"Introduction". In: Fiber-Optic Communication Systems

APPENDIX C. GENERAL FORMULA FOR PULSE BROADENING 525

a frequency-dependent phase. The propagation constantβdepends on frequency be-
cause of dispersion. It can also depend onzwhen dispersion management is used or
when fiber parameters such as the core diameter are not uniform along the fiber.
If we substitute Eq. (C.4) in Eqs. (C.1) and (C.2), perform the derivatives as indi-
cated, and calculateσ^2 =〈t^2 〉−〈t〉^2 , we obtain

σ^2 =σ 02 +[〈τ^2 〉−〈τ〉^2 ]+ 2 [〈τθω〉−〈τ〉〈θω〉], (C.5)

where the angle brackets now denote average over the input pulse spectrum such that

〈f〉=

1

2 π

∫∞

−∞

f(ω)|S(ω)|^2 dω. (C.6)

In Eq. (C.5),σ 0 is the root-mean-square (RMS) width of input pulses,θω=dθ/dω,
andτis the group delay defined as

τ(ω)=

∫L

0

∂β(z,ω)

∂ω

dz (C.7)

for a fiber of lengthL. Equation (C.5) can be used for pulses of arbitrary shape, width,
and chirp. It makes no assumption about the form ofβ(z,ω)and thus can be used for
dispersion-managed fiber links containing fibers with arbitrary dispersion characteris-
tics.
As a simple application of Eq. (C.5), one can use it to derive Eq. (2.4.22). Assuming
uniform dispersion and expandingβ(z,ω)to third-order inω, the group delay is given
by
τ(ω)=(β 1 +β 2 ω+^12 β 3 ω^2 )L. (C.8)

For a chirped Gaussian pulse, Eq. (2.4.13) provides the following expressions forSand
θ:

S(ω)=

√

4 πT 02

1 +C^2

exp

[

−

ω^2 T 02

2 ( 1 +C^2 )

]

, θ(ω)=

Cω^2 T 02

2 ( 1 +C^2 )

−tan−^1 C. (C.9)

The averages in Eq. (C.5) can be performed analytically using Eqs. (C.8) and (C.9) and
result in Eq. (2.4.22).
As another application of Eq. (C.5), consider the derivation of Eq. (2.4.23) that
includes the effects of a wide source spectrum. For such a pulse, the input field can
be written asA( 0 ,t)=A 0 (t)f(t), wheref(t)represents the pulse shape andA 0 (t)is
fluctuating because of the partially coherent nature of the source. The spectrumS(ω)
now becomes a convolution of the pulse spectrum and the source spectrum such that

S(ω)=

1

2 π

∫∞

−∞

Sp(ω−ω 1 )F(ω 1 )dω 1 , (C.10)

whereSpis the pulse spectrum andF(ωs)is the fluctuating field spectral component at
the source with the correlation function of the form

〈F∗(ω 1 )F(ω 2 )〉s=G(ω 1 )δ(ω 1 −ω 2 ). (C.11)