Observation of resonant doublet and variable finesse in a tabletop meter-scale linear three-mirror cavity

Ang\'elique Lartaux-Vollard; Aymeric van de Walle; Fran\c{c}ois Glotin; Manuel Andia; Nicolas Leroy; Paul Stevens; Pierre-Emmanuel Bonningues; Th\'eo Lesieur; Vincent Loriette

REVIEW 2 major objections 4 minor 15 references

Observation of resonant doublet and variable finesse in a tabletop meter-scale linear three-mirror cavity

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper reports experimental confirmation of the resonant doublet and variable finesse in a tabletop meter-scale linear three-mirror cavity, validating the virtual-mirror model.

desk verdict A credible but quantitatively soft experimental confirmation of two known three-mirror-cavity effects; worth refereeing, not yet a design basis. read the letter →

arxiv 2505.03416 v2 pith:BW3AA7FB submitted 2025-05-06 physics.optics

Paul Stevens , Pierre-Emmanuel Bonningues , Théo Lesieur , François Glotin , Vincent Loriette , Manuel Andia , Angélique Lartaux-Vollard , Nicolas Leroy

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Aymeric van de Walle

This is my paper · ORCID

classification physics.optics

keywords linearthree-mirrorcavityresonantdoubletvariablefinesseFabry-Perotfrequency-dependentsqueezinggravitational-wavedetectorpiezolengthtuning

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 aims to show experimentally that the two unusual behaviors predicted for linear three-mirror cavities—resonance splitting into a doublet and an effective finesse that changes with mirror tuning—appear in a compact, tabletop optical device. The authors build a 1 m long cavity split into two equal 0.5 m sub-cavities, with three mirrors of nominally equal reflectivity and piezoelectric control of the outer mirrors. Synchronized length scans produce the resonant doublet, with measured intra-doublet spacing of 42 ± 6 nm, in line with the 54 ± 6 nm expected from the nominal 90 ± 2% reflectivity. A dual-scan procedure maps finesse against slow sub-cavity detuning, yielding a fit with a common mirror reflectivity of 86 ± 1% and a maximum finesse near 40, confirming the variable-finesse equivalence. These observations matter because a cavity with in-situ tunable finesse could offer more flexible filtering of squeezed light in future gravitational-wave detectors.

What carries the argument

The central object is the overall amplitude transmission coefficient of the three-mirror stack, $t = -t_1t_2t_3e^{ik(L_1+L_2)}/[e^{2ik(L_1+L_2)} - r_1r_2e^{2ikL_2} - r_2r_3e^{2ikL_1} + r_1r_3(r_2^2+t_2^2)]$, which couples the two sub-cavities and produces the resonant doublet when both sub-cavities are near the same resonance. The variable finesse is carried by a different viewpoint: the first sub-cavity is treated as a virtual mirror $M'_1$ whose reflection and transmission coefficients depend on its length $L_1$, so the entire three-mirror system is optically equivalent to a two-mirror Fabry-Perot with an adjustable effective finesse. Experimentally, the machinery is a fully symmetric cavity of total length 1 m, with two equal sub-cavities, three mirrors of nominal reflectivity $R = 90\%$, and two piezo-mounted outer mirrors; synchronized scans map the doublet, and a 100 Hz/10 mHz dual scan maps FWHM into finesse through the free spectral range.

What would settle it

At a fixed slow detuning, replace the width-based finesse value with a cavity-ringdown decay measurement on the transmitted signal; if the ringdown finesse systematically disagrees with the width-derived value, or if removing the observed higher-order peaks changes the fitted $R$ outside $86 \pm 1\%$, the finesse interpretation as presented would fail.

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Extended reading notes

Core claim

The central claim is that a linear three-mirror cavity behaves, in practice, as its theoretical model predicts: when both sub-cavities are near resonance the single transmission peak splits into a symmetric doublet, and when one sub-cavity is detuned the whole device acts like a two-mirror cavity whose effective mirror reflectivity—hence finesse—can be varied. The doublet is observed by scanning the two outer mirrors together along a diagonal in the two-detuning plane; the measured spacing between the two maxima, about 42 ± 6 nm, agrees with the 54 ± 6 nm predicted for mirrors with $R = 90 \pm 2\%$. The finesse variation is observed by sweeping one sub-cavity fast while slowly moving the other; the height and width of the transmitted peaks change periodically with the slow scan, with maximum finesse at antiresonance of the virtual mirror and near-zero finesse at resonance. Fitting these data with a single reflectivity for all three mirrors gives $R = 86 \pm 1\%$, and a maximum finesse of about 39.7 against a theoretical 58 for the nominal reflectivity. The authors read the agreement as confirmation that the three-mirror cavity is equivalent to a two-mirror cavity with tunable finesse, with residual asymmetry and extra peaks attributed to mechanical drift, imperfect mode matching, and misalignment.

Load-bearing premise

The measured finesse values assume the width of each transmitted resonance belongs to the cavity's fundamental mode alone, so if hidden higher-order modes or misalignment broaden those peaks, the fitted reflectivity of 86% and the claimed variable-finesse behavior would be biased.

Editorial extensions

If this is right

A three-mirror cavity can supply frequency-dependent squeezing filters whose finesse is adjustable during operation, changing the squeezing bandwidth without replacing optics.
The resonant doublet provides two nearby transmission peaks that can be spaced at the sub-nanometer level by relative sub-cavity detuning, suggesting a tunable two-frequency filter.
Maximum finesse occurs at antiresonance of the virtual sub-cavity and minimum near resonance, so sub-wavelength piezo travel covers the full finesse range, including the theoretically lossless full-transmission point.
The match between measured and predicted finesse supports the plane-wave, lossless-mirror model as an adequate design tool for meter-scale three-mirror cavities, despite real misalignment and higher-order modes.

Reading between the lines

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

If the same tuning principle survives in larger, suspended, vacuum-enclosed cavities, a three-mirror filter could be re-optimized for different squeezing frequencies in real time—something a fixed two-mirror filter cannot do without swapping hardware.
The observed amplitude asymmetry of the doublet peaks, though treated as a nuisance, is a sensitive indicator of the central mirror's drift; a control loop could lock the doublet symmetry to keep the cavity at its designed operating point.
A ringdown-based finesse measurement, or spatial filtering of the transmitted beam, would separate the advertised variable finesse from the higher-order-mode contamination acknowledged in the paper and would test whether the extracted $R = 86\%$ is the true mirror reflectivity.

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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

Quick take: the paper does what it says—it shows the resonant doublet and variable finesse in a meter-scale three-mirror cavity—but the quantitative case is shakier than the text implies, and the finesse numbers are likely biased by higher-order mode contamination. Worth reading, worth refereeing, but not a design basis yet.

The genuinely new thing here is experimental. The theory of three-mirror cavity doublets and variable finesse is established in the cited literature (refs 9–15, including the group's own CQG 2025 paper), and this is the first tabletop meter-scale demonstration of both effects together. The setup is sensible: convex–plano–convex geometry, piezo stages with calibrated displacement, and a dual-scan method (fast L1 scan, slow L2 modulation) that keeps the measurement time short. The qualitative signatures are visible in Fig. 5 and Fig. 6. The authors are candid about environmental noise, drift, and misalignment, which is more than many short letters do.

Where it gets soft: the numbers do not agree as well as claimed. The doublet spacing is 42±6 nm against an expected 54±6 nm; that's overlap at the edge of the error bars, not a strong confirmation. More importantly, the variable-finesse evidence rests on converting the FWHM of the transmitted peak into finesse, and the photodiode sees all transmitted light. The paper itself acknowledges a residual higher-order mode and smaller finesse drops from it, plus imperfect mode matching. If a higher-order mode is even weakly excited, the measured FWHM is not the fundamental-mode linewidth, and the conversion in Sec. 5A is invalid. The stress-test note is right on this: the fitted R=86% sits outside the mirror spec of 90±2%, and the fitted max finesse of 39.7 is well below the theoretical 58.1, both in the direction that non-fundamental content would push. Since R is the only free parameter and is fit to the same FWHM data, the claimed agreement with the variable-finesse model is not an independent validation.

The paper would be stronger with a mode-cleaned signal or a measured mode decomposition, a proper error budget for the FWHM extraction, and the raw data made available. The authors flag the residual mode but do not quantify it. The citation pattern looks appropriate; the references are relevant and the self-citation to the companion theory paper is justified.

Bottom line: for someone working on frequency-dependent squeezing filters for next-generation gravitational-wave detectors, this is a useful proof-of-principle with clear limitations. I'd send it to peer review, but I'd ask for revision before treating any of the quantitative values as reliable.

Referee Report

2 major / 4 minor

Summary. The paper reports an experimental implementation of a meter-scale linear three-mirror cavity, with two 0.5 m sub-cavities and nominally identical mirrors (R_th = 90 ± 2%), and claims the first tabletop observation of two theoretically predicted effects: the splitting of the cavity resonance into a doublet, and the equivalence of the three-mirror system to a two-mirror cavity with variable finesse. The doublet is observed by synchronously scanning the two outer mirrors and recording the transmitted power; the measured intra-doublet spacing is 42 ± 6 nm versus an expected 54 ± 6 nm. The variable finesse is studied by scanning one sub-cavity slowly to vary an effective virtual-mirror reflectivity while rapidly scanning the other sub-cavity, converting the FWHM of transmitted resonance peaks into finesse values. A single-parameter fit with equal mirror reflectivities yields R = 86 ± 1%, compared with the manufacturer specification of 90 ± 2%. The paper concludes that the observations confirm the theoretical predictions and support the use of such cavities for frequency-dependent squeezing applications.

Significance. If the central claims are accepted, this would be a useful experimental demonstration that resonant peak splitting and variable finesse, previously studied theoretically, can be produced in a compact meter-scale configuration with straightforward piezoelectric control. The setup is carefully built: the piezo stages are calibrated, the cavity is fully symmetric, and the two-dimensional transmission map in Fig. 4 provides a clear qualitative picture of the doublet structure. The relevance to frequency-dependent squeezing in gravitational-wave detectors gives the work clear motivation. However, the quantitative support for the variable-finesse claim is currently not persuasive: the finesse values are extracted from unfiltered transmitted light despite acknowledged higher-order mode contamination, and the single fitted reflectivity is outside the manufacturer specification and is fit to the same data used to display the agreement. The paper therefore establishes the qualitative phenomena but does not yet rigorously validate the quantitative equivalence to a variable-finesse two-mirror cavity.

major comments (2)

[§5A–5B, Fig. 6] The finesse measurement assumes that the FWHM of the transmitted fundamental-mode peak divided by the FSR equals the cavity finesse. In §5A the FWHM is measured with photodiode PD_Trans without spatial filtering, and §5B explicitly acknowledges a residual higher-order mode that produces additional finesse drops, as well as misalignment and imperfect mode matching. When a higher-order mode is present, the FWHM of the total transmitted power is not the fundamental-mode linewidth, so the conversion in §5A is invalid for the stated finesse. The fitted R = 86% lies outside the manufacturer specification R_th = 90 ± 2%, and the fitted maximum finesse (39.7) is far below the theoretical value (58.1); both discrepancies are in the direction expected if the FWHM is broadened by non-fundamental content. The quantitative variable-finesse claim therefore needs either spatially filtered detection of the TEM00 mode, an independent finesse measurement such as a ringdown, or a model that explicitly includes the higher-order modes.
[§5B, fitted R value] The only free parameter of the finesse model, the common mirror reflectivity R, is fit to the same finesse-versus-position data that is then displayed as the model curve. This procedure cannot by itself validate the equivalence to a two-mirror cavity with variable finesse, because the agreement is partly obtained by absorbing all systematic broadening into the fitted R. The extracted R = 86 ± 1% should be checked against an independent measurement of the mirror reflectivities or cavity losses. Without such a check, the reported agreement may reflect systematic bias from higher-order modes and imperfect mode matching rather than confirmation of the theoretical model.

minor comments (4)

[§4B, intra-doublet spacing] The measured spacing of 42 ± 6 nm is described as being in agreement with the expected 54 ± 6 nm, but these values agree only marginally (about 1.4σ); please quantify the comparison and discuss systematic uncertainties in the piezo calibration and the possible influence of higher-order resonance peaks on the peak-position determination.
[Fig. 5] The x-axis label contains corrupted LaTeX ('/uni0394L 1,2 Δ[nm]') and should be rendered as ΔL1,2 [nm].
[§5B, maximum finesse comparison] The sentence stating that the fitted maximum finesse 'closely aligns with the theoretical value' is difficult to reconcile with the quoted numbers F_max ≈ 39.7 and F_th,max ≈ 58.1; please rephrase and quantify the disagreement.
[Fig. 4] The figure contains typographical errors in the axis and colorbar labels ('F om initial length', 'Mi o s displacement', 'T ansmission'); these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the observed doublet and finesse variation are compared with explicit, externally grounded theory; the fitted reflectivity is an in-sample consistency check, not a prediction.

full rationale

The paper's headline claims are experimental observations, not first-principles derivations. The resonant-doublet prediction rests on Eq. (1), which is written out explicitly in Sec. 2A; reference [12] supplies the same formula, but the expression is parameter-free and fully stated, so no load-bearing content is hidden in the self-citation. The variable-finesse interpretation is attributed to Refs. [13-15], which are external to the authors. The finesse measurement is operational: the FWHM of the transmitted peak divided by the cavity FSR (Sec. 5A). This is a standard proxy, and Sec. 5B explicitly acknowledges that residual higher-order modes, misalignment, and imperfect mode matching can broaden the measured widths and produce small additional finesse drops; that is a measurement-limitation issue, not a circular derivation. The only fitted parameter is R=86%, obtained by fitting the finesse data in Sec. 5B. The paper does not present the fitted curve as an out-of-sample prediction; it uses R as a consistency check against the manufacturer specification Rth=90±2% and the associated maximum finesse. The fit is in-sample and the agreement language is somewhat generous given the discrepancy, but this does not make the observed variable finesse equivalent by construction to the model input: the periodicity and minima/maxima are present in the raw FWHM data, and the model parameter does not reproduce the data unless the periodic structure is actually there. No step was found in which a defined quantity reduces to its own input, and no load-bearing uniqueness or ansatz is imported from the authors' prior work.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The analysis uses the standard plane-wave transfer function of a three-mirror cavity and the virtual-mirror equivalence from the cited literature. The only fitted free parameter is a common mirror reflectivity R equals 86 percent, inferred from the same finesse data used for the comparison. No new physical entities are introduced. The FWHM-to-finesse conversion assumes the transmitted fundamental mode width is dominated by cavity loss rather than by higher-order mode contamination or misalignment.

free parameters (1)

Common mirror reflectivity R = 86 percent, plus or minus 1 percent
Fitted to the finesse versus displacement curve in Fig. 6 assuming identical mirrors; used to claim agreement with the theoretical variable-finesse model. The manufacturer specification is 90 plus or minus 2 percent, so the fitted value is outside the stated range.

assumptions (4)

domain assumption The three-mirror transmission is described by Eq. (1) under plane-wave, lossless-mirror assumptions, and the same equation produces the expected doublet and finesse values.
Used in Section 4A and Fig. 4 to predict the resonance map and in Section 5B to set the theoretical finesse scale; losses and mode effects are treated as negligible or as perturbations.
domain assumption The length scanned by each piezoelectric stage is linearly related to applied voltage with calibrated constants, and the nominal 1 micrometer per volt value is stable during the measurement.
All length detunings and finesse periods in Sections 4B and 5B are derived from piezo displacements; the paper mentions possible nonlinearity only as a post hoc explanation for discrepancies.
domain assumption The measured FWHM of the transmitted peaks, divided by the free spectral range, gives the fundamental-mode finesse.
Used in Section 5A to convert oscilloscope traces into finesse; residual higher-order modes are acknowledged as causing extra smaller finesse drops, so this assumption is load-bearing.
domain assumption All three mirrors have identical reflectivity, represented by a single common R in the fit.
The fit in Section 5B assumes a single reflectivity R for all three mirrors, despite the manufacturer's 90 plus or minus 2 percent range allowing inter-mirror differences.

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Cite this review

Pith. "Pith review of Observation of resonant doublet and variable finesse in a tabletop meter-scale linear three-mirror cavity." pith.science (2026). https://pith.science/paper/BW3AA7FB

@misc{pith2026250503416,
  author       = {Pith},
  title        = {Pith review of: Observation of resonant doublet and variable finesse in a tabletop meter-scale linear three-mirror cavity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BW3AA7FB}},
  note         = {Machine review of arXiv:2505.03416}
}

read the original abstract

Fabry-Perot cavities are widely used in current gravitational-wave detectors. In particular, they play a key role in frequency-dependent squeezing systems, enabling broadband quantum noise reduction. However, their ability to precisely control squeezing properties may be insufficient for the next generation of detectors. In this context, theoretical studies on linear three-mirror cavities have revealed promising features, such as resonance peak splitting and their equivalence to two-mirror cavities with variable finesse. In this paper, we report experimental observations of both the resonant doublet and variable finesse using a meter-scale implementation of a linear three-mirror cavity.

Figures

Figures reproduced from arXiv: 2505.03416 by the authors.

**Figure 1.** Simulation of the transmission through a three-mirror cavity as a function of the length detuning in both sub-cavities. The condition ∆L1,2 = 0 corresponds to resonance in each subcavity. Simulation parameters: L1 = L2 = 0.5 m, R1 = R2 = R3 = 0.9 and λ = 1064 nm, assuming lossless mirrors. the first or second sub-cavity can act as a virtual mirror. However, the choice may not be equivalent for two reasons. First, … view at source ↗

**Figure 3.** presents the schematic of our three-mirror cavity, the key components of the experimental setup, and the labeling conventions used in the following sections. We implemented a fully symmetric three-mirror cavity with a total length of 1 m, divided into two equal-length sub-cavities: L1 = L2 = 0.5 m. All three mirrors from Laseroptiks are specified to have the same theoretical reflectivity of R1 = R2 = R3 = Rth = 90 … view at source ↗

**Figure 2.** Equivalence between a three-mirror cavity (top) and a virtual two-mirror cavity with variable finesse (bottom). In the three-mirror configuration, mirrors M1 and M2 form a subcavity parameterized by (r1 , t1), (r2, t2), and length L1 . This sub-cavity can be modeled as a virtual mirror M′ 1 with tunable reflection and transmission coefficients (r ′ 1 , t ′ 1 ). Both configurations share the same second cavity leng… view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Observation of resonant doublets. Voltage signal from the photodiode PDTrans as a function of piezoelectric stages displacement. 5. EXPERIMENTAL OBSERVATION OF FINESSE VARIATION A. Method To measure the variation of finesse, one of the sub-cavities can be kept at a fi…

**Figure 6.** Figure 6: shows the variable finesse, reconstructed from our measurement of FWHM, as a function of the displacement of mirror M3. The transition between low and high finesse regimes, as well as the corresponding minimum and maximum finesse values, are governed by the reflectiv…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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