Nonlinear Fiber Optics - 4 ed. Agrawal

6.4. Vector Modulation Instability 197
Second, shear strain induces circular birefringence in proportion to the twist rate. When
both of these effects are included, Eqs. (6.3.1) and (6.3.2) take the following form [45]:
dA +
dz
dA −
dz
= ib c A + + iΔβ
2 e2ir tz A − + 2iγ
3 (|A +| 2 + 2|A − | 2 )A + , (6.3.24)
= ib c A − + iΔβ
2 e−2ir tz A + + 2iγ
3 (|A −| 2 + 2|A + | 2 )A − , (6.3.25)
where b c = hr t /2¯n is related to circular birefringence, r t is the twist rate per unit length,
and ¯n is the average mode index. The parameter h has a value of ∼0.15 for silica fibers.
The preceding equations can be used to find the fixed points, as done in Section 6.3.1
for an untwisted fiber. Above a critical power level, we again find four fixed points.
As a result, polarization instability still occurs along the fast axis but the critical power
becomes larger.
Birefringence modulation can also be included by making the parameter Δβ in
Eqs. (6.3.1) and (6.3.2) a periodic function of z such that Δβ = Δβ 0 [1 − iε cos(b m z)],
where ε is the amplitude and b m is the spatial frequency of modulation [42]. The
resulting equations can not be solved analytically but one can use the phase-space or
the Poincaré-sphere approach to study evolution of the polarization state approximately
[39]–[42]. This approach shows that the motion of the Stokes vector on the Poincaré
sphere becomes chaotic in the sense that polarization does not return to its original
state after each successive period of modal birefringence Δβ. Such studies are useful
for estimating the range of parameter values that must be maintained to avoid chaotic
switching if the fiber was to be used as an optical switch.
6.4 Vector Modulation Instability
This section extends the scalar analysis of Section 5.1 to the vector case in which a
CW beam, when launched into a birefringent fiber, excites both polarization components
simultaneously. Similar to the scalar case, modulation instability is expected to
occur in the anomalous-GVD region of the fiber. The main issue is whether the XPMinduced
coupling can destabilize the CW state even when the wavelength of the CW
beam is in the normal-GVD regime of the fiber. Vector modulation instability in an
isotropic nonlinear medium (no birefringence) was predicted as early as 1970 using
the coupled NLS equations [49]. In the context of birefringent fibers, it has been studied
extensively since 1988, both theoretically and experimentally [50]–[69]. Since the
qualitative behavior is different for weakly and strongly birefringent fibers, we consider
the two cases separately.
6.4.1 Low-Birefringence Fibers
In the case of low-birefringence fibers, one must retain the coherent coupling term in
Eqs. (6.1.11) and (6.1.12) in a study of modulation instability [50]. As before, it is
easier to use Eqs. (6.1.15) and (6.1.16), written in terms of the circularly polarized
components of the optical field. The steady-state or CW solution of these equations
is given in Section 6.3 but is quite complicated to use for the analysis of modulation