Unified description of stimulated-Raman-scattering and

four-wave mixing in wavelength-division-multiplexed

systems
Frédérique Vanholsbeeck, G. Van Simaeys, Philippe Emplit, Member, IEEE,
Marc Haelterman, and Thibaut Sylvestre
Service d’Optique et d’Acoustique, Université Libre de Bruxelles,
CP 194/5, Avenue F.D. Roosevelt 50, B-1050 Brussels, Belgium.

A detailed theoretical analysis is presented that illustrates the crosstalk induced simul-
taneously by stimulated-Raman-scattering and four-wave mixing in wavelength-division
multiplexed systems. This analysis is based on numerical simulations of coupled-mode
equations that allow to predict accurately the Raman-induced spectral power tilt for un-
limited optical bandwidth. By means of a realistic example, we show that this power tilt
can be drastically reduced in a distributed dual-pump Raman amplification scheme. It is
also shown that multiple four-wave mixing interactions induce strong parametric oscilla-
tions whose dynamics is essentially ruled by the channel spacing and the group-velocity
dispersion.

Introduction
Third-order nonlinear effects such as stimulated Raman scattering (SRS) and four-wave
mixing (FWM) limit the performance of wavelength-division-multiplexed (WDM) trans-
mission systems [1]. On one hand, SRS induces power transfer from lower to higher
wavelength channels resulting in a tilt in the power distribution among channels. On
the other hand, for small channel spacings, four-wave mixing leads to the generation of
undesirable parametric sidebands that can interfere with system operation [2]. Though
these two effects have been thoroughly investigated, they have never been studied simul-
taneously in WDM systems. Moreover, SRS has only to date been considered through a
triangular approximation of the Raman gain curve [3, 4, 5].
In this letter, we present a coupled-mode theory that fully describes the dynamics of
equally-spaced WDM channels interacting through Kerr and Raman effects simultane-
ously. In order to get a realistic insight into the involved phenomena, we model the Ra-
man response with the measured Raman gain curve of silica fibers [6]. We also take into
account the dispersion and the attenuation of the fiber, as well as the self-and cross-phase
modulations. In addition, we perform the simulations at 1.55µm for different channel
spacings ∆λ, initial mean powers Pm (0), dispersion regimes and propagation lengths. On
the basis of our theory, we propose a new simple scheme to compensate the Raman-
induced power tilt between the channels.

Characterization and compensation of the Raman-induced power tilt

Let us first study the Raman-induced power tilt for various channel spacings in an unam-
plified link. Studies based on the linear approximation of the Raman gain curve, suggest
that the tilt grows exponentially as the channel spacing as far as the total WDM band-
width is smaller than the Raman shift (ΩR /2π = 13.2THz) [3, 4]. Figure 1(a) represents
the power evolution for 6 WDM channels propagating through a 10km-long single-mode
fiber (SMF). It clearly shows that the power of all channels decreases due to the fiber loss.
The difference of loss between each WDM channel result from the SRS-induced power
transfers, which causes the tilt in the output power spectrum distribution as can be viewed
in the inset of Fig. 1(a). From a Lorentzian approximation of the Raman response curve,
we derive a general analytical formula for the power ratio (P1 /PN )dB that reads as

ωl 2 N−1 1
(P1 /PN )dB = 4.34 · Pm(0) · 2g(ΩR)( ) ∑ · Leff (1)
2 j=1 ((N − j)∆ω − ΩR )2 + ( ω2l )2

where g(ΩR ) is the maximum of the Raman gain curve, ωl is its full width at half max-
imum (7.2THz), N is the number of channels, ∆ω is the channel spacing, and L e f f =
(1 − eαz )/α is the effective length that accounts for the loss. In Fig. 1(b), we plot the
power ratio as a function of the total bandwidth (N-1)∆ω for the linear approximation of
Ref. [4] (dotted line), for our analytical Lorentzian approximation (solid line) as well as
for our simulation results (dashed-dotted line). Compared with the usual triangular ap-
proximation, these two last curves clearly demonstrate that the detrimental impact of SRS
strongly decrease for a total bandwidth slightly greater than the Raman shift.
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(a) Normalized power evolu- (b) Power ratio (dashed- (c) Normalized power evo-
tion of 6 WDM channels at dotted line) as a function of lution of the same WDM
1.55µm, D = 16.7ps/nm/km, the total bandwidth. The system as Fig.1(a) with two
L = 10km, Pt = 48mW dotted and the solid lines pumps: λP1 = 1422nm, λP2 =
(17dBm) and ∆λ = 8nm. are, respectively, the trian- 1448nm, PP1 = 28.8mW and
The insets show the output gular and the Lorentzian ap- PP2 = 24mW . The in-
spectral power distribution at proximations for the power set shows the output spec-
10km. (P1 /P6 )dB = 1.36dB, ratio. trum. (P1 /P6)dB = 0.23dB,
∆P = 47 × 10−4. ∆P = 11 × 10−4.
Figure 1: Characterization and compensation of the Raman-induced power tilt.

As the Raman-induced tilt is one of the major limitations of transmission performances,

its suppression would allow for increased total bandwidth, power per channel and ampli-
fier spacing along the transmission line [7]. We suggest to exploit the particular shape
of the Raman gain curve to compensate the tilt through a forward dual-pumped Raman
amplification scheme. With a proper choice of the pump wavelengths, the SRS-induced
tilt could be balanced by the resulting gain curve. The lower-wavelength pump amplifies
the lower-wavelength channels while the second pump power is transferred to the mid-
dle channels. As a result, the upper-wavelength channels, which fall outside the pumps
bandwidth, get essentially amplified through the SRS from the lower-wavelength chan-
nels. This principle of Raman tilt suppression is illustrated in Fig. 1(c) that shows the
same power evolution as in Fig. 1(a) but now in the presence of two pumps at 1422nm
and 1448nm respectively. For the convenience of the analysis, we introduced the stan-
dard deviation of the total power ∆P = ∑Nj=1 ((Pj (L) − Pm (L))2 )/Pm2 (0)). With our Raman
compensating technique, the tilt is reduced by 83% and the power deviation ∆P by 77%.
As we can see in Fig. 1(c), the forward pumping scheme presents however some im-
portant drawbacks for longer amplifier spacing. Indeed, the pump waves are completely
depleted before the output of the fiber while the SRS-induced transfer goes on all over the
fiber span, leading to an asymmetry in the power distribution between the WDM chan-
nels. Increasing the initial pump power does not make sense as pumps depletion would
happen faster. We expect that this difficulty can be alleviated in a backward pumping con-
figuration. We believe that this two-pump Raman amplification technique for achieving
Raman-induced power tilt suppression constitutes a simple and promising solution that
could be implemented in future telecommunication systems.

Influence of the four-wave mixing

To complete this study, we also investigated the power exchanges induced by FWM be-
tween channels in WDM systems in the presence of SRS. The amplitude of the FWM
products increases as the minimum linear phase mismatch (∆k = β2 ∆ω2 ) decreases. There-
fore, we consider a channel spacing as low as 1nm corresponding to a total frequency
bandwidth of 0.6THz, which is much smaller than the Raman shift. In this configuration,
SRS effect remains small in comparison with the FWM processes.
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(a) D=16.7ps/nm/km. (b) D=-17.6ps/nm.km. (c) D=1.17ps/nm.km.

Figure 2: Normalized power evolution of 6 WDM channels at 1.55µm, L = 1km, Pm =
8mW, ∆λ = 1nm in different dispersion regimes. The dotted lines indicate where the
spectral power distributions shown in the insets are measured.

The simulations whose results are shown in Fig. 2(a) and 2(b) and performed, respectively,
in the anomalous and normal dispersion regime. The power exchanges exhibit a recurrent
dynamics over a large fiber length that corresponds to the maximum coherence length
(Lc = 2π/∆k) associated with the most efficient FWM processes. As a result, the power
tends to concentrate in the central channels in the anomalous dispersion regime whereas
the opposite behavior is observed in the normal dispersion regime as shown in the insets.
In Fig. 2(c) are depicted the results obtained in the nearly zero-dispersion regime. In this
case, the power exchanges are no longer recurrent because the FWM processes are nearly
phase-matched giving rise to strong parametric power transfers (L c → ∞ as ∆k → 0). It
is worth noting that the dynamics are entirely determined once the dispersion regime and
the initial phases of the channels are fixed.
In Fig. 3(a) that differs from Fig.2(a) only by the initial phase mismatches between chan-
nels, i.e. the anomalous dispersion parameter is D = 16.7ps/nm.km, we can observe that
the amplitude of parametric oscillations increases. These results suggest that, by a proper
control of initial phases and dispersion regime, it should be possible to reduce the ampli-
tude of the FWM processes. In Fig. 3(b) and Fig. 3(c), we show the resulting output power
spectra for a channel spacing of 1.5 and 2nm respectively. With such channel spacings
we emphasize the mutual influence of SRS and FWM. As can be seen in Fig. 3(c), the tilt
due to SRS is periodically compensated by the FWM process for anomalous dispersion
(∆P = 2.6 × 10−6 at 600 m and 6.2 × 10−6 at 700 m).
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(a) ∆λ = 1nm. (b) ∆λ = 1.5nm. (c) ∆λ = 2nm.

Figure 3: Normalized power evolution of 6 WDM channels at 1.55µm, L = 1km, Pm =
8mW in a SMF for different channel spacings. For (a), the channels are mismatched at
the input of the fiber. The dotted lines indicate where the spectral power distributions
shown in the insets are measured.
Conclusion
By introducing the measured Raman gain curve in our model we have shown that the
impact of SRS-induced impairment in WDM systems dramatically decreases with respect
to the triangular approximation. We have also demonstrated that two-pump distributed
Raman amplification can be exploited to control and reduce the Raman-induced power
tilt. Moreover, we have characterized the periodic power exchanges between channels
due to four-wave mixing and its influence on the Raman-induced power tilt.

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