Nonlinear Fiber Optics - 4 ed. Agrawal

5.1. Modulation Instability 123
2
L NL
= 1 km
Instability Gain (km −1 )
1.5
1
0.5
2 km
5 km
0
−150 −100 −50 0 50 100 150
Frequency Shift (GHz)
Figure 5.1: Gain spectra of modulation instability for three values of the nonlinear length, L NL =
(γP 0 ) −1 when a CW beam with power P 0 is launched into a fiber with β 2 = −5ps 2 /km.
(5.1.8) is replaced by Ω c exp(−αz/2). Modulation instability still occurs as long as
αL NL < 1. The effect of higher-order dispersive and nonlinear effects such as selfsteepening
and intrapulse Raman scattering can also be included using Eq. (2.3.43) in
place of Eq. (5.1.1) as the starting point [14]–[16]. The third-order dispersion β 3 (or any
odd-order dispersive term) does not affect the gain spectrum of modulation instability.
The main effect of self-steepening is to reduce the growth rate and the frequency range
over which gain occurs from the values seen in Figure 5.1.
As discussed in Chapter 10, modulation instability can be interpreted in terms of
a four-wave-mixing process that is phase-matched by SPM. If a probe wave at a frequency
ω 1 = ω 0 +Ω were to copropagate with the CW beam at ω 0 , it would experience
a net power gain given by Eq. (5.1.8) as long as |Ω| < Ω c . Physically, the energy of
two photons from the intense pump beam is used to create two different photons, one
at the probe frequency ω 1 and the other at the idler frequency 2ω 0 − ω 1 . The case in
which a probe is launched together with the intense pump wave is referred to as induced
modulation instability.
Even when the pump wave propagates by itself, modulation instability can lead to
spontaneous breakup of the CW beam into a periodic pulse train. Noise photons (vacuum
fluctuations) act as a probe in this situation and are amplified by the gain provided
by modulation instability [27]. As the largest gain occurs for frequencies ω 0 ± Ω max ,
where Ω max is given by Eq. (5.1.9), these frequency components are amplified most.
Thus, clear-cut evidence of spontaneous modulation instability at the fiber output is
provided by the appearance of two spectral sidebands located symmetrically at ±Ω max
on each side of the central line at ω 0 . In the time domain, the CW beam is converted
into a periodic pulse train with a period T m = 2π/Ω max .