Nonlinear Fiber Optics - 4 ed. Agrawal

7.2. XPM-Induced Modulation Instability 229
where k ≠ j, β 1 j = 1/v gj , v gj is the group velocity, β 2 j is the GVD coefficient, and α j
is the loss coefficient. The overlap integral f jk is defined as
∫∫ ∞
−∞
f jk =
|F j(x,y)| 2 |F k (x,y)| 2 dxdy
(∫∫ ∞
−∞ |F j(x,y)| 2 dxdy )(∫∫ ∞
−∞ |F ).
k(x,y)| 2 (7.1.14)
dxdy
The differences among the overlap integrals can be significant in multimode fibers
if the two waves propagate in different fiber modes. Even in single-mode fibers,
f 11 , f 22 , and f 12 differ from each other because of the frequency dependence of the
modal distribution F j (x,y). The difference is small, however, and can be neglected in
practice. In that case, Eq. (7.1.13) can be written as the following set of two coupled
NLS equations [7]–[10]
∂A 1
∂z + 1 ∂A 1
+ iβ 21 ∂ 2 A 1
v g1 ∂t 2 ∂t 2 + α 1
2 A 1 = iγ 1 (|A 1 | 2 + 2|A 2 | 2 )A 1 , (7.1.15)
∂A 2
∂z + 1 ∂A 2
+ iβ 22 ∂ 2 A 2
v g2 ∂t 2 ∂t 2 + α 2
2 A 2 = iγ 2 (|A 2 | 2 + 2|A 1 | 2 )A 2 , (7.1.16)
where the nonlinear parameter γ j is defined similar to Eq. (2.3.29) as
γ j = n 2 ω j /(cA eff ), ( j = 1,2), (7.1.17)
and A eff is the effective mode area (A eff = 1/ f 11 ), assumed to be the same for both
optical waves. Typically, A eff is 50 μm 2 in the 1.55-μm wavelength region. The corresponding
values of γ 1 and γ 2 are close to 2 W −1 /km depending on the frequencies ω 1
and ω 2 . Generally, the two pulses not only have different GVD coefficients but also
propagate at different speeds because of the difference in their group velocities. The
group-velocity mismatch plays an important role because it limits the XPM interaction
as pulses separate from each other. One can define the walk-off length L W using Eq.
(1.2.13). Physically, it is a measure of the fiber length over which two overlapping
pulses separate from each other as a result of the group-velocity mismatch.
7.2 XPM-Induced Modulation Instability
This section extends the analysis of Section 5.1 to the case in which two CW beams
of different wavelengths propagate inside a fiber simultaneously. Similar to the singlebeam
case, modulation instability is expected to occur in the anomalous-GVD region
of the fiber. The main issue is whether XPM-induced coupling can destabilize the CW
state even when one or both beams experience normal GVD [11]–[20].
7.2.1 Linear Stability Analysis
The following analysis is similar to that of Section 6.4.2. The main difference is that
XPM-induced coupling is stronger and the parameters β 2 and γ are different for the
two beams because of their different wavelengths. As usual, the steady-state solution