Research on Shock Acceleration Limit of an Ultra-Stable Optical Cavity for Space Applications Based on the Finite Element Methodology

: Ultra-stable optical cavities (USOCs) as fragile precision instruments have many important applications in space. In order to protect them from being damaged during a rocket launch, we analyzed a USOC by means of ﬁnite element methodology. The shock acceleration limits that the USOC can withstand in different directions and under various conditions are given. To increase the shock acceleration limit, the midplane thickness and the ﬁxed hole diameter should be selected to be as high as possible. It is worth noting that the launch direction of the USOC should be selected as the horizontal direction, for which the shock acceleration limit that the USOC can withstand is approximately two times that of the vertical direction. In this paper, results provide guidance for the design of USOCs for space applications, especially the design to prevent the damage caused by a shock. The method could then be applied to other space optical cavities, providing a tool to improve the effect of shock at high accelerations.


Introduction
Ultra-stable lasers are key elements in many domains, including frequency metrology [1][2][3][4][5], gravitational wave detection [6], fundamental physics tests [7,8], and coherent optical links [9,10]. Ultra-stable lasers can be produced by locking lasers onto USOCs with the Pound-Drever-Hall (PDH) technique, and the instability of an ultra-stable laser frequency can be defined with the stability of the optical length of a USOC [11][12][13][14], which is generally machined with ultra-low-expansion glass (ULE). However, most ultra-stable lasers have been constrained to operate in well-controlled laboratory environments. There is growing interest in frequency-stable lasers capable of operating outside a laboratory for applications such as space optical clocks, geodesy, tests of fundamental physics in space, and the generation of ultra-stable microwaves for radar [15][16][17][18]. Hence, it is important to investigate transportable USOCs to determine whether an ultra-stable laser can operate in a non-laboratory environment [17][18][19].
To this end, different groups have designed a variety of shapes and support modes for USOCs . In 2011, Leibrandt et al. designed a spherical USOC that was rigidly supported at two points on a diameter with a squeeze-insensitive angle [15,16,20]. Acceleration sensitivities below 3 × 10 −10 /g were achieved. In the same year, Webster et al. constructed a cubic USOC with four supports placed in a tetrahedral configuration [18]. An acceleration sensitivity better than 2.5 × 10 −11 /g was achieved. In 2012, Argence et al.
reported a cylinder USOC rigidly held at its mid-plane [19]. Acceleration sensitivities below 4 × 10 −10 /g were achieved. In 2014, Chen et al. constructed a cylinder USOC rigidly held at ten pairs of points on the cavity spacer [21] and acceleration sensitivities below 4 × 10 −10 /g were achieved. In 2018, a similar cuboid USOC was mounted rigidly and tested by dropping from a height of 25 cm, corresponding to an acceleration of 100 g [28]. In 2020, Chen et al. [31] designed and developed a cubic optical cavity with a side length of 100 mm based on the 50 mm cubic cavity designed by Webster et al. [18]. The vibration sensitivity of the three orthogonal directions was 10 −10 /g. Compared with lab applications, USOCs with low vibration sensitivity used in the space face challenges, such as being rigidly held for transport, enduring larger vibration, and shock acceleration [15][16][17][18][19][20][21][22]27,28,[31][32][33][34][35][36]. In the rocket launching stage, the shock acceleration loaded on a USOC can be hundreds of times greater than gravitational acceleration (9.8 m/s 2 ) [19,28,37,38]. Due to this, as a fragile precision instrument, the USOC may have unrecoverable structural failures and fractures [28,35,[39][40][41]. To realize the space applications of the USOC, it must withstand the large shock of the rocket launching stage without damage. However, there is no in-depth study on this at present. In this paper, we focus on the effect of shock acceleration on USOCs for space applications, including Space Optical Clocks (SOCs) in Europe [18,19,25] and China [29]. It can provide design guidance for preventing damage caused by a shock in the space applications of USOCs.
The objective of this work was to protect the USOCs from being damaged by shock vibration during the rocket launching stage by giving the limit of shock acceleration. In this paper, we considered the cylindrical optical cavity rigidly mounted at its mid-plane designed by Argence et al. [19] and used finite element analysis (FEA) to study the effect of shock acceleration on the USOC for space applications. Firstly, the elastic deformation induced by the shock acceleration with frequency variation is simplified as the quasistatic elastic deformation based on the vibration dispersion relationship of the USOC. The failure criterion of the USOC, based on the maximum von Mises stress and the maximum tensile strength of the material was presented. Secondly, to calculate the maximum von Mises stress of the USOC induced by the shock acceleration, the FEA model of quasi-static mechanics was established. Thirdly, we calculated the maximum acceleration that the USOC could bear under different conditions, including the thickness T of the mid-plane (used for fixing the optical cavity), the diameter ϕ1 of the fixed hole, the shock acceleration directions, and the positions of the constraint.
The paper was organized as follows. In Section 2, the fundamental theories were presented, including the vibration dispersion relationship and failure criterion of the USOC. The FEA model of quasi-static mechanics is established in Section 3. The simulation results and the discussion are presented in Section 4. Finally, in Section 5, we present our discussions and conclusions.

Duasi-Static Mechanics
The geometry model of the USOC (Ref. [19]) is described in Figure 1a, and the cylindrical USOC is rigidly mounted at its mid-plane by the fixed holes. The USOC can be simplified as a one-dimensional elastic molecular chain. Then, based on the vibration dispersion relationship of the solid materials, the vibration dispersion relationship of the USOC can be expressed by Equation (1) [14]: where E and ρ are the elastic modulus and the density of the USOC, respectively, while f and λ are the vibration frequency and the wavelength, respectively. For the ULE USOC, according to Equation (1) and Table 1, when the frequency is f < 5000 Hz, the wavelength is λ > 1100 mm (which is ten times the length of the USOC). In this work, the wavelength in the USOC is 100 mm, which is larger than the size of the cavity, and all the particles in is λ > 1100 mm (which is ten times the length of the USOC). In this work, the wavelength in the USOC is 100 mm, which is larger than the size of the cavity, and all the particles in the USOC vibrate in the same phase, so the quasi-static mechanic analysis is sufficient to study its motion.

Failure Criterion of the Ultra-Stable Optical Cavity
According to the widely used shape-change strength ratio theory (the fourth strength theory) in mechanical design [42], we checked the strength of the USOC, as shown in where σvon-Mises is the maximum von Mises stress, σmax is the ultimate tensile stress of the material, and K is the safety coefficient. As a matter of convenience, K = 1 in this work. Based on Equation (2), when σvon-Mises is larger than σmax, the failure criterion of the USOC is met and it is thought to have fractured. Figure 1a shows the USOC described in [19], that was mounted at its midplane with a mechanical interface and with a complex design involving vibration. The geometrical dimensions of the USOC are described (Figure 1a) in detail. The spacer was made of Ultra-Low Expansion (ULE) glass. To reduce the thermal noise, Fused Silica (FS) was used to make two mirror substrates with 25.4 mm diameters and 6.3 mm thicknesses.

Model of the Finite Element Analysis
In Table 1, the material properties of the ULE and FS are listed, including the elastic properties and the ultimate tensile stress of the structural failure. As described in Figure  1a, three out of six holes located at the midplane are used to fix the USOC, and these three holes are symmetrical at 120 degrees. In the FEA model, there are two kinds of positions

Failure Criterion of the Ultra-Stable Optical Cavity
According to the widely used shape-change strength ratio theory (the fourth strength theory) in mechanical design [42], we checked the strength of the USOC, as shown in Figure 1. The failure criterion of the USOC can be expressed as Equation (2) [42]: where σ von-Mises is the maximum von Mises stress, σ max is the ultimate tensile stress of the material, and K is the safety coefficient. As a matter of convenience, K = 1 in this work. Based on Equation (2), when σ von-Mises is larger than σ max , the failure criterion of the USOC is met and it is thought to have fractured. Figure 1a shows the USOC described in [19], that was mounted at its midplane with a mechanical interface and with a complex design involving vibration. The geometrical dimensions of the USOC are described (Figure 1a) in detail. The spacer was made of Ultra-Low Expansion (ULE) glass. To reduce the thermal noise, Fused Silica (FS) was used to make two mirror substrates with 25.4 mm diameters and 6.3 mm thicknesses.

Model of the Finite Element Analysis
In Table 1, the material properties of the ULE and FS are listed, including the elastic properties and the ultimate tensile stress of the structural failure. As described in Figure 1a, three out of six holes located at the midplane are used to fix the USOC, and these three holes are symmetrical at 120 degrees. In the FEA model, there are two kinds of positions of constraint (A and B), which are presented in Figure 1a. Constraint A: six degrees freedoms of the inner surface of the three holes are fully constrained. Constraint B: six degrees freedoms of the surface of the six pads for the three holes are fully constrained. As shown in Figure 2b, the USOC meshes have approximately 200,000 tetrahedral elements, for which each tetrahedron has four nodes. In this work, it was assumed that there was only elastic deformation and no plastic deformation. The shock acceleration loaded on the USOC varied between 10 g (9.8 m/s 2 ) and 600 g (9.8 m/s 2 ). The directions of the shock acceleration included the vertical direction (Z-axis), horizontal direction (X-axis or Y-axis), and three orthogonal directions (X-axis, Y-axis, and Z-axis).

Deformation and Von Mises Stress Characteristics
The deformations and the von Mises stress of the USOC (Ref. [19]) are shown in Figure 2, which is loaded by a 100 g shock acceleration following by the three previously mentioned different directions. In general, the maximum von Mises stress is located at the three holes that fix the cavity and at the edge between the midplane and the spacer. The main reason for this may be that these locations are prone to stress concentration. The von Mises stresses that were located at the spacer and the two mirrors were approximately zero. Moreover, the deformation direction is almost the same as that of the shock acceleration. In that sense, and under the same conditions, it was determined that σvon-Mises along the vertical direction (Z-axis) was greater than that along the horizontal direction (Y-axis).

Deformation and Von Mises Stress Characteristics
The deformations and the von Mises stress of the USOC (Ref. [19]) are shown in Figure 2, which is loaded by a 100 g shock acceleration following by the three previously mentioned different directions. In general, the maximum von Mises stress is located at the three holes that fix the cavity and at the edge between the midplane and the spacer. The main reason for this may be that these locations are prone to stress concentration. The von Mises stresses that were located at the spacer and the two mirrors were approximately zero. Moreover, the deformation direction is almost the same as that of the shock acceleration. In that sense, and under the same conditions, it was determined that σ von-Mises along the vertical direction (Z-axis) was greater than that along the horizontal direction (Y-axis). Figure 3 shows the variation of σ von-Mises in the three directions of loading as a function of the shock acceleration and the thickness T of the mid-plane. Generally, for different thicknesses T of the midplane, a linear relationship will exist between σ von-Mises of the USOC and the shock acceleration with the different shock directions. As the thickness T of the mid-plane increases, σ von-Mises decreases. To protect the ULE USOC from being damaged during a rocket launch, the results show that the thickness T of the mid-plane should be the largest possible without affecting the performance of the USOC. The σ von-Mises loaded by the shock acceleration along the three orthogonal directions (X-axis, Y-axis, and Z-axis) was the largest, compared to the other two directions of loading. The main reason for this may be that the equivalent shock acceleration is the maximum under the same conditions. The σ von-Mises loaded by the shock along the vertical direction was three times that along the horizontal direction under the same conditions, therefore indicating that the launch direction of the USOC should be horizontal. Three orthogonal directions (X-axis, Y-axis, and Z-axis) (the deformation has been amplified by a factor of 5 × 10 3 for demonstration). Figure 3 shows the variation of σvon-Mises in the three directions of loading as a function of the shock acceleration and the thickness T of the mid-plane. Generally, for different thicknesses T of the midplane, a linear relationship will exist between σvon-Mises of the USOC and the shock acceleration with the different shock directions. As the thickness T of the mid-plane increases, σvon-Mises decreases. To protect the ULE USOC from being damaged during a rocket launch, the results show that the thickness T of the mid-plane should be the largest possible without affecting the performance of the USOC. The σvon-Mises loaded by the shock acceleration along the three orthogonal directions (X-axis, Y-axis, and Z-axis) was the largest, compared to the other two directions of loading. The main reason for this may be that the equivalent shock acceleration is the maximum under the same conditions. The σvon-Mises loaded by the shock along the vertical direction was three times that along the horizontal direction under the same conditions, therefore indicating that the launch direction of the USOC should be horizontal.   Figure 4 shows the variation of the σvon-Mises in the three directions of loading as a function of the shock acceleration as well as the diameter φ1 of the holes. Generally, the relationship between σvon-Mises and the shock acceleration with different shock directions with  Figure 4 shows the variation of the σ von-Mises in the three directions of loading as a function of the shock acceleration as well as the diameter ϕ1 of the holes. Generally, the relationship between σ von-Mises and the shock acceleration with different shock directions with different ϕ1 is linear, similar to the relationship found when the T of the mid-plane was varied. The σ von-Mises with ϕ1 = 4.1 mm is larger than that with ϕ1 = 6.1 mm. To protect the ULE USOC from being fractured during a rocket launch with a large shock acceleration, the diameter ϕ1 = 4.1 mm of the holes should be as high as possible. The σ von-Mises loaded Crystals 2021, 11, 998 6 of 10 by the shock acceleration along the three orthogonal directions (X-axis, Y-axis, and Z-axis) was also the largest, compared to the other two directions. We believe that the main reason for this is similar to that of the thickness T of the mid-plane. The σ von-Mises loaded by the shock along the horizontal direction is about one-third of that along the vertical direction under the same condition, which also indicates that the launch direction of the USOC should be horizontal. 2021, 11, 998 6 of 10 different φ1 is linear, similar to the relationship found when the T of the mid-plane was varied. The σvon-Mises with φ1 = 4.1 mm is larger than that with φ1 = 6.1 mm. To protect the ULE USOC from being fractured during a rocket launch with a large shock acceleration, the diameter φ1 = 4.1 mm of the holes should be as high as possible. The σvon-Mises loaded by the shock acceleration along the three orthogonal directions (X-axis, Y-axis, and Z-axis) was also the largest, compared to the other two directions. We believe that the main reason for this is similar to that of the thickness T of the mid-plane. The σvon-Mises loaded by the shock along the horizontal direction is about one-third of that along the vertical direction under the same condition, which also indicates that the launch direction of the USOC should be horizontal.

Effect of the Positions of Constraint on the Maximum von Mises Stress
Two kinds of positional constraints were investigated, and the results of the σvon-Mises loaded with different directions are shown in Figure 5. Generally, for constraints A and B, there was also a linear relationship between σvon-Mises and the shock acceleration with different shock directions, and this relationship was similar to that of the thickness T of the mid-plane and the diameter φ1. When the shock direction is vertical (Z-axis), σvon-Mises of constraint A is larger than that of constraint B. For the horizontal direction (Y-axis), the opposite is true. However, when the shock directions are along the three orthogonal directions (X-axis, Y-axis, and Z-axis), the gap between constraints A and B can be almost negligible. The σvon-Mises loaded by the shock accelerations along the three orthogonal direc-

Effect of the Positions of Constraint on the Maximum von Mises Stress
Two kinds of positional constraints were investigated, and the results of the σ von-Mises loaded with different directions are shown in Figure 5. Generally, for constraints A and B, there was also a linear relationship between σ von-Mises and the shock acceleration with different shock directions, and this relationship was similar to that of the thickness T of the mid-plane and the diameter ϕ1. When the shock direction is vertical (Z-axis), σ von-Mises of constraint A is larger than that of constraint B. For the horizontal direction (Y-axis), the opposite is true. However, when the shock directions are along the three orthogonal directions (X-axis, Y-axis, and Z-axis), the gap between constraints A and B can be almost negligible. The σ von-Mises loaded by the shock accelerations along the three orthogonal directions (X-axis, Y-axis, and Z-axis) was also the largest, compared to the other two directions. The main reason for this was the similarity to the thickness T of the mid-plane and the diameter ϕ1. For constraint A, σ von-Mises loaded by the shock along the horizontal direction was about one-third than along the vertical direction under the same conditions. However, for constraint A, the σ von-Mises loaded by the shock along the horizontal direction tions (X-axis, Y-axis, and Z-axis) was also the largest, compared to the other two directions. The main reason for this was the similarity to the thickness T of the mid-plane and the diameter φ1. For constraint A, σvon-Mises loaded by the shock along the horizontal direction was about one-third than along the vertical direction under the same conditions. However, for constraint A, the σvon-Mises loaded by the shock along the horizontal direction was almost equal to that along the vertical direction under the same conditions. Overall, this also indicates that the launch direction of the USOC should be horizontal.

Shock Acceleration Limit
The maximum shock acceleration that the USOC could withstand is presented in Table 2, where the safety factor is defined as K = 1. Generally, the maximum shock acceleration along the horizontal direction (X-axis or Y-axis) that the USOC can withstand is about two times that of the vertical direction (Z-axis) and three times that of the three orthogonal directions (X-axis, Y-axis, and Z-axis). As the thickness T of the mid-plane and the diameters of the holes increase, the maximum shock accelerations also increase. When T = 10 mm, φ1 = 6.1 mm, and K = 1, the maximum shock acceleration that the optical cavity can withstand is 508 g along the horizontal direction. In summary, for a USOC designed for space applications, based on the results of this work, the optimal launch direction should be the horizontal direction, and the values of T and φ1 should be selected to be as high as possible. (c) Three orthogonal directions (X-axis, Y-axis, and Z-axis). (Figure 1). The cyan dashed line with circles and the magenta dashed line with stars represent σ von-Mises corresponding to constraints A and B, respectively. The red line represents the ultimate tensile stress σ max of the material ULE of the USOC.

Shock Acceleration Limit
The maximum shock acceleration that the USOC could withstand is presented in Table 2, where the safety factor is defined as K = 1. Generally, the maximum shock acceleration along the horizontal direction (X-axis or Y-axis) that the USOC can withstand is about two times that of the vertical direction (Z-axis) and three times that of the three orthogonal directions (X-axis, Y-axis, and Z-axis). As the thickness T of the mid-plane and the diameters of the holes increase, the maximum shock accelerations also increase. When T = 10 mm, ϕ1 = 6.1 mm, and K = 1, the maximum shock acceleration that the optical cavity can withstand is 508 g along the horizontal direction. In summary, for a USOC designed for space applications, based on the results of this work, the optimal launch direction should be the horizontal direction, and the values of T and ϕ1 should be selected to be as high as possible.

Discussions and Conclusions
USOCs as fragile precision instruments are important elements in many domains of space applications. To protect the USOCs from being damaged by shock vibration during the rocket launching stage, this research investigates the effect of the shock on a USOC for space applications in detail through FEA. Compared with previous studies, we give the shock acceleration limit that the USOC can withstand and methods to improve the shock resistance for the first time.
The maximum von Mises stresses of the optical cavity under different conditions were studied, including the thickness T of the mid-plane, the constraint positions, the shock acceleration directions, and the diameter ϕ1 of the holes. Results show that there is a linear relationship between σ von-Mises and the shock acceleration along different directions. As the thickness T and the diameter ϕ1 increase, σ von-Mises decreases. The different constraint positions affect σ von-Mises along the vertical direction and the horizontal direction. However, the gap for the three orthogonal directions can be ignored. Results of FEA simulations also showed that the optimal rocket launch direction of the USOC is horizontal, for which the maximum shock acceleration that the USOC can withstand is about two times that of a launch along the vertical direction and three times that of a launch along the three orthogonal directions.
When T = 10 mm, ϕ1 = 6.1 mm, and K = 1, the maximum shock acceleration that the optical cavity can withstand is about 508 g along the horizontal direction. To protect the USOC in the rocket launch stage, the thickness T and the diameter ϕ1 should be selected to be as high as possible. Noting that the results of our research show that changing the launch direction is also one of the effective methods to improve the shock resistance, which has not been reported in previous studies. Our method can be readily extended to other space optical cavities, thus providing a powerful tool for improving the robustness for shock, which is particularly important for space environments with larger shock accelerations.