Near-perfect measuring of full-field transverse-spatial modes of light

Markus Hiekkam\"aki; Robert Fickler; Shashi Prabhakar

REVIEW 3 major objections 4 minor 38 references

Near-perfect measuring of full-field transverse-spatial modes of light

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Three phase-only wavefront modulations, computed by wave-front matching, convert any full-field transverse spatial mode into a Gaussian so a single-mode fiber projects the mode near-perfectly and with low loss.

desk verdict Three-plane wave-front matching gives a genuinely useful low-loss full-field spatial-mode projector, but the 'any mode' universality claim rests only on nine low-order modes. read the letter →

arxiv 1909.01685 v1 pith:OVMM33S2 submitted 2019-09-04 quant-ph physics.optics

Markus Hiekkamäki , Shashi Prabhakar , Robert Fickler This is my paper · ORCID

classification quant-phphysics.optics

keywords transversespatialmodesmodeconversionwave-frontmatchinglightmodulatorsingle-modefiberprojectionLaguerre-GaussHermite-Gausshigh-dimensionalquantuminformation

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 asserts that any transverse spatial mode of light, including both its radial and azimuthal structure, can be measured almost perfectly by first converting it into a Gaussian beam with only three consecutive phase-only modulations, then letting a single-mode fiber act as the filter. The conversion is designed by wave-front matching and is in principle lossless and error-free. In experiment, the authors reach visibilities of 95.5% for nine Laguerre-Gauss modes and 96.2% for nine Hermite-Gauss modes, with coupling efficiencies up to 70%, and they demonstrate the projection in a 7-dimensional quantum cryptography protocol and in quantum state tomography.

What carries the argument

The central mechanism is multi-plane mode conversion driven by the wave-front matching algorithm. For a chosen target mode $M$ and the Gaussian fiber mode $G$, the algorithm propagates $M$ forward and $G$ backward through the chain of phase planes, and at each plane adjusts the phase so that the overlap between the two fields is maximized. The phase update is $\Delta\Phi_t(x,y)=-\arg\bigl(o_t(x,y)e^{-i\varphi}\bigr)$, where $o_t(x,y)=M(x,y,t)G(x,y,t)e^{i\Phi_t(x,y)}$ is the field overlap at plane $t$ and $\varphi$ is a mean-phase offset that speeds convergence. With three planes, the simulated overlap reaches 99.9%; because the operation is unitary, orthogonality of modes is preserved and only the target mode couples into the single-mode fiber.

What would settle it

Simulate or measure the conversion of a clearly harder mode, such as a high-order Laguerre-Gauss mode with large radial index, a random speckle field, or a superposition of many modes, using exactly three phase planes; if the Gaussian overlap drops well below 99.9% or the cross-talk visibility falls noticeably, the universality claim is false.

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Extended reading notes

Core claim

The central claim is that a unitary mode conversion, implemented as three consecutive transverse phase modulations separated by free-space propagation and followed by a single-mode fiber, performs a near-perfect projective measurement of any full-field transverse-spatial mode. The wave-front matching algorithm finds the phase patterns by iteratively matching the backward-propagated Gaussian fiber mode to the forward-propagated target mode; with three planes the simulated overlap between the converted mode and a Gaussian reaches 99.9%. The operation is unitary, so it preserves mode orthogonality and keeps the measurement in principle lossless and error-free. Experimentally, projecting onto the nine lowest-order Laguerre-Gauss modes gives a visibility of 95.5 ± 0.9%, and onto the nine Hermite-Gauss modes 96.2 ± 1.0%, with fiber-coupling efficiencies between 50% and 72%; across the full set the average error is 4.2%.

Load-bearing premise

The central assumption is that three wavefront-shaping steps are enough to convert any mode to a Gaussian, but only nine low-order Laguerre-Gauss and Hermite-Gauss modes were tested.

Editorial extensions

If this is right

Because the conversion is unitary, the same device can be run in reverse to generate arbitrary full-field spatial modes as efficiently as it measures them.
The method measures radial and azimuthal mode structure together, so it can decompose a light field into any chosen orthogonal mode basis, not just Laguerre-Gauss or Hermite-Gauss modes.
In a 7-dimensional BB84 protocol, the measured error rate of 4.98 ± 2.81% yields a secret key rate of 1.98 bits per sifted photon.
In quantum state tomography, measuring all eight mutually unbiased bases in dimension 7 reconstructs the target state with fidelity 96.4 ± 0.5%.
The technique is not limited to the nine modes demonstrated: any mode whose wave-front matching conversion converges with three planes is measurable with the same setup and comparable efficiency.

Reading between the lines

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

If three-plane sufficiency fails for higher-order or random modes, the method would still work but would require more phase planes; the practical limit of wave-front matching convergence is the key unknown to probe next.
The same unitary-conversion principle could be extended to other fiber modes or waveguides by substituting the Gaussian target with a different mode profile, potentially enabling all-optical mode add-drop multiplexers.
Replacing the spatial light modulator with custom diffractive elements could eliminate the 75% modulation efficiency penalty and push the device close to the theoretical zero-loss limit, making it attractive for photon-starved quantum experiments.

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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

Short version: this is a solid experimental demonstration that three phase-modulation planes computed by wave-front matching can convert low-order LG and HG modes into a Gaussian with 95.5–96.2% visibility, and that the resulting projection works for QKD and state tomography. That is real and useful. The headline claim that it measures 'any full-field transverse-spatial mode' is not supported. They tested the nine lowest-order LG and HG modes, plus seven LG modes in the applications. Nothing in the paper gives a reason to expect three planes to keep converging as radial and azimuthal complexity grows. The unitarity argument is fine for a fixed mode, but the number of planes needed for arbitrary modes is an open question. This is an overclaim in the abstract and conclusion, not a fatal flaw in the experiments.

What is actually new: applying the WFM unitary conversion idea to projective measurement that includes the radial degree of freedom. Phase-flattening fails for p>0, and the two-plane phase-retrieval method of Choudhary et al. is indirect. Here the conversion is direct, and the tomography result with 96.4% fidelity shows that superpositions are handled properly. The theory is standard, the implementation with a single SLM and a mirror is clever, and the experiments seem carefully done.

Soft spots, in order of importance. First, the 'any mode' claim: the engine is demonstrated only for low-order modes, and the paper lacks a scaling argument or tests at higher order. This is fixable in revision by qualifying the claim or adding higher-order data. Second, the efficiency reporting is easy to misread: the abstract says up to 70%, but that is the coupling efficiency of the converted mode before counting the three 75%-efficient SLM modulations. The text later says about 25% of input light reaches the fiber, which is the practical number. The abstract should say that. Third, there are no error bars on the coupling efficiencies in Fig. 2c, and no raw data or code shipped. For an experimental methods paper, publishing the WFM algorithm and the measured cross-talk tables would make it much easier to build on. That is minor relative to the main result.

The citation pattern is fine. Ref [27] is a companion paper on gates and is not load-bearing. The QKD demonstration is modest but proves the concept. Who is this for? People in spatial-mode quantum information who need a low-loss projection approach for radial modes. It deserves a serious referee: the result is important enough and the experiments careful enough that the overclaim can be fixed in revision. I would recommend acceptance after the abstract and conclusion are qualified and the efficiency numbers are clarified.

Referee Report

3 major / 4 minor

Summary. The paper proposes a method for measuring full-field transverse-spatial modes of light by unitary mode conversion: the mode of interest is transformed into a Gaussian via three phase-modulation planes (designed by wavefront matching), and then coupled into a single-mode fiber. The authors test the method experimentally on the nine lowest-order LG and HG modes, reporting visibilities of 95.5% and 96.2%, respectively, and efficiencies up to 70%. They further demonstrate the use of the projector in a 7-dimensional BB84 quantum key distribution protocol and in quantum state tomography, with a reconstructed state fidelity of 96.4%. The central claim is that three phase modulations suffice for near-perfect projection of any full-field transverse-spatial mode.

Significance. If the universality claim holds, the method is a useful and simple tool for high-dimensional quantum information and classical mode analysis, because it addresses both azimuthal and radial degrees of freedom with low crosstalk and potentially low loss. The experimental data for the tested modes are convincing: cross-talk matrices, efficiency measurements, QKD secret key rate, and tomography fidelity are all reported. The method is based on a standard unitary transformation principle, and the implementation is straightforward. However, the evidence for 'any' mode is limited to low-order modes, and the simulation supporting the 99.9% overlap is not presented. These gaps must be addressed before the paper's central claim can be accepted.

major comments (3)

[Abstract and Section 6 (Conclusion)] The abstract and conclusion claim that the method measures 'any full-field transverse-spatial mode' and that 'three phase modulations are enough' for this. The experimental evidence, however, is restricted to the nine lowest-order LG and HG modes in Section 4 and the seven LG modes in Section 5. No theoretical argument (e.g., a scaling law with radial/azimuthal order) or numerical simulation for higher-order or arbitrary modes is provided. The wavefront-matching algorithm is known to require more planes as the mode structure becomes more complex, so the demonstrated three-plane sufficiency for low-order modes does not establish universality. Please either restrict the claims to the demonstrated mode families and orders, or provide additional evidence (e.g., WFM simulations for high-order/random modes) that three planes suffice.
[Section 2 (Multi-plane mode conversion)] The statement 'for all the modes we investigated, three phase modulations were enough to achieve an overlap of 99.9 %' is a key quantitative claim that is never substantiated. The paper gives no simulation details: no propagation distances, grid sizes, beam parameters, convergence criteria, or a table/figure of the achieved overlaps for each mode. Since this 99.9% figure is the basis for the 'near-perfect' and 'in principle error-free' assertions, please include the simulation methodology and results, or temper the claim to reflect that 99.9% is an optimization target.
[Section 4.1 and Abstract] The efficiency reporting is ambiguous and potentially misleading. In Section 4.1 the authors state that modes couple into the SMF with an efficiency between 55% and 72%, but then note that 'around 25 % of the input light was detected after the fiber.' The abstract cites 'an efficiency of up to 70%' without clarifying that this is the fiber-coupling efficiency of the converted mode, not the end-to-end system efficiency. Please define the efficiency metric clearly and state both values in the abstract or conclusions if 'up to 70%' is to be used.

minor comments (4)

[Section 2] The notation M(x,y,t) and G(x,y,t) uses t for the plane index, which conflicts with the common use of t for time; consider using k or n.
[Section 5.2] In the direct inversion formula, the projector index m should be n, and the expression 'Π(k)m − 1' should be 'Π(k)n − I' or the subtraction of the identity matrix should be explicitly defined.
[Figure 2 caption] The caption states 'Diagonal = 0.955 0.013' and 'Diagonal = 0.962 0.013' without the plus-minus sign; these should read '0.955 ± 0.013' and '0.962 ± 0.013'.
[Section 5.1] The secret key rate formula R = log2(d) - 2h(d)(eb) would benefit from a citation to the d-dimensional BB84 security proof, as this is not a universally known result.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the WFM phases are computed from field propagation, not fitted to measured outputs, and the mode-projection claim follows from unitarity and orthogonality.

full rationale

The derivation chain is self-contained. The phase modulations are obtained by the wave-front matching iteration described in Eqs. (1)-(2): the forward-propagated input mode M is compared with the backward-propagated Gaussian G at each plane, and ΔΦ_t is set to the phase of their overlap. This is a calculation from the mode fields and the propagation model, not a fit to the experimental visibility, efficiency, or cross-talk data. The claim that the transformation is a projection onto mode M rests on unitarity preserving orthogonality, which is a standard mathematical property and is not defined in terms of the experimental results. The experimental visibilities (95.5% and 96.2%) and fidelities (96.4%) are measured consequences, not inputs. The only self-citation, ref. [27], is cited for the fact that WFM has been used with multiple output modes; that citation is contextual and not load-bearing, since the present paper describes the relevant optimization explicitly. The strongest caveat, that only nine low-order LG/HG modes were tested while the abstract claims arbitrary full-field modes, is a correctness/extrapolation risk rather than a circularity: the insufficiency of three planes for higher-order modes would falsify the universality claim but would not mean any prediction was constructed from its own target. No equation in the paper reduces to a fitted parameter or to a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Fourier optics, the orthogonality of the chosen mode families, and the paper-specific assumption that three phase planes achieve near-unitary conversion for arbitrary modes. No free parameters are fitted to the measured data; experimental settings such as beam waist and pixel pitch are fixed inputs to the WFM algorithm.

assumptions (5)

domain assumption Scalar diffraction theory and paraxial propagation accurately describe the field transformations between phase planes.
The WFM algorithm computes forward and backward propagations using free-space propagation; if the model is inaccurate, the computed phase patterns will not produce the claimed near-unitary conversion. Invoked throughout Section 2.
standard math The set of LG modes and the set of HG modes are complete orthonormal bases of paraxial spatial modes.
The projection onto a Gaussian after unitary conversion is equivalent to projection onto the target mode only if mode orthogonality holds; this is standard for LG/HG modes.
domain assumption A single-mode fiber couples only to a Gaussian transverse profile.
The detection relies on the fiber acting as a Gaussian mode filter; this is a standard property of single-mode fibers and is used in the mode projection argument.
domain assumption The input modes are generated close enough to ideal LG and HG modes that the measured cross-talk primarily reflects the projection system.
The paper uses SLM-based amplitude and phase holograms to generate modes and notes generation is 'good but not perfect'; if generation errors dominated, attribution of the 4.2% error to the measurement would be uncertain.
ad hoc to paper Three phase-modulation planes are sufficient for the WFM algorithm to reach 99.9% conversion overlap for all modes of interest.
The paper states in Section 2 that for all investigated modes three planes reached 99.9% overlap, but gives no proof or simulation details; this is the basis for the 'any mode' claim and is only validated for the tested low-order modes.

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Pith. "Pith review of Near-perfect measuring of full-field transverse-spatial modes of light." pith.science (2026). https://pith.science/paper/OVMM33S2

@misc{pith2026190901685,
  author       = {Pith},
  title        = {Pith review of: Near-perfect measuring of full-field transverse-spatial modes of light},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVMM33S2}},
  note         = {Machine review of arXiv:1909.01685}
}

read the original abstract

Along with the growing interest in using the transverse-spatial modes of light in quantum and classical optics applications, developing an accurate and efficient measurement method has gained importance. Here, we present a technique relying on a unitary mode conversion for measuring any full-field transverse-spatial mode. Our method only requires three consecutive phase modulations followed by a single mode fiber and is, in principle, error-free and lossless. We experimentally test the technique using a single spatial light modulator and achieve an average error of 4.2% for a set of 9 different full-field Laguerre-Gauss and Hermite-Gauss modes with an efficiency of up to 70%. Moreover, as the method can also be used to measure any complex superposition state, we demonstrate its potential in a quantum cryptography protocol and in high-dimensional quantum state tomography.

Figures

Figures reproduced from arXiv: 1909.01685 by the authors.

**Figure 2.** Modal cross talk matrices for the projection measurements of the 9 low [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Modal cross talk matrix of the 14 modes used in a 7-dimensional quantum [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Real part of the density matrix reconstructed using quantum state to [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [27]

High-dimensional quantum gates using full-field spatial modes of photons

F. Brandt, M. Hiekkam¨ aki, F. Bouchard, M. Huber, and R. Fickler, “High-dimensional quantum gates using full-ﬁeld spatial modes of photons,” arXiv: 1907.13002 , 2019. [Online]. Available: https://arxiv.org/abs/1907.13002

work page Pith review arXiv 1907

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