Reconstructing the amplitude and phase of structured light from a single speckle pattern using a deep neural network

Abstract

Abstract Directly measuring an optical beam’s full complex field is foundational in optics, but conventional methods typically require interferometric setups. An alternative is to infer the field from an intensity-only speckle pattern produced by a scattering medium, however inverting this scattering process is ill-posed. While deep learning has been applied to reconstruct smooth amplitude and phase information, and to classify orbital angular momentum modes from speckle, recovering the intrinsic amplitude and phase topology of structured light, including helical phase singularities and 2 π jump discontinuities, remains an open challenge. Here, we examine single-shot reconstruction of both the amplitude and phase of Laguerre–Gaussian modes and their superpositions from intensity-only speckle using a U-Net. While single-shot classical linear inversion is fundamentally ill-suited to this task, the network can memorize a continuous inverse mapping from speckle to complex field with high fidelity (amplitude structural similarity index measure (SSIM) 0.9879, phase SSIM 0.9058). We further test generalization to held-out mode combinations against predicting the mean training label for every member of the test set, which scored an amplitude SSIM of 0.3233 (0.6994 PCC) and a phase SSIM of 0.2022 (0.0318 PCC). Amplitude generalization attempts ranged from 0.4413 (mean+0.1180) SSIM to 0.5043 (mean+0.1383) SSIM and 0.7523 (mean+0.0529) Pearson correlation coefficient (PCC) to 0.7851 (mean+0.0857) PCC, depending on the architecture and training budget. Phase generalization was unsuccessful with the strongest achieving an SSIM of 0.2645 (mean+0.0623) and PCC of 0.2569 (mean+0.2251). This work establishes that single-shot reconstruction of amplitude and phase inherent to complex structured light is a materially more difficult problem than the amplitude reconstruction and modal classification tasks reported to date. We suggest architectural, loss-function, and dataset improvements for future work. Finally, we conclude that while deep learning is clearly a path forward in this endeavor, the smooth convolutional nature of the popular U-Net is likely not the best candidate architecture.