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arXiv:2512.19072

A fast, large-scale optimal transport algorithm for holographic beam shaping

Andrii Torchylo, Hunter Swan, Lucas Tellez, Jason Hogan

physics.opticsphysics.atom-phquant-phSubmitted Dec 22, 2025• Announced Dec 23, 2025
v1 · Dec 22, 2025v2 · Feb 19, 2026
arXivHTMLPDF

Abstract

Optimal transport methods have recently established state of the art accuracy and efficiency for holographic laser beam shaping. However, use of such methods is hindered by severe $\mathcal{O}(N^2)$ memory and $\mathcal{O}(N^2)$ time requirements for large scale input or output images with $N$ total pixels. Here we leverage the dual formulation of the optimal transport problem and the separable structure of the cost to implement algorithms with greatly reduced $\mathcal{O}(N)$ memory and $\mathcal{O}(N\log N)$ to $\mathcal{O}(N^{3/2})$ time complexity. These algorithms are parallelizable and can solve megapixel-scale beam shaping problems in tens of seconds on a CPU or seconds on a GPU.

Journal reference
Opt. Express 34, 6796-6803 (2026)
DOI
10.1364/OE.588337
Comments
8 pages, 4 figures, plus supplement

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Optimal transport methods have recently established state of the art accuracy and efficiency for holographic laser beam shaping. However, use of such methods is hindered by severe O(N2)\mathcal{O}(N^2)O(N2) memory and O(N2)\mathcal{O}(N^2)O(N2) time requirements for large scale input or output images with NNN total pixels. Here we leverage the dual formulation of the optimal transport problem and the separable structure of the cost to implement algorithms with greatly reduced O(N)\mathcal{O}(N)O(N) memory and O(Nlog⁡N)\mathcal{O}(N\log N)O(NlogN) to O(N3/2)\mathcal{O}(N^{3/2})O(N3/2) time complexity. These algorithms are parallelizable and can solve megapixel-scale beam shaping problems in tens of seconds on a CPU or seconds on a GPU.
Retrieving the phase of a complex-valued field from the measurements of its amplitude is a crucial problem with a wide range of applications in microscopy and ultracold atomic physics. In particular, obtaining an accurate and efficient solution to this problem is a key step in shaping laser beams for trapping atoms in optical tweezer arrays and applying high-fidelity entangling gates on a neutral atom quantum computer. Current approaches to this problem fail to converge on the optimal solution due to a phenomenon known as vortex formation. In this work, we present an efficient optimization algorithm using Optimal Transport. Our approach completely bypasses the creation of phase vortices and allows for a state-of-the-art solution both in terms of accuracy and efficiency. Furthermore, we show a deep theoretical connection between the Optimal Transport plan and the ray-optics limit of the Wigner distribution of the unknown complex-valued field, and show that our method can be used to retrieve the phase-space transformation of any unknown quadratic phase system. Finally, we reinterpret this problem in the modern quantum learning framework. The techniques we develop provide both useful intuition and practical tools for advancing the frontiers of phase retrieval and laser beam shaping.
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