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REVIEW 3 major objections 4 minor 47 references

Mismatched wavefront curvature between interfering beams makes quadrant-photodiode heterodyne efficiency peak at nonzero tilt on each segment, creating a new, measurable tilt-dependent phase-noise coupling in LISA-type interferometers.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 21:58 UTC pith:5UJ6SJRX

load-bearing objection A genuinely useful closed-form model of tilt-dependent heterodyne efficiency for LISA, but the headline experimental claim is stronger than the data support: the key parameter ρ is imported from prior work and fixed because the fit would not converge, so the 'first measurement' is really a consistency check. the 3 major comments →

arxiv 2602.18239 v2 pith:5UJ6SJRX submitted 2026-02-20 astro-ph.IM

Mathematical derivation and verification of the amplitude of LISA's interferometric signals on an ultra-stable interferometer testbed

Alvise Pizzella , Lennart Wissel , Miguel Dovale-Alvarez , Pablo Martinez Cano , Rodrigo Garcia Alvarez , Christoph Bode , Juan Jose Esteban Delgado , Gerhard Heinzel This is my paper
classification astro-ph.IM
keywords heterodyne efficiencyquadrant photodiodedifferential wavefront sensingwavefront curvature mismatchLISAinterferometric readout noisetilt-to-length couplingGaussian beam interference
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that the amplitude of a heterodyne signal in a tilted-beam interferometer can be computed analytically, and that one neglected detail—the curvature mismatch between the two interfering wavefronts—leaves a measurable fingerprint in quadrant photodiode readouts. The authors derive closed-form expressions for the heterodyne efficiency of Gaussian and flat-top beam pairs on single-element and quadrant photodiodes, for both infinite and finite detectors, and validate them numerically and on an ultra-stable optical bench representative of LISA. The central new claim is that when the wavefronts have different curvatures, the individual segments of a quadrant photodiode do not lose signal symmetrically under tilt: each half's efficiency peaks at a nonzero tilt angle, an effect predicted by the model and reported as measured for the first time. From that, they derive compact coupling coefficients that capture how much tilt multiplies the uncorrelated and correlated phase noise of the QPD's longitudinal and angular signals. If correct, the result means LISA's test-mass interferometer must control wavefront curvature mismatch or pay about 10% extra readout phase noise.

Core claim

On the paper's own terms, the discovery is that the previously ignored relative wavefront curvature mismatch ρ between two interfering beams breaks the tilt symmetry of a quadrant photodiode. On an infinite single-element detector the heterodyne efficiency is an even function of tilt and peaks at zero tilt regardless of ρ; on a QPD segment, by contrast, the efficiency peaks at a nonzero angle whenever ρ is nonzero (eqs. 56–57). Because the top and bottom halves then have different signal amplitudes for the same tilt, the phase readout noise of the QPD's LPS and DWS signals contains a new coupling term that rises with ρ, captured by the functions k_unc, k_cor,Σ, and k_cor,Δ (eqs. 117, 123, 12

What carries the argument

The central object is the heterodyne efficiency √η_het(θ), defined as the magnitude of the normalized overlap of the two beam fields on the detector, together with the curvature-mismatch parameter ρ = (w_eff² k)/(4 R_rel), which measures how much the two wavefront curvatures differ in units of the effective beam spot. The derivation linearizes the tilted Gaussian beam field in the tilt angle, integrates the resulting complex intensity over circular or half-circular detector domains, and expresses the HE in closed form for infinite detectors and as Maclaurin series in tilt for finite detectors (eqs. 147–148). For QPD segments the key mechanism is that the integration domain is not symmetric a

Load-bearing premise

The experimental confirmation rests on the curvature-mismatch parameter being fixed from an earlier characterization at ρ = 0.14 ± 0.1, and the fit would not converge if it were free; if that imported number or the beam-size model behind it is wrong, the measured segment asymmetry could be a fitting artifact rather than the predicted effect.

What would settle it

On a bench where both beams' wavefront radii and spot sizes at the photodiode are measured independently, compute ρ and predict the HE peak angle from eq. (57); then tilt the beams and locate the HE peak of each QPD half. If the peak offset does not track the predicted value—or if it appears when the beams are independently verified to be mode-matched (ρ = 0)—the central claim falls.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • LISA's test-mass interferometer readout noise under tilt is now described by closed-form expressions; the highest expected curvature mismatch ρ ≈ 0.47 would add about 10% phase noise, so beam mode matching in the TMI and SCI is recommended.
  • When ρ is nonzero, QPD segments have different heterodyne efficiencies at the same tilt, so segment-level noise must be combined through the k_unc and k_cor coefficients rather than modeled from one segment alone.
  • A benchmark of ρ > 0.2 marks where treating all four QPD segments as identical becomes inadequate for phase-noise modeling.
  • Operating LISA's TMI at DWS-zero to follow a misaligned local-oscillator beam gives only sub-percent noise reduction in the ±100 µrad tilt range, suggesting the tilt-to-length penalty likely outweighs the gain.
  • The HE model can be used to detect phasemeter cycle slips and to extend DWS operation beyond its first sign-reversal point.

Where Pith is reading between the lines

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

  • A direct extension the authors do not pursue: measure the peak-angle offset θ_max as a function of an externally controlled curvature mismatch (for example, by moving a lens in one arm of a tabletop heterodyne interferometer) and test the predicted scaling; this would isolate the mechanism without relying on an imported value of ρ.
  • The empirical constant c required in fits of finite-QPD data suggests that diffraction fringes from the finite detector aperture bias the infinite-QPD formula; a finite-QPD closed form that includes fringes could remove the 23% width discrepancy and the need for c.
  • If the segment-asymmetry effect is real, DWS calibrations that assume identical segment amplitudes will carry a ρ-dependent bias; conversely, the HE asymmetry could be turned into an in-situ wavefront-matching monitor for LISA's optical benches.
  • The same mechanism should appear in any segmented-detector heterodyne interferometer with a curvature mismatch, so future space interferometers with tilted beams should expect an analogous segment-asymmetric noise coupling.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper derives closed-form and Maclaurin-expansion expressions for the heterodyne efficiency (HE) of Gaussian-Gaussian and Gaussian-top-hat beam pairs on infinite and finite circular single-element and quadrant photodiodes, including the effect of wavefront curvature mismatch ρ (Eqs. 41–50, 147–148). It then uses these expressions to construct tilt-dependent noise-coupling coefficients k_unc, k_cor,Σ, k_cor,Δ for correlated and uncorrelated QPD readout noises (Eqs. 117, 123, 125) and applies them to LISA's TMI noise budget, concluding that curvature mismatch should be controlled and that operating at DWS-zero gives negligible optical-noise benefit (Section 7). The central new physical claim is that, for a QPD with tilt about the detector center, a nonzero ρ breaks the symmetry of the segment HEs, shifting the HE peak of individual segments away from θ=0 (Eqs. 56–57), and that this asymmetry is 'predicted and measured for the first time' in the SCIQPD1 data of Section 5.2.

Significance. If the derivation stands, the paper provides a valuable analytical toolkit for LISA and other tilted-beam interferometers: the HE expressions are not fitted but derived, and the comparison against independent numerical integration (Figs. 13–19) is a genuine strength. The newly identified segment-level HE asymmetry from curvature mismatch is physically plausible and, if confirmed, would matter for LISA's error budget and for DWS/LPS noise modeling. The derivation of closed-form noise-coupling coefficients is a useful contribution. However, the experimental confirmation of the headline new feature is weaker than the abstract suggests: the key measurement uses an externally fixed ρ, an empirical offset constant, a free width parameter, and shows unexplained 17–23% discrepancies in the fit parameters and a 20% discrepancy in the magnification-ratio cross-check. The central derivation is not circular, but the central experimental claim is not yet established to the standard claimed.

major comments (3)
  1. [§5.2, Eq. (100), Table 3] The central experimental evidence for the 'new noise feature' is the segment-HE peak shift in the SCIQPD1 data (Table 3: x0−x̄0 ≈ ±0.123 mrad), which matches the model to 2.4%. But this agreement is conditional on fixing ρ = 0.14 ± 0.1, taken from the authors' earlier characterization [25], and the text states the fit 'would not converge otherwise' if ρ were free. Since the uncertainty in ρ is comparable to its central value, the predicted peak shift has a very broad range; the data therefore do not independently measure ρ or confirm the ρ-dependence. Please add an in-situ determination of ρ, or a likelihood scan over ρ, or clearly reframe the claim as a consistency check with an external prior rather than a first measurement of the effect.
  2. [§5.2, Table 3 and Eqs. (101)–(102)] Even after fixing ρ, the finite-QPD data are fit with the infinite-QPD functional form plus an empirical additive constant c and a free width s. The fit returns s 23% below the model, c differing by −17%, and the independent REFQPD/SCIQPD s-ratio is measured as 0.826 ± 0.006 versus a modeled 0.663 ± 0.004 — roughly a 20% discrepancy. These are not small residuals compared to the claimed 2.4% agreement on the peak shift. The empirical flexibility (c, s, and the fit normalization) could absorb a large part of the observed asymmetry, so the 'first measurement' claim is premature. Please either constrain c and s by independent beam-width and magnification calibrations, or show that the asymmetry persists when the finite-QPD expression (148) is used instead of the infinite-QPD form with an ad hoc constant.
  3. [§5.1, Table 1] The absolute HE validation is only partially successful: only the Rx-GB & LO on SCIQPDs case is compatible with the model; the two REFQPD cases disagree by roughly 14% and 7% and are attributed to unmodelled lateral offsets. This is explicitly acknowledged in the text, but the abstract and conclusions state that the model is verified against experimental measurements. Given that Table 1 is the only absolute test and two of three cases fail, the paper should either provide a quantitative model of the lateral-offset sensitivity, add data with controlled beam centering, or qualify the validation claim as partial. As it stands, the experimental support for the absolute HE model is weaker than the narrative suggests.
minor comments (4)
  1. [Abstract and §5] The phrase 'validated against experimental measurements' should be softened in light of Table 1 and the residual discrepancies in Section 5.2. A more precise formulation such as 'partially validated, with the remaining discrepancies attributed to known systematic effects' would better match the content.
  2. [§5.2, Figs. 23–24] The comparison between data and model would be much more informative with residual plots. The current figures show the fits but do not make the 23% width discrepancy or the 17% offset discrepancy visually transparent.
  3. [§7] The conclusion that 'operation at a DWS offset is preferable' is based only on the optical-noise analysis and the text explicitly states that the TTL contribution is not modeled. Please make the conditional nature of this recommendation explicit in the summary/conclusion, not only in the body.
  4. [General] There are several typographical and consistency issues: 'ingium gallium arsenide' should be 'indium gallium arsenide'; 'developement' should be 'development'; 'infinte' in Table 8 should be 'infinite'; and the sentence 'An reference interferometer' in the introduction should read 'A reference interferometer'.

Circularity Check

1 steps flagged

Central HE derivation is self-contained; the first-measurement support is weakened by an imported, self-cited ρ and by free fit parameters, but not reduced to its inputs.

specific steps
  1. self citation load bearing [Section 5.2, eq. (100), Table 3]
    "The parameter ρ, which is known from previous measurements to be equal to ρ=0.14±0.1 in both analyzed cases [25, Figure 3.9], is fed as a fixed parameter of eq. (100) to the fit routine, as it would not converge otherwise."

    The new experimental feature — the differential segment-HE peak shift Δx0≈0.246 mrad in Table 3 — is presented as a prediction measured for the first time. But the model curves used to compute the 'predicted' x0 difference are generated with ρ fixed to the authors' earlier characterization, and the fit would not converge if ρ were free. With a, x0, s, and c also free, the data cannot independently determine ρ; the measured asymmetry is therefore a consistency check conditional on an imported, self-reported parameter rather than an out-of-sample confirmation of the ρ-dependence.

full rationale

The analytical derivation in Section 3 is not circular: eqs. (41)–(50) follow from explicit integration of the Gaussian-beam overlap intensity (eq. 24) under stated linearization assumptions, and are validated against independent numerical integration (Subsection 3.8) with discrepancies ~1e-6 to 1e-3. The k_unc and k_cor coupling functions (eqs. 117–126) are algebraic consequences of those HE expressions, not fitted results. The experimental confirmation of the curvature-mismatch feature, however, depends on ρ=0.14±0.1 imported from the authors' own PhD thesis [25] and fixed because the fit 'would not converge otherwise'. The fit additionally requires an empirical constant c and leaves the width s free, with s deviating 23% from the model and the magnification-ratio cross-check disagreeing by ~20% (eqs. 101–102). These are genuine independence/transparency weaknesses in the validation, but they do not make the derivation equivalent to its inputs: the central claim has substantive first-principles content. Score 3 reflects this partial, not total, circularity.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

No new physical entities are postulated. The k_unc, k_cor functions are modeling constructs derived from the HE, not entities; the 'new feature' is a predicted and measured effect of the existing curvature-mismatch parameter ρ. The load-bearing free parameters are the imported ρ and the fit constants (s, c, a) used in eq. (100) to close the gap between the infinite-QPD formula and finite-QPD data. The axioms are standard paraxial optics and domain assumptions that are mostly explicit in the paper; the least-controlled one is the DPS-based angle calibration with 7% uncertainty.

free parameters (4)
  • ρ (effective-spot-radius-normalized wavefront curvature mismatch) = 0.14 ± 0.1, imported from ref. [25]; fixed in fits (eq. 100)
    All segment-asymmetry and peak-shift predictions scale with ρ; not fitted in this paper, but its origin in self-cited prior work and large uncertainty make it a load-bearing input to the validation.
  • fit width s in eq. (100) = 1.50/1.49 mrad (SCIQPD data); 1.950 ± 0.007 mrad (model)
    Left free in the fit although theory fixes s∞ = m/(k weff) = 0.137 ± 0.002 mrad; 23% discrepancy between data and model.
  • empirical additive constant c (eq. 100) = 0.237 ± 0.005 / 0.201 ± 0.005 (data); 0.187 ± 0.003 (model)
    Added to the infinite-QPD fit curve to absorb finite-QPD diffraction fringes; required for both data and simulation but absent from the analytic HE expression.
  • fit amplitude/normalization a (eqs. 99–100) = 0.76–1.00 (data); 1.00/0.817 (model)
    Normalizes HE curves because the absolute scale G(f) is unknown; carries the absolute-HE calibration problem into the normalized comparison.
axioms (5)
  • domain assumption Paraxial Gaussian-beam propagation and the linearized tilted-beam form (eq. 19): w_m, R_m, Gouy phase independent of transverse coordinate; tilt enters only as exp(ikθy)
    Introduced between eqs. (18)–(19); numerically validated to ~1e-6 (SEPD) and ~1e-4 (QPD within ±0.5 mrad, Fig. 20), but is the structural basis of all closed forms.
  • domain assumption Rotation pivot at the detector center
    Justified for LISA by imaging systems (§2.4, Fig. 4); in TDOBS the tilt angle is recovered from DPS on an auxiliary QPD (eq. 98) with ~7% systematic uncertainty.
  • domain assumption Measurement-beam power on the detector is tilt-independent (P_PD,m neglected in θ)
    Stated in §3.1; exact for infinite detectors, 'negligible otherwise' — but the validations live at β≈0.74–1.87 where this is approximate.
  • domain assumption Noise model restricted to shot, electronic, and 1f/2f-RIN, with vertical tilt only and two-segment symmetry
    §6.1 and §6.4; the paper explicitly flags vertical-only as a simplification (noise increase 'well within one order of magnitude') and leaves the azimuthal analysis to future work.
  • standard math Standard Fourier-domain PSD/coherence formulas for sum/difference of correlated noises (Appendix A)
    Textbook result (Bendat & Piersol), used to derive eqs. (120)–(121) for correlated segment noises.

pith-pipeline@v1.3.0-alltime-deepseek · 57465 in / 20261 out tokens · 185412 ms · 2026-08-02T21:58:04.246575+00:00 · methodology

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read the original abstract

The Laser Interferometer Space Antenna (LISA) mission aims to detect gravitational waves by interferometrically measuring the change of separation between free-falling test masses (TMs). LISA's interferometers must deliver pm/rtHz sensitivity while accommodating beam tilts up to 1 mrad at the photodiodes, which degrade the interferometric amplitude and increase the induced readout noise coupling. This paper uses an analytical framework developed by the authors in a previous work, based on minimal and justified approximations, that relates beam tilt to the resulting heterodyne signal amplitude in a generic two-beam interferometer with circular-area photodiodes (PDs). A set of interferometric topologies is analyzed, all of high relevance for LISA. We derive the exact amplitude response for an infinite detector and a closed-form approximation for finite detectors, and we validate both against numerical simulations and experimental measurements on an ultra-stable LISA-representative testbed. We then use this model to quantify the phase-noise amplification arising from reduced signal-to-noise ratio (SNR) under tilt, showing that curvature mismatches between the interfering beams substantially enhance this effect. Finally, we introduce a compact function that captures the angular dependence of correlated and uncorrelated phase noises in quadrant photodiode (QPD)-based readouts. Here, a new noise feature, caused by wavefront curvature mismatch, is predicted and measured for the first time. These results indicate that controlling wavefront curvature mismatch in the test mass interferometer (TMI) is essential to limit excess phase noise. The models and results derived in this paper, although originating in the context of LISA, are general and can be applied to any interferometric topology undergoing tilts with pivot on the detector plane.

Figures

Figures reproduced from arXiv: 2602.18239 by Alvise Pizzella, Christoph Bode, Gerhard Heinzel, Juan Jose Esteban Delgado, Lennart Wissel, Miguel Dovale-Alvarez, Pablo Martinez Cano, Rodrigo Garcia Alvarez.

Figure 1
Figure 1. Figure 1: Optical layout of the testbed. Left: The telescope simulator (TS) provides the optical bench (OB) (right) with laser beams (Rx-GB and Rx-FT, both in green) to simulate LISA’s TMI and SCI. The first beam is a TEM00 Gaussian beam provided by a fiber injector optical sub-assembly (FIOS) with a width of ∼2 mm. The second is a flat-top beam generated by clipping a 9 mm radius beam Gaussian beam on an apodized a… view at source ↗
Figure 2
Figure 2. Figure 2: Right: Picture of a large commercial QPD. Left: definition of the used segment-nomenclature throughout the paper. The area between the segments is called slit. Depending on the specific QPD model, the slit can be either insensitive, partially sensitive, or even more sensitive than the active area itself. Note that the QPDs in this experiment use a diameter of 1 mm. 4 [PITH_FULL_IMAGE:figures/full_fig_p004… view at source ↗
Figure 3
Figure 3. Figure 3: Measurement with three different light intensities, SPICE simulation, and fit of the electronic noise from the TIA used for the photodiodes in TDOBS. The fit function is f(x) = q p 2 0 + (p1 x) 2 , reproducing eq. (1). 2.4 Imaging Systems Pupil-to-pupil imaging systems are used to mitigate TTL coupling, as well as beam walk, defined as the lateral displacement of the beam spot on the PD surface along one o… view at source ↗
Figure 4
Figure 4. Figure 4: Description of the lever arm effect between two beams, which dominates the geometrical TTL. Two beams, the reference beam and the measurement beam, interfere. The resulting intensity is measured by a PR located on the z = 0 plane. The measurement beam can rotate about the pivot point (0, 0, −d) T , representing either the Rx-clip or the TM’s surface in LISA. Such a tilt causes the measurement beam to propa… view at source ↗
Figure 5
Figure 5. Figure 5: Principle of DWS. A reference beam (red) and measurement beam (blue) with a MHz-scale frequency differ￾ence produce a heterodyne beat tone on each QPD segment. When aligned (left), all beat notes share the same phase. When the measurement beam is tilted (right), the wave fronts reach the top segments earlier than the bottom ones, pro￾ducing a phase difference proportional to the tilt angle. Figure from [26… view at source ↗
Figure 6
Figure 6. Figure 6: Framework of the used reference frames. The QPD defines the lab’s RF, with the QPD’s center being located at ⃗x = ⃗0. The reference beam’s RF shares the same coordinates with the lab’s RF, but is shifted along the z-axis. The measurement beam’s RF is rotated and shifted along the z axis with respect to the lab’s RF. A positive tilt of the beam is defined as an anticlockwise tilt. The measurement beam’s RF … view at source ↗
Figure 7
Figure 7. Figure 7: HE on a SEPD as a function of the measurement beam’s tilt angle for two same spot-radius beams resulting in wr = wm = weff = 1 mm, λ = 1064 nm, plotted for four different values of the effective-spot-radius-normalized relative wavefront curvature mismatch parameter ρ. The plotted curves are derived from eq. (41). The presence of wavefront curvature mismatch on one side strongly reduces the maximum achievab… view at source ↗
Figure 8
Figure 8. Figure 8: Plot of the HE of the top segments of an infinite QPD as a function of the measurement beam’s tilt angle for two same spot-radius beams resulting in wr = wm = weff = 1 mm, λ = 1064 nm, plotted for four different values of the effective-spot-radius-normalized relative wavefront curvature mismatch parameter ρ. The plotted curves are derived analytically from eq. (47). If eq. (49) was used, the same effect wo… view at source ↗
Figure 9
Figure 9. Figure 9: Plot of the peak angle θmax, QPD, top(ρ), defined as the angle at which the HE for a given value of ρ √ ηhet, QPD, top, ∞(θ) is maximum, and of the HE evaluated at θmax, QPD, top, for two interfering beams with wr = wm = weff = 1 mm. These are two non-linear functions of the effective-spot-radius-normalized relative wavefront curvature mismatch parameter ρ, which must be, respectively, even and odd. One ca… view at source ↗
Figure 10
Figure 10. Figure 10: Plot of the HE for an infinite SEPD in eq. (41) and the whole-QPD HE for an infinite QPD (calculated numerically, approximated in eq. (60) ). The used beam parameters are weff = wr = wm = 1 mm and ρ = 0. The HE for a SEPD degrades noticeably quicker as a function of the beam tilt than the whole-QPD HE. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Geometric interpretation of the shift in the HE’s maximum angle on a QPD. The principle followed in this explanation can be found also in [35, [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Plot of the HE and heterodyne phase shift for two GBs impinging on a finite SEPD in eqs. (41) and (42). The used beam parameters are weff = wr = wm = 1 mm. Both figures manifest a large variation of the plotted quantity around β ∼ 1. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Plot of the HE as a function of the beam tilt calculated for a finite SEPD for the parameters wr = wm = weff = 1 mm, rQPD = 1 mm and ρ = 0. The HE is calculated both numerically and analytically at various orders. Only one value of ρ is shown, as this method is exact in ρ. Dashed lines indicate where the analytic value deviates from the numerical result by more than 0.1. Note that, in the numeric curve, d… view at source ↗
Figure 14
Figure 14. Figure 14: Plot of the HE as a function of the beam tilt numerically calculated for a GB-GB beam pair impinging on a finite SEPD. The beam parameters are wr = wm = weff = 1 mm, rSEPD = [0.5, 1, 2]/ √ 2 mm and ρ = 0.2. Note that, in the β = 0.5 and β = 1 plots, diffraction fringes are visible outside the main peak. These are the causes of the DWS sign reversal discussed in [26, [PITH_FULL_IMAGE:figures/full_fig_p021… view at source ↗
Figure 15
Figure 15. Figure 15: Plots of the t0, t2, t4 and t6 coefficients as a function of the ratio between SEPD radius and effective beam spot radius β = √ 2rSEPD/weff calculated both analytically using eq. (147), and numerically (see subsection 3.8). The used beam parameters are wr = wm = weff = 1 mm and ρ = 0.2. The numeric parameters are obtained by numerically integrating eq. (24) and linearly fitting the resulting HE, as explai… view at source ↗
Figure 16
Figure 16. Figure 16: Plot of the HE as a function of the beam tilt calculated for the top segments of a finite QPD for the parameters weff = 1 mm, rQPD = 1 mm and ρ = 0.2. The HE is calculated both numerically and analytically at various orders. The analytic line is dashed when its value departs from the numerical one by more than 0.1. The overlap power is hence described with the same equations eqs. (76) and (82) in the case… view at source ↗
Figure 17
Figure 17. Figure 17: Comparison between the analytically obtained parameters t0, t1 and t2 from eq. (148) and the numerically calculated ones for four specific values of ρ and as a function of the parameter β = rQPD/weff. t1 and t2 are normalized respectively by the first and second power of kweff. The asymptotic analytical value for an infinite QPD β → ∞ is added as reference. In figs. 17a and 17b, for ρ = 0, the numerical s… view at source ↗
Figure 18
Figure 18. Figure 18: Plot of the HE as a function of the beam tilt numerically calculated for a GB-TH beam pair impinging on a finite SEPD. The beam parameters are weff = 1 mm, rTH = rSEPD = [0.5, 1, 2]/ √ 2 mm and ρ = 0.2. Note that to a larger PD size corresponds a lower HE. Note that in the β = 0.5 and β = 1 plots diffraction fringes are visible outside the main peak. These are the causes of the DWS sign reversal discussed… view at source ↗
Figure 19
Figure 19. Figure 19: Plots of the t0, t2, t4 and t6 coefficients as a function of the ratio between SEPD radius and effective beam spot radius β = √ 2rSEPD/weff calculated for a GB-TH beam interference both analytically using eq. (147), and numerically (see subsection 3.8). The used beam parameters are wr = weff = 1 mm, ρ = 0.2 and rTH = rSEPD. Note that all coefficients tend to zero for β → ∞ due to the normalization to an i… view at source ↗
Figure 20
Figure 20. Figure 20: b), which is well within the target precision of this model. From the numerically computed power Pnum(θ), the HE is extracted by evaluating its absolute value and normalizing it by the power of the two beams according to eq. (35). Finally, the parameters t0, t1 ... tn are obtained by performing a linear fit of the resulting HE. We stress that this process is sensitive to the used angular fitting range, wh… view at source ↗
Figure 21
Figure 21. Figure 21: Drawing of the two systems compared in this section. Above: two beams impinging on a QPD, with the measurement beam (in blue) rotated about the center of the QPD by an angle θ. Below: same system as above, with an ensemble consisting of an imaging system plus a QPD being placed instead of the QPD. Note that in this case the measurement beam’s angle at the QPD is magnified to mθ. Now, consider introducing … view at source ↗
Figure 22
Figure 22. Figure 22: Drawing of the beam rotation process. The REFQPD is located on the left, while the AUXQPD is located on the right. The virtual position of the REFQPD along the beam path is also shown in transparency. The Rx beam’s rotation pivot is indicated by the green cross. The lateral displacement ∆x is measured using the DPS signal (see [26, Section V]) of the AUXQPD. The tilt angle of the Rx beam θ can be recovere… view at source ↗
Figure 23
Figure 23. Figure 23: Left: Measurement of the HE as a function of the beam tilt angle using the Rx-GB&LO beams on REFQPD1. We remind that the HE has been normalized to its maximum value. The Rx-GB beam was tilted vertically, hence we show the HE measured on the top and bottom segments of the QPD. The parameters of the two interfering beams are wr = 0.874 ± 0.006 mm, wm = 1.038 ± 0.008 mm, weff = 0.95 ± 0.02 mm, ρ = 0.14 ± 0.0… view at source ↗
Figure 24
Figure 24. Figure 24: Left: Measurement of the HE as a function of the beam tilt angle in TDOBS using the Rx-GB&LO beams on SCIQPD1. We remind that the HE has been normalized to its maximum value. The Rx-GB beam was tilted horizontally, hence we show the HE measured on the left and right segments of the QPD. The parameters of the two interfering beams are wr = 0.350 ± 0.002 mm, wm = 0.415 ± 0.003 mm, weff = 0.377 ± 0.002 mm, ρ… view at source ↗
Figure 25
Figure 25. Figure 25: Plot of the reciprocal of QPD-signal tilt-coupling coefficient for uncorrelated noises in eq. (117) for an infinite QPD as a function of the beam tilt, compared with the HEs of an infinite SEPD in eq. (41) and the whole-QPD HE of an infinite √ ηhet∀QPD, ∞ QPD in eq. (59). The used beam parameters are wr = wm = weff = 1 mm, and an infinite QPD radius, and four different values of the effective-spot-radius-… view at source ↗
Figure 26
Figure 26. Figure 26: Plot of the reciprocal of the QPD-signal tilt-coupling coefficient for correlated noises for sum k −1 cor, Σ (θ) (a) and difference k −1 cor, ∆ (θ) (b) for an infinite QPD from eqs. (123) and (125). The used beam parameters are weff = 1 mm, and an infinite QPD radius, and four different values of the effective-spot-radius-normalized wavefront curvature mismatch parameter ρ. Note that a smaller value of k … view at source ↗
Figure 27
Figure 27. Figure 27: Plot of the best- and worst-case scenario HEs of the TMI in LISA. Note that the x axis is the Tx beam’s tilt angle; the TM’s angle can be obtained by rescaling the x axis by a factor of mIS/2, where mIS is the magnification of the imaging system in LISA. As previously argued, the top and bottom HEs are coincident in the best-case scenario, as ρ = 0. noise on the right. The noise parameters for LISA, which… view at source ↗
Figure 28
Figure 28. Figure 28: Calculated LPS and WFA noise as a function of the Tx beam’s tilt angle for the TMI in LISA. These curves are based on equations eqs. (132) and (133), which assume an infinite QPD. This approximation is justified by the values of β in [PITH_FULL_IMAGE:figures/full_fig_p040_28.png] view at source ↗

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Works this paper leans on

47 extracted references · 22 canonical work pages

  1. [1]

    Pizzella G 2016The European Physical Journal H41267–302 URL https://doi.org/10.1140/epjh/e2016-70036-8

    work page doi:10.1140/epjh/e2016-70036-8

  2. [2]

    Opt.102495–2498 URL https://opg.optica.org/ao/abstract.cfm?URI=ao-10-11-2495

    Moss G E, Miller L R and Forward R L 1971Appl. Opt.102495–2498 URL https://opg.optica.org/ao/abstract.cfm?URI=ao-10-11-2495

  3. [3]

    Weiss R 2022General Relativity and Gravitation54153 ISSN 1572-9532 URL https://doi.org/10.1007/s10714-022-03021-3

    doi:10.1007/s10714-022-03021-3

  4. [4]

    Abbott B Pet al.(LIGO Scientific Collaboration and Virgo Collaboration) 2016Phys. Rev. Lett.116(6) 061102 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.116.061102

    doi:10.1103/physrevlett.116.061102

  5. [5]

    Branchesi M, Maggiore M, Alonso D, Badger C, Banerjee B, Beirnaert F, Belgacem E, Bhagwat S, Boileau G, Borhanian S, Brown D D, Leong Chan M, Cusin G, Danilishin S L, Degallaix J, De Luca V , Dhani A, Dietrich T, Dupletsa U, Foffa S, Franciolini G, Freise A, Gemme G, Goncharov B, Ghosh A, Gulminelli F, Gupta I, Kumar Gupta P, Harms J, Hazra N, Hild S, Hin...

    2023

  6. [6]

    Evans M, Corsi A, Afle C, Ananyeva A, Arun K G, Ballmer S, Bandopadhyay A, Barsotti L, Baryakhtar M, Berger E, Berti E, Biscoveanu S, Borhanian S, Broekgaarden F, Brown D A, Cahillane C, Campbell L, Chen H Y , Daniel K J, Dhani A, Driggers J C, Effler A, Eisenstein R, Fairhurst S, Feicht J, Fritschel P, Fulda P, Gupta I, Hall E D, Hammond G, Hannuksela O ...

    Pith/arXiv arXiv 2023

  7. [7]

    Cahillane C and Mansell G 2022Galaxies1036 URL https://doi.org/10.3390/galaxies10010036

    work page doi:10.3390/galaxies10010036

  8. [8]

    Colpi M, Danzmann K, Hewitson M, Holley-Bockelmann K, Jetzer P, Nelemans G, Petiteau A, Shoemaker D, Sopuerta C, Stebbins R, Tanvir N, Ward H, Weber W J, Thorpe I, Daurskikh A, Deep A, N´u˜nez I F, Marirrodriga C G, Gehler M, Halain J P, Jennrich O, Lammers U, Larra˜naga J, Lieser M, L¨utzgendorf N, Martens W, Mondin L, Ni˜no A P, Amaro-Seoane P, Sedda M ...

    Pith/arXiv arXiv 2024

  9. [9]

    ESA 2024 Capturing the ripples of spacetime: Lisa gets go-ahead URL https://www.esa.int/Science_Exploration/Space_Science/LISA/Capturing_the_ ripples_of_spacetime_LISA_gets_go-ahead

    2024

  10. [10]

    Armano M, Audley H, Auger G, Baird J T, Bassan M, Binetruy P, Born M, Bortoluzzi D, Brandt N, Caleno M, Carbone L, Cavalleri A, Cesarini A, Ciani G, Congedo G, Cruise A M, Danzmann K, de Deus Silva M, De Rosa R, Diaz-Aguil´o M, Di Fiore L, Diepholz I, Dixon G, Dolesi R, Dunbar N, Ferraioli L, Ferroni V , Fichter W, Fitzsimons E D, Flatscher R, Freschi M, ...

  11. [11]

    Armano M, Audley H, Baird J, Binetruy P, Born M, Bortoluzzi D, Castelli E, Cavalleri A, Cesarini A, Cruise A M, Danzmann K, de Deus Silva M, Diepholz I, Dixon G, Dolesi R, Ferraioli L, Ferroni V , Fitzsimons E D, Freschi M, Gesa L, Gibert F, Giardini D, Giusteri R, Grimani C, Grzymisch J, Harrison I, Heinzel G, Hewitson M, Hollington D, Hoyland D, Hueller...

  12. [12]

    Wanner G 2019Nature Physics15200–202 ISSN 1745-2481 URL https://doi.org/10.1038/s41567-019-0462-3

    work page doi:10.1038/s41567-019-0462-3

  13. [13]

    Danzmann K, Prince T and the Lisa internatinal science team 2011 Assessment study report (yellow book) URLhttps://sci.esa.int/documents/35005/36499/1567258681608-LISA_ YellowBook_ESA-SRE-2011-3_Feb2011.pdf

    2011

  14. [14]

    Chwalla M, Danzmann K, Alvarez M D, Delgado J E, Barranco G F, Fitzsimons E, Gerberding O, Heinzel G, Killow C, Lieser M, Perreur-Lloyd M, Robertson D, Rohr J, Schuster S, Schwarze T, Tr¨obs M, Wanner G and Ward H 2020Physical Review Applied14URL https://doi.org/10.1103%2Fphysrevapplied.14.014030

  15. [15]

    Brzozowski W, Robertson D, Fitzsimons E, Ward H, Keogh J, Taylor A, Milanova M, Perreur-Lloyd M, Ali Z, Earle A, Clarkson D, Sharman R, Wells M and Parr-Burman P 2022 The LISA optical bench: an overview and engineering challengesSpace Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wavevol 12180 ed Coyle L E, Matsuura S and Perrin M...

    work page doi:10.1117/12.2627465 2022

  16. [16]

    van Veggel A M A and Killow C J 2014Advanced Optical Technologies3293–307 URL https://doi.org/10.1515/aot-2014-0022

    work page doi:10.1515/aot-2014-0022 2014

  17. [18]

    Fitzsimons E, Martino J and Vetrugno D 2024LISA Performance Model and Error Budget

  18. [19]

    Tr ¨obs M, Schuster S, Lieser M, Zwetz M, Chwalla M, Danzmann K, Barr´anco G F, Fitzsimons E D, Gerberding O, Heinzel G, Killow C J, Perreur-Lloyd M, Robertson D I, Schwarze T S, Wanner G and Ward H 2018Classical and Quantum Gravity35105001 URL https://dx.doi.org/10.1088/1361-6382/aab86c

    work page doi:10.1088/1361-6382/aab86c

  19. [20]

    Hartig M S, Paczkowski S, Hewitson M, Heinzel G and Wanner G 2025Phys. Rev. D111(4) 043048 URLhttps://link.aps.org/doi/10.1103/PhysRevD.111.043048

    work page doi:10.1103/physrevd.111.043048

  20. [21]

    Wu Y L, Luo Z R, Wang J Y , Bai M, Bian W, Cai R G, Cai Z M, Cao J, Chen D J, Chen L, Chen L S, Chen M W, Chen W B, Chen Z Y , Cong L X, Deng J F, Dong X L, Duan L, Fan S Q, Fan S S, Fang C, Fang Y , Feng K, Feng P, Feng Z, Gao R H, Gao R L, Guo Z K, He J W, He J B, Hou X, Hu L, Hu W R, Hu Z Q, Huang M J, Jia J J, Jiang K L, Jin G, Jin H B, Kang Q, Lei J ...

  21. [22]

    Luo J, Chen L S, Duan H Z, Gong Y G, Hu S, Ji J, Liu Q, Mei J, Milyukov V , Sazhin M, Shao C G, Toth V T, Tu H B, Wang Y , Wang Y , Yeh H C, Zhan M S, Zhang Y , Zharov V and Zhou Z B 2016 Classical and Quantum Gravity33035010 URL https://dx.doi.org/10.1088/0264-9381/33/3/035010

    doi:10.1088/0264-9381/33/3/035010 2016

  22. [23]

    Sheard B S, Heinzel G, Danzmann K, Shaddock D A, Klipstein W M and Folkner W M 2012Journal of Geodesy861083–1095 ISSN 1432-1394 URLhttps://doi.org/10.1007/s00190-012-0566-3

    work page doi:10.1007/s00190-012-0566-3

  23. [24]

    Chwalla M, Danzmann K, Barranco G F, Fitzsimons E, Gerberding O, Heinzel G, Killow C J, Lieser M, Perreur-Lloyd M, Robertson D I, Schuster S, Schwarze T S, Tr¨obs M, Ward H and Zwetz M 2016 Classical and Quantum Gravity33245015 URL https://dx.doi.org/10.1088/0264-9381/33/24/245015

    work page doi:10.1088/0264-9381/33/24/245015 2016

  24. [25]

    thesis Leibniz Universit ¨at Hannover URL https://doi.org/10.15488/18487

    Pizzella A 2025LISA optical bench development: experimental investigation of differential-wavefront sensing for a spaceborne gravitational wave detectorPh.D. thesis Leibniz Universit ¨at Hannover URL https://doi.org/10.15488/18487

    work page doi:10.15488/18487

  25. [26]

    Pizzella A, Dovale- ´Alvarez M, Cano P M, ´Alvarez R G, Bode C, Delgado J J E and Heinzel G 2025 Phys. Rev. D112(8) 082005 URLhttps://link.aps.org/doi/10.1103/sqgt-plx6

    work page doi:10.1103/sqgt-plx6 2025

  26. [27]

    Huarcaya V , ´Alvarez M D, Penkert D, Gozzo S, Cano P M, Yamamoto K, Delgado J J E, Mehmet M, Danzmann K and Heinzel G 2023Phys. Rev. Appl.20(2) 024078 URL https://link.aps.org/doi/10.1103/PhysRevApplied.20.024078

    work page doi:10.1103/physrevapplied.20.024078

  27. [28]

    1063/1.1930095/16000353/063112_1_online.pdf) URL https://doi.org/10.1063/1.1930095

    Donley E A, Heavner T P, Levi F, Tataw M O and Jefferts S R 2005Review of Scientific Instruments76 063112 ISSN 0034-6748 (Preprinthttps://pubs.aip.org/aip/rsi/article-pdf/doi/10. 1063/1.1930095/16000353/063112_1_online.pdf) URL https://doi.org/10.1063/1.1930095

    work page doi:10.1063/1.1930095

  28. [29]

    thesis Leibniz Universit¨at Hannover URLhttps://doi.org/10.15488/3866

    Fernandez Barranco G 2017Photodetection in intersatellite laser interferometersPh.D. thesis Leibniz Universit¨at Hannover URLhttps://doi.org/10.15488/3866

    work page doi:10.15488/3866

  29. [30]

    Fern ´andez Barranco G, Sheard B S, Dahl C, Mathis W and Heinzel G 2018IEEE Sensors Journal18 7414–7420

  30. [31]

    Siegman A 1986Lasers(University Science Books) ISBN 9780685055953 URL https://books.google.de/books?id=qBBaDQEACAAJ

  31. [32]

    Gerberding O, Dahl C, Kullmann J, Tr ¨obs M, Bykov I, Barke S, Brause N, Esteban Delgado J J, Schwarze T, Reiche J, Danzmann K, Rasmussen T, Hansen T, Engaard A, Pedersen S, Jennrich O, Suess M, Sodnik Z and Heinzel G 2015The Review of scientific instruments86

  32. [33]

    thesis Leibniz Universit ¨at Hannover URLhttps://doi.org/10.15488/8255

    Gerberding O 2014Phase readout for satellite interferometryPh.D. thesis Leibniz Universit ¨at Hannover URLhttps://doi.org/10.15488/8255

    work page doi:10.15488/8255

  33. [34]

    thesis Leibniz Universit ¨at Hannover URL https://doi.org/10.15488/17997

    Bode C 2024Noise in the LISA phasemeterPh.D. thesis Leibniz Universit ¨at Hannover URL https://doi.org/10.15488/17997

    work page doi:10.15488/17997

  34. [35]

    Hartig M S, Schuster S, Heinzel G and Wanner G 2023Journal of Optics25055601 URL https://doi.org/10.1088/2040-8986/acc3ac

    work page doi:10.1088/2040-8986/acc3ac 2040

  35. [36]

    Opt.335041–5049 URL https://opg.optica.org/ao/abstract.cfm?URI=ao-33-22-5041

    Morrison E, Meers B J, Robertson D I and Ward H 1994Appl. Opt.335041–5049 URL https://opg.optica.org/ao/abstract.cfm?URI=ao-33-22-5041

  36. [37]

    Opt.335037–5040 URL https://opg.optica.org/ao/abstract.cfm?URI=ao-33-22-5037 49 IOP PublishingJournalvv(yyyy) aaaaaa Authoret al

    Morrison E, Meers B J, Robertson D I and Ward H 1994Appl. Opt.335037–5040 URL https://opg.optica.org/ao/abstract.cfm?URI=ao-33-22-5037 49 IOP PublishingJournalvv(yyyy) aaaaaa Authoret al

  37. [38]

    Hechenblaikner G 2010J. Opt. Soc. Am. A272078–2083 URL https://opg.optica.org/josaa/abstract.cfm?URI=josaa-27-9-2078

    2083

  38. [39]

    Heinzel G, ´Alvarez M D, Pizzella A, Brause N and Delgado J J E 2020Phys. Rev. Appl.14(5) 054013 URLhttps://link.aps.org/doi/10.1103/PhysRevApplied.14.054013

    work page doi:10.1103/physrevapplied.14.054013

  39. [40]

    Wissel L, Wittchen A, Schwarze T S, Hewitson M, Heinzel G and Halloin H 2022Phys. Rev. Appl. 17(2) 024025 URLhttps://link.aps.org/doi/10.1103/PhysRevApplied.17.024025

    work page doi:10.1103/physrevapplied.17.024025

  40. [41]

    Wanner G, Heinzel G, Kochkina E, Mahrdt C, Sheard B S, Schuster S and Danzmann K 2012Optics Communications2854831–4839 ISSN 0030-4018 URL https://www.sciencedirect.com/science/article/pii/S0030401812008528

  41. [42]

    Wissel L, Hewitson M and Heinzel G 2024Phys. Rev. Appl.22(4) 044048 URL https://link.aps.org/doi/10.1103/PhysRevApplied.22.044048

    work page doi:10.1103/physrevapplied.22.044048

  42. [43]

    Wanner G, Karnesis N and on behalf of the LISA Pathfinder collaboration 2017Journal of Physics: Conference Series840012043 URLhttps://doi.org/10.1088/1742-6596/840/1/012043

    work page doi:10.1088/1742-6596/840/1/012043

  43. [44]

    thesis Leibniz Universit ¨at Hannover URL https://doi.org/10.15488/7550

    Wanner G 2010Complex optical systems in space : numerical modelling of the heterodyne interferometry of LISA Pathfinder and LISAPh.D. thesis Leibniz Universit ¨at Hannover URL https://doi.org/10.15488/7550

    work page doi:10.15488/7550

  44. [45]

    Armano M, Audley H, Baird J, Binetruy P, Born M, Bortoluzzi D, Castelli E, Cavalleri A, Cesarini A, Cruise A M, Danzmann K, de Deus Silva M, Diepholz I, Dixon G, Dolesi R, Ferraioli L, Ferroni V , Fitzsimons E D, Freschi M, Gesa L, Giardini D, Gibert F, Giusteri R, Grimani C, Grzymisch J, Harrison I, Hartig M S, Heinzel G, Hewitson M, Hollington D, Hoylan...

  45. [46]

    Mistry T e a 2024 The eyes of lisa URLhttps://app.oxfordabstracts.com/content/events/ 5189/submitters/405298/submissions/fe-412377de-ee7f-408e-849d-84ca70d81af8/ questions/91860/file/e18cb94a-84d5-4284-b64f-39c13cb08bb4.pdf

    2024

  46. [47]

    thesis Australian National University

    Francis S P 2017Multi-link laser interferometer architecture for a next generation GRACEPh.D. thesis Australian National University

  47. [48]

    Bendat J S and Piersol A G 2000Random Data: Analysis and Measurement Procedures3rd ed (USA: John Wiley & Sons, Inc.) ISBN 0471317330 50