Laser beams can carry multi-scale properties in space and time that ultimately impact their quality. The study of their evolution along complex optical sequences is of crucial interest, especially in high-intensity laser chains. For such analysis, results obtained with standard numerical methods strongly depend on the sampling. In this paper, we develop an analytic model for a sinusoidal phase modulation inside a sequence of first-order optics elements based on the Wigner matrix formalism. A Bessel decomposition of the Wigner function gives pseudo-Wigner functions that obey the general ABCD matrix law transformation without approximations and sampling considerations. Applied to a Gaussian beam, explicit expressions are obtained for the projections of the Wigner function in the sub-spaces and give a powerful tool for analyzing the laser beam properties. The formalism is established in the spatial and temporal domains and can be used to evaluate the impact of the phase noise on the beam properties and is not limited to small modulation depths. For the sake of illustration, the model is applied to the Talbot effect with the analysis of the propagation in the spatial and phase-space domains. A comparison with full numerical calculations evidences the high accuracy of the analytic model that retrieves all the features of the diffracted beam.

中文翻译：

---

相位调制信号的Wigner矩阵形式

激光束可以在空间和时间上具有多尺度特性，最终会影响其质量。研究它们沿复杂光学序列的演化至关重要，特别是在高强度激光链中。对于此类分析，使用标准数值方法获得的结果在很大程度上取决于采样。在本文中，我们基于Wigner矩阵形式主义，开发了一系列一阶光学元件内部正弦相位调制的解析模型。Wigner函数的Bessel分解给出了伪Wigner函数，该函数遵循一般的ABCD矩阵定律变换，而没有近似值和采样注意事项。应用于高斯光束 Wigner函数在子空间中的投影得到了明确的表达式，并为分析激光束特性提供了强大的工具。形式主义是在空间和时间域中建立的，可用于评估相位噪声对光束特性的影响，并且不限于较小的调制深度。为了说明起见，将该模型应用于Talbot效应，并分析了空间和相空间域中的传播。与全部数值计算的比较证明了解析模型的高精度，该解析模型可获取衍射光束的所有特征。为了说明起见，将该模型应用于Talbot效应，并分析了空间和相空间域中的传播。与全部数值计算的比较证明了解析模型的高精度，该解析模型可获取衍射光束的所有特征。为了说明起见，将该模型应用于Talbot效应，并分析了空间和相空间域中的传播。与全部数值计算的比较证明了解析模型的高精度，该解析模型可获取衍射光束的所有特征。