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.

中文翻译：

---

PMD影响任意长度的光纤中Mueller矩阵的高阶统计

Mueller矩阵元素的较高阶（例如2阶和4阶）矩对于估计偏振模色散（PMD）引起的前向传播和后向散射信号的功率波动（例如拉曼增益的波动）很重要。和布里渊收益）。关于穆勒矩阵的当前知识仅限于足够长的光纤中其元素的二阶矩。在这项工作中，对任意长度的纤维的穆勒矩阵元素的高阶矩进行了分析研究。推导了Mueller矩阵元素矩的随机微分方程（SDE），并将其转换为相关的常微分方程（ODE）。由于ODE的系数恒定，因此可以获得解析解。在足够长的光纤中预测的二阶矩与现有结果非常吻合。通过数值模拟验证了任意长光纤中Mueller矩阵元素的4阶矩的结果，具有很好的一致性。