Nonlinear Fiber Optics - 4 ed. Agrawal

3.3. Third-Order Dispersion 67
6
5
C = 1
β 2
> 0
Broadening Factor
4
3
2
β 2
< 0
β 2
= 0
1
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Distance, z/L’ D
Figure 3.8: Broadening factor as a function of z/L D ′ for a chirped Gaussian pulse in the vicinity
of λ D such that L D = L D ′ . Dashed curve corresponds to the case of λ 0 = λ D (β 2 = 0).
ω 0 [28]. If the source bandwidth δω is much smaller than the pulse bandwidth Δω, its
effect on the pulse broadening can be neglected. However, for many light sources such
as light-emitting diodes (LEDs) this condition is not satisfied, and it becomes necessary
to include the effects of a finite source bandwidth. In the case of a Gaussian pulse and
a Gaussian source spectrum, the generalized form of Eq. (3.3.14) is given by [9]
σ 2
σ0
2
(
= 1 + Cβ ) 2
2z
2σ0
2 +(1 +Vω)
2
( )
β2 z 2
+(1 +C 2 +Vω) 2 2 1 ( )
β3 z 2
, (3.3.16)
2
σ 2 0
where V ω = 2σ ω σ 0 and σ ω is the RMS width of the Gaussian source spectrum. This
equation describes broadening of chirped Gaussian pulses in a linear dispersive medium
under quite general conditions. It can be used to discuss the effect of GVD on the performance
of lightwave systems [29].
4σ 3 0
3.3.3 Arbitrary-Shape Pulses
The formal similarity of Eq. (3.2.1) with the Schrödinger equation can be exploited
to obtain an analytic expression of the RMS width for pulses of arbitrary shape while
including the third- and higher-order dispersive effects [30]. For this purpose, we write
Eq. (3.3.1) in an operator form as
i ∂U
∂z
= ĤU, (3.3.17)