Nonlinear Fiber Optics - 4 ed. Agrawal

4.2. Effect of Group-Velocity Dispersion 95
Figure 4.11: (a) Temporal and (b) spectral evolution of an initially unchirped Gaussian pulse
with N = 30 at a distance z = 0.1L D in the normal-dispersion regime of an optical fiber.
travels down the fiber. However, as seen from Eqs. (3.2.14) and (4.1.9), although the
GVD-induced chirp is linear in time, the SPM-induced chirp is far from being linear
across the entire pulse. Because of the nonlinear nature of the composite chirp, different
parts of the pulse propagate at different speeds [52]. In particular, in the case of normal
GVD (β 2 > 0), the red-shifted light near the leading edge travels faster and overtakes
the unshifted light in the forward tail of the pulse. The opposite occurs for the blueshifted
light near the trailing edge. In both cases, the leading and trailing regions of
the pulse contain light at two different frequencies that interfere. Oscillations near the
pulse edges in Figure 4.11 are a result of such interference.
The phenomenon of optical wave breaking can also be understood as a four-wave
mixing process (see Chapter 10). Nonlinear mixing of two different frequencies ω 1
and ω 2 in the pulse tails creates new frequencies at 2ω 1 − ω 2 and 2ω 2 − ω 1 . The
spectral sidelobes in Figure 4.12 represent these new frequency components. Temporal
oscillations near pulse edges and the spectral sidelobes are manifestations of the same
phenomenon. It is interesting to note that optical wave breaking does not occur in the
case of anomalous GVD. The reason is that the red-shifted part of the pulse cannot take
over the fast-moving forward tail. Instead, the energy in the pulse tail spreads out, and
the pulse acquires a pedestal [52].
The results shown in Figures 4.11 and 4.12 are obtained for an unchirped pulse
(C = 0). Pulses emitted from practical laser sources are often chirped and may follow
quite a different evolution pattern depending on the sign and magnitude of the chirp
parameter C [49]. Figure 4.13 shows the pulse shape and the spectrum for a chirped