Asymptotic Phase Synthesis by Transport Maps -Part II Optimal Transport Problem

In a companion paper the authors have shown that the problem of phase synthesis of apertures with assigned amplitude can be asymptotically reconducted to a Monge-Amp & egrave;re partial differential equation (PDE) which can be solved identifying an irrotational transport map from the source aperture to the target beam. In this contribution we address the general problem of synthesizing beams with shapes that cannot be obtained as a linear transformation of the aperture. According to a fundamental result of Brenier, the irrotational transport map satisfying the Monge-Ampere PDE can be obtained as a solution of Monge's Optimal Transport Problem (OTP) with quadratic cost. The phase profile can then be obtained by direct integration. A numerical method based on Von Neumann's alternating projections is adopted with an original discretization of the involved differential equations and boundary conditions. The numerical results demonstrate the effectiveness and accuracy of the proposed approach.