Nonlinear Fiber Optics - 4 ed. Agrawal

4.1. SPM-Induced Spectral Changes 85
Figure 4.4: Evolution of SPM-broadened spectra for fiber lengths in the range 0 to 10L NL for
unchirped (C = 0) Gaussian (m = 1) and super-Gaussian (m = 3) pulses.
4.1.3 Effect of Pulse Shape and Initial Chirp
As mentioned earlier, the shape of the SPM-broadened spectrum depends on the pulse
shape, and on the initial chirp if input pulse is chirped [11]–[13]. Figure 4.4 compares
the evolution of pulse spectra for Gaussian (m = 1) and super-Gaussian (m = 3)
pulses over 10L NL using Eq. (3.2.24) in Eq. (4.1.13) and performing the integration
numerically. In both cases, input pulses are assumed to be unchirped (C = 0) and fiber
losses are ignored (α = 0). The qualitative differences between the two spectra can be
understood by referring to Figure 4.1, where the SPM-induced chirp is shown for the
Gaussian and super-Gaussian pulses. The spectral range is about three times larger for
the super-Gaussian pulse because the maximum chirp from Eq. (4.1.10) is about three
times larger in that case. Even though both spectra in Figure 4.4 exhibit multiple peaks,
most of the energy remains in the central peak for the super-Gaussian pulse. This is so
because the chirp is nearly zero over the central region in Figure 4.1 for such a pulse,
a consequence of the nearly uniform intensity of super-Gaussian pulses for |T | < T 0 .
The frequency chirp occurs mainly near the leading and trailing edges. As these edges
become steeper, the tails in Figure 4.4 extend over a longer frequency range but, at the
same time, carry less energy because chirping occurs over a small time duration.
An initial frequency chirp can also lead to drastic changes in the SPM-broadened
pulse spectrum. This is illustrated in Figure 4.5, which shows the spectra of a Gaussian
pulse for both positive and negative chirps using φ max = 4.5π. It is evident that the sign
of the chirp parameter C plays a critical role. For C > 0, spectral broadening increases