Nonlinear Fiber Optics - 4 ed. Agrawal

6.3. Evolution of Polarization State 193
Figure 6.6: Trajectories showing motion of the Stokes vector on the Poincaré sphere. (a) Linear
birefringence case (upper left); (b) nonlinear case with Δβ = 0 (upper right); (c) mixed case with
Δβ > 0 and P 0 > P cr (lower row). Left and right spheres in the bottom row show the front and
back of the Poincaré sphere. (After Ref. [33]; c○1986 OSA.)
In the purely nonlinear case of isotropic fibers (Δβ = 0), W L = 0. The Stokes
vector now rotates around the S 3 axis with an angular velocity 2γS 3 /3 (upper right
sphere in Figure 6.6). This rotation is referred to as self-induced ellipse rotation, or as
nonlinear polarization rotation, because it has its origin in the nonlinear birefringence.
Two fixed points in this case correspond to the north and south poles of the Poincaré
sphere and represent right and left circular polarizations, respectively.
In the mixed case, the behavior depends on the power of incident light. As long as
P 0 < P cr , nonlinear effects play a minor role, and the situation is similar to the linear
case. At higher power levels, the motion of the Stokes vector on the Poincaré sphere
becomes quite complicated because W L is oriented along the S 1 axis while W NL is
oriented along the S 3 axis. Moreover, the nonlinear rotation of the Stokes vector along
the S 3 axis depends on the magnitude of S 3 itself. The bottom row in Figure 6.6 shows
motion of the Stokes vector on the front and back of the Poincaré sphere in the case
P 0 > P cr . When input light is polarized close to the slow axis (left sphere), the situation
is similar to the linear case. However, the behavior is qualitatively different when input
light is polarized close to the fast axis (right sphere).
To understand this asymmetry, let us find the fixed points of Eqs. (6.3.17) and
(6.3.18) by setting the z derivatives to zero. The location and number of fixed points
depend on the beam power P 0 launched inside the fiber. More specifically, the number
of fixed points changes from two to four at a critical power level P cr defined in Eq.
(6.3.10). For P 0 < P cr , only two fixed points, (S 0 ,0,0) and (−S 0 ,0,0), occur; these
are identical to the low-power case. In contrast, when P 0 > P cr , two new fixed points