"Introduction". In: Fiber-Optic Communication Systems

526 APPENDIX C. GENERAL FORMULA FOR PULSE BROADENING

The quantityG(ω)represents the source spectrum. The subscriptsin Eq. (C.11) is a
reminder that the angle brackets now denote an ensemble average over the field fluctu-
ations.
The moments〈t〉and〈t^2 〉are now replaced by〈〈t〉〉sand〈〈t^2 〉〉swhere the outer
angle brackets stand for the ensemble average over field fluctuations. Both of them
can be calculated in the special case in which the source spectrum is assumed to be
Gaussian, i.e.,

G(ω)=

1

σω

√

2 π

exp

(

−

ω^2

2 σω^2

)

, (C.12)

whereσωis the RMS spectral width of the source. For example,

〈〈t〉〉s=

∫∞

−∞

τ(ω)〈|S(ω)|^2 〉sdω−i

∫∞

−∞

〈S∗(ω)Sω(ω〉sdω

=L

∫∫∞

−∞

(β 1 +β 2 ω+^12 β 3 ω^2 )|Sp(ω−ω 1 )|^2 G(ω 1 )dω 1 dω (C.13)

Since both the pulse spectrum and the source spectrum are assumed to be Gaussian, the
integral overω 1 can be performed first, resulting in another Gaussian spectrum. The
integral overωis then straightforward in Eq. (C.13) and yields

〈〈t〉〉s=L

[

β 1 +

β 3

8 σ 02

( 1 +C^2 +Vω^2 )

]

, (C.14)

whereVω= 2 σωσ 0. Repeating the same procedure for〈〈t^2 〉〉s, we recover Eq. (2.4.13)
for the ratioσ/σ 0.