Abstract
We find exact time-periodic solutions to the nonlinear equations that describe the polarization interaction of two intense counter-propagating beams in a randomly birefringent fiber, expressed in terms of the forward and backward beam Stokes vectors. The overlap function between the forward signal beam and the backward pump beam has an exact expression in terms of an elliptic integral and varies periodically along the fiber. For specific parameters, this overlap function takes the form of a soliton, with an almost constant value at all points on the fiber except for a pulse located at the soliton center. In the soliton configuration, the system displays polarization attraction for any beam intensities and arbitrary fiber lengths. We show how to solve the vector equations for the forward and backward beam Stokes vectors, using a shooting method to implement the endpoint boundary conditions, and plot the trajectories on the Poincare sphere.
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