Inverse diffraction in phase space

Inverse diffraction refers to recovering the input field in the plane z=0 from the knowledge of the field in some plane z>0 of the free half-space to which the input field propagates. With the rapid development of computational optics, inverse diffraction is increasingly used in optimization and imaging simulations involving round-trip wave propagation. This increasing usage makes it necessary and valuable to revisit this old, important problem and clarify some existing ambiguities. In this study, an exhaustive inverse diffraction analysis is presented from the perspective of phase space. With the help of the Wigner distribution function, it is shown that the forward and inverse diffraction processes in phase space are geometrically equivalent to the deformation and recovery of the phase space diagram (PSD). The symmetries of PSD transformations corresponding to different inverse diffraction forms are revealed. The ambiguities between the conjugation of diffraction kernels and the negative diffraction distance are clarified. The physical pictures of inverse diffraction are further given. This phase space analysis provides an intuitive view on problems involving inverse diffraction and a more concise approach for understanding propagation behaviors of three-dimensional wave fields, including the phase conjugation in holography.