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arxiv: 2604.14119 · v1 · submitted 2026-04-15 · ⚛️ physics.optics · eess.SP· physics.app-ph· quant-ph

Single Plane Spatial Mode Sorter

Khen Cohen , Yoav Yosif-Or , Yaron Oz , Ady Arie This is my paper

Pith reviewed 2026-05-10 12:16 UTC · model grok-4.3

classification ⚛️ physics.optics eess.SPphysics.app-phquant-ph
keywords spatial mode sorterHermite-Gaussian modesLaguerre-Gaussian modesBessel-Gaussian modesorbital angular momentumphase maskmode multiplexingsingle-plane optics
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The pith

A single phase mask in one plane sorts orthogonal spatial modes from multiple families into separate outputs with near-zero crosstalk.

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

The paper derives and tests a phase-only mask placed in a single plane that takes a superposition of orthogonal modes in one input beam and directs each mode to its own output position. It works for Hermite-Gaussian, Laguerre-Gaussian, and Bessel-Gaussian families alike. The same mask, when illuminated from the other side by a Gaussian beam, generates arbitrary modes. Transmission efficiency scales as 1/M for M modes and the authors prove this bound is optimal for typical detector placements.

Core claim

A single-plane phase pattern can be analytically derived that maps any chosen set of M orthogonal spatial modes to M spatially separated output locations with almost no crosstalk. For the special case of OAM modes with zero radial index the pattern reduces to the known fork grating. In the limit of large M the design yields an analytic expression for sorting every member of a given mode family. The power reaching each output is 1/M, shown to be the maximum possible under the constraint of fixed detector positions.

What carries the argument

The single-plane phase mask whose transmission function is obtained by solving for the phases that steer each input mode to a distinct output coordinate while preserving orthogonality.

If this is right

  • The device sorts HG, LG, and BG modes in a single plane without needing multiple elements.
  • Reversing the sorter turns a Gaussian input into any desired mode from the family.
  • For orbital angular momentum modes with zero radial index the design reproduces the standard fork grating.
  • Power efficiency is bounded by 1/M and this bound is tight for standard output geometries.
  • The sorter remains functional across a modest wavelength range before diffraction and phase errors degrade performance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may allow compact mode-multiplexed links in both classical and quantum optics by replacing bulk mode sorters.
  • Because the mask is analytic, it can be fabricated once and used for any chosen subset of a mode family without redesign.
  • Sensitivity analysis in the paper implies that random phase noise on the mask or input must stay below a few percent of a radian to keep crosstalk low.
  • Operating the device in the large-M limit could provide a continuous spatial-mode demultiplexer for beams that contain many modes.

Load-bearing premise

A fixed phase pattern chosen in advance can map every orthogonal input mode to a unique non-overlapping output spot without residual interference that grows with wavelength or noise.

What would settle it

An experiment in which measured crosstalk between any two sorted modes exceeds a few percent or the measured power per channel falls significantly below the predicted 1/M value.

Figures

Figures reproduced from arXiv: 2604.14119 by Ady Arie, Khen Cohen, Yaron Oz, Yoav Yosif-Or.

Figure 1
Figure 1. Figure 1: A schematic diagram of the setup. Distinct spatial modes traverse a single [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sorting results for three groups of modes. From top to bot￾tom: (a) Hermite-Gaussian modes 𝐻𝐺0,0, 𝐻𝐺1,0, 𝐻𝐺0,1, 𝐻𝐺1,1 ; (b) Radial La￾guerre–Gaussian modes 𝐿𝐺0,0, 𝐿𝐺1,0, 𝐿𝐺2,0, 𝐿𝐺3,0; and (c) Bessel-Gaussian modes 𝐵𝐺0,0, 𝐵𝐺1,1, 𝐵𝐺2,−2, 𝐵𝐺3,3. The modes and the sorter size are about three times the beam waist. The detection matrices report the experimental result in red, and simulation results in white. The… view at source ↗
Figure 3
Figure 3. Figure 3: Experimental results of the relationship between the power transmission [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MUB confusion matrix. The function 𝐷(.) denotes the detector linked to each transmitted mode. In each cell, the value at the top (red) indicates the experimental data, while the value at the bottom (white/black) reflects the simulation outcome. Refer to the Supplementary Materials for the projection figures. 3.6. Mode Generation Another direct application of our sorter is spatial mode generation. By projec… view at source ↗
Figure 5
Figure 5. Figure 5: Using the sorter in reverse for spatial mode generation. Top: Schematic of the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The analytical mode sorter used for distinguishing between spatial modes of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Random sampling of different detector positions for HG and LG random modes (up to mode 10). Each histogram contains 50,000 randomly chosen detector positions and mode families to sort. The detectors were placed randomly following a normal distribution around the optical axis, ensuring good coverage of the entire output plane [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Optimization over the detectors location, can still improve the results but this is very limited. An example of an optimization over HG modes: 𝐻𝐺00, 𝐻𝐺01, 𝐻𝐺10, 𝐻𝐺11, 𝐻𝐺02, 𝐻𝐺20. Left: Evolution of the detector locations over the optimization epochs. Right: Output intensity coefficient as a function of the optimization epoch; the red horizontal dashed line indicates the 𝑀−1 baseline [PITH_FULL_IMAGE:figur… view at source ↗
Figure 9
Figure 9. Figure 9: Simulation results showing the projection of the true sorter (top), [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Simulation results of the intensities differences between the projection of the [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Experimental results for sorting four orthogonal modes, with mixed type: LG [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Experimental results for sorting four non-orthogonal modes (MUB): LG modes [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Phase mask of a sorter for four Laguerre–Gaussian beams with different [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Experimental results of sorting four Laguerre–Gaussian modes with different [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Noise analysis. Average cross-talk versus noise standard deviation for [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Wavelength analysis. Panel (a) illustrates the projected mask corresponding [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Rayleigh-like separation criteria for spectroscopy. Assuming some mode [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
read the original abstract

A mode sorter separates a set of M orthogonal spatial modes in a shared input channel into M different output channels. Here we present an analytic derivation and experimental validation of a single plane device for sorting spatial modes from a diverse variety of mode families, including Hermite-Gaussian (HG), Laguerre-Gaussian (LG), Bessel-Gaussian (BG), with almost no cross-talk. This sorting capability is required for a wide range of applications that employ classical or quantum light. We also show that applying this design in order to sort a set of Orbital Angular Momentum (OAM) modes with zero radial index reproduces the well-known Fork grating configuration. Furthermore, by taking the limit of M -> inf, we present an analytical expression for sorting all the modes of a given family. By operating this device in reverse, it can be used to generate arbitrary modes, by illuminating it with a Gaussian beam. The power transmission coefficient for this sorter goes as 1/M and we provide a mathematical proof that this is optimal for any typical arrangement of the detector positions. We further study the sorter sensitivity to wavelength and random phase noise.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript presents an analytic derivation of a single-plane phase mask that sorts M orthogonal spatial modes from families including HG, LG, BG, and OAM into spatially separated outputs with near-zero crosstalk. It includes experimental validation for these families, shows that the design reproduces the fork grating for OAM modes with zero radial index, derives an analytic expression for the M to infinity limit, demonstrates reverse operation for arbitrary mode generation from a Gaussian input, proves that the 1/M power transmission is optimal under typical detector geometries, and quantifies sensitivity to wavelength shifts and random phase noise.

Significance. If the central claims hold, the work offers a compact, analytically grounded single-plane solution for spatial mode sorting that applies across multiple mode bases and is supported by a tight optimality bound. This is relevant for quantum and classical applications in structured-light communications, imaging, and information processing, where multi-plane sorters are currently common. The explicit construction, mathematical proof of optimality, and experimental crosstalk data are strengths that could influence practical device implementations.

minor comments (3)
  1. The abstract states 'almost no cross-talk' and 'near-zero crosstalk' but does not quote the measured values or the exact threshold used; adding quantitative figures (e.g., from the experimental section) would improve precision.
  2. The wavelength and phase-noise sensitivity analysis is mentioned but the manuscript would benefit from a brief statement of the maximum acceptable phase-noise variance or wavelength detuning before crosstalk exceeds a stated level.
  3. In the reverse-operation section, the claim that arbitrary modes can be generated should specify the input beam waist and alignment tolerances required to maintain the reported fidelity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their detailed summary of the manuscript, positive assessment of its significance, and recommendation to accept. We are pleased that the central claims, analytic construction, optimality proof, and experimental results were found to be of interest for applications in structured light.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper supplies an explicit analytic construction for the single-plane phase mask derived from mode orthogonality and Fourier optics, together with an independent mathematical proof that the 1/M transmission bound is tight for the stated detector geometry. This proof does not reduce to a fitted parameter or self-referential definition. Reproduction of the known Fork grating for OAM modes functions as a consistency check rather than a load-bearing justification. The M to infinity limit and noise-sensitivity analysis are direct extensions of the same construction. No self-citation chain, ansatz smuggling, or renaming of known results is required for the central claims, which remain self-contained against standard Fourier optics and orthogonality.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of linear optics and mode orthogonality; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Spatial modes of the listed families are mutually orthogonal.
    Required for crosstalk-free separation; invoked implicitly when claiming almost no cross-talk.
  • domain assumption A single-plane phase mask can implement the required mode-to-position mapping.
    Core design premise of the analytic derivation.

pith-pipeline@v0.9.0 · 5503 in / 1286 out tokens · 26993 ms · 2026-05-10T12:16:11.420781+00:00 · methodology

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Reference graph

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    The separation should be larger than the bandwidth which can be approximated using M-squared. B. Sorter Optimality for one layer - a proof Given some arbitrary sorter𝑆(𝑥), we express the output field as: 𝐸𝑚, 𝜇 =𝐹𝑇 { 𝑓𝑚(𝑥, 𝑦)𝑆(𝑥, 𝑦) }𝜈 𝑥 = 𝛼𝜇 𝜆𝑧 ,𝜈 𝑦 = 𝛽𝜇 𝜆𝑧 .(24) Wewanttomaximizethisvalueasmuchaspossiblefor 𝑚=𝜇 andminimizefor 𝑚≠𝜇 . Focusing on the case wh...

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