Fast analytical models for FWM-impaired
mixed-fiber transparent optical networks
Stephan Pachnicke1, Stefan Spälter2 and Edgar Voges1
1 University of Dortmund, High Frequency Institute, 44221 Dortmund, Germany,
Tel: +49-231-755-6675, Fax: +49-231-755-4631, E-mail: [email protected].
2 Siemens ICN, Carrier Products, Optical Solutions, 81359 Munich, Germany.
different channels is taken into account by ∆βijk. If the
Abstract
The paper describes the impact of four-wave mixing on
phases of the different mixing products are assumed
The
independent from each other, the powers of the different
impairments due to FWM are related to a Q-factor and the
mixing products generated at the same frequency can be
effects of dispersion management are covered.
summed up to find the total power of the FWM crosstalk
1 Introduction
generated at a certain frequency. It has to be taken into
NRZ-modulated
Four-wave
WDM
mixing
mixed-fiber
is
one
of
systems.
the
dominating
account that the probability for a three-tone product in the
degradation effects in WDM systems with dense channel
modulated case is only 1/8 – compared to the CW-case –
spacing and low chromatic dispersion on the fiber. If in a
because on an average 50% of all bits are zeros. Similarly,
WDM system the channels are equally spaced, the new
the probability for a two-tone product is 1/4. The variance
waves generated by FWM will fall at the channel
of a channel, which is distorted by FWM crosstalk, is
frequencies and thus will give rise to crosstalk. In this
given by
µ − µ0
Q= 1
with σ 2 = 2 Pchannel ∑ PFWM ,ijk
σ1 + σ 0
ijk
paper the signal distortions due to FWM have been
isolated from the other nonlinear effects and a Q-factor
(2)
due to the FWM degradations is computed. Special
In the case of a high extinction ratio, the mean value of
attention is payed to mixed fiber systems, which are very
the zeros will be very low, and thus the variance of the
common in European networks [1]. In transparent optical
zeros due to FWM is negligible. The Q-factor
networks it is mandatory to consider the physical
approximation assumes a Gaussian distribution of the PDF,
constraints such as the nonlinear fiber effects. It is
which is very good if the number of FWM products is
therefore necessary to define fast but still accurate
greater than 10 [3]. The Q-factor approximation in eq. (2)
analytical models to find out whether the desired path can
does not consider ASE-noise. The noise term can be easily
be set up.
included, if desired, by summing the variances of the
2 Analytical Model for FWM Evaluation
FWM crosstalk and the ASE-noise.
The amplitude of an FWM product generated at
frequency m by the three generating waves i, j and k is
3 Full-Inline Dispersion Compensation Scheme
In the case of full-inline dispersion compensation [4]
given in the general case by [2]
(FOCS), the same conditions exist at the beginning of each
l2
d
(1)
⎛D⎞
exp(− αz − j∆β ijk z )dz
Am, ijk (l2 ) = jγ ⎜ ⎟ Ai (0) A j (0) Ak* (0) ∫
dz
⎝3⎠
z = l1
span. This means that the amplitudes of the FWM
where A are the amplitudes of the different waves, γ is the
crosstalk power is rising with a square law. For this
nonlinearity constant of the fiber, D=3 for two-tone
situation it is possible to significantly simplify the formula
products and D=6 for three-tone products and α is the
for the crosstalk arising from all possible combinations of
attenuation constant. The phase mismatch between the
frequencies. In the following formula it is assumed that the
products add-up in phase from span to span, and the FWM
frequency spacing between the different channels and the
an
input powers of all WDM channels are constant. The
undercompensation scheme has been utilized with
dispersion slope has been neglected. Furthermore, perfect
-20ps/(nm⋅span). At the end of the system the residual
phase matching is assumed
dispersion has been tuned to zero. In the split-step Fourier
[
Pxtalk , FWM = γ 2 P 3 (1 − e −αL ) + 4e −αL
2
] (4π
WT
2
β ∆f
2
2
2
)
2
(3)
automatically
switched
transport
network.
An
(SSF) simulations the Q-factor has been derived from the
ratio of the mean value and standard deviation of the
with the GVD constant β2 [ps²/km] and the frequency
marks measured with an eye analyzer. The coupled
spacing ∆f. The amplitude change due to GVD is
nonlinear Schrödinger equation has been employed and
neglected, which is suitable for low dispersion fibers (e.g.
solely the nonlinearity stemming from FWM has been
NZDSFs and DSFs) and moderate bit rates (e.g. 10 Gbit/s).
considered. From Fig. 1 it can be derived that the signal
These fiber types on the other hand impose major
quality significantly drops at the 4th span, which
impairments due to FWM and analytical models are
incorporates a DSF. The overall system performance is
needed to assess the signal degradation. The weighting
limited by this span. Further DSF spans do not impose a
factor WT is calculated as:
significant additional impairment. Although the FOCS
Dijk
⎛
⎞
⎟⎟
WT = ∑ ⎜⎜
ijk ⎝ 3 ⋅ (i − k )( j − k ) ⎠
≈ −7.13 + 22.22(1 − e
−
N
4.57
curve is obtained for a pure TWRS system it shows
2
) + 4.05(1 − e
(4)
−
N
31.01
)
The total number of WDM channels is denoted by N. Eq.
significantly worse impairments than the mixed fiber
system with DUCS. For both setups the analytical model
shows excellent agreement with the values obtained from
SSF simulations (compare Fig. 1).
(4) reaches a plateau for N ≈ 20. This arises from the fact,
that if an increasing number of WDM channels is added,
the channel separation to the considered center channel is
so high, that the additional contribution to the FWM
crosstalk is negligible. If there is a multitude of spans with
the same dispersion parameters, the FWM mixing power
in eq. (3) increases with the square of the number of spans.
4 Distributed Under-Compensation Scheme (DUCS)
Another common dispersion compensation scheme is
DUCS [4]. In this scheme in each span a certain amount of
residual dispersion remains uncompensated. This means
that there is a phase mismatch from span to span and the
FWM power is not increasing with a square law as in the
FOCS case. In the most general case, mixed-fiber
Fig. 1 Q-factor of the center channel (λ0 =1570nm) for a mixed fiber
system consisting of 9 NRZ channels, TWRS (D=4.9 ps/(nm⋅km),
S=0.045 ps/(nm⋅km²)) and DSF (D=1.12 ps/(nm⋅km), S=0.056
ps/(nm⋅km²)) fibers with -20 ps / (nm⋅span) DUCS (3 dBm/ch) and a pure
TWRS system with FOCS (3 dBm/ch).
transmission links are considered. It is quite common in
5
today’s systems that there are single spans of DSF in a
1 D. Breuer, et al, “WDM-Transmission over mixed-fiber
transmission system otherwise consisting of NZDSF fibers.
References
infrastructures”, ECOC 2002, P3.28, Copenhagen, 2002.
To reduce the impairments due to FWM such a system is
2 G. P. Agrawal, “Nonlinear Fiber Optics”, 3rd ed., 2001.
usually operated in the L-band, where DSFs have a small
3 4 I. B. Djordjevic, “Probability Density Function of Four
amount of local dispersion. In the system considered here
Wave Mixing Crosstalk in WDM Systems”, J. Opt. Commun.,
a DSF has been deployed in the 4th, 10th, 16th and 22th span.
vol. 22, no. 6, pp. 236-238, December 2001.
An input power of 3 dBm/ch has been used and 9 channels
4 A. Färbert, et al, “Optimised dispersion management scheme
have been placed on a 50 GHz grid. A 10 Gbit/s NRZ
for long-haul optical communication systems”, El. Lett., vol.
modulation scheme has been assumed, which is likely for
35, no. 21, pp. 1865-1866, October 1999.