evaluation of ber degradation and power limits in wdm networks...
TRANSCRIPT
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Evaluation of BER degradation and power limits in WDM
networks due to Four-Wave Mixing by Monte-Carlo
simulations
John Neokosmidis, Thomas Kamalakis, Aris Chipouras and Thomas Sphicopoulos
Department of Informatics and Telecommunications, University of Athens
Panepistimiopolis Ilissia, Athens, Greece, GR-15784
Abstract: Fiber non-linearities can degrade the performance of a Wavelength Division
Multiplexing optical network. For high input power, low chromatic dispersion coefficient
or low channel spacing the severest penalties are due to Four Wave Mixing (FWM). To
compute the Bit Error Rate (BER) due to the FWM noise, the probability density function
(pdf) of both the space and mark states must be accurately evaluated. In this paper, an
accurate evaluation of the pdf of the FWM noise in the space state is given for the first
time using Monte-Carlo (MC) simulations. Additionally, it is shown that the pdf in the
mark state is not symmetric as assumed in previous studies. Diagrams are presented which
allow the estimation of the pdf given the number of channels in the system. The accuracy
of the previous models is also investigated and finally the results of this study are used to
estimate the power limits of a WDM system.
©2003 Optical Society of America
OCIS codes: (060.4230) Multiplexing; (190.4380) Nonlinear optics, four-wave mixing
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I. INTRODUCTION
Wavelength Division Multiplexing (WDM) is a promising technology for the realization of
all-optical networks, allowing the full utilization of the fiber bandwidth. However, the
performance of a WDM system is strongly influenced by both linear and non-linear phenomena
determining the signal propagation inside the optical fiber. Although linear propagation effects
may be compensated, using optical amplifiers and chromatic dispersion compensators, there is a
class of non-linear effects, such as Self Phase Modulation (SPM), Cross Phase Modulation
(XPM) and Four Wave Mixing (FWM) that pose additional limitations in dense WDM systems.
Both XPM and FWM cause interference between the different wavelength channels resulting in
an upper power limit for each WDM channel. However, the severest problems are imposed by
FWM since the power of the FWM product is inversely proportional to the square of the channel
spacing while the influence of the XPM is approximately inversely proportional to the channel
spacing [1]-[2].
In this paper an accurate statistical description of the FWM noise will be given and its
impact on the system performance will be investigated. The properties of the FWM noise are
investigated using numerical Monte-Carlo simulations both for the mark and the space state. The
results show that in the case of the space state the pdf of the FWM-FWM beating noise, which
constitutes the main noise source of the system, exhibits an exponential decay. The computed
pdf for the mark state is shown to be asymmetric, a fact which is confirmed theoretically as well.
Estimations of the pdf for the mark and the space states is accomplished by relevant diagrams in
terms of the number of channels within the system.
A comparison is also carried out between the values of the error probabilities obtained by
the present method and the two other methods previously proposed [3]-[4]. The first one assumes
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a Gaussian statistics for the FWM noise in the optical domain, allowing the computation of the
system’s Bit Error Rate (BER) in closed form [3]. Although this might provide a first insight on
the implications of the FWM noise, its validity must be examined since the assumption of
Gaussian statistics is somewhat arbitrary. Indeed, the FWM noise is a sum of a large number of
components, dependent on each other. Hence, the central limit theorem [5], on which the
Gaussian approximation is based, may not be valid in this case. In the other approach, the
probability density function (pdf) of the FWM noise in the mark state can be approximated with
a symmetrical double-sided exponential distribution [4]. However, as mentioned before the
distribution of the FWM noise in the mark state is asymmetric and hence the symmetrical
exponential approximation may prove inaccurate. It is shown that in many cases, there are
significant differences between the two approaches. Finally, the proposed model is used to assess
the implications of the FWM in a WDM system employing nonzero dispersion or standard single
mode fibers as well.
The rest of the paper is organized as follows: In section II.A, some basic considerations
are given concerning the origin of the FWM phenomenon which will be used in section II.B to
derive an expression for the photocurrent at the receiver. This expression will be used in section
III in order to compute numerically the pdfs of the FWM noise in the mark and space states. The
asymmetry of the pdf in the mark state, observed by the simulations, is also theoretically
justified. The accuracy of the symmetrical double-sided pdf is investigated in section IV.A while
the validity of the Gaussian approximation is examined in section IV.B. The implications of the
FWM effect in a WDM system is discussed in section V. In section VI, guidelines for the
incorporation of the other noises (thermal and Amplified Spontaneous Emission – ASE) are
given. The work is concluded in section VII.
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II. BASIC ASSUMPTIONS AND THEORETICAL BACKGROUND
A) Four Wave Mixing
In this work, a WDM system with equally spaced channels and ASK modulation is considered
which is the most frequently used modulation scheme. All signals are assumed co-polarized and
synchronized, which represents a worst-case scenario [6]. Only the pdf of the photocurrent for
the central channel has to be derived for the calculation of the BER, since, in this case the energy
conservation requirement is satisfied by the largest number of frequency combinations [7].
Provided that the photocurrent depends on the bit values and the optical phases of all channels in
a rather complicated random manner, a closed form of the pdf of the FWM noise is not possible.
Consequently, Monte-Carlo (MC) simulations are used for its determination.
The origin of the FWM effect is the existence of the third-order non-linear polarization
vector WNL. Considering optical waves oscillating at frequencies ωi and linearly polarized along
the same axis x, WNL is expressed in the following form:
( )[ ] ccexccex iii ji
tzkji +=+= ∑∑ − θω WW
21ˆ
21ˆNLW (1)
where Wi is the amplitude of the third order polarization at the frequency ωi. For i=n, where n is
the assumed channel, it can be shown that [8]:
( )[ ]( ) ....
23
243
43
)3(
222)3(2)3(
++
++++=
∑
∑−−+∗
pqr
jrqpxxxx
o
pqrnrqpxxxx
onnxxxx
on
nrqpeEEE
EEEEEE
θθθθχε
χεχεW
(2)
where Ei is the electric field at frequency ωi, εo is the vacuum permittivity and )3(xxxxχ is the third-
order nonlinear susceptibility. The third sum is the contribution of the FWM noise. This sum
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runs for all integers p, q, r that satisfy the conditions p+q-r=n (which is imposed by the energy
conservation requirement) and r≠p,q (which guarantees that the corresponding term is not due to
SPM or XPM).
The output power Ppqr of the FWM product is given by [9]:
ηγ 2-22
9 effaL
rqppqrpqr LePPPdP = (3)
where Pi (i = p, q, r) represents the input peak power at the frequencies fi=ωi/2π in the mark
state. Assuming a perfect extinction ratio, the average input power is Pav=Pi/2. It should be noted
that equation (3) is an approximation that holds since the power of the FWM components is very
small compared with each channel’s power [10]. In a WDM system it can be assumed that all the
peak powers at the mark state are equal (Pi=Pin for i=1,2,…,N). In (3) γ is the nonlinear
coefficient of the fiber [9], a is the fiber loss coefficient, L is the total fiber length,
( ) aeL aLeff /-1 -= is the effective length of the fiber, dpqr is the degeneracy factor (dpqr=3 when
p=q, dpqr=6 when p≠q) and η is the mixing efficiency given by:
[ ] ⎭⎬⎫
⎩⎨⎧
−
Δ+
Δ+=
−
−
2
2
22
2
1
)2/(sin41)( aL
aL
e
Lea
a ββ
η (4a)
In (4a), Δβ represents the phase mismatch and may be expressed in terms of the channel
frequencies fi :
( )( ) ( ) ( ) ( ) ( )( )
( )( ) ( ) ( ) ( ) ( )( )⎭⎬⎫
⎩⎨⎧
−+−Δ⎟⎟⎠
⎞⎜⎜⎝
⎛+−−Δ=
⎭⎬⎫
⎩⎨⎧
−+−⎟⎟⎠
⎞⎜⎜⎝
⎛+−−=Δ
oqopfcd
dDDrqrpf
c
ffffcd
dDDffff
c
o
oqopo
orqrp
22
22
22
2
22
λλλ
λπλ
λλλ
λπλβ
ο
(4b)
or approximately,
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( )( )rqrpfc
D−−Δ≈Δ 2
22πλβ (4c)
In (4b), D is the fiber chromatic dispersion coefficient, λ is the wavelength of the signal
and c is the speed of light in vacuum. Equation (4c) is derived using the fact that for typical
values of D, dD/dλ and Δf, the second term in the brackets is much smaller than the first one [4].
Contrary to previous studies, the statistics of the space state is also considered. The
amplitude of the optical fields E(m) and E(s), in the mark and the space state respectively, at a
given channel n is written as [3]:
)(]exp[]exp[ )()(
)()(
)( markjPjePE mIMF
mIMFn
aLn
m θθ += − (5a)
)(]exp[ )()(
)()(
)( spacejPE sIMF
sIMF
s θ= (5b)
where Pn and θn are the input peak power and the phase in the mark state, respectively, of the
given channel n and
∑∑∑≠==≠≠≠≠≠
++=rqp
jpprrp
nrqp
jpqnqp
nrqp
jpqrrqp
mIMF
mIMF
pprpqnpqr ePBBePBBePBBBjP θθθθ ]exp[ )()(
)()(
(6a)
∑∑≠=≠≠≠
+=rqp
jpprrp
nrqp
jpqrrqp
sIMF
sIMF
pqrpqr ePBBePBBBjP θθθ ]exp[ )()(
)()( (6b)
while Ppqr and θpqr=θp+θq-θr are the peak power and the phase of the FWM noise generated from
a channel combination (p, q, r). Furthermore, Bi=0 or Bi=1 is the bit value of channel i. These
expressions for the electric field will be used in the next section in order to derive an expression
for the photocurrent at the receiving photodiode.
B) Calculation of the Photocurrent
At the receiver, the photocurrent is proportional to the optical power and hence to |E|2 where
E=E(m) or E=E(s) [11]. In practical applications, it can be assumed that Δβ>>a which generally
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holds for D≥2ps/nm/Km and channel spacing Δf≥10GHz. For large L one can also use the fact
that exp(-aL)<<1. Assuming a single fiber span without optical amplification, all other noises at
the receiver except FWM can be ignored. This is especially true for high input powers and in this
case the photocurrent at the detector is written as:
maL
naL
nmm IePkekPEkS −− +≈= δ2
2)()( (7a)
sss IkEkS 22)()( δ≈= (7b)
where k is the receiver responsivity and
2/22
23
2aL
in ePfD
c −
Δ=
πλγδ (8a)
( )∑ −−−
=pqr
npqrpqr
rqpm nqnpd
BBBI θθcos31
(8b)
22
sin31cos
31
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−= ∑∑
≠≠ nrpqr
pqrpqr
rqp
nrpqr
pqrpqr
rqps nqnpd
BBBnqnp
dBBBI θθ (8c)
Equations (7a) and (7b) provide an expression for the photocurrent in the mark and space
state in terms of two new variables Im and Is given by (8b) and (8c) respectively. It is interesting
to note that for a given number of channels, these new variables depend only on the bits and the
phases of the optical signals.
III. STATISTICAL BEHAVIOR OF THE PHOTOCURRENTS S(m) AND S(s)
Computation of the pdf by using Monte Carlo (MC) Simulation
In this section the pdf of Im and Is will be computed using Monte Carlo (MC) simulations. The
optical phases of all channels are assumed to be uniformly distributed within [0, 2π], due to
phase noise [11], and the data bits are assumed to be in the mark and space state with equal
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probability, P(Bi=0)=P(Bi=1)=1/2. Hence, the statistics of Im and Is will depend only on the total
number of channels N and the channel number n, which is assumed to be the central channel
n= ⎡ ⎤2/N . The variables Im and Is are related to the optical phases and the values of the optical
bits in a rather complicated manner and consequently, their pdf can not be derived in closed
form. In order to obtain the pdf of Im and Is, through which the pdf of S(m) and S(s) will be
determined, a series of v Monte Carlo (MC) experiments were performed. v=1011 for the cases
N=8, 16 channels and v=1010 for the case N=32 channels (due to the increased computation time
required). The obtained results, both for the space and mark states, are plotted in Figures 1(a) and
1(b).
It is easily deduced that in all cases the pdf exhibits an almost exponential behavior of the
form y = Aebx (solid lines in Figures 1(a)-(b)). Specifically, the pdf in the space state follows an
exponential decay while in the mark state the pdf can be approximated by an asymmetrical
double-sided exponential distribution. Hence, the exponential approximation can be used away
from Is,m=0, in the region of the tails in order to provide an accurate estimation for the pdf. In
order to evaluate the parameters A and b with respect to N, a series of Monte Carlo (MC)
experiments were performed for various values of N<32. The values of the parameters A and b
calculated for each N are shown in Figures 2(a)-(d). To reduce the number of diagrams, the
parameter A is taken equal to the peak of the pdf of Im and is the same for Im>0 and Im<0. Given
this value of A, the value of b for Im>0 is calculated so that the exponential Aebx approximates the
right tails of the pdf. The same is done for Im<0. In the case of the pdf of Is the parameters A and
b are calculated so that the tail of the pdf of Is is accurately approximated. As shown in the
figures 2(a)-(d), A and b approximately exhibit a yo+y1e-N/t dependence. The values of y0, y1 and t
in each case are given in the figures. Given the number of channels N, Figures 2(a)-(d) are useful
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in determining A and b for the mark and the space states, and consequently the shape of the pdfs
of Im and Is without having to perform MC simulations. For N>32, the parameters A and b do not
vary significantly and hence their values obtained for N=32 can be used.
Once the pdf of Im and Is is determined, the pdf of S(m) and S(s) can also be determined,
using the theorem of transformation of random variables [5]. Applying this theorem and using
(7a) and (7b):
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −=
−
−
− aLn
aLn
m
IaLn
mS ePk
ekPSfePk
Sfmm
δδ 221)(
)()(
)( (9a)
⎟⎟⎠
⎞⎜⎜⎝
⎛= 2
)(
2)( 1)()( δδ k
Sfk
Sfs
Is
S ss (9b)
where fX is the pdf of the variables X=Im, Is, S(m) and S(s) respectively. Hence, given the number of
channels N in the system the pdf of Im and Is can be determined using Figures 2(a)-(d). The
computation of the pdf of S(m) and S(s) can then be carried out using equations (7a)-(7b). These
pdfs will be used in the following sections in order to estimate the performance of the system.
From figure (1b), it can be seen that the pdf for the mark state is not symmetric around
x=0. In order to justify the asymmetry of the pdf, the odd order moments
∫+∞
∞−
++ = dxxfxImI
uum )(1212 , of Im can be examined. If one of the odd order moments is nonzero,
then the pdf )(xfmI is not symmetric. Indeed if )(xf
mI were symmetric around x=0, then the
product of the odd function x2u+1 with mIf would be an odd function of x for every integer u, and
hence, the integral of x2u+1mIf from –∞ to ∞, which gives 12 +u
mI would be equal to zero.
Therefore in order to show that mIf is not symmetric, it is sufficient to show that there exists one
u for which 012 ≠+umI .
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As it can be seen from the Appendix, it is easy to expand the terms of the odd order
moments into a positive linear superposition of terms of type cosθ, whose mean value is either 1
if θ≡0 or 0 if θ≠0. Following the above method, one can obtain 0=mI for u=0 and 03 >mI for
u=1. Hence, at least one odd moment of Im is non-zero and it can be concluded that the pdf of Ιm
is asymmetric. Since both the symmetric exponential and the Gaussian pdf are even functions,
they may not provide an accurate description for the statistical behavior of Im. This will be
further illustrated in the next sections.
IV. COMPARISON WITH OTHER MODELS
A) The Symmetrical Exponential pdf Approximation
The difference between the symmetrical exponential approximation and the numerically
computed pdf can be demonstrated by calculating the probability Pe1 that an error occurs in the
mark state, which is written as:
∫∞−
=Q
Se dfP m ξξ )()(1 (10)
where Q is the receiver decision threshold. The pdf )( mSf of the photocurrent S(m) in the mark
state can be evaluated from (9a) using the symmetrical exponential approximation [4] for the
pdf:
⎟⎟⎠
⎞⎜⎜⎝
⎛ −
=σ
σ
mI
m eIf2
exp 21)( (11)
where σ2 is the noise variance of Im given in [4]. We have chosen to compute Pe1 and not the
average BER because in [4] no pdf was given for the space state that allows the direct
computation of the error probability Pe0 in this state.
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In the remainder of the paper the parameters used in the calculations are as follows:
λ=1.55μm, c=3×108m/s, L=80Km, a=0.2dB/Km, γ=2.4 (Km×W)-1 and k=1.28 A/W.
Figure 3(a) shows the pdf of the photocurrent for N=16, D=10ps/nm/Km, Pin= 5dBm,
Δf=25GHz and gives a first glance of the effectiveness of the symmetrical exponential
approximation in describing the statistics of the photocurrent. It should be noticed that only the
left part of the pdf, is shown in the figure. This happens because for practical values of the BER,
only this part of the pdf is needed to compute Pe1 using (10). In Figs. 3(b) and 3(c), Pe1 is plotted
as a function of the receiver threshold for N=16, D=10ps/nm/Km, Pin=5dBm and Δf=25GHz and
N=8, D=5ps/nm/Km, Pin=18dBm and Δf=100GHz respectively. For the numerically calculated
pdfs, Pe1 was computed using numerical integration of (10). In figures 3(b) and 3(c) there is an
obvious deviation between the two models, which is several orders of magnitude in the examined
cases, and implies that the average BER will be much higher for the exponential approximation.
For example as shown in Figure 3(b), for a threshold Q=60μA the values of Pe1 are 2×10-5 and
7×10-7 in the case of symmetrical exponential pdf and the numerically computed pdf
respectively. Hence, it is evident that the symmetrical exponential model may overestimate the
system bit error rate by about two orders of magnitude for error probabilities of the order 10-9.
However, it can be used to easily estimate an upper bound of the BER.
To further illustrate the difference between the symmetrical exponential and the
numerically computed pdf, the values of σ obtained in the case of the symmetrical exponential
pdf using the closed form formula of [4] is plotted (with solid lines) in Fig. 3(d). Also plotted are
the values of σ΄ obtained by fitting the numerical pdf with 1/σ΄/21/2exp(21/2Im/σ΄) for Im<0. As
observed by the figure there is some difference between the values of σ and σ΄ and this is a
further indication that the symmetrical exponential pdf fails to describe the statistical behavior of
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Im. It should also be noted that Fig 3(d) provides an alternative approximation for the numerically
computed pdf of Im for Im<0. Once the value of σ΄ is obtained from Fig 3(d) then the pdf of [4]
(i.e. equation (11)) can be used to approximate the pdf for Im<0 by replacing σ with σ΄.
B) The Gaussian pdf Approximation
The most common approach in the calculation of the BER in the presence of FWM is to assume
that the FWM noise is Gaussian. According to the Gaussian approximation [3] the error
probability is written as:
⎟⎟⎠
⎞⎜⎜⎝
⎛== ∫
∞⎟⎟⎠
⎞⎜⎜⎝
⎛−
221
21 2
2
g
Q
t
e
QerfcdteP
gπ
(12)
where Qg is the Q-factor given by
⎭⎬⎫
⎩⎨⎧
++
⎭⎬⎫
⎩⎨⎧
+−
≈+
−=
∑∑∑
∑∑
≠==≠≠≠≠≠
−
≠=≠≠≠
−
rqpppr
nrqppqn
nrqppqr
aLn
rqpppr
nrqppqr
aLn
sm
sm
g
PPPePk
PPkekPSS
Q
41
41
812
41
81
2
)()(
)()(
σσ (13)
To compute the value of the BER obtained by the actual pdfs of S(m) and S(s) the
following formula is used.
∫∫∞−
∞
+=Q
SQ
Se dfdfP ms ξξξξ )(21)(
21
)()( (14)
where the first and second terms are the probabilities that an error occurs in the space and mark
states, respectively, while the decision level Q is chosen so as to minimize the error probability
Pe. This can be done by solving the equation dPe/dQ=0 using well-known numerical methods.
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The accuracy of the Gaussian approximation in predicting Pe1 and the BER is examined
in Figure 4(a) for N=16, D=10ps/nm/Km and Δf=25GHz and in Figure 4(b) for the above case
and for N=32, D=15ps/nm/Km and Δf=10GHz.
From Fig. 4 it is evident that the Gaussian model cannot be used to estimate the BER
accurately. The error in the estimation of the Bit Error rate (BER) is more pronounced for small
error rates (<10-9). For example for Pin=2dBm, Pe=10-18 for the Gaussian approximation and
Pe=10-5 in the numerically calculated case as shown in Figure 4(b).
The results of Fig 4 are obtained in the case where D ≥ 2ps/nm/Km. If the WDM system
operates near the zero dispersion wavelength the formulation of equations (7) and (8) is not
applicable. In order to investigate the validity of the Gaussian approximation in such systems,
the pdf of the photocurrents S(m) and S(s) must be estimated through MC simulations applying
S(m)=k|E(m)|2 and S(s)=k|E(s)|2 where E(m) and E(s) are given by (5) and (6) and Ppqr is obtained using
(3), (4a) and (4b) setting D=0 (Note that (4c) does not apply in this case). The results obtained by
the MC simulations (dots) and the Gaussian approximation (lines) in the case of D=0 (for the
central channel) are illustrated in Figure 5 for N=16 channels, Pin=0dBm, dD/dλ=0.07ps/nm2/Km
and Δf=100GHz. Inspecting Figure 5, it is understood that even though the difference between
the pdfs of the present and the Gaussian model is reduced in the mark state, it remains
observable since the FWM products are correlated and the central limit theorem is not valid in
this case as well.
V. ESTIMATION OF POWER LIMITS DUE TO FWM
In practice, in order to evaluate the performance of the system, it is useful to have a relation
between the BER and the input power. With the method presented here, it is quite easy to
14
calculate the BER, using numerical integration, given the characteristics of the system. Once the
BER is determined one may calculate other useful system performance measures such as the
Packet Error Rate in IP over WDM systems [12].
Figures 6(a) to 6(c) depict the variations of the BER with respect to the variations of the
channel spacing, signal power and fiber chromatic dispersion in a DWDM system. The error
probability was calculated using numerical integration of (14). For the calculations of the BER,
the optimum decision threshold was chosen, i.e the threshold, which minimized the error
probability Pe. The diagrams show that Pe tends to diminish rapidly below a certain power value.
This happens due to the fact that since |cosθ|≤1, the random variables |Im| and Is cannot exceed a
certain value Im,max and Is,max obtained by setting θi=0 and Bi=1 in (8b) and (8c) respectively.
Hence, for very low input powers the distributions of the mark and space state may not overlap
at all which implies an error free transmission [13]. However, in practical systems the existence
of other noises (e.g. thermal noise) will force the two distributions to overlap, preventing the
BER from becoming zero.
When the dispersion or the channel spacing is increased, lower values of Pe are achieved
at a given input power. This can be explained by equations (4a) and (4b) which give the mixing
efficiency η and phase mismatch Δβ. From these equations it is easy to see that the mixing
efficiency increases when the dispersion D and/or the channel spacing Δf is reduced. Since the
power Ppqr of the FWM product given by (3), is proportional to the mixing efficiency η, it is
expected that the noise power will be increased and that the performance of the system will
degrade. A similar behavior is observed when the number of channels is reduced. This takes
place because the number of the neighboring channels on each side of the central channel is
reduced and therefore, the number of channel combinations (p,q,r) that satisfy the condition
15
p+q-r=n is also reduced. Hence, the power of the FWM noise will decrease when the number of
channels is decreased. It is interesting to note that an increase in the number of channels above a
certain value does not significantly change the system performance. For example for
D=5ps/nm/Km, Δf=25GHz and Pin=4.5dBm a BER=10-9 is achieved for N=8 channels, a
BER=7×10-7 is achieved for N=16 channels and a BER=10-6 is achieved for N=32 channels. This
happens because the channels, which are far away from the central one, do not play a significant
role in the production of the FWM noise.
From figures 6(a)-6(c), it is confirmed that a simple way to eliminate the impact of FWM
is to use nonzero dispersion fibers and WDM systems with large channel spacing or with fewer
number of channels. For example for N=16 channels, Δf=25GHz and Pin=8dBm, an increase of
D from 5 to 10ps/nm/Km causes a reduction of BER from 2×10-3 to 3.8×10-6. So graphs like 6(a)
to 6(c) are useful in determining the maximum input power allowed in a WDM link, given its
characteristics, i.e. the channel spacing Δf, the fiber dispersion coefficient D and the total number
of channels. For example for N=32, D=2ps/nm/Km and Δf=50GHz, the input power of each
channel must not exceed 4.9dBm if the BER is not to exceed 10-9. Note that this result is
obtained for a fiber length of L=80Km. Since the FWM influence is not much different in
systems with a fiber length longer than the effective length defined as (1-exp(-aL))/a, these
results are also valid when the total fiber length is longer than the effective length.
Note, that the analysis carried out so far, assumes a single span system. If the system
consists of multiple spans and optical amplifiers are used to compensate the optical losses, then
additional FWM noise products are generated in each span. The phase of these products depends
on the phase of the channels at the input of the span. It can be assumed [4] that the dispersion of
each fiber span differs slightly and that the length of the fibers used in the span is different.
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Hence, it is possible to assume that the phases of the channels at the beginning of each span are
independent of each other. This implies that the phases of the products in different spans are
independent as well. Also, the net dispersion in each span causes a walk-off of neighbouring
channels by at least one bit period. Hence, the FWM noise products in each span can be assumed
independent and the pdf of the total FWM noise can be approximated by the convolution of the
individual FWM noise pdfs. These assumptions may not be valid in all cases. If for example, the
dispersion is completely compensated in each span, then the phases of the channels can not be
considered independent. In such cases, the pdf of Sm and Ss can be computed by altering the
FWM efficiency η in equation (4a) in order to take account the number of spans as in [14]. This
however prohibits the use of the auxiliary variables Im and Is whose applicability assumes the
FWM efficiency given by (4a). Consequently, the MC simulations must be performed again, this
time for Sm and Ss directly.
VI. INCORPORATION OF OTHER NOISES
In actual systems, receivers can also suffer from other noises such as the thermal and the
Amplified Spontaneous Emission (ASE) noise. In order to include these noises, a technique
based on the Moment Generating Function (MGF) of the decision variable can be used [11], [15]
while the error probability can be estimated using the saddle-point approximation through the
MGF.
To illustrate the above technique, a simple optically preamlified receiver with a
rectangular optical filter and an integrate-and-dump electrical filter will be assumed. The
Moment Generating Function (MGF) MZ(s) of the decision variable Z at the receiver is defined
as the expected value of esZ. Under the assumption that the quantum efficiency of the
17
photodetector equals to unity, the MGF MZ(s) of Z is related to the MGF MX(s) of the energy X
of the incident optical field at the input of the amplifier through [15]:
{ } ⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
==sN
GsMsN
sMEsMo
Xo
XZZ 111)()( |
μ
(15)
In (15), No=nsp(G-1) is the power spectral density of the amplified spontaneous emission (ASE)
noise, while G and nsp are the gain and the spontaneous emission parameter of the optical
amplifier respectively. μ+1 is equal to the product BTb of the bandwidth B of the optical filter
and the bit duration Tb (μ is assumed to be an integer). Hence, the MGF of Z can be directly
computed from the MGF of X by a simple change of variables and multiplication by (1-N0s)-μ.
Assuming rectangular pulses in all channels for the mark state it is easy to show that X=STb,
where S=S(m) and S=S(s) for the mark and space states of the central channel respectively. To
estimate the MGF of X=STb one can use the approximate exponential formulas for the pdf of S.
For example, in the space state where X=kδ2ΤbIs, MX(s) is approximately written as:
( )( )( )1exp)( max, −′+′+
≅ sX Isbsb
AsM (16)
where s΄=kδ2Τb and the exponential approximation Aebx for the pdf of Is is used (see section III).
Note that a more accurate estimation of the MGF of X could be performed by directly using the
pdf computed from the Monte-Carlo simulations and numerical integration. Using (15) and (16)
the MGF of the decision variable at the receiver including the influence of the FWM and ASE
noises can be computed. A similar expression as in (16) can be derived for the mark state. In
order to take into account the electrical thermal noise, the MGF of the decision variable Z is
multiplied by exp(σth2s2 / 2) which is the MGF of the thermal noise. The power of the thermal
noise is equal to σth2=2KBTKTb / (q2RL), where KB is Boltzmann’s constant, RL the load resistor of
the photodetector, TK the temperature (in Kelvin), while q is the charge of the electron. Using the
18
above remarks the MGF of the decision variable can be approximately computed and the saddle
point approximation can be used to estimate the error probabilities from the MGF [11]. Equation
(15) also provides an indication of the validity of the Gaussian model for the description of the
statistics of the decision variable in the presence of FWM and ASE noises. The shape of the
MGF of Z is related to the respective one of X, which as indicated by (16) can not be assumed
Gaussian. Consequently, it is expected that the MGF of Z will also not be Gaussian, especially if
the FWM noise (amplified by the optical amplifier) is dominant. This implies that the Gaussian
model may not give accurate values for the error probability. However, if the receiver is
dominated by the Gaussian distributed thermal noise (e.g. in the absence of optical
preamplification), the pdf of the decision variable can be approximated by a Gaussian
distribution, accurately enough.
VII. CONCLUSIONS
In this paper the statistical behaviour of the FWM noise was investigated and its implications in
the performance of a WDM system were analyzed as well. Using numerical MC simulations, the
pdf of the FWM noise in the space state was calculated for the first time. It was also shown both
by MC simulations and theoretical considerations that the pdf in the mark state is not symmetric
as previously assumed in the literature. Hence, the proposed model is considered more accurate
compared to the models used so far. By fitting the data obtained from numerical simulations,
approximate expressions of both pdfs are derived and diagrams are given that allow the
computation of these expressions given the number of channels. Using these diagrams the pdfs
can be estimated without having to perform any MC simulations. The model is then used to
estimate the performance of a WDM system for various values of its parameters (fiber
19
dispersion, channel spacing and input power). The obtained results can be very useful in the
design of practical systems. A comparison between the present model and two models previously
proposed was carried out and it was shown that these models may not provide an accurate value
for the BER. Finally, some guidelines for the incorporation of other noises (thermal and
Amplified Spontaneous Emission – ASE) are presented.
APPENDIX
CALCULATION OF THE ODD ORDER MOMENTS
In this appendix the first two odd order moments <Im2u+1> of the photocurrent Im in the
mark state will be calculated. For u=0, mu
m II =+12 is given by:
( )
( ) 0cos31
cos31
31
=⎥⎦
⎤⎢⎣
⎡−
−−=
=−−−
==
∑
∑∑
pqrnpqrrqp
pqr
pqrnpqr
rqppqr
pqrpqrm
BBBnqnp
d
nqnpBBBd
eI
θθ
θθ
(A1)
since ( ) ( ) 0coscos =−−+=− nrqpnpqr θθθθθθ . However for u=1, the third moment 3mI of Im
is:
( )
[ ] [ ]∑∑∑
∑∑
≠≠≠
++=
⎥⎦
⎤⎢⎣
⎡=⎟
⎟⎠
⎞⎜⎜⎝
⎛−
−−=
''''''''''''''''''
''''''
23
33
3
92
91
271
31cos
31
rqprqppqrrqprqppqr
rqppqrrqppqr
pqrpqr
pqrpqr
pqrnpqr
rqppqrm
eeeeee
enqnp
BBBdI θθ
(A2)
where ( ) nqnpBBBde npqrrqppqrpqr −−−= //cos θθ .
Using simple mathematical operations one can show that the terms in the sums of (A2)
have mean value either 1 or 0. Indeed, it is easy to show that every term of the sums of (A2) can
be expanded into a linear superposition of terms of the type cosθ=cos(apθp+aqθq+arθr-
20
anθn+…+ap΄΄θp΄΄+aq΄΄θq΄΄+ar΄΄θr΄΄+anθn΄΄) whose mean value will be either 1 if θ≡0 or 0 if θ≠0. The
coefficient of every term of the above type, in the superposition is positive. For example,
working with the terms of the third sum of (A2) one can write:
θ=θpqr-n+θp΄q΄r΄-n-θp΄΄q΄΄r΄΄-n=θp+θq-θr-θn+θp΄+θq΄-θr΄-θn-θp΄΄-θq΄΄+θr΄΄+θn (A3)
For the terms for which r΄΄=n, p=r΄, p΄=r, q=p΄΄ and q΄=q΄΄ one obtains θ≡0. Since θ≡0, one has
1cos =θ and the mean value of the corresponding terms will be 1. For the terms for which all
the phases θi in θ do not cancel out (θ≠0), θ will be given by a linear combination of the
independent random variables θk each of which is uniformly distributed in [0,2π] and hence
<cosθ>=0. Therefore, for these terms the mean value vanishes. From the above remarks one can
easily conclude that 03 >mI .
ACKNOWLEDGEMENTS
The authors would like to thank Mr. George Kakaletris for his valuable programming assistance.
REFERENCES
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over Installed Fiber,” IEEE Photonics Technology Letters 12, 669-671 (2000).
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in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10, 979-981 (1998).
3. K. Inoue, K. Nakanishi, K. Oda and H. Toba, “Crosstalk and Power Penalty Due to Fiber
Four-Wave Mixing in Multichannel Transmissions,” J. Lightwave Technol. 12, 1423-1439
(1994).
21
4. M. Eiselt, “Limits on WDM Systems Due to Four-Wave Mixing: A Statistical Approach,” J.
Lightwave Technol. 17, 2261-2267 (1999).
5. J. G. Proakis, Digital Communications (4th ed. McGraw-Hill, New York, 2000).
6. K. Inoue, “Polarization Effect on Four-Wave Mixing Efficiency in a Single-Mode Fiber,”
IEEE J. Quantum Electron. 28, 883-894 (1992).
7. N. Shibata, K. Nosu, K. Iwashita and Y. Azuma, “Transmission Limitations Due to Fiber
Nonlinearities in Optical FDM Systems,” IEEE J. Select. Area Commun. 8, 1068-1077 (1990).
8. G. P. Agrawal, Nonlinear Fiber Optics (2nd ed., Academic, New York, 1995).
9. S. Song, C. T. Allen, K. R. Demarest and R. Hui, “Intensity-Dependent Phase-Matching
Effects on Four-Wave Mixing in Optical Fibers,” J. Lightwave Technol. 17, 2285-2290 (1999).
10. K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, “CW three-wave mixing in
single-mode optical fibers,” J. Appl. Phys. 49, 5098-5106 (1978).
11. G. H. Einarsson, Principles of Lightwave Communications (John Wiley & Sons, Chichester,
1996).
12. J. Tang, C. K. Siew and L. Zhang, “Optical nonlinear effects on the performance of IP traffic
over GMPLS-based DWDM networks,” Computer Communications 26, 1330-1340 (2003).
13. P.E. Green, Fiber Optic Networks (Prentice-Hall, New Jersey, 1993).
14. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with
multistage optical amplifiers,” Optics Letters 17, 801-802 (1992).
15. T. Kamalakis and T. Sphicopoulos, “Asymptotic Behavior of In-Band Crosstalk Noise in
WDM Networks,” IEEE Photon. Technol. Lett. 15, 476-478 (2003).
22
Figures
0 50 100 150 20010-10
10-8
10-6
10-4
10-2
1
(a)
N=8 channels N=16 channels N=32 channels
Is
Figure 1a
23
-15 -10 -5 0 5 10 15 2010-12
10-10
10-8
10-6
10-4
10-2
1
N=32Ch
N=16Ch
N=8Ch
(b)
Im
Figure 1b
24
5 10 15 20 25 30 350
1x10-3
2x10-3
3x10-3
4x10-3
5x10-3
6x10-3
A fo
r the
cen
tral
cha
nnel
N
A=A(N) A=3,9x10-4+0,19824e-N / 1,94173
(a)
Figure 2a
25
5 10 15 20 25 30 35-0,13
-0,12
-0,11
-0,10
-0,09
-0,08
-0,07
-0,06
-0,05
b fo
r the
cen
tral
cha
nnel
N
b=b(N)b=-0,05949-0,78167e-N / 2,87922
(b)
Figure 2b
26
5 10 15 20 25 30 350,46
0,48
0,50
0,52
0,54
0,56
0,58
0,60
0,62
A fo
r the
cen
tral
cha
nnel
N
A=A(N) A=0,47783+0,323e-N / 6,85235
(c)
Figure 2c
27
5 10 15 20 25 30 35-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
2,0
2,5
b fo
r the
cen
tral
cha
nnel
N
b=b(N) for Im<0 b=b(N) for Im>0
b= -1,03091-0,90364e-N / 3,93013
b=1,49027+9,93815e-N / 2,6189
(d)
Figure 2d
28
0,0 4,0x10-5 8,0x10-5 1,2x10-41E-8
1E-6
1E-4
0,01
1
100
10000
(a)
pdf (
Amp-1
)
Photocurrent (Amp)
Symmetrical exponential Numerically computed
Figure 3a
29
3,0x10-5 6,0x10-5 9,0x10-5 1,2x10-41E-101E-91E-81E-71E-61E-51E-41E-30,010,1
1
(b)
P e1
Threshold (Amp)
Symmetrical exponential Numerically computed
Figure 3b
30
5,0x10-4 1,0x10-3 1,5x10-3 2,0x10-310-6
10-5
10-4
10-3
10-2
10-1
1
(c)
P e1
Threshold (Amp)
Symmetrical exponential Numerically computed
Figure 3c
31
8 12 16 20 24 28 320,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
σ fo
r cen
tral
cha
nnel
N
Std (Numerically Computed) Std (Symmetrical Exponential)
(d)
Figure 3d
32
3,0x10-5 6,0x10-5 9,0x10-5 1,2x10-410-3010-2710-2410-2110-1810-1510-1210-910-610-3
1
P e1
Threshold (Amp)
Numerically computed Gaussian distribution
(a)
Figure 4a
33
-2 0 2 4 6 8 10 12 1410-2010-1810-1610-1410-1210-1010-810-610-410-2
1N=32 D=15ps/nm/KmΔf=10GHz
N=16 D=10ps/nm/KmΔf=25GHz
P e
Pin(dBm)
Numerically computed Gaussian distribution
(b)
Figure 4b
34
-4x10-5 0 4x10-5 8x10-5 1x10-410-2
10-1
110
102
103
104
105
106
107
pdf (
Amp-1
)
Photocurrent (Amp)
MC (space) MC (mark) Gaussian (mark) Gaussian (space)
Figure 5
35
-10 -5 0 5 10 15 20 2510-10
10-8
10-6
10-4
10-2
(a)
D1 D2 D3D1 D2 D1D3 D2 D1D3 D2 D3
P e
Pin(dBm)
Δf=10GHz Δf=25GHz Δf=50GHz Δf=100GHz
Figure 6a
36
-15 -10 -5 0 5 10 15 20 2510-14
10-12
10-10
10-8
10-6
10-4
10-2
(b)
D1 D
2 D
3D
1 D
2 D
1D
3 D
2 D
1D
3 D
2 D
3
P e
Pin(dBm)
Δf=10GHz Δf=25GHz Δf=50GHz Δf=100GHz
Figure 6b
37
-15 -10 -5 0 5 10 15 20 2510-40
10-35
10-30
10-25
10-20
10-15
10-10
10-5
1
(c)
D1 D2 D3D1 D2 D1D3 D2 D1D3 D2 D3
P e
Pin(dBm)
Δf=10GHz Δf=25GHz Δf=50GHz Δf=100GHz
Figure 6c
38
Figure captions
Fig. 1. Pdf of a) the space and b) the mark state. The squares represent the case where N=8, the
circles the case where N=16 and the triangles correspond to N=32. The exponential fittings for
each pdf are also shown with solid lines.
Fig. 2. (a)-(b) The parameters A and b in the space state, for the central channel, with respect to
the number of channels N and (c)-(d) the parameters A and b in the mark state, for the central
channel, with respect to the number of channels N.
Fig. 3. a) The pdf of the photocurrent for N=16, D=10ps/nm/Km, Pin=5dBm and Δf=25GHz, b)
The error probability Pe1 for the mark state, with respect to the receiver threshold for the same
parameters, c) Pe1 as a function of receiver threshold for N=8, D=5ps/nm/Km, Pin=18dBm and
Δf=100GHz and d) Standard deviation of the sum Im for the central channel with respect to the
number of channels N for the symmetrical distribution (solid line) and the numerically computed
(Dashed line).
Fig. 4. a) Error probability in the mark state with respect to the receiver threshold in the case
N=16, D=10ps/nm/Km, Pin=5dBm and Δf=25GHz for the Gaussian and numerical models, b)
BER dependence with respect to the input power for the Gaussian and the numerical models.
Fig. 5. The pdf of the photocurrent in the mark and the space state for N=16, D=0ps/nm/Km,
dD/dλ=0.07ps/nm2/Km, Pin=0dBm and Δf=100GHz. The solid lines represent the Gaussian
distribution.
39
Fig. 6. BER as a function of the input peak power Pin in the mark state, for a) N=8, b) N=16 and
c) N=32. The values of the chromatic dispersion used are KmnmpsD //21 = ,
KmnmpsD //52 = and KmnmpsD //103 = .