Analytical probability density function for the Q factor due to PMD and noise

The statistics of the signal quality due to first-order polarization mode dispersion (PMD) in uncompensated systems is analyzed. First, the probability density function (pdf) of the eye-opening penalty (EOP) due to first-order PMD is presented. Based on an analytical relation between Q factor and EOP, the pdf of the Q factor is determined. By integration of this pdf, the outage probability (OP) is derived. The OP is determined by including the optical signal-to-noise ratio (OSNR) and the differential group delay (DGD). Finally, the OSNR penalty for the required OP and a certain average DGD is derived, which is an important result for system designers. All analytical results are verified by simulations.