Profiling the Mirror Surface from the Fraunhofer Amplitude Distribution that Can be Generated Under Coherent Illumination

TLDR: In this article, a procedure is proposed to profile the mirror surface needed to produce under coherent illumination, a pre-specified Fraunhofer amplitude distribution U(ξ, ψ).

Abstract: A procedure is proposed to profile the mirror surface needed to produce under coherent illumination, a pre-specified Fraunhofer amplitude distribution U(ξ, ψ). The unknown optical path function ϕ(x y) in the diffraction integral is reconstructed from its stationary points {(x i, y i)}. The points {(x i, y i)} correspond to locations where the first Fresnel zone distribution F(x, y) is an extremum. The reconstruction utilizes the equivalence of the method of stationary points and the method of Fresnel zones. In Fraunhofer diffraction, the F(x, y) has the functional distribution of the inverse Fourier transform of U(ξ, ψ). We consider only Hermitian U(ξ, ψ)'s because they are generated by real F(x, y) distributions. The ϕ(x, y) is deduced from the behaviour of its first derivatives, which are expressed as product series expansions of {(x i, y i)}. The path function ϕ(x, y) which describes all the possible paths taken by the light waves from the point source to the observation plane via the reflecti...