Open Access
Add to BibSonomy
Abstract
In order to increase the number of detectable gravitational-wave sources, future gravitational-wave detectors will operate with cryogenically cooled mirrors. However, recent studies showed that cryogenic mirrors can suffer from the molecular layer formation, which introduces an additional optical loss, and the detector’s performance degrades. In order to evaluate the impact of the molecular layer on future cryogenic gravitational-wave detectors, we built a cryogenic folded-cavity setup and developed an ellipsometric measurement method. The optical loss induced by the cryogenic molecular layer shows a large value even at a few nanometer thickness and can deteriorate the performance of the future cryogenic gravitational-wave detectors.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The first direct detection of gravitational wave (GW) opened a new window to the Universe [1]. There have been a number of detections of GW events, which brought us remarkable astrophysical discoveries such as the evidence of r-process nucleosynthesis in a neutron star merger, and the existence of an intermediate mass black hole [2,3]. Improvements in the detector’s sensitivity will bring forth a larger number of detections with possibly wider variety of GW sources, which in turn will give us further insights into the Universe. The Einstein Telescope (ET), a 3rd-generation gravitational-wave detector (GWD) in Europe, and LIGO Voyager, a substantial upgrade of aLIGO, are planning to employ cryogenic silicon mirrors to reduce thermal noise [4,5]. It is expected that such future cryogenic GWDs will enable us to constrain the formation process of such massive black holes or discover new GW sources [6].
In cryogenic GWDs such as the ET, a molecular layer can be formed on a cryogenic test mass mirror surface. This is caused by continuous molecular transportation from room-temperature beam ducts where residual gas molecules can be easily detached from surfaces [7]. When the gas molecules hit a cryogenic mirror surface, they lose their kinetic energy and adhere onto the surface, which is called the cryopumping effect. Therefore, the continuous collisions of gas molecules create a molecular layer on top of a cryogenic mirror surface, and the thickness of the layer grows over time. We call such a layer a cryogenic molecular layer (CML) in this paper. Figure 1 shows a schematic picture of how a molecular layer is formed on a cryogenic mirror surface.
Fig. 1. Schematic of how the residual gas molecules from the room temperature environment adhere to the cryogenic mirror surface (left), and the formed CML (right).
The ET is planning to employ silicon substrates for $10\, \textrm{K}$ cryogenic test mass because its properties are suited for use at cryogenic temperatures [8,9]. As silicon is opaque for wavelengths shorter than $1100\, \textrm{nm}$ and has small enough absorption only for wavelengths longer than $1400\, \textrm{nm}$, the wavelength of the main laser should be within the range of $1400\, \textrm{nm}$ to $2100\, \textrm{nm}$ [10]. Therefore, the wavelength of the main laser of the ET is chosen to be $1550\, \textrm{nm}$. Amorphous ice, which is assumed to be the major element of CMLs, has a large absorption coefficient above $\sim 1300\, \textrm{nm}$ [11]. A previous study pointed out that such large absorption in CMLs can introduce large heat input to cryogenic mirrors [12].
In order to investigate the impact of CMLs on the optical loss, we developed a $10\, \textrm{K}$ cryogenic folded optical cavity and measured the optical loss induced by CMLs. Using this folded cavity, we developed a new ellipsometric measurement method to measure the thickness of CMLs in situ. We call such an ellipsometric measurement a cavity enhanced ellipsometry (CEE) in this paper. This method is capable of estimating the thickness of a CML with a nanometer level resolution.
In this article, we report on the measurements of optical absorption of CMLs using our folded-cavity setup. First, we introduce the experimental setup. Second, we derive the principle of CEE, and then we show the obtained results. Finally, we discuss the impact of optical loss induced by the CML and possible applications to future cryogenic GWDs.
2. Experimental setup
2.1 Folded cavity
The main component of our setup is a cryogenic folded cavity, which consists of input and output mirrors and one folding mirror [13]. Here, the input and output mirrors are made of fused silica, and the folding mirror is silicon. The parameters of the folded cavity are summarized in Table 1. This configuration enables us to perform cavity ringdown measurements and ellipsometric measurements of molecular layer thickness simultaneously.
Table 1. Parameters of the folded cavity.
Figure 2 shows a schematic of the folded cavity. Both input and output mirrors have the same radius of curvature $\mathrm {RoC}=50\,{\textrm{mm}}$, and the folding mirror is a flat mirror. The cavity length, $L$, becomes
Thus, this folded cavity is a symmetric optical cavity. Each mirror has $\mathrm {SiO_2 / Ta_2O_5}$ dielectric multilayer coatings for high reflectivity. The optical thickness of the coating layers except for the top layer is $\lambda /4$, where $\lambda$ is the wavelength of the laser. The top layer has $\lambda /2$ optical thickness, which is called a $\lambda /2$ cap. The reflectivity of the folding mirror, $R$, is higher than that of the input and output mirrors, $R_1$ and $R_2$. These mirrors are attached to a rigid spacer. The spacer material is an invar, IC-DX developed by Shinhokoku Steel Corporation, which can be used at cryogenic temperatures.
Fig. 2. Schematic of the folded cavity, which is composed of three mirrors with a folding angle $\gamma$.
The resonant condition, and related parameters of the folded cavity are defined in the same manner as the case of a Fabry-Perot cavity [14]. The round trip phase shift of the laser is expressed as
where $\lambda$, $\nu$, $c$ are the laser wavelength, the laser frequency, and the speed of light. The cavity is on resonance when the round trip phase satisfies the following condition, where $n$ is an arbitrary integer. When the cavity is on resonance, the intra-cavity power becomes maximum. The finesse of the folded cavity is defined as2.2 Optical layout
Figure 3 shows the optical layout of this experiment. A commercial laser module (named ORION), built by RIO, is used as the laser source. The wavelength of the laser is $1550\, \textrm{nm}$ and output power is $\sim 20\, \textrm{mW}$. The AOM (Gooch and Housego 3080) is employed in this measurement to interrupt the input beam for cavity ringdown measurements. The input beam can be turned off at a timescale of $\sim 10\, \textrm{ns}$ by shutting down the drive signal into the AOM. The transmitted beam power of the cavity is monitored by a high-bandwidth photodetector. Transient signal of the transmitted power is recorded by a digital oscilloscope, Tektronix DPO 2024B.
Fig. 3. Overview of the optical layout. The spacer of the folded cavity has a slit, which enables the CML formation on the folding mirror surface. The AOM is used to cut the input beam. A half-wave ($\lambda /2$) plate is used to adjust the polarization of the input beam.
A pulse tube cryo-cooler is used for cooling the cavity. A pure aluminum rod is connected to a table in the cryostat chamber from the cold head of the cryo-cooler to cool down the table. The folded cavity is fixed on the table through aluminum pedestals which extract heat from the cavity. The cryostat chamber has three radiation shields; $300\, \textrm{K}$, $80\, \textrm{K}$, and $4\, \textrm{K}$ shields. Two of them except for the outer most shield ($300\, \textrm{K}$ shield) are covered by super insulator (SI) for efficiently shielding the inner components from thermal radiations. Temperature of the folding mirror is monitored by a silicon temperature sensor, DT-670 manufactured by Lake Shore Cryotronics Inc. [15]. Figure 4 shows the cooling curve of the folding mirror. The silicon folding mirror is cooled down to approximately $10\, \textrm{K}$, which is the same temperature as the ET test mass mirror. During the cryogenic operation, the temperature of the folding mirror was highly stable; the temperature fluctuation was less than $0.02\, \textrm{K}$. Therefore, the cavity length fluctuation induced by the temperature fluctuation is assumed to be negligible.
We achieved the vacuum level of $6\times 10^{-4}\, \textrm{Pa}$ with a combination of a turbomolecular pump and a scroll pump at room temperature. The pressure inside the chamber decreases due to the cryopumping effect when cooled down. After reaching about $8\, \textrm{K}$, the pressure of room temperature part, i.e., between the room temperature and $80\, \textrm{K}$ shields, becomes the order of $10^{-5}\, \textrm{Pa}$. The cryogenic part inside the $4\, \textrm{K}$ shield is supposed to have a better vacuum level than that of the room temperature part because once molecules hit a cryogenic wall, most of them are adsorbed to it. Therefore, freely flying molecules contributing to the formation of CMLs are assumed to be mostly coming from the room temperature part of the vacuum chamber. In fact, in a real cryogenic GWDs, a cryogenic mirror is always exposed to a room temperature part through a hole to let a laser beam in and out of the cryostat as shown in Fig. 1. This hole provides a continuous flow of residual gas molecules to the surface of a cryogenic mirror leading to the formation of a CML.
In order to simulate the situation of a real cryogenic GWD, we made a slit in the cavity spacer to expose the folding mirror to the room temperature part with a limited solid angle. The slit is oriented in a way that the input and output mirrors are not exposed to the room temperature part, thus limiting the CML formation to only on the folding mirror.
2.3 Ringdown measurement
Cavity ringdown measurement is a common method to measure the optical loss inside a cavity [16]. Here we consider a steady-state resonant condition. When the input beam is cut by switching off the AOM, the power stored inside the cavity decays over time, resulting in the transmitted beam power to decay. The transient of the transmitted beam power can be written as
where $P_{\mathrm {cav}}$ is the intra-cavity power, and $T_2$ is the transmittance of the output mirror. This transient signal is measured by a photodetector at transmitted port of the folded cavity. By fitting the measured transmitted power change with the above formula, one can obtain the storage time of the cavity $\tau _{\mathrm {storage}}$. The storage time is related to the finesse of the cavity by the following formula,Therefore, one can estimate the finesse of the folded cavity by a ringdown measurement.
The optical properties of a mirror satisfies the conservation of energy as
where $R=r^2$ is the reflectance, $T=t^2$ is the transmittance, and $L$ is the power loss. Then, the reflectance of the mirror can be written asHere we consider the case that the loss of the folding mirror grows over time from the initial loss value $L_0$ due to the formation of the CML. The loss of the folding mirror can be expressed as
where $L_{\mathrm {CML}}$ is the accumulated optical loss induced by the CML. Then, Eq. (5) becomesAs the loss induced by the CML grows with time, the finesse of the folded cavity decreases over time. By monitoring the finesse of the folded cavity for a certain period, one can observe the finesse drop, and estimate the optical loss of growing CML, $L_{\mathrm {CML}}$.
3. Cavity enhanced ellipsometry
Ellipsometry is a technique to investigate the properties of thin films [17]. We adopt this technique to the cryogenic folded cavity to estimate the thickness of the CML. This technique is called a cavity enhanced ellipsometry (CEE) in this article.
3.1 Characteristic matrix
A characteristic matrix is a useful tool to calculate the reflectance or transmittance of a multilayer coating thin film [18]. Not only the reflectance and transmittance, but also the phase shift between the P- and S-polarizations can be computed by using a characteristic matrix. Here we consider a multilayer structure of $N (>0)$ layers. For the $i$-th layer with a complex refractive index $N_i$ and mechanical thickness $d_i$, the characteristic matrix is expressed as
The phase shift induced by a coating on the folding mirror can be calculated from the characteristic matrix. The phase of the reflected light is expressed as
Here $\eta _{\mathrm {m}}$ is the effective refractive index of the substrate material written as
where $N_{\mathrm {m}}$ is the refractive index of the substrate material. We denote the phase difference between the two polarization asWhen the angle of incidence is $0\, \textrm{deg}$, $\eta _i$ is same for both polarization. Therefore, the phase difference does not exist as long as the beam is injected normally incident to the mirror.
3.2 Frequency split
For an S-polarized beam with frequency $\nu$, the round trip phase $\phi _{\mathrm {rtp, S}}$ is given as follows,
This $\phi _{\mathrm {rtp, S}}$ needs to be an integral multiple of $2\pi$ for the S-polarized beam to be resonant in the cavity. For a P-polarized beam, it receives an extra phase of $2\Delta \phi$ during a round trip because of the oblique incidence onto the folding mirror. Therefore, the round trip phase becomes
where $\Delta \nu$ represents the frequency difference between S- and P-polarized beams. Assuming that $\nu$ satisfies the resonant condition for the S-polarized beam, for $\phi _{\mathrm {rtp, P}}$ to be also an integer multiple of $2\pi$, $\Delta \nu$ must satisfy the following condition,This expression gives us the amount of resonant frequency split between S- and P-polarizations. Figure 5 shows an example of resonant frequency split.
Fig. 5. Example of the resonant frequency spit between S-polarization (solid curve) and P-polarization (dashed curve) . Here we assumed $\Delta \phi =10\, \textrm{mrad}$ phase shift, $\mathscr {F}\sim 1.5\times 10^{4}$, and $L=82\,{\textrm{mm}}$ case.
3.3 Drift of the resonant frequency split
In this experiment, the folding mirror is exposed to the room temperature vacuum as shown in Fig. 3. Therefore, the molecular layer is formed on the folding mirror, resulting in a change in the characteristic matrix of the folding mirror as
where $M_{\mathrm {CML}}$ is the characteristic matrix of the CML defined asSuch a change in the characteristic matrix leads to a drift in the phase shift, $\Delta \phi$, hence the resonant frequency split, $\Delta \nu$.
The frequency split becomes function of the thickness of the CML, $\Delta \nu (d_{\mathrm {CML}})$. Figure 6 shows how the frequency split changes with the thickness of the CML. In this calculation, we assumed the refractive indices of $\mathrm {SiO_2}$, and $\mathrm {Ta_2O_5}$ to be $n_{\mathrm {SiO_2}}=1.44$ and $n_{\mathrm {Ta_2O_5}}=2.2$, respectively [19], and the number of $\mathrm {SiO_2 / Ta_2O_5}$ pairs is set to $N=20$. When the thickness of the CML is much thinner than $\lambda /4$, the frequency split changes linearly with the thickness. Within this region, the relation between the frequency split and the thickness can be approximated as
which is shown in Fig. 6 as a dashed line. The offset, $-20.28\, \textrm{MHz}$, is induced by the $\mathrm {SiO_2 / Ta_2O_5}$ layers. The actually measured frequency split at room temperature, i.e., $d_{\mathrm {CML}}=0$, was $-20.29\pm 0.24\, \textrm{MHz}$, which is consistent with this model.Fig. 6. How the resonant frequency split changes over the thickness of the CML (solid line), and linearly approximated line (dashed line).
It should be noted that the refractive index of the CML is assumed to be $n_{\mathrm {CML}}=1.2$ in this calculation. A previous study reported that the measured refractive index was $n_{\mathrm {CML}}=1.26$ at $1064\, \textrm{nm}$ [7]. Using the Lorentz-Lorenz equation, we estimated the refractive index of CML at $1550\, \textrm{nm}$ to be about 1.2 in the same manner as the Ref. [12].
In this experiment, the resonant frequency split was measured by using the transmitted beam. The input laser frequency is scanned by using a current tuning mechanism of the laser module, and a frequency to maximize the transmitted power is searched to find a resonant frequency of the cavity. The polarization of the input beam can be changed with the half-wave plate shown in Fig. 3. By adjusting the half-wave plate, we made sure that both P- and S-polarized beams are injected to the cavity simultaneously. Under this condition, we can observe two neighboring peaks in the transmitted power as shown in Fig. 5. By measuring the separation between the two peaks, the amount of resonant frequency split can be obtained.
4. Results and discussion
We monitored the finesse degradation and the drift of the resonant frequency split for about $10$ days by the ringdown and the CEE measurements. The cavity ringdown was performed with an S-polarization beam. Figure 7 shows the results of these measurements. The finesse monotonically decreases during the cryogenic operation period, and recovered after raising the temperature back to room temperature. The amount of the resonant frequency split also changes over time at cryogenic temperature, and recovered after raising the temperature. Therefore, these changes in the finesse and the resonant frequency split are considered to be induced by the formation of a CML.
From Eqs. (11) and (26), one can compute the accumulated optical loss, and thickness of the CML, $L_{\mathrm {CML}}$ and $d_{\mathrm {CML}}$, respectively. Figure 8 shows the growing optical loss $L_{\mathrm {CML}}$, and thickness of the CML $d_{\mathrm {CML}}$. The loss of $7.3\pm 1.1\, \textrm{ppm}$, measured on the first day, is generated by the adsorption of molecules during the cooling down phase. In the following $10\, \textrm{days}$, the optical loss and the thickness of the CML grew over time, and became $L_{\mathrm {CML}} = 18.1\pm 2.0\, \textrm{ppm}$ and $d_{\mathrm {CML}}=4.13\pm 0.31\, \textrm{nm}$ at the end. After raising the temperature, the CML was removed, and the optical loss induced by the CML became zero.
Fig. 8. Accumulated loss, $L_{\mathrm {CML}}$ (blue circle), and thickness, $d_{\mathrm {CML}}$ (red triangle), of the CML.
We estimate the absorption coefficient of the CML, $\alpha _{\mathrm {CML}}$, based on the obtained results. The Lambert-Beer law is assumed to calculate $\alpha _{\mathrm {CML}}$. In addition, we assume that the optical loss induced by the scattering from a CML is negligible based on the previous study [12], i.e., the obtained optical loss in this experiment is assumed to be induced only by the optical absorption. The heat input induced by the CML, $Q_{\mathrm {CML}}$, is expressed by the Lambert-Beer law as
where $P_{\mathrm {CML}}$ is the laser power impinging on to the CML, $\alpha _{\mathrm {CML}}$ is the absorption coefficient, and $d_{\mathrm {CML}}$ is the thickness of the CML. When the CML is very thin, Eq. (27) is approximated asThe heat input scales with the absorption coefficient, $\alpha _{\mathrm {CML}}$. Therefore, the optical loss induced by a thin CML becomes
The following function is assumted as a fitting curve to estimate the absorption coefficient,
where $\beta$ is an offset. Here, $d_{\mathrm {eff}}=d_{\mathrm {CML}}/\cos \gamma '$ is the effective path length induced by the folding angle $\gamma$. The angle of incidence, $\gamma '$, is calculated by Snell’s law asThe red dashed line shown in Fig. 9 represents the results of the fitting by assuming Eq. (30), and shaded band represents the $1\sigma$ uncertainty. The obtained parameters from the fitting are $\alpha _{\mathrm {CML}} = 2.7 \pm 0.2\, \textrm{ppm/nm}$ and $\beta =-6.3\pm 1.2\, \textrm{ppm}$.
Fig. 9. Measured data and a fitted curve. Red dashed line is the fitted curve and shaded band represents the $1\sigma$ uncertainty.
Lastly, we estimate a heat input to a test mass for the case of the ET based on the obtained results. Here, we assume that the power inside an arm-cavity of the ET is $18\, \textrm{kW}$ and constant as long as the thickness of the CML is very thin ($< 10\, \textrm{nm}$). In addition, the optical loss induced by the scattering from a CML is assumed to be negligible. Figure 10 shows projected, and theoretical estimates of heat input to the test mass induced by the CML. The black dashed line represents a theoretical prediction based on the previous study where the absorption coefficient is assumed to be $\alpha =2.0\, \textrm{ppm/nm}$ [12].
Fig. 10. Heat input to the test mass for the case of the ET. Black solid line shows the theoretical expectation of heat input [12]. Blue dashed-dotted line represents the cooling capacity [4]. Red dashed line is the fitting curve and shaded band is $1\sigma$ uncertainty.
While there is a slight difference between the theoretical prediction and the experimental result, both the experimental and theoretical estimates exceed the cooling capacity of the mirror, $\sim 100\, \textrm{mW}$, when the thickness is above $2\, \textrm{nm}$. Therefore, the tolerance of the ET for the thickness of CML is only $2\, \textrm{nm}$.
The slope of the experimental estimate is steeper than the theoretical prediction in which the CML is assumed to be composed of amorphous ice. This implies that the absorption coefficient of the actual CML in this experiment is larger. In practice, not only water molecules, but also other molecular species are adsorbed by the cryogenic mirror, and it is not surprising to have a different absorption coefficient from pure amorphous ice. Therefore, the CML formed in our experimental setup has a larger absorption coefficient than the literature value of amorphous ice absorption and the tolerance on its thickness may be severer. A simple extrapolation of the experimental data does not cross zero at thickness $d_{\mathrm {CML}}=0$, corresponding to $\beta$ of Eq. (30) being non-zero, which is unphysical. A possible explanation of this is that the Lambert-Beer law does not hold for very thin layers and the absorption coefficient becomes smaller near zero thickness, deviating from the linear fit. Further study is necessary on this issue.
The methods and device developed in this study can be useful for the development of future cryogenic GWDs. For example, in situ measurement of the CML thickness and loss is one possible application of this device. By forming a folded cavity with a test mass mirror as the folding mirror and two auxiliary mirrors as input and output mirrors, we may be able to monitor the growth of a CML on the test mass through optical loss and CEE measurements. It can provide not only more realistic optical loss value than the tabletop measurements, but also the actual molecular layer formation rate. Such in situ measurement in a prototype interferometer may be helpful in optimizing the design of cryogenic systems such as the length and diameter of cryogenic beam ducts to limit the solid angle of room temperature part exposed to a cryogenic mirror. Development of a molecular layer desorption system may also be necessary if the formation rate of CMLs is too large. A laser induced desorption system has been proposed to solve the problem of the CML [20]. The experimental setup developed in this study can be used as a test bench for such a desorption system.
5. Conclusion
Cryogenic GWD is a promising way to enhance the GW astronomy by increasing the number of detectable GW events. A cryogenic mirror, however, can suffer from a technical problem caused by adsorption of residual gas molecules, which leads to a large heat absorption and can hinder cryogenic operation. In order to experimentally evaluate the impacts of the optical loss induced by CMLs on future cryogenic GWDs, we developed a cryogenic folded cavity setup and an ellipsometric measurement scheme (CEE). By combining the ringdown measurement and the CEE measurement, we characterized the thickness and the optical loss of CMLs formed in our folded cavity. The optical absorption induced by the CML shows a large value even at a thickness of several nanometers, which can exceed the cooling capacity of the ET. Therefore, the CML can potentially prevent the cryogenic operation of future GWDs. The cryogenic folded cavity and the CEE method developed in this study can be used for future R&D experiments of cryogenic systems such as a test bench for a CML desorption system, or in situ measurements of CML in cryogenic GWDs.
Funding
Japan Society for the Promotion of Science (26000005, JP18K03681); Research Fund for Students (2017) of the Department of Astronomical Science, The Graduate University for Advanced Studies, SOKENDAI.
Acknowledgments
The authors appreciate Shinhokoku Steel Corporation for developing an invar which enabled us to develop a cryogenic folded cavity.
Disclosures
The authors declare no conflicts of interest.
References
1. LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016). [CrossRef]
2. LIGO Scientific Collaboration and Virgo Collaboration, “Multi-messenger observations of a binary neutron star merger,” Astrophys. J., Lett. 848(2), L12 (2017). [CrossRef]
3. LIGO Scientific Collaboration and Virgo Collaboration, “Gw190521: A binary black hole merger with a total mass of $150\,M_{\ensuremath {{\bigodot} }}$,” Phys. Rev. Lett. 125(10), 101102 (2020). [CrossRef]
4. M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Andersson, K. Arun, F. Barone, B. Barr, M. Barsuglia, M. Beker, N. Beveridge, S. Birindelli, S. Bose, L. Bosi, S. Braccini, C. Bradaschia, T. Bulik, E. Calloni, G. Cella, E. C. Mottin, S. Chelkowski, A. Chincarini, J. Clark, E. Coccia, C. Colacino, J. Colas, A. Cumming, L. Cunningham, E. Cuoco, S. Danilishin, K. Danzmann, G. D. Luca, R. D. Salvo, T. Dent, R. D. Rosa, L. D. Fiore, A. D. Virgilio, M. Doets, V. Fafone, P. Falferi, R. Flaminio, J. Franc, F. Frasconi, A. Freise, P. Fulda, J. Gair, G. Gemme, A. Gennai, A. Giazotto, K. Glampedakis, M. Granata, H. Grote, G. Guidi, G. Hammond, M. Hannam, J. Harms, D. Heinert, M. Hendry, I. Heng, E. Hennes, S. Hild, J. Hough, S. Husa, S. Huttner, G. Jones, F. Khalili, K. Kokeyama, K. Kokkotas, B. Krishnan, M. Lorenzini, H. Luck, E. Majorana, I. Mandel, V. Mandic, I. Martin, C. Michel, Y. Minenkov, N. Morgado, S. Mosca, B. Mours, H. Muller-Ebhardt, P. Murray, R. Nawrodt, J. Nelson, R. Oshaughnessy, C. D. Ott, C. Palomba, A. Paoli, G. Parguez, A. Pasqualetti, R. Passaquieti, D. Passuello, L. Pinard, R. Poggiani, P. Popolizio, M. Prato, P. Puppo, D. Rabeling, P. Rapagnani, J. Read, T. Regimbau, H. Rehbein, S. Reid, L. Rezzolla, F. Ricci, F. Richard, A. Rocchi, S. Rowan, A. Radiger, B. Sassolas, B. Sathyaprakash, R. Schnabel, C. Schwarz, P. Seidel, A. Sintes, K. Somiya, F. Speirits, K. Strain, S. Strigin, P. Sutton, S. Tarabrin, A. Thering, J. van den Brand, C. van Leewen, M. van Veggel, C. van den Broeck, A. Vecchio, J. Veitch, F. Vetrano, A. Vicere, S. Vyatchanin, B. Willke, G. Woan, P. Wolfango, and K. Yamamoto, “The einstein telescope: a third-generation gravitational wave observatory,” Classical Quantum Gravity 27(19), 194002 (2010). [CrossRef]
5. R. X. Adhikari, K. Arai, A. F. Brooks, C. Wipf, O. Aguiar, P. Altin, B. Barr, L. Barsotti, R. Bassiri, A. Bell, G. Billingsley, R. Birney, D. Blair, E. Bonilla, J. Briggs, D. D. Brown, R. Byer, H. Cao, M. Constancio, S. Cooper, T. Corbitt, D. Coyne, A. Cumming, E. Daw, R. deRosa, G. Eddolls, J. Eichholz, M. Evans, M. Fejer, E. C. Ferreira, A. Freise, V. V. Frolov, S. Gras, A. Green, H. Grote, E. Gustafson, E. D. Hall, G. Hammond, J. Harms, G. Harry, K. Haughian, D. Heinert, M. Heintze, F. Hellman, J. Hennig, M. Hennig, S. Hild, J. Hough, W. Johnson, B. Kamai, D. Kapasi, K. Komori, D. Koptsov, M. Korobko, W. Z. Korth, K. Kuns, B. Lantz, S. Leavey, F. Magana-Sandoval, G. Mansell, A. Markosyan, A. Markowitz, I. Martin, R. Martin, D. Martynov, D. E. McClelland, G. McGhee, T. McRae, J. Mills, V. Mitrofanov, M. Molina-Ruiz, C. Mow-Lowry, J. Munch, P. Murray, S. Ng, M. A. Okada, D. J. Ottaway, L. Prokhorov, V. Quetschke, S. Reid, D. Reitze, J. Richardson, R. Robie, I. Romero-Shaw, R. Route, S. Rowan, R. Schnabel, M. Schneewind, F. Seifert, D. Shaddock, B. Shapiro, D. Shoemaker, A. S. Silva, B. Slagmolen, J. Smith, N. Smith, J. Steinlechner, K. Strain, D. Taira, S. Tait, D. Tanner, Z. Tornasi, C. Torrie, M. V. Veggel, J. Vanheijningen, P. Veitch, A. Wade, G. Wallace, R. Ward, R. Weiss, P. Wessels, B. Willke, H. Yamamoto, M. J. Yap, and C. Zhao, “A cryogenic silicon interferometer for gravitational-wave detection,” Classical Quantum Gravity 37(16), 165003 (2020). [CrossRef]
6. LIGO Scientific Collaboration and Virgo Collaboration, “Binary black hole population properties inferred from the first and second observing runs of advanced LIGO and advanced virgo,” Astrophys. J., Lett. 882(2), L24 (2019). [CrossRef]
7. K. Hasegawa, T. Akutsu, N. Kimura, Y. Saito, T. Suzuki, T. Tomaru, A. Ueda, and S. Miyoki, “Molecular adsorbed layer formation on cooled mirrors and its impacts on cryogenic gravitational wave telescopes,” Phys. Rev. D 99(2), 022003 (2019). [CrossRef]
8. R. Nawrodt, A. Zimmer, T. Koettig, C. Schwarz, D. Heinert, M. Hudl, R. Neubert, M. Therk, S. Nietzsche, W. Vodel, P. Seidel, and A. Tunnermann, “High mechanical q-factor measurements on silicon bulk samples,” J. Phys.: Conf. Ser. 122, 012008 (2008). [CrossRef]
9. C. A. Swenson, “Recommended values for the thermal expansivity of silicon from 0 to 1000 k,” J. Phys. Chem. Ref. Data 12(2), 179–182 (1983). [CrossRef]
10. M. J. Keevers and M. A. Green, “Absorption edge of silicon from solar cell spectral response measurements,” Appl. Phys. Lett. 66(2), 174–176 (1995). [CrossRef]
11. B. Schmitt, E. Quirico, F. Trotta, and W. M. Grundy, Optical Properties of Ices From UV to Infrared (Springer, 1998), pp. 199–240.
12. S. Tanioka, K. Hasegawa, and Y. Aso, “Optical loss study of molecular layer for a cryogenic interferometric gravitational-wave detector,” Phys. Rev. D 102(2), 022009 (2020). [CrossRef]
13. S. Gras, H. Yu, W. Yam, D. Martynov, and M. Evans, “Audio-band coating thermal noise measurement for advanced ligo with a multimode optical resonator,” Phys. Rev. D 95(2), 022001 (2017). [CrossRef]
14. A. Freise and K. Strain, “Interferometer techniques for gravitational-wave detection,” Living Rev. Relativ. 13(1), 1 (2010). [CrossRef]
15. S. Courts, “One year stability of Cernox and DT-670-SD silicon diode cryogenic temperature sensors operated at 77 K,” Cryogenics 107, 103050 (2020). [CrossRef]
16. T. Isogai, J. Miller, P. Kwee, L. Barsotti, and M. Evans, “Loss in long-storage-time optical cavities,” Opt. Express 21(24), 30114–30125 (2013). [CrossRef]
17. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).
18. H. MacLeod, Thin-Film Optical Filters (CRC, 2010).
19. J. Steinlechner, I. W. Martin, J. Hough, C. Krüger, S. Rowan, and R. Schnabel, “Thermal noise reduction and absorption optimization via multimaterial coatings,” Phys. Rev. D 91(4), 042001 (2015). [CrossRef]
20. K. Hasegawa, “Optical and thermal study of molecular thin layers on cryogenic mirrors in next-generation gravitational wave telescopes,” Ph.D. thesis, The University of Tokyo (2020).
