Open Access
Add to BibSonomy
Abstract
State-of-the-art laser frequency stability has been pushed to the 10−17 level. The laser reference cavity is typically nested in a multi-layer thermal enclosure to increase vacuum thermal time constant and thermally controlled at the zero-thermal-expansion temperature to reduce the external temperature fluctuation effect. It is rather time consuming to accurately determine the zero-thermal-expansion temperature for a large thermal time constant system. Here we develop a fast method for measuring the zero-thermal-expansion temperature of the cavity by relying on just one single temperature scan. We first develop a theoretical model to predict the performance of the laser locked to the reference cavity, and then construct an evaluation system for verification of the model. The zero-thermal-expansion temperature of a 30-cm cavity is measured to be 4.3±0.5 °C. The fast and high precision method for determining the zero-thermal-expansion temperature will be valuable in improving long-term frequency stabilities of cavity stabilized lasers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ultra-stable lasers based on optical reference cavities play an important role in modern precision measurement physics, such as optical clocks, gravitational wave detection, and optical generation of low phase noise microwave signals [1–3]. State-of-the-art, ultra-stable lasers using longer cavities or cryogenic cavities show fractional frequency instabilities on the order of $10^{-17}$ under strict environmental control [4–7]. As the cavity length stability is the key factor in determining the stability of the locked laser, great efforts have been made to reduce the influence of environmental perturbations on cavity length. One of the most important factors affecting the cavity stability is the external temperature fluctuation.
Cavity spacer materials such as ultra low expansion glass (ULE), Zerodur, fused silica (FS), silicon, and sapphire show very low coefficient of thermal expansion (CTE) or zero CTE at certain temperatures. Those temperatures shall be the optimal setting points of thermostats. The frequency instability $\sigma _y$ induced by temperature fluctuation of the cavity $\sigma _T^c$ can be approximated as $\sigma _y \approx A \left |\delta T\right |\sigma _T^c$ with a small offset $\delta T$ from the zero-CTE temperature, where $A$ is a constant related to the changes of the cavity’s thermal expansion property. Therefore, an accurate measurement of the zero-CTE temperature can help reduce the cavity’s sensitivity to the temperature fluctuation. It can greatly improve the long-term performance of cavity stabilized ultra-stable lasers, expanding their application scopes like the space-based cavity stabilized lasers of Laser Interferometer Space Antenna (LISA), Gravity Recovery and Climate Experiment (GRACE) Follow-On mission, and TianQin, etc. [8–10]. For frequency noise less than 30 $\textrm{Hz}/\sqrt{\textrm{Hz}}$ at 1 mHz, assuming $\sigma _T^c$ of a cavity made of ULE is at the level of 1 $\mu {\textrm K}$, then the deviation $\delta T$ from the set point temperature to the zero-CTE temperature should be less than 1 ${\textrm K}$.
Given the importance of the zero-CTE temperature, many methods have been developed to measure the thermal properties of cavities. Thermal properties of raw materials used for manufacturing cavities can be measured through dilatometer measurements, Fizeau interferometry, photoelastic measurements, ultrasonic speed measurements, and phase measuring interferometers [11–14]. But taking into account of the inhomogeneity of materials and the development of hybrid cavities [15], it is more appropriate to measure the thermal expansion properties of the whole cavity. Multiple-beam Fabry-Pérot (FP) interferometry has been used to measure the CTE of a sample FP etalon spacer before, and the measurement precision of ${\textrm 1}\times 10^{-9}/\textrm{K}$ is limited by the frequency instability of the laser used [16]. With the development of high-finesse cavities and improved frequency instability of cavity stabilized lasers, the most widely used method of measuring the zero-CTE temperature of cavities is to record the frequency beat note between a laser stabilized on the cavity under test and a frequency reference. The frequency reference can be another stable reference laser, or a frequency comb referenced to an atomic clock, or an atomic transition line [17–19]. These methods usually need to change the temperatures of cavities several times or to scan the temperature periodically in a wide-range to obtain the relationship of the cavity resonant frequency and the cavity temperature. Unfortunately, for ultra-stable lasers with very large time constant thermal shields, the process of changing and waiting for temperature to be settled down can be very time-consuming. Faster measurements have been realized using lower vacuum or better thermal contact, but the thermal time constant can still be several hours long [15,18,20]. Furthermore, the temperature gradient of the vacuum chamber and the cavity may be different between the testing environments and the actual running environment.
In addition to the zero-CTE temperature, thermal time constant of the vacuum housing of the ultrastable cavity is also a very important parameter that needs to be determined. In fact, zero-CTE temperature and thermal time constant decide different features of the cavity frequency response to temperature change curve. Here we propose a new fast method to measure the thermal properties of cavities. Suppose we have a rough knowledge of the cavity zero-CTE temperature, we can then induce a step temperature change of the cavity which crosses the cavity zero-CTE temperature, and it is possible to find all the thermal parameters simultaneously with only one single step measurement. In addition, the thermal model based on the measured data can be used to predict frequency drift rate of the cavity-stabilized lasers caused by temperature changes. Compensation of the frequency drift by using only the real-time temperature data will be very useful in applications like space-based ultra-stable lasers.
In this paper, we investigate the relationship between frequency and temperature of a cavity-stabilized laser and present a fast method of measuring the zero-CTE temperatures of optical reference cavities. The theoretical model is discussed in Sec. 2. We then give a description of the experimental setup including two 1070 nm ultra-stable lasers and a frequency comb referenced to a hydrogen maser in Sec. 3. The experimental processes and results of the zero-CTE temperature and other thermal parameters measurements are discussed in Sec. 4. The applicability of our method is discussed in Sec. 5. We conclude in Sec. 6.
2. Thermal model of the system
2.1 Thermal properties of the cavity
The cavity length will deviate from a certain value while sensing the temperature fluctuation, resulting in a fluctuation of the laser frequency. When the temperature of the cavity alters from $T_i$ to $T_j$, the variation of the laser frequency can be expressed as
2.2 Relationship between frequency and the setting temperature
In an ultra-stable laser system, an active temperature control is usually applied to the vacuum chamber, and several internal thermal shields are designed as a low pass thermal filter of the temperature fluctuation. The vacuum thermal time constant $\tau$ characterizes the process that the temperature response of the cavity to the temperature fluctuation of the control point. To keep good thermal isolation, no temperature sensor is installed on the cavity directly. Therefore, the temperature of the cavity is estimated by the temperature fluctuation of the active control and the thermal time constant $\tau$. For a step change from $T_1^{set}$ to $T_2^{set}$ on the active temperature control of the outer layer thermal shield, the instantaneous temperature of the cavity $T^{cav}$ can be expressed as
where $\Delta T=T_2^{set}-T_1^{set}$. For systems with multiple layers of thermal shields the temperature change of the cavity can be expressed as more complex forms with multiple thermal time constants [21], and we discuss it in Sec. 5.2. The single effective $\tau$ approach used here is a simplification in a complicated thermal shields system. Substituting Eq. (4) into Eq. (3), the instantaneous fractional frequency change of the cavity-stabilized laser can be expressed as
Fig. 1. Fractional frequency response to a step change of the temperature set point $T^{set}$. (a) Response curves in a heating process, the red dotted line represents instantaneous temperature of the cavity, the solid blue line corresponds to a case when the zero-CTE temperature is below the temperature change range, while the blue dotted-dashed line corresponds to a case when the zero-CTE temperature is above the temperature change range. (b) Response curves in a cooling process, the red dotted line represents instantaneous temperature of the cavity, the solid blue line corresponds to a case when the zero-CTE temperature is below the temperature change range, while the blue dotted-dashed line corresponds to a case when the zero-CTE temperature is above the temperature change range. (c) A bell shape curve like the solid blue line can be observed in a heating process when the zero-CTE temperature is within the temperature change range. The cooling process has a similar response, and is not shown here.
3. Experimental setup
The optical reference cavity under test is 30 cm long, and the estimated thermal noise limit of the cavity is $7.2\times 10^{-17}$ (expressed with modified Allan deviation). The 30 cm long cavity is housed in a vacuum chamber at pressure below $2\times 10^{-6}$ Pa. The conceptual sketch of the vacuum system is shown in Fig. 2. Two layers of thermal shields made of oxygen-free copper are used in the vacuum chamber. They are designated as outer layer and inner layer, respectively. The outer layer is actively temperature controlled using a thermistor as temperature sensor and three Peltiers as actuators. Glass balls are placed between different layers to get a better thermal isolation. All the inner surfaces of the vacuum system are gold-plated to reduce emissivity of materials.
Fig. 2. Schematic diagram of the experimental setup. The red solid lines represent lasers operating at 1070 nm and the black dotted lines represent electrical paths. Vacuum system of the 10 cm long cavity is not shown in this figure.
The optical setup for measuring the zero-CTE temperature of the optical reference cavity is also shown in Fig. 2. An external cavity diode laser (Toptica, DL pro) operating at 1070 nm is locked to TEM$_{00}$ mode of the 30 cm cavity with the Pound-Drever-Hall (PDH) method. Another diode laser is locked to a 10 cm cavity as the reference laser. The frequency instability of the reference laser is evaluated to be $6\times 10^{-16}$ [22], with a drift rate of 30 mHz/s due to temperature effects and material aging. We name these two ultra-stable lasers as Laser 1 and Laser 2, respectively. The beat note frequency of Laser 1 and Laser 2 is recorded by a frequency counter (FXE-m, K+K). An optical frequency comb referenced to a hydrogen maser is used to beat with Laser 2 to correct the long-term frequency drift of Laser 2, and gives absolute frequencies of Laser 1 and Laser 2.
4. Zero-CTE temperature measurement
The precision to determine the zero-CTE temperature is affected by temperature and frequency fluctuations of the cavity. We measure the frequency-temperature relationship in an actual running ultra-stable laser system with a vacuum level of $10^{-6}$ Pa. The high vacuum allows a good thermal isolation and thermal uniformity. Measured temperature and frequency fluctuations are presented in Fig. 3. The temperature fluctuation of the active temperature controlled layer is within $\pm$0.3 mK in 12 hours. The temperature fluctuation of the 30 cm cavity is calculated using a digital low pass filter algorithm. With a designed vacuum thermal time constant around $6.5\times 10^{5}$ s from the temperature controlled layer to the 30 cm cavity [23], the calculated temperature fluctuation of the cavity is less than $\pm$0.6 $\mu {\textrm K}$ in 12 hours (red line in Fig. 3(a)). The frequency fluctuation of the beat note between Laser 1 and Laser 2 is $\pm$2 Hz with a gate time of 1 s, and shows a linear drift of 7 mHz/s caused by relative frequency drift of the 30 cm cavity and the 10 cm cavity. Limited by the short term frequency instability of the frequency comb, the frequency fluctuation of the beat note between Laser 2 and the frequency comb is less than $\pm$200 Hz with a gate time of 1 s, a linear drift of 30 mHz/s is the absolute drift rate of the 10 cm cavity. The small beat frequency fluctuation between the two ultra-stable lasers makes it possible to observe more details of the frequency response to temperature changes. However, the long-term frequency drift of the optical reference cavities need to be corrected by the frequency comb, which limits the frequency uncertainty of our method to about 1 kHz. For a frequency uncertainty of 1 kHz, we can estimate the uncertainty of the zero-CTE temperature to be 0.05 $^{\circ }$C.
Fig. 3. (a) Temperature fluctuations of the temperature controlled layer (blue line) and the 30 cm cavity (calculated, red line). (b) Frequency fluctuations of beat notes between Laser 1 and Laser 2 (orange line), Laser 2 and the frequency comb (green line).
For the first attempt to identify the zero-CTE temperature, we cool down the 30 cm cavity from 17.5 $^{\circ }$C to 15 $^{\circ }$C. The frequency response curve is shown in Fig. 4(a), and it has a similar shape as the blue solid line in Fig. 1(b), which indicates that the zero-CTE temperature is below 15 $^{\circ }$C. We then cool down the 30 cm cavity with a bigger temperature range, from 15 $^{\circ }$C to 1.3 $^{\circ }$C, designated as Proc. 1. The response curve shows a predicted turning point (see Fig. 4(b)), which indicates that the zero-CTE temperature is within the temperature change range. To further identify the zero-CTE temperature, we heat up the cavity with smaller temperature steps, from 1.3 $^{\circ }$C to 2.6 $^{\circ }$C (Fig. 4(c)), and from 2.6 $^{\circ }$C to 6 $^{\circ }$C (Fig. 4(d)), the response curves show that the zero-CTE temperature is within the range from 2.6 $^{\circ }$C to 6 $^{\circ }$C, designated as Proc. 2. It costs several hours in a typical temperature change of the active controlled layer while the whole measurement costs several days.
Fig. 4. Frequency response of the 30 cm cavity to different temperature change ranges. The blue dots show fractional frequency deviation recorded by a frequency counter, the red lines show nonlinear fitting performed on the datasets. The error bar of the fractional frequency due to the laser frequency measurement using the frequency comb is $4\times 10^{-12}$, which is too small to be shown in the figures. (a) Cooling down from 17.5 $^{\circ }$C to 15 $^{\circ }$C. (b) Cooling down from 15 $^{\circ }$C to 1.3 $^{\circ }$C, Proc. 1. The inset is a zoom of the data. (c) Heating up from 1.3 $^{\circ }$C to 2.6 $^{\circ }$C. (d) Heating up from 2.6 $^{\circ }$C to 6 $^{\circ }$C, Proc. 2.
We apply a nonlinear fitting of Eq. (5) using Levenberg-Marguardt (L-M) algorithm to the two datasets of Proc. 1 and 2, respectively. The fitting results together with the fitting uncertainties are shown in Table 1. The fitting standard errors are presented in parentheses. When the temperature changes in a large range, as shown in Proc. 1, the fitting results show the zero-CTE temperature is 3.306(8) $^{\circ }$C, which agree with the conclusion that the zero-CTE temperature is within the range from 2.6 $^{\circ }$C to 6 $^{\circ }$C. The first order temperature coefficient is $a=3.59(1)\times 10^{-9} /{\textrm K}^2$, which is of the same order compared to cavities using similar structures [15], though the second order temperature coefficient $b=-2.99(7)\times 10^{-10} /{\textrm K}^3$ seems larger. In addition, a vacuum thermal time constant of $5.91(1)\times 10^5$ s and a linear frequency drift of 46(2) mHz/s are inferred. When the temperature changes near the zero-CTE temperature, as shown in Proc. 2, the fitting results show the zero-CTE temperature is 4.341(4) $^{\circ }$C, and the first order temperature coefficient is $a=2.944(3)\times 10^{-9} /{\textrm K}^2$, which are close to the results we obtain in Proc. 1. The second order temperature coefficient $b=1.4(1)\times 10^{-10} /{\textrm K}^3$, which is in the same order but has a sign reversal compared to the result we obtained from Proc. 1. The thermal time constant is $6.52(2)\times 10^5$ s, 10% more than the thermal time constant we get in Proc. 1. A linear frequency drift of 16(1) mHz/s is also inferred, close to the value of 10 mHz/s measured independently by a frequency comb when temperature of the cavity is controlled at 4.8 $^{\circ }$C.
Table 1. Fitting results of the thermal parameters of the cavity. Numbers in parentheses are fitting uncertainties.
5. Discussion
Using nonlinear fitting with multiple unknown parameters, it is very sensitive to the initial values of the corresponding parameters. We estimate the goodness of our fit in Sec. 5.1. To discuss the effect of the simplification of the theoretical model, we discuss the fitting result using a two-layer thermal shields model and other factors in Sec. 5.2. For comparison, we measure the zero-CTE temperature using the regular method and discuss the results in Sec. 5.3.
5.1 Estimation of the goodness of fit
The operation process described in Sec. 4 indicates that we can find the zero-CTE temperature in a single step with a wide range temperature change. However, the precision of the fitting results will be influenced by the ranges of the temperature change. Therefore, we adjust the zero-CTE temperature $T_0$ in fittings and evaluate the consistency between the fitting curves and the collected data using the adjusted R-square. For $n$ sample size and $p$ number of predictor variables, the adjusted R-square can be expressed as
where $RSS$ is the residual sum of squares, and $TSS$ is the total sum of squares [24]. As shown in Fig. 5, we choose the adjusted R-square higher than 0.999 as the evaluation criterion of fittings. For Proc. 1, the criterion can always be satisfied when $T_0$ is set in a range of -1.0 $^{\circ }$C to +3.5 $^{\circ }$C of the best fitting result. In contrast, for Proc. 2, the allowable deviation range of $T_0$ is within $\pm$0.5 $^{\circ }$C which means the precision of our method is related to the temperature change range. The fitted zero-CTE temperature of $4.3\pm 0.5$ $^{\circ }$C from Proc. 2 is consistent with the one from Proc. 1, $3.3^{+3.5}_{-1.0}$ $^{\circ }$C.Fig. 5. (a,c) Fitting curves of the frequency response to step temperature changes of Proc. 1 and 2. The black dots represent measured data. The red solid line is the best fitting result. The red dashed line and the red dotted line represent fitting results when the adjusted R-square is below 0.999 as $T_0$ changes. (b,d) Adjusted R-square of fitting when $T_0$ deviates from the best fitting result of Proc. 1 and 2, the grey bars indicate adjusted R-square less than 0.999.
The zero-CTE temperature is expected to occur in the top of the bell curve in Fig. 1, but the known information does not include the temperature of the cavity. The effective vacuum thermal time constant is a key parameter that determines the transformation from the measured temperature data to the actual temperature of the cavity, see Eq. (4). Therefore, the smaller the temperature step, the less sensitive the zero-CTE temperature is to the thermal time constant. The relationships between the CTE curves of the cavity and the various thermal time constants in fittings are shown in Fig. 6. The zero crossing points of the CTE curves indicate the zero-CTE temperature. For Proc. 1, the deviation of the fitted zero-CTE temperature is $\pm$0.4 $^{\circ }$C when the thermal time constant varies $\pm 10\%$. For a smaller temperature change of Proc. 2, the fitted zero-CTE temperature is even less sensitive to the thermal time constant.
Fig. 6. Relationship of CTE curve of the cavity and the thermal time constant. (a) Fitting result when temperature changes from 15 $^{\circ }$C to 1.3 $^{\circ }$C. If the thermal time constant is varied by $\pm 1\%$, the fitted zero-CTE temperature varies by $\pm$0.04 $^{\circ }$C. If the thermal time constant is varied by $\pm 5\%$, the fitted zero-CTE temperature varies by $\pm$0.2 $^{\circ }$C. If the thermal time constant is varied by $\pm 10\%$, the fitted zero-CTE temperature varies by $\pm$0.4 $^{\circ }$C. (b) Fitting result when temperature changes from 2.6 $^{\circ }$C to 6 $^{\circ }$C. If the thermal time constant is varied by $\pm 1\%$, the fitted zero-CTE temperature varies by $\pm$0.01 $^{\circ }$C. If the thermal time constant is varied by $\pm 5\%$, the fitted zero-CTE temperature varies by $\pm$0.08 $^{\circ }$C. If the thermal time constant is varied by $\pm 10\%$, the fitted zero-CTE temperature varies by $\pm$0.16 $^{\circ }$C.
5.2 Effect of different theoretical models
As mentioned in Sec. 2, we use a single exponential expression and an effective thermal time constant for simplification. For systems with two layers of thermal shields as shown in Fig. 2, assuming the thermal conduction is negligible with specially designed mounting structures and only thermal radiation dominates, we can modify Eq. (4) as [21]
Fig. 7. (a) Fitting curves of one $\tau$ model (the blue solid line) and two $\tau$ model (the red solid line). The grey dots represent measured data. (b) The residuals of fittings applied to Proc. 2. The blue squares represent the residual of fitting using the one $\tau$ model. The red dots represent the residual of fitting using the two $\tau$ model.
Though the fitting using two $\tau$ model shows a lower residual than the one using one $\tau$ model, the two $\tau$ model is a simplification, too. When the thermal conduction of the mounting structures can not be neglected and the coupling of thermal conduction and thermal radiation is included, the accurate theoretical model is more complicated [21,25]. Besides, the optical windows on the thermal shields introduce bypass of heat transfer, which also influence the model [26]. However, for ultra-stable laser systems with multi-layer thermal shields, an effective thermal time constant is widely used to estimate the ability of temperature filtering [21,23]. The complicated model including various thermal effects will make the fitting result of zero-CTE temperature more sensitive to any small variations.
Temperature gradient will influence both the model of thermal expansion of the cavity and the thermal time constant of heat transfer. The maximum static temperature gradient in the cavity is 0.01 $^{\circ }$C evaluated using the finite element analysis software (COMSOL Multiphysics), much smaller than the zero-CTE temperature uncertainty of this method. The time delay of heat transfer in the cavity is included in the effective thermal time constant in our model so this part will not introduce a significant error.
5.3 Comparison with the regular method
To prove the correctness of our method, we also determine the zero-CTE temperature with the regular method that needs to change the temperature several times and wait for the temperature of cavity to be settled down. A nonlinear fitting using Eq. (3) is applied to data measured when the cavity temperature is changed. The data and fitting line are shown in Fig. 8. It takes us around four months to obtain the four points in Fig. 8 due to the large thermal time constant. The second order temperature coefficient $b$ and the linear frequency drift $\gamma$ in Eq. (3) are omitted in the fitting to reduce unknown parameters. The fitting results show a zero-CTE temperature of 4.27(2) $^{\circ }$C, which agrees well with the result we obtained in Proc. 2. The first order temperature coefficient $a=2.30(4)\times 10^{-9} /{\textrm K}^2$, which agrees with the result we obtained in Proc. 2.
Fig. 8. Relationship of fractional frequency change and the cavity temperature. The blue dots represent data measured when the cavity temperature is settled down and the red line represents the fitting result. The error bars are magnified by 100 times to be observable.
Consequently, depending on the precision we required for the determined zero-CTE temperature, we can get a rough estimate of the zero-CTE temperature with one single large step temperature change; for better precision of the determined zero-CTE temperature, a smaller step temperature change around the zero-CTE temperature determined in the previous step can be introduced. So minimum one step and maximum two steps temperature changes are needed to determine the zero-CTE temperature of an ultra-stable cavity even for a large thermal time constant vacuum system. Even considering the time costs of one wide range temperature change to locate the zero-CTE temperature and one smaller temperature change to improve the measurement precision, this method is significantly faster than the regular method which needs to wait for temperature to be settled down several times [27]. For example, with a thermal time constant of 7.5 days like ours, it takes around 40 days for one step and 80 days for two steps in our method, while it takes 200 days to record five points for fitting using the regular method. In comparison with the zero-CTE temperature measurement in rough vacuum environments, experimental conditions in this method are closer to the actual running conditions so that influences like temperature gradients can be more realistic.
6. Conclusion
In conclusion, a fast method of measuring the zero-CTE temperatures of optical reference cavities is developed. The method measures the full response process of the cavity with a single, wide range temperature step change, and processes the collected data with a simple fitting algorithm, resulting in a reasonable accurate determination of the zero-CTE temperature. Better precision can be obtained with a smaller step temperature change. We measure the thermal property of a 30 cm long cavity and obtain a zero-CTE temperature of $4.3\pm 0.5$ $^{\circ }$C with a simple test. The method is applicable for the zero-CTE temperature measurements of different materials and complicated shape cavities. The frequency uncertainty is limited to around 1 kHz by the frequency comb that is referenced to a hydrogen maser, leading to a temperature uncertainty of 50 mK. The experimental setup can be simplified and the uncertainty of the beat frequency can be reduced by using reference lasers with better long-term frequency stabilities, such as iodine stabilized lasers. Thermal parameters obtained by fitting can be used to predict the long-term frequency change of the cavity-stabilized lasers and a real-time compensation algorithm can be realized. A long-term frequency stable laser will have great values in time and frequency metrology and space applications.
Funding
National Key Research and Development Program of China (2017YFA0304400); Key-Area Research of GuangDong Province (2019B030330001); National Natural Science Foundation of China (11774108, 61875065, 91336213, 91536116); China Postdoctoral Science Foundation (2017M622400).
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.
References
1. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015). [CrossRef]
2. A. De Michele, A. Weinstein, and D. Ugolini, “The pre-stabilized laser for the LIGO Caltech 40m interferometer: stability controls and characterization,” LIGO document LIGO-T010159-00-R (2001).
3. E. Lucas, P. Brochard, R. Bouchand, S. Schilt, T. Südmeyer, and T. J. Kippenberg, “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based transfer oscillator,” Nat. Commun. 11(1), 374–378 (2020). [CrossRef]
4. S. Häfner, S. Falke, C. Grebing, S. Vogt, T. Legero, M. Merimaa, C. Lisdat, and U. Sterr, “8×10−17 fractional laser frequency instability with a long room-temperature cavity,” Opt. Lett. 40(9), 2112–2115 (2015). [CrossRef]
5. D. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. Robinson, J. Ye, F. Riehle, and U. Sterr, “1.5 µm lasers with sub-10 mHz linewidth,” Phys. Rev. Lett. 118(26), 263202 (2017). [CrossRef]
6. W. Zhang, J. Robinson, L. Sonderhouse, E. Oelker, C. Benko, J. Hall, T. Legero, D. Matei, F. Riehle, U. Sterr, and J. Ye, “Ultrastable silicon cavity in a continuously operating closed-cycle cryostat at 4 K,” Phys. Rev. Lett. 119(24), 243601 (2017). [CrossRef]
7. J. M. Robinson, E. Oelker, W. R. Milner, W. Zhang, T. Legero, D. G. Matei, F. Riehle, U. Sterr, and J. Ye, “Crystalline optical cavity at 4 K with thermal-noise-limited instability and ultralow drift,” Optica 6(2), 240–243 (2019). [CrossRef]
8. G. Mueller, P. McNamara, I. Thorpe, and J. Camp, “Laser frequency stabilization for LISA,” NASA Technical Publication TP-2005–212790 (2005).
9. K. Abich, A. Abramovici, B. Amparan, A. Baatzsch, B. B. Okihiro, D. C. Barr, M. P. Bize, C. Bogan, C. Braxmaier, M. J. Burke, K. C. Clark, C. Dahl, K. Dahl, K. Danzmann, M. A. Davis, G. De Vine, J. A. Dickson, S. Dubovitsky, A. Eckardt, T. Ester, G. F. Barranco, R. Flatscher, F. Flechtner, W. M. Folkner, S. Francis, M. S. Gilbert, F. Gilles, M. Gohlke, N. Grossard, B. Guenther, P. Hager, J. Hauden, F. Heine, G. Heinzel, M. Herding, M. Hinz, J. Howell, M. Katsumura, M. Kaufer, W. Klipstein, A. Koch, M. Kruger, K. Larsen, A. Lebeda, A. Lebeda, T. Leikert, C. C. Liebe, J. Liu, L. Lobmeyer, C. Mahrdt, T. Mangoldt, K. McKenzie, M. Misfeldt, P. R. Morton, V. Müller, A. T. Murray, D. J. Nguyen, K. Nicklaus, R. Pierce, J. A. Ravich, G. Reavis, J. Reiche, J. Sanjuan, D. Schütze, C. Seiter, D. Shaddock, B. Sheard, M. Sileo, R. Spero, G. Spiers, G. Stede, M. Stephens, A. Sutton, J. Trinh, K. Voss, D. Wang, R. T. Wang, B. Ware, H. Wegener, S. Windisch, C. Woodruff, B. Zender, and M. Zimmermann, “In-orbit performance of the GRACE Follow-on laser ranging interferometer,” Phys. Rev. Lett. 123(3), 031101 (2019). [CrossRef]
10. J. Luo, L. Chen, H. Duan, Y. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C. Shao, V. T. Toth, H. Tu, Y. Wang, Y. Wang, H. Yeh, M. Zhan, Y. Zhang, V. Zharov, and Z. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quantum Grav. 33(3), 035010 (2016). [CrossRef]
11. H. E. Hagy, “High precision photoelastic and ultrasonic techniques for determining absolute and differential thermal expansion of titania–silica glasses,” Appl. Opt. 12(7), 1440–1446 (1973). [CrossRef]
12. M. J. Edwards, E. H. Bullock, and D. E. Morton, “Improved precision of absolute thermal-expansion measurements for ULE glass,” in Advanced Materials for Optical and Precision Structures, vol. 2857 (International Society for Optics and Photonics, 1996), pp. 58–63.
13. J. Kushibiki and M. Arakawa, “Precise evaluation of zero-CTE temperature of EUVL-grade TiO2-SiO2 ultra-low-expansion glass using the line-focus-beam ultrasonic material characterization system,” in Alternative Lithographic Technologies, vol. 7271 (International Society for Optics and Photonics, 2009), p. 72713D.
14. K. E. Hrdina, B. G. Ackerman, A. W. Fanning, C. E. Heckle, D. C. Jenne, and W. D. Navan, “Measuring and tailoring CTE within ULE glass,” in Emerging Lithographic Technologies VII, vol. 5037 (International Society for Optics and Photonics, 2003), pp. 227–235.
15. T. Legero, T. Kessler, and U. Sterr, “Tuning the thermal expansion properties of optical reference cavities with fused silica mirrors,” J. Opt. Soc. Am. B 27(5), 914–919 (2010). [CrossRef]
16. S. Jacobs, J. Berthold III, and J. Osmundsen, “Ultraprecise measurement of thermal expansion coefficients—recent progress,” in AIP Conference Proceedings, vol. 3 (American Institute of Physics, 1972), pp. 1–12.
17. C. Hagemann, “Ultra-stable laser based on a cryogenic single-crystal silicon cavity,” Ph.D. thesis, Gottfried Wilhelm Leibniz Universität Hannover (2013).
18. S. Amairi ep Pyka, “A long optical cavity for sub-Hertz laser spectroscopy,” Ph.D. thesis, Gottfried Wilhelm Leibniz Universität Hannover (2014).
19. C. Wang, Z. Ji, T. Gong, D. Su, Y. Zhao, L. Xiao, and S. Jia, “A simple, low cost and robust method for measurement of the zero-crossing temperature of an ultralow expansion cavity,” J. Phys. D: Appl. Phys. 52(45), 455104 (2019). [CrossRef]
20. Y. Jiang, A. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L. Ma, and C. W. Oates, “Making optical atomic clocks more stable with 10−16-level laser stabilization,” Nat. Photonics 5(3), 158–161 (2011). [CrossRef]
21. X. Dai, Y. Jiang, C. Hang, Z. Bi, and L. Ma, “Thermal analysis of optical reference cavities for low sensitivity to environmental temperature fluctuations,” Opt. Express 23(4), 5134–5146 (2015). [CrossRef]
22. X. Zeng, Y. Ye, X. Shi, Z. Wang, K. Deng, J. Zhang, and Z. Lu, “Thermal-noise-limited higher-order mode locking of a reference cavity,” Opt. Lett. 43(8), 1690–1693 (2018). [CrossRef]
23. J. Zhang, W. Wu, X. Shi, X. Zeng, K. Deng, and Z. Lu, “Design verification of large time constant thermal shields for optical reference cavities,” Rev. Sci. Instrum. 87(2), 023104 (2016). [CrossRef]
24. A. Akossou and R. Palm, “Impact of data structure on the estimators R-square and adjusted R-square in linear regression,” Int. J. Math. Comput 20, 84–93 (2013).
25. J. Sanjuan, N. Gürlebeck, and C. Braxmaier, “Mathematical model of thermal shields for long-term stability optical resonators,” Opt. Express 23(14), 17892–17908 (2015). [CrossRef]
26. J. Sanjuan, K. Abich, M. Gohlke, A. Resch, T. Schuldt, T. Wegehaupt, G. P. Barwood, P. Gill, and C. Braxmaier, “Long-term stable optical cavity for special relativity tests in space,” Opt. Express 27(25), 36206–36220 (2019). [CrossRef]
27. H. Liu, K. Jiang, J. Wang, Z. Xiong, L. He, and B. Lü, “Precise calibration of zero-crossing temperature and drift of an ultralow expansion cavity with a clock transition spectrum,” Chin. Phys. B 27(5), 053201 (2018). [CrossRef]
