Linear systems formulation of non-paraxial scalar diffraction theory

Abstract

Goodman's popular linear systems formulation of scalar diffraction theory includes a paraxial (small angle) approximation that severely limits the conditions under which this elegant Fourier treatment can be applied. In this paper a generalized linear systems formulation of non-paraxial scalar diffraction theory will be discussed. Diffracted radiance (not intensity or irradiance) is shown to be shift-invariant with respect to changes in incident angle only when modeled as a function of the direction cosines of the propagation vectors of the usual angular spectrum of plane waves. This revelation greatly extends the range of parameters over which simple Fourier techniques can be used to make accurate diffraction calculations. Non-paraxial diffraction grating behavior (including the Woods anomaly phenomenon) and wide-angle surface scattering effects for moderately rough surfaces at large incident and scattered angles are two diffraction phenomena that are not limited to the paraxial region and benefit greatly from this extension to Goodman's Fourier optics analysis. The resulting generalized surface scatter theory has been shown to be valid for rougher surfaces than the Rayleigh-Rice theory and for larger incident and scattered angles than the classical Beckman-Kirchhoff theory. This has enabled the development of a complete linear systems formulation of image quality, including not only diffraction effects and geometrical aberrations from residual optical design errors, but surface scatter effects from residual optical fabrication errors as well. Surface scatter effects can thus be balanced against optical design errors, allowing the derivation of optical fabrication tolerances during the design phase of a project.