Nonlinear Fiber Optics - 4 ed. Agrawal

110 Chapter 4. Self-Phase Modulation
0.4
0.3
(a)
0.1
(b)
z/L D
= 0.2
Intensity
0.2
Intensity
0.05
0.1
0
0
−5 0 5
−4 −2 0 2 4
Time, T/T 0
Frequency, (ν − ν 0
)T 0
Intensity
0.2
0.15
0.1
(c)
Intensity
0.1
0.08
0.06
0.04
(d)
z/L D
= 0.4
0.05
0.02
0
0
−10 −5 0 5 10 −4 −2 0 2 4
Time, T/T Frequency, (ν − ν )T 0 0 0
Figure 4.21: Pulse shapes and spectra at z/L D = 0.2 (upper row) and 0.4 (lower row) for a
Gaussian pulse propagating in the normal-dispersion regime of the fiber. The other parameters
are α = 0, β 3 = 0, s = 0.01, and N = 10.
β 3 = 0. The parameter N, defined in Eq. (4.2.3), is taken to be 10 and corresponds to
L D = 100L NL . In the absence of GVD (β 2 = 0), the pulse shape and the spectrum shown
in the upper row of Figure 4.21 reduce to those shown in Figures 4.19 and 4.20 in the
case of sz/L NL = 0.2. A direct comparison shows that both the shape and spectrum are
significantly affected by GVD even though the propagation distance is only a fraction
of the dispersion length (z/L D = 0.2). The lower row of Figure 4.21 shows the pulse
shape and spectrum at z/L D = 0.4; the qualitative changes induced by GVD are selfevident.
For this value of z/L D , the propagation distance z exceeds the shock distance
z s given by Eq. (4.4.10). It is the GVD that dissipates the shock by broadening the
steepened trailing edge, a feature clearly seen in the asymmetric pulse shapes of Figure
4.21. Although the pulse spectra do not exhibit deep oscillations (seen in Figure 4.20
for the dispersionless case), the longer tail on the blue side is a manifestation of selfsteepening.
With a further increase in the propagation distance z, the pulse continues
to broaden while the spectrum remains nearly unchanged.
The effect of self-steepening on pulse evolution has been seen experimentally in
liquids and solids as a larger spectral broadening on the blue side compared with that
on the red side [4]. In these early experiments, GVD played a relatively minor role, and
the spectral structure similar to that of Figure 4.20 was observed. In the case of optical