Nonlinear Fiber Optics - 4 ed. Agrawal

4.2. Effect of Group-Velocity Dispersion 99
2
1.5
z/L’ D
= 0.1
(a)
0.15
(b)
Intensity
1
Intensity
0.1
0.5
0.05
0
0
−4 −2 0 2 4 −4 −2 0 2 4
Time, T/T 0
Frequency, (ν − ν 0
)T 0
Figure 4.16: (a) Shape and (b) spectrum under conditions identical to those of Figure 4.15 except
that ¯N = 10 and z/L ′ D = 0.1.
the number of oscillations seen near the trailing edge of the pulse. At the same time,
the intensity does not become zero at the oscillation minima. The effect of TOD on the
spectrum is also evident in Figure 4.15. In the absence of TOD, a symmetric two-peak
spectrum is expected (similar to the one shown in Figure 4.2 for the case φ max = 1.5π)
since φ max = 5 for the parameter values used in Figure 4.15. The effect of TOD is to
introduce spectral asymmetry without affecting the two-peak structure. This behavior
is in sharp contrast with the one shown in Figure 4.8 for the normal-dispersion case
where GVD hindered splitting of the spectrum.
Pulse evolution exhibits qualitatively different features for large values of N. Asan
example, Figure 4.16 shows the shape and spectrum of an initially unchirped Gaussian
pulse at ξ ′ = 0.1 for the case ¯N = 10. The pulse develops an oscillatory structure
with deep modulation. Because of rapid temporal variations, the third derivative in
Eq. (4.2.7) becomes large locally, and the TOD effects become more important as the
pulse propagates inside the fiber. The most noteworthy feature of the spectrum is that
the pulse energy becomes concentrated in two spectral bands, a feature common for all
values of ¯N ≥ 1. As one of the spectral bands lies in the anomalous-dispersion regime,
pulse energy in that band can form a soliton [62]. The energy in the other spectral
band, lying in the normal-GVD regime of the fiber, disperses with propagation. The
soliton-related features are discussed later in Chapter 5. The important point to note is
that, because of SPM-induced spectral broadening, the pulse does not really propagate
at the zero-dispersion wavelength even if β 2 ≈ 0 initially. In effect, the pulse creates its
own β 2 through SPM. Roughly speaking, the effective value of β 2 is given by
|β 2 |≈β 3 |δω max /2π|, (4.2.9)
where δω max is the maximum chirp given in Eq. (4.1.10). Physically, β 2 is set by the
dominant outermost spectral peaks in the SPM-broadened spectrum.
In dispersion-managed fiber links, β 2 is large locally but nearly vanishes on average.
The effects of TOD play an important role in such links, especially for short op-