"Introduction". In: Fiber-Optic Communication Systems

8.3. SYSTEM PERFORMANCE ISSUES 367

to estimate the limiting value of the channel power. A simple model considers the
depletion of the highest-frequency channel in the worst case in which 1 bits of all
channels overlap completely simultaneously [165]. The amplification factor for each
channel isGm=exp(gmLeff), whereLeffis the effective interaction length as defined
in Eq. (2.6.2) andgm=gR(Ωm)Pch/Aeffis the Raman gain atΩm=ω 1 −ωm.For
gmLeff1, the shortest-wavelength channel atω 1 is depleted by a fractiongmLeffdue
to Raman amplification of themth channel. The total depletion for aM-channel WDM
system is given by

DR=

M

∑

m= 2

gR(Ωm)PchLeff/Aeff. (8.3.9)

The summation in Eq. (8.3.9) can be carried out analytically if the Raman gain
spectrum (see Fig. 2.18) is approximated by a triangular profile such thatgRincreases
linearly for frequencies up to 15 THz with a slopeSR=dgR/dνand then drops to
zero. UsinggR(Ωm)=mSR∆νch, the fractional power loss for the shortest-wavelength
channel becomes [165]
DR=^12 M(M− 1 )CRPchLeff, (8.3.10)

whereCR=SR∆νch/( 2 Aeff). In deriving this equation, channels were assumed to have
a constant spacing∆νchand the Raman gain for each channel was reduced by a factor
of 2 to account for random polarization states of different channels.
A more accurate analysis should consider not only depletion of each channel be-
cause of power transfer to longer-wavelength channels but also its own amplification
by shorter-wavelength channels. If all other nonlinear effects are neglected along with
GVD, evolution of the powerPnassociated with thenth channel is governed by the
following equation (see Section 6.3):

dPn

dz

+αPn=CRPn

M

∑

m= 1

(n−m)Pm, (8.3.11)

whereαis assumed to be the same for all channels. This set ofMcoupled nonlinear
equations can be solved analytically. For a fiber of lengthL, the result is given by [172]

Pn(L)=Pn( 0 )e−αL

Ptexp[(n− 1 )CRPtLeff]

∑Mm= 1 Pm( 0 )exp[(m− 1 )CRPtLeff]

, (8.3.12)

wherePt=∑Mm= 1 Pm( 0 )is the total input power in all channels. This equation shows
that channel powers follow an exponential distribution because of Raman-induced cou-
pling among all channels.
The depletion factorDRfor the shorter-wavelength channel (n=1) is obtained
usingDR=(P 10 −P 1 )/P 10 , whereP 10 =P 1 ( 0 )exp(−αL)is the channel power expected
in the absence of SRS. In the case of equal input powers in all channels,Pt=MPchin
Eq. (8.3.12), andDRis given by

DR= 1 −exp

[

−

1

2

M(M− 1 )CRPchLeff

]

Msinh(^12 MCRPchLeff)

sinh(^12 M^2 CRPchLeff)

. (8.3.13)