## 1 Introduction  
At present, optical Fabry-Perot cavities are used for different applications and for precise measurements in fundamental physics. In particular, they are used to address the problem of optical frequency standards [1] and for gravitational wave detection using large-scale laser interferometers [2; 3]. Atomic transitions used as reference frequencies in optical clocks usually have a quality factor in the region of \(10^{15}\), which corresponds to a line width of about 1 Hz or less. To explore such narrow transitions, an optical local oscillator is required that has a similar or narrower line width. Available solid-state and semiconductor diode lasers have a line width of tens of kHz to tens of MHz. Reducing the laser line width to the required level is most often achieved using the Pound-Drever-Hall stabilization method [4], which binds the optical generator to the ultra-stable optical Fabry-Perot cavity with a narrow line (high finesse). Such reference resonator (spacer) used for this purpose consists of two concave mirrors that are attached to each end of a solid-state hollow cylindrical specimen by optical contact. Both the sample and the mirror substrates are usually made of ultra-low thermal expansion glass (ULE), such that the coefficient of thermal expansion is of the order (3-0.3) \(\times 10^{-7}\)\(\,{}^{\circ}\)C in the temperature range of 5-40\(\,{}^{\circ}\)C, crossing zero near room temperature. The typical size of the reference resonator (spacer) is about 10 cm in length with a high value of the number of reflected raysand finesse of the order of \(10^{5}\), which corresponds to the width of the optical resonator line in the range 5-10 kHz. With good temperature control at the point of zero thermal expansion and suppression of the influence of mechanical vibrations, the frequency stability of the reference optical resonator (spacer) is limited only by dimensional changes due to thermal vibrations of the substrates of mirrors and their coatings [5]. Currently, ULE glasses are used for optical reference resonators, since they have zero expansion near room temperature. A significant reduction in thermal noise can be achieved by switching to silica, which has much lower mechanical losses. Then, the thermal noise will be determined mainly by coating (sprayed) and will be at a lower level. However, silica has a relatively large thermal expansion, and therefore very good temperature control is required. An alternative would be to use a body of a reference resonator (spacer) made of ULE glass with low thermal noise and silica for substrates of mirrors [6]. Mirror coatings are alternating layers of silica (SiO\({}_{2}\)) and tantalum (Ta\({}_{2}\)O\({}_{5}\)), with tantalum layers making the largest contribution to thermal noise [7]. The cooling of the cavity itself will reduce the thermal noise, since it decreases as the square root of the temperature. However, in the case of a ULE, this would mean working away from the zero point of thermal expansion, which can lead to unacceptably high drift velocities. Silicon with zero expansion at \(\sim\)120 K [8] is proposed as an alternative material. Nevertheless, the problem of noise in the coating films remains. It becomes even more serious at low temperature, since the coefficient of mechanical loss of tantalum has a resonance at 70 \({}^{\circ}\)[9]. In addition, silicon is transparent only for wavelengths above 1.2 microns. Currently, by scientific groups involved in the detection of gravitational waves, a significant work is being done to develop coatings with lower losses, and, for example, a halving has been achieved by doping tantalum layers with titanium (TiO\({}_{2}\)) [10]. The prospects for increasing the sensitivity of laser interferometric gravitational-wave antennas [11; 12] of the third generation are closely related to the idea of suppressing thermal noise of mirrors (test bodies) through their deep cooling to liquid nitrogen or helium temperature [13].