Asymptotic Phase Synthesis by Transport Maps—Part I: Theory and Irrotational Linear Maps

The paper addresses the problem of phase synthesis of apertures with assigned amplitude. Applying the method of stationary phase it will be shown that the asymptotic solution satisfies a Monge-Amp & egrave;re partial differential equation (PDE) with appropriate boundary value conditions. In agreement with Chu's energy mapping principle for reflector shaping, the Monge-Ampere PDE can be solved identifying an irrotational transport map from the source aperture to the target beam. After a description of the general theory, the paper focuses on irrotational linear maps associated to quadratic phase solutions demonstrating the possibility of obtaining beams of the same shape of the source aperture, a result only observed for circular and square apertures. Exploiting a polar decomposition of the affine transformation matrix, it will be also demonstrated the possibility of rotating the beam, a result reported (to the best knowledge of the authors) for the first time. In a companion paper, the general problem of finding a solution to the Monge-Amp & egrave;re PDE via irrotational transport maps will be addressed by mean of the theory of "optimal transport".