1636IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 11, NOVEMBER 2003
Numerical Simulation of the SPM Penalty in a
10-Gb/s RZ-DPSK System
Xing Wei, Xiang Liu, and Chris Xu
Abstract—The impact of self-phase modulation-induced non-
linear phase noise in a 10-Gb/s return-to-zero differential phase-
shift keying system is studied by numerical simulation. We show
that the simple differential phase
approximation for the phase noise provides a relatively good esti-
mate of the nonlinear penalty.
method based on the Gaussian
Index Terms—Bit-error rate (BER), differential phase-shift
keying (DPSK), optical communication, self-phase modulation
much attention for high-speed long-haul and ultralong-haul
optical transmission. Experiments have been carried out at
40-Gb/s ,  and 10-Gb/s – bit rates. Numerical
simulations have also been reported –. One limiting
factor in long-haul and ultralong-haul DPSK systems is the
Gordon–Mollenauer nonlinear phase noise (or phase jitter)
which arises from the interaction between amplified sponta-
neous emission (ASE) noise and self-phase modulation (SPM)
, . This effect has been studied experimentally ,
. In numerical simulations, however, a simple and reliable
method to assess the phase-noise penalty in DPSK systems
remains an unsolved issue. In this letter, we study the SPM
penalty in a single-channel 10-Gb/s RZ-DPSK system by nu-
merical simulation. Furthermore, we propose a simple method
for bit-error-rate (BER) estimation, which takes into account
the effect of phase noise.
(RZ-DPSK) modulation format has recently attracted
II. SIMULATION DETAILS
The RZ-DPSK system used for the simulation is illustrated
in Fig. 1. The transmitter consists of a continuous-wave laser,
a 33% duty cycle optical pulse generator, a 10-Gb/s input
pseudorandom bit sequence (PRBS) of length
ential encoder, and a Mach–Zehnder modulator to modulate
the phases of the optical pulses. Predispersion compensation
100 ps nm is used before transmission. The RZ-DPSK
signal is then transmitted over an optical link made of 40 fiber
spans separated by erbium-doped fiber amplifiers (EDFAs) and
, a differ-
Manuscript received March 10, 2003; revised June 18, 2003.
X. Wei is with Bell Laboratories, Lucent Technologies, Murray Hill, NJ
07974 USA (e-mail: firstname.lastname@example.org).
X. Liu is with Bell Laboratories, Lucent Technologies, Holmdel, NJ 07733
C. Xu is with the Applied and Engineering Physics Department, Cornell Uni-
versity, Ithaca, NY 14853 USA.
Digital Object Identifier 10.1109/LPT.2003.818664
dispersion compensating fibers (DCFs). Each span consists
of an EDFA to boost the optical power to the desired launch
power, 100 km of nonzero dispersion-shifted fiber (NZDSF)
with a dispersion of 4 ps/nm km, a DCF to compensate 95%
of the dispersion of the NZDSF (20-ps/nm residual dispersion
per span), and a second EDFA before the DCF to partially
compensate for the span loss. The last DCF has an additional
350 ps nm to bring the total residual dispersion
of the link to 350 ps/nm. The loss of the NZDSF is 0.23 dB/km,
and the nonlinear coefficient is 1.6 rad/W km. For simplicity,
the fiber birefringence and the nonlinearity of the DCF are
neglected since they would not change our general conclusion
in a significant way. On the receiver side, the signal is filtered
by a Gaussian optical bandpass filter (BPF) with a 3-dB
bandwidth of 15 GHz to remove the ASE noise power outside
the signal bandwidth. The filtered optical signal is detected by
a delay interferometer and a balanced receiver. All the receiver
filtering is performed in the optical domain and no electrical
filter is used. All components in the transmitter and the receiver
are assumed to be ideal.
When the ASE is included in the simulations, we use a noise
figure (NF) of 7 dB for the EDFAs. We note that the NF of
state-of-the-art EDFAs can be less than 5 dB. We intentionally
use a larger NF because this allows for direct counting of the
errors in the simulation (for BER higher than
sumptions on the noise probability density function (pdf) at the
receiver. Depending on the BER level, it is sometimes neces-
ASE noise to achieve sufficient accuracy for one BER value. It
is alsocrucial that theASE is loaded alongthetransmission link
rather than only at the receiver. It is known that the noise distri-
bution at the receiver can be distorted due to nonlinear interac-
tion between the signal and noise, and this may have a more no-
ticeable effect in DPSK systems than in ON–OFF keying (OOK)
systems because of the Gordon–Mollenauer phase-noise effect
) without as-
III. RESULTS AND DISCUSSION
Fig. 2 shows the BER as a function of the launch power. The
solid square symbols are obtained from direct error counting in
thesimulation, and theopen symbols are obtained from two dif-
ferent models that will be explained later. The directly counted
BER results in Fig. 2 show that the optimum launch power of
this RZ-DPSK system is around
power, the system performance degrades due to fiber nonlin-
earity. Note that if the transmission system were linear, a 6-dB
increase in power (from
3 dBm. With higher launch
1 dBm) would bring the BER
. The fact that the BER with
1041-1135/03$17.00 © 2003 IEEE
WEI et al.: NUMERICAL SIMULATION OF THE SPM PENALTY IN A 10-Gb/s RZ-DPSK SYSTEM1637
Fig. 1.Schematic diagram of the RZ-DPSK transmission system.
(b) estimation using the conventional ? factor method, and (c) estimation
using the differential phase ? method.
BER obtained from different methods: (a) direct error counting,
1-dBm launch power is on the order of
nonlinear penalty is very large at this power level. Since the
pulse overlap is not significant in this system, the dominating
source of nonlinear penalty is SPM.
Based on a simplifiedmodel,  predicted that theoptimum
power in a DPSK system corresponds to a total SPM phase
1 radian. This can now be verified directly with our
simulation. The loss of the fiber 0.23 dB/km corresponds to
an effective nonlinear length of approximately 19 km in each
span. With the optimum launch power
the average SPM phase shift at the end of the transmission is
rad W kmkm
SPM phase shift at the center of a pulse should be somewhat
larger. This agrees quite well with the prediction of .
Although the direct error counting is proven effective to esti-
mate the SPM penalty, it is rather time-consuming and still not
practical for a BER on the order of
tive methods which can provide a quick estimate of the system
performance under different conditions will be useful. In sim-
ulations of OOK systems, it is a common practice to exclude
ASE and evaluate the eye-closure penalty as an indication of
nonlinear effects. We shall now show that this approach may
lead to erroneous result for DPSK systems. We have simulated
the balanced electrical eye diagrams (Fig. 3) at different launch
powers with the ASE noise excluded. With
power, no eye-closure degradation is observed. From these eye
diagrams, it appears that SPM may actually help “open up” the
eye by counteracting the effect of the residual chromatic disper-
sion, which is known as the “soliton” effect. Here, the eye-clo-
sure methodhas completelyfailed topredicttheSPM penaltyin
this RZ-DPSK system. A simple explanation for this is that the
Gordon–Mollenauer phase noise arises from the interaction be-
tween the ASE noise and SPM, and therefore, it does not show
up in a simulation without ASE.
shows that the
3 dBm or 0.5 mW,
mWrad, and the
. Therefore, alterna-
?7 dBm (left), ?4 dBm (middle), and ?1 dBm (right). The vertical scale is
normalized with the launch power.
Eye diagrams from the balanced receiver with launch powers of
One may also consider to plot these eye diagrams with the
ASE noise included (not shown) and estimate the BER from the
It is known that this
estimate of the BER in OOK system simulations although the
We find, however, this method has also failed to predict the
SPM penalty, as shown by the open triangles in Fig. 2. This
method underestimates the system performance in the linear
regime and overestimates the performance in the highly non-
linear regime. No significant nonlinear penalty is observed even
with the highest launch power
dicts the result from direct error counting. This large discrep-
ancy is due to the fundamentally non-Gaussian nature of the
pdf of the receiver output signal. Therefore, the conventional
factor method should also be avoided in numerical simulations
of RZ-DPSK systems.
We find, however, that a slightly modified model can be used
to estimate the SPM penalty in such an RZ-DPSK system. We
still use the
factor concept but it is now definedfor theoptical
phase of the received signal. Fig. 4 shows a “differential phase”
eye diagram of the received RZ-DPSK signal after the 15-GHz
optical BPF. The “differential phase” is the phase difference be-
tween two sampling points separated by one bit period mapped
to the range of
to. In Fig. 4, the center of the bit
slot is located at 50 ps (or 150 ps) and the divergence of the dif-
ferential phase at 100 ps is due to the fact that the intensity of
the light signal is nearly zero. We then assume a Gaussian dis-
tribution for the noise at the center of each bit slot and define a
factor method usually gives a fairly good
1 dBm, which strongly contra-
differential phase on the 0 and
andrepresent the standard deviations of the
1638IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 11, NOVEMBER 2003
Fig. 4.Differential phase eye diagram with a launch power of ?3 dBm.
The open circles in Fig. 2 are obtained from the expression
which can be derived from the Gaussian approximation for the
differential phase noise. We find that (3) qualitatively repro-
duces the result obtained from the direct error counting method,
and in particular, (3) predicts the optimum launch power quite
between the BER values obtained from the direct error counting
and the differential phase
methods. This is understandable
because the pdf of the differential phase noise in the received
signal is not exactly Gaussian.
We have shown that the Gordon–Mollenauer nonlinear phase
noise must be taken into account in numerical simulations of
RZ-DPSK systems in which such phase noise is the dominating
source of nonlinear penalty. A simplified model based on the
appears to provide a relatively good esti-
mate of such nonlinear penalty. Compared with the direct error
counting method, the differential phase
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