OPTICS LETTERS / Vol. 28, No. 23 / December 1, 2003
Analysis of intrachannel four-wave mixing
in differential phase-shift
keying transmission with large dispersion
Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974
Bell Laboratories, Lucent Technologies, Holmdel, New Jersey 07733
Received May 23, 2003
Intrachannel four-wave mixing (IFWM) in highly dispersed differential phase-shift keying (DPSK) transmis-
sion systems is studied with a simple analytical model and numerical simulation.
effect in DPSK systems is smaller and less dependent on the bit pattern than in on–off-keying systems with
the same average power.The reduced pulse energy in DPSK and correlation between the nonlinear phase
shifts of two adjacent pulses contribute to the robustness of DPSK versus IFWM.
060.2330, 190.4370, 190.4380, 260.2030.
It is found that the IFWM
© 2003 Optical Society of
Intrachannel nonlinear effects, such as intrachannel
cross-phase modulation (IXPM) and intrachannel
four-wave mixing (IFWM), are the dominating sources
of nonlinear penalties in the so-called pseudolinear
high-speed transmission systems in which the optical
pulses are highly dispersed and overlap strongly
during transmission. These nonlinear effects have
been studied extensively.1–7
IXPM- and IFWM-induced timing and amplitude jit-
ters of the ones can be greatly reduced in a symmetric
dispersion-managed link, whereas the IFWM-induced
ghost pulses on the zeros do not cancel.6
with proper dispersion management, the ghost pulses
are the major remaining nonlinear impairment.
eral methods to suppress the ghost pulses have been
Recently, superior transmission performance has
been demonstrated with return-to-zero (RZ) differen-
tial phase-shift keying (DPSK) in highly dispersed
major advantage of DPSK over on–off keying (OOK) is
the improved receiver sensitivity in the linear regime.
In addition, DPSK greatly reduces the nonlinear
IXPM effect in highly dispersed transmissions by use
of pulses of equal energy in all time slots.
IFWM introduces pattern-dependent nonlinear phase
shifts to the DPSK signal.12
performance degradation that is due to IFWM in
DPSK systems has not been investigated in detail.
In this Letter we study this effect by theoretical
analysis and numerical simulation.
IFWM causes less performance degradation in DPSK
than it does in OOK with the same average power
because of the lower peak power of DPSK and a partial
cancellation of the nonlinear phase shifts.
We use a simple model similar to what has been
previously used for the study of IFWM in RZ OOK
It is known that the
To our knowledge, the
We show that
systems3,6and extend the analysis to RZ DPSK.
consider the transmission of a 40-Gbit?s signal over
L ? 100 km of standard single-mode fiber with a
dispersion of D ? 17 ps?nmkm at the wavelength of
1550 nm (or b00? 222 ps2?km).
ficient of the single-mode fiber is g ? 1.2 rad?Wkm.
Here we neglect the fiber loss and focus on the non-
linear effects. For simplicity the fiber birefringence
and the nonlinearity of the dispersion-compensating
fiber are also neglected.The predispersion compen-
sation is 2850 ps?nm, exactly one half of the total
dispersion of the transmission fiber.
dispersion map fully symmetric to minimize the IXPM
and IFWM effects.6
In this study we also neglect
the nonlinear interaction between the signal and the
amplified spontaneous emission (ASE) noise.
interaction could produce additional phase noise in
DPSK, which is known as the Gordon–Mollenauer ef-
Fortunately, this is not critical here since this
self-phase modulation- (SPM-) induced phase noise
is suppressed in the highly dispersed transmission
In our model we consider a RZ DPSK optical pulse
train that consists of 2N Gaussian pulses.
tical field of each pulse before transmission can be
The nonlinear coef-
This makes the
uk?0,t? ? Akexp?2?t 2 kT?2?2t2?,
where k ? 1,2,...2N denotes the kth pulse, T ? 25 ps
is the bit period, t ? 5.0 ps corresponds to an 8.3-ps
width (FWHM) of the pulse intensity profile, and Ak?
6A0 is the complex amplitude of the DPSK signal.
Here, A0 is a constant related to the average power
P0 through P0 ?pp jA0j2t?T.
tive approach, Mecozzi et al.6showed that nonlinearity
produces the following term in the center of the kth
Using the perturba-
0146-9592/03/232300-03$15.00/0 © 2003 Optical Society of America
December 1, 2003 / Vol. 28, No. 23 / OPTICS LETTERS
Duk?L,t ? kT? ?
?1 2 2ib00z?t21 3?b00z?t2?2?1?2
t2?1 1 3ib00z?t2?
?l 2 m?2T2
t2?1 2 2ib00z?t21 3?b00z?t2?2?
23?l 2 n??m 2 n?T2
where ?l,m,n? are the indices of the IFWM contribut-
ing pulses. In this study, terms with l ? k or m ? k
are excluded because they are from IXPM and SPM
and produce only a constant phase shift to all pulses
in DPSK.We note here that jb00jL?2t2? 43, therefore
we can use the approximation based on jb00jz?t2¿ 1
to obtain the following much simplified expression:
∑2?l 2 k??m 2 k?T2
dt is the cosine integral func-
It should be noted that the IFWM-induced pulse
might not be exactly centered in the kth time slot, and
it might also have a larger pulse width.
however, are neglected in the following analysis since
our goal is to determine general features of the IFWM
effect in DPSK and compare them with OOK.
The fact that Duk has no real part, but only an
imaginary part (assuming A0 is real) as shown in
approximation (3), indicates that the IFWM distorts
mainly the phase of the original light pulse.
mation (3) allows us to compute efficiently such a
nonlinear phase shift of the kth pulse:
where Ci?x? ? 2R`
For each bit pattern we can then calculate the dif-
ferential phase error DfN2 DfN11for the Nth and
the ?N 1 1?th pulses positioned in the middle of the
pulse train and define the following differential phase
eye-closure factor (ECF):
This is used to quantify the penalty in DPSK based on
the observation that the differential phase eye14closes
(ECFDPSK? 1) when the jDfN2 DfN11j reaches p?2.
We use approximation (3) and Eqs. (4) and (5) to
compute ECFDPSK for a pulse train that contains
14 pulses (N ? 7).Since only the relative phase
matters, we choose A1 to be positive and exhausted
all 213? 8192 polarity (phase) combinations for the
remaining 13 pulses. The optical power used in
this study is P0? 5 mW. The calculated ECFs are
presented in Fig. 1(a) as a histogram on a logarithmic
scale.The histogram appears to decay rapidly and
the worst-case ECFDPSK is less than 0.2.
increase of N (and the number of patterns) in the
calculation does not change the result significantly,
which indicates that N ? 7 is sufficient for this study.
We perform the same calculation for RZ OOK.
Approximation (3) is still valid for OOK.
difference is that Akis now either 0 orp2A0to main-
tain the same average power P0.
bit to be a zero (AN? 0) and exhaust all the 22N21
combinations for the other 2N 2 1 time slots.
ECF for OOK is defined in the field domain (we do so
because the ASE noise can be better approximated by
a Gaussian random variable in the field domain than
in the intensity domain), i.e.,
We choose the Nth
Here we have neglected the fluctuations on ones.
result for N ? 7 is shown in Fig. 1(b).
trast to DPSK, the histogram for OOK has a long tail
that extends beyond 0.4.
to the well-known worst-case ghost pulse problem, i.e.,
a single zero surrounded by successive ones on both
These worst-case ghost pulses could give rise
to error floors in transmission experiments when long
pseudorandom bit sequence (PRBS) patterns are used.
There are two main reasons that the ECF in DPSK
is smaller and also less dependent on the bit pattern.
The first reason is obvious.
power, each pulse that represents a one in OOK car-
ries twice as much energy compared with a pulse in
DPSK.Approximation (3) shows that the magnitude
of each contributing term for Duk is proportional to
OOK can be larger than its counterpart in DPSK
by a factor of 2p2 if l, m, and n are all ones for
most combinations, which corresponds to the worst
case of OOK.The second reason is more subtle,
that is, the phase shifts DfN and DfN11 are par-
tially correlated.This can be clearly seen in Fig. 2.
We can prove that IFWM processes that involve
both the Nth and the ?N 1 1?th pulses, for ex-
ample, ?l? 1 ?N 1 1? 2 ?l 1 1? ! ?N? and ?l 1 1? 1
?N? 2 ?l? ! ?N 1 1?, contribute an identical phase
shift to DfN and DfN11 and cancel out in the
In strong con-
This long tail corresponds
With the same average
Therefore, each contributing term in
sampled over 8192 different bit patterns.
ECF histograms for (a) DPSK and (b) OOK, each
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OPTICS LETTERS / Vol. 28, No. 23 / December 1, 2003
calculated for the 14-bit-long DPSK pulse train with
8192 different bit patterns.
Distribution of the IFWM-induced Df7 and Df8
and the differential phase error Df72 Df8.
curves are simply guides to the eye.
Histograms of the nonlinear phase shift Df7
DPSK (left) and amplitude (square root of intensity) eye
diagram for RZ OOK (right).
Simulated differential phase eye diagram for RZ
The PRBS length is 292 1.
differential phase error regardless of the bit pattern.
Figure 3 shows histograms of the calculated Df7and
Df72 Df8 for the DPSK pulse train that contains
14 pulses. If Df7 and Df8 were uncorrelated, the
standard deviation of Df72 Df8would be
larger than that of Df7. Because of the correlation,
the differential phase error appears to have the same
distribution width as the IFWM-induced phase shift
of a single pulse.
To verify the effectiveness of the simple model we
perform numerical simulations using a PRBS of length
292 1 at 40 Gbits?s with the same transmission pa-
rameters that we used in the above analysis.
nonlinear effects, not only IFWM but also IXPM and
SPM, are included in the simulation.
still neglected. The results shown in Fig. 4 are in good
agreement with our model prediction.
In conclusion, our model analysis and numerical
simulation show that IFWM introduces phase noise
in strongly dispersion-managed RZ DPSK systems,
but the penalty appears to be smaller than that in RZ
OOK systems with the same average power because
the worst cases are significantly suppressed with
DPSK. We caution, however, that this analysis is
mainly to understand the underlying physics, and the
model used here is much too simplified compared with
practical systems. Nonetheless, this simplified model
might provide useful guidance for the design of future
high-speed optical transmission systems.
ASE noise is
We thank A. H. Gnauck, A. R. Chraplyvy, R. E.
Slusher, and R. C. Giles for inspiring discussions.
Xing Wei’s e-mail address is firstname.lastname@example.org.
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