Nonlinear Fiber Optics - 4 ed. Agrawal

86 Chapter 4. Self-Phase Modulation
Spectral Intensity
1
0.8
0.6
0.4
0.2
(a)
1
(b)
C = 0 C = 10
0.8
0.6
0.4
0.2
Spectral Intensity
0
−4 −2 0 2 4
Normalized Frequency
0
−4 −2 0 2 4
Normalized Frequency
Spectral Intensity
1
0.8
0.6
0.4
0.2
(c)
C = −10
Spectral Intensity
1
0.8
0.6
0.4
0.2
(d)
C = −20
0
−4 −2 0 2 4
Normalized Frequency
0
−4 −2 0 2 4
Normalized Frequency
Figure 4.5: Comparison of output spectra for Gaussian pulses for four values of chirp parameter
C when fiber length and peak powers are chosen such that φ max = 4.5π. Spectrum broadens for
C > 0 but becomes narrower for C < 0 when compared with that of the input pulse.
and the oscillatory structure becomes less pronounced, as seen in Figure 4.5(b). However,
a negatively chirped pulse undergoes spectral narrowing through SPM, as seen
clearly in parts (c) and (d) of Figure 4.5. The spectrum contains a central dominant
peak for C = −20 and narrow further as C decreases. This behavior can be understood
from Eq. (4.1.10) by noting that the SPM-induced phase shift is linear and positive
(frequency increases with increasing T ) over the central portion of a Gaussian pulse
(see Figure 4.1). Thus, it adds with the initial chirp for C > 0, resulting in an enhanced
oscillatory structure. In the case of C < 0, the two chirp contributions are of opposite
signs (except near the pulse edges), and the pulse becomes less chirped. If we employ
the approximation that φ NL (t) ≈ φ max (1 − t 2 /T0 2 ) near the pulse center for Gaussian
pulses, the SPM-induced chirp is nearly cancelled for C = −2φ max . This approximation
provides a rough estimate of the value of C for which narrowest spectrum occurs
for a given value of φ max .
The SPM-induced spectral narrowing has been observed in several experiments
[11]–[13]. In a 1993 experiment, 100-fs pulses, obtained from a Ti:sapphire laser operating
near 0.8 μm, were first chirped with a prism pair before launching them into
a 48-cm-long fiber [11]. The 10.6-nm spectral width of input pulses was nearly unchanged
at low peak powers but became progressively smaller as the peak power was