436 CHAPTER 9. SOLITON SYSTEMS
Fiber losses do not appear in Eq. (9.5.3) because of the transformationB=A/
√
pmade
in deriving Eq. (9.4.5); the actual pulse energy is given bypE, wherep(z)is obtained
by solving Eq. (9.3.11).
The physical meaning of the moment equations is clear from Eqs. (9.5.3)–(9.5.5).
BothEandΩremain constant while propagating inside optical fibers but change in a
random fashion at each amplifier location. Equation (9.5.5) shows how frequency fluc-
tuations induced by an amplifier become temporal fluctuations because of the GVD.
Physically speaking, the group velocity of the pulse depends on frequency. A ran-
dom change in the group velocity results in a shift of the soliton position by a random
amount within the bit slot. As a result, frequency fluctuations are converted into timing
jitter by the GVD. The last term in Eq. (9.5.5) shows that ASE also shifts the soliton
position directly.
Fluctuations in the position and frequency of a soliton at any amplifier vanish on
average but their variances are finite. Moreover, the two fluctuations are not indepen-
dent as they are produced by the same physical mechanism (spontaneous emission).
We thus need to consider how the optical fieldB(z,t)is affected by ASE and then cal-
culate the variances and correlation functions ofE,Ω, andq. At each amplifier, the
fieldB(z,t)changes byδB(z,t)because of ASE. The fluctuationδB(z,t)vanishes on
average; its second-order moment can be written as
〈δB(za,t)δB(za,t′)〉=Sspδ(t−t′), (9.5.6)wherezadenotes the location of an amplifier and
Ssp=nsphν 0 (G− 1 ) (9.5.7)is the spectral density of ASE noise assumed to be constant (white noise) by treating
the ASE process as a Markovian stochastic process [9]. This is justified in view of
the independent nature of each spontaneous-emission event. The angle brackets in Eq.
(9.5.6) denote an ensemble average over all such events. In Eq. (9.5.7),Grepresents
the amplifier gain,hν 0 is the photon energy, and the spontaneous emission factornspis
related to the noise figureFnof the amplifier asFn= 2 nsp.
The moments ofδEn,δqn, andδΩnare obtained by replacingBin Eqs. (9.5.1) and
(9.5.2) withB+δBand linearizing inδB. For an arbitrary pulse shape, the second-
order moments are given by [122]
〈(δE)^2 〉= 2 Ssp∫∞−∞|B|^2 dt, 〈(δq)^2 〉=2 Ssp
E^20∫∞−∞(t−q)^2 |B|^2 dt, (9.5.8)〈(δΩ)^2 〉=2 Ssp
E^20∫∞−∞∣∣
∣
∣
∂B
∂t∣∣
∣
∣
2
dt, 〈δEδq〉=2 Ssp
E 0∫∞−∞(t−q)|B|^2 dt, (9.5.9)〈δEδΩ〉=iSsp
E 0∫∞−∞(
V∗
∂V
∂t−V
∂V∗
∂t)
dt, (9.5.10)〈δΩδq〉=iSsp
2 E^20∫∞−∞(t−q)(
V∗
∂V
∂t−V
∂V∗
∂t)
dt, (9.5.11)whereV=Bexp(iΩt). The integrals in these equations can be calculated ifB(za,t)
is known at the amplifier location. The variances and correlations of fluctuations are