Nonlinear Fiber Optics - 4 ed. Agrawal

94 Chapter 4. Self-Phase Modulation
components should be solved numerically, it can be solved analytically in some cases
if the pulse shape is assumed not to change significantly.
4.2.3 Optical Wave Breaking
Equation (4.2.1) suggests that the effects of SPM should dominate over those of GVD
for values of N ≫ 1, at least during the initial stages of pulse evolution. In fact, by
introducing a new distance variable as Z = N 2 ξ = z/L NL , Eq. (4.2.1) can be written as
i ∂U
∂Z − d ∂ 2 U
2 ∂τ 2 + |U|2 U, (4.2.5)
where fiber losses are neglected and d = β 2 /(γP 0 T0 2 ) is a small parameter. If we use
the transformation
U(z,T )= √ ( ∫ T
)
ρ(z,T ) exp i v(z,T )dT
(4.2.6)
0
in Eq. (4.2.5), we find that the pulse-propagation problem reduces approximately to a
fluid-dynamics problem in which the variables ρ and v play, respectively, the role of
density and velocity of a fluid [43]. In the optical case, these variables represent the
power and chirp profiles of the pulse. For a square-shape pulse, the pulse-propagation
problem becomes identical to the one related to “breaking of a dam” and can be solved
analytically. This solution is useful for lightwave systems employing the NRZ format
and provides considerable physical insight [44]–[46].
The approximate solution, although useful, does not account for a phenomenon
termed optical wave breaking [47]–[53]. It turns out that GVD cannot be treated as a
small perturbation even when N is large. The reason is that, because of a large amount
of the SPM-induced frequency chirp imposed on the pulse, even weak dispersive effects
lead to significant pulse shaping. In the case of normal dispersion (β 2 > 0), the
pulse becomes nearly rectangular with relatively sharp leading and trailing edges and
is accompanied by a linear chirp across its entire width [23]. It is this linear chirp that
can be used to compress the pulse by passing it through a dispersive delay line.
The GVD-induced pulse shaping has another effect on pulse evolution. It increases
the importance of GVD because the second derivative in Eq. (4.2.1) becomes large
near the pulse edges. As a consequence, the pulse develops a fine structure near its
edges. Figure 4.11 shows the temporal and spectral evolution for N = 30 in the case
of an initially unchirped Gaussian pulse. The oscillatory structure near pulse edges is
present already at z/L D = 0.06. Further increase in z leads to broadening of the pulse
tails. The oscillatory structure depends considerably on the pulse shape. Figure 4.12
shows the pulse shape and the spectrum at z/L D = 0.08 for an unchirped “sech” pulse.
A noteworthy feature is that rapid oscillations near pulse edges are always accompanied
by the sidelobes in the spectrum. The central multipeak part of the spectrum is
also considerably modified by the GVD. In particular, the minima are not as deep as
expected from SPM alone.
The physical origin of temporal oscillations near the pulse edges is related to optical
wave breaking [47]. Both GVD and SPM impose frequency chirp on the pulse as it