Phase Retrieval via Gain-Based Photonic XY-Hamiltonian Optimization

TL;DR

This work reformulates Coded Diffraction Pattern phase retrieval as a continuous-variable XY Hamiltonian minimization problem and solves it with gain-dissipative photonic networks. By deriving the coupling matrix $J_{ij}$ and mapping the phase-recovery task to minimizing $H_{XY}$, the authors implement a simulated hardware-inspired dynamics, $rac{d oldsymbol{ m \\psi}}{dt}=(oldsymbol\gamma-|oldsymbol{ m \\psi}|^2)oldsymbol{ m \\psi}$ with $rac{d oldsymbol\gamma}{dt}=\\epsilon(1-|oldsymbol{ m \\psi}|^2)$, and extract the phase configuration as the solution. They generate CDP problems via stacked DFT blocks and random phase masks, evaluate using ROE and RSE, and show that the gain-based solver consistently outperforms the Relaxed-Reflect-Reflect (RRR) algorithm in medium-noise regimes while maintaining performance as problem size grows. The results span real-valued and complex-valued samples, including large-scale 2D images and 3D vortex rings, highlighting robustness to noise and scalability. The work argues for hardware implementations on photonic/exciton-polariton lattices or digital-SPIM with feedback, promising fast, energy-efficient phase retrieval for real-time imaging tasks.

Abstract

Phase-retrieval from coded diffraction patterns (CDP) is important to X-ray crystallography, diffraction tomography and astronomical imaging, yet remains a hard, non-convex inverse problem. We show that CDP recovery can be reformulated exactly as the minimisation of a continuous-variable XY Hamiltonian and solved by gain-based photonic networks. The coupled-mode equations we exploit are the natural mean-field dynamics of exciton-polariton condensate lattices, coupled-laser arrays and driven photon Bose-Einstein condensates, while other hardware such as the spatial photonic Ising machine can implement the same update rule through high-speed digital feedback, preserving full optical parallelism. Numerical experiments on images, two- and three-dimensional vortices and unstructured complex data demonstrate that the gain-based solver consistently outperforms the state-of-the-art Relaxed-Reflect-Reflect (RRR) algorithm in the medium-noise regime (signal-to-noise ratios 10--40 dB) and retains this advantage as problem size scales. Because the physical platform performs the continuous optimisation, our approach promises fast, energy-efficient phase retrieval on readily available photonic hardware. uch as two- and three-dimensional vortices, and unstructured random data. Moreover, the solver's accuracy remains high as problem sizes increase, underscoring its scalability.

Figures (7)

• Figure 1: A schematic diagram for the CDP experiment framework. A light source is diffracted by the sample under investigation, and the diffracted complex-valued signal is split into $L$ identical beams, each of which is directed towards a phase filter that modifies the phase of the incident signal at each spatial location. The phase-modified beams are then directed through a lens system, and their intensities are finally captured.
• Figure 2: Comparison of phase retrieval problems with different number of phase masks. (a) Time evolution of errors for the phase retrieval problem under CDP experiment framework with 2 phase masks. The inset gives the final reconstructed image produced by the phase retrieval algorithm. (b) Time evolution of errors for the phase retrieval problems with 5 phase masks. In both cases the problems were solved with the gain-dissipative system with a random initial condition. The original image, which is visually identical to the image shown in inset of (b), is from the labelled face in the wild (LFW) dataset.
• Figure 3: Comparison of GS method and gain-based system performance in recovering a real-valued image. (a) Error evolution and final reconstructed image with GS method starting from a complex-valued random initial condition whose phase is uniformly randomly distributed in the range $[0, 2\pi)$ and whose amplitude is the known observation vector $\mathbf{b}$. (b) Error evolution and final reconstructed image with GS method starting from an initial condition whose phase is obtained by multiplying $\mathbf{A}$ with a random real-valued vector $\mathbf{\tilde{x}}$, and whose amplitude is the known observation vector $\mathbf{b}$. (c) Error evolution and final reconstructed image with gain-based method starting from the exact same initial condition as used in (a).
• Figure 4: Phase retrieval with a large-scale sample vector using the gain-based system. (a) A $180 \times 180$ pixel grayscale image of an astronaut (sourced from NASA’s Great Images Database, public domain). This image is used as the sample vector in a CDP-based phase retrieval setup with 8 phase filters, yielding an observation vector $\mathbf{b}$ of length $259{,}200$. (b) The final reconstruction after the gain-based system evolves for $t=5$ from a random initial condition. The resulting RSE is $-9.4$ and the ROE is $-12$.
• Figure 5: Phase retrieval of a two-dimensional vortex in the presence of noise, comparing RRR and the gain-based system. (a) Each panel displays the reconstructed sample vector $\mathbf{\tilde{x}}$, where the grayscale image encodes amplitude and the color image encodes phase. For each noise level, both RRR and the gain-based method start from the same initial condition. Here, RRR runs for 10,000 iterations, while the gain-based system is evolved to $t=1{,}000$. (b) The ground-truth sample vector $\mathbf{x}$ that describes a 2D vortex, showing amplitude (left) and phase (right). (c) Phase retrieval error (RSE) versus the signal-to-noise ratio (SNR). Each data point represents the average of 20 random instances, where the vortex core is placed at different positions and each algorithm is initialized randomly. Error bars denote the standard deviation of the final RSE values.