Nonlinear Fiber Optics - 4 ed. Agrawal

6.3. Evolution of Polarization State 191
Figure 6.5: Phase-space trajectories representing evolution of the polarization state along fiber
for (a) p ≪ 1 and (b) p = 3. (After Ref. [34]; c○1986 OSA.)
where cn(x) is a Jacobian elliptic function with the argument
K(m) is the quarter period, and m and q are defined as
x = √ |q| (Δβ)z + K(m), (6.3.12)
m = 1 2 [1 − Re(q)/|q|], q = 1 + pexp(iψ 0). (6.3.13)
Here ψ 0 is the value of ψ at z = 0. Both p − (z) and ψ(z) can be obtained in terms of
p + (z) using Eq. (6.3.9). The ellipticity and the azimuth of the polarization ellipse at
any point along the fiber are then obtained from Eq. (6.3.4) after noting that θ = ψ/2.
It is useful to show evolution of the polarization state as trajectories in a ellipticity–
azimuth phase plane. Figure 6.5 shows such phase-space trajectories in the cases of
(a) low input power (p ≪ 1) and (b) high input power (p = 3). In the low-power case,
all trajectories close, indicating oscillatory evolution of the polarization state [see Eq.
(6.3.3)]. However, at power levels such that p > 1, a “seperatrix” divides the phase
space into two distinct regions. In the region near e p = 0 and θ = 0 (light polarized
close to the slow axis), trajectories form closed orbits, and polarization evolution is
qualitatively similar to the low-power case. However, when light is polarized close to
the fast axis, nonlinear rotation of the polarization ellipse leads to qualitatively different
behavior because the fast axis corresponds to an unstable saddle point.
One can use the analytic solution to find the “fixed points” in the phase space.
A fixed point represents a polarization state that does not change as light propagates
inside the fiber. Below the critical power (p < 1), light polarized linearly (e p = 0)
along the slow and fast axes (θ = 0 and π/2) represents two stable fixed points. At the
critical power (p = 1), the fast-axis fixed point exhibits a pitchfork bifurcation. Beyond
this power level, the linear-polarization state along the fast axis becomes unstable, but
two new elliptically polarized states emerge as fixed points. These new polarization
eigenstates are discussed next using the Poincaré-sphere representation.
6.3.2 Poincaré-Sphere Representation
An alternative approach to describe evolution of the polarization state in optical fibers
is based on the rotation of the Stokes vector on the Poincaré sphere [33]. In this case,