Nonlinear Fiber Optics - 4 ed. Agrawal

4.4. Higher-Order Nonlinear Effects 109
Intensity
0.2
z/L NL
= 20
0.15
s = 0.01
0.1
0.05
0
−4 −3 −2 −1 0 1 2 3 4 5 6
Figure 4.20: Spectrum of a Gaussian pulse at a distance z = 0.2L NL /s, where s = 0.01 and L NL
is the nonlinear length. Self-steepening is responsible for the asymmetry in the SPM-broadened
spectrum. The effects of GVD are neglected.
spectrum using
S(ω)=
∣
∫ ∞
−∞
[I(z,τ)] 1/2 2
exp[iφ(z,τ)+i(ω − ω 0 )τ]dτ
∣ . (4.4.11)
Figure 4.20 shows the calculated spectrum at sz/L NL = 0.2 for s = 0.01. The most
notable feature is spectral asymmetry—the red-shifted peaks are more intense than
blue-shifted peaks. The other notable feature is that SPM-induced spectral broadening
is larger on the blue side (called the anti-Stokes side in the terminology used for
stimulated Raman scattering) than the red side (or the Stokes side). Both of these features
can be understood qualitatively from the changes in the pulse shape induced by
self-steepening. The spectrum is asymmetric simply because pulse shape is asymmetric.
A steeper trailing edge of the pulse implies larger spectral broadening on the blue
side as SPM generates blue components near the trailing edge (see Figure 4.1). In the
absence of self-steepening (s = 0), a symmetric six-peak spectrum is expected because
φ max ≈ 6.4π for the parameter values used in Figure 4.20. Self-steepening stretches the
blue portion. The amplitude of the high-frequency peaks decreases because the same
energy is distributed over a wider spectral range.
4.4.2 Effect of GVD on Optical Shocks
The spectral features seen in Figure 4.20 are considerably affected by GVD, which
cannot be ignored when short optical pulses propagate inside silica fibers [100]–[107].
The pulse evolution in this case is studied by solving Eq. (4.4.1) numerically. Figure
4.21 shows the pulse shapes and the spectra at z/L D = 0.2 and 0.4 in the case of an
initially unchirped Gaussian pulse propagating with normal dispersion (β 2 > 0) and