Nonlinear Fiber Optics - 4 ed. Agrawal

6.4. Vector Modulation Instability 203
where a ± (z,t) is a small perturbation. By using Eq. (6.4.24) in Eqs. (6.4.21) and
(6.4.22) and linearizing in a + and a − , we obtain a set of two coupled linear equations.
These equations can be solved assuming a solution in the form of Eq. (6.4.3).
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 [49]
where
and the XPM coupling parameter C X is now defined as
(K − H + )(K − H − )=C 2 X, (6.4.25)
H ± = 1 2 β 2Ω 2 ( 1 2 β 2Ω 2 + γP ± ), (6.4.26)
C X = 2β 2 γΩ 2√ P + P − . (6.4.27)
A necessary condition for modulation instability to occur is C 2 X > H +H − .AsC X
depends on √ P + P − and vanishes for a circularly polarized beam, we can conclude that
no instability occurs in that case. For an elliptically polarized beam, the instability gain
depends on the ellipticity e p defined in Eq. (6.3.4).
6.4.4 Experimental Results
The vector modulation instability was first observed in the normal-dispersion region
of a high-birefringence fiber [52]–[54]. In one experiment, 30-ps pulses at the 514-
nm wavelength with 250-W peak power were launched into a 10-m fiber with a 45 ◦ -
polarization angle [53]. At the fiber output, the pulse spectrum exhibited modulation
sidebands with a 2.1-THz spacing, and the autocorrelation trace showed 480-fs intensity
modulation. The observed sideband spacing was in good agreement with the value
calculated theoretically. In another experiment, 600-nm input pulses were of only 9-ps
duration [52]. As the 18-m-long fiber had a group-velocity mismatch of ≈1.6 ps/m, the
two polarization components would separate from each other after only 6 m of fiber.
The walk-off problem was solved by delaying the faster-moving polarization component
by 25 ps at the fiber input. The temporal and spectral measurements indicated that
both polarization components developed high-frequency (∼3 THz) modulations, as expected
from theory. Moreover, the modulation frequency decreased with an increase in
the peak power. This experiment also revealed that each polarization component of the
beam develops only one sideband, in agreement with theory. In a later experiment [54],
modulation instability developed from temporal oscillations induced by optical wave
breaking (see Section 4.2.3). This behavior can be understood from Figure 4.13, noting
that optical wave breaking manifests as spectral sidebands. If these sidebands fall
within the bandwidth of the modulation-instability gain curve, their energy can seed
the instability process.
Although vector modulation instability is predicted to occur even in the anomalous-
GVD regime of a high-birefringence fiber, its experimental observation turned out to
be more difficult [67]. The reason is that the scalar modulation instability (discussed in
Section 5.1) also occurs in this regime. If the input field is polarized along a principal
axis, the scalar one dominates. In a 2005 experiment, clear-cut evidence of vector