"Introduction". In: Fiber-Optic Communication Systems

9.5. IMPACT OF AMPLIFIER NOISE 437

the same for all amplifiers because ASE in any two amplifiers is not correlated (the
subscriptnhas been dropped for this reason).
The pulse shape depends on whether the GVD is constant along the entire link or is
changing in a periodic manner through a dispersion map. In the case of DM solitons,
the exact form ofB(za,t)can only be obtained by solving Eq. (9.4.5) numerically.
The use of Gaussian approximation for the pulse shape simplifies the analysis without
introducing too much error because the pulse shape deviates from Gaussian only in
the pulse wings (which contribute little to the integrals because of their low intensity
levels). However, Eq. (9.4.6) should be modified as

B(z,t)=aexp[−( 1 +iC)(t−q)^2 / 2 T^2 −iΩ(t−q)+iφ] (9.5.12)

to include the frequency shiftΩand the position shiftqexplicitly, both of which are
zero in the absence of optical amplifiers. The six parameters (a,C,T,q,Ω, andφ) vary
withzin a periodic fashion. Using Eq. (9.5.12) in Eqs. (9.5.8)–(9.5.11), the variances
and correlations of fluctuations are found to be

〈(δE)^2 〉= 2 SspE 0 , 〈(δΩ)^2 〉=(Ssp/E 0 )[( 1 +C 02 )/T 02 ], (9.5.13)

〈(δq)^2 〉=(Ssp/E 0 )T 02 , 〈δEδq〉= 0 , (9.5.14)

〈δΩδq)〉=(Ssp/E 0 )C 0 , 〈δEδΩ〉=( 2 π−^1 /^2 )SspC 0 /T 0. (9.5.15)

The input pulse parameters appear in these equations because the pulse recovers its
original form at each amplifier for DM solitons.
In the case of constant-dispersion fibers or DDFs, the soliton remains unchirped
and maintains a “sech” shape. In this case, Eq. (9.5.12) should be replaced with

B(z,t)=asech[(t−q)/T]−iΩ(t−q)+iφ]. (9.5.16)

Using Eq. (9.5.16) in Eqs. (9.5.8)–(9.5.11), the variances are given by

〈(δE)^2 〉= 2 SspE 0 , 〈(δΩ)^2 〉=

2 Ssp

3 E 0 T 02

, 〈(δq)^2 〉=

π^2 Ssp

6 E 0

T 02 , (9.5.17)

but all three cross-correlations are zero. The presence of chirp on the DM solitons is
responsible for producing cross-correlations.

9.5.2 Energy and Frequency Fluctuations

Energy fluctuations induced by optical amplifiers degrade the optical SNR. To find
the SNR, we integrate Eq. (9.5.3) between two neighboring amplifiers and obtain the
recurrence relation
E(zn)=E(zn− 1 )+δEn, (9.5.18)

whereE(zn)denotes energy at the output of thenth amplifier. It is easy to solve this
recurrence relation for a cascaded chain ofNAamplifiers to obtain

Ef=E 0 +

NA

∑

n= 1

δEn, (9.5.19)