Analytic beam shaping for flattened output irradiance profile

A flattened Lorentzian irradiance profile, which was introduced by Brenner in 2003, offers prospects for analytic design of beam shaping optics when transforming a Gaussian beam into a soft-flat-top profile. It is shown that for the flattened Lorentzian profile the output power integral and the ray mapping function are analytic functions. This paper presents a systematic derivation of the flattened Lorentzian profile from the super-Lorentzian profile and exploration of properties of the flattened Lorentzian profile and compares its behavior to the super-Gaussian, flattened Gaussian, and Fermi-Dirac profiles. Details are presented for evaluation of the FL profile normalization, profile matching conditions, ray mapping function, and M 2.