  The realm of classical phase retrieval concerns itself with the arduous task
of recovering a signal from its Fourier magnitude measurements, which are
fraught with inherent ambiguities. A single-exposure intensity measurement is
commonly deemed insufficient for the reconstruction of the primal signal, given
that the absent phase component is imperative for the inverse transformation.
In this work, we present a novel single-shot phase retrieval paradigm from a
fractional Fourier transform (FrFT) perspective, which involves integrating the
FrFT-based physical measurement model within a self-supervised reconstruction
scheme. Specifically, the proposed FrFT-based measurement model addresses the
aliasing artifacts problem in the numerical calculation of Fresnel diffraction,
featuring adaptability to both short-distance and long-distance propagation
scenarios. Moreover, the intensity measurement in the FrFT domain proves highly
effective in alleviating the ambiguities of phase retrieval and relaxing the
previous conditions on oversampled or multiple measurements in the Fourier
domain. Furthermore, the proposed self-supervised reconstruction approach
harnesses the fast discrete algorithm of FrFT alongside untrained neural
network priors, thereby attaining preeminent results. Through numerical
simulations, we demonstrate that both amplitude and phase objects can be
effectively retrieved from a single-shot intensity measurement using the
proposed approach and provide a promising technique for support-free coherent
diffraction imaging.
