A Spaceborne Mounting Method for Fixing a Cubic Fabry–P é rot Cavity in Ultra-Stable Lasers

: A spaceborne mounting method for ﬁxing a 100 × 100 × 100 mm cubic Fabry–P é rot (FP) cavity is presented. The method constrains the FP cavity in eight directions with a titanium bracket, PEEK gaskets, and vacuum glue. Three criteria were proposed for judging whether the FP cavity is offset or not during aerospace vibration tests. Results indicate that the proposed method is a reliable, robust, and stable way to ﬁx a cubic FP cavity in ultra-stable lasers (USLs). This approach paves the way for the use of USLs in space.

USLs use the Pound-Drever-Hall (PDH) technique to lock continuous-wave lasers to the resonance frequency of a high-finesse Fabry-Pérot (FP) cavity via large-bandwidth and high-speed electronic control systems [11,12]. The fractional frequency instability of a laser is determined by the FP cavity's length instability [13]. Differently shaped FP cavities, including cylindrical, spherical, cubic, notched, multiple-bore, midplane, etc., have been proposed [14][15][16][17][18]. Among them, the symmetrical cubic FP cavity has lowest forceand vibration-insensitive design, which seems to be the best choice for space missions. Based on cubic FP cavities, a four-point tetrahedral mounting method was proposed for nonlaboratory applications [19]. With this configuration, a few ground-transportable USLs have been developed [13,20,21]. However, to the best of our knowledge, no studies describing USLs that can withstand aerospace vibration tests have been reported.
In the present work, a spaceborne mounting method for fixing a 100 × 100 × 100 mm cubic FP cavity is proposed since the FP cavity is the weakest part of a USL during periods of vibration. Based on the proposed method, aerospace vibration tests of the FP cavity were carried out. To judge the performance of the FP cavity during vibration tests, three criteria are also proposed.

Model for Mounting the FP Cavity
To meet the requirements of the space-based strontium optical clock in the China Space Station, the frequency instability of the ultra-stable laser should be lower than 2 × 10 −15 /s. Therefore, a homemade FP cavity with a cubic shape was first fabricated, as shown in Figure 1. The spacer of the FP cavity, which is made of ultralow-expansion (ULE) glass, has a size of 100 × 100 × 100 mm. To obtain minimum acceleration sensitivity, the spacer's eight vertices were all truncated to a depth of 18.3 mm toward the center of the spacer. Two fused silica mirrors with radius curvatures of infinity and 1000 mm were optically attached to the spacer, along with Two mirrors, both with a diameter of 25.4 mm and a thickness of 6 mm. Additionally, two ULE rings, with an outer diameter of 25.4 mm, an inner diameter of 10 mm, and a thickness of 6 mm, were optically attached to the mirrors to compensate for the thermal mismatch between the ULE spacer and the fused silica mirrors. been reported.
In the present work, a spaceborne mounting method for fixing a 100 × 100 × 100 mm cubic FP cavity is proposed since the FP cavity is the weakest part of a USL during periods of vibration. Based on the proposed method, aerospace vibration tests of the FP cavity were carried out. To judge the performance of the FP cavity during vibration tests, three criteria are also proposed.

Model for Mounting the FP Cavity
To meet the requirements of the space-based strontium optical clock in the China Space Station, the frequency instability of the ultra-stable laser should be lower than 2 × 10 −15 /s. Therefore, a homemade FP cavity with a cubic shape was first fabricated, as shown in Figure 1. The spacer of the FP cavity, which is made of ultralow-expansion (ULE) glass, has a size of 100 × 100 × 100 mm. To obtain minimum acceleration sensitivity, the spacer's eight vertices were all truncated to a depth of 18.3 mm toward the center of the spacer. Two fused silica mirrors with radius curvatures of infinity and 1000 mm were optically attached to the spacer, along with Two mirrors, both with a diameter of 25.4 mm and a thickness of 6 mm. Additionally, two ULE rings, with an outer diameter of 25.4 mm, an inner diameter of 10 mm, and a thickness of 6 mm, were optically attached to the mirrors to compensate for the thermal mismatch between the ULE spacer and the fused silica mirrors. As shown in Figure 2a, the FP cavity was first fixed into a titanium bracket with eight titanium screws supporting its eight vertices. Gaskets, which were made of PEEK, were installed between the screws and the spacer to isolate heat conduction. The contact faces between the gaskets and the spacer and between the gaskets and the screws were glued together. The squeeze forces of all of the screws were set to 100 Nm. To ensure the reliability of the cavity, the screws and the titanium bracket were laser welded. Subsequently, the bracket was fixed to the inner heat shield with four titanium screws. As depicted in Figure 2b, three heat shields were used to avoid the influence of temperature fluctuation of environment. Gaskets made of PEEK were placed between the different heat shields to isolate heat conduction. Finally, heat shields were fixed to the vacuum house using titanium screws. Two ion pumps and a getter pump were used to maintain a high vacuum (<2 × 10 −6 Pa) in the vacuum hose. A total of 12 vibration isolation pads were also attached to the vacuum hose to isolate high-frequency vibrations, as shown in Figure 2c. As shown in Figure 2a, the FP cavity was first fixed into a titanium bracket with eight titanium screws supporting its eight vertices. Gaskets, which were made of PEEK, were installed between the screws and the spacer to isolate heat conduction. The contact faces between the gaskets and the spacer and between the gaskets and the screws were glued together. The squeeze forces of all of the screws were set to 100 N·m. To ensure the reliability of the cavity, the screws and the titanium bracket were laser welded. Subsequently, the bracket was fixed to the inner heat shield with four titanium screws. As depicted in Figure 2b, three heat shields were used to avoid the influence of temperature fluctuation of environment. Gaskets made of PEEK were placed between the different heat shields to isolate heat conduction. Finally, heat shields were fixed to the vacuum house using titanium screws. Two ion pumps and a getter pump were used to maintain a high vacuum (<2 × 10 −6 Pa) in the vacuum hose. A total of 12 vibration isolation pads were also attached to the vacuum hose to isolate high-frequency vibrations, as shown in Figure 2c.

Experimental Results and Discussion
To verify the feasibility of the developed FP cavity for use in space applications, aerospace environmental tests were carried out at Hangzhou Saibo Mechanics Environment Test Co., Ltd. (Gongshu District, Hangzhou, China), since the alignment requirements of the high-finesse FP cavity are extremely high. Additionally, since the ULE

Experimental Results and Discussion
To verify the feasibility of the developed FP cavity for use in space applications, aerospace environmental tests were carried out at Hangzhou Saibo Mechanics Environment Test Co., Ltd. (Gongshu District, Hangzhou, China), since the alignment requirements of the high-finesse FP cavity are extremely high. Additionally, since the ULE glass was fixed inside the vacuum hose, it was difficult to judge whether the FP cavity was offset or not. Therefore, optical systems were attached to the vacuum hose, both in front of and behind the FP cavity. Characterizing the optical parameters of the whole system (WS) can reveal information about the FP cavity. Figure 3 shows the optical systems in front of and behind the FP cavity. Figure 3a illustrates how the laser beam is aligned with the FP cavity in the optical system. A collimating lens (CL) first collimates the laser beam using a polarization-maintaining singlemode (PMSM) fiber (PM630-HP, Coherent). Since the laser radiates from the PMSM fiber, it possesses a mode distribution of TEM 00 and an M 2 of less than 1.1 [22]. Besides collimation, the CL can also guarantee a good mode-match between laser beam and the FP cavity. After collimation, a 90:10 beam splitter (BS1) is used to split the laser beam. A 10% proportion of the laser beam is reflected into a photodiode detector (PD1) to monitor the laser power. The transmitted laser beam is then precisely aligned by four wedged prisms (WP1 and WP2 each hold two prisms). The four wedged prisms all have a wedge angle of 0.1 • . The rotation of each wedged prism leads to a fine adjustment of the laser-beam pointing. Subsequently, the laser is transmitted through a polarization beam splitter (PBS) and a quadra-wave (QW) plate that is aligned perpendicularly to the center of the FP cavity. A photodiode detector (PD2) collects the reflected laser beam from the FP cavity to monitor error signals during laser frequency tuning. When the laser frequency is consistent with the resonance frequency of the FP cavity, the laser beam will be transmitted through the cavity. Figure 3b shows the optical system used to collect the transmitted laser beam. As shown in Figure 3b, the transmitted laser beam is split into two parts by a 50:50 beam splitter. The charge-coupled device (CCD) can monitor the mode distribution of the transmitted laser beam, whereas, the photodiode detector (PD3) can monitor the amplitude of the intensity of the transmitted laser beam. All of the optical elements in the optical systems, both in front of and behind the FP cavity, are glued to the titanium mirror mounts. Theses mounts are then screwed and glued to two titanium substrates (in front of and behind the FP cavity). Two substrates are screwed and glued to the vacuum hose. Thus, there is no spring-adjustment structure anywhere in the optical system.  Two vibration tests, including sinusoidal and random tests, were carried out. The parameters of the above-mentioned tests are shown in Table 1. All of these parameters are based on the Long March 5 rocket platform [23]. It should be noted that the unit of the amplitude of the sinusoidal vibration is different in different frequency ranges. When the vibration frequency is in the range of 4-10 Hz, the unit "mm" is used to describe the peakto-peak distance of the sinusoidal vibration, whereas the unit "g" is used to describe the acceleration of the sinusoidal vibration in other frequencies. The parameters shown in Table 1 are stricter than the actual conditions during a rocket launch so as to ensure the reliability of the payloads. Sinusoidal and random vibration tests were carried out sequentially. During each vibration test, the FP cavity was tested on three mutually Two vibration tests, including sinusoidal and random tests, were carried out. The parameters of the above-mentioned tests are shown in Table 1. All of these parameters are based on the Long March 5 rocket platform [23]. It should be noted that the unit of the amplitude of the sinusoidal vibration is different in different frequency ranges. When the vibration frequency is in the range of 4-10 Hz, the unit "mm" is used to describe the peak-to-peak distance of the sinusoidal vibration, whereas the unit "g" is used to describe the acceleration of the sinusoidal vibration in other frequencies. The parameters shown in Table 1 are stricter than the actual conditions during a rocket launch so as to ensure the reliability of the payloads. Sinusoidal and random vibration tests were carried out sequentially. During each vibration test, the FP cavity was tested on three mutually perpendicular axes (X, Y, and Z).  Figure 4 shows the WS (including the FP cavity, vacuum hose, ion pumps, and optical systems in front of and behind the cavity) during the vibration tests. In order to analyze whether the WS could withstand the vibration tests, three criteria were proposed.
Two vibration tests, including sinusoidal and random tests, were carried out. The parameters of the above-mentioned tests are shown in Table 1. All of these parameters are based on the Long March 5 rocket platform [23]. It should be noted that the unit of the amplitude of the sinusoidal vibration is different in different frequency ranges. When the vibration frequency is in the range of 4-10 Hz, the unit "mm" is used to describe the peakto-peak distance of the sinusoidal vibration, whereas the unit "g" is used to describe the acceleration of the sinusoidal vibration in other frequencies. The parameters shown in Table 1 are stricter than the actual conditions during a rocket launch so as to ensure the reliability of the payloads. Sinusoidal and random vibration tests were carried out sequentially. During each vibration test, the FP cavity was tested on three mutually perpendicular axes (X, Y, and Z).  Figure 4 shows the WS (including the FP cavity, vacuum hose, ion pumps, and optical systems in front of and behind the cavity) during the vibration tests. In order to analyze whether the WS could withstand the vibration tests, three criteria were proposed. First, we monitored the direct current (DC) component of the voltage of the PD2. The PD2 is based on a silicon PIN photodiode (S9055, Hamamatsu). The width of the photosensitive area of the PD2 is 0.2 mm. In order to capture more laser power, a focus lens with a focal length of 2.75 mm was used in front of the photodiode chip of the PD2. Consequently, the chip was fixed to the focal point of the lens. The diameter of the laser beam radiated on the PD2 (before the focus lens) is ca. 400 µm. Based on the Gaussian distribution and above-measured diameter of the laser beam, the diameter of the laser spot at the focal point of the lens is ca. 9.34 µm (calculated using Wyrowski VirtualLab Fusion software). Considering the focal length of the lens, the voltage of the PD2 can remain the same when the tilt angle of the laser beam is within range of Therefore, monitoring the voltage of the PD2 can roughly determine whether the optical system, especially the fixed FP cavity, has a severe tilt or not during aerospace vibration tests. This is important for protecting the system because of the high expense of the FP cavity. During the vibration tests, a frequency tunable laser was connected to the fiber from the CL. The laser power at the position of the PD2 was ca. 55 µW. The voltage of the PD2 was always within the range of 2.13 ± 0.01 V during the vibration tests, indicating that the FP cavity did not have a severe offset.
Second, we monitored the voltage of the PD3 while performing laser frequency tuning over 1 free spectral range (FSR). The FSR of the cavity can be determined by the following equation: where c is light speed and L is the length of the cavity. Therefore, the FSR of the 100 × 100 × 100 mm cubic cavity is 1.5 GHz. Because the voltage of the PD2 can only indicate whether the FP cavity has a severe offset or not, the voltage of the PD3 was used to judge the offset of the FP cavity more precisely. When the laser is aligned well with the FP cavity, modes are excited. Tuning the laser frequency causes the laser to resonate with the FP cavity at certain frequencies. At these frequencies, the laser is transmitted through the FP cavity. Then, the voltage will be generated on the PD3. using this criterion, a few milliradians of tilt can be identified (as simulated in the OSCAR toolbox in MATLAB) [24]. The WS was first adjusted to a condition under which only the TEM00 mode could resonate with the FP cavity in 1 FSR before the vibration tests. Figure 5 shows the voltage of the PD3 and the frequency of the laser when the laser is being tuned at a certain speed. The wavelength of the laser was around 698.447 nm (429.227 THz in the frequency domain). As shown in Figure 5, the frequency of laser was tuned over an FSR 25 times. Because the tuning range was bigger than 1 FSR, two voltages could be recorded by the PD3 during each FSR tuning. Thus, 50 voltage data sets for the PD3 were recorded. It should be noted that the voltage of the PD3 in Figure 5 was not always the same; this was caused by the sampling and responding rate of the PD3 and the oscilloscope. A total of 10 sets of voltage data for PD3, which was acquired at the same time as the data shown in Figure 5, before and after the vibrations were recorded. Based on these data, the values of the average, standard deviation, and variance of each set of voltage data were calculated, which are shown Table 2.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 8 Figure 5. Voltage of PD3 during laser frequency tuning.  From Table 2, it can be noted that the values of the average, standard deviation, and variance are basically the same before and after the variation tests. The differences are due to the sampling and responding rates of the PD3 and the oscilloscope. Additionally, only the fundamental mode (TEM00) resonated in the cavity, which could be observed using the charge-coupled device (CCD) camera. Therefore, there is no obvious offset of the cavity that could influence the mode resonating in the cavity. Monitoring the voltage of the PD3 during the vibration tests can also indicate whether the FP cavity has a slight offset or not. If the voltage of the PD3 drops obviously or disappears, the FP cavity is definitely offset.
Third, we monitored the transmission and error signals of the FP cavity while the frequency-tuning laser was locked to the FP cavity. Although the first and second criteria can help one to judge whether the FP cavity is offset or not during the vibration tests, this is limited to the sampling and responding rate of the PD3 and the oscilloscope. So, these judgment criteria are still not clearly quantified. To solve this problem, a Pound-Drever-Hall (PDH)-technique-based ultra-stable laser system was applied. After modulation using an electro-optic modulator (EOM), the frequency tuning laser was locked onto the FP cavity with a high-speed PID electronic control system. Before and after the vibration tests, the transmission and error signals of the FP cavity were recorded to judge whether the FP cavity had an offset or not. The transmission signal can be directly obtained from the PD3, whereas the error signal can be obtained by mixing the alternating current (AC) signal of the PD2 with local oscillating signal of the EOM.  Figure 6 shows the transmission and error signals before and after the vibration tests. The insets show the mode distribution of the transmitted laser beam as captured by the CCD camera. Based on the data shown in Figure 6, the values of the average, standard deviation, and variance of the transmission and the error signals before and after the vibration tests were also calculated. The results are shown in Table 3, and the values from before and after the vibration tests are basically the same. This indicates that the aerospace vibration tests did not have obvious influence on the FP cavity or the entire optical system.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 8 CCD camera. Based on the data shown in Figure 6, the values of the average, standard deviation, and variance of the transmission and the error signals before and after the vibration tests were also calculated. The results are shown in Table 3, and the values from before and after the vibration tests are basically the same. This indicates that the aerospace vibration tests did not have obvious influence on the FP cavity or the entire optical system.  From the above-mentioned results of the three criteria, the proposed mounting method successfully fixed a 100 × 100 × 100 mm cubic FP cavity during aerospace vibration tests. The performance levels of the USLs were not influenced by the vibration tests, indicating that this method could be a good way to fix cubic cavities to space-based USLs.  From the above-mentioned results of the three criteria, the proposed mounting method successfully fixed a 100 × 100 × 100 mm cubic FP cavity during aerospace vibration tests. The performance levels of the USLs were not influenced by the vibration tests, indicating that this method could be a good way to fix cubic cavities to space-based USLs.

Conclusions
In summary, a spaceborne mounting method for fixing a 100 × 100 × 100 mm cubic FP cavity to USLs was proposed. A system, including an FP cavity in front of and behind the FP-cavity optical system and vacuum pump, was built based on the proposed method. Aerospace vibration tests of the whole system were carried out. To validate whether there was an offset or not during the vibration tests, three criteria were also proposed. The differences in the results for the whole system before and after the aerospace vibration tests were negligible for the three criteria. Thus, the proposed mounting method could be a good way to fix cubic FP cavities to USLs for space applications.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.