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Abstract
Many factors still need to be evaluated to fully understand the physical mechanisms determining optical cavity vibration, which are crucial for designing and constructing a transportable ultra-stable laser. Herein, a detailed dynamic analysis is used to characterize the vibration modes of a transportable optical cavity. The first five resonance modes are identified in the presence of a cavity support, and the guidelines to achieve a high first-order resonance frequency are proposed; thereby, high robustness is described using a 50 mm length optical cavity. The results demonstrate that the first-order resonance of up to 681 Hz with a gross weight of 2.51 kg can be achieved using an optimization strategy for given cavity support. The theoretical results are consistent with simulation and experiment. With the optimal group, a 1.34 Hz linewidth transportable ultra-stable laser at 1550 nm is established, and a linewidth of 1.5 Hz and frequency instability of 9.5×10−15@1s are obtained after the test including 100 km actual road transportation and 34 min continuous vibration. The test condition is equivalent to ∼ 1000 km road transportation. Our method can be readily extended to other transportable optical cavities, providing a powerful tool for improving the robustness of vibration, particularly important for transportable environments.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical cavities have many applications, such as frequency metrology [1–5], gravitational wave detection [6], fundamental physics tests [7,8], and coherent optical links [9,10], where they are used as laser resonators and for laser stabilization, spectroscopy, and power buildup. Ultra-stable lasers can be achieved by locking the lasers onto optical cavities with the Pound-Drever-Hall (PDH) technique, and the ultra-stable laser's frequency instability can be defined by the stability of the optical length of the optical cavity [11,12]. Nevertheless, most ultra-stable lasers have been constrained to operate in well-controlled laboratory environments. There is growing interest in frequency-stable lasers capable of operation outside the laboratory for applications such as space optical clocks, geodesy, tests of fundamental physics in space, and the generation of ultra-stable microwaves for radar. Hence, it is significant to investigate transportable optical cavities to determine whether an ultra-stable laser can operate in a non-laboratory environment [13–30].
However, compared with laboratory applications, all the degrees of freedom of optical cavities for transportable applications must be constrained by the support system to withstand a wide vibrational frequency spectrum and large vibrational inertial forces [16–20]. For example, the highly accurate ground-based time service system (HAGTSS) in China and the Chinese space optical clock (CSOC) project require an optical cavity that can withstand the vibration of 2 g acceleration (1 Hz to 500 Hz) and 15 g acceleration (10 Hz to 2000 Hz), respectively [27,31]. Such vibrations can cause resonance in an optical cavity and result in the fracture of optical cavities machined by ultra-low expansion (ULE) glass [26]. Simultaneously, resonance phenomena will also lead to the deterioration of the ultra-stable laser's frequency noise under normal operation [18]. To avoid the adverse effects of resonance and improve the robustness of anti-vibration, the first-order resonance frequency of an optical cavity needs to be maximized under the condition of limited weight for the requirements of transportation [28,32–33]. Recently, there has been significant progress in simulations of optical cavities’ vibrational modes [17–18,34–35]. For space applications, in 2012, B. Argence et al. simulated the vibration modes of a 100 mm vertical optical cavity with mid-plane support, and the first resonance frequency of the cavity system reached 300 Hz [17]. In 2013, D. R. Leibrandt et al. simulated the vibration modes of a 50 mm spherical optical cavity by finite element analysis (FEA) simulation [18], and the first four resonance frequencies and vibration shapes were given, in which the first resonance frequency was 280 Hz. However, experiments on the vibration modes of transportable optical cavities have not been reported in detail. Herein, the vibration modes of the cavities with their supporting structures are simulated and tested to adapt to the non-lab requirements more effectively to satisfy the urgent needs of transportable lasers of CSOC and HAGTSS in China.
In this work, a strategy is presented for increasing the resonance frequency of an optical cavity mainly by optimizing the support system. The vibrational modes are studied utilizing experiments and FEA simulation. A dynamic analytical model of a 50 mm-long optical cavity made of ULE is established based on the system's vibration transmission characteristics from the vacuum flange to the optical cavity. The first-order resonance frequency is determined by FEA and well agrees with the experimentally measured results. It is necessary to design an optical cavity support system with a high first-order resonance frequency and a low weight required for transportable applications. An orthogonal experiment design (OED) is employed to optimize the cavity support system's performance; this method is more efficient and cost-effective as it only needs a small number of experiments to obtain an optimal group [36]. The modal shapes and vibration transfer function of the optimal group for an optical cavity support system are investigated, and the results reveal that the vibration transfer function obtained by FEA is very close to the theoretical value in this paper. Based on the optimal group, a 1.34 Hz linewidth transportable ultra-stable laser at 1550 nm is established, and the linewidth of 1.5 Hz and frequency instability of 9.5×10−15@1s are obtained after testing for ∼ 1000 km of road transportation, which includes about 100 km actual road transportation followed by continuous vibration for 34 min in the laboratory, equal to approximately 900 km of road transportation. With these results, guidelines are presented for the material selection and the structural design of the transportable cavity that can significantly reduce the experimental time and effectively increase the first-order resonance frequencies under a certain weight for portable systems and space applications. Note that the developed optical cavity with support systems mainly meets the needs of CSOC and HAGTSS.
The paper is organized as follows: In Sec. 2, an optical cavity with the support system is presented, and a simplified dynamic model of an optical cavity with a support system is established. In Sec. 3, we present the simulation and experimental results of an optical cavity's vibrational modes with a support system and provide a strategy and method to improve an optical cavity's resonance frequency. The modal shapes and the transfer function of the optimal group are presented in Sec. 4. The linewidth and stability of a transportable 1550 nm ultra-stable laser based on the optimal group are measured in Sec. 5. Finally, in Sec. 6, we present our conclusions.
2. Design and dynamic model
2.1 Design
Figure 1 shows a transportable optical cavity with a support system [37]. To adapt to a non-laboratory environment with large vibrations and impacts, the optical cavity needs to be fixed and protected by the support system to avoid damage or motion of the optical cavity [13–22,26,28–30]. In Fig. 1, the optical cavity's support system includes thermal shields, a bracket, and a vacuum flange, all of which are fixed to each other by screws. Gaskets are installed between the bracket, the two thermal shields, the vacuum flange, and the screws to isolate vibrations and heat.
Fig. 1. Diagram of a transportable optical cavity with a support system [37], including the bracket, two thermal shields, and the vacuum flange. (a) Cross-sectional view. (b) Photograph of the cavity mounted on the support system. (c) Front view of the cavity. (d) Side view of the cavity.
As shown in Fig. 1, the 50 mm-length optical cavity with a 50 mm diameter is mounted horizontally, and the two mirror substrates with a 12.7 mm diameter have a thickness of 6.3 mm. The fineness of the optical cavity is about 80,000. The radius of curvature is 50 cm, and the coupling efficiency is about 5%. Both the cavity spacer and the mirror substrate are made of ULE glass, which has a very low thermal expansion coefficient. Its optical axis is along the Y-axis for this cavity, and it is fixed to the bracket with four screws. As shown in Fig. 1, the four fixing points’ positions are calculated through extensive simulations to design a cavity with very low vibration sensitivity that is squeeze force-insensitive. The definition of vibration sensitivity of the cavity is referred to in Refs. [38,39]. Finally, the following parameters are selected: Xc = 18.00 mm, Zc = 3.25 mm, and Yp = 5.18 mm, which is a complex trade-off between the vibration sensitivity, fastening interface dimensions, and machining constraints. In this work, the material properties are listed in Table 1, including the density, Poisson ratio, modulus of elasticity, and specific stiffness.
Table 1. Material properties
2.2 Dynamic model
2.2.1 Dynamic equation
Generally, the vibration mainly comes from the Z-axis contribution of the base. Figure 2 shows a one-dimensional dynamic model established to analyze the vibration transmission mechanism based on the vibration transmission path along the Z-axis. The vacuum flange, first thermal shield, second thermal shield, bracket, and optical cavity are equivalent to mass blocks with masses M1, M2, M3, M4, and M5, respectively. The gaskets in Fig. 1 are simplified as springs with damping, where k is the spring stiffness and c is the damping.
The parameter${x_2}$, ${x_3}$, ${x_4}$, and ${x_5}$represent the displacements of the first thermal shield, second thermal shield, bracket, and optical cavity, respectively. ${x_1}$represents the disturbance displacement transmitted from the environmental disturbance to the vacuum flange. The dynamic equation can be obtained as follows [44,45]:
2.2.2 Transfer function
Performing a Laplace transform on Eq. (1), the transfer function $G(s)$ between the vibrational acceleration of the optical cavity after the support system and the vibration acceleration transmitted from the environmental disturbance to the vacuum flange is [44,45]
According to Eq. (2), the resonance frequencies and the acceleration magnification of the optical cavity with the support system in the Z-axis direction are obtained. Equation (2) can guide the design of anti-vibration of the transportable optical cavity. To reduce the effect of the vibration along the Z-axis, the resonance frequency and the acceleration magnification in the Z-axis direction can also be optimized by using the theoretical formula.
3. Optimization
3.1 General scheme and strategy
The optical cavity with the support system should have high first-order resonance frequencies, low weight, and better robustness to the strong vibrations [28,32–33]. Thus, experiments and FEA simulations are conducted to provide guidelines for designing an optical cavity with a support system with high first-order resonance frequencies and low weights. In this work, the influence of the vacuum flange materials, thermal shields, bracket, and gaskets on the first-order resonance frequencies and weights is of primary consideration. Since the OED method has the advantage of simultaneously considering multiple factors and objectives and reducing the number of experiments, it is used to obtain an optimum scheme between first-order resonance frequency and low weight for optical cavity systems. Furthermore, we perform a statistical analysis of the sum of squared deviations to determine the most significant parameters and each parameter's contribution rate. With this method and statistical analysis, the optimal parameter combination of the system is predicted. The experimental factors and their magnitudes are listed in Table 2. For a problem with a factor of 4 and a level of 3, it takes 81 repetitions to perform all experiments. However, using the 4-level orthogonal table L9 to carry out the system's orthogonal experimental design only requires nine repetitions under the premise of ensuring the objective results of the experiment, where 4-level and L9 present four variables and nine orthogonal experimental combinations, respectively.
Table 2. Factors and levels of orthogonal experimental design
3.2 Experiment setup
The block diagram and site of the test system are shown in Fig. 3. There are two accelerometers, one for the response and the other for the excitation. Accelerometer 1 is located on the top of the optical cavity with a strong adhesive and yellow adhesive tape, which protects the optical cavity from damage and facilitates the accelerometer's removal. The other is located on the top of the vibration table and fixed with screws to measure the excitation. This cavity system is fixed to the vibration table through four clamps. Signal wire and anti-static stickers are used to reduce the influence of vibrations on the accelerometers. The torque on the M3 screws is 0.5 N.m to prevent the M3 screws from crushing and breaking the cavity, and the torque on the M6 screws is 2.0 N.m to prevent excessive deformation of gaskets. The frequency-domain of the test sweep is 10 to 2000 Hz, and the acceleration amplitude is 1 g. The error between the actual amplitude and the set amplitude of the vibration table excitation amplitude is controlled within 10%. The total sweep time is 7 min and 39 s.
3.3 Optimization results
To obtain an optical cavity with a higher first-order resonance frequency and lower weight, it is necessary to perform FEA and experiments to optimize the various parameters. The OED method is used to arrange the experiments for two different optimization indices, including first-order resonance frequencies f, and weights m. The orthogonal experiment scheme and its experimental results are shown in Table 3 and Fig. 4. I, II, and III represent the FEA results by fixing the underside, upside, and installing grooves of the vacuum flange in Fig. 1, respectively. In Table 3, the specific stiffness is the ratio of the elastic modulus to the density, which is referenced in Table 1. Based on OED, the contributions of the four factors to the first-order resonance frequency and weight of the system are shown in Table 4, where the four factors are A, B, C, and D from Table 2.
Fig. 4. Results of the first-order resonance frequency including FEA calculation and measurement. Blue solid line with squares, cyan dashed line with triangles, and magenta solid line with stars represent the FEA results by fixing the underside, upside, and installing grooves of the vacuum flange in Fig. 1, respectively. Red solid line with circles represents the results of experimental measurement.
Table 3. Orthogonal experiment scheme and experiment results
Table 4. Range and contribution analysis of orthogonal test results
As shown in Fig. 4, the first-order resonance frequencies calculated by the FEA simulations strongly agree with the experimental values. The average difference of first-order resonance frequencies between the FEA simulation and experimental is approximately 10%. The group with the largest and smallest differences is No. 9 and No. 3, respectively, with differences of about 19% and 2%. There are four possible explanations for these differences. First, in the experiments, four clamps are used to fix the vacuum flange and the vibration table, and the fixing configuration for the FEA calculations is not entirely equivalent. Second, there are errors in the assembly and machining and differences in materials’ mechanical properties between the FEA simulations and experiments. Third, there is a specific difference in the pre-tightening force between installing each component using screws in the experiment and the process of finite element analysis. Fourth, an ideal friction is adopted in the process of the FEA model, while that between the various devices in the actual model is very complex.
The orthogonal experiment results show that when one index is optimized, the other index is not optimal, revealing a contradiction between the two optimization conditions. The range and contribution analysis of the orthogonal test results are shown in Table 4.
In Table 4, Rf1, Rf2, and Rf3 show the ranges of first-order resonance frequencies f calculated by FEA simulation including I, II, and III. Rf is the range of first-order resonance frequencies measured by experiments. Pf1, Pf2, and Pf3 are the contribution percent of four factors to the first-order resonance frequencies f calculated by FEA, and Pf is that measured by experiments. Pm is the contribution percent of four factors to weight m. It can be seen from Table 4 that when the optimization index is the first-order resonance frequencies f, the optimal combination calculated by FEA is A2B2C3D2, and the orders of Pf1, Pf2, and Pf3 are C > B > A > D, C > B > A > D, and C > A > B > D, respectively. The order of Pf is C > A > B > D, which is in accordance with the result Pf3 of FEA III, and it verifies that the FEA III model should be used as far as possible for similar simulation analysis. When the optimization index is weight m, the optimal combination is A3B3C2D3 or A3B3C3D3, and the order of Pm is A > B > D > C. Based on the contributions of four factors to first-order resonance frequencies f and weight m, the optimal combination A2B2C3D2 is finally chosen as a trade-off.
As the optimized combination A2B2C3D2 is not arranged in the simulation and measurement test in Table 3, we need to arrange an experiment to verify the optimized combination. The experiment results are shown in Table 5.
Table 5. Optimization group and experiment results
4. Modal shape and transfer function of the optimal group
4.1 Modal shape
From the modal shapes of FEA in Table 6, we can know the optical cavity's deformation trend at a particular natural resonance frequency, which can help avoid the direction of mechanical vibration in the working environment coincides with that of resonance vibration for the optical cavity. Table 6 shows that the optical cavity vibrates along the X-axis with the YZ plane as a symmetry plane for the first and the fifth modes. For the second mode, it vibrates along Y-axis with the XZ plane as a symmetry plane. For the third mode, the cavity twists along the Z-axis and vibrates up and down along the Z-axis for the fourth mode. The results show the stiffness of the cavity in the X-axis direction is weakest and should be strengthened as much as possible.
Table 6. First five vibrational modal shapes
4.2 Transfer function
Generally, compared with other directions, the vibration amplitude along the Z-axis is significant. It is also important to calculate the resonance frequencies and the acceleration magnification along the Z-axis. The transfer functions of the vibration along the Z-axis of the optimal group in Table 5 are calculated by Eq. (2) and simulated by FEA, which are presented by the red and blue dotted lines, respectively, in Fig. 5, and it shows the two results are in agreement. There is also a deficiency between the FEA simulation and analytical calculation: the magnitude has an approximate difference of 18 dB, and the average difference of frequency is approximately 20% at the resonance peaks. There are two possible explanations for these differences. First, structural damping is not considered in the analytical calculation. Second, the optical cavity is considered an elastic solid in the FEA model, whereas it is simplified as a mass block in the analytical calculation. Equation (2) can guide the design of anti-vibration of the transportable optical cavity to reduce the effect of vibration along the Z-axis.
Fig. 5. Results of the transfer function of the optimal group. The blue dash line and red dot line represent Eq. (2) and FEA, respectively.
5. Development of a transportable ultra-stable laser based on the optimal group
As shown in Fig. 6, a transportable ultra-stable laser system is established using the optimal group. A commercial fiber laser (NKT Photonics Koheras Adjustik-E15) operating at 1550.12 nm with a linewidth of ∼ 2.7 kHz and the homemade electronics of PDH frequency stabilization are used here. To verify the vibration reliability of this ultra-stable laser, it is transported 100 km roundtrip by a car from the National Time Service Center (NTSC) of the Chinese Academy of Sciences to AVIC Aircraft Strength Research Institute (ASRI) and then vibrated continuously for 34 min on the vibration table in the AVIC-ASRI equaling to about 900 km road transportation. The data of this vibration test is shown by an inset in Fig. 6, where the red line represents the acceleration power spectral density (PSD) on the box of the ultra-stable laser. The results show that After the vibration test, this transportable ultra-stable laser still maintains excellent performance without any adjustment. The measurement results are shown in Fig. 7.
Fig. 7. Measurements of linewidth and fractional frequency instability of the transportable ultra-stable laser.
The linewidth and frequency instability was measured by comparing it with a 0.3 Hz linewidth ultra-stable laser in our laboratory to evaluate this transportable laser's performance. The beat note between two ultra-stable lasers is directly measured to obtain the linewidth using a fast Fourier transformer (FFT), and the single measurement time is 1 s allowing for a sufficiently high-frequency resolution bandwidth (RBW) of 1 Hz. In Fig. 7(a), the linewidth of this transportable ultra-stable laser is 1.50 Hz (blue line), which was 1.34 Hz (green line) before the vibration test. The linewidth is obtained by Lorentz fitting. The beat note is directly measured by a frequency counter working on Λ-mode (Agilent 53230a) to estimate the frequency instability of this transportable ultra-stable laser. After removing a linear drift of about 1.0 Hz/s, the red line in Fig. 7(b) shows that the fractional instability is approximately 8.0×10−15 at the 1 s averaging time. The fractional instability is approximately 9.5×10−15 at the 1 s averaging time, and this is shown by the black line in Fig. 7(b) after the vibration test, and the linear drift removed is 0.93 Hz/s. The above results for linewidth and stability are consistent with those before the vibration test, indicating that this transportable ultra-stable laser has extremely high reliability. Since the optical cavity with a low coupling efficiency for this transportable ultra-stable laser results in a considerable residual amplitude (RAM) noise, the contribution of the RAM noise on frequency instability is around = 6×10−15, averaging time from 0.4 s to 10 s shown by the blue line in Fig. 7(b). The cavity's thermal noise floor-induced frequency instability equals 1.6×10−15 (gray line in Fig. 7(b)). For long averaging times, the instability is mainly subjected to the locking point drift caused by temperature fluctuation and lower efficiency coupling leading to a more sensitive locking point.
6. Conclusion
In this work, we characterized the vibration modes of a transportable optical cavity in detail through dynamic analysis, FEA, and experiments. The theoretical results were consistent between the simulation and experiment. The reasons for the differences between the FEA and experiment results are provided. Based on the orthogonal experiment design, the optimal group A2B2C3D2 with high first-order resonance frequencies and low weights is obtained. The factors that contributed to the first-order resonance frequencies and weight the most and the least are provided. We investigated the transfer function of the optimal group by analytical calculation and FEA simulation. The transfer functions determined by the FEA simulations agree with the analytical calculation values by about 80%. Based on the optimal group, a 1.34 Hz linewidth transportable ultra-stable laser at 1550 nm was established and tested by about 1000 km road transportation. In this work, we presented the guide for the material selection and the structural design of the transportable cavity, which can save experimental time cost and effectively increase the first-order resonance frequencies under the condition of a certain weight, such as in portable and space applications. Our method can be readily extended to other transportable optical cavities, providing a powerful tool for improving the robustness of vibration, which is particularly important for space and transportable environments.
Funding
Youth Innovation Promotion Association of the Chinese Academy of Sciences; National Natural Science Foundation of China (11903041); Young Innovative talents of the National Time Service Center of the Chinese Academy of Sciences.
Acknowledgment
We would like to acknowledge contributions of the vibration transmission theory of the optical cavity to this paper from Prof. X. Zhang, Dr. J. Liang, Dr. Q. Li, and Prof. L. Liu of Northwestern Polytechnical University, and Prof. F. Sun of National Time Service Center. The project is partially supported by the Youth Innovation Promotion Association of the Chinese Academy of Sciences, the Chinese National Natural Science Foundation (Grant No. 11903041), and the Young Innovative talents of the National Time Service Center of the Chinese Academy of Sciences.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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