Nonlinear Fiber Optics - 4 ed. Agrawal

4.1. SPM-Induced Spectral Changes 81
1
(a)
2
(b)
0.8
1
Phase, φ NL
0.6
0.4
Chirp, δωT 0
0
−1
0.2
0
−2 −1 0 1 2
Time, T/T 0
−2
−2 −1 0 1 2
Time, T/T 0
Figure 4.1: Temporal variation of SPM-induced (a) phase shift φ NL and (b) frequency chirp δω
for Gaussian (dashed curve) and super-Gaussian (solid curve) pulses.
where the minus sign is due to the choice of the factor exp(−iω 0 t) in Eq. (2.3.2). The
time dependence of δω is referred to as frequency chirping. The chirp induced by SPM
increases in magnitude with the propagated distance. In other words, new frequency
components are generated continuously as the pulse propagates down the fiber. These
SPM-generated frequency components broaden the spectrum over its initial width at
z = 0 for initially unchirped pulses.
The qualitative features of frequency chirp depend on the pulse shape. Consider,
for example, the case of a super-Gaussian pulse with the incident field U(0,T ) given
in Eq. (3.2.24). The SPM-induced chirp δω(T ) for such a pulse is
δω(T )= 2m ( ) [
L eff T 2m−1 ( ) ]
T
2m
exp − , (4.1.9)
T 0 L NL T 0 T 0
where m = 1 for a Gaussian pulse. For larger values of m, the incident pulse becomes
nearly rectangular with increasingly steeper leading and trailing edges. Figure 4.1
shows variation of (a) the nonlinear phase shift φ NL and (b) the induced frequency
chirp δω across the pulse at L eff = L NL in the cases of a Gaussian pulse (m = 1) and
a super-Gaussian pulse (m = 3). As φ NL is directly proportional to |U(0,T )| 2 in Eq.
(4.1.5), its temporal variation is identical to that of the pulse intensity.
The temporal profile of the SPM-induced frequency chirp δω shown in Figure
4.1(b) has several interesting features. First, δω is negative near the leading edge (a
red shift) and becomes positive near the trailing edge (a blue shift) of the pulse. Second,
the chirp is linear and positive (up-chirp) over a large central region of the Gaussian
pulse. Third, the chirp is considerably larger for pulses with steeper leading and trailing
edges. Fourth, super-Gaussian pulses behave differently than Gaussian pulses because
the chirp occurs only near pulse edges and does not vary in a linear fashion. The main
point is that chirp variations across an optical pulse depend considerably on the exact
pulse shape.