Theory of polarization attraction in parametric amplifiers based on telecommunication fibers

We develop from first principles the coupled wave equations that describe polarization-sensitive parametric amplification
based on four-wave mixing (FWM) in standard (randomly birefringent) optical fibers. We show that in
the small-signal case these equations can be solved analytically, and permit us to predict the gain experienced by
the signal beam as well as its state of polarization (SOP) at the fiber output. We find that, independently of its initial
value, the output SOP of a signal within the parametric gain bandwidth is solely determined by the pump SOP. We
call this effect of pulling the polarization of the signal towards a reference SOP the polarization attraction, and we
call the parametric amplifier the FWM polarizer (which can equivalently be called the fiber-optic parametric
amplifier polarizer). Our theory is valid beyond the zero polarization mode dispersion (PMD) limit, and it takes
into account moderate deviations of the PMD from zero. In particular, our theory is capable of analytically predicting
the rate of degradation of the efficiency of the parametric amplifier, which is caused by the detrimental
PMD effect.