Nonlinear Fiber Optics - 4 ed. Agrawal

5.1. Modulation Instability 121
of ultrashort pulses [8]–[27]. This section discusses modulation instability in optical
fibers as an introduction to soliton theory.
5.1.1 Linear Stability Analysis
Consider propagation of CW light inside an optical fiber. The starting point is the
simplified propagation equation (2.3.45). If fiber losses are ignored, this equation takes
the form
i ∂A
∂z = β 2 ∂ 2 A
2 ∂T 2 − γ|A|2 A, (5.1.1)
and is referred to as the nonlinear Schrödinger (NLS) equation in the soliton literature.
As discussed in Section 2.3, A(z,T ) represents the amplitude of the field envelope, β 2
is the GVD parameter, and the nonlinear parameter γ is responsible for SPM. In the
case of CW radiation, the amplitude A is independent of T at the input end of the fiber
at z = 0. Assuming that A(z,T ) remains time independent during propagation inside
the fiber, Eq. (5.1.1) is readily solved to obtain the steady-state solution
Ā = √ P 0 exp(iφ NL ), (5.1.2)
where P 0 is the incident power and φ NL = γP 0 z is the nonlinear phase shift induced
by SPM. Equation (5.1.2) implies that CW light should propagate through the fiber
unchanged except for acquiring a power-dependent phase shift (and for reduction in
power in the presence of fiber losses).
Before reaching this conclusion, however, we must ask whether the steady-state solution
(5.1.2) is stable against small perturbations. To answer this question, we perturb
the steady state slightly such that
A =( √ P 0 + a)exp(iφ NL ) (5.1.3)
and examine evolution of the perturbation a(z,T ) using a linear stability analysis. Substituting
Eq. (5.1.3) in Eq. (5.1.1) and linearizing in a, we obtain
i ∂a
∂z = β 2 ∂ 2 a
2 ∂T 2 − γP 0(a + a ∗ ). (5.1.4)
This linear equation can be solved easily in the frequency domain. However, because
of the a ∗ term, the Fourier components at frequencies Ω and −Ω are coupled. Thus,
we should consider its solution in the form
a(z,T )=a 1 exp[i(Kz− ΩT )] + a 2 exp[−i(Kz− ΩT )], (5.1.5)
where K and Ω are the wave number and the frequency of perturbation, respectively.
Equations (5.1.4) and (5.1.5) provide a set of two homogeneous equations for a 1 and a 2 .
This set has a nontrivial solution only when K and Ω satisfy the following dispersion
relation
K = ± 1 2 |β 2Ω|[Ω 2 + sgn(β 2 )Ω 2 c] 1/2 , (5.1.6)