8.3. SYSTEM PERFORMANCE ISSUES 373
both cases, the system performance is degraded because of a loss in the channel power,
but the coherent crosstalk degrades system performance much more severely.
The FWM process in optical fibers is governed by a set of four coupled equations
whose general solution requires a numerical approach [59]. If we neglect the phase
shifts induced by SPM and XPM, assume that the three channels participating in the
FWM process remain nearly undepleted, and include fiber losses, the amplitudeAFof
the FWM component at the frequencyωFis governed by
dAF
dz=−
α
2
AF+dFγAiAjA∗kexp(−i∆kz), (8.3.16)whereAm(z)=Am( 0 )exp(−αz/ 2 )form=i,j,kanddF= 2 −δijis the degeneracy
factor defined such that its value is 1 wheni=jbut doubles wheni =j. This equation
can be easily integrated to obtainAF(z). The power transferred to the FWM component
in a fiber of lengthLis given by [200]
PF=|AF(L)|^2 =ηF(dFγL)^2 PiPjPke−αL, (8.3.17)wherePm=|Am( 0 )|^2 is the launched power in themth channel andηFis a measure of
the FWM efficiency defined as
ηF=∣
∣
∣∣^1 −exp[−(α+i∆k)L]
(α+i∆k)L∣
∣
∣∣
2. (8.3.18)
The FWM efficiencyηFdepends on the channel spacing through the phase mis-
match governed by
∆k=βF+βk−βi−βj≈β 2 (ωi−ωk)(ωj−ωk), (8.3.19)where the propagation constants were expanded in a Taylor series aroundωc=(ωi+
ωj)/2 andβ 2 is the GVD parameter at that frequency. If the GVD of the transmission
fiber is relatively large, (|β 2 |>5ps^2 /km),ηFnearly vanishes for typical channel spac-
ings of 50 GHz or more. In contrast,ηF≈1 close to the zero-dispersion wavelength
of the fiber, resulting in considerable power in the FWM component, especially at high
channel powers. In the case of equal channel powers,PFincreases asPch^3. This cubic
dependence of the FWM component limits the channel powers to below 1 mW if FWM
is nearly phase matched. Since the number of FWM components for anM-channel
WDM system increases asM^2 (M− 1 )/2, the total power in all FWM components can
be quite large.
A simple scheme for reducing the FWM-induced degradation consists of design-
ing WDM systems with unequal channel spacings [167]. The main impact of FWM
in this case is to reduce the channel power. This power depletion results in a power
penalty that is relatively small compared with the case of equal channel spacings. Ex-
perimental measurements on WDM systems confirm the advantage of unequal channel
spacings. In a 1999 experiment, this technique was used to transmit 22 channels, each
operating at 10 Gb/s, over 320 km of dispersion-shifted fiber with 80-km amplifier
spacing [201]. Channel spacings ranged from 125 to 275 GHz in the 1532- to 1562-nm
wavelength region and were determined using a periodic allocation scheme [202]. The