Finite element analysis of the zero-crossing temperature of a long Fabry–Perot cavity

An ultra-stable Fabry–Perot (F–P) cavity is designed, whose spacer is made of an ultra-low-expansion (ULE) glass and mirrors made of fused silicon (FS). The spacer aperture is composed of three holes, among which a long and small hole is in the middle, and a short and large hole is on each side. The fused silicon (FS) mirrors are placed in the large holes of the spacer aperture, and the FS rings are mounted on the end face of the spacer. FS mirrors, aperture and FS rings are coaxial. The thermal noise limit of the cavity is 9.1×10-17\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$9.1\times 10^{-17}$$\end{document}. By finite element analysis, the specific FS rings are choose to successfully tune the zero-crossing temperature from −10 to 27 K, which enables the cavity zero-crossing temperature easily be tuned above room temperature. In the meantime, this design makes the machining error has little effect on the zero-crossing temperature of the cavity, and it is convenient to manufacture a long F–P cavity.


Introduction
Recent years, it's more and more intriguing to obtain ultrastable lasers by the Pound-Drever-Hall (PDH) technique. Ultra-stable lasers are essential in optical atomic clocks [1], quantum information [2] and high-resolution laser spectroscopy [3,4]. A common method to obtain a highly stable laser is locking the laser to a highly stable Fabry-Perot (F-P) cavity by the PDH technique [5]. The dependence of the nominal resonant optical frequency on the cavity length is Dm=m ¼ ÀDL=L; Dm is the change of frequency and DL is the jitter of the cavity length. So, the stability of the laser frequency depends on the stability of F-P cavity length. The thermal noise limit is usually reduced by means of increasing the F-P cavity length, increasing the laser spot size, and using high mechanical quality (Q) materials [6]. Among these methods, highquality materials have the most significant impact on the thermal noise limit. Conventional F-P cavity is made from ultra-low expansion glass, which Q is 6:1 Â 10 4 . Its thermal noise limit is at the level between 10 À15 to 10 À16 . In order to achieve a cavity with a thermal noise limit of 10 À17 magnitude, the fused silicon (FS) with higher Q (10 6 ) than that of ULE is selected as the substrate material of the cavity mirror.
When the cavity mirror substrate is FS, the thermal noise limit can be reduced to half of that when ULE is used. However, the difference in thermal expansion coefficient (CTE) between FS and ULE is two orders of magnitude, which may cause stress due to temperature changes in the cavity, bending the mirror substrate and introducing greater effective CTE into the F-P cavity [7]. This problem can be solved by operating the F-P cavity at zero crossing temperature (ZCT) of the effective CTE [8]. However, the ZCT of the cavity is 10 C lower than that of the ULE cavity spacer [7], which is in the range of 5-35 C. So, the ZCT is lower than room temperature. This means that in the experiment, the whole cavity needs to be cooled to ZCT. But cooling the cavity is more complex than heating it, and the cooling will reduce the robustness of the cavity. So, this kind of cavity is not suitable for miniaturized portable optical clocks which needs to be as simple as possible in structure. Thus, the zero-crossing temperature should be adjusted above room temperature.
In order to raise the ZCT, a series of studies have been taken. Richard W. Fox et al. used the negative CTE of the ULE at room temperature and the positive CTE of FS to make the ZCT of the cavity higher than room temperature [9]. Thomas Legero et al. compensated for the ZCT by sticking a ULE ring on the FS mirror [7]. The ZCT of this F-P cavity was equivalent to that of the ULE spacer. Stephen Webster et al. put the FS cavity mirror into the optical hole of the cavity and fixed it on the ULE ring by optically contacting. At the same time, the ULE ring was optically glued to the ULE cavity, so that its tuning range was achieved at about 30 K [10]. Similar to Stephen Webster's work, Zhang Jie et al. used the FS ring instead of the ULE ring to achieve the tuning range between À10 to 23 K [11]. These approaches have some limitations. The first method relies on the availability of ULE materials with negative CTE at room temperature, which requires special selection and testing. This greatly increases the cost. The second method can only make the ZCT close to that of ULE spacer, which is not applicable when the cavity ZCT is lower than room temperature. In principle, the third method can be used to adjust the ZCT to about 30 C. However, the cavity hole need to be larger than the mirror diameter (25.4 mm), which is unsuitable for the long cavity machining. In this paper, the F-P cavity is redesigned with three stepped optical holes, which has following characteristics. First, the cavity can be easily manufactured with low cost. Second, the tuning range of ZCT is -10 K to 27 K, which means the cavity's ZCT can be adjusted to higher than room temperature, no matter what its value is.
The cavity we designed is used to stabilize the clock laser in the aluminum ion optical clock. The aluminum ion clock laser at 267 nm is obtained by quadrupling the frequency 1068 nm laser. And the stability of the 267 nm laser is achieved by stabilizing the double frequency laser (534 nm) onto the F-P cavity. It is proved that this meets the experimental requirements. In this paper, the second part introduces the structure of the F-P cavity, and the third part is the zero-crossing temperature compensation by finite element analysis. Finally, the fourth part concludes this work.

Low-thermal noise F-P cavity
Thermal fluctuation is a fundamental phenomenon of F-P cavity. At nonzero temperatures, the cavity has an inevitable mechanical thermal fluctuation which limits the frequency stability fundamentally, even if the CTE is zero. The Poisson's ratio, coating thickness, curvature radius of the concave mirror, cavity length, Young's modulus of the spacer and mechanical losses / of the spacer, mirror substrate, and reflective coating are the basic parameters for evaluating the thermal noise of the cavity. The thermal fluctuation spectrum Allan deviation of the F-P cavity can be described as Eq. (1) [6].
Here, k B is the Boltzmann constant, T is the temperature, L is the cavity length, E is Young's modulus of the spacer, is the beam radius, R is the curvature radius of the concave mirror, r is Poisson's ratio, d is the mirror coating thickness, / sub and / coat are the mechanical losses of mirror substrate and mirror coating, respectively. Because the mechanical loss of FS (10 À6 ) is lower than that of ULE (1:6 Â 10 À5 ) [6], according to Eq. (1), it is better to choose FS as the cavity mirror substrate to obtain the cavity with low thermal noise.
Equation (1) shows that the longer the cavity is, the smaller the thermal noise limit is. And the thermal noise limit is negatively correlated with curvature radius of the concave mirror. The thermal noise limits are listed in Table 1 The thermal noise limit of the F-P cavity with different dimensions and materials. R is the curvature radius of the concave mirror, L is the cavity length. r them is the Allan deviation of cavity thermal noise  Table 1 with different materials and parameters. In order to obtain the stable laser of aluminum ion clock transition detection, the length and diameter of the cavity is selected as 300 mm and 150 mm, the curvature radius of the concave mirror is 1 m. In addition, the substrate material of the cavity mirror is FS. The structure of the cavity is shown in Fig. 1, and the cutting sizes is C x ¼ 77:96 mm, C y ¼ 6 mm. Thus, the thermal noise limit of the cavity will reach 9:1 Â 10 À17 when the ZCT is at 30 C.

Zero-crossing temperature compensation
The length L of the F-P cavity varies with temperature due to thermal expansion: dL=L ¼ adT. The thermal expansion coefficient of ULE (a ULE ðTÞÞ is a quadratic function of temperature T [7], where a is 2:4 Â 10 À9 =K 2 , b is À10 À11 =K 3 , T is cavity temperature, and T 0 is the zero-crossing temperature of the ULE. When T ¼ T 0 , the thermal expansion coefficient is 0, and the temperature jitter does not affect the length of the cavity. So that the temperature of cavity is usually stabilized to T 0 in the experiment. The thermal expansion coefficient of FS is a FS ¼ 5 Â 10 À7 =K 2 , which is two orders of magnitude larger than that of ULE. When the cavity spacer material is ULE, and the cavity mirror material is FS, called composite cavity, the radial deformation of the cavity mirror caused by the temperature variation (dT) is Since there is no relative movement between the spacer and the mirror, radial pressure causes the axial deformation of the mirror to be Here, d is a coefficient relating to the geometry, size and material of the cavity. Therefore, the cavity length deformation dL of the composite cavity is given by the thermal expansion La ULE dT and the axial deformation of the two mirrors. It can be described by an effective cavity CTE as Let b=0, from Eqs. (2), (3), (4) and (5), we can obtain In this case, a eff ¼ 0 corresponds to the effective zerocrossing temperature T 0 0 , and the offset of T 0 0 and T 0 is Since d depends on the geometry, size and material of the cavity, it is difficult to get the analytical solution through theoretical calculation. The d values under different conditions are obtained by finite element analysis (FEA). Because the cavity is axially symmetric (Fig. 1), half of the cavity is selected as the research object. The displacement reference is set as the central cross section of the cavity (Fig. 2), and the temperature variation is set as 1 K. Moreover, the contact surfaces of the FS mirror, ULE spacer and the FS ring are perfectly bonded such that there is no relative displacement. The cavity length variation dL is obtained by probing the axial displacement at the center of the cavity mirror's inner surface (Fig. 1). The equivalent thermal expansion coefficient a eff is calculated by the formula (6), and the d is obtained by substituting a eff into formula (7). The deformation of the composite cavity simulated by finite element analysis is shown in Fig. 2, and the cavity mirror is obviously bow. By calculating, we obtain d ¼ 0:0077, a eff ¼ 2:8 Â 10 À8 and DT 0 ¼ À10:19K. It means that the effective CTE of the composite cavity is  Finite element analysis of the zero-crossing temperature relatively larger than that of ULE at room temperature, and the zero point of the effective CTE still exists, but it is shifted to a even lower temperature. T 0 ranges from 5 to 35 C, so the effective ZCT T 0 0 ranges from À5 to 25 C, which is close to or below room temperature. In this case, the cavity needs to be cooled. But cooling the cavity may cause additional problems, such as water condensation on the cavity shell. It is best to adjust its zero-crossing temperature to slightly higher than room temperature, such as 30 C.
In order to tune the zero-crossing temperature to 30 C, DT 0 should be adjustable from À5 to 25 C. The following describes the search for a cavity structure by finite element analysis. With this structure, the appropriate compensation ring size can be conveniently selected to achieve a cavity zero-crossing temperature at about 30 C. At the same time, the cavity structure selected is convenient for processing and manufacturing.
One solution for T 0 0 below room temperature is to fix a ULE ring behind each FS cavity mirror to tune the zerocrossing temperature (Fig. 3(a)). Figure 3

(b), (c) and (d)
shows the results of the simulation. The temperature difference DT 0 changes with the external radius R, inner radius r, thickness h of the ring and diameter of the optical aperture bore. However, no matter what the parameters are, DT 0 is mostly about 0 K, which means that the zerocrossing temperature of the cavity is equivalent to that of the ULE spacer. The simulation result is in good agreement with the reference [7]. Therefore, if the ZCT of the ULE spacer is lower than room temperature, the zero crossing temperature of the whole cavity is below room temperature. Obviously, this scheme cannot satisfy the requirements.
We propose a scheme of a ultra-stable cavity, in which the spacer aperture is replaced by stepped holes. Meantime, the mirror is fixed on the ring, and the mirror is inserted into the larger one of stepped holes. At last, the ring is glued to one end of the spacer.
The rings can be ULE or FS material (Fig. 4). In this way, the thermal expansion direction of mirrors is opposite to that of the spacer. Figure 5 shows the simulation results of zerocrossing temperature when ULE ring is used in cavity. Ring Fig. 3 (a) The structure of cavity. (b), (c) and (d) are the results of FEM simulations of the axial mirror displacement along a radial line on the mirror surface when additional ULE rings are contacted behind the FS cavity mirrors. R: ring external radius; h: thickness of ring; r: the ring inner radius, bore: diameter of the optical aperture thicknesses (h), external and inside radius (R, r), and large optical aperture's diameter and depth (bore, h 1 ) of the spacer are used as the independent variables. Of all these conditions, h 1 and R have almost no effect on DT 0 . DT 0 decreases with the decreases of the bore, and it decays exponentially with h. The inside radius of r affects the variation range of DT 0 . However, no matter what the size of the rings are, DT 0 is always higher than 15 K. That is to say, this configuration is only suitable for cavities with a ZCT below 15 C. Figure 6 shows the simulation results of zero-crossing temperature when FS ring is used in cavity. It is proved that, the FS ring can compensate the deformation of the ULE spacer. Firstly, the influence of h 1 on DT 0 is calculated, and the result is shown in Fig. 6(a). When h 1 changes from 7 to 140 mm, DT 0 only changes by about 1 K. Thus, h 1 has little influence on DT 0 . In order to facilitate processing, the h 1 is fixed at 7 mm. Secondly, We study the influence of both ring external radius R and large aperture bore diameter on DT 0 . DT 0 increases logarithmically with R, and its variation range and rise rate increase with the bore diameter, as shown in Fig. 6(b). When the bore diameter is 28 mm (40 mm), the variation range of DT 0 is 15 K (30 K). In order to satisfy the adjustment range and reduce the impact of ring sizes errors, we select bore diameter as 40 mm. Thirdly, the influence of ring thickness h on DT 0 is studied, as shown in Fig. 6(c). Thickness of ring (h) mainly affects the variation range of DT 0 . For example, when h is 2 mm, the value ranges is À5 to 27 K. If h is 3 mm, the value ranges is À15 to 20 K. In order to improve the practicality of the scheme, the sizes of ring with a wide tuning range should be selected as far as possible for considering the processing error of the ring, so the h is selected as 3 mm. In this case, the adjustment range is À14 to 22 K, slightly lower than the request. According to Fig. 6(d), the variation range can be improved by increasing r, and the adjustment range is changed to À10 to 27 K when r=6 mm, which meets the requirements. In summary, the cavity has a diameter of 150 mm and length of 300 mm, respectively, the depth and diameter of the large clear aperture are 7 mm and 40 mm, the cutting size is 77.96 mmÂ6 mm, and the ring has a thickness and inner radius of h=3 mm, r=6 mm, while outer radius is chosen according to the ULE spacer ZCT. At this point, the thermal noise limit of the designed cavity is 9:1 Â 10 À17 . Fig. 4 The geometry of the cavity we designed

Conclusions
A composite cavity with thermal noise limit of 9:1 Â 10 À17 and zero-crossing temperature point above room temperature was designed. The optical aperture is composed of a series of stepped holes. The cavity mirrors are placed in the large optical aperture and fixed on the spacer through the FS ring. As long as you know the zero crossing temperature of the selected ULE cavity, no matter what the value is, it is easy to get the effective zero crossing temperature of the cavity above room temperature by selecting the ring of the appropriate size. This design adopts a stepped holes structure, which is convenient for processing and cost less. This design is suitable for different lengths of cavity, especially for long cavity. To furtherly reduce the thermal noise limit, the following methods can be used: increasing the cavity length, reducing the cavity temperature, and building the cavity with single crystal silicon(or sapphire).
Funding National Natural Science Foundation of China (11704099); Science and Technology Department of Henan Province (222102320449,212102310527); Natural Science Foundation of Zhejiang Province (LQ21A040001).

Declarations
Conflict of interest The authors have no relevant financial or nonfinancial interests to disclose