its derivatives in single mode optical fibers with power law profiles in the presence of a Gaussian axial index dip, and hence, studied the effect of a dip on the dispersion characteristics of the fibers.

derivatives in single mode optical fibers in the presence of an axial dip, and hence, studied the effect of the dip on the dispersion characteristics of the fiber. PROCEDURE

For an optical fiber with refractive index profile given by n2(R) = n: - (n: - n;)&f(R)

INTRODUCTION

I

T is now well known that the scalar wave equation can be used to determine the propagation characteristics of graded index optical fibers in most regions of practical interest. It may, however, be mentioned that analytical expressions for the propagation constants and their derivatives with respect to frequency are available only for an infinitely extended parabolic profile [ I ] . For a step profile or a cladded parabolic profile one has transcendental equations determining the propagation constant; the derivatives, however, can be expressed as analytical expressions in terms of the propagation constant. For all other profiles one has to numerically solve the wave equation to calculate the propagation constant as a function of frequency and then calculate the first and second derivatives required to evaluate the group velocities and dispersion coefficient. Such a numerical calculation of the derivatives requires the calculation of the propagation constant to a considerable accuracy. We may point out that the various approximate and semianalytical techniques usually give sufficient accuracy in the calculation of the propagation constant, but are, in general, not sufficiently accurate to obtain its first and second derivatives [2].

In this paper we present a direct numerical procedure to calculate the propagation constant and its first and second derivatives accurately in single mode optical fibers with any arbitrary index profile; the numerical method is similar to that used in [3] for the calculation of the cutoff frequency for single mode operation. As an illustration of the procedure, we have also used it to evaluate the propagation constant and its Manuscript received March 2,1982;revised May 27,1982. This work was supported in part by the Council of Scientific and Industrial Research, India. E. K. Sharma and A. Sharma are with the Department of Physics, Indian Institute of Technology, New Delhi, India. I. C. Goyal is with the Department of Physics, Indian Institute of Technology, New Delhi, India, on leave at the Institut fur Hochfrequenztechnik, Technische Universitat, Braunschweig, Germany.

= n;

R < 1 R>l

(1)

(where R = r/a, a being the core radius of the fiber,f(R) defines the profile shape, and 6 defines the "index jump" at the core cladding interface) the scalar wave equation for the fundamental mode can be written as

where v and b are normalized parameters defined as (31

and 6 = 1- u2/v2;

u=k,a(n? - p'/k;)'/'

(4)

(3 being the propagation constant and k, the free space wave number. The boundary conditions on +(R) atR = 0 andR = 1 are given by [4] d\jj\ —

=0 and

\ d\p\ -

U dR R = l

(5)

where

Following the Ricatti transformation as in [3], we can reduce (2) to the following first order differential equation dG

dG-~'(1-b)-G/R-G2

(6)

dR

where \p dR

(7)

and the boundary conditions transform to G(R = 0) = 0

(8a)

SHARMA et al.: SINGLE MODE OPTICAL FIBERS WITH ARBITRARY INDEX PROFILES

and WK ! (w)

(8b)

Ko(w) '

Further, by differentiating (6) and (8), we can write the following differential equations to be solved along with the associated boundary conditions to obtain the derivatives of the propagation constant, i.e., b' and b" (the prime denotes differentiation with respect to u).

Forb' (9) with the boundary conditions (10a)

G'(R = 0) = 0 and

For b" dG dR

r"

Sf(R)]

-,>2 "inr*"

r -2GG"- - — + 4vb' + v2b"

calculation' of the propagation constants and its derivatives are summarized below. 1) The transcendental equation (8b) is solved to obtain b; the LHS is obtained by a numerical solution of the first order differential equation (6) with boundary condition (sa) at R = 0 and step size h = 1/N (i.e., N is the number of divisions into which the domain R = 0 to R = 1 is divided). 2) With b known, (6) is solved with step size h = 1 /4N and the numerical values of G(R) are stored at each step at 4N discrete points (R = 1/4N, 2/4N, • . • , 1) for subsequent calculations in steps 3), 4), and 5). 3) The transcendental equation (lob) is solved for b'; the LHS is now obtained by solving (9) with boundary condition (loa), step size h = l /N and values of G(R) stored at step 2). 4) Again with the known value of b', (9) is solved and numerical values of G'(R) stored at 2N discrete points R = 1/2N, 2/2N, - . . , 1 for use in step 5). (Step size h = 1/2N.) 5) Equation (12b) is solved to obtain 6"; the LHS is now obtained by solving(1 1)with boundary condition (12a), stored values of G(R) and G'(R) from steps 2) and 4) and step size h = l /N. Further, we also used the propagation constants so calculated to evaluate the dispersion characteristics of single mode fibers in terms of the dispersion coefficient s, defined as [6]

(11)

-b)v2

s=

with the boundary conditions

1485

cond

(12a)

G"(R =0) = 0

tl)

J_

and ,,,2

A:?(w) !

"

K2o(w)+

ATtCv

wKo{w)

where

h]

(17) (18)

n2n2

2ub'

(19)

=nt-n$ The procedure to obtain the propagation constant and its derivatives accurately now requires the solution of the three boundary value problems in sequence. A close look at (6) shows that the last term on the RHS is indeterminate at R = 0 and hence, one has to take the limiting form of the equation at R = 0. Similar terms also occur in (9) and (10) and it can be easily shown [3] that the limiting forms are

and the dot denotes differentiation with respect to A; b and b can be related to b' and b" as (20) *»2

' dG\ \ dR IR

T - T -

(21)

(13)

dG'\

(14)

and

f

(15)

2

NUMERICAL EXAMPLES AND DISCUSSION

To illustrate the use of the procedure and test its convergence, we carried out numerical calculations for single mode fibers with various refractive index profiles. The steps in the

It may be noted that (9)-(12) are so normalized that the solutions depend only on normalized frequency v and normalized profile shape f(R). Hence, once b, b', and b" are known as functions of u for any profile, s as a function of A can be calculated directly using the algebraic expression (16) without any further computation. All the calculations carried out correspond to silica fibers with Ge0 2 doping in the core; the refractive index n corresponding to a 13.8 percent Ge0 2 doping and the cladding index n2 corresponding to silica. The 'We used the fourth order Runge-Kutta procedure for the numerical solution of the differential equation [5]. The number of points, step size, storage, etc., are hence in reference to this procedure.

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-18, NO. 10, OCTOBER 1982

1486

TABLE I CONVERGENCE OF THE NUMERICAL RESULTS FOR A FEW TYPICAL REFRACTIVE INDEX PROFILES Profile

Parameters

N

u2

2 3 = 2

4

= 1 • 75 ^.m

5

a = 2.5 p m

6

= 2.2371

8 1c 12 2

ii

3

£"

o< = CO

4

X = 1 . 4 pm

5

a = 2 . 5 pm

6

= 2.7743

8

V

10 12 2 3 "0 1

^