236
Journal of Optical Communications
22 (2001) 6
Summary
The probability density function of four-wave-mixing
crosstalk in wavelength-division-multiplexing systems is
derived analytically in a closed-form formula. For dis-
persion shifted fiber the plots were given for different
number of channels, different channel indexes and dif-
ferent channel spacings. The comparison with Gaussian
approximation is given, and the limits of its application
are pointed out.
1 Introduction
Optical networks and long-haul communications using
wavelength-division-multiplexing (WDM) systems have
experienced the rapid researching attention in recent
years [1–6]. For the number of channels, the channel
spacings and the bit rates we are interested today the
fiber nonlinearities becomes unavoidable factor in per-
formance degradation. The most important fiber non-
linearities in WDM environment, four-wave mixing
(FWM), cross-phase modulation (XPM) and stimulated
Raman scattering (SRS) were considered in a number of
papers [2–6]. The probability density function of SRS
crosstalk was found to be lognormal [3], and parameters
of this distribution were also derived. The statistical
properties of XPM induced phase noise was considered
in [4]. Unfortunately, the probability density function
(PDF) of FWM crosstalk has not derived yet, in an ana-
lytical closed form.
In this paper the PDF of FWM crosstalk in WDM sys-
tems is derived analytically in a closed-form. For disper-
sion shifted fiber (DSF) the plots are given for different
number of channels, different channel indexes and dif-
ferent channel spacings. The comparison with Gaussian
approximation is given, and the limits of its application
are pointed out.
2 PDF of FWM crosstalk derivation
The photodiode input signal in the presence of FWM for
IM/DD systems can be written in the form
Et a P t t
P t t
S n n
ijk n ijk
n
() = + () [ ]
+ + ()
[ ]
∑
cos
cos '
,, ,,
ω φ
ω φ
A
, (1)
where P
s,
w
n
and f
n
are the observed channel power,
frequency and phase respectively; P
i,j,k
, w
n
and f '
i,j,k
are
the mixing product peak power, frequency and phase.
Those mixing products having the same frequency f
n
=
w
n
/2p = f
i
+ f
j
– f
k
, (k π i,j) as observed channel are taken
into consideration. A
n
= {(i,j,k)
Í
i,j,k = 1, 2, …, N; k π i,j ;
k = i + j – n} is the set of all triples of channel numbers
that generate a mixing product of observed channel n,
and N is the number of channels. With a Π{0,1} is
denoted information content. d
i,j,k
= 3 for two tone prod-
ucts (i = j) and 6 for three tone products. The peak power
of the mixing product of frequency f
n
can be written as
follows
P p
d
L PPPe
ijk ij
ijk
e i j k
L
ijk
=
−
3
2
γ η
α
, (2)
where g = 2p n
2
/(lA
e
) (with n
2
being the intensity-de-
pendent refractive index, A
e
the effective cross-sectional
area, and l being the wavelength) is the nonlinear coef-
ficient, h
ijk
is the FWM efficiency defined in [1, p. 245],
L
e
the effective length of fiber with attenuation coef-
ficient a and length L, and P
i
, P
j
and P
k
are the input
peak powers of corresponding channels. The factor p
ij
comes from the fact that FWM light is generated when
two (i = j) or three channels (i π j) creating the crosstalk
are simultaneously in “mark” condition, probability of
which is (1/2)
2
= 1/4 for i = j that is (1/2)
3
= 1/8 for
i π j. Under assumption that all input peak powers are
the same P
i
= P, (i = 1, 2, …, N) the photodiode input
power, in the absence of FWM, can be written as P
S
=
Pe
- aL
.
Probability Density Function of Four Wave Mixing Crosstalk
in WDM Systems
Ivan B. Djordjevic, Alexandros Stavdas
Address of author:
National Technical University of Athens
Department of Electrical & Computer Engineering
Telecommunications Laboratory
Heroon Polytechniou 9,
157 73 Zographou, Athens, Greece
E-mail: icadj@yahoo.com
Received 12 February 2001
J. Opt. Commun. 22 (2001) 6, 236–238
© by Fachverlag Schiele & Schön 2001