Ultra-stable laser has excellent characteristics such as extremely low frequency noises and extremely high coherence, and it is widely used in cold atomic light clocks, geodesy, gravitational wave detection, and optical frequency transmission. When the laser frequency is locked on the Fabry-Pérot (FP) cavity using Pound-Driver-Hall (PDH) frequency stabilization technology, the frequency stability of the laser depends entirely on the cavity length stability of the cavity. The temperature fluctuation of the FP cavity is one of the main factors that affect the cavity length. How to quickly analyze its temperature characteristics has been the research focus of ultra-stable lasers. The cavity length change of the FP cavity is mainly affected by the temperature fluctuation of the external environment. To suppress this effect, researchers both in China and abroad usually place the cavity in a vacuum chamber with multi-layer thermal shields to obtain a larger thermal time constant and a lower temperature sensitivity. Therefore, to quickly and accurately analyze the influence of the thermal shield parameters in the vacuum chamber on the temperature of the FP cavity, global researchers have carried out corresponding research based on various thermal analysis methods, such as the transfer function method, finite element analysis method, and direct differential method. At present, most of the thermal analysis methods of the FP cavity system only focus on the law of the temperature of the cavity changing with time, and the research results of the temperature sensitivity of the FP cavity required by the actual working conditions cannot meet the urgent needs of the actual working conditions. Therefore, this paper proposes a thermal analysis method of the FP cavity after comprehensively considering heat conduction and radiation and establishes the relationship between temperature sensitivity and corresponding physical parameters of the system, which can guide the design of the FP cavity system in practical engineering.

In this paper, a typical FP cavity vacuum system is taken as the research object. Through theoretical analysis, the differential equations between the FP cavity's temperature and the external temperature under heat conduction and radiation are derived. According to the differential equations, the transfer function relationship between the FP cavity's temperature and the external temperature is derived by using a reasonable simplified approximation method. The correctness of the transfer function relationship is proved by numerical calculation and finite element simulation. On the basis of the temperature sensitivity curve obtained from the Bode plot of the transfer function, a simplified approximate formula for calculating the temperature sensitivity of the FP cavity is proposed. By combining the overall trend of the curve and considering the wide applicability of the approximate formula, the proposed approximate formula is improved and revised. Finally, the curve obtained from the approximate formula is compared with that obtained from the theoretical formula by numerical calculation. The results show that the overall trend of the curves obtained by the two methods is completely consistent, although there are some small errors.

In this paper, the transfer function relationship between the FP cavity's temperature and the external temperature is derived by reasonably simplifying the approximation method. Through the analysis and comparison of numerical calculation and finite element analysis methods, it can be concluded that the temperature curve of the FP cavity obtained by this transfer function is completely consistent with the one obtained by the theoretical formula, and it is very close to the calculation result of ANSYS software under both the condition of only considering heat radiation [Figs. 2(a) and 2(c)] or considering heat conduction and radiation comprehensively [Fig. 3(a)]. The residual curves [Figs. 2(b), 2(d), and 3(b)] given in the paper show that the difference between the curves is very small, and it can be approximately considered that the curves are completely consistent. The calculation results of the two formulas for the thermal time constant are consistent and very close to those of the ANSYS software. Based on the temperature sensitivity curve of the FP cavity (Fig. 4), this paper gives a simplified approximate formula for calculating the temperature sensitivity of the cavity. The comparison between the curve obtained from the approximate formula and that obtained from the theoretical formula shows (Fig. 5) that the overall trend of the curves is completely consistent. Although there are some errors, the sensitivity approximate formula has the characteristics of simple form, intuitional parameters, and convenient calculation, and it is of important guiding significance for the design of the ultra-stable laser system.

In this paper, the FP cavity vacuum system is taken as the research object, and the temperature characteristics of the FP cavity under multi-layer thermal shields are analyzed theoretically. The transfer function relationship between the FP cavity's temperature and the external temperature is simplified and deduced. The simulation results show that the transfer function formula is correct. According to the transfer function, the approximate formula for fitting the temperature sensitivity curve of the FP cavity is obtained. The results show that although there are some differences between the approximate formula and the theoretical value, the approximate formula has the characteristics of simple form, intuitional parameters, and convenient calculation, and it is of strong guiding significance for the preliminary design of the FP cavity vacuum system.