Nonlinear Fiber Optics - 4 ed. Agrawal

112 Chapter 4. Self-Phase Modulation
Figure 4.23: (a) Temporal and (b) spectral evolution of an unchirped Gaussian pulse over 5
dispersion lengths in the anomalous-dispersion regime. Parameters used for solving the NLS
equation were N = 2, τ R = 0.03, s = 0, and β 3 = 0.
However, even in the case of normal GVD, the inclusion of both self-steepening and
intrapulse Raman scattering is essential for an agreement between theory and experiments.
The generalized NLS equation (4.4.1) should be solved numerically to study the
impact of intrapulse Raman scattering on the evolution of ultrashort pulses. Figure
4.23 shows the temporal and spectral evolution on the anomalous-dispersion regime
of an optical fiber over five dispersion lengths for an input Gaussian pulse with a peak
power and width such that N = 2 using τ R = 0.033 fs. To isolate the effects of intrapulse
Raman scattering, we have set s = 0 and β 3 = 0 for these numerical simulations. Most
noteworthy features of Figure 4.23 compared with Figure 4.9 are: (i) a large temporal
shift of the pulse position and (ii) a Raman-induced frequency shift (RIFS) in the pulse
spectrum toward longer wavelengths. Both of them are a direct consequence of intrapulse
Raman scattering. As discussed in Section 2.3.2, when the input pulse spectrum
is relatively broad, high-frequency components of an optical pulse can pump the lowfrequency
components of the same pulse through stimulated Raman scattering, thereby
transferring energy to the red side. As the pulse spectrum shifts through the RIFS,
pulse slows down because the group velocity of a pulse changes with wavelength.
In the case of normal dispersion, both the spectral and temporal shifts are reduced
considerably because of a rapid dispersion-induced broadening of the input pulse. It is
possible to extend the moment method of Section 4.3 to obtain an approximate semianalytic
expression of the spectral shift induced by intrapulse Raman scattering [109].