Amorphous optical coatings of present gravitational-wave interferometers^*

We measured the film refractive index and thickness, by transmission spectrophotometry through coated fused-silica witness samples and by reflection spectroscopic ellipsometry on coated silicon wafers.

Spectrophotometric measurements have been carried out with a Perkin Elmer Lambda 1050 spectrophotometer. Spectra have been acquired at normal incidence in the 400–1400 nm range. Film refractive index and thickness were first evaluated using the envelope method [26], then these results were used as initial values in a numerical least-square regression analysis. In the model, the adjustable parameters were the thickness and the (B_i, C_i) coefficients of the Sellmeier dispersion equation:

$$\begin{equation}{n}^{2}=1+\sum _{i=1}^{3}\frac{{B}_{i}{\lambda }^{2}}{{\lambda }^{2}-{C}_{i}}.\end{equation} \tag{ 1 }$$

We used used two J.A. Woollam Co. instruments for the ellipsometric analysis, a VASE for the 190–1100 nm range and an M-2000 for the 245–1680 nm range. The wide wavelength range swept with both ellipsometers allowed us to extend the analysis from ultraviolet to infrared (0.7–6.5 eV). The optical response of the substrates has been characterized with prior dedicated measurements. To maximize the response of the instruments, coating spectra have been acquired for three different angles of incidence of the light (θしーた = 50°, 55°, 60°), chosen to be close to the Brewster angle of each coating material (θしーた_B ∼ 55° for silica and ∼64° for tantala, for instance). The refractive index and thickness of the films have been derived by comparing the experimental data with simulations based on realistic optical models [27] describing the properties of amorphous oxides, the Tauc–Lorentz [28] and Cody–Lorentz [29] models, providing a Kramers–Kronig-consistent determination of the complex refractive index. More details about our ellipsometric analysis can be found in a dedicated article [30].

Finally, we measured the coating optical absorption at λらむだ₀ = 1064 nm through photo-thermal deflection [31].

To measure the coating mass, we have used an analytical balance and measured the mass of the disks before and after the deposition, as well as after the annealing. To estimate the surface area coated in the deposition process, we have measured the diameter and the flat spacing of the disks with a Vernier caliper. As the coating thickness was known with high accuracy from the optical characterization, the coating density ρろー_c could be straightforwardly estimated as the mass-to-volume ratio.

To measure the coating loss, we have applied the ring-down method [32] to the disks and measured the ring-down time of their vibrational modes. In each sample, for the kth mode of frequency f_k and ring-down time τたう_k, the measured loss is ${\phi }_{k}={\left(\pi {f}_{k}{\tau }_{k}\right)}^{-1}$. The coating loss ${\phi }_{k}^{\mathrm{c}}$ can be written

$$\begin{equation}{\phi }_{k}^{\mathrm{c}}=\left[\;{\phi }_{k}+\left({D}_{k}-1\right){\phi }_{k}^{\mathrm{s}}\;\right]/{D}_{k},\end{equation} \tag{ 2 }$$

where ${\phi }_{k}^{\mathrm{s}}$ is the measured loss of the bare substrate. D_k is the dilution factor, defined as the ratio of the elastic energy of the coating, E_c, to the elastic energy of the coated disk, E = E_c + E_s, where E_s is the elastic energy of the substrate. This ratio actually depends on the mode shape, which in turn is determined by the number of r radial and a azimuthal nodes [33], r and a respectively, denoted by the pair (r,a)_k. D_k is thus mode-dependent and can be written as a function of the frequencies ${f}_{k}^{\mathrm{s}}$, f_k and of the masses m^s, m of the sample before and after the coating deposition, respectively [34]:

$$\begin{equation}{D}_{k}=1-{\left(\frac{{f}_{k}^{\mathrm{s}}}{{f}_{k}}\right)}^{2}\frac{{m}^{\mathrm{s}}}{m}.\end{equation} \tag{ 3 }$$

We have used a gentle nodal suspension (GeNS) [35] to suspend the disks from the center, in order to avoid systematic damping from suspension. The system was placed inside a vacuum enclosure at p ⩽ 10⁻⁶ mbar to prevent residual-gas damping. We have measured up to 16 modes on each disk, sampling the coating loss in the 1–30 kHzきろへるつ band. This sampling partially overlaps with the detection band of ground-based gravitational-wave interferometers (10–10⁴ Hz).

For each coated disk, we have estimated the Young's modulus Y_c and the Poisson's ratio νにゅー_c of the coating materials by adjusting finite-element simulations to match the measured dilution factors from equation (3): we found the set of values (Y_c, νにゅー_c) minimizing the least-squares figure of merit

$$\begin{equation}{m}_{D}=\sum _{k}{\left[\frac{{D}_{k}^{\text{meas}}-{D}_{k}^{\text{sim}}}{{\sigma }_{k}^{\text{meas}}}\right]}^{2},\end{equation} \tag{ 4 }$$

where ${D}_{k}^{\text{sim}}$ and (${D}_{k}^{\text{meas}}{\pm}{\sigma }_{k}^{\text{meas}}$) are the simulated and measured dilution factors, respectively (${\sigma }_{k}^{\text{meas}}$ is the measurement uncertainty). In this method, knowledge of the substrate parameters is critical: dimensions have been assigned measured values, values of density (ρろー = 2202 g cm⁻³), Young's modulus (Y = 73.2 GPa) and Poisson's ratio (νにゅー = 0.17) have been taken from the literature [36]; in a dedicated subset of simulations of the bare substrates, thickness t has been adjusted to minimize a merit function m_t of the same form as equation (4) for simulated and measured mode frequencies (${f}_{k}^{\text{sim}}$ and ${f}_{k}^{\text{meas}}$, respectively). For a few disks, we independently measured t with a micrometer and found that the discrepancy with the fitted values is less than 2%. Figure 1 shows the results of the overall process, comparing measured and least-squares simulated dilution factors of representative samples of tantala and silica. Dilution factors of modes with a different number of radial nodes lie on distinct curves, which we have called mode families; thus, for example, modes with (0,a)_k and modes with (1,a)_k belong to two different families (called butterfly and mixed modes, respectively). For a given substrate with (Y, νにゅー), dilution factors are determined by the coating elastic constants: their average value depends on the Y_c/Y ratio, as in the case of flexural modes of cantilevers [37], whereas the separation between mode families increases as the difference |νにゅー_c − νにゅー| grows.

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Further details about our GeNS system and finite-element simulations are available elsewhere [38].

Before annealing, the tantala films had similar refractive index (table 3) and equal Young's modulus and Poisson's ratio, whereas their density and loss were different (table 4 and figure 3).

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Interestingly, the loss seems to be related to the coating deposition rate, determined by the energy and the flux of particles of the sputtering beam and by the configuration of the coating chamber. Whereas we found no correlation between the loss values and the distances of targets and substrates, the GC provided the slowest rate and lowest loss, the SPECTOR the fastest rate (3 Å s⁻¹) and the highest loss; thus, according to our results, the faster the deposition rate, the higher the loss.

The situation changed radically after the annealing, when all the films exhibited equal and significantly lower loss, as if their deposition history had been completely erased. This outcome seems to suggest that 500 °C in-air annealing for 10 h brings the structure of tantala coatings down to a stable optimal configuration for lowest loss, in agreement with observations that higher annealing temperatures or longer duration do not decrease loss further [43]. The coating structure depends on the chemical composition of the sputtered particles and on their energy distribution and electrical charge, however so far there is no model available nor there are any measurements for any of our coaters about these data. A research project has been recently funded to address this issue (project ViSIONs, Grant No. ANR-18-CE08-0023-01 of the French Government operated by the National Research Agency, ANR). The same erasing effect was observed later on in an independent experiment, where sputtered tantala coatings (IBS, magnetron) had been annealed after being deposited on heated substrates [44].

Finally, refractive index and density also slightly decreased after annealing, as the physical thickness of all the films increased.

By now, there have been many experimental studies on the internal friction of tantala coatings [18, 20, 23, 25, 34, 45–47]. However, the comparison with our results is not straightforward: previous works used coatings deposited with different conditions [25, 45] and/or stacked with other layers [18, 25, 46]; moreover, their analyses relied on assumptions made on the value of the coating Young's modulus [18, 20, 23, 34, 45–47]. Probably, the fairest comparison would be with annealed tantala coatings deposited with the GC on cantilever blades [20] under identical conditions, featuring a constant loss of (3.0 ± 0.1) × 10⁻⁴ rad in the 60–1100 Hzへるつ band; by extrapolating our results of the annealed samples to lower frequencies, we obtain (2.8 ± 0.1) × 10⁻⁴ rad at 60 Hzへるつ and (3.8 ± 0.2) × 10⁻⁴ rad at 1100 Hzへるつ, matching the value from cantilever blades at 110 Hzへるつ. By correcting the cantilever blades results by our value of coating Young's modulus we would obtain 3.9 × 10⁻⁴ rad, so that the match with our results would be shifted to 1260 Hzへるつ. At 8.9 kHzきろへるつ, our results also match the value of (4.7 ± 0.6) × 10⁻⁴ rad from quadrature-phase interferometry [34] on our SPECTOR films, obtained assuming constant loss but with no further assumptions on the coating Young's modulus; as a matter of fact, those results already pointed to an actual Young's modulus of 118 GPa, very close to our value (121 ± 2 GPa).

If measured via nano-indentation, the Young's modulus of IBS tantala coatings appears to be about 140 GPa [48, 49], i.e. about 18% higher than our value. This difference could be explained by the nature of the films, deposited with different conditions, and by the fact that results from nano-indentation are model dependent and rely on assumptions about the coating Poisson's ratio. Furthermore, nano-indentations of the same coating deposited on different substrates might give different results: our tantala SPECTOR coatings yielded a reduced coating Young's modulus of 130 ± 3 GPa on silica witness samples and of 100 ± 3 GPa on silicon wafers.

Titania features a very high refractive index (n ∼ 2.3, table 5), making it particularly well suited to increase the index contrast within the HR stack and thus consequently to allow for thinner high-index dissipative layers. Unfortunately, it becomes poly-crystalline when annealed at T ⩾ 300 °C [50, 51], yielding scattering and absorption losses far from meeting the stringent requirements of current gravitational-wave detectors [10]. Because of crystallization, we could not measure its refractive index and some mechanical parameters after annealing.

Before annealing, the density of our film (table 6) is slightly higher than that of typical sputtered titania films [52] and close to that of the crystalline anatase phase (3.9 g cm⁻³). The loss increases significantly after the annealing, very likely due to crystallization (figure 4).

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The titania content in HR coatings for gravitational-wave detectors was initially determined by analyzing a set of Bragg reflectors with λらむだ/4 layers [22], produced in the DIBS and in the GC with different titania-to-tantala mixing ratios; Ti/Ta = 0.27 yielded minimum loss and had thus been chosen as the optimal ratio. Since then, the design of the HR coatings of Advanced LIGO and Advanced Virgo has evolved [10, 38], while the Ti/Ta ratio in the titania-doped tantala layers remained the same. This choice is now confirmed by our latest loss measurements of single titania-doped tantala films produced with the GC, which we also characterized through Rutherford back-scattering (RBS) and energy-dispersive x-ray (EDX) spectrometry (tables 7 and 8 and figures 5 and 6).

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By comparing Ti/Ta = 0.27 titania-doped tantala to undoped GC tantala, we observe that both coating materials feature similar loss before annealing, whereas after annealing the loss of titania-doped tantala is ∼25% lower in the whole sampled band, suggesting that the lower loss of titania-doped tantala is the result of a combined effect of mixing and annealing. The ∼25% loss reduction has been confirmed by independent coating thermal noise measurements [53]. Except for the sample with Ti/Ta = 0.04, all the annealed titania-doped tantala coatings showed a lower loss than that of annealed undoped tantala; this is in fairly good agreement with molecular dynamics simulations showing that even small amounts of doping could decrease the coating internal friction [54].

Several results of mechanical loss measurements are already available for titania-doped tantala coatings [20, 21, 23, 25, 38, 46, 55, 56], and a correlation between coating internal friction and microscopic structure has also been found [57].

The fairest comparison with our results would be with annealed titania-doped tantala coatings deposited with the GC on cantilever blades [38] under identical conditions (deposition rate, Ti/Ta ratio), featuring a constant coating loss of (2.4 ± 0.3) × 10⁻⁴ rad in the 50–900 Hzへるつ band; by extrapolating our results of the annealed samples to lower frequencies, we obtain (2.2 ± 0.1) × 10⁻⁴ rad at 50 Hzへるつ and (3.0 ± 0.2) × 10⁻⁴ rad at 900 Hzへるつ, matching the value from cantilever blades at 120 Hzへるつ. By correcting the cantilever blade results by our value of coating Young's modulus we would obtain 3.01 × 10⁻⁴ rad, so that the match with our results would be shifted to 920 Hzへるつ.

The refractive index of the Ta₂O₅–TiO₂ coating with Ti/Ta = 0.27 is in between those of tantala and titania, as expected [50]. Its optical absorption is lower than that of tantala, as we could measure on single films as well as on the HR coatings of Advanced LIGO and Advanced Virgo [10]; this in agreement with our previous measurements [23] and with previous studies of coatings deposited in similar conditions [50], showing that the lowest absorption is obtained with low titania content.

The annealing increased the physical thickness of the coating, which in turn resulted in a decrease of density and of refractive index, as demonstrated by previous studies as well [50]. Finally, we observed that the crystallization temperature is the same of that of tantala, i.e. between 600 °C and 650 °C [43], and hence much higher than that of titania [50, 51].

During the mechanical characterization of silica coatings, we observed a branching of the coating loss as a function of mode shape, closely following a separation in mode families (figure 7). This loss branching might be caused by a spurious contribution coming from the edge of the disks [39], which was also coated during deposition (masking the edge would have induced uncontrolled, undesired shadowing effects on the main surfaces). Such edge effect might be included in the frequency-dependent coating loss model by introducing an additional term dϕ_e, where ϕ_e is a constant edge energy loss, is a mode-dependent dilution factor linear density (with dimensions of m⁻¹) and d is the effective thickness of the coating deposited at the disk edge, so that the coating loss of silica can be written

$$\begin{equation}{\phi }_{\mathrm{c}}\left(f\right)=a{f}^{b}+{\epsilon}d{\phi }_{\mathrm{e}}.\end{equation} \tag{ 5 }$$

The dilution factor density is calculated for a homogeneous disk as the ratio of the elastic energy per unit length of the edge to the total elastic energy of the disk. increases as the mode order (r,a)_k grows but is substantially larger for butterfly modes which mainly vibrate at the edge, so that in the end their actual loss is larger. More details about the formal definition of an edge term dϕ_e are available elsewhere [39].

If neglected, the edge effect could lead to a poor estimation of coating loss; this is particularly evident for the GC and DIBS samples (tables 9 to 11), for which the af^b term is hidden by the overall measured trend. When considering the af^b term only, the loss of the GC and the DIBS samples appeared fairly constant (b ∼ 0) before annealing, whereas the loss of the SPECTOR sample showed a weak decreasing trend (b < 0); after the annealing, the loss values of the SPECTOR and GC samples moved closer to those of the DIBS sample but kept their distinctive features. Indeed, once again, we observed that the SPECTOR sample had the fastest deposition rate (2 Å s⁻¹) and the highest loss values; however, while having the same deposition rate (within 25% experimental uncertainty) than the GC sample, the DIBS sample had the lowest loss values. This result might be related to the unusually small measured density value for the DIBS sample, and will be subject to further investigation.

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Previous studies have been made of the loss [18, 20, 25, 38, 46, 58–60] and the elastic constants [48] of silica coatings. Recently we found a correlation between loss and microscopic structure, which holds for the silica coatings discussed here and for bulk fused silica as well [61].

We may compare our results with those obtained from clamped cantilever blades [38] and free-standing micro-cantilevers made of coatings [60], all produced in the GC under identical conditions. The cantilever blades yielded a constant coating loss of (4.5 ± 0.3) × 10⁻⁵ rad in the 50–900 Hzへるつ band; by considering the af^b term only and extrapolating the GeNS results of the annealed sample to lower frequencies, we obtain (2.2 ± 0.7) × 10⁻⁵ rad at 50 Hzへるつ and (2.4 ± 1.0) × 10⁻⁵ rad at 900 Hzへるつ, eventually matching the value from cantilever blades only at very high frequency. In other words, there is an irreconcilable discrepancy between our latest results obtained with disks and a GeNS system and our previous results obtained with clamped cantilever blades. One reason for this discrepancy may be that the loss measured with clamped cantilever blades was affected by spurious suspension losses (in the clamp and/or in the weld [38]), yielding to its systematic overestimation. Another reason may be that the edge effect was not included in the analysis of the cantilevers, as it was unknown at that time. In principle, the edge effect might have affected the measurement of clamped cantilevers as well; however, it would have been hardly possible to isolate it anyway, as for cantilevers has the same value for all the flexural modes.

The free-standing micro-cantilevers had been measured before annealing, yielding a variable loss of (0.8–1.8) × 10⁻³ rad in the 1.2–21.8 kHzきろへるつ band [60]; again, by considering the af^b term only and extrapolating our GeNS results of the as-deposited sample to the same band, we obtain (1.6 ± 0.5) × 10⁻⁴ rad at 1.2 kHzきろへるつ and (1.7 ± 0.5) × 10⁻⁴ rad at 21.8 kHzきろへるつ, in clear disagreement with the results from free-standing micro-cantilevers. A likely explanation for this disagreement might be that the fabrication process of micro-cantilevers (with its steps of mask deposition, photolithography patterning and repeated etching [60]) may had changed the properties of the coating material they were made of. Otherwise, since in free-standing cantilevers the intrinsic stress is relieved (they only have a stress gradient along the thickness), this might be the first evidence of stress-dependent loss.

Remarkably, the SPECTOR sample is significantly denser and stiffer than the GC one, whose properties closely resemble those of bulk fused silica [36] (table 10).

The reference loss values of the Advanced LIGO and Advanced Virgo input (ITM) and end (ETM) mirror HR coatings (table 12) had been previously estimated [38] by assuming Y_c = 140 GPa for titania-doped tantala layers [49]. These values may now be updated by using our measured value of Young's modulus of titania-doped tantala layers, Y_c = 120 GPa: the new estimations, listed in table 13, are about 10% higher than the previous ones [38] for both ITM and ETM coatings.

Table 13. Mechanical loss of Advanced LIGO and Advanced Virgo input (ITM) and end (ETM) mirror HR coatings. (r,a)_k is the pair denoting the kth mode with r radial and a azimuthal nodes, ${\phi }_{k}^{\mathrm{c}}$ is the coating loss of the kth mode defined in equation (2).

	f (Hz)	(r,a)k	${\phi }_{k}^{\mathrm{c}}$ (10−4 rad)	a (10−4 rad Hz−b)	b
ITM	2708.1	(0,2)1	1.6 ± 0.1	1.1 ± 0.3	0.05 ± 0.03
	16 092.6	(0,5)10	1.7 ± 0.1
	16 283.9	(1,2)12	1.8 ± 0.1
	22 423.4	(0,6)15	1.7 ± 0.1
ETM	2708.6	(0,2)1	2.4 ± 0.1	2.2 ± 0.6	0.01 ± 0.03
	6168.3	(0,3)4	2.3 ± 0.1
	16 088.1	(0,5)10	2.5 ± 0.1
	16 297.9	(1,2)12	2.4 ± 0.1
	22 414.5	(0,6)15	2.3 ± 0.1

It is commonly accepted that the loss of an HR coating stack ϕ_HR is the linear combination of the measured loss of its constituent layers [18],

$$\begin{equation}{\phi }_{\text{HR}}=\frac{{t}_{\text{H}}{Y}_{\text{H}}{\phi }_{\text{H}}+{t}_{\text{L}}{Y}_{\text{L}}{\phi }_{\text{L}}}{{t}_{\text{H}}{Y}_{\text{H}}+{t}_{\text{L}}{Y}_{\text{L}}},\end{equation} \tag{ 6 }$$

where t_H, Y_H and ϕ_H (t_L, Y_L and ϕ_L) are the thickness, the Young's modulus and the loss of the high-index Ta₂O₅–TiO₂ (low-index SiO₂) layers. Figure 8 shows the comparison between the expected loss of equation (6), calculated using the values of loss and Young's modulus of table 2, and the updated loss of table 13, obtained from direct loss measurements of the HR coatings [38]. The measured loss is fairly constant over the sampled band (2.7–22.4 kHzきろへるつ), whereas the expectations have the same frequency dependence of their dominant contribution, the titania-doped tantala layers. When extrapolating down to 0.1 kHzきろへるつ, i.e. the region of the Advanced LIGO and Advanced Virgo detection band limited by coating thermal noise, the expectations underestimate the actual measured loss of the HR coatings, by about 30% for the ITM coating and about 43% for the ETM coating. Though well-known [38], this discrepancy remains unexplained to date and will be subject to further investigation.

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Measurements of coating internal friction performed with our GeNS system [43, 64] showed weak but clear frequency-dependent trends, i.e. ϕ_c(f) = af^b with −0.208 < b < 0.140, depending on the sample considered; the frequency dependence observed for Ta₂O₅ and Ta₂O₅–TiO₂ layers has been later confirmed by independent measurements of coating thermal noise on the end mirror (ETM) HR coatings of Advanced LIGO and Advanced Virgo [53]. Our method for loss characterization is exclusively based on measured quantities (quality factors, frequencies and masses) and, unlike other experimental setups based on the ring-down method, does not require prior knowledge of the coating Young's modulus and thickness.

With our tantala films, we observed that the level of coating internal friction is determined by the deposition rate: the slower the rate, the lower the loss; the same rule fairly holds for our silica films too. Thus, special care should be taken when comparing the loss of coating samples, even if composed of the same coating material.

Post-deposition heat treatment makes the thickness increase and hence density decrease, while optical absorption, refractive index and internal friction decrease. Depending on the initial values considered, the reduction of internal friction is of a factor 1.5 to 2.5 for films Ta₂O₅, 1.8 for the Ta₂O₅–TiO₂ layer with Ti/Ta = 0.27, 5 to 6.5 for SiO₂ films. As the disks are annealed at 900 °C before deposition, the observed loss change upon 500 °C annealing is due to the coating only. Once annealed, all the Ta₂O₅ films exhibited equal loss, as if their deposition history had been erased, whereas the gap between the loss values of SiO₂ films decreased. For SiO₂ layers, it has been possible to establish a correlation between their structural change upon annealing and their internal friction [61]; for Ta₂O₅ and Ta₂O₅–TiO₂ films, we carried out analogous studies of structure and internal friction as a function of annealing temperature and duration [43, 65] but—despite the large variation of internal friction—we observed very limited structural change and eventually found no correlations.

Our GeNS system allows the estimation of coating Young's modulus and Poisson's ratio, via the measurement of dilution factors [66]. Our values of Young's modulus for Ta₂O₅ and Ta₂O₅–TiO₂ coatings are 14% lower than those commonly accepted [48, 49] by the scientific collaborations running gravitational-wave interferometers. Two reasons may explain this discrepancy: likely the different nature of coating samples used in previous works, determined by the different deposition parameters used, and the method used, i.e. nano-indentations. The analysis of nano-indentation measurements relies on the prior knowledge of the coating Poisson's ratio and is model dependent. Moreover, nano-indentations of our tantala SPECTOR coatings suggest that the outcome of this technique may depend on the nature of the substrate: we obtained a reduced Young's modulus of 100 GPa from the coating sample deposited on a fused silica witness substrate, of 130 GPa from the coating sample deposited on a silicon wafer; both samples were from the same deposition run. While the discussion of the details about those measurements and their analysis is beyond the scope of this article, the important point we would like to stress here is that the outcomes of nano-indentations should be considered cautiously. Unlike nano-indentations, our estimation method is a non-destructive technique which does not require prior knowledge of the coating Poisson's ratio; it has been tested also on tantala SPECTOR coatings deposited at the same time on fused-silica disks and on silicon wafers, yielding consistent results. The agreement between finite-element simulations and measured dilution factors vouches for the reliability of our method; however, larger residuals for silica coatings seem to point out that either our model or our simulations might need further fine tuning, and will be be subject to further investigation.

SiO₂ films showed a mode-dependent loss branching, which may be accounted for by including a term of spurious loss from the coated disks' edge in the coating loss model, as it is already the case for bare substrates [39].

The reference loss values of the Advanced LIGO and Advanced Virgo input (ITM) and end (ETM) mirror HR coatings [38] have been updated by using our estimated value of Young's modulus of titania-doped tantala layers (Y_c = 120 GPa) and are about 10% higher than previous estimations. By using our latest measurements of SiO₂ and Ta₂O₅–TiO₂ films deposited on fused silica disks and the updated loss values of the Advanced LIGO and Advanced Virgo HR coatings, we demonstrated that the loss of an HR coating stack is not equal to the linear combination of the measured loss of its constituent layers: a different frequency dependence makes the linear combination smaller than the measured loss at lower frequencies (below 2 kHzきろへるつ). This is a confirmation of what had already been observed with the same coatings (deposited with the same conditions) measured on clamped cantilever blades [38].

The frequency dependence of coating loss of Ta₂O₅, Ta₂O₅–TiO₂ and SiO₂ coatings had not been observed previously with clamped cantilever blades [20, 23, 38]. Our latest loss measurements of Ta₂O₅ and Ta₂O₅–TiO₂ layers are compatible with previous estimations at least in some specific frequency band, whereas in the case of SiO₂ coatings there is a discrepancy that might be explained by considering the presence of spurious suspension losses and of the edge effect in the clamped cantilevers.