Nonlinear Fiber Optics - 4 ed. Agrawal

126 Chapter 5. Optical Solitons
NLS equation (5.1.1). A simple approach solves this equation in the frequency domain
as a four-wave mixing problem [23]; it is discussed in detail in Chapter 10. The main
disadvantage of this approach is that it cannot treat generation of higher-order sidebands
located at ω 0 ±mΩ (m = 2,3,...) that are invariably created when the first-order
sidebands (m = 1) become strong.
The time-domain approach solves the NLS equation directly. Numerical solutions
of Eq. (5.1.1) obtained with the input corresponding to a CW beam with weak sinusoidal
modulation imposed on it show that the nearly CW beam evolves into a train of
narrow pulses, separated by the period of initial modulation [8]. The fiber length required
to realize such a train of narrow pulses depends on the initial modulation depth
and is typically ∼5 L D . With further propagation, the multipeak structure deforms and
eventually returns to the initial input form. This behavior is found to be generic when
Eq. (5.1.1) is solved by considering arbitrary periodic modulation of the steady state
[32]. The foregoing scenario suggests that the NLS equation should have periodic solutions
whose forms change with propagation. Indeed, it turns out that the NLS equation
has a multiparameter family of periodic solutions [32]–[39]. In their most general
form, these solutions are expressed in the form of Jacobian elliptic functions. In some
specific cases, the solution can be written in terms of trigonometric and hyperbolic
functions [36].
From a practical standpoint, modulation instability is useful for generating a train
of short optical pulses whose repetition rate can be externally controlled. As early as
1989, 130-fs pulses at a 2-THz repetition rate were generated through induced modulation
instability [40]. Since then, this technique has been used to create optical sources
capable of producing periodic trains of ultrashort pulses at repetition rates higher than
those attainable from mode-locked lasers. Several experiments have used dispersiondecreasing
fibers for this purpose [41]–[43]. Initial sinusoidal modulation in these experiments
was imposed by beating two optical signals. In a 1992 experiment [42], the
outputs of two distributed feedback (DFB) semiconductor lasers, operating continuously
at slightly different wavelengths near 1.55 μm, were combined in a fiber coupler
to produce a sinusoidally modulated signal at a beat frequency that could be varied in
the 70–90 GHz range by controlling the laser temperature. In a later experiment [43],
250-fs pulses at a tunable repetition rate of 80 to 120 GHz were generated. The beat
signal from two DFB lasers was amplified to power levels ∼0.8 W by using a fiber amplifier
and then propagated through a 1.6-km-long dispersion-decreasing fiber, whose
GVD decreased from 10 to 0.5 ps/(km-nm) over its length. Figure 5.4 shows the output
pulse train (width 250 fs) at a 114-GHz repetition rate and the corresponding optical
spectrum. The spectrum was shifted toward red because of intrapulse Raman scattering
that becomes important for such short pulses (see Section 5.5.4).
The use of a dispersion-decreasing fiber is not essential for producing pulse trains
through modulation instability. In an interesting experiment, a comb-like dispersion
profile was produced by splicing pieces of low- and high-dispersion fibers [44]. A
dual-frequency fiber laser was used to generate the high-power signal modulated at a
frequency equal to the longitudinal-mode spacing (59 GHz). When such a modulated
signal was launched into the fiber, the output consisted of a 2.2-ps pulse train at the 59-
GHz repetition rate. In another experiment [45], a periodic train of 1.3-ps pulses at the
123-GHz repetition rate was generated by launching the high-power beat signal into a