Hamiltonian phase reconstruction enables quantitative prediction of diffraction in inhomogeneous media

Abstract We introduce a phase-consistent framework that bridges geometric and wave descriptions of light propagation in inhomogeneous media. By reconstructing the optical phase directly from Hamiltonian optics, we establish a first-principle connection between local refractive-index variations and the emergence of diffraction patterns. The recovered Hamiltonian phase is subsequently propagated using scalar diffraction within Fourier optics, providing a unified description of ray trajectories and interference within a single physical framework. We validate this approach experimentally using thermal-lens diffraction, a paradigmatic system in which continuously distributed refractive-index gradients generate complex interference patterns beyond the scope of thin-lens or ray-optical models. The reconstructed phase quantitatively reproduces the observed diffraction patterns across different transport regimes, demonstrating that diffraction arises directly from accumulated Hamiltonian phase. Beyond thermal lensing, the framework applies broadly to optical and other wave systems governed by an eikonal equation, establishing diffraction as a direct manifestation of Hamiltonian phase accumulation.