Phase-gradient information from vortex encoders.

Coded diffraction enables the reconstruction of complex fields from phase-less sensor images. Such reconstruction is possible with diffractive encoders; however, the reconstruction quality depends significantly on the object-encoder pair and structure, as well as the sampling of the intensity-only sensor-plane images. Here, we derive the Fisher information content of the image phase gradient when additional information is captured from a single topological phase singularity. This diffractive mask provides sensor-plane patterns that depend on the unencoded phase gradient. We numerically validate the machine-learned phase-gradient retrieval with simple neural networks. Our results identify issues of sampling and the traceability of phase-gradient information in the Fourier-plane diffractive encoder scheme and point to information-guided strategies for hybrid computing.