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Ultra-high sensitivity temperature sensing and stable thermal control are crucial for many science experiments testing fundamental theories to high precision. Here we report the first pico-kevin scale thermometer operating at room temperature with an exceptionally low theoretical noise figure of at 1 Hz and a high dynamic range of ~500 K. We have experimentally demonstrated a temperature sensitivity of at 1 Hz near room temperature, which is an order of magnitude improvement over the state of the art. We have also demonstrated an ultra-high stability thermal control system using this thermometer, achieving 3.7 nK stability at 1 s and ∼ 120 pK at 104 s, which is 10–100 times more stable than the state of the art. With some upgrades to this proof-of-principle device, we can expect it to be used for very high resolution tests of special relativity and in critical point phenomena.
© 2017 Optical Society of America
1. Introduction
Temperature measurement and control are ubiquitous but generally routine aspects of many precision experiments. When a high degree of control is needed common practice is to use a multi-layer thermal enclosure and high sensitivity thermistors as the control sensors. Due to their inherent Johnson noise these sensors typically demonstrate a temperature resolution of a few µK at 1 s integration time [1]. This can be improved by increasing the power to the device, but soon self-heating becomes a limiting factor. The best reported resolution with thermistors we are aware of is ~ 1.7 µK sensitivity achieved with a 5 s time constant [2].
In environments that are strongly influenced by electromagnetic effects such as from a microwave field, where measurements with traditional electrical temperature sensors are not effective, optical thermometers (OTs) are a good alternative. These date back to at least 1979, when a thermometer was built based on a Mach-Zehnder interferometer [3]. Since then, numerous types of OTs have been developed. Among them waveguide based OTs show good performance in sensitivity, dynamic range and response time [4–6].
The largest subset of OTs is comprised of fiber optical thermometers [5, 7–10] One of the highest temperature sensitivities was achieved by resolving a temperature induced spectral shift, reaching up to 6.6 nm/K [10]. Assuming performance for state-of-the-art devices, this can be converted to an equivalent temperature sensitivity of ∼ 0.3 µK. The recent development of OTs based on whispering gallery mode (WGM) resonators has resulted in a 1 s resolution in the 30–100 nK range [11,12]. These devices are based on the thermal refractivity effect in a dielectric resonator, which can cause a temperature dependent frequency shift between two modes with orthogonal polarizations. The fundamental resolution is set by thermal and Kerr effect noise, typically on the order of 1 Hz [12].
For experiments that require higher resolution, new measurement devices are needed. An application we are pursuing is the reduction of thermal effects on cavities in low Earth orbit where day/night transitions occur on a nominal 90 min orbital cycle [13]. In this case high fidelity thermal performance at 2800 and 5600 second measurement times is critical, but to effectively suppress thermal cycling effects at these periods a bandwidth of ~0.01 Hz is needed. To resolve potential changes of the velocity of light at the level 1 part in 1018, a thermal stability on the order of 10 nK is needed. The device described here appears adequate for this case.
Another application is the high resolution study of critical phenomena where measurements of fluid properties asymptotically close to a transition are of great importance [14]. Unfortunately hydrostatic compression limits the value of ground-based measurements in this region. The problem can be greatly reduced in the space environment [15] where a number of experiments are now in development or have been completed. Examples are the DECLIC/ALI package [16] for studying the critical point of sulfur hexafluoride on the International Space Station and earlier experiments conducted on the Shuttle and MIR [17]. To reach the resolution obtained near the lambda transition of helium, for example [18], temperature changes must be resolved to ~30 nK.
Here we report on the development of a resonant cavity thermometer (RCT) that uses commonly available technology to obtain resolutions of at 1 Hz. Detailed analysis of the fundamental limits to its resolution indicates that a ×50 improvement can be obtained under optimal conditions. In the following sections we describe the device, detail its operation and analyze the factors that contribute to its ultimate noise performance. We also show the calibration procedure and the general performance of the device compared to other devices such as thermistors. Lastly, we show a thermal control system using the RCT, achieving a temperature stability of 3.7 nK with 1 s integration time and ∼ 120 pK stability at 104 s.
2. Resonant cavity thermometry
2.1. Operating principle
Our approach to ultra high resolution thermometry is based on the use of a stable spacer material with a high coefficient of thermal expansion (CTE) in a resonant Fabry-Perot (FP) cavity to act as a sensor for temperature changes. Since the resonant frequencies of the optical modes in such a cavity are extremely high, it is necessary to beat the FP resonant frequency against a proximal optical frequency standard. We have used a second cavity with a spacer of ultra-low expansion (ULE) glass and a transition in molecular iodine as frequency standards. A conceptual diagram of the configuration using the cavity as the reference is shown in Fig. 1.
Fig. 1 Conceptual diagram of resonant cavity thermometer. Orange: temperature sensor; blue: reference system. Detected signal frequency at photodetector (PD) is | f1 − f2|.
Two laser sources are independently locked to the cavities residing in thermal enclosures in separate vacuum systems. A small amount of light is picked off from each optical chain and the two beams are brought together to interfere on a high-speed photo-detector producing a beat note in the RF range. A temperature change of the high CTE cavity induces a shift in the beat frequency between the two cavities. Based on this shift and the material properties of the high CTE cavity, the temperature change can easily be calculated.
In the second configuration the frequency-doubled light from a Nd:YAG laser is locked to the a10 hyperfine transition in the 1110 line of molecular iodine near 532 nm [19,20]. The 1064 nm stabilized light from this laser is used as the frequency standard in place of the ULE cavity light.
In general, the resonant frequencies ω of the TEM00 modes of a FP cavity are given by:
where c is the velocity of light, n is the refractive index of the cavity medium, L is the cavity length, q is the mode number and ΔνFSR is the free spectral range (FSR) of the cavity. The cavity temperature sensitivity can then be expressed as: where T is the temperature and αn, αL are the temperature coefficients of the refractive index and cavity spacer material respectively. Here we report results with an evacuated cavity so it is reasonable to set αn = 0, although we do consider the small effect of the gas below. Many materials could be used for the spacer, depending on the application. We used 310 stainless steel (SS) that has a nominal CTE αSS ≈ 14.4 × 10−6 K−1, and a reference cavity of ULE operating near its null CTE point with an estimated CTE of 1 × 10−9 K−1. The resulting conversion factor from frequency to temperature is d(ω/2π)/dT = αSSc/λ ≈ 4.06 Hz/nK, where λ is the laser wavelength.2.2. Experimental implementation
The FP cavity was constructed by clamping two mirrors to the ends of the SS spacer using Viton o-rings as compression springs on their external surfaces. The dielectric-coated fused-silica mirrors were configured as a flat for the input and a radius of curvature Rcc of 1 m for the output mirror, and were held in place with retaining rings external to the o-rings [21]. The optical finesse was measured to be ∼ 60, 000 using the cavity ring-down method [22]. The 10 cm long cavity was soft-mounted to a SS V-block using SS coil springs on both sides. A heater and thermistor were mounted directly on the spacer. The device was placed on the fourth stage of an evacuated thermal enclosure that could be servo-controlled using thermistor sensors on each stage. The outer three stages were massive aluminum cans designed to attenuate temperature gradients. The thermal time constant of each layer was ∼ 2 to 4 hours, attenuating room temperature variations. The base pressure in the vacuum can was 4 × 10−5 Pa, maintained with an ion pump. The three stages were used to stabilize the cavity environment to ± 50 µK, using conventional thermistor bridges and heaters. With the servos engaged, no sign of a 24-hour day/night cycle could be detected on the cavity stage. A CAD rendering of the test system is shown in Fig. 2(a).
Fig. 2 (a): Cutaway CAD view of the thermal test system. (b): Schematic of experimental setup. Optical paths are shown as solid lines, electrical signals as dashed lines. EOM is electro-optic modulator, LPF is low pass filter, PI is proportional integral controller and λ/4 is quarter wave plate. The lasers are stabilized to the ULE cavity using the PDH locking technique, and to the steel cavity with the dither lock technique [23]. Inputs 1, 2 can be the frequency reference from cavities for normal operation, or from the Iodine setup for calibration of each cavity.
The optical setup based on the conceptual diagram in Fig. 1 is shown in Fig. 2(b). For some tests we replaced the entire ’laser 1’ setup with a third laser system locked to an iodine reference cell after frequency doubling as mentioned in section 2.1. The ULE cavity and iodine setups used free space coupling, while the SS cavity optics used fiber coupling except for the final mode-matching stage into the cavity. For this cavity, the transmitted light was also coupled into a fiber for external processing. The miniaturized optics for this section were located on the innermost stage of the enclosure. The laser was locked to the SS cavity using the dither locking method [23] to avoid etalon effects in the fibers. The dither frequency was ~9 kHz and the transmission signal was demodulated with a digital lock-in amplifier. The resulting error signal was processed by a simple proportional/integral controller and fed back to both the laser piezo input and its crystal temperature for noise suppression from DC up to the dither frequency.
The ULE cavity was operated near its CTE null at ∼ 7 °C. It was stabilized to ~10−5 K using a 2-stage thermal enclosure. Its laser was locked to a cavity mode using the Pound-Drever-Hall (PDH) method [24]. An intensity servo was used to stabilize the cavity input power.
The a10 hyperfine transition of the 1110 absorption line near 532 nm of molecular iodine 127I2 was interrogated using the modulation transfer spectroscopy method [20]. The vapor pressure in the cell was controlled using a cold finger embedded in a thermo-electric cooler operating at −20 °C.
3. Noise analysis
The temperature of a finite sized system will fluctuate due to heat exchange with its surroundings, often modeled as an infinite heat reservoir. This effect is inversely proportional to the heat capacity of the system and usually sets the fundamental limit for measurement on a given time scale. In addition, various practical factors set a device-specific noise floor that limit the resolution of a given design. Here we consider these device-specific factors that contribute to the noise floor of our device.
3.1. Thermal noise
At room temperature, performance of optical cavities is in general thermal noise limited. This noise (both homogeneous thermal noise caused by Brownian motion, and inhomogeneous thermal noise from thermoelastic dissipation) has been extensively studied [25–30]. We have neglected inhomogeneous thermal noise since at low frequency it is small in fused silica substrates [29] and in any case, the noise budget is dominated by the spacer noise. The power spectral density (PSD) of the spacer contribution, Gspacer, to the noise is given by [26]:
where kB is Boltzmann’s constant, R is the spacer outer radius, Es is its Young’s modulus, f is the Fourier frequency, and ϕspacer is the loss angle of the spacer material defined as 1/Q, where Q is the quality factor of the material.Since stainless steel has a low Q on the order of 104 [31], compared to fused-silica (Q ~ 106) used in the mirrors [26], it becomes the dominant source of cavity thermal noise.
The thermal noise contribution from a mirror substrate is given by:
where σ is Poisson’s ratio, w0 is the beam radius, Em is its Young’s modulus and ϕsub = 1 × 10−6 is the loss angle of the fused silica mirror substrate.The coating contribution can be evaluated by scaling the mirror substrate thermal noise [28,32], and is given by:
where d = 2 µm is the typical coating thickness and the typical value for coating loss angle ϕcoat is 4 × 10−4.The above formulae summarize the thermal noise contributions from the spacer, mirror substrates and coatings. The corresponding noise spectral densities are compared in section 3.6. The total thermal noise converts to an equivalent temperature noise of at 1 Hz.
3.2. Mechanical noise
Mechanical noise is induced by vibration in the laboratory, building movement and the seismic motion of the earth, and is the main source of environmental noise. It causes mechanical deformation of the cavity and produces small length fluctuations. Due to its location dependence, we performed a direct measurement of the vertical acceleration spectrum by mounting an accelerometer on the cavity platform. Laboratory conditions were as close as possible to those during the thermometry measurements. The amplitude spectral density of the measurement is shown in Fig. 3. For comparison, a typical seismic displacement spectrum based on a model of Earth deformation [33] is converted to acceleration spectrum, and is also plotted in Fig. 3.
Fig. 3 Measured and theoretical vertical acceleration spectrum. Insert: schematic showing the test arrangement with the accelerometer mounted inside the thermal enclosure.
It is apparent that the general trend of the measurements is dominated by other effects than seismic disturbances. The lower measured noise in the frequency region > 1 Hz is due to the seismic isolation provided by various damping structures between stages and the laboratory floor. It can be seen that the measurement noise exceeds the typical spectrum in the frequency region < 0.1 Hz. This is mostly due to the electronics noise of the accelerometer system, hence it is a worst case estimate.
The acceleration noise spectrum was converted to a frequency noise spectrum by a finite element analysis (FEA) calculating the horizontal displacement noise of the cavity mirror center when the measured vertical acceleration spectrum was applied. This displacement noise can then be converted to an equivalent laser beat frequency noise, as plotted in Fig. 4.
Fig. 4 Amplitude spectrum of noise sources. Left axis is expressed in frequency noise amplitude, right axis is the equivalent temperature noise amplitude.
3.3. Shot noise
Shot noise is due to the photon counting statistics as seen by the photodetectors used for laser frequency stabilization. Phase modulation of the laser light and subsequent demodulation of the reflected or transmitted light can yield an error signal with a null at line center. The error signal after signal conditioning can be used to drive actuators that push the laser frequency to the cavity line center as is done in the PDH method. Shot noise on the detected signal will appear as laser frequency or cavity length noise. The spectral density of shot noise can be converted to apparent frequency noise after demodulation using the discriminant D which is in units of V/Hz [34] and can be calculated from the error signal. In the case of PDH locking, the central slope of the error signal assuming a high finesse cavity is given as:
where Pc and Ps are the powers in the carrier and each of the 1st order sidebands, respectively, δν is the linewidth and δ f is the frequency deviation from line center. DPDH is the discriminant for the error signal obtained in reflection from the cavity.The expression for shot noise can be written as:
where h is Planck’s constant and Pref ≈ 2Ps is the reflected power incident on the photodetector assuming ideal mode matching.The apparent laser frequency noise can be calculated by dividing Se by DPDH and converting linewidth δν to finesse , giving:
In the case of dither locking, the transmission signal is used instead of reflection to generate the error signal [21]
where ω is the carrier frequency, Ω is the dither frequency, and t is the transmission coefficient for a symmetrical FP cavity with lossless mirrors, given as [35]: where R is the power reflection coefficient of the mirrors.The last two terms in Eq. (9) can be Taylor expanded around ±Ω for small frequency deviation δ f = δω/2π, and the resulting error signal in the limit of R = ±1 and Ω ≪ ΔνFSR is:
As can be seen, both techniques yield the same frequency discriminant D near resonance. Hence the noise spectrum for shot noise given in Eq. (8) is the same for both techniques. Therefore, for a 500 µW input power at a wavelength of 1.06 µm and a cavity finesse of 60, 000, the shot noise spectrum is , and the equivalent temperature noise is .
3.4. Residual gas noise
The residual gas in a cavity causes optical pathlength (OPL) variations from the density fluctuations in the beam at fixed pressure. Theoretical and experimental work [36,37] has shown that the noise spectrum can be estimated by modeling the OPL change from a molecule as a function of time, and performing a Fourier transform after summing the changes over all molecules in the beam. The PSD of the length fluctuations is given by:
Where α is the molecular polarizability, ϵ0 is vacuum permittivity, v0 is the most probable speed of the molecule, w(z) is the beam radius and η0 is the number density of the molecules.
For our setup the corresponding temperature noise is in the frequency band from 0.1 mHz to 10 kHz and the spectrum is almost flat.
3.5. Quantum noise
Quantum noise is introduced into the measurement by the position fluctuation of the cavity mirrors due to shot noise induced forces from the incident laser light [38].
For an input power to the cavity of ∼ 500 µW, the resulting quantum noise is negligible. Above 0.01 Hz it asymptotically approaches an equivalent thermal noise level of . At lower frequencies radiation pressure dominates and the noise scales as 1/f2, resulting in a equivalent temperature noise level of at 0.1 mHz. In our measurement frequency band, this number is four to twelve orders of magnitude smaller than the dominant noise sources, and hence is not plotted in Fig. 4.
3.6. Summary of noise analysis
The major noise sources discussed above are plotted as a function of Fourier frequency in Fig. 4 considering both technical and fundamental noise. They represent various limits on the ultimate sensitivity of the RCT.
It can be seen that, for room temperature operation, the noise is limited by thermal noise from the cavity components of the RCT. The root sum of squares thermal noise is at 1 Hz. Other noise sources such as mechanical noise, shot noise, residual gas noise and quantum noise are significantly lower. In particular, the low shot noise makes it possible to build a control loop that suppresses the major noise source within the bandwidth of the laser frequency control servo. This will be discussed in detail below.
4. Results
4.1. General behavior of RCT
In order to characterize the general behavior of the device, we need to calibrate the RCT output with a reference thermometer, for which we use a thermistor located on the spacer. For these measurements the RCT was used in a control loop that drove a heater mounted on the spacer to maintain a constant cavity resonant frequency. A schematic of the arrangement is shown in Fig. 5
Fig. 5 RCT thermal control setup. Optical signals are indicated by solid lines, electrical signals by dashed lines. Blue dashed lines show the laser stabilization control loop and black dashed lines indicate the thermal control loop. Green circle is the thermistor on the cavity spacer.
The set frequency of the RCT was stepped up and down by increasing amounts, until the beat frequency generated as in Fig. 2(b) covered one cavity FSR. This is equivalent to a maximum step size of 0.38 K. The corresponding thermistor readings were also logged as shown in Fig. 6. The calibration procedure consisted of adjusting the assumed CTE of the SS spacer until the difference between the RCT output and that of the thermistor was minimized. The best fit of the CTE was 14.38 × 10−6 K−1, which agrees well with the nominal value for 310 SS [39].
Fig. 6 Comparison of RCT and thermistor outputs for various sized temperature steps. (a): A 52-hour data set with various step sizes. (b): A 4-hour section of the data showing detailed features of RCT and thermistor readings.
It can be seen that the RCT has a rapid response to temperature changes notwithstanding the low thermal conductivity of the spacer. This is because the RCT uses the cavity resonant frequency to measure the total length change of the spacer, so any local thermal expansion will be detected within a fraction of a millisecond by the high-speed optical servo. The thermal feedback to the heater is also wideband, allowing rapid changes in the local length of the spacer. The thermistor on the other hand, is located a short distance from the heater and responds primarily to the distribution of heat along the spacer as it comes to equilibrium. This is the origin of the small overshoots seen in the thermistor data in Fig. 6.
The useful temperature range of the RCT is ultimately limited by the operating range of its constituent parts. In our configuration, the limiting components are the Viton o-rings that press the mirrors against the spacer. These are not recommended for use over ~470 K. If they were replaced by SS springs, for example, the limit would be significantly higher, but the SS spacer will start showing creep above ~500 K [40].
The continuous dynamic range can be defined as the temperature range over which the RCT can be operated without requiring a change of the resonant cavity mode. In our setup, we are limited by the bandwidth of the photodetector and the frequency counter, requiring us to lock to a different mode every 0.74 K. Since the FSR is equivalent to 0.38 K, there is no need for gaps in the accessible temperatures. We note that electronics are available with a larger bandwidth to extend this continuous dynamic range to a few Kelvin. It is also possible to expand it to cover a dynamic range of ∼ 500 K, using special construction materials and a frequency comb to avoid the bandwidth limitations of the electronics [21].
The temperature sensitivity of the RCT is defined as the beat frequency shift per unit temperature change, which in our case is determined by the temperature dependent CTE of SS. The calculated sensitivity in units of Hz/nK is shown in Fig. 7 over a wide temperature range. It is clear that our approach to high-resolution thermometry can be used in many experiments where precision thermal control is needed. We note that thermal noise falls with temperature and as the mechanical Q of the spacer increases. In this regard the use of a quartz or silicon spacer may provide a similar noise performance even though its temperature sensitivity would be reduced.
Fig. 7 Expected temperature sensitivity of the RCT in Hz/nK over a wide range. Note that the sensitivity drops to zero at 23.6 K as the CTE of SS reaches a null point. Below the null it increases again to 1 Hz/nK near 4 K. The blue curve is calculated from a cryogenics model of the CTE for the temperature range of 4K to 293 K [41], and the orange curve is from a model spanning 311 K to 922 K [42].
4.2. Resolution and stability
A metric of great interest is the resolution of the RCT as a function of measurement time or frequency. This was determined from the spectral density of the beat note, which is shown in Fig. 8(a), where two cases are plotted: with and without thermal control. In the former case the beat note was stabilized with a digital servo described earlier feeding back to the heater on the spacer. For the case without temperature control the spectral density is at 1 Hz. This noise has contributions from the RCT and the reference cavity. Since the reference cavity noise performance was not optimized, it is quite possible the RCT noise is significantly lower. Assuming equal noise contributions from the two sources, the RCT sensor noise would be at 1 Hz.
Fig. 8(a) where the Fig. 8. (a): Amplitude spectral density of RCT, with and without thermal feedback control activated; (b): Allan deviations of RCT. Grey line indicates a slope of where τ is the averaging time.
Above ∼ 0.05 Hz the spectral densities are roughly the same in both the thermal controlled and open loop cases. However at lower frequencies we observe a significant divergence between the two cases. It can be seen that below ∼ 0.04 Hz the open loop noise is strongly suppressed by the thermal control system. The frequency where servo suppression becomes effective is related to the thermal time constant of the RCT cavity spacer, which is ~20–30 s. Note that at low frequencies the spectral density of the RCT noise could be below the thermal noise floor as the intrinsic measurement noise is lower. If the sensor noise is smaller than the thermal noise then the control system can suppress the latter in the frequency band where the servo gain is > 1. An indication of this behavior can be seen on the left of the upper chart in controlled spectral density shows the trend of dipping slightly below the total noise estimate. This behavior is expected to continue to even lower frequencies. Similar behavior has been seen in a thermal control experiment based on a different sensing mechanism [12].
An alternative measure of RCT noise in the time domain is given by the Allan deviation (AD), and is shown in Fig. 8(b) for both the closed and open loop cases. The 1 s AD for the controlled case is 3.67 nK, and it falls to ∼ 500 pK at 1000 s trending to ∼ 120 pK at 104 s. For the open loop case, the AD starts to rise above 0.5 s averaging time and continues to increase approximately linearly due to thermal and other sources of instability affecting the system. For the closed loop case there is no drift contribution to the AD because of the action of the thermal servo. An estimate of the potential drift of the device is made in the Appendix part 1. The limit on the drift can be improved when a highly stable temperature standard with nano-Kelvin repeatability becomes available.
4.3. Linearity compared to thermistor
Another issue of interest is the linearity of RCT compared to other standard sensors, such as thermistors, over a large temperature range. Nonlinear responses such as hysteresis can occur in a material if it is cycled beyond its yield point. In our case the thermal-mechanical stress induced over the dynamic range of thermometer is well below the yield point of the spacer and the support springs, so nonlinear behavior is not expected. Another source of non-reproducibility could be attributed to the cavity mounting mechanism, where localized stress may build up. Poor reproducibility has been reported for some mounting structures [43]. We have looked for such effects by sweeping the cavity temperature many times, as shown in Fig. 6, and looking at residuals with respect to the thermistor located on the spacer. The steady state values of the two measurements are plotted against each other for different temperatures, both when cooling and heating, as shown in Fig. 9(a).
Fig. 9 (a): Linear correlation measurement between thermistor and RCT. The cooling and heating processes are differentiated by blue circles and orange dots. TR and TRCT are the relative temperature readings of thermistor and RCT, respectively. The uncertainties in the insert equation are obtained from the 95% confidence intervals of the fit. (b): Residuals of data shown in (a) after subtracting the linear fit. Error bars are obtained as described in the appendix.
We show a linear fit through the data, and as can be seen, within fitting uncertainty, the two measurements show a linear correlation with a unity coefficient. Fig. 9(b) shows the residuals of the data shown in panel (a) after subtracting the linear fit. The error bars are estimated from noise analysis considering various effects from both the thermistor and the RCT, as described in the Appendix. It is clear from the figure that so far no non-linearity or hysteresis is observed to the level of measurement uncertainties within a temperature range of ~0.4 K. Over a very wide temperature range up to a few hundred K, the RCT will show some repeatable non-linearity due to the temperature dependence of the CTE that can easily be modeled.
5. Discussion
Many improvements come to mind for our device which was developed as a proof of principle. Significant issues are its size and thermal mass. The resolution of our thermometer is determined by the 10 cm cavity length and the Q of the spacer. A miniaturized version with a high Q material appears possible. The temperature resolution of the RCT can be expressed as:
where ηνFWHM is the apparent reduced laser linewidth after frequency stabilization, with η typically on the order of 10−4 being the factor we can obtain with standard electronics after laser frequency stabilization feedback control is activated, and α is the lumped CTE expressed as α = αn + αLThere are several routes for improvement of the RCT’s resolution, depending on specific requirements of spatial and temporal resolutions. For applications that require a measurement volume of ∼ 10 mm3 (comparable to a thermistor) and a temporal resolution of a few nK at 1 s, we can use a miniaturized version of the free-space resonator as described above. One of the main constraints for the size of the sensor is the resonating mode volume for a stable (and often times fundamental) mode. Typically for this type of miniature RCT, the mode size limits the cavity diameter to ∼ 1 mm, for a flat and Rcc = 0.1 m mirror configuration and a spacer length of 10 mm.
For applications that require a 1 s resolution of a few µK but a better spatial resolution of 10–100 µm3, a high Q nanocavity based on a double-heterostructure photonic crystal can be a good candidate [44]. It has a projected Q of 2 × 107, and given the silicon CTE of 2.6 × 10−6 and a 104 reduction in laser linewidth using frequency stabilization, a temperature resolution of < 2 µK at 1 s can be expected.
Note that for a 1 s resolution < 100 nK, a free space RCT is preferred over the dielectric based RCT, because of the inherent Kerr effect [12] in the latter, which adds a noise floor of ∼ 100 nK/Hz at 1 Hz, ultimately limiting the resolution and the long term stability.
The device we have developed is capable of giving very high temperature resolution in a relatively simple and robust package. No specialized items were used in the instrument beyond what can easily be purchased or fabricated in a small machine shop. We have demonstrated the highest temperature resolution to date ( at 1 Hz) near room temperature using well-established techniques for locking a laser to an optical cavity with a high CTE. We expect the theoretical resolution of the device to be , which may be achieved under optimum conditions, such as a more stable laser, better controllers and less noisy reference devices. We have shown the highest temperature stability (3.7 nK at 1 s) using the device as the sensor in a control system. With longer averaging, a stability of ~120 pK has been obtained. We have also indicated a number of ways such a device can be put to use in various applications.
Appendix: uncertainty analysis
In this appendix, we discuss the sources of uncertainties considered for the thermistor-RCT linearity measurement.
RCT measurement uncertainties
The two optical cavities used in the RCT setup are expected to have long term drift due to creep and stress relaxation. The drift rates for both cavities were characterized individually.
For the ULE cavity, the drift rate can be measured directly by beating its resonance frequency against the iodine stabilized standard. The drift rate was found to be equivalent to 0.1 nK/s by a linear fit to 200 hours of data, and it is expected that this drift is due to the slow stress relaxation of the glass.
For the SS cavity, it is hard to measure the drift rate by beating it against a reference directly, since the environmental temperature instabilities will dominate due to the high CTE of the spacer. Instead, we thermally control the RCT using the iodine stabilized laser as the reference, and record the apparent temperature of the thermistor as a function of time. The latter will show the overall drift rate from the thermistor and the RCT, which can be used as a rough estimate of the RCT SS cavity drift, assuming they do not cancel. The resulting linear drift rate from a cubic fit to the data converts to 0.2 nK/s.
Thermistor measurement uncertainties
There are two main sources contributing to the uncertainties for the thermistor. First, there is the lumped apparent measurement noise consisting mostly of electronics noise from the thermistor bridge circuit, affecting noise and stability of the measurement on both short and long time scales. Second, there is the thermistor long-term drift.
For the former effect, we measured the bridge circuit stability by replacing the thermistor with a low temperature coefficient (< 50 ppb/K) resistor, to separate out the effects caused by the thermistor and the electronics. The AD plot of the electronics noise is shown in Fig. 10.
This data is similar to that obtained with the thermistor when the cavity is servo controlled, indicating that the stability is mostly dominated by the bridge circuit electronics, which yields an uncertainty of ∼ 20 µK at long integration times.
For the thermistor long term drift, previous work [45] has shown measurements of the drift rate of a similar type of thermistor over ∼ 800 days. The drift rate obtained from a linear fit to this data is 0.12(5) nK/s where the uncertainty is 1σ. In our case, it is a small effect compared to the uncertainty caused by the thermistor electronics.
Systematic uncertainties
One of the systematic uncertainties is the error caused by the different spatial resolution of the RCT and the thermistor. The RCT measures the average temperature along the cavity spacer, while the thermistor measures the temperature at a specific location on the cavity within a small volume. The temperature distribution of the RCT depends on the heat input and the thermal exchange with its environment. A theoretical model of the cavity temperature distribution including both conductive and radiative heat transfer was developed, using both a simplified model of a radiating fin and a more detailed finite element model. The results are shown in Fig. 11.
Fig. 11 Temperature distribution along one half of the RCT cavity assuming the heater is located at the center. The x axis is the normalized cavity length, with 0 being cavity center. Orange: 2D axisymmetrical FEA simulation, with symmetrical boundary condition at x/L = 0, and radiation rate equals conduction rate at x/L = 1. Distributed heat source with 25 mW is applied to 1/4 of the length of cavity. Blue: simplified radiation fin model. A nominal heat flux of 25 mW was applied at x/L = 0, and the standard tip condition was used, where radiation rate equals conduction rate at x/L = 1. Insert: FEA simulation of cavity temperature distribution. x = 0 indicates cavity center, heater power is applied to the left 1/4 section. Color map is the standard heat map, where white indicates hot.
As can be seen, the simplified model agrees with the more realistic FEA simulation in the bulk part, with slight differences near cavity center due to the different boundary conditions. The thermistor is located at x/L = 0.4, where the simplified model yields a good estimate of the temperature, and hence can be used to calculate the temperature difference dT between thermistor and RCT as a function of power. For a known heater power change, the corresponding change in dT can easily be estimated. The linear slope was calculated to be 169 µK/mW, hence for a given heater power change, the corresponding temperature difference between the two methods could be calculated. We can treat it as an uncertainty or a correction caused by the difference in spatial resolution. We note that in most practical applications, it is not necessary to apply heat directly to the cavity.
Funding
This work was supported by King Abdulaziz City for Science and Technology funding #1169603.
Acknowledgments
We thank R. Byer and S. Buchman (Stanford University) for supporting the project. We also thank G. Cutler (Stanford University) for helping with the construction of the RCT thermal enclosure.
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