Noise robust Zernike phase retrieval via learning based algorithm only with 2-step phase shift measurements

We present a noise robust deep learning based aberration analysis method using 2-step phase shift measurement data. We first propose a realistic aberration pattern generation method to synthesize a sufficient amount of real-world-like aberration patterns for training a deep neural network by exploiting the asymptotic statistical distribution parameters of the real-world Zernike coefficients extracted from a finite number of experimentally measured real-world aberration patterns. As a result, we generate a real-world-like synthetic dataset of 200,000 different aberrations from 15 sets of real-world aberration patterns obtained by a Michelson interferometer under a variety of measurement conditions using the 4-step derivative fitting method together with the exploitation of the Gaussian density estimation. We then train the deep neural network with the real-world-like synthetic dataset, using two types of network architectures, GoogLeNet and ResNet101. By applying the proposed learning based 2-step aberration analysis method to the analysis of numerically generated aberrations formed under 100 different conditions, we verify that the proposed 2-step method can clearly outperform the existing 4-step iterative methods based on 4-step measurements, including the derivative fitting, transport of intensity equation (TIE), and robust TIE methods, in terms of noise robustness, root mean square error (RMSE), and inference time. By applying the proposed 2-step method to the analysis of the real-world aberrations experimentally obtained under a variety of measurement conditions, we also verify that the proposed 2-step method achieves compatible performance in terms of the RMSE between the reconstructed and measured aberration patterns, and also exhibits qualitative superiority in terms of reconstructing more realistic fringe patterns and phase distributions compared to the existing 4-step iterative methods. Since the proposed 2-step method can be extended to an even more general analysis of aberrations of any higher order, we expect that it will be able to provide a practical way for comprehensive aberration analysis and that further studies will extend its usefulness and improve its operational performance in terms of algorithm compactness, noise robustness, and computational speed.