Nonlinear Fiber Optics - 4 ed. Agrawal

66 Chapter 3. Group-Velocity Dispersion
From the convolution theorem
Ĩ(z,ω)=
∫ ∞
−∞
Ũ(z,ω − ω ′ )Ũ ∗ (z,ω ′ )dω ′ . (3.3.11)
Performing the differentiation and limit operations indicated in Eq. (3.3.9), we obtain
〈T n 〉 = (i)n
N c
∫ ∞
−∞
Ũ ∗ (z,ω) ∂ n
∂ωnŨ(z,−ω)dω. (3.3.12)
In the case of a chirped Gaussian pulse, Ũ(z,ω) can be obtained from Eqs. (3.2.16)
and (3.3.2) and is given by
( 2πT
2
) 1/2
Ũ(z,ω)= 0 iω
2
exp[ (β 2 z + iT 2 )
0
+ i ]
1 + iC 2 1 + iC 6 β 3ω 3 z . (3.3.13)
If we differentiate Eq. (3.3.13) two times and substitute the result in Eq. (3.3.12), we
find that the integration over ω can be performed analytically. Both 〈T 〉 and 〈T 2 〉 can
be obtained by this procedure. Using the resulting expressions in Eq. (3.2.26), we
obtain [9]
σ
σ 0
=
[ (
1 + Cβ 2z
2σ 2 0
) 2 ( )
β2 z 2
+
2σ0 2 +(1 +C 2 ) 2 1 ( ) ]
β3 z 2 1/2
2 4σ0
3 , (3.3.14)
where σ 0 is the initial RMS width of the chirped Gaussian pulse (σ 0 = T 0 / √ 2). As
expected, Eq. (3.3.14) reduces to Eq. (3.2.19) for β 3 = 0.
Equation (3.3.14) can be used to draw several interesting conclusions. In general,
both β 2 and β 3 contribute to pulse broadening. However, the dependence of their contributions
on the chirp parameter C is qualitatively different. Whereas the contribution
of β 2 depends on the sign of β 2 C, the contribution of β 3 is independent of the sign of
both β 3 and C. Thus, in contrast to the behavior shown in Figure 3.2, a chirped pulse
propagating exactly at the zero-dispersion wavelength never experiences width contraction.
However, even small departures from the exact zero-dispersion wavelength
can lead to initial pulse contraction. This behavior is illustrated in Figure 3.8 where
the broadening factor σ/σ 0 is plotted as a function of z/L ′ D for C = 1 and L D = L ′ D .
The dashed curve shows, for comparison, the broadening expected when β 2 = 0. In the
anomalous-dispersion regime the contribution of β 2 can counteract the β 3 contribution
in such a way that dispersive broadening is less than that expected when β 2 = 0 for
z ∼ L ′ D . For large values of z such that z ≫ L D/|C|, Eq. (3.3.14) can be approximated
by
σ/σ 0 =(1 +C 2 ) 1/2 [1 +(1 +C 2 )(L D /2L ′ D) 2 ] 1/2 (z/L D ), (3.3.15)
where we have used Eqs. (3.1.5) and (3.3.6). The linear dependence of the RMS pulse
width on the propagation distance for large values of z is a general feature that holds
for arbitrary pulse shapes, as discussed in the next section.
Equation (3.3.14) can be generalized to include the effects of a finite source bandwidth
[9]. Spontaneous emission in any light source produces amplitude and phase
fluctuations that manifest as a finite bandwidth δω of the source spectrum centered at