Higher order statistics of the Mueller matrix in a fiber with an arbitrary length impacted by PMD

The higher order (such as the 2nd order and the 4th order) moments of the Mueller matrix elements are important to estimate the polarization mode dispersion (PMD) induced power fluctuations for the forward propagation and the backward scattered signals (e.g. fluctuation of the Raman gain and the Brillouin gain). Current knowledge about the Mueller matrix is limited to the 2nd order moments of its elements in a sufficiently long fiber. In this work, the higher order moments of the Mueller matrix elements of a fiber with arbitrary length is studied analytically. The stochastic differential equations (SDEs) for the moments of the Mueller matrix elements are derived and converted to the related ordinary differential equations (ODEs). Since the ODEs are with the constant coefficients, it is possible to obtain the analytical solutions. The predicted 2nd order moments in a sufficiently long fiber agree well with the existing results. The results of the 4th order moments of the Mueller matrix elements in an arbitrarily long fiber are validated by the numerical simulations with excellent agreement.