## 3 Substrate Material for Mirrors  
The choice of material for the mirror substrates for use gravitational-wave interferometers is primarily dictated by the requirement for small mechanical (acoustic) losses [2]. This is due to the need to suppress the level of thermal (Brownian) noise, not only from the mirror's pendulum degree of freedom but also from their own acoustic modes. Experience has shown that some optical glasses, fused silica, sapphire, and a number of crystals, in particular silicon and fluorites (calcium fluoride CaF\({}_{2}\)), satisfy this requirement [26]. Optical glasses in astronomical instruments (telescopes) use materials with a low coefficient of thermal expansion (so-called zerodur, or Russian variant still). In our test experiments, we used mirrors with substrates made of stall, sapphire, and calcium fluoride (CaF\({}_{2}\)). All of them have a wide transmission spectrum in the optical frequency range and can withstand low temperatures without irreversible changes during warming. Experimental estimates of mechanical losses for materials are \(\phi\sim 10^{-8}\)[26] under cryogenic temperatures but the crystalline quartz has a jump to \(10^{-5}\) in the temperature region of 40 K. Although silicon is considered to be a promising material when operating at sub-nitrogen temperatures (\(\sim\)120 K), it is opaque for the visible frequency range. Its use implies a transition to the wavelength \(\lambda=1.55\,\upmu\)m, which leads to a serious reconstruction of all elements of the optical path of the gravitational-wave interferometers. Our experiments were carried out in infrared light with \(\lambda=1.064\,\upmu\)m. At this wavelength, all of the laser gravitational antennas work, including the OGRAN resonant optoacoustic detector [24]. Among the key parameters describing the physical properties of the substrate material and determining their behavior under low temperature conditions with high optical pump powers, a number of thermodynamic, mechanical, and optical characteristics are important [28]. In particular, their heat capacity and thermal conductivity depend on the heating temperature and the uniformity of its distribution over the sample volume. Young's modulus and thermo elasticity (coefficient of thermal expansion) affect the measurement of the mirror's deformation, then optical refractive index and its nonlinear component, and also the coefficients of optical reflection and absorption of incident radiation. The latter parameters are mainly determined by the quality and number of reflective layers of sprayed mirror coatings. Our mirrors were manufactured (Polyus research institute of M.F.Stelmakh, Moscow, Russia [3]) using multilayer dielectric interference coatings of alternating layers of oxides TiO\({}_{2}\)/SiO\({}_{2}\) and Ta\({}_{2}\)O\({}_{5}\)/SiO\({}_{2}\), which were applied by ion beam spraying. The average parameters are: scattering \(K_{s}<10\) ppm, absorption coefficient \(K_{a}<10\) ppm, reflection coefficient \(R\geq 99.998\%\), and laser strength \(\sim\)1-2 GW/cm\({}^{2}\). Table 1 provides a summary of parameters for the substrates in our study. An important point in the experiment on cooling mirrors on substrates of these materials was to ensure sufficient thermal contact of the mirrors with the cryostat, so that the experiment could be carried out in a finite time. Working through various mounting options led to the design shown in Figure 1. The comments to the picture are as follows (the numbers and arrows mark particular details): the lower parts of the holder (2) are pressed (inserted on friction) into both bar ends of central tunnel of the PM (1), so that