Empirical experimental scattering data from conventional optical surfaces is shown to exhibit shift-invariant behavior with respect to incident angle when plotted in direction cosine space. This implies the existence of a surface transfer function that completely characterizes the scattering properties of the surface, and permits the application of linear systems theory and Fourier techniques in modeling the scattering effects of optical surfaces. A theoretical basis for this behavior is illustrated by showing that scalar diffraction phenomena (conical diffraction from gratings) is shift-invariant with respect to incident angle only in direction cosine space, and surface roughness can be considered to be composed of a superposition of sinusoidal phase gratings. The fact that many optical surfaces of interest deviate from this shift-invariant behavior does not invalidate the usefulness of the linear systems formalism. The ideal behavior of a shift-invariant scattering process can still be used for making engineering calculations and retained as the reference from which scattering from real surfaces is compared. This is completely analogous to the universally accepted transfer function characterization of imaging systems in spite of the fact that few real imaging systems are isoplanatic (no field-dependent aberrations).
James E. Harvey,
"Surface Scatter Phenomena: A Linear, Shift-Invariant Process", Proc. SPIE 1165, Scatter from Optical Components, (2 January 1990); doi: 10.1117/12.962839; https://doi.org/10.1117/12.962839