Direct object recovery from the Fraunhofer diffraction integral

In this Letter, we present a novel, to the best of our knowledge, approach for recovering objects directly from the Fraunhofer diffraction integral, where the diffraction field of an object is approximated by the Fourier transform of this object augmented by an additional phase factor. This phase factor at the observation plane is universal for the diffraction fields generated by objects located at the same plane and illuminated by the same monochromatic plane wave. It can be first extracted from dividing the Fraunhofer diffraction field by the Fourier transform of an object reference. Rapid recovery for unknown objects is then enabled after applying a two-dimensional inverse Fourier transform to the ratio of the Fraunhofer diffraction fields to the phase factor. This approach is verified experimentally by constructing a modified Mach-Zehnder interferometer, with a digital micromirror device (DMD) generating the objects of desired structures. To record the Fraunhofer diffraction and interference patterns on a finite-size CCD camera, a convex lens is introduced with the CCD sensing surface positioned at the focal plane of the lens. The strategy described in [Nat. Commun.7, 10820 (2016)10.1038/ncomms10820] is applied to extract the phase of the diffraction field from the interference pattern. The results demonstrate the efficiency of our approach in swiftly and accurately recovering small objects with elimination of zero-order and conjugate images.