Statistical Characterization of the Jones Matrix of Long Fibers Affected by Polarization Mode Dispersion (PMD)

The unitary transfer matrix of a fiber affected by polarization mode dispersion (PMD) is analyzed using the Stokes representation of its eigenmodes and its retardation angle, or equivalently through its Pauli coordinates.We develop a statistical theory applied to these parameters and relate it to the extensive existing literature on the statistics of the PMD vector \vec{\Omega}_o(\omega). Dynamical equations are established for the Pauli coordinates. Assuming a standard “white Gaussian” model for the local birefringence, and using the tools of stochastic calculus, we derive the distributions of the eigenmodes, the retardation angle, the Pauli coordinates, and of the frequency derivatives of all these parameters. The evolution in space of the Pauli coordinates is also characterized as a standard Brownian motion on the unit sphere in \Re^4. An expression for the frequency autocorrelation function of the Pauli coordinates, the eigenmodes and the retardation angle is derived and their coherence bandwidth is compared to that of the PMD vector. All theoretical results are supported by simulation over an ensemble of 10 000 fibers, using the standard retarder plate model.