Three-Dimensional Complex Point Spread Function Reconstruction via Multi-Image Phase Retrieval in the Frequency Domain

Accurate measurement of the 3-D complex point spread function (PSF) greatly benefits the aberration analysis of optical systems and beam engineering. In this work, a multi-image phase retrieval algorithm is proposed to retrieve the wavefront of either a scalar or vector beam in the frequency domain. The gradients with respect to the angular spectrum are analytically derived by reformulating the beam propagation into a neural network framework, grounded in rigorous electromagnetic theory. The angular spectrum is optimized to minimize the deviation between measured and numerically reconstructed intensity by the stochastic gradient descent algorithm. Numerical simulation is first performed to examine the applicability of the proposed method in a scalar Gaussian beam and radially polarized Bessel-Gaussian vector beam. Experimental measurements of a beam with astigmatism are then demonstrated. Given the method’s simplicity, ease of implementation, and robustness against noise, we anticipate its broad application in the quantitative characterization of optical systems and structured beams.