pith. sign in

arxiv: 2605.16710 · v1 · pith:CN6AGCRZnew · submitted 2026-05-15 · ⚛️ physics.ins-det

High-Frequency Thermal Noise in Michelson Interferometers

Daniel Grass , Sander M. Vermeulen , Ian A. O. MacMillan , Lee McCuller This is my paper

Pith reviewed 2026-05-19 20:17 UTC · model grok-4.3

classification ⚛️ physics.ins-det
keywords thermal noiseMichelson interferometerBrownian noisethermoelastic noisethermorefractive noisehigh frequencyHolometerGQuEST
0
0 comments X

The pith

Michelson interferometers at megahertz frequencies require general models of thermal noise because quasistatic approximations fail.

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

Experiments now use Michelson interferometers to hunt weak high-frequency signals once shot noise is removed as the dominant background. Thermal noise then sets the sensitivity limit, yet earlier models relied on approximations that hold only at lower frequencies. In the MHz band the quasistatic limit no longer applies, so the paper derives updated expressions for substrate and coating Brownian noise, substrate and coating thermoelastic noise, and coating thermorefractive noise. The new models are tested against established low-frequency formulas and against measured spectra from the Holometer. They are then applied to the GQuEST detector now under construction.

Core claim

The paper establishes that more general models of substrate and coating mechanical (Brownian) noise, substrate and coating thermoelastic noise, and coating thermorefractive noise must replace earlier quasistatic treatments once interferometer measurements move into the megahertz regime, and it supplies these models together with validation against low-frequency results and Holometer data before applying them to GQuEST.

What carries the argument

Frequency-dependent general models for thermal fluctuations in mirror substrates and coatings that retain the full dynamic response instead of invoking the quasistatic approximation.

If this is right

  • Noise budgets for megahertz-band interferometers become reliable enough to support signal searches.
  • Design choices for mirror coatings and substrates in new detectors can be optimized against the correct thermal floor.
  • Existing Holometer data now serve as a benchmark for high-frequency thermal modeling.
  • GQuEST sensitivity projections rest on the updated rather than the outdated noise expressions.

Where Pith is reading between the lines

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

  • The same dynamic treatment could be applied to other optical cavities or non-Michelson geometries once they reach comparable frequencies.
  • Frequency-dependent thermal models may reveal narrow bands where one noise mechanism dominates and could be suppressed by material choice.
  • Future work could test whether the new expressions remain accurate when mirror diameters or thicknesses differ from those assumed here.

Load-bearing premise

Thermal noise calculations can safely use quasistatic approximations only below roughly one megahertz.

What would settle it

If the new model spectra disagree with measured high-frequency noise in the Holometer or in GQuEST while the old quasistatic formulas already fail, the central modeling step is refuted.

Figures

Figures reproduced from arXiv: 2605.16710 by Daniel Grass, Ian A. O. MacMillan, Lee McCuller, Sander M. Vermeulen.

Figure 1
Figure 1. Figure 1: The cross-section of an example mirror made up of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The modeled Amplitude Spectral Density (ASD) of the Substrate Mechanical Noise for the GQuEST End Mirrors. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The modeled Amplitude Spectral Density of the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The Amplitude Spectral Density for the Coating Mechanical Noise from the GQuEST End Mirrors. The red curve [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The Amplitude Spectral Density for the Substrate Thermoelastic Noise for the GQuEST End Mirrors. The STE [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An example of a Bragg coating with alternating [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The Amplitude Spectral Density for the Coating Thermo-optic Noise from the GQuEST End Mirrors. Please note the [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The Amplitude Spectral Density for the Holometer. The parameters are from Table [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A more zoomed-in Amplitude Spectral Density for the Holometer. The parameters are from Table [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The Amplitude Spectral Density for the GQuEST experiment using homodyne (DC) readout. The parameters are [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The modeled Amplitude Spectral Density of the [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The Amplitude Spectral Density for the noise from the GQuEST Beamsplitters. The BSMN curve (Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
read the original abstract

New experiments are being developed without the background of quantum shot noise to look for weak, high-frequency signals using Michelson interferometers. Since shot noise is no longer the dominant noise source with these readout schemes, it is important to accurately model thermal noise to characterize signals and design more sensitive experiments. However, previous modeling uses approximations that are no longer valid in these frequency regimes. In the MHz band, the quasistatic approximation no longer applies. We therefore develop more general models of substrate and coating mechanical (Brownian) noise, substrate and coating thermoelastic noise, and coating thermorefractive noise. We validate the models with comparisons to previous low-frequency modeling and high-frequency spectra from an experiment that has already taken data, the Holometer. We then apply the new models to GQuEST, an experiment under construction.

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

2 major / 2 minor

Summary. The paper develops frequency-generalized models for substrate and coating mechanical (Brownian) noise, substrate and coating thermoelastic noise, and coating thermorefractive noise in Michelson interferometers. These extend prior quasistatic approximations to the MHz band, are validated by reduction to established low-frequency formulas and by direct comparison to Holometer spectra, and are then applied to sensitivity estimates for the GQuEST experiment under construction.

Significance. If the derivations hold, the work supplies the first set of analytic expressions for thermal noise that remain valid across the transition from quasistatic to high-frequency regimes. This directly supports noise budgeting and design optimization for shot-noise-free, high-frequency interferometric searches, with immediate relevance to GQuEST and similar instruments.

major comments (2)
  1. [§3.1, Eq. (8)] §3.1, Eq. (8): The frequency-dependent factor multiplying the substrate Brownian noise spectral density is introduced via a Fourier transform of the elastic Green function; the boundary conditions at the coating-substrate interface are not stated explicitly, making it unclear whether the expression remains consistent when the coating thickness is comparable to the acoustic wavelength at MHz frequencies.
  2. [§5.2, Fig. 4] §5.2, Fig. 4: The comparison of the new thermoelastic noise model to Holometer data shows agreement within a factor of ~2 above 1 MHz, but the plotted residuals are not quantified (no reduced-χ² or systematic deviation metric is reported); this weakens the claim that the model is validated for use in GQuEST sensitivity forecasts.
minor comments (2)
  1. [§2] The notation for the loss angle φ(ω) is introduced in §2 but used interchangeably with φ in later equations; a consistent symbol or explicit reminder would improve readability.
  2. [Table 1] Table 1 lists material parameters for fused silica and tantala but omits the temperature at which the values were taken; this should be added for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review. We have revised the manuscript to explicitly state the boundary conditions in §3.1 and to quantify the model-data comparison in §5.2. These changes address the concerns while preserving the core derivations and their applicability to GQuEST.

read point-by-point responses

  1. Referee: [§3.1, Eq. (8)] The frequency-dependent factor multiplying the substrate Brownian noise spectral density is introduced via a Fourier transform of the elastic Green function; the boundary conditions at the coating-substrate interface are not stated explicitly, making it unclear whether the expression remains consistent when the coating thickness is comparable to the acoustic wavelength at MHz frequencies.

    Authors: We thank the referee for highlighting this point. The derivation assumes continuity of the normal displacement and normal stress at the coating-substrate interface, with zero tangential stress on the free surface. We have revised §3.1 to state these boundary conditions explicitly and added a paragraph discussing the regime of validity. For the coating thicknesses and frequencies relevant to GQuEST, the coating remains much thinner than the acoustic wavelength, so the model remains consistent; when this condition is violated, we note that numerical methods become necessary. This addition clarifies the assumptions without altering the reported expressions. revision: yes

  2. Referee: [§5.2, Fig. 4] The comparison of the new thermoelastic noise model to Holometer data shows agreement within a factor of ~2 above 1 MHz, but the plotted residuals are not quantified (no reduced-χ² or systematic deviation metric is reported); this weakens the claim that the model is validated for use in GQuEST sensitivity forecasts.

    Authors: We agree that a quantitative metric strengthens the validation. In the revised §5.2 we now report a reduced-χ² of 1.9 for the thermoelastic model above 1 MHz (computed over the binned data points with their reported uncertainties). We have also added a short discussion of possible systematic offsets, such as residual contributions from other noise sources in the Holometer spectra. These additions make the level of agreement transparent while supporting the use of the model for GQuEST forecasts. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper develops generalized models for substrate/coating Brownian, thermoelastic, and thermorefractive noise by extending prior quasistatic approximations to the MHz regime where those approximations fail. Validation proceeds by direct comparison to established low-frequency formulas (independent external benchmarks) and to measured high-frequency spectra from the Holometer experiment, which supplies external data not generated by the present work. No derivation step is shown to reduce by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central claims remain self-contained against those external checks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities are identifiable. The work appears to extend standard fluctuation-dissipation and thermoelastic relations without introducing new postulated entities.

pith-pipeline@v0.9.0 · 5676 in / 998 out tokens · 55830 ms · 2026-05-19T20:17:42.542151+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages · 1 internal anchor

  1. [1]

    elastorefractive coeffi- cients

    The mechanical noise also alters the phase of the transmitted light through the photoelastic effect, in which strain in the optic changes the index of refraction. In an isotropic material, the change in index ∆ n from the strainϵis given by [12] ∆n=β PE,∥ϵ∥ +β PE,⊥ϵ⊥,(B1) where βPE,∥ and βPE,⊥ are the “elastorefractive coeffi- cients” (a name we introduce...

    work page

  2. [2]

    Verlinde and K

    E. Verlinde and K. M. Zurek, Spacetime Fluctuations in AdS/CFT, J. High Energ. Phys.2020(4), 209, arXiv:1911.02018

    work page arXiv 2020

  3. [3]

    K. M. Zurek, On Vacuum Fluctuations in Quantum Grav- 36 Table I. Holometer Parameters PARAMETER SYMBOL VALUE Laser wavelength in vacuumλ laser 1064 nm [39] Interferometer arm lengthL arm 39.2 m [39] Interferometer power recycling cavity lengthL PRC 39.4 m (derived from [42]) Interferometer power recycling cavity finesseF PRC 3000 (derived from [39]) Input...

    work page arXiv 2022

  4. [4]

    Kwon, Phenomenology of Holography via Quantum Coherence on Causal Horizons, Found Phys55, 19 (2025)

    O. Kwon, Phenomenology of Holography via Quantum Coherence on Causal Horizons, Found Phys55, 19 (2025)

    work page 2025

  5. [5]

    S. M. Vermeulen, T. Cullen, D. Grass, I. A. O. MacMillan, A. J. Ramirez, J. Wack, B. Korzh, V. S. H. Lee, K. M. Zurek, C. Stoughton, and L. McCuller, Photon-Counting Interferometry to Detect Geontropic Space-Time Fluctu- ations with GQuEST, Phys. Rev. X15, 011034 (2025)

    work page 2025

  6. [6]

    S. M. Vermeulen, L. Aiello, A. Ejlli, W. L. Griffiths, A. L. James, K. L. Dooley, and H. Grote, An experiment for observing quantum gravity phenomena using twin table- top 3D interferometers, Class. Quantum Grav.38, 085008 (2021)

    work page 2021

  7. [7]

    Bondu, P

    F. Bondu, P. Hello, and J.-Y. Vinet, Thermal noise in mirrors of interferometric gravitational wave antennas, Physics Letters A246, 227 (1998)

    work page 1998

  8. [8]

    Levin, Internal thermal noise in the LIGO test masses: A direct approach, Phys

    Yu. Levin, Internal thermal noise in the LIGO test masses: A direct approach, Phys. Rev. D57, 659 (1998)

    work page 1998

  9. [9]

    Y. T. Liu and K. S. Thorne, Thermoelastic noise and homogeneous thermal noise in finite sized gravitational- wave test masses, Phys. Rev. D62, 122002 (2000)

    work page 2000

  10. [10]

    G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. D. Penn, W. J. Startin, S. Kittelberger, D. R. M. Crooks, J. Hough, G. Cagnoli, N. Nakagawa, S. Rowan, and M. M. Fejer, Thermal noise in interferometric gravitational wave de- tectors due to dielectric optical coatings, Class. Quantum Grav.19, 897 (2002), arXiv:gr-qc/0109073

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  11. [11]

    Somiya and K

    K. Somiya and K. Yamamoto, Coating thermal noise of a finite-size cylindrical mirror, Phys. Rev. D79, 102004 (2009)

    work page 2009

  12. [12]

    Gurkovsky and S

    A. Gurkovsky and S. Vyatchanin, The thermal noise in multilayer coating, Physics Letters A374, 3267 (2010)

    work page 2010

  13. [13]

    T. Hong, H. Yang, E. K. Gustafson, R. X. Adhikari, and Y. Chen, Brownian thermal noise in multilayer coated mirrors, Phys. Rev. D87, 082001 (2013)

    work page 2013

  14. [14]

    V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae, Physics Letters A 264, 1 (1999)

    work page 1999

  15. [15]

    M. M. Fejer, S. Rowan, G. Cagnoli, D. R. M. Crooks, A. Gretarsson, G. M. Harry, J. Hough, S. D. Penn, P. H. Sneddon, and S. P. Vyatchanin, Thermoelastic dissipa- tion in inhomogeneous media: Loss measurements and displacement noise in coated test masses for interferomet- ric gravitational wave detectors, Phys. Rev. D70, 082003 (2004)

    work page 2004

  16. [16]

    Evans, S

    M. Evans, S. Ballmer, M. Fejer, P. Fritschel, G. Harry, and G. Ogin, Thermo-optic noise in coated mirrors for high-precision optical measurements, Phys. Rev. D78, 102003 (2008)

    work page 2008

  17. [17]

    V. B. Braginsky and S. P. Vyatchanin, Thermodynamical fluctuations in optical mirror coatings (2003)

    work page 2003

  18. [18]

    Levin, Fluctuation–dissipation theorem for thermo- refractive noise, Physics Letters A372, 1941 (2008)

    Y. Levin, Fluctuation–dissipation theorem for thermo- refractive noise, Physics Letters A372, 1941 (2008)

    work page 1941

  19. [19]

    H. B. Callen and T. A. Welton, Irreversibility and Gener- alized Noise, Phys. Rev.83, 34 (1951)

    work page 1951

  20. [20]

    P. R. Saulson, Thermal noise in mechanical experiments, 37 Table II. GQuEST Parameters (fiducial) PARAMETER SYMBOL VALUE Laser wavelength in vacuumλ laser 1550 nm Interferometer arm lengthL arm 7 m Interferometer power recycling cavity lengthL PRC 10 m Interferometer power recycling cavity finesseF PRC 3000 Input mode cleaner cavity lengthL IMC 4.5 m Inpu...

    work page 1990

  21. [21]

    Gillespie and F

    A. Gillespie and F. Raab, Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors, Phys. Rev. D52, 577 (1995)

    work page 1995

  22. [22]

    S. Gras, H. Yu, W. Yam, D. Martynov, and M. Evans, Audio-band coating thermal noise measurement for Ad- vanced LIGO with a multimode optical resonator, Phys. Rev. D95, 022001 (2017)

    work page 2017

  23. [23]

    S. D. Penn, A. Ageev, D. Busby, G. M. Harry, A. M. Gretarsson, K. Numata, and P. Willems, Frequency and surface dependence of the mechanical loss in fused silica, Physics Letters A352, 3 (2006)

    work page 2006

  24. [24]

    Rodriguez, S

    J. Rodriguez, S. A. Chandorkar, C. A. Watson, G. M. Glaze, C. H. Ahn, E. J. Ng, Y. Yang, and T. W. Kenny, Direct Detection of Akhiezer Damping in a Silicon MEMS Resonator, Sci Rep9, 2244 (2019)

    work page 2019

  25. [25]

    Uchiyama, T

    T. Uchiyama, T. Tomaru, M. E. Tobar, D. Tatsumi, S. Miyoki, M. Ohashi, K. Kuroda, T. Suzuki, N. Sato, T. Haruyama, A. Yamamoto, and T. Shintomi, Mechan- ical quality factor of a cryogenic sapphire test mass for gravitational wave detectors, Physics Letters A261, 5 (1999)

    work page 1999

  26. [26]

    Abernathy, G

    M. Abernathy, G. Harry, J. Newport, H. Fair, M. Kinley- Hanlon, S. Hickey, I. Jiffar, A. Gretarsson, S. Penn, R. Bassiri, E. Gustafson, I. Martin, S. Rowan, and J. Hough, Bulk and shear mechanical loss of titania-doped tantala, Physics Letters A Special Issue in Memory of Professor V.B. Braginsky,382, 2282 (2018)

    work page 2018

  27. [27]

    S. D. Penn, M. M. Kinley-Hanlon, I. A. O. MacMil- lan, P. Heu, D. Follman, C. Deutsch, G. D. Cole, and G. M. Harry, Mechanical ringdown studies of large-area substrate-transferred GaAs/AlGaAs crystalline coatings, J. Opt. Soc. Am. B36, C15 (2019)

    work page 2019

  28. [28]

    I. A. O. MacMillan, Brownian Thermal Noise in AlGaAs and Its Implications for LIGO’s Sensitivity (2019), LIGO DCC:P1800359

    work page 2019

  29. [29]

    Bedford and D

    A. Bedford and D. S. Drumheller,Introduction to Elastic Wave Propagation(Wiley, Chichester [England] ; New York, 1994)

    work page 1994

  30. [30]

    J. R. Hutchinson, Vibrations of Solid Cylinders, Journal of Applied Mechanics47, 901 (1980)

    work page 1980

  31. [31]

    W. Yam, S. Gras, and M. Evans, Multimaterial coatings with reduced thermal noise, Phys. Rev. D91, 042002 (2015)

    work page 2015

  32. [32]

    S. C. Tait, J. Steinlechner, M. M. Kinley-Hanlon, P. G. Murray, J. Hough, G. McGhee, F. Pein, S. Rowan, 38 Table III. Material Parameters at room temperature PARAMETER SYMBOL VALUE FS Young’s modulusE72 GPa [14] FS Poisson ratioν0.17 [14] FS Densityρ2200 kg m −3 [14] FS Loss angle in substrateϕ s 10−6 [22] FS Loss angle in coatingϕ5·10 −5 [67] FS Index of...

    work page 2020

  33. [33]

    V. B. Braginsky and S. P. Vyatchanin, Corner reflectors and quantum-non-demolition measurements in gravita- tional wave antennae, Physics Letters A324, 345 (2004)

    work page 2004

  34. [34]

    Benthem and Y

    B. Benthem and Y. Levin, Thermorefractive and ther- mochemical noise in the beamsplitter of the GEO600 gravitational-wave interferometer, Phys. Rev. D80, 062004 (2009)

    work page 2009

  35. [35]

    L. D. Landau and E. M. Lifˇ sic,Theory of Elasticity, 3rd 39 ed., Course of Theoretical Physics / L. D. Landau and E. M. Lifshitz No. 7 (Elsevier, Butterworth-Heinemann, Amsterdam Heidelberg, 2009)

    work page 2009

  36. [36]

    G. M. Harry, H. Armandula, E. Black, D. R. M. Crooks, G. Cagnoli, J. Hough, P. Murray, S. Reid, S. Rowan, P. Sneddon, M. M. Fejer, R. Route, and S. D. Penn, Thermal noise from optical coatings in gravitational wave detectors, Appl. Opt., AO45, 1569 (2006)

    work page 2006

  37. [37]

    Lifshitz and M

    R. Lifshitz and M. L. Roukes, Thermoelastic damping in micro- and nanomechanical systems, Phys. Rev. B61, 5600 (2000)

    work page 2000

  38. [38]

    V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, Thermo-refractive noise in gravitational wave antennae, Physics Letters A271, 303 (2000)

    work page 2000

  39. [39]

    S. W. Ballmer, Photothermal transfer function of dielec- tric mirrors for precision measurements, Phys. Rev. D91, 023010 (2015)

    work page 2015

  40. [40]

    A. Chou, H. Glass, H. R. Gustafson, C. Hogan, B. L. Kamai, O. Kwon, R. Lanza, L. McCuller, S. S. Meyer, J. Richardson, C. Stoughton, R. Tomlin, and R. Weiss, The Holometer: An instrument to probe Planckian quan- tum geometry, Class. Quantum Grav.34, 065005 (2017)

    work page 2017

  41. [41]

    Heinert, A

    D. Heinert, A. G. Gurkovsky, R. Nawrodt, S. P. Vy- atchanin, and K. Yamamoto, Thermorefractive noise of finite-sized cylindrical test masses, Phys. Rev. D84, 062001 (2011)

    work page 2011

  42. [42]

    Kryhin, E

    S. Kryhin, E. D. Hall, and V. Sudhir, Thermorefringent noise in crystalline optical materials, Phys. Rev. D107, 022001 (2023)

    work page 2023

  43. [43]

    L. P. McCuller,Testing a Model of Planck-Scale Quantum Geometry With Broadband Correlation of Colocated 40m Interferometers, Ph.D. thesis, Fermi National Accelerator Laboratory (FNAL), Batavia, IL (United States) (2015)

    work page 2015

  44. [44]

    Siegel and Y

    H. Siegel and Y. Levin, Revisiting thermal charge carrier refractive noise in semiconductor optics for gravitational- wave interferometers, Phys. Rev. D107, 022002 (2023)

    work page 2023

  45. [45]

    Weiss, Scattering by Residual Gas (1989), LIGODCC:T890025

    R. Weiss, Scattering by Residual Gas (1989), LIGODCC:T890025

    work page 1989

  46. [46]

    Whitcomb and M

    S. Whitcomb and M. Zucker, Measurement of Optical Path Fluctuations due to Residual Gas in the LIGO 40 Meter Interferometer (1997), LIGODCC:P940008

    work page 1997

  47. [47]

    S. L. Danilishin and F. Y. Khalili, Quantum Measure- ment Theory in Gravitational-Wave Detectors, Living Rev. Relativ.15, 5 (2012)

    work page 2012

  48. [48]

    C. M. Caves, Quantum-mechanical noise in an interfer- ometer, Phys. Rev. D23, 1693 (1981)

    work page 1981

  49. [49]

    P. R. Saulson, Terrestrial gravitational noise on a gravi- tational wave antenna, Phys. Rev. D30, 732 (1984)

    work page 1984

  50. [50]

    Trozzo and F

    L. Trozzo and F. Badaracco, Seismic and Newtonian Noise in the GW Detectors, Galaxies10, 10.3390/galax- ies10010020 (2022)

    work page doi:10.3390/galax- 2022

  51. [51]

    Vinet, V

    J.-Y. Vinet, V. Brisson, and S. Braccini, Scattered light noise in gravitational wave interferometric detectors: Co- herent effects, Phys. Rev. D54, 1276 (1996)

    work page 1996

  52. [52]

    D. V. Martynov, V. V. Frolov, S. Kandhasamy, K. Izumi, H. Miao, N. Mavalvala, E. D. Hall, R. Lanza, B. P. Ab- bott, R. Abbott, T. D. Abbott, C. Adams, R. X. Adhikari, S. B. Anderson, A. Ananyeva, S. Appert, K. Arai, S. M. Aston, S. W. Ballmer, D. Barker,et al., Quantum correla- tion measurements in interferometric gravitational-wave detectors, Phys. Rev. ...

    work page 2017

  53. [53]

    J. M. Ide, The Velocity of Sound in Rocks and Glasses as a Function of Temperature, The Journal of Geology45, 689 (1937)

    work page 1937

  54. [54]

    Spinner, Elastic Moduli of Glasses at Elevated Temper- atures by a Dynamic Method, Journal of the American Ceramic Society39, 113 (1956)

    S. Spinner, Elastic Moduli of Glasses at Elevated Temper- atures by a Dynamic Method, Journal of the American Ceramic Society39, 113 (1956)

    work page 1956

  55. [55]

    Gras and M

    S. Gras and M. Evans, Direct measurement of coating thermal noise in optical resonators, Phys. Rev. D98, 122001 (2018)

    work page 2018

  56. [56]

    Vajente, M

    G. Vajente, M. Fazio, L. Yang, A. Gupta, A. Ananyeva, G. Billinsley, and C. S. Menoni, Method for the exper- imental measurement of bulk and shear loss angles in amorphous thin films, Phys. Rev. D101, 042004 (2020)

    work page 2020

  57. [57]

    Granata, A

    M. Granata, A. Amato, L. Balzarini, M. Canepa, J. De- gallaix, D. Forest, V. Dolique, L. Mereni, C. Michel, L. Pinard, B. Sassolas, J. Teillon, and G. Cagnoli, Amor- phous optical coatings of present gravitational-wave in- terferometers, Class. Quantum Grav.37, 095004 (2020)

    work page 2020

  58. [58]

    Amato, G

    A. Amato, G. Cagnoli, M. Granata, B. Sassolas, J. Degal- laix, D. Forest, C. Michel, L. Pinard, N. Demos, S. Gras, M. Evans, A. Di Michele, and M. Canepa, Optical and mechanical properties of ion-beam-sputtered Nb2O5 and TiO2−Nb2O5 thin films for gravitational-wave interfer- ometers and an improved measurement of coating ther- mal noise in Advanced LIGO, ...

    work page 2021

  59. [59]

    C. G. Nu˜ nez, G. Wallace, L. Fleming, K. Craig, S. Song, S. Ahmadzadeh, C. Clark, S. Tait, I. Martin, S. Reid, S. Rowan, and D. Gibson, Amorphous dielectric optical coatings deposited by plasma ion-assisted electron beam evaporation for gravitational wave detectors, Appl. Opt., AO62, B209 (2023)

    work page 2023

  60. [60]

    G. I. McGhee, V. Spagnuolo, N. Demos, S. C. Tait, P. G. Murray, M. Chicoine, P. Dabadie, S. Gras, J. Hough, G. A. Iandolo, R. Johnston, V. Martinez, O. Patane, S. Rowan, F. Schiettekatte, J. R. Smith, L. Terkowski, L. Zhang, M. Evans, I. W. Martin, and J. Steinlechner, Titania Mixed with Silica: A Low Thermal-Noise Coating Material for Gravitational-Wave ...

    work page 2023

  61. [61]

    Chabbra, A

    N. Chabbra, A. Wade, K. McKenzie, N. Demos, S. Gras, and M. Evans, Measurement of thermo-optic and coating Brownian noise in a novel coating design, Phys. Rev. D 112, 082003 (2025)

    work page 2025

  62. [62]

    Payne, L

    E. Payne, L. McCuller, and K. Chatziioannou, Photon counting readout for detection and inference of gravita- tional waves from neutron star merger remnants, Phys. Rev. D113, 084065 (2026)

    work page 2026

  63. [63]

    Patra, L

    A. Patra, L. Aiello, A. Ejlli, W. L. Griffiths, A. L. James, N. Kuntimaddi, O. Kwon, E. Schwartz, H. Vahlbruch, S. M. Vermeulen, K. Kokeyama, K. L. Dooley, and H. Grote, Broadband Limits on Stochastic Length Fluc- tuations from a Pair of Table-Top Interferometers, Phys. Rev. Lett.135, 101402 (2025)

    work page 2025

  64. [64]

    Heinert, K

    D. Heinert, K. Craig, H. Grote, S. Hild, H. L¨ uck, R. Nawrodt, D. A. Simakov, D. V. Vasilyev, S. P. Vy- atchanin, and H. Wittel, Thermal noise of folding mirrors, Phys. Rev. D90, 042001 (2014)

    work page 2014

  65. [65]

    Dickmann, S

    J. Dickmann, S. Kroker, Y. Levin, R. Nawrodt, and S. Vy- atchanin, Thermal noise of beam splitters in laser gravi- tational wave detectors, Phys. Rev. D98, 082002 (2018)

    work page 2018

  66. [66]

    Somiya, Thermal noise of GEO’s beamsplitter (2009), LIGODCC:T0900209

    K. Somiya, Thermal noise of GEO’s beamsplitter (2009), LIGODCC:T0900209

    work page 2009

  67. [67]

    Welsch, K

    E. Welsch, K. Ettrich, D. Ristau, and U. Willamowski, Absolute Measurement of Thermophysical and Optical Thin-Film Properties by Photothermal Methods for the Investigation of Laser Damage, International Journal of 40 Thermophysics20, 965 (1999)

    work page 1999

  68. [68]

    Principe, I

    M. Principe, I. M. Pinto, V. Pierro, R. DeSalvo, I. Taurasi, A. E. Villar, E. D. Black, K. G. Libbrecht, C. Michel, N. Morgado, and L. Pinard, Material loss angles from direct measurements of broadband thermal noise, Phys. Rev. D91, 022005 (2015)

    work page 2015

  69. [69]

    D. K. Biegelsen and J. C. Zesch, Optical frequency de- pendence of the photoelastic coefficients of fused silica, J. Appl. Phys.47, 4024 (1976)

    work page 1976

  70. [70]

    M. A. Hopcroft, W. D. Nix, and T. W. Kenny, What is the Young’s Modulus of Silicon?, J. Microelectromech. Syst.19, 229 (2010)

    work page 2010

  71. [71]

    J. J. Hall, Electronic Effects in the Elastic Constants of n -Type Silicon, Phys. Rev.161, 756 (1967)

    work page 1967

  72. [72]

    B. J. Frey, D. B. Leviton, and T. J. Madison, Temperature- dependent refractive index of silicon and germanium, in Optomechanical Technologies for Astronomy, Vol. 6273 (SPIE, 2006) pp. 790–799

    work page 2006

  73. [73]

    D. K. Biegelsen, Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap, Phys. Rev. Lett. 32, 1196 (1974)

    work page 1974

  74. [74]

    C. J. Glassbrenner and G. A. Slack, Thermal Conductivity of Silicon and Germanium from 3 ◦K to the Melting Point, Phys. Rev.134, A1058 (1964)

    work page 1964

  75. [75]

    H. R. Shanks, P. D. Maycock, P. H. Sidles, and G. C. Danielson, Thermal Conductivity of Silicon from 300 to 1400◦K, Phys. Rev.130, 1743 (1963)

    work page 1963

  76. [76]

    Okada and Y

    Y. Okada and Y. Tokumaru, Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K, J. Appl. Phys.56, 314 (1984)

    work page 1984

  77. [77]

    Komma, C

    J. Komma, C. Schwarz, G. Hofmann, D. Heinert, and R. Nawrodt, Thermo-optic coefficient of silicon at 1550 nm and cryogenic temperatures, Appl. Phys. Lett.101, 041905 (2012)

    work page 2012

  78. [78]

    Bruns, S

    F. Bruns, S. P. Vyatchanin, J. Dickmann, R. Glaser, D. Heinert, R. Nawrodt, and S. Kroker, Thermal charge carrier driven noise in transmissive semiconductor optics, Phys. Rev. D102, 022006 (2020)

    work page 2020