A new derivation of the Dickey-Romero-Holswade phase function

Abstract

The method of Kreuzer for irradiance redistribution with aspheric lenses can be used to derive a differential equation for the sag curve of a single aspheric surface which transforms the irradiance profile of a beam without regard to the phase of the wavefront of the shaped beam. It is shown that in the paraxial approximation the sag curve naturally splits into a quadratic term and an integral expression that depends on the shapes of the input and output profiles. For the special case where the input beam is Gaussian and the output profile is uniform, the integral term that appears in the sag curve is identical to the phase function derived by Dickey, Romero, and Holswade using the formalism of Fourier optics. The identity derived here demonstrates the equivalence, in the paraxial limit, of two apparently different methods of solving the beam shaping problem.