An efficient algorithm for determining the dispersion characteristics of single-mode optical fibers

Abstract

An efficient algorithm is proposed for calculating the dispersion
characteristics of optical fibers with radially arbitrary
refractive-index profiles. It is based on a variational finite-element
formulation. Formulas for the derivatives of the normalized propagation
constant w with respect to the normalized frequency V
up to the second order are derived by using a reaction formula. These
formulas contain the modal field ψ and its derivative
d ψ/ dV . Two variational problems are then formed
and solved as matrix equations by using the finite-element method. The
first one is a conventional eigenvalue problem with eigen-solution
{ w ,ψ} and dominates the computing time. The second one is a
direct problem for d ψ/ dV and can be solved by a
few simple matrix manipulations. The proposed algorithm turns out to be
a rapidly convergent one and careful arrangement results in saving for
both storing memory and computing time

Full text

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 10, NO. 6, JUNE 1992
705
An Efficient Algorithm for Determining
the Dispersion Characteristics
of Single-Mode Optical Fibers
Hoang Yan Lin, Ruey-Beei Wu, and Hung-Chun Chang, Member, IEEE
Abstract-An eficient algorithm is proposed for calculating the
dispersion characteristics of optical fibers with radially arbitrary
refractive-index profiles. It is based on a variational finite-element
formulation. Formulas for the derivatives of the normalized
propagation constant w with respect to the normalized frequency
V up to the second order are derived by using a reaction formula.
These formulas contain the modal field $ and its derivative
d$/dV. Two variational problems are then formed and solved as
matrix equations by using the finite-element method. The first one
is a conventional eigenvalue problem with eigen-solution {w. $}
and dominates the computing time. The second one is a direct
problem for d$/dV and can be solved by a few simple matrix
manipulations. The proposed algorithm turns out to be a rapidly
convergent one and careful arrangement results in saving for both
storing memory and computing time.
I. INTRODUCTION
ALCULATION of the propagation constant as a function
C of wavelength and the modal field distribution for optical
fibers is a well-established problem and many different solu-
tion methods have been proposed and studied. In transmission
applications, one important quantity of an optical fiber other
than the two mentioned above is the dispersion coefficient
S in ps/nm-km. Understanding and control of the variation
of S versus wavelength is essential in the design of optical
fibers with more sophisticated refractive-index profiles, such as
dispersion-shifted and dispersion-flattened fibers which have
been under extensive study in recent years [l].
To calculate the propagation constant and the modal field
distribution, one can formulate a variational problem and then
solve it by using Rayleigh-Ritz method or finite-element
method (FEM) [2, ch. 51. The FEM has proved to be an
efficient technique for solving variational problems, so we
adopt it to solve the two variational problems which we shall
formulate later. The definition of S involves the first and
second derivatives of the propagation constant with respect
to wavelength, thus theoretical evaluation of S requires the
determination of these derivatives first. However, direct nu-
merical calculation of the first and second derivatives from
the propagation constant versus wavelength data based on
bimple finite differences can result in great errors [3]. Different
procedures have then been proposed, aiming at obtaining
Manuscript received July 11, 1991. This work was supported by Telecom-
munication Laboratories, Ministry of Communications, Republic of China.
The authors are with the Department of Electrical Engineering, National
Taiwan University, Taipei, Taiwan 10764, Republic of China.
IEEE Log Number 9107685.
0733-8724/92$03
good accuracy in calculation of the dispersion coefficient
[4]-[6]. Mammel and Cohen [4] used the Rayleigh quotient
to obtain the first derivative of the propagation constant, but
used direct numerical differentiation in the calculation of the
second derivative. E. K. Sharma et al. [5] avoided numerical
differentiations by solving three differential equations for the
propagation constant and its first and second derivatives,
respectively. Recently, A. Sharma and S. Banerjee [6] reported
another method based on a matrix perturbation theory and
showed that computational effort can be reduced compared
with the method of [5].
In this paper we present a novel method based on a varia-
tional finite-element formulation for calculating the various
characteristics, including S, of optical fibers with radially
arbitrary refractive-index profiles. First, we derive the formulas
for the derivatives of the normalized propagation constant
w with respect to the normalized frequency V. By using
a reaction formula, we show that for a given V , the first
derivative dw/dV can be expressed in terms of w and modal
field $ at that frequency, while the second derivative d2w/dV2
can be expressed in terms of w, $, dw/dV, and d$/dV. Then,
by formulating two variational problems, we are able to solve
for w, $, and d$/dV at the specified frequency using the
FEM. Once w, dw/dV, and d2w/dV2 along with the material
dispersion information are given, the dispersion coefficient S
can be accurately calculated from its definition. Numerical
results show that our method offers better convergence speed
when computing S, as compared with the methods given in
[5] and [6]. One nice feature in the present method is that
although two matrix equations are formed corresponding to the
two variational problems, only one of them needs to be solved
as an eigenvalue problem, leading to considerable saving in
the computation.
The main body of mathematical formulation, including
the derivation of formulas for w and w from a reaction
formula and two variational problems for {w, $} and d$/dV,
is presented in Section 11. For understanding how to save
storing memory and computing time, the solution procedure
is described in Section I11 where some careful arrangement is
tailored to find w, w, and finally S. For specific refractive-
index profiles, the modal fields are known explicitly, then
the formulas for W and w can be used to obtained results
directly. In Section IV, analytical results for step-index fibers
are demonstrated as an example and used to examine the
validity of these formulas. For an arbitrary refractive-index
8.00 0 1992 IEEE
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706
profile, solution of the propagation constant and the modal field
resorts to numerical methods. A computer program has been
developed to implement the solution procedure described in
Section 111. We shall investigate numerical convergence of the
present algorithm and analyze a type of dispersion-flattened
fibers in Section V.
11. FORMULATION
Consider an optical fiber with circularly symmetrical
refractive-index profile defined as
where R is the radial coordinate normalized with respect to the
radius U of the solution region beyond which it is a uniform
cladding, n1 is the maximal refractive index within the solution
region, 712 is the refractive index of the outer uniform cladding,
and f(R) defines the shape of profile within the solution
region.
As the index difference tends to zero, it is well known
that the fundamental mode of the optical fiber satisfies the
following scalar wave equation:
]
= o
where L is a linear operator defined as above. The normalized
frequency V and the eigenvalue w are related to the free
space wavenumber k-0 and the propagation constant @ by
I’ = koa(n: - n;)1’2 and w = k o a [ ( @ / k ~ ) ~
paper. Note that a may be larger than the real core radius. The
boundary condition associated with (2) at R = 1 requires that
dd
Ld,(R) E - R-$(R)
dR[ dR
+ [V2 f ( R ) - w2] &$(I?)
(2)
- 4 1
in this
where K,, denotes the 71th order Bessel function of the second
kind.
Obviously, both the eigenvalue w and the eigenfunction $
vary as the frequency V changes. Here, we assume that the
profile shape function f(R) is independent of V. Let {wl, $I}
and {wp, $ 2 ) be the eigensolutions corresponding to VI and
L>, respectively. Consider the reaction between these two
systems, we obtain
1
(4)
0
U here L, (/ = 1.2) is the operator L defined at frequency
L I . Taking integration by parts and imposing the boundary
condition (3), we get a reaction formula
1
JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 10, NO. 6, JUNE 1992
which is valid for any VI and V2.
For convenience, we u5e dot to denote the differentiation
with respect to V. Choose VI and V2 in (5) to be V + dV
and V , respectively, where dV is a differential element for V .
Then, the eigensolution of system 1 can be written as
’U1 = w + lhdV + -w(dV)2
$1 = $ + ?jdV + ,$(dV)2 + ’. .
1 ..
2!
1 ”
+ . . ‘
while that of system 2 is {w2,+2} = {w, $}. Substituting
these solutions into (5) and after some algebraic manipulations
involving merging the terms having the same power in dV, we
obtain from the terms of (dV)l
vc2
w =
wc3 - +%Cl
where
1
c 1 = ?J12(l).
c:~ = J’ f$’RdR,
and
0
1
C3 = 1 $RdR.
0
Similarly, we obtain from the terms of (dV)2
w =
(7)
where
I
D1 = +(1)4(1),
0 2 = J’f+dRdR,
and
0
1
0 3 = J’ $?jRdR.
0
Note that w can be obtained from (6) once the solution {w,
is obtained for a. given V. However, the determination of G
by (7) requires $. One of the key points of this paper is that
to calculate the higher order derivative of w, one needs only
the derivative of w and $ of lower orders.
By differentiating (2) and (3) with respect to V , we get the
governing equation of $
$1
(V2f(R) - w2)R?j(R) +
2(V f (R) - wzii)R+(R) = 0
which can be expressed as
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LIN i? U/.: DISPERSION CHARACTERISTICS OF SINGLE-MODE OPTICAL FIBERS
707
and the boundary condition at R = I
It can be shown that (9) can be derived from (2), (3), and (8),
that is, (8) and (9) are not sufficient to determine 4 uniquely.
For simplicity, we choose di(1) = 0 as the essential boundary
condition (EBC). In other words, the value of yi, at R = 1
is taken to be the same for every V , which is allowed for a
linear problem.
The work remaining to do is to solve {tu. p } and 4.
with optical fibers with arbitrary index profile, the differential
equations together with the boundary conditions are trans-
formed into the variational-equation (VE) formulations, which
are to be solved by the FEM [2], [7]. For the original eigen-
solution {w.
is
To deal
Ji} which satisfies (2) and (3), the VE formulation
For ii which satisfies (8) and the EBC, vi(1) = 0, the VE
formulation is
1
- 2 . I gdi dR + 2 E 7 A $ ( l)di( 1)
0
EBC : ,$I( 1) = 0.
It should be recalled that the function g depends on the
eigensolution {w. 41) and iii.
Once w, tii, and ui are obtained, the normalized propagation
constant h and its derivatives can be found by
W2
V2
2 [ ('UJW + tu2) v2 - 4WWV + 3UJ2]
V4
The dispersion coefficient S can be calculated from the for-
mula PI, PI
x d2n
s _ - - t
('71e dx2
.
b =
2(WliIV-W2)
b=--.
.
and
h =
v3
and the prime denotes the differentiation with respect to the
wavelength A.
111. SOLUTION
PROCEDURE
For clarity, we summarize our algorithm as shown at the
bottom of the page.
It is worthy to note that the error-susceptible numerical
differentiation is fully circumvented in the evaluation of w
and ,w by the present approach.
The steps of our solution procedure are detailed in the
following with the saving of the memory and computation
efforts emphasized:
Step 1: Obtain the matrix eigen-equation. Based on the
FEM, (10) is transformed to a matrix equation
[M(w)l[*l = [OI
(13)
where [M(w)] is a symmetrical, banded matrix, [*[I]
column vector of nodal unknowns, and [O] is a null vector.
This is an eigenvalue problem where the matrix elements are
in general nonlinear functions of the eigenvalue w.
Step 2: Search for the eigenvalue 7u. The eigenvalue w
is such that the determinant of the matrix [M(w)] is zero.
The bisection method serves to locate the desired root w.
During each trial in the bisection root search, we apply the
Gauss elimination method to obtain the LU decomposition
of the matrix [MI [7, ch. 71. Since the matrix [MI is both
symmetrical and banded, the memory and computation time
required can be saved significantly. It is one of the important
features of the FEM. Note that the LU decomposition can
be executed in place without further requirement of computer
storage. Storing the decomposed matrices is essential for
economy of the computation efforts as will become evident
soon.
is a
givm
l7
1 ' . {w.$}
\r. {w.
* solve (10) =+
3
use
=+ solve (11) +
{w.$}
w
7 j
then with
then with
(6) +
then with
Lr. {w.$}. { h..} * use
(7)
=+
iu
$}. .Li/
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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 10, NO. 6, JUNE 1992
Step 3: Solve the eigenvector [e].
bottom diagonal element is very small such that the determi-
nant tends to zero. The eigenvector [!PI] can be obtained by
imposing its bottom element being unity and solving its other
elements by back-substitution [7, ch. 71.
Step 4: Find the first derivative of the eigenvalue w, w. Once
the eigenvector [!PI is solved in Step 3, the eigen-function
can be expressed in terms of the nodal values by known
basis function in each element. Taking element integrals and
summing them together, the constants C2 and C3 in (6)
can be obtained and then the derivative U, can be calculated
accordingly. The term dZ/dw in (6) can be calculated more
efficiently by the relation
In the final trial, the
Step 5: Solve the first derivative of the eigenvector [!PI, [&I.
By applying the FEM again and choosing the same nodes and
basis functions as in Step I, (11) is transformed to a matrix
equation
Here, the matrix [MI is exactly the same as that in (14), [&I
is a column vector of nodal unknowns for 4,
found from w,
by (8). To cope with the EBC that
'(1 at the boundary node is zero, we delete the bottom row and
the rightmost column of the matrix [M] and force the bottom
element of the column vector [&] to be zero. Since the matrix
[MI has already been LU-decomposed and stored during
Step 3, the other elements of the column vector [&] in (15) can
be obtained directly from a forward reduction process followed
by a back-substitution process [7, ch. 71.
Step 6: Find the second derivative of the eigenvalue w,
ui. Once the lower order eigen-solutions, w, [!PI, w, and
[&I are obtained, we can calculate w using (7) and then the
dispersion coefficient S using (12). In the calculation of w
by (7), the constant 0 2 and 0 3 are evaluated by summations
of element integrals, while the term d2 Z/dw2 can be found
more efficiently by the formula
and [g] can be
20, and [e]
In this solution procedure, almost all the computation time
is spent in the first two steps searching for the eigenvalue. The
operations involve matrix generation and the time-consuming
Gauss elimination process. The number of multiplications is
approximately proportional to the number of trials in the
bisection search multiplied by mB2 where m is the total
number of nodes and B is the half-bandwidth of banded
matrices [7, ch. 71. Once the eigenvalue is solved, the vectors
[e], [&] and the derivatives 20, w can be obtained very
efficiently by the last four steps. The operations involve
element integrals, vector generation, and direct substitution
process, in all of which the number of multiplications is
approximately proportional to mB only!
Iv. DISPERSION
IN STEP-INDEX FIBERS
For the cases in which modal fields of optical fibers are
known, the integrals in (6) and (7), and hence w and W, can
be obtained directly. Since the modal fields in step-index fibers
are well known in terms of Bessel functions, (6) and (7) lead
to closed-form expressions. So we consider step-index fibers
in this section.
For step-index fibers, f(R) = 1. The modal field of the
LPol mode can be expressed as
where J, is the nth order Bessel function of the first kind and
U2 = v 2 - w2.
(18)
This field satisfies (2) (or (lo)), and +(1) = 1. Then, in (6),
C1 = 1 and C 2 and C3 can be integrated analytically to be
by using the properties of Bessel functions, where Y(u)
-uJl(u)/Jo(u). Substituting (19) into (6), we obtain
_ _
V d Y
Similarly, by differentiating (17) with respect to V, we
obtain
which can be shown to satisfy (8) (or (11)) and the EBC,
d(1) = 0. It is not difficult to show that the constants in (7)
are such that 0 1 = 0 and
Substituting (19) and (22) into (7), we obtain
-_ - -_
6 = 2 d w 2
2u d u
-__ - _ _
W 2 d 2 Z
1 d Y -I W 2 dl'
g g u2 d2Y
2u d u
1 d Z
2 d w
duZ .
(23)
2u d u
w d I '
221 d u
To check the validity of (20) and (23), we start from the
characteristic equation of the LPol mode given as
Y(u) = Z(w).
(24)
By differentiating (18) and (24) with respect to V , we obtain
.dY
du
and
uu = v - ww.
. dZ
dw
U-
=w-
From (25) and (26), we obtain the same formula as (20).
Similarly, by differentiating (25) and (26) with respect to V ,
we obtain
(26)
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LIN er al.: DISPERSION CHARACTERISTICS OF SINGLE-MODE OPTICAL FIBERS
709
TABLE I
CONVERGENCE OF THE NUMERICAL
SOLUTIONS AS -1-
IS INCREASED
o = x,
core radius = 2..j pin, X = 1.4 pm
core: 13.jGe02: 8G.3SiOa
cladding: pure Si02
-1-
1
i
3
4
5
6
7
Section IV
[4], (40 points)
[5], (25 x 25)
u 2 p - 2 = 1 - h
0.388955
0.388521
0.388475
0.388464
0.388464
0.388463
0.388463
0.388463
0.388452
0.388560
b
-/I
S (ps/nm-km)
2.762393
2.78408 1
2.786983
2.787481
2.787790
2.787720
2.787831
2.787476
2.7832
2.7916
0.190640
0.190809
0.190825
0.190828
0.190829
0.190829
0.190830
0.190830
0.190824
0.130230
0.130421
0.130442
0.130446
0.130448
0.130447
0.130448
0.130448
0.130438
TABLE I1
CONVERGENCE OF THE NUMERICAL
SOLUTIONS AS Ay IS INCREASED
n = 2, core radius = 2.3 pin, X = 1.75 iriii
core: 13.5Ge02: 86.5Si02
cladding: pure Si02
-1-
1
2
3
4
5
6
7
[4], (40 points)
(51, (35 x 35)
([2/\.2 = 1 -
0.797078
0.793724
0.793513
0.793474
0.793462
0.793458
0.793457
0.793554
0.793458
/I
b
0.224872
0.229491
0.229689
0.229731
0.229743
0.229747
0.229749
0.229750
- b
0.042655
0.042060
0.042240
0.042257
0.042261
0.042262
0.042261
0.042261
~
S (ps/nm-km)
3.395296
2.5 17000
2.516869
2.5 12909
2.5 1 1744
2.51 1172
2.510856
2.5108
2.5169
and
U'& + 'k2 = 1 - 7lJG - tu2.
(28)
From (27) and (28), we obtain the same formula as (23). We
thus prove that (6) and (7) yield the exact results.
v. NUMERICAL RESULTS AND DISCUSSION
For optical fibers with other profiles, analytical eigen-
functions are in general unavailable such that the numerical
solutions should be resorted to. A Fortran program has been
tailored to implement the above mentioned solution procedure
on an IBM/PC. The program can calculate the eigenvalue w
and its derivatives w and .Lij for arbitrary shape function f(R).
In the finite element analysis, the solution region is divided
into _li
elements, inside each of which the quadratic basis
functions are employed to interpolate the eigenfunction and
its derivatives and the four-point Gauss-Legendre quadrature
is applied to calculate the element integrals.
Before presenting numerical results, we make some remarks
on the general features of the formulations in [SI, [6], and
this paper. The solution method proposed in [SI needs to
solve three differential equations. Solution of each differential
equation takes about the same amount of computing time,
i.e., computation effort for obtaining S is three times that for
the original eigenvalue problem. It is rather time-consuming.
The formulation in [6] is based on the perturbation theory
which is basically an approximate method. Although it avoids
the repeated solution of differential equations, it deals with
an infinite region R E [0, m) and requires theoretically
the whole set of eigensolutions which cannot be included
practically. Meanwhile, the involved matrix is full which
implies larger storing memory and larger computing time. The
present algorithm, however, is based on an exact formulation.
The only eigenvalue problem for the guided LPol. mode
dominates the computation time and the evaluation of b and b
is quite simple. It is also advantageous to deal with banded
matrices. This algorithm seems more economical for both
computer storage and computation time.
A. Convergence of the Algorithm
To examine the convergence of our algorithm, the computer
program is applied first to analyze two examples which have
been considered previously in [5] and [6]. In Table I and 11,
we list the operating frequency, the parameters of .the fibers,
the convergence of our algorithm in calculating b, b, 6, and S
as the number of elements ( N ) in the FEM is increased, and
certain results adopted from [ S I and [6] for comparison. In
Table I, we also list the exact results obtained from the explicit
formula in Section IV. The material dispersion information is
calculated by using the Sellmeier's coefficients given in [9].
We tabulate the data of a step-index fiber corresponding
to an cy-power index profile with a = 03 in Table I. In the
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