Poisson wavefront imaging in photon-starved scenarios

TL;DR

Low-photon phase imaging is essential but limited by Poisson noise. The authors introduce Poisson Wavefront Imaging (PWI), which uses phase diversity from multiple SLM patterns and solves a Poisson-likelihood objective with total-variation regularization via ADMM to reconstruct the wavefront. Theoretical analysis with Fisher information and the Cramer-Rao lower bound shows PWI can surpass Shack–Hartmann performance under the same photon budget, while experiments with a USAF phase target validate substantial improvements: up to 1.6x reduction in phase RMSE and up to 1.8x enhancement in resolvable spatial frequency in photon-starved conditions. This approach enables robust, data-efficient, photon-limited wavefront sensing with potential extensions to broadband operation and data-driven regularizers for broader imaging applications.

Abstract

Low-photon phase imaging is essential in applications where the signal is limited by short exposure times, faint targets, or the need to protect delicate samples. We address this challenge with Poisson Wavefront Imaging (PWI), an optimization-based method that incorporates Poisson photon statistics and a smoothness prior to improve wavefront reconstruction. By using multiple spatial light modulator's phase patterns, PWI enhances Fisher information, boosting theoretical accuracy and regularizing the retrieval process effectively. In simulations, PWI approaches the theoretical phase error limit, and in experiments it reduces phase error by up to 1.6x compared to the Gerchberg-Saxton algorithm, achieving 1.8x higher resolution wavefront imaging in low photon regime. This method advances photon-limited imaging with applications in astronomy, semiconductor metrology, and biological systems.

Figures (4)

• Figure 1: Overview of PWI frameworks. A. Simplified diagram of coded-detection setup. B. PWI using multiple phase diversity patterns $\{\Phi_i\}_{i=1, \dots, M}$ and corresponding measured images $\{I_i\}_{i=1, \dots, M}$. C. Iterative reconstruction of the complex e-fields through forward and backward propagation.
• Figure 2: Theoretical accuracy vs. reconstruction performance for SHWFS and PWI. A. Simulation setups for PWI and SHWFS, illustrating their detection architectures. B. Dashed curves show the square root of the Cramer–Rao lower bound, representing the theoretical phase precision for the SHWFS (blue) and for PWI using either random phase diversity (red) or a flat phase pattern (black, serving as the no-diversity reference). Solid curves show the corresponding reconstruction phase RMSE: SHWFS (blue) and PWI with (orange) and without (red) TV regularization. The PWI reconstruction curves (solid orange and red) should both be compared to the dashed red CRLB. Note that due to phase wrapping, the maximum attainable RMSE is constrained to approximately 0.28 (green dash–dot line). C. Reconstructed phase images at 10,000, 158, and 1.58 mean photons per pixel for PWI with TV and SHWFS; bottom-row insets show SHWFS intensity measurements.
• Figure 3: Comparative analysis of phase reconstruction across varied photon levels. A. Phase retrieval methods; GS, Poisson, and Poisson with TV prior are shown at total exposure times of 240, 26.4, 7.0 and 1.9 ms, correlating with cumulative mean photons per pixel and total photons from 24 measurements. B. Correlation plots between ground truth and reconstructed phases demonstrate fidelity across different photon settings. C-D. Phase cross sections along elements 2–6 in Group 6 of the USAF resolution target, shown for mean photon levels of 257.8 (C) and 66.0 (D). Each panel displays, from top to bottom, GS, PWI, and PWI+TV results with ground truth profile (black). Reconstructions used 24 SLM phase patterns and 400 iterations.
• Figure 4: Phase retrieval tested across varying photon levels using a Purdue logo and stained tissue images displayed on the SLM. Phases are retrieved from 16 images with different SLM patterns. Total photon count from all images has an uncertainty of ±13%. Each retrieval process involves 700 iterations. Ground truth phases and corresponding RMSE errors are shown in (B).