Twin-stagnation-free phase retrieval with vortex phase illumination.

The recovery of a complex-valued exit wavefront from its Fourier transform magnitude is challenging due to the stagnation problems associated with iterative phase retrieval algorithms. Among the various stagnation artifacts, the twin-image stagnation is the most difficult to address. The upright object and its inverted and complex-conjugated twin correspond to the identical Fourier magnitude data and hence appear simultaneously in the iterative solution. We show that the twin stagnation problem can be eliminated completely if a coherent beam with charge-1 vortex phase is used for illumination. Unlike the usual plane wave illumination case, a charge-1 vortex illumination intentionally introduces an isolated zero near the zero spatial frequency region, where maximal energy in the Fourier space is usually concentrated for most natural objects. The early iterations of iterative phase retrieval algorithms are observed to develop a clockwise or anti-clockwise vortex in the vicinity of this isolated zero. Once the Fourier transform of the solution latches onto a specific vortex profile in the neighborhood of this intentionally introduced intensity zero in early iterations, the solution quickly adjusts to the corresponding twin (upright or inverted) and further iterations are not observed to bring the other twin into the reconstruction. Our simulation studies with the well-known hybrid input-output (HIO) algorithm show that the solution always converges to one of the twins within a few hundred iterations when vortex phase illumination is used. Using a clockwise or anti-clockwise vortex phase as an initial guess is also seen to deterministically lead to a solution consisting of the corresponding twin. The resultant solution still has some faint residual artifacts that can be addressed via the recently introduced complexity guidance methodology. There is an additional vortex phase in the final solution that can simply be subtracted out to obtain the original test object. The near guaranteed convergence to a twin-stagnation-free solution with vortex illumination as described here is potentially valuable for deploying practical imaging systems that work based on the iterative phase retrieval algorithms.