Nonlinear Fiber Optics - 4 ed. Agrawal

194 Chapter 6. Polarization Effects
emerge. The components of the Stokes vector at the location of the new fixed points
on the Poincaré sphere are given by [47]
√
S 1 = −P cr , S 2 = 0, S 3 = ± P0 2 − P2 cr. (6.3.21)
These two fixed points correspond to elliptically polarized light and occur on the back
of the Poincaré sphere in Figure 6.6 (lower right). At the same time, the fixed point
(−S 0 ,0,0), corresponding to light polarized linearly along the fast axis, becomes unstable.
This is equivalent to the pitchfork bifurcation discussed earlier. If the input beam
is polarized elliptically with its Stokes vector oriented as indicated in Eq. (6.3.21), the
polarization state will not change inside the fiber. When the polarization state is close
to the new fixed points, the Stokes vector forms a close loop around the elliptically
polarized fixed point. This behavior corresponds to the analytic solution discussed earlier.
However, if the polarization state is close to the unstable fixed point (−S 0 ,0,0),
small changes in input polarization can induce large changes at the output. This issue
is discussed next.
6.3.3 Polarization Instability
The polarization instability manifests as large changes in the output state of polarization
when the input power or the polarization state of a CW beam is changed slightly
[33]–[35]. The presence of polarization instability shows that slow and fast axes of a
polarization-preserving fiber are not entirely equivalent.
The origin of polarization instability can be understood from the following qualitative
argument [34]. When the input beam is polarized close to the slow axis (x axis if
n x > n y ), nonlinear birefringence adds to intrinsic linear birefringence, making the fiber
more birefringent. By contrast, when the input beam is polarized close to the fast axis,
nonlinear effects decrease total birefringence by an amount that depends on the input
power. As a result, the fiber becomes less birefringent, and the effective beat length
increases. At a critical value of the input power nonlinear birefringence can cancel
LB
eff
intrinsic birefringence completely, and LB
eff becomes infinite. With a further increase
in the input power, the fiber again becomes birefringent but the roles of the slow and
fast axes are reversed. Clearly large changes in the output polarization state can occur
when the input power is close to the critical power necessary to balance the linear and
nonlinear birefringences. Roughly speaking, the polarization instability occurs when
the input peak power is large enough to make the nonlinear length L NL comparable to
the intrinsic beat length L B .
The period of the elliptic function in Eq. (6.3.11) determines the effective beat
length as [34]
L eff
B
= 2K(m)
π √ |q| L B, (6.3.22)
where L B is the low-power beat length, K(m) is the quarter-period of the elliptic function,
and m and q are given by Eq. (6.3.13) in terms of the normalized input power
defined as p = P 0 /P cr . In the absence of nonlinear effects, p = 0, q = 1, and we recover
L eff
B = L B = 2π/|Δβ|. (6.3.23)