Phase Space Modeling of Extended Sources Based on Wigner Distribution and Hamiltonian Optics

Abstract

Precise modeling of extended sources is a central challenge in modern optical engineering, laser physics, and computational lithography. Unlike ideal point sources or completely incoherent thermal radiation sources, real-world light sources -- such as high-power laser diode arrays, superluminescent diodes (SLD), extreme ultraviolet (EUV) lithography sources, and beams transmitted through atmospheric turbulence -- typically exhibit partial spatial coherence. Traditional geometric optics based on ray tracing ignores diffraction and interference effects; while classical wave optics is accurate, the computational cost of handling four-dimensional correlation functions for partially coherent fields is enormous. To balance computational efficiency and physical accuracy, phase space optics provides a unified theoretical framework. By introducing the Wigner distribution function (WDF), we can map the light field into a joint space-time-spatial frequency domain $(\bm{r}, \bm{p})$. This description not only retains all the information of wave optics (including interference terms) but also naturally transitions to the ray description of Hamiltonian optics in the short-wavelength limit, governed by Liouville's theorem of phase space volume conservation. This report aims to establish optimal modeling methods based on phase space and Hamiltonian optics for different types of extended sources such as partially coherent light, fully coherent light, and quasi-homogeneous light. The report will derive in detail the mathematical models for each source type and provide strict criteria for the applicability of geometric optics models using mathematical tools such as the Moyal expansion and generalized Fresnel number.