Nonlinear Fiber Optics - 4 ed. Agrawal

7.2. XPM-Induced Modulation Instability 231
Figure 7.1: Gain spectra of XPM-induced modulation instability in the normal-dispersion
regime of a fiber for (i) different power ratios with δ = 0 and (ii) different values of δ with
equal beam powers. (After Ref. [11]; c○1987 APS.)
are so close to each other that the group-velocity mismatch is negligible, or they are
located on opposite sides of the zero-dispersion wavelength such that (v g1 ≈ v g2 ), the
four roots are given by [11]
K = Ω/v g1 ±{ 1 2 ( f 1 + f 2 ) ± [( f 1 − f 2 ) 2 /4 +C XPM ] 1/2 } 1/2 . (7.2.10)
It is easy to verify that K can become complex only if C XPM > f 1 f 2 . Using Eqs. (7.2.8)
and (7.2.9), the condition for modulation instability to occur can be written as
where Ω c1 and Ω c2 are defined as
[Ω 2 /Ω 2 c1 + sgn(β 21 )][Ω 2 /Ω 2 c2 + sgn(β 22 )] < 4, (7.2.11)
Ω cj =(4γ j P j /|β 2 j |) 1/2 (7.2.12)
with j = 1 or 2. When the condition (7.2.11) is satisfied, the gain spectrum of modulation
instability is obtained from g(Ω)=2Im(K).
The modulation-instability condition (7.2.11) shows that there is a range of Ω over
which the gain g(Ω) exists. The steady-state solution (7.2.3) is unstable to perturbations
at those frequencies. The most important conclusion drawn from Eq. (7.2.11) is
that modulation instability can occur irrespective of the signs of the GVD coefficients.
Thus, whereas modulation instability requires anomalous GVD in the case of a single
beam (see Section 5.1), it can occur in the two-beam case even if both beams experience
normal GVD. The frequency range over which g(Ω) > 0 depends on whether
β 21 and β 22 are both positive, both negative, or one positive and the other negative.
The smallest frequency range corresponds to the case in which both beams are in the
normal-dispersion regime of the fiber. Because modulation instability in that case is
due solely to XPM, only this case is discussed further.
Figure 7.1 shows the gain spectra of XPM-induced modulation instability for silica
fibers in the visible region near 0.53 μm choosing β 2 j = 60 ps 2 /km and γ j = 15
W −1 /km in Eq. (7.2.8). The group-velocity mismatch is neglected in the left graph