Nonlinear Fiber Optics - 4 ed. Agrawal

198 Chapter 6. Polarization Effects
instability as it involves elliptic functions. The problem becomes tractable when the
polarization state of the incident CW beam is oriented along a principal axis of the
fiber.
Consider first the case in which the polarization state is oriented along the fast
axis (A x = 0). This case is especially interesting because the polarization instability
discussed in Section 6.3 can also occur. If fiber losses are neglected by setting α = 0,
the steady-state solution becomes
Ā ± (z)=±i √ P 0 /2exp(iγP 0 z), (6.4.1)
where P 0 is the input power. Following the procedure of Section 5.1, stability of the
steady state is examined by assuming a solution in the form
A ± (z,t)=±[i √ P 0 /2 + a ± (z,t)]exp(iγP 0 z), (6.4.2)
where a ± (z,t) is a small perturbation. Using Eq. (6.4.2) in Eqs. (6.1.15) and (6.1.16)
and linearizing in a + and a − , we obtain a set of two coupled linear equations. These
equations can be solved by assuming a solution of the form
a ± = u ± exp[i(Kz− Ωt)] + iv ± exp[−i(Kz− Ωt)], (6.4.3)
where K is the wave number and Ω is the frequency of perturbation. We then obtain
a set of four algebraic equations for u ± and v ± . This set has a nontrivial solution only
when the perturbation satisfies the following dispersion relation [50]
where
[(K − β 1 Ω) 2 −C 1 ][(K − β 1 Ω) 2 −C 2 ]=0, (6.4.4)
C 1 = 1 2 β 2Ω 2 ( 1 2 β 2Ω 2 + 2γP 0 ), (6.4.5)
C 2 =( 1 2 β 2Ω 2 + Δβ − 2γP 0 /3)( 1 2 β 2Ω 2 + Δβ). (6.4.6)
As discussed in Section 5.1, the steady-state solution becomes unstable if the wave
number K has an imaginary part for some values of Ω, indicating that a perturbation
at that frequency would grow exponentially along the fiber with the power gain
g = 2Im(K). The nature of modulation instability depends strongly on whether the
input power P 0 is below or above the polarization-instability threshold P cr given in Eq.
(6.3.10). For P 0 < P cr , modulation instability occurs only in the case of anomalous
dispersion, and the results are similar to those of Section 5.1. The effect of XPM is to
reduce the gain from that of Eq. (5.1.9) but the maximum gain occurs at the same value
of Ω (see Figure 5.1).
It is easy to deduce from Eq. (6.4.4) that modulation instability can occur even in
the normal-dispersion regime of the fiber (β 2 > 0) provided C 2 < 0. This condition is
satisfied for frequencies in the range 0 < |Ω| < Ω c1 , where
Ω c1 =(4γ/3β 2 ) 1/2√ P 0 − P cr . (6.4.7)