2520 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999
Polarization Evolution Due to the Kerr
Nonlinearity and Chromatic Dispersion
D. Wang and C. R. Menyuk
Abstract— This paper numerically investigated the evolution
of the degree of polarization of individual channels and their
Stokes parameters in a wavelength division multiplexed (WDM)
system in which Kerr nonlinearity and chromatic dispersion are
taken into account but in which polarization mode dispersion
as well as polarization-dependent loss and gain are neglected.
We compared the results to a mean field model which assumes
that the channels are strongly dispersion-managed so that each
channel is only affected by the Stokes parameters of the others.
This model predicts no change in the degree of polarization of
each of the channels so that an initially polarization-scrambled
channel does not repolarize. The full simulations showed that
the repolarization of a polarization-scrambled signal is small for
parameters corresponding to realistic communication systems,
validating the use of the mean field model. However, we also
found that the repolarization can become significant for low data
rates and a small number of channels in a dispersion-managed
system with a short length map, thus setting limits on the model’s
validity and indicating operating regimes that should be avoided
in real communication systems.
Index Terms—Chromatic dispersion, evolution, modeling, non-
linearity, optical fiber communication, polarization, repolari-
zation, Stokes parameters, wavelength division multiplexing
(WDM).
I. INTRODUCTION
P
OLARIZATION effects in optical fiber transmission sys-
tems are due to birefringence in both the optical fiber
itself and the components like the WDM couplers that are
used in current amplifier systems. Birefringence in the optical
fiber is due to the accidental loss of degeneracy of the two
orthogonal polarization modes that exist in a single-mode
fiber and is weak and randomly varying. By contrast the
materials like LiNbO
3
that are used in optical components
often have a strong polarization dependence in which case
these components are strongly birefringent along a fixed axis.
Polarization effects in both optical components and optical
fiber have become important in recent years because the
advent of the erbium-doped fiber amplifiers (EDFA’s) implies
that these effects accumulate over hundreds of kilometers in
terrestrial systems and thousands of kilometers in undersea
systems.
It is generally assumed that the principal polarization ef-
fects that lead to transmission impairments are polarization-
Manuscript received November 10, 1998; revised August 9, 1999. This
work was supported by AT&T, NSF, DOE, and AFOSR.
The authors are with the Department of Computer Science and Electrical
Engineering, University of Maryland, Baltimore County, Baltimore, MD
21250 USA (e-mail: dwang1@gl.umbc.edu).
Publisher Item Identifier S 0733-8724(99)09676-0.
dependent loss, polarization-dependent gain, and polariza-
tion mode dispersion (PMD) [1]. Polarization-dependent loss
is caused by the strong polarization dependence in optical
components such as WDM couplers, isolators, and optical
switches. Polarization-dependent gain is caused by polarization
hole-burning in the Er-doped fiber amplifiers. Finally, polar-
ization mode dispersion is principally caused by the randomly
varying birefringence in the optical fiber, although the differ-
ential group delay of the optical components can contribute
under some circumstances. The combination of these effects
contribute to channel fading [2] which may sometimes lead to
the loss of an entire channel in a long-distance, high-data-rate
wavelength division multiplexed (WDM) system. Polarization
scrambling greatly ameliorates the effects of polarization im-
pairments [3], but under some circumstances the signal can
repolarize [4], [5], leading once more to polarization effects.
These effects couple in turn to the nonlinearity and disper-
sion in a complex way that is described by the Manakov-PMD
equation [6], [7]. There is a general misconception that initially
polarization-scrambled channels cannot repolarize unless some
combination of polarization mode dispersion, polarization-
dependent loss, and polarization-dependent gain are present.
However, nothing could be farther from the truth! As we shall
describe in detail in this article, nonlinearity and dispersion
all by themselves can led to channel interactions that in turn
cause significant repolarization when the channels are closely
spaced. These interactions set limits on how closely channels
can be spaced and other system parameters. These limits are
of increasing importance in modern-day WDM systems.
It is now commonplace to study fiber system impairments
using the coupled nonlinear Schr
¨
odinger equation that has been
modified to include loss, amplification, spontaneous emission
noise [8], and other effects. When polarization effects, includ-
ing polarization mode dispersion, can be neglected and the
signal is launched in a single state of polarization, then it is
possible to study optical fiber impairments using the scalar
nonlinear Schr
¨
odinger equation and its modifications [6], [7].
In fact, this scalar approach is far more commonly used than
the complete vector equation [8].
There is a misconception that the nonlinear Schr
¨
odinger
equation is more fundamental than the Manakov equation in
which polarization mode dispersion is neglected [5]. In fact,
just the opposite is true since polarization mode dispersion
must be negligible and in addition the signal must be launched
in a single polarization state for the nonlinear Schr
¨
odinger
equation to hold. Polarization mode dispersion is a significant
issue in terrestrial systems with optical fiber that is more
0733–8724/99$10.00
1999 IEEE
WANG AND MENYUK: KERR NONLINEARITY AND CHROMATIC DISPERSION 2521
than a decade old, but it plays a nearly negligible role in
undersea systems or terrestrial systems that use fiber that
has been laid within the last five years. For example, if
we assume a value for the polarization mode dispersion of
0.1 ps/(km)
1/2
, then over 10 000 km, the differential group
delay is only 10 ps. This value is not large enough to
significantly distort individual channels, although it does lead
to some relative drift of the polarization states of channels that
are spaced far apart [6], [7]. We will show explicitly, later
in this paper, that the effect of this additional polarization
mode dispersion on signal repolarization is small. Modern-
day undersea systems also use polarization scrambling or
orthogonal polarization when launching their signals. In these
systems, the Manakov equation is the correct equation to use
to study of nonlinear and dispersive effects. Solving either the
scalar nonlinear Schr
¨
odinger equation or vector equations such
as the Manakov equation or the Manakov-PMD equation can
be very computationally time-consuming—particularly when
WDM systems that include more than ten channels are studied.
A full simulation of a terrestrial system can take many central
processor unit (CPU) hours for a single parameter set and a
full simulation of an undersea system can actually take many
CPU days [9]! To appropriately study polarization effects, one
must additionally simulate many different realizations of the
randomly varying polarization orientations in the optical fiber.
Thus, it is not realistic to study the impairments introduced
by polarization effects in modern-day WDM systems that
may have many tens of channels using complete numerical
solutions of the Manakov equation and its modifications. There
is thus an urgent need for effective reduced models.
In previous work, we introduced a mean field approach in
which one merely follows the Stokes parameters of the indi-
vidual channels neglecting their detailed temporal evolution
[5]. The potential computational saving in this approach is
vast. We note that this approach presupposes that polarization
effects like repolarization are dominated by the interplay
of polarization mode dispersion, polarization-dependent loss,
and polarization-dependent gain and that nonlinearity and
dispersion play no significant role in these effects. This as-
sumption is reasonable in modern-day dispersion-managed
systems in which the local dispersion is large, although the
average dispersion is kept small. So, the bits in two channels
slide rapidly through one another, and each channel is only
affected by the average properties of the others. However,
this assumption will fail when channels are spaced too closely
together. At this point nonlinearity and dispersion can lead
to significant polarization impairment. Given the interest in
spacing channels ever closer together in dense WDM system,
it is of practical importance to determine this limit.
It is the purpose of this article to explore the conditions
under which the mean field approach is valid for studying
polarization effects. We will be focusing on the extent to which
the combination of chromatic dispersion and the Kerr effect
can induce changes in the polarization states of the individual
channels beyond what the mean field model predicts. Our
procedure will be to compare full split-step simulations with
realistic temporal pulse profiles and with up to seven channels
to the mean field model. Our purposes are threefold. First,
we will determine when the system parameter are such that
nonlinearity and dispersion alone lead to significant polariza-
tion effects. These effects limit the channel spacing and other
parameters in real systems. Second, we will set a baseline
for further studies in which neglected effects like polarization
mode dispersion, polarization-dependent loss, polarization-
dependent gain and amplified spontaneous emission noise are
included. Given the complex interplay between the different
effects, it is important to study them in isolation before com-
bining them. We will however determine the effect of adding
moderate polarization in this paper. Third, we will determine
when the mean field approach can be safely used to study
polarization effects. This approach yields vast computational
savings [5].
The remainder of this paper is organized as follows:
Section II reviews the derivation of the mean field equations
governing the evolution of the Stokes parameters. We add to
previous work [5] by taking into account the effect of fiber
attenuation and amplifier gain. In Section III, we compare the
results of the mean field model to full split-step simulations
for a typical set of parameters and up to seven channels.
In Section IV, we examine the effects of varying the channel
spacing, varying the dispersion, and adding filtering. Section V
contains the conclusions.
II. D
ERIVATION OF THE MEAN FIELD EQUATION
Our starting point is the Manakov equation
(1)
where
represents the complex envelope of
the two polarizations, where
is the dispersion coefficient,
is the nonlinear coefficient, and and are distance
along the fiber and retarded time. Earlier experimental [9]
and theoretical [6], [7] work has shown that this equation
accurately describes nonlinear and dispersive light propagation
in standard communication fiber with rapidly and randomly
varying birefringence when polarization mode dispersion can
be neglected. There are several reasons for neglecting po-
larization mode dispersion in most of this study. First, with
high-quality optical fibers and a limited number of channels,
polarization mode dispersion is negligible. For example, as-
suming a value of 0.1 ps/(km)
1/2
, polarization mode dispersion
can be neglected over a bandwidth of about 4 nm over 500 km
and about 1 nm over 10 000 km. These numbers are sufficient
to include several channels with single channel rates of 2.5
Gb/s. Even at higher data rates and with channels spread
over a larger bandwidth, the effect is small, as we will show
explicitly in Section III, as long as the polarization mode
dispersion is moderate. Second, it is of practical importance
to determine when nonlinearity and chromatic dispersion lead
to significant polarization effects on their own. Third, it is of
critical importance in validating the mean field approach to
determine its ability to accurately reproduce the impact of the
Kerr effect and chromatic dispersion on the polarization states
of the channels in a WDM systems in the absence of other
2522 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999
effects. Thus, this work will serve as a baseline against which
the importance of polarization mode dispersion and other
polarization effects in combination can be determined. We also
note that (1) does not include the spatially varying gain and
loss that is present in real systems; however, this effect is easily
included and we will show how to do that at the end of this
section. Finally, we note that
, where
is the light’s central wavenumber, is the Kerr coefficient,
and
is the optical fiber’s effective area. The extra factor
of 8/9 appears due to rapid mixing of the central frequency’s
polarization state on the Poincar
´
e sphere [6], [7]. In principle,
there can be an additional contribution to (1) due to incomplete
mixing of the electric field on the Poincar´e sphere that is
referred to as nonlinear polarization mode dispersion [6], [7].
However, in standard communication fiber in which the scale
length on which the field mixes on the Poincar
´
e sphere is 100
m or less while the scale on which the Kerr nonlinearity and
chromatic dispersion act is many hundreds of kilometers, this
effect is completely negligible [7].
We next write
as a sum of contributions over channels,
obtaining
(2)
where
and are the central wavenumber and
frequency of the
-th channel with respect to the central
wavenumber and frequency of
, and is the cor-
responding wave envelope. Substituting (2) into (1), we
find
(3)
where, consistent with our assumption that the dispersion
between channels is large, we neglect four wave mixing terms.
We now define the Stokes parameters for each of the channels
as
(4a)
(4b)
(4c)
(4d)
where we assume that
contains a very large number
of individual bits. Using (3) to determine the evolution of the
Stokes parameters, we find
, and we find for
that
(5)
In a highly dispersive system, the channels for which
rapidly pass through the -th channel in the time domain.
Consequently, the evolution of the
-th channel is only
affected by the averaged time variation of the
channels
so that we may effectively treat them as continuous waves.
This assumption is the heart of the mean field approximation.
We thus replace
(6)
from which we conclude
(7)
We can find similar expressions for
and ,
so that we finally obtain
(8)
The effect of dispersion does not appear in (8); only the
nonlinearity appears, and the equations are analogous to the
equations that govern nonlinear polarization rotation of contin-
uous wave beams [10]. However, the local dispersion plays a
critical role since it must be large enough so that each channel
only affects its neighbors through its mean field. We note as
well that there is no change in
, i.e., .
From (8), it follows that the only effect of combining the
Kerr effect with large dispersion is to rotate the polarization
states of the different WDM channels. In particular, there is no
change in the degree of polarization! This result is significant
because polarization scrambling which reduces the degree of
polarization of a single channel to a small value is often used to
combat fading. However, under some circumstances the signal
can repolarize. We earlier showed that polarization-dependent
loss can account for this repolarization [4], but it is important
to verify that chromatic dispersion and the Kerr effect cannot
also account for it. If the mean field approximation is valid,
i.e., (6) is an accurate approximation, then these effects lead
to no repolarization. Conversely, if chromatic dispersion and
the Kerr effect do contribute significantly to repolarization,
which can happen even in the absence of polarization mode
dispersion and other polarization effects when the mean field
approximation fails, then there is a large nonlinear interaction
between the channels which is unattractive for communication
WANG AND MENYUK: KERR NONLINEARITY AND CHROMATIC DISPERSION 2523
systems. Thus, examining the evolution of the degree of
polarization in full simulations will allow us to determine the
parameter regimes in which the mean field approximation is
valid and in which chromatic dispersion and the Kerr effect
do not contribute significantly to repolarization. We will do
that in the following sections.
It is interesting to note that even though (8) is nonlinear, a
complete analytical solution may be found
(9)
where
, and are the initial
values of the Stokes vector in the channels. This result is in-
trinsically significant because the number of large-dimensional
nonlinear systems for which exact solutions can be found is
small.
When fiber loss and lumped gain at the amplifier are
introduced into the transmission line, (1) becomes
(10)
where
is the loss coefficient, is the amplifier gain,
is the amplifier spacing, and is the total number of
amplifiers. Since the loss and gain occur periodically in real
systems on a length scale that is short compared to the scale
on which the nonlinearity and average chromatic dispersion
operate, they will induce periodic oscillations in the amplitude
without changing the state of polarization. Since the Stokes
vector components also periodically oscillate as a result,
in (8) will be smaller than its initial value right after an
amplifier during most of the propagation, and
will
evolve more slowly than if
always had this initial value.
Defining
in the interval
, so that we normalize the amplitude
throughout the interval between amplifiers to its value at the
beginning of the interval, and defining
from
analogous to (4), we find that
(11)
Defining now,
where is a constant,
we conclude
(12)
which has the same form as (8). If we let
then varies continuously, and
(12) can be interpreted as an evolution equation-dependent
upon this new variable. The variation of
is somewhat
complicated by the exponential variation, but in most realistic
settings, the nonlinear evolution is slow compared to the
periodic evolution due to the fiber loss and lumped gain so
that it is sufficient to approximate (12)
(13)
We will validate this approximation in Section IV.
When we include the effect of polarization mode dispersion,
(1) becomes
(14)
where
indicate the inverse group velocity difference
between the fast and slow axes while
is a rapidly varying
unitary matrix that takes into account the changes in orienta-
tion of the birefringent axes [8]. We solve this equation using
the coarse step method [8].
III. B
ASIC NUMERICAL MODEL
In our simulations, we used standard split-step methods to
solve (1). In each channel, we used synchronous phase mod-
ulation and in some cases synchronous amplitude modulation
of nonreturn-to-zero (NRZ) signals, much as described by
Bergano et al. [3], to both polarization scramble the signals
and minimize pulse distortion. The functional form that we
used for the initial field at the entry to the fiber is
(15a)
(15b)
when the signal has no amplitude modulation and
(16a)
(16b)
when the signal is amplitude-modulated, where
and .
Here, we let
and , where
and are constant coefficients, while in the time
slots of the marks and
in the time slots of the spaces
except when making a transition from a space to a mark or
from a mark to a space. When making a transition from a
space to a mark, we set
,
where
is the boundary between the space and the mark and
ps for a 5 Gb/s signal and ps for a 10 Gb/s
signal. When making a transition from a mark to a space, we
2524 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 12, DECEMBER 1999
define . The coefficients
and give the strength of the phase modulation. For all the
simulations in this paper, we set
and so
that the difference
nearly equals to 2.405, the first zero
of the zeroth Bessel function. With this choice, setting
,
an ideal square pulse would be completely depolarized. In our
simulations, since the pulses are never perfectly square, even
without amplitude modulation, there is always a small residual
degree of polarization like in the experiments of Bergano et al.
The sum
was chosen consistent with the experiments
of Bergano et al. [3]. The phase modulation frequency
corresponds to the bit rate, and and describe the relative
phase between the phase modulation and the data bits. By
varying
, , , and , we can adjust the degree of
polarization to any desired value. We used a 64 bit pattern in
each channel, chosen so that the number of marks and spaces is
identical and so that strings of marks of varying sizes are in the
pattern. We experimented with a number of different strings
in several cases and verified that our results are insensitive to
this choice. We used 8192 node points for each channel in all
cases which experimentation showed was adequate.
We chose the system parameters as follows: the central
wavelength
1.58 m, the Kerr coefficient
m W, and a total fiber length km.
For our basic simulations, we used a dispersion map with
a map length of 1000 km like that of Bergano et al. [3],
but we also investigated the effect of shortening the map,
and we will report these results in Section IV. In every map
period, there is a span of length
that consists of standard
fiber with dispersion value
and a span of length
that consists of dispersion-shifted fiber with dispersion value
. We chose the lengths of these two portions so that
. At the central frequency m, we
choose the corresponding dispersion values so that they are in
the ratio 8.5:
1, and the net cumulative dispersion is zero.
Typical experimental systems that model undersea systems
would have a value of
ps/nm-km and
ps/nm-km, but we often varied these values significantly in our
simulations to compare our results to the mean field approach.
We note that values in terrestrial systems are often substantially
higher. When
, third-order dispersion implies that
the net dispersion is nonzero. In our simulations, we used
ps/nm
2
-km. In our basic simulations, we did
not use amplitude modulation, and the signaling rate is 5 Gb/s,
corresponding to a bit period of 200 ps. The channel spacing
is 0.5 nm.
In Fig. 1, we show a two-channel simulation. We note
that because
, and and are different for
the two channels, the two channels have slightly different
values of
and and hence slightly different degrees
of polarization. The degree-of-polarization is nonzero in this
case because
. Fig. 1(a) shows the theoretically
predicted result from (9), and Fig. 1(b) shows the evolution of
the Stokes parameters with standard values of the dispersion.
There are significant quantitative discrepancies between the
mean field theory and the simulation, and these discrepancies
only completely disappear when the local dispersion becomes
quite large as shown in Fig. 1(c). Nonetheless, the Stokes
(a)
(b)
(c)
Fig. 1. Evolution of the Stokes vector components as a function of distance
in a 5-Gb/s system. The dispersion map length is 1000 km, and the channel
spacing is 0.5 nm. The solid lines are the Stokes components of channel one;
the dashed lines are the Stokes components of channel two. (a) Analytical
result. (b) Simulation result,
D
1
=
0
2
ps/nm-km,
D
2
=17
ps/nm-km.
(c) Simulation result,
D
1
=
0
20
ps/nm-km,
D
2
= 170
ps/nm-km. Other
simulation parameters are
= 1550
nm for channel one,
= 1550
:
5
nm
for channel two;
x
=0
:
7
and
y
=0
for channel one,
x
=0
and
y
=0
:
7
for channel two; the peak power in the
x
-polarization is 0.24
mW for channel one and is 0.2 mW for channel two; the peak power in the
y
-polarization is 0.2 mW for channel one and 0.24 mW for channel two.
Fig. 2. Evolution of the degree of polarization as a function of distance with
D
1
=
0
2
ps/nm-km,
D
2
=
17 ps/nm-km. Other parameters are the same
as in Fig. 1.
parameters still oscillate around their initial values in Fig. 1(b),
although with somewhat different frequencies and amplitudes
than in Fig. 1(a). There are no long-term drifts in the Stokes
parameters from the theoretically-predicted values. Thus, we
would anticipate that there is little change in the degree of
polarization of the channels, and this expectation is borne out
as shown in Fig. 2 where we show the degree of polarization,
, for each channel over 10 000 km.
The change is only about 0.02. In particular, if the degree-
of-polarization is initially near zero for both channels, which
we obtain by setting
in this case, then they undergo
little repolarization as shown in Fig. 3.
