JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 22, NO. 4, APRIL 2004
1155
PMD-Induced Fluctuations of Bit-Error Rate in
Optical Fiber Systems
Vladimir Chernyak, Michael Chertkov, Ildar Gabitov, Igor Kolokolov, and Vladimir Lebedev
Invited Paper
Abstract—This paper presents a method that allows evaluating
the performance of an optical fiber system where bit errors result
from a complex interplay of spontaneous noise generated in optical
amplifiers and birefringent disorder of the transmission fiber. We
demonstrate that in the presence of temporal fluctuations of bire-
fringence characteristics, the bit-error rate (BER) itself is insuffi-
cient for characterizing system performance. Adequate character-
ization requires introducing the probability distribution function
(PDF) of the BER obtained by averaging over many realizations
of birefringent disorder. Our theoretical analysis shows that this
PDF has an extended tail indicating the importance of anomalously
large values of BER. We present the results of comprehensive anal-
ysis of the following issues: 1) The dependence of the PDF tail shape
on detection details, such as filtering and regular temporal shift ad-
justment; 2) the changes in the PDF of BER that occur when the
first- or higher order polarization mode dispersion (PMD) com-
pensation techniques are applied; 3) an alternative PMD compen-
sation method capable of providing more efficient suppression of
extreme outages.
Index Terms—Bit-error rate (BER), optical fiber telecommuni-
cation systems, polarization mode dispersion (PMD) compensa-
tion, probability distribution function (PDF).
I. INTRODUCTION
P
OLARIZATION mode dispersion (PMD) constitutes one
of the main limiting factors for reliable optical fiber system
performance at transmission rates of 40 Gb/s or higher. PMD
causes broadening of initially compact pulses in a data stream
that eventually leads to bit-pattern corruption [1]–[4]. This ef-
fect can be characterized in terms of the PMD vector [5]–[8],
[37]. It has been also recognized that the PMD vector does
Manuscript received June 1, 2003; revised September 12, 2003. This work
was supported by LDRD ER on “Statistical Physics of Fiber Optics Commu-
nications” at Los Alamos National Laboratory and by a personal grant as well
as under Grant 03-02-16 147a from the Russian Foundation for Basic Research
(IK).
V. Chernyak is with Corning Inc., SP-DV-02-8, Corning, NY 14831 USA
(e-mail: ChernyakV@corning.com).
M. Chertkov is with the Theoretical Division, Los Alamos National Labora-
tory (LANL), Los Alamos, NM 87545 USA.
I. Gabitov is with the Theoretical Division, Los Alamos National Laboratory
(LANL), Los Alamos, NM 87545 USA, and with the Department of Mathe-
matics, University of Arizona, Tucson, AZ 87521 USA, and with the Landau
Institute for Theoretical Physics, Moscow, 117334, Russia.
I. Kolokolov and V. Lebedev are with the Theoretical Division, Los Alamos
National Laboratory (LANL), Los Alamos, NM 87545 USA, and with the
Landau Institute for Theoretical Physics, Moscow, 117334, Russia.
Digital Object Identifier 10.1109/JLT.2004.825237
not provide a complete description of the PMD phenomenon
and some more sophisticated approaches that take into account
“higher order” PMD effects, have been recently discussed in the
literature [9]–[12].
It is well known from experiment that birefringence in optical
fiber systems is slowly, but substantially changing with time
under the influence of fluctuations in environmental conditions
(stresses, temperature, etc.), see, e.g., [13], [14]. Thus, dynam-
ical PMD compensation becames a major issue in modern high-
speed optical-fiber telecommunication technology [15], [16].
Development of new techniques capable of first- [15], [19],
[20] and higher order [20], [21] PMD compensation has raised
a question of how to evaluate the compensation success (or
failure). Traditionally, the statistics of the PMD vectors of first
[5], [6], [37], [8] and higher orders [9]–[11] are considered as
a measure for any particular compensation method’s perfor-
mance. However, these objects are only indirectly related to
fiber system reliability.
In this paper, that develops the ideas briefly described in a
series of recent publications [22]–[25], we clearly demonstrate
that PMD effects should be considered together with impair-
ments due to amplifier (and other types of) noise. Indeed, the
system performance for a given realization of birefringent dis-
order is characterized by a certain value of BER, i.e., proba-
bility of detecting an error, which is nonzero because of the
noise. However, the value of the bit-error rate (BER) is varying
together with the temporal variations of the birefringent dis-
order. The characteristic time scale of such variations is much
longer than the times related to signal transmission, however,
it can substantially exceed the overall system operation time.
Therefore, evaluating system performance should be based on
the analysis of fluctuations in the value of BER. We show that
fluctuations of BER caused by variations of the birefringent dis-
order are substantial. Large fluctuations of BER originate from
the very different nature (temporal correlations) of the ampli-
fied spontaneous emission (ASE) noise compared to that of the
birefringent disorder. Birefringent disorder is practically frozen
(i.e., it does not vary at least on the time scales related to the op-
tical signal propagation). Optical noise originating from ASE
constitutes an impairment of a different nature: the amplifier
noise is short correlated on the time scale of the signal width.
We demonstrate that the probability of extreme outages (i.e.,
such situations or, stated differently, realizations of birefringent
disorder when the BER substantially exceeds its typical value)
0733-8724/04$20.00 © 2004 IEEE
1156 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 22, NO. 4, APRIL 2004
is much larger than one could expect from naive estimates based
on singling out effects of either of these two impairments. This
phenomenon is a consequence of a complex interplay between
these two different impairments. It may not be rationalized in
terms of just an average value of BER, or statistics of any PMD
vectors of different orders. Complete description of this phe-
nomena requires studying the probability distribution function
(PDF) of BER, and specifically its tail.
The emergence of an extremely extended (algebraic or
algebraic-like) tail in the PDF of BER constitutes the major
result of the paper. The result is general, i.e. it applies to a
whole variety of transmission regimes (linear, nonlinear, and
quasi-linear), various signal modulation formats [return-to-zero
(RZ), nonreturn-to-zero (NRZ), differential phase-shift
keying (DPSK)-RZ, differential quadrature phase-shift keying
(DQPSK)-RZ, etc.], and detection techniques (optical and
electrical filtering, decision threshold choice, etc.). However,
for the sake of simplicity and clarity and also to be specific, we
consider the following situation of major practical interest in
optical fiber communications: 1) the modulation format is RZ
(on–off keying); 2) transmission is linear, i.e. Kerr, Raman, and
other nonlinear terms are not taken into account. Besides (and
less importantly), the other two model assumptions (which
do not restrict the generality of the model independent results
reported in the paper) are that: a) the electrical filter (window)
is represented by a symmetric step function; b) the optical filter
is of Lorentzian shape.
After brief technical introduction into the problem given in
Section II, our theoretical analysis starts in Section III with eval-
uating the signal BER due to the amplifier noise for a given real-
ization of birefringent disorder. We next study the PDF of BER,
where the statistics are collected over different fibers or, equiv-
alently, over the birefringence states of a given fiber at different
times. At the second step, we focus on evaluating the proba-
bility of anomalously large BER. This general scheme will be
applied to four situations of interest. We start with the basic (no
compensation) case in Section IV-A and compare it with the
case of the simplest compensation scheme known as “setting
the clock” in Section IV-B, and also with the cases of first- and
higher order PMD compensations schemes in Sections V-A and
V-B, respectively. Finally, in Section V-C, we discuss a com-
pensation scheme referred to as (quasi)-periodic that appears to
be more efficient in reducing the extreme outages compared to
the traditional high-order compensation scheme with the same
number of compensating degrees of freedom. Section VI is re-
served for discussions and conclusions.
II. T
ECHNICAL INTRODUCTION
In this section, we introduce the basic relations and termi-
nology that describe data transmission (signal propagation) in
an optical fiber system. The goal here is to set the problem in
formal terms, introduce the objects of interest, and also make
some preliminary evaluations.
A. Amplifier Noise and Birefringent Disorder
The envelope of the optical field propagating in a given
channel in the linear regime (i.e., at relatively low optical
power), which is subject to PMD distortion and amplifier noise,
satisfies the following equation [26]–[28]:
(1)
Here
, , , as well as , , and are the position along the fiber,
retarded time (i.e., time associated with the reference frame
moving with group velocity of a chosen frequency channel), the
amplifier noise, and the chromatic dispersion, attenuation, and
gain coefficients, respectively. (We assume that neither gain nor
damping are polarization dependent, leaving the more general
problem for future investigation.) The envelope
is a two-com-
ponent complex field, the two components represent two states
of the optical signal polarization. Our approach allows us to
treat discrete (erbium) and distributed (Raman) amplification
schemes within the same framework. The birefringent disorder
is characterized by two random
traceless matrix fields
related to the zero-,
, and first-, , orders in the frequency ex-
pansion with respect to the deviation from the channel carrier
frequency
. Birefringence that affects the light polarization
is practically frozen (
-independent) on all propagation-related
time scales.
The matrix
as well as the attenuation and gain coefficients
and can be excluded from consideration by the following
transformation
, and . Here
is a -dependent number, and the unitary matrix
is the ordered exponential defined as a formal solution of the
equation
with . We assume that the
gain coefficient is properly chosen to perfectly compensate for
damping, so that
, where is the total system length.
The renormalized quantity
satisfies the equation
(2)
The solution of (2) can be represented as
(3)
(4)
(5)
(6)
where
stands for the initial pulse shape and denotes
the so-called ordered exponential operator
that can be formally described as the solution of the operator
equation
with the initial condition
(note that when the operators commute for all the ordered
exponential coincides with the usual one). Solving the operator
CHERNYAK et al.: PMD-INDUCED FLUCTUATIONS OF BER IN OPTICAL FIBER SYSTEMS 1157
equation iteratively leads to a very useful representation of the
ordered exponential in a form of a functional infinite series
that will be used in this manuscript for performing perturbative
computations.
We consider a situation when the pulse propagation distance
substantially exceeds the inter-amplifier separation (the system
consists of a large number of spans). The additive noise
, gen-
erated by optical amplifiers, is zero on average. The statistics of
are Gaussian with spectral properties determined solely by the
steady-state features of amplifiers (gain and noise figure) [30].
The noise correlation time is much shorter than the pulse tem-
poral width, and, therefore,
can be treated as -correlated in
time. Equations (4) and (5) imply that the noise contribution to
the output signal
is a zero mean Gaussian field characterized
by the following pair correlation function:
(7)
with the product
being the ASE spectral density accumu-
lated along the system. The coefficient
is introduced into
(7) to reveal the linear growth of the ASE factor with the total
line length
[30]. Provided the noise is short-correlated in
space (that is, correct for both erbium and Raman amplifiers),
the factor
in (7) is statistically independent of both and
, as immediately follows from (4)–(6).
The matrix of birefringence
can be parameterized using a
three-component real field
, , with being a
set of three Pauli matrices. The field
is zero on average and
short-correlated in
. Therefore, in accordance with the central
limit theorem (see, e.g., [31]) the integral
has
Gaussian statistics (with zero average) characterized by the pair
correlation function
(8)
where the average is taken over the birefringent disorder realiza-
tions (corresponding to different fibers or, equivalently, states of
birefringence in a single fiber at different times). The isotropy of
the pair correlation function (8) is guaranteed by the above trans-
formation
since the presence of leads to fast
rotations of the vector
along . In the case of weak birefringent
disorder
represents the PMD vector. Thus, ,
with
being the so-called PMD coefficient that is usually mea-
sured in units of picoseconds per square root of a kilometer
(ps
km) and has the following meaning. In a system of length
short enough so that effects of PMD are typically weak,
represents a typical time splitting between the two principle po-
larization components of a pulse accumulated along the system.
The factor of
is obtained in the following way: is twice
the typical value of the differential group delay (DGD) vector
resulting in
to be four times the typical (defined as the av-
erage) value of its square, the latter being naturally given by
. As we will see later, some observables contain the field
in a more sophisticated form than just the integral . Statis-
tical properties of these more sophisticated objects can be estab-
lished by using the relation
(9)
instead of (8).
B. BER as a Functional of Birefringent Disorder
We consider the RZ modulation format when the pulses are
well separated in time. The signal detection at the line output,
, corresponds to measuring the output pulse intensity
(10)
where
is a convolution of the electrical (current) filter func-
tion with the sampling window function. The linear operator
in (12) stands for an optical filter and a variety of engineering
“tricks” applied to the output signal
. These tricks con-
sist of the optical filter
, and the compensation parts, re-
spectively, assuming the compensation is applied first followed
by filtering, i.e.,
(11)
We can replace
by in (10) since and is a
unitary matrix. Upon substituting the representation (3) into (10)
we obtain
(12)
Compensating options, coded in specific form(s) of the operator
, are discussed in Sections II-C and II-D. Filtering options,
formalized by specific choices of the function
and oper-
ator
are described in Appendix I, where we also discuss the
specific form of the initial pulse
used for the modeling anal-
ysis.
Ideally,
takes two distinct values corresponding to the bits
“
” and “ ,” respectively. However, the impairments enforce
deviations of
from the ideal values. The output signal (bit of
information) is identified by introducing a threshold (decision
level)
and declaring that the signal encodes “ ” if
and “ ” otherwise. Sometimes, the information is lost, i.e., an
initial “
” is detected as “ ” at the output or vice versa. BER
is the probability to detect a false event measured by counting
many pulses coming through a fiber with a given realization of
the PMD (birefringent) disorder
. For successful system
performance BER should be extremely small, i.e., typically both
impairments (noise and disorder driven) can cause only a small
distortion of a pulse or, stated differently, the optical signal-to-
noise ratio (OSNR) and the ratio of the squared pulsewidth to
the mean squared value of the PMD vector are both large. OSNR
can be estimated as
where, according to (10)
is the intensity of the unperturbed signal, being the input
signal normalized to one. Therefore, the two small parameters
of our theory are represented by
(13)
(14)
1158 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 22, NO. 4, APRIL 2004
being the pulsewidth, and the condition (14) is assumed to hold
for all cases considered in the paper except for the one discussed
in Section V-C.
1
We distinguish events associated with the
transition
(loss of the signal), and with the
transition (false pulse de-
tection) and designate the corresponding probabilities as
and . These two objects are defined as
(15)
and according to (12), the PDF
of the output signal intensity
which can be written as
(16)
where
is the PDF of , is normalization constant, and
is an arbitrary functional of . Also the subscript in the defi-
nition of
corresponds to the “zero” input bit , while
the subscript
corresponds to the “one” input bit .In
(16), averaging is performed over the noise statistics. Experi-
mentally, such an average is measured by collecting the statistics
over many pulses propagating along a fiber with the same bire-
fringent disorder realization
since different pulses experi-
ence different realizations of the noise the latter being stochastic
in nature. Formally, this constitutes averaging an observable that
can be represented by any functional
over all possible re-
alizations of the noise
with the probability distribution ,
whereas a specific form of
also given in (16) corresponds
to the situation under consideration, i.e., Gaussian statistics of
the noise fully determined by the pair correlation function of (7).
Since realizations of noise are represented by functions
of
time that represent the field
at . (16) constitutes a
path (functional)-integral representation for
(see, e.g., [32],
[33] for an introduction to path-integral techniques). This, how-
ever, does not constitute a major conceptual problem since path
integrals can (and strictly speaking should) be considered as fi-
nite-dimensional integrals where functions are represented by
sets of values at a large but finite set of points
.
An important difference between
and , defined
by (15), is in the strong dependence of the first case and inde-
pendence of the second one on the
realization (i.e., the bire-
fringence profile along the fiber). This difference stems from
the fact that
in the case of whereas is a non-
trivial functional of
, as well as from statistical independence
of
and . One concludes that even though and are
of the same order in the absence of birefringence, anomalously
large values of BER (which is the focus of this paper) originate
solely from the “
” transitions. Therefore, in what follows
1
Section V-C is devoted to a special case where without compensation,
the condition (14) is essentially violated, however, a weaker condition,
D Z=b
1
=N
still holds. It is shown then that by using a new quasi-peri-
odic compensation strategy of
N
th order one can still get an operable system,
i.e., a system with typical BER essentially smaller than one.
we concentrate primarily on the analysis of
thus dropping
the
subscript to simplify notation.
The PDF
has a maximum near and decays
quickly as
departs from . The tail of the transition proba-
bility at
is exponential (see Appendix II for
details). This implies that the integration in the right-hand side
(RHS) of (15) is actually concentrated near
thus yielding
the following estimate for BER,
:
(17)
C. “Setting the Clock” Compensation
An essential part of the signal loss can be compensated using
a simple procedure, in the fiber-optics jargon usually called
“setting the clock.” This procedure accounts for adjusting the
overall time shift which is a functional of the birefringent dis-
order. (We are not discussing here an important engineering
problem of how to make this dynamical adjustment, simply as-
suming that a device capable of doing this operation does exist.)
Formally, the “setting the clock” procedure can be described by
the following modification of (12):
(18)
or returning to the notation of (11), (18) corresponds to the fol-
lowing form of the compensation operator:
.
As discussed in Section IV-B, the one-parameter flexibility one
gains through
can be used to minimize system outage. The
important question to be addressed is: What is dependence of
the “optimal” shift on the birefringent disorder?
D. PMD Compensator
Effects of PMD can be reduced by using a device usually
called a PMD compensator (PMDC). Any optical PMDC con-
sists of two parts: a compensating (optical) part responsible for
the compensation itself, and a measuring part that extracts (mea-
sures) relevant information on the transmission fiber birefrin-
gence. We start by considering the optical part of the compen-
sator that usually consists of a set of relatively short elements.
Each element includes a piece of polarization-maintaining fiber
(this is a fiber characterized by uniform, i.e., position-indepen-
dent, birefringence vector) usually surrounded by two polariza-
tion controllers that allow rotation of the polarization state [15] .
This implies that the optical part of a PMDC (hereafter referred
to as a PMDC itself when it does not lead to confusion) is char-
acterized by its transfer function that can be parameterized by a
finite number of parameters (degrees of freedom). Additionally,
one would naturally distinguish between i) describing a com-
pensator in terms of available transfer functions (the subject of
this subsection), and ii) compensating strategy, i.e., a prescrip-
tion of how to fix the compensating degrees of freedom based
on the measured data. The compensating strategy part of the
problem is discussed in Sections V-A, V-B, and V-C.
The so-called, first-order PMDC corresponds to
(19)
with
. Such a form of the compensating operator
offers richer adjustment options compared to the “setting the
CHERNYAK et al.: PMD-INDUCED FLUCTUATIONS OF BER IN OPTICAL FIBER SYSTEMS 1159
clock” compensation as it actually contains three compensating
degrees of freedom, i.e., the three components of the compen-
sating vector
, instead of one.
2
Note also, that the transfer
matrix
of the transmission fiber is defined as an ordered ex-
ponential (6), whereas the compensating operator
is defined
in terms of the usual exponential (19). This important difference
stems from the fact that the birefringence profile along the trans-
mission fiber
is a random function of , while the birefrin-
gence of the compensating part is flat, as it is accurately con-
trolled to be
-independent, with being the position marker
along the polarization maintaining piece.
A compensation strategy that allows for more compensating
options (more degrees of freedom) is potentially better. Thus, a
compensator, hereafter referred to as an
th-order PMD com-
pensator consists of
concatenated PMD compensators of the
first order [15]. Each of the
compensators is characterized
by its own three-component compensating vector
, where
, so that the compensating operator generalizes
that of (19)
(20)
and the set of vectors introduces compensating degrees
of freedom that are at our disposal for outage optimization.
Once the set of compensation options, described by (19), (20)
is fixed, the next task, addressed in Sections V-A, V-B, and
V-C is about how to use the compensating degrees of freedom
offered by the compensators (19) or (20) to minimize the effects
of the system outages. Or rephrasing the question in more formal
terms: What are the optimal values of the
compensating
degrees of freedom
that correspond to a given realization
of the transmission fiber birefringence profile?
III. A
MPLIFIER NOISE AVERAGING
In this short section, we present only the basic results, while
all derivations can be found in the Appendixes.
Since the OSNR is large, the expression for the transition
probability (16) allows for an asymptotic saddle-point evalua-
tion. The details of an analytical calculation, resulting in an alge-
braic system of equations that implicitly relate the saddle-point
value of the transition probability to the inhomogeneous part of
the measured signal are given in Appendix II.
2
A three-parameter compensator with the transfer function given by (19) can
be implemented by surrounding a polarization-sensitive delay line with two
polarization controllers (PC). The transfer function of such a device has the
form
K
=
U
K
(
t
)
U
where
U
and
U
are frequency independent of
the PC located after and before the delay line, respectively, and
K
(
t
)=
exp(
0
t
^
@
)
is the delay-line transfer function with
t
being the relative time
delay. If the time delay
t
is controlled, such a compensator provides the transfer
function of (19) since with a proper choice of
U
provided by the PC we can
obtain any vector
MM
M
with
j
MM
M
j
=
t
. A possible implementation of a delay line
with adjustable
t
involves a free-space optical–mechanical device that achieves
a relative time delay by varying the relative value of the optical paths for two
polarization states. Such a device naturally offers three-parametric compensa-
tion, as it is capable of dynamically generating any value of the three-compo-
nent vector
MM
M
. A less expensive and maybe more practical option is to use a
piece of polarization-maintaining fiber instead of a complicated free-space op-
tical–mechanical device. In this case, only two parameters (the components of
MM
M
) are dynamically adjusted with the value of
j
MM
M
j
=
t
being determined
by the polarization-maintaining fiber length. The dynamical adjustment is only
two-parametric, as it originates from changes in the polarization controllers ori-
entation, while the birefringence of the polarization-maintaining fiber is fixed.
Fig. 1. Dependence of the dimensionless coefficients
0 =
0
D Z
ln
B =I
,
,
,
=
, and
, entering (21), (26), (28), (33), and (34), on the electric
filter width
T
, and the optical filter width
(both measured in units of the
pulsewidth
b
) for the model introduced in Appendix I. Details of calculations
resulted in the dependencies shown in the figure are explained in Appendix II.
The bottom line of these calculations, accounting for aver-
aging with respect to the stochastic noise (i.e., many pulses) in
(15), (16) is the saddle-point [i.e. asymptotic, applied whenever
the condition (13) holds] expression for the loss probability
that is obtained (see Appendixes I and II for the details) by
first representing the
-function in (16) as a Fourier trans-
form of an imaginary exponent [see, e.g., (46)] which yields
as an integral over , with the integrand in a form
. Note that depends parametrically on
through [see (3), (6), and (16)]. Evaluating the path integral
(16) using the saddle-point approximation (see, e.g., [32], [33],
for the general description of the method) and making use of
(17) we obtain with the exponential accuracy
(21)
where
is the saddle-point value of the
action
. In particular, corresponds to zero PMD, ,
value of
and is a dimensionless quantity with a smooth de-
pendence on
. By definition, tends to with . The
quantity
, that determines a typical value of BER, is a dimen-
sionless parameter of order unity. The dependence of
on the
electric filter width
and the optical filter width calculated
numerically for the model introduced in Appendix I is displayed
in Fig. 1.
The dependence of
on the birefringence profile is the
key subject of the analysis presented in Sections IV and V.
IV. PDF
OF BER AND EXTREME OUTAGES
This section constitutes the core of the paper. The bottom line
here is that fluctuations of BER from one realization of birefrin-
gence to another are strong. To demonstrate that, we study the
extended (toward larger values of BER,
) tail of the
PDF (histogram) of BER,
.
