Bridging Statistical Scattering and Aberration Theory: Ray Deflection Function -- I: Theoretical Framework

Abstract: This paper introduces a new conceptual framework that recasts surface roughness effects as a "ray deflection function" (RDF) which can be statistically represented through a modified Zernike-Fourier hybrid approach that directly connects the PSD with statistical aberration coefficients through spectral overlap integration. By establishing a direct mathematical relationship between the power spectral density (PSD) of surface imperfections and the statistical distribution of aberration coefficients, we develop a formalism that bridges known probabilistic scattering theory with deterministic aberration analysis. This transformation allows surface roughness to be seamlessly integrated with other optical aberrations by expressing its effects through equivalent modifications to the ideal mirror shape. This framework provides computational advantages for ray-tracing simulations while maintaining statistical fidelity to established scattering models, particularly for predicting the three-dimensional structure of imperfect focal bodies in optical systems.