General solution of two-dimensional beam-shaping with two surfaces

Beam shaping is a technique, by which a known input irradiance is transformed into a desired output irradiance by changing the local propagation vector of the wave front. Unlike one-dimensional beam-shaping, which leads to a simple differential equation which can be integrated in a straight forward manner, the two-dimensional beam shaping problem leads to a Monge-Ampere type equation, which is difficult to solve. In this paper, we generalize the problem to refractive and reflective systems and use to shifted-base-function approach to obtain a general solution.