Transient evolution of the polarization-dispersion vector's probability distribution

We determine the transient evolution of the probability distribution of the polarization dispersion vector both analytically and numerically, using a physically reasonable model of the fiber birefringence. We show that, for all practical birefringence parameters, the distribution of the differential group delay (DGD), which is the magnitude of the polarization dispersion vector, becomes Maxwellian in just a few kilometers, except in the tail region, where the DGD is large. In this limit, the approach to a Maxwellian distribution takes much longer, of the order of tens of kilometers. In addition, we show that in the transient regime the DGD distribution is very different from Maxwellian. We also find that the probability-distribution function for the polarization-dispersion vector at the output of the fiber depends upon the angle between it and the local birefringence vector on the Poincaré sphere, showing that the DGD remains correlated with the orientation of the local birefringence axes over arbitrarily long distances.