Ray Optics Approach to Holography

Andrii Torchylo

REVIEW 4 major objections 5 minor 61 references

Ray Optics Approach to Holography

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The optimal-transport plan between two measured intensities is the ray-optics limit of the light field's Wigner distribution, and seeding phase-retrieval algorithms with it eliminates vortex stagnation.

desk verdict A credible OT-based phase initialization with real computational improvements, but the abstract overstates the scope of the Wigner–OT identity and several headline claims are explicitly left unfinished in the body. read the letter →

arxiv 2506.11352 v1 pith:BBYTTI6E submitted 2025-06-12 physics.optics math-phmath.MPphysics.atom-phquant-ph

Andrii Torchylo This is my paper · ORCID

classification physics.opticsmath-phmath.MPphysics.atom-phquant-ph

keywords phaseretrievaloptimaltransportWignerdistributionholographybeamshapingGerchberg-SaxtonSinkhornalgorithmfractionalFouriertransform

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the phase-retrieval problem behind laser beam shaping can be approximated, in the ray-optics limit, by an optimal transport (OT) problem, and that the OT plan is exactly the zeroth-order term of the Wigner distribution of the unknown field: $W_f^{(0)} = \Gamma$. This matters because OT is convex and vortex-free, whereas standard iterative solvers such as Gerchberg-Saxton stagnate by forming phase vortices. The paper reports that using the entropically regularized OT solution as an initialization for Gerchberg-Saxton or MRAF gives an order-of-magnitude improvement in intensity loss with no vortex formation and near-total laser efficiency. The same machinery is extended to fractional Fourier transforms, giving a two-measurement beam estimation protocol and a way to retrieve the phase-space effect of an unknown quadratic-phase optical system. The solver's memory cost is reduced from $O(n^4)$ to $O(n^2)$.

What carries the argument

The load-bearing object is the optimal transport plan $\Gamma$ between the two measured intensities, obtained by entropy-regularized optimal transport. In the ray-optics limit the transport plan coincides with the zero-order Wigner term $\Gamma(u,\mu) = g(u)^2\delta(\mu-\nabla\phi(u))$, a bundle of rays; the gradient of the phase $\nabla\phi$ is the transport map, and integrating it gives the convex, unwrapped phase that seeds the iterative refinement. The practical engine is the Sinkhorn-Knopp algorithm in the form of repeated Gaussian convolutions, which enforces the marginal constraints with a single hyperparameter $\epsilon$ and avoids storing the $n^4$ transport plan.

What would settle it

Choose a target phase $\phi(u,v)=\alpha(u^2+v^2)+\beta u^3$ with $\beta$ comparable to $\alpha$, propagate exactly to get $G^2$, run OT initialization followed by Gerchberg-Saxton refinement, and count vortices and measure intensity loss; the paper's Figures 3.8-3.10 already show the loss growing with $\beta$, so a regime where the vortex-free improvement disappears would falsify the claim outside the convex/ray-optics domain.

Watch

Extended reading notes

Core claim

The central claim is that the optimal transport plan $\Gamma$ between the squared input intensity $g^2$ and the squared target intensity $G^2$, under quadratic cost, is exactly the zeroth-order stationary-phase term of the Wigner distribution of the unknown field, $W_f^{(0)} = \Gamma$, with $\Gamma(u,\mu)=g(u)^2\delta(\mu-\phi'(u))$ in one dimension. Solving the entropy-regularized OT problem therefore produces a convex, unwrapped phase with no $2\pi$ branch cuts, and using that phase to initialize Gerchberg-Saxton or MRAF eliminates the vortex formation that otherwise stalls convergence. The paper demonstrates numerically that the reconstructed Wigner distribution from the OT-seeded phase matches the ground truth for flat and quadratic phases, and shows how the same reduction applies when the two planes are related by a fractional Fourier transform instead of a plain Fourier transform. On the algorithmic side, writing the Sinkhorn-Knopp updates as Gaussian convolutions reduces the memory cost from $O(n^4)$ to $O(n^2)$ and the runtime from $O(n^4/\epsilon^2)$ to $O(n^2\log n/\epsilon^2)$. The same ray-optics reduction yields a two-shot beam-estimation protocol from two fractional Fourier intensity measurements, and the Hermite-Gaussian expansion of the problem is reinterpreted as a quantum state tomography task.

Load-bearing premise

The load-bearing premise is that the unknown phase is a single smooth, bowl-shaped surface (strictly convex or concave) and that the beam features are large enough relative to focal length and wavelength that diffraction can be neglected; the paper's own cubic-phase simulation shows degraded Wigner estimation and increased intensity loss when a non-convex term is present.

Editorial extensions

If this is right

OT-initialized Gerchberg-Saxton produces vortex-free phases on smooth targets and reaches an intensity loss roughly an order of magnitude below random or inverse-Fourier initialization, with no efficiency sacrifice.
Composing OT with MRAF gives a controllable accuracy-efficiency trade-off: near-zero error in a chosen signal region at the cost of a tunable fraction of laser power.
The convolution form of Sinkhorn-Knopp reduces memory from $O(n^4)$ to $O(n^2)$ and runtime from $O(n^4/\epsilon^2)$ to $O(n^2\log n/\epsilon^2)$, making $1024\times1024$ and larger hologram computations feasible.
Two diversity images at different fractional Fourier angles are sufficient for a closed OT-based estimate of an unknown beam's amplitude and phase, with error set by the angle separation.
The same OT/Wigner machinery can retrieve the phase-space (ABCD) transformation of an unknown quadratic-phase optical system from projective intensity measurements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The identity $W_f^{(0)}=\Gamma$ suggests that the difference $W_\psi-\Gamma$ between the reconstructed Wigner distribution and the transport plan directly measures diffraction and interference corrections, so the norm of that difference could serve as a validity diagnostic for when the ray-optics initialization is trustworthy.
The convexity limitation suggests a windowed or locally convex extension: decompose a non-convex target into convex patches, solve OT on each patch, and stitch the phases; the paper's cubic-phase experiments indicate exactly where such a scheme would be stress-tested.
The Hermite-Gaussian/quantum learning formulation implies a concrete algorithmic relaxation not proven in the paper: promote the pure-state vector to a rank-$N$ density matrix in the same constraints, which should make the phase-generation problem feasible even when no pure state exists, at the cost of producing a mixed beam.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: this is a solid methods thesis, not a breakthrough. The core idea—use optimal transport to compute a convex, unwrapped phase that seeds Gerchberg–Saxton or MRAF—is genuinely useful and supported by the numerical experiments shown. The derivation of the Monge–Ampere equation from stationary phase is legitimate, and identifying the zero-order Wigner term with the OT plan is a nice observation, but it only holds for strictly convex phases in the ray-optics limit (beta >> 1). The paper itself admits this in Section 3.2.3 and demonstrates the breakdown on a cubic phase in Figures 3.8–3.10. The abstract, however, states the identity unconditionally and promises 'completely bypasses' vortices and 'state-of-the-art' accuracy without those caveats. That is a scoping gap, not an internal contradiction, but it is load-bearing because it affects how the headline contribution should be read.

What is genuinely new and good: the fractional Fourier transform generalization is a real extension of prior OT phase generation, and the O(n^2) memory reduction via the convolution trick is a practical improvement that matters for high-resolution holograms. The Sinkhorn–Knopp convolution formulation is clearly explained and the complexity analysis is honest. The numerical comparison of OT, GS, and MRAF is also credible for smooth targets, and the paper demonstrates an order-of-magnitude loss improvement when OT seeds GS.

Soft spots, in proportion: the quantum learning chapter is explicitly unfinished—the author says the finite-N proof is not done and calls the vortex-free claim for that approach 'hopeful.' The ABCD retrieval section is also a sketch, not a proven method, with the exact implementation left to future work. The abstract's claim about retrieving any quadratic phase system goes beyond what the body supports. There are no error bars or a linked code repository with a commit hash, which makes the 'state-of-the-art' claim hard to verify independently.

Who this is for: people working on beam shaping for neutral atom arrays or ultracold atom experiments will get real value from the OT initialization and the memory-efficient implementation. The theory is also a clean exposition of ray-optics limits of Wigner distributions.

My recommendation: this deserves a serious referee. It should not be desk-rejected. But it needs major revision to scope the abstract correctly, move the unfinished quantum and ABCD claims to clearly marked speculation, and provide reproducible code and benchmarks. I would not block publication on the theory—the central derivation holds under its stated assumptions—but the claims need to match the evidence.

Referee Report

4 major / 5 minor

Summary. The manuscript presents a ray-optics reduction of phase retrieval. It derives the Monge-Ampere equation from stationary phase, maps that equation to optimal transport, identifies the zero-order term of the Wigner distribution with the OT transport plan, generalizes the argument to fractional Fourier transforms, and uses entropy-regularized OT with a Gaussian-kernel convolution to reduce the memory cost of hologram computation from O(n^4) to O(n^2). The same machinery is applied to beam estimation, and a final chapter sketches a quantum-learning reinterpretation of the phase-generation problem. The text is written as a Stanford BS thesis and includes substantial review material on wave optics, Wigner distributions, and iterative phase-retrieval algorithms.

Significance. If restricted to its demonstrated regime, the paper is a valuable contribution: the stationary-phase derivation of Monge-Ampere, the explicit error parameter beta = 2*pi*R_in*R_out/(f*lambda), and the identification of the OT plan with the zero-order Wigner term are original and pedagogically useful bridges between holography and optimal transport. The O(n^2)-memory Sinkhorn implementation with Gaussian convolutions is a concrete algorithmic improvement over the earlier O(n^4) formulation, and the open-source implementation makes the claims checkable. However, the significance is bounded by the conditions of the theory (strictly convex or concave phase, beta >> 1) and by the fact that the headline performance claims rest on a small number of favorable numerical examples rather than systematic benchmarks. The central theoretical chain is coherent, but the scope of several claims in the abstract and conclusions exceeds what the manuscript proves.

major comments (4)

[Abstract, Secs. 3.2.3 and 3.4.3] The identity W_f^(0) = Gamma (Eq. 3.58) and the associated vortex-free claim are established only in the ray-optics regime: the stationary-phase reduction assumes a strictly convex or concave phase and beta = 2*pi*R_in*R_out/(f*lambda) >> 1 (Secs. 3.2.2-3.2.3). The paper's own cubic-phase experiment (Eq. 3.72; Figs. 3.8-3.10) shows visibly degraded retrieval and increased intensity loss for a non-convex phase. The abstract, however, presents these results for 'the unknown complex-valued field' without these restrictions. This is a scoping gap rather than an internal inconsistency, but it is load-bearing because the abstract's unconditional phrasing is the headline of the paper; the scope conditions should be stated there and in the conclusions.
[Sec. 4.6 and Fig. 4.13] The claim that OT-initialized GS/MRAF yields state-of-the-art accuracy and efficiency and 'completely bypasses' vortex formation is supported by a small set of numerical examples: the Gaussian-to-ring experiment of Sec. 4.2.2 and the 128x128 comparison from [13] reproduced in Fig. 4.13. In Fig. 4.13(j) the MRAF mixing parameter m is hand-tuned to 0.48 to obtain epsilon = 5.95e-16 at 85.15% efficiency, while OT+GS gives 2.58% at 99.91%; this is a favorable demonstration, not a general benchmark. To support the stated generality, the paper should either explicitly restrict the performance claims to smooth, convex, ray-optics-valid targets or provide systematic sweeps over input-output pairs and the two free parameters (epsilon, m).
[Sec. 3.4.4] The claim in Sec. 3.4.4 and the abstract that the method 'can be used to retrieve the phase-space transformation of any unknown quadratic phase system' is not demonstrated. The section derives only the coefficient A/(2B) for a single collimated input and then states that one or two additional diversity images should suffice, with the 'exact implementations of this idea' left to future work. No full ABCD-matrix retrieval algorithm, numerical experiment, or error analysis is given. This statement should be reframed as a conjecture or the missing construction should be supplied.
[Sec. 5, Problem 5] The quantum-learning equivalence in Problem 5 is proved only in the infinite-N, zero-error, one-dimensional setting, and the text explicitly concedes that the finite-N, error-tolerant version 'we have not proven it yet.' The abstract's mention of a quantum learning framework and the chapter's framing as a 'deep theoretical connection' therefore outrun the proved content. This secondary claim should be explicitly labeled as a conjecture, or the finite-N proof and the tomographic algorithm with guarantees should be added.

minor comments (5)

[Theorem 3.2.1] The stationary-phase formula in Eq. (3.14) omits the standard nondegeneracy condition det(nabla^2 psi) != 0; strict convexity alone does not exclude a vanishing Hessian at an isolated point (e.g., x^4), so the theorem's assumptions should be stated precisely.
[Eq. (3.41)] There is a missing parenthetical grouping in the term 'cot alpha pi (x^2 + y^2)'; it should read pi cot(alpha) (x^2 + y^2) or an equivalent unambiguous expression.
[Appendix A.5] The text refers to 'Parallel's identity' in Eq. (A.5); this should be 'Parseval's identity.'
[Sec. 4.2.1] The footnote stating that the convergence conditions were 'summarized from the paper [15] by GPT-o3' is not an appropriate citation style for a scientific article and should be rewritten as a normal scholarly summary with proper attribution.
[Sec. 4.6] The bullet list in Sec. 4.6 says the method has a single hyperparameter epsilon, but when OT is combined with MRAF, the mixing parameter m is also user-selected, as shown in Fig. 4.13(j); the hyperparameter discussion should acknowledge this coupling.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the OT/Wigner identity is a derived mathematical equivalence, not a definitional restatement, and the main claims are tested against ground truth with explicit convexity and ray-optics caveats.

full rationale

The paper's central derivation chain is self-contained. In Sections 3.2.1–3.2.4, the stationary phase approximation is applied to the diffraction integral, and under the stated strict-convexity/concavity assumption it produces the Monge-Ampere equation (3.22). Optimal transport with quadratic cost is then invoked through the standard Brenier theorem (3.24), which gives the same Monge-Ampere equation. This is a derivation, not an assumed equivalence. In Section 3.4.1, the zero-order Wigner approximation W_f^(0) is independently computed by Taylor-expanding the phase and amplitude in the Wigner integral, yielding g^2 δ(φ'-μ). The transport plan Γ for a convex potential satisfying the Monge-Ampere equation has exactly the same form, so Eq. (3.58), W_f^(0)=Γ, is an algebraic identity between two independently defined objects, not a fitted parameter renamed as a prediction. The numerical experiments in Section 3.4.3 compare the retrieved Wigner distribution and propagated intensity against ground truth, and the cubic-phase experiment (Figs. 3.8–3.10) explicitly demonstrates the degradation when the phase is non-convex; this is an honest scoping limitation, not circularity. The abstract's unqualified phrasing is broader than the proven convex case, but that is a scoping gap, not a circular step. Self-citations to [13] are used to identify prior publication of the original algorithm and to point to code and supplementary materials, but the thesis itself contains the derivations and its own numerical validations, with open-source code at [44]. No load-bearing argument reduces to an unverified self-citation, and no external uniqueness theorem is imported from the authors' prior work. The only hyperparameter, ε, is an entropy-regularization parameter, not a quantity fitted to the target and then reported as a prediction. Therefore the paper does not exhibit circular reasoning.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard Fourier analysis, stationary phase, and optimal transport theory. No new physical entities are introduced. The most fragile inputs are the convexity assumption and the beta >> 1 ray-optics condition, both of which the paper itself identifies as limits. The quantum learning reformulation is an additional speculative layer whose assumptions are not yet discharged.

free parameters (2)

entropic regularization epsilon = 1e-3 or 1e-4
Sinkhorn-OT hyperparameter chosen by hand in Section 4.6; controls transport plan smearing and numerical stability. It is not fitted to data, but the central algorithm depends on its choice.
MRAF mixing parameter m = 0.48
Set by hand in Figure 4.13 for the baseline comparison; affects the efficiency-accuracy trade-off of MRAF, not the OT method itself.

assumptions (6)

standard math Stationary phase approximation for highly oscillatory integrals (Theorem 3.2.1).
Used in Section 3.2.1 to reduce the Fresnel/Fourier integral to the Monge-Ampere equation. The theorem is stated and referenced to [30], not proved in the thesis.
domain assumption Strict convexity or concavity of the unknown phase phi.
Invoked in Section 3.2.2 to make the map T from output to input single-valued. The OT solver returns a convex phase, so this assumption limits the method to targets whose ray-optics solution is convex.
standard math Existence and uniqueness of the Monge-Ampere boundary value problem under mild regularity and convexity assumptions.
Cited from [31,32] in Section 3.2.2 to justify that the OT solution is well-posed.
standard math Sinkhorn theorem and generalized Sinkhorn theorem for matrix scaling.
Used in Sections 4.6.5 and 4.6.6 to solve entropy-regularized OT; the theorem is stated and cited to [48,49].
domain assumption Ray optics validity requires beta = 2*pi*R_in*R_out/(f*lambda) >> 1.
Established in Section 3.2.3 as the scaling limit where the stationary phase error vanishes. The paper acknowledges that this is not always controllable in experiments.
ad hoc to paper The finite-N quantum learning equivalence (Problem 5) is asserted with only a sketch.
Section 5.1 states the proof is a sketch for infinite N and that the finite-N, noisy version has not been proven. This is a load-bearing part of the quantum tomography reinterpretation but is unproven in the preprint.

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Cite this review

Pith. "Pith review of Ray Optics Approach to Holography." pith.science (2026). https://pith.science/paper/BBYTTI6E

@misc{pith2026250611352,
  author       = {Pith},
  title        = {Pith review of: Ray Optics Approach to Holography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BBYTTI6E}},
  note         = {Machine review of arXiv:2506.11352}
}

read the original abstract

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.

Figures

Figures reproduced from arXiv: 2506.11352 by the authors.

**Figure 1.1.** Model optical system. An input laser beam with intensity [PITH_FULL_IMAGE:figures/full_fig_p013_1_1.png] view at source ↗

**Figure 2.1.** First five Hermite polynomials Hn(u) [PITH_FULL_IMAGE:figures/full_fig_p020_2_1.png] view at source ↗

**Figure 2.2.** First five Hermite-Gaussian functions hn(u) 2.3.1 Hermite-Gaussian Properties Complete Basis of L 2 (R) Notice that each hl is square-integrable, because it is suppressed by e −πu2 , so it decays sufficiently fast. They have norm 1, according to the standard inner product of L 2 (R), and they are orthogonal to each other. In fact, one can view Hermite polynomials as a Gram-Schmidt orthogonalization of the basis {1, … view at source ↗

Figures from the paper (43 more)

**Figure 2.3.** Figure 2.3: Pictorial representation of a fractional Fourier transform [PITH_FULL_IMAGE:figures/full_fig_p024_2_3.png]

**Figure 2.4.** Figure 2.4: Wigner distributions of exp(i2πuξ) (left), δ(u − ξ) middle, and exp(iπ(χu2 + 2ξu + ζ)) (right). The values ξ and χ are set to 1. Let’s interpret some of the signals. The Wigner distribution of exp(i2πuξ), δ(u − ξ), and exp(iπ(χu2 + 2ξu + ζ)) are just straight lines t…

**Figure 2.5.** Figure 2.5: Wigner distribution of rect(u). 2.5.2 Hermite-Gaussian Decomposition Recall that Hermite Gaussian basis (hl)∞ l=0 is a complete orthonormal basis for L 2 (R 2 ). So, we can express any function as f = P∞ n=0 Cnfn. This allows us to express the Wigner distribution as …

**Figure 2.6.** Figure 2.6: Wigner distribution of first five Hermite Gaussians [PITH_FULL_IMAGE:figures/full_fig_p029_2_6.png]

**Figure 2.7.** Figure 2.7: Relationship between the Wigner distribution, fractional Fourier Transform, and Radon [PITH_FULL_IMAGE:figures/full_fig_p031_2_7.png]

**Figure 2.8.** Figure 2.8: Effect of the free space propagation on the Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p035_2_8.png]

**Figure 2.9.** Figure 2.9: Effect of the propagation through the lens on the Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p036_2_9.png]

**Figure 2.10.** Figure 2.10: Effect of the propagation through the quadratic-graded index medium on the Wigner [PITH_FULL_IMAGE:figures/full_fig_p037_2_10.png]

**Figure 2.11.** Figure 2.11: Spherical reference surface [PITH_FULL_IMAGE:figures/full_fig_p037_2_11.png]

**Figure 2.12.** Figure 2.12: Optical system composed of a section of free space, followed by a lens, followed by [PITH_FULL_IMAGE:figures/full_fig_p038_2_12.png]

**Figure 3.1.** Figure 3.1: Schematic representation of the Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p044_3_1.png]

**Figure 3.2.** Figure 3.2: Optical set-up for the beam estimation. If the SLM on the input plane applies a quadratic [PITH_FULL_IMAGE:figures/full_fig_p045_3_2.png]

**Figure 3.3.** Figure 3.3: Input distributions p(x), q(x), and the optimal transport plan T ∗ (in our notation Γ). The “trick" is to optimize R Γ(x, y)c(x, y)dxdy over all possible transport plans Γ(x, y) subject to conditions that R Γ(x, y)dxdy = 1, R Γ(x, y)dy = µ(x), and R Γ(x, y)dx = ν(y).…

**Figure 3.4.** Figure 3.4: Marginal intensities for α = 0, β = 0, σ = 2 [PITH_FULL_IMAGE:figures/full_fig_p060_3_4.png]

**Figure 3.5.** Figure 3.5: Target Wigner distribution Wf (left), transport plan Γ = W (0) f (middle), and retrieved Wigner distribution Wψ (right) for α = 0, β = 0, σ = 2 Quadratic Phase Experiment α = 2, β = 0 For this experiment, we fix α = 2, β = 0, and σ = 2. This is equivalent to placing …

**Figure 3.6.** Figure 3.6: Marginal intensities for α = 2, β = 0, σ = 2 [PITH_FULL_IMAGE:figures/full_fig_p061_3_6.png]

**Figure 3.7.** Figure 3.7: Target Wigner distribution Wf (left), transport plan Γ = W (0) f (middle), and retrieved Wigner distribution Wψ (right) for α = 2, β = 0, σ = 2 Cubic Phase Experiment α = 2, β = −2 For this experiment, we fix α = 2, β = −2, and σ = 2. We note that this choice of the …

**Figure 3.8.** Figure 3.8: Marginal intensities for α = 2, β = −2, σ = 2 [PITH_FULL_IMAGE:figures/full_fig_p062_3_8.png]

**Figure 3.9.** Figure 3.9: Target Wigner distribution Wf (left), transport plan Γ = W (0) f (middle), and retrieved Wigner distribution Wψ (right) for α = 2, β = −2, σ = 2 Parameter Sweep Experiment In this experiment, we sweep through the range of parameters α and β, recording the intensity l…

**Figure 3.10.** Figure 3.10: Intensity loss between the target and predicted distributions for [PITH_FULL_IMAGE:figures/full_fig_p063_3_10.png]

**Figure 4.1.** Figure 4.1: Convergence of the method of iterated projections depending on the geometry of sets [PITH_FULL_IMAGE:figures/full_fig_p067_4_1.png]

**Figure 4.2.** Figure 4.2: Input is a Gaussian intensity pattern with [PITH_FULL_IMAGE:figures/full_fig_p070_4_2.png]

**Figure 4.3.** Figure 4.3: Optimized phase over first 16 iterations. We start with a flat phase of all zeros, and [PITH_FULL_IMAGE:figures/full_fig_p071_4_3.png]

**Figure 4.4.** Figure 4.4: Visual diagram depicting vortex formation (courtesy to [41]). (a) Two adjacent points [PITH_FULL_IMAGE:figures/full_fig_p071_4_4.png]

**Figure 4.5.** Figure 4.5: Intensity Loss as defined in (B.25) as a function of number of iterations. [PITH_FULL_IMAGE:figures/full_fig_p072_4_5.png]

**Figure 4.6.** Figure 4.6: The phase solution at iteration 500 of the Gerchberg-Saxton algorithm. The left panel [PITH_FULL_IMAGE:figures/full_fig_p072_4_6.png]

**Figure 4.7.** Figure 4.7: Accuracy vs. Efficiency trade-off. By controlling the value of the parameter [PITH_FULL_IMAGE:figures/full_fig_p073_4_7.png]

**Figure 4.8.** Figure 4.8: MRAF solutions. The top left is the m-value of 0.1, and the top right is 0.5 — both [PITH_FULL_IMAGE:figures/full_fig_p074_4_8.png]

**Figure 4.9.** Figure 4.9: Overview of Deep Learning Based Approaches. 1a shows approach (1) and 1b shows [PITH_FULL_IMAGE:figures/full_fig_p076_4_9.png]

**Figure 4.10.** Figure 4.10: Deep learning approach when one starts from the unknown weights [PITH_FULL_IMAGE:figures/full_fig_p077_4_10.png]

**Figure 4.11.** Figure 4.11: The optimization loss of the GS algorithm if we start from iFFT phase (blue line), or [PITH_FULL_IMAGE:figures/full_fig_p078_4_11.png]

**Figure 4.12.** Figure 4.12: Final phase, produced from OT initialization and GS refinement (left) and the corre [PITH_FULL_IMAGE:figures/full_fig_p079_4_12.png]

**Figure 4.13.** Figure 4.13: Comparison of phases and output beams from various phase generation algorithms. [PITH_FULL_IMAGE:figures/full_fig_p079_4_13.png]

**Figure 4.14.** Figure 4.14: Input and output distributions for the experiment with the Sinkhorn-Knopp algorithm. [PITH_FULL_IMAGE:figures/full_fig_p085_4_14.png]

**Figure 4.15.** Figure 4.15: Optimal transport using the Sinkhorn-Knopp algorithm for various values of [PITH_FULL_IMAGE:figures/full_fig_p086_4_15.png]

**Figure 4.16.** Figure 4.16: Beam estimates using various phase diversity algorithms. The top row (a-d) is the beam [PITH_FULL_IMAGE:figures/full_fig_p095_4_16.png]

**Figure 4.17.** Figure 4.17: Beam estimation error metrics, (a) in absence of noise, (b) with additive noise, and [PITH_FULL_IMAGE:figures/full_fig_p096_4_17.png]

**Figure 1.** Figure 1: Overview of Deep Learning Based Approaches. 1a shows [PITH_FULL_IMAGE:figures/full_fig_p138_1.png]

**Figure 2.** Figure 2: Input and output amplitudes for our experiment [PITH_FULL_IMAGE:figures/full_fig_p140_2.png]

**Figure 3.** Figure 3: Phase maps and amplitudes for selected methods. Each [PITH_FULL_IMAGE:figures/full_fig_p141_3.png]

**Figure 4.** Figure 4: L2 Distance of Predicted Amplitudes for all Baseline [PITH_FULL_IMAGE:figures/full_fig_p142_4.png]

**Figure 6.** Figure 6: L2 Norm of all Deep Learning Methods While the ResUNet does not substantially alter performance over a base UNet, we observe a longer period to convergence. For completeness, we ran OT + ResUNet (scheduler) to 8000 iterations and the converged L2 norm was nearly ident…

**Figure 7.** Figure 7: All L2 norm curves in Hyperparameter Sweep of OT + [PITH_FULL_IMAGE:figures/full_fig_p144_7.png]

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Works this paper leans on

61 extracted references · 56 canonical work pages

[13]

Van de Graaff, Jan Rudolph, and Jason M

Hunter Swan, Andrii Torchylo, Michael J. Van de Graaff, Jan Rudolph, and Jason M. Hogan. High-fidelity holographic beam shaping with optimal transport and phase diversity, 2024

work page 2024
[1]

A high-accuracy algorithm for designing arbitrary holographic atom traps.Opt

Matthew Pasienski and Brian DeMarco. A high-accuracy algorithm for designing arbitrary holographic atom traps.Opt. Express, 16(3):2176–2190, Feb 2008

work page 2008
[2]

Ribet, Steven E

Georgios Varnavides, Stephanie M. Ribet, Steven E. Zeltmann, Yue Yu, Benjamin H. Savitzky, Dana O. Byrne, Frances I. Allen, Vinayak P. Dravid, Mary C. Scott, and Colin Ophus. Iterative phase retrieval algorithms for scanning transmission electron microscopy, 2024

work page 2024
[3]

Kim, Xudong Pei, Chen Huang, Mark Boyce, Luiza Men- donça, Daniel Clare, Alistair Siebert, Christopher S

Liqi Zhou, Jingdong Song, Judy S. Kim, Xudong Pei, Chen Huang, Mark Boyce, Luiza Men- donça, Daniel Clare, Alistair Siebert, Christopher S. Allen, Emanuela Liberti, David Stuart, Xiaoqing Pan, Peter D. Nellist, Peijun Zhang, Angus I. Kirkland, and Peng Wang. Low-dose phase retrieval of biological specimens using cryo-electron ptychography.Nature Communica...

work page 2020
[4]

M. D. Barrett, J. A. Sauer, and M. S. Chapman. All-optical formation of an atomic bose-einstein condensate. Phys. Rev. Lett., 87:010404, Jun 2001

work page 2001
[5]

Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond

Maciej Lewenstein, Anna Sanpera, Veronica Ahufinger, Bogdan Damski, Aditi Sen(De), and Ujjwal Sen. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Advances in Physics, 56(2):243–379, 2007

work page 2007
[6]

Hogan, Alex Sugarbaker, Susannah M

Tim Kovachy, Jason M. Hogan, Alex Sugarbaker, Susannah M. Dickerson, Christine A. Don- nelly, Chris Overstreet, and Mark A. Kasevich. Matter wave lensing to picokelvin temperatures. Physical Review Letters, 114(14), April 2015

work page 2015
[7]

Manetsch, Gyohei Nomura, Elie Bataille, Kon H

Hannah J. Manetsch, Gyohei Nomura, Elie Bataille, Kon H. Leung, Xudong Lv, and Manuel Endres. A tweezer array with 6100 highly coherent atomic qubits, 2024

work page 2024

Show all 61 references

[8]

Thompson

Bichen Zhang, Pai Peng, Aditya Paul, and Jeff D. Thompson. Scaled local gate controller for optically addressed qubits.Optica, 11(2):227, February 2024

2024
[9]

Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D

Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. Probing many-body dynamics on a 51-atom quantum simulator.Nature, 551(7682):...

2017
[10]

Sepehr Ebadi, Tout T. Wang, Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Dolev Bluvstein, Rhine Samajdar, Hannes Pichler, Wen Wei Ho, Soonwon Choi, Subir Sachdev, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. Quantum phases of matter on a 256-atom p...

2021
[11]

Shaw, Ran Finkelstein, and Manuel Endres

Richard Bing-Shiun Tsai, Xiangkai Sun, Adam L. Shaw, Ran Finkelstein, and Manuel Endres. Benchmarking and fidelity response theory of high-fidelity rydberg entangling gates. PRX Quantum, 6:010331, Feb 2025

2025
[12]

Shack-hartmann wavefront sensor optical dynamic range.Opt

Vyas Akondi and Alfredo Dubra. Shack-hartmann wavefront sensor optical dynamic range.Opt. Express, 29(6):8417–8429, Mar 2021

2021
[14]

R. W. Gerchberg and W. O. Saxton. A practical algorithm for the determination of plane from image and diffraction pictures.Optik, 35(2):237–246, 1972

1972
[15]

Alternating projections with applications to gerchberg-saxton error reduction, 2021

Dominikus Noll. Alternating projections with applications to gerchberg-saxton error reduction, 2021

2021
[16]

Bruce, Jonathan Keeling, and Donatella Cassettari

Tiffany Harte, Graham D. Bruce, Jonathan Keeling, and Donatella Cassettari. Conjugate gra- dient minimisation approach to generating holographic traps for ultracold atoms.Opt. Express, 22(22):26548–26558, Nov 2014

2014
[17]

Epstein, Leslie Greengard, and Jeremy Magland

Alexander Barnett, Charles L. Epstein, Leslie Greengard, and Jeremy Magland. Geometry of the phase retrieval problem, 2020

2020
[18]

Gutiérrez.Optimal Transport and Applications to Geometric Optics

Cristian E. Gutiérrez.Optimal Transport and Applications to Geometric Optics. SpringerBriefs on PDEs and Data Science. Springer Singapore, 1 edition, 2023

2023
[19]

Light in power: a general and parameter- free algorithm for caustic design.ACM Transactions on Graphics (TOG), 37(6):1–13, 2018

Jocelyn Meyron, Quentin Mérigot, and Boris Thibert. Light in power: a general and parameter- free algorithm for caustic design.ACM Transactions on Graphics (TOG), 37(6):1–13, 2018

2018
[20]

Optical design of single reflector systems and the monge– kantorovich mass transfer problem.Journal of Mathematical Sciences, 117(3):4096–4108, 2003

Tilmann Glimm and Vladimir Oliker. Optical design of single reflector systems and the monge– kantorovich mass transfer problem.Journal of Mathematical Sciences, 117(3):4096–4108, 2003

2003
[21]

On the design of a reflector antenna II

Xu-Jia Wang. On the design of a reflector antenna II. Calculus of Variations and Partial Differential Equations, 20(3):329–341, 2004

2004
[22]

High-contrast computational caustic design.ACM Transactions on Graphics (TOG), 33(4):1–11, 2014

Yuliy Schwartzburg, Romain Testuz, Andrea Tagliasacchi, and Mark Pauly. High-contrast computational caustic design.ACM Transactions on Graphics (TOG), 33(4):1–11, 2014

2014
[23]

The Refractor Problem in Reshaping Light Beams

Cristian E Gutiérrez and Qingbo Huang. The Refractor Problem in Reshaping Light Beams. Arch. Ration. Mech. Anal., 193(2):423–443, 2009

2009
[24]

Dickey.Laser Beam Shaping: Theory and Techniques

Fred M. Dickey.Laser Beam Shaping: Theory and Techniques. CRC Press, Boca Raton, FL, 2nd edition, 2014. BIBLIOGRAPHY 138

2014
[25]

Murat Ozaktas, M

H. Murat Ozaktas, M. Alper Kutay, and Zeev Zalevsky.The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, 2001

2001
[26]

A.E. Siegman. Lasers. G - Reference,Information and Interdisciplinary Subjects Series. Uni- versity Science Books, 1986

1986
[27]

Young.An Introduction to Hilbert Space

N. Young.An Introduction to Hilbert Space. Cambridge University Press, 1988

1988
[28]

Laguerre–gaussian modes and the wigner transform

Michael VanValkenburgh. Laguerre–gaussian modes and the wigner transform. Journal of Modern Optics, 55(21):3537–3549, December 2008

2008
[29]

Psycharakis, Evangelos Liapis, Jorge Ripoll, and Giannis Zacharakis

Daniele Ancora, Diego Di Battista, Georgia Giasafaki, Stylianos E. Psycharakis, Evangelos Liapis, Jorge Ripoll, and Giannis Zacharakis. Optical projection tomography via phase retrieval algorithms. Methods, 136:81–89, 2018. Methods in Quantitative Phase Imaging in Life Science

2018
[30]

Handelsman

Norman Bleistein and Richard A. Handelsman. Asymptotic Expansions of Integrals. Dover Publications, New York, corrected dover reprint of the 1975 original edition, 1986. See Chap.8, §8.4, eq.(8.4.10) for the 2-D stationary-phase formula

1975
[31]

Caffarelli

Luis A. Caffarelli. Interiorw2,p estimates for solutions of the monge–ampère equation.Annals of Mathematics, 131(1):135–150, 1990

1990
[32]

On the second boundary value problem for equations of monge–ampère type

John Urbas. On the second boundary value problem for equations of monge–ampère type. Journal für die reine und angewandte Mathematik, 487:115–124, 1997

1997
[33]

Optimaltransport.jl

Tim Matsumoto, David Widmann, Davi Barreira, and Stephen Zhang. Optimaltransport.jl. https://github.com/JuliaOptimalTransport/OptimalTransport.jl, commit: 9da044c, 2022

2022
[34]

Pot: Python optimal transport.The Journal of Machine Learning Research, 22(1):3571–3578, 2021

Rémi Flamary, Nicolas Courty, Alexandre Gramfort, Mokhtar Z Alaya, Aurélie Boisbunon, Stanislas Chambon, Laetitia Chapel, Adrien Corenflos, Kilian Fatras, Nemo Fournier, et al. Pot: Python optimal transport.The Journal of Machine Learning Research, 22(1):3571–3578, 2021

2021
[35]

Optimal Transport: Old and New, volume 338

Cédric Villani. Optimal Transport: Old and New, volume 338. Springer, 2009

2009
[36]

Topics in optimal transportation, volume 58

Cédric Villani. Topics in optimal transportation, volume 58. American Mathematical Soc., 2021

2021
[37]

Décomposition polaire et réarrangement monotone des champs de vecteurs.CR Acad

Yann Brenier. Décomposition polaire et réarrangement monotone des champs de vecteurs.CR Acad. Sci. Paris Sér. I Math., 305:805–808, 1987

1987
[38]

Polar factorization and monotone rearrangement of vector-valued functions

Yann Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Communications on Pure and Applied Mathematics, 44(4):375–417, 1991

1991
[39]

The Monge–Ampère equation and its link to optimal transportation

Guido De Philippis and Alessio Figalli. The Monge–Ampère equation and its link to optimal transportation. Bulletin of the American Mathematical Society, 51(4):527–580, 2014. BIBLIOGRAPHY 139

2014
[40]

Goldstein

Ward Cheney and Allen A. Goldstein. Proximity maps for convex sets. Proceedings of the American Mathematical Society, 10(3):448–450, 1959

1959
[41]

Bijnen, van.Quantum engineering with ultracold atoms

R.M.W. Bijnen, van.Quantum engineering with ultracold atoms. Phd thesis 1 (research tu/e / graduation tu/e), Applied Physics and Science Education, 2013

2013
[42]

Kaiqiang Wang, Li Song, Chutian Wang, Zhenbo Ren, Guangyuan Zhao, Jiazhen Dou, Jianglei Di, George Barbastathis, Renjie Zhou, Jianlin Zhao, and Edmund Y. Lam. On the use of deep learning for phase recovery.Light: Science & Applications, 13(1), January 2024

2024
[43]

Phase imaging with an untrained neural network

Fei Wang, Yaoming Bian, Haichao Wang, Meng Lyu, Giancarlo Pedrini, Wolfgang Osten, George Barbastathis, and Guohai Situ. Phase imaging with an untrained neural network. Light: Science & Applications, 9(1):77, May 2020

2020
[44]

SLMTools.https://github.com/ hoganphysics/SLMTools, commit: d7db9fd, 2024

Hunter Swan, Michael Van de Graaff, and Andrii Torchylo. SLMTools.https://github.com/ hoganphysics/SLMTools, commit: d7db9fd, 2024

2024
[45]

Computational optimal transport, 2020

Gabriel Peyré and Marco Cuturi. Computational optimal transport, 2020

2020
[46]

Cambridge University Press, 2004

Stephen Boyd and Lieven Vandenberghe.Convex Optimization. Cambridge University Press, 2004

2004
[47]

Sinkhorndistances: Lightspeedcomputationofoptimaltransportationdistances, 2013

MarcoCuturi. Sinkhorndistances: Lightspeedcomputationofoptimaltransportationdistances, 2013

2013
[48]

A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices

Richard Sinkhorn. A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices. The Annals of Mathematical Statistics, 35(2):876 – 879, 1964

1964
[49]

On sinkhorn’s representation of nonnegative matrices.Journal of Mathematical Analysis and Applications, 54(3):674–677, 1976

Helge Tverberg. On sinkhorn’s representation of nonnegative matrices.Journal of Mathematical Analysis and Applications, 54(3):674–677, 1976

1976
[50]

A review of matrix scaling and sinkhorn’s normal form for matrices and positive maps, 2016

Martin Idel. A review of matrix scaling and sinkhorn’s normal form for matrices and positive maps, 2016

2016
[51]

Computational optimal trans- port: Complexity by accelerated gradient descent is better than by sinkhorn’s algorithm, 2018

Pavel Dvurechensky, Alexander Gasnikov, and Alexey Kroshnin. Computational optimal trans- port: Complexity by accelerated gradient descent is better than by sinkhorn’s algorithm, 2018

2018
[52]

Stabilized sparse scaling algorithms for entropy-regularized transport problems

Bernhard Schmitzer. Stabilized sparse scaling algorithms for entropy-regularized transport problems. SIAM Journal on Scientific Computing, 41(3):A1443–A1481, 2019

2019
[53]

Overrelaxed sinkhorn– knopp algorithm for regularized optimal transport.arXiv preprint, 2017

Alexis Thibault, Lénaïc Chizat, Charles Dossal, and Nicolas Papadakis. Overrelaxed sinkhorn– knopp algorithm for regularized optimal transport.arXiv preprint, 2017. v2, March 2021

2017
[54]

Robert J. Berman. The sinkhorn algorithm, parabolic optimal transport and geometric monge– ampère equations. Numerische Mathematik, 145(4):771–836, 2020. BIBLIOGRAPHY 140

2020
[55]

Rethinking initialization of the sinkhorn algorithm

James Thornton and Marco Cuturi. Rethinking initialization of the sinkhorn algorithm. In Proc. 26th Int. Conf. on Artificial Intelligence and Statistics (AISTATS), pages 7728–7754. PMLR, 2023

2023
[56]

A Survey of Gaussian Convolution Algorithms.Image Processing On Line, 3:286–310, 2013

Pascal Getreuer. A Survey of Gaussian Convolution Algorithms.Image Processing On Line, 3:286–310, 2013. https://doi.org/10.5201/ipol.2013.87

2013 doi
[57]

Kick-off seminar — optimal transport, statistics and learning

Gabriel Peyré. Kick-off seminar — optimal transport, statistics and learning. https: //kantorovich.org/event/kickoff_peyre/, 2023. Accessed: 2025-05-15

2023
[58]

Andrews, Richard Askey, and Ranjan Roy.Special Functions, volume 71 ofEncyclo- pedia of Mathematics and its Applications

George E. Andrews, Richard Askey, and Ranjan Roy.Special Functions, volume 71 ofEncyclo- pedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999

1999
[59]

Madalin Guta, Jonas Kahn, Richard Kueng, and Joel A. Tropp. Fast state tomography with optimal error bounds, 2018

2018
[60]

Santhanam

Balu Santhanam and Thalanayar S. Santhanam. Discrete gauss-hermite functions and eigen- vectors of the centered discrete fourier transform. In2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP ’07, volume 3, pages III–1385–III–1388, 2007

2007
[61]

Claasen and W

T. Claasen and W. Mecklenbrauker. The aliasing problem in discrete-time wigner distributions. IEEE Transactions on Acoustics, Speech, and Signal Processing, 31(5):1067–1072, 1983

1983

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [13]

Van de Graaff, Jan Rudolph, and Jason M

Hunter Swan, Andrii Torchylo, Michael J. Van de Graaff, Jan Rudolph, and Jason M. Hogan. High-fidelity holographic beam shaping with optimal transport and phase diversity, 2024

work page 2024

[2] [1]

A high-accuracy algorithm for designing arbitrary holographic atom traps.Opt

Matthew Pasienski and Brian DeMarco. A high-accuracy algorithm for designing arbitrary holographic atom traps.Opt. Express, 16(3):2176–2190, Feb 2008

work page 2008

[3] [2]

Ribet, Steven E

Georgios Varnavides, Stephanie M. Ribet, Steven E. Zeltmann, Yue Yu, Benjamin H. Savitzky, Dana O. Byrne, Frances I. Allen, Vinayak P. Dravid, Mary C. Scott, and Colin Ophus. Iterative phase retrieval algorithms for scanning transmission electron microscopy, 2024

work page 2024

[4] [3]

Kim, Xudong Pei, Chen Huang, Mark Boyce, Luiza Men- donça, Daniel Clare, Alistair Siebert, Christopher S

Liqi Zhou, Jingdong Song, Judy S. Kim, Xudong Pei, Chen Huang, Mark Boyce, Luiza Men- donça, Daniel Clare, Alistair Siebert, Christopher S. Allen, Emanuela Liberti, David Stuart, Xiaoqing Pan, Peter D. Nellist, Peijun Zhang, Angus I. Kirkland, and Peng Wang. Low-dose phase retrieval of biological specimens using cryo-electron ptychography.Nature Communica...

work page 2020

[5] [4]

M. D. Barrett, J. A. Sauer, and M. S. Chapman. All-optical formation of an atomic bose-einstein condensate. Phys. Rev. Lett., 87:010404, Jun 2001

work page 2001

[6] [5]

Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond

Maciej Lewenstein, Anna Sanpera, Veronica Ahufinger, Bogdan Damski, Aditi Sen(De), and Ujjwal Sen. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Advances in Physics, 56(2):243–379, 2007

work page 2007

[7] [6]

Hogan, Alex Sugarbaker, Susannah M

Tim Kovachy, Jason M. Hogan, Alex Sugarbaker, Susannah M. Dickerson, Christine A. Don- nelly, Chris Overstreet, and Mark A. Kasevich. Matter wave lensing to picokelvin temperatures. Physical Review Letters, 114(14), April 2015

work page 2015

[8] [7]

Manetsch, Gyohei Nomura, Elie Bataille, Kon H

Hannah J. Manetsch, Gyohei Nomura, Elie Bataille, Kon H. Leung, Xudong Lv, and Manuel Endres. A tweezer array with 6100 highly coherent atomic qubits, 2024

work page 2024