Nonlinear Fiber Optics - 4 ed. Agrawal

202 Chapter 6. Polarization Effects
of birefringence, the phase-matching condition can only be satisfied if the nonlinear
phase shifts remain below a certain level. This is the origin of the critical power level
in Eq. (6.4.19). An interesting feature of the four-wave-mixing process is that spectral
sidebands are generated such that the low-frequency sideband at ω 0 − Ω appears along
the slow axis whereas the sideband at ω 0 + Ω is polarized along the fast axis. This can
also be understood in terms of the phase-matching condition of Section 10.3.3.
6.4.3 Isotropic Fibers
It is clear that modal birefringence of fibers plays an important role for modulation
instability to occur. A natural question is whether modulation instability can occur in
isotropic fibers with no birefringence (n x = n y ). Even though such fibers are hard to
fabricate, fibers with extremely low birefringence (|n x − n y | < 10 −8 ) can be made by
spinning the preform during the drawing stage. The question is also interesting from a
fundamental standpoint and was discussed as early as 1970 [49].
The theory developed for high-birefringence fibers cannot be used in the limit Δβ =
0 because the coherent-coupling term has been neglected. In contrast, theory developed
for low-birefringence fibers remains valid in that limit. The main difference is that
P cr = 0 as polarization instability does not occur for isotropic fibers. As a result, Ω c2 =
0 while Ω c1 = Ω c3 ≡ Ω c . The gain spectrum of modulation instability in Eq. (6.4.8)
reduces to
√
g(Ω)=|β 2 Ω| Ω 2 c − Ω 2 , (6.4.20)
irrespective of whether the input beam is polarized along the slow or fast axis. This is
the same result obtained in Section 5.1 for the scalar case. It shows that the temporal
and spectral features of modulation instability should not depend on the direction in
which the input beam is linearly polarized. This is expected for any isotropic nonlinear
medium on physical grounds.
The situation changes when the input beam is circularly or elliptically polarized.
We can consider this case by setting Δβ = 0 in Eqs. (6.1.15) and (6.1.16). Using
α = 0 for simplicity, these equations reduce to the following set of two coupled NLS
equations [49]:
∂A +
∂z + iβ 2 ∂ 2 A +
2 ∂T 2 + ( iγ′ |A + | 2 + 2|A − | 2) A + = 0, (6.4.21)
∂A −
∂z + iβ 2 ∂ 2 A −
2 ∂T 2 + ( iγ′ |A − | 2 + 2|A + | 2) A − = 0, (6.4.22)
where T = t −β 1 z is the reduced time and γ ′ = 2γ/3. The steady-state solution of these
equations is obtained easily and is given by
Ā ± (z)= √ P ± exp(iφ ± ), (6.4.23)
where P ± is the input power in the two circularly polarized components and φ ± (z)=
γ ′ (P ∓ + 2P ± )z is the nonlinear phase shift.
As before, we perturb the steady-state solution using
A ± (z,t)=[ √ P ± + a ± (z,t)]exp(iφ ± ), (6.4.24)