Allan Variance Research Articles

Powering an autonomous clock with quantum electromechanics

We theoretically analyse an autonomous clock comprising a nanoelectromechanical system, which undergoes self-oscillations driven by electron tunnelling. The periodic mechanical motion behaves as the clockwork, similar to the swinging of a pendulum, while induced oscillations in the electrical current can be used to read out the ticks. We simulate the dynamics of the system in the quasi-adiabatic limit of slow mechanical motion, allowing us to infer statistical properties of the clock’s ticks from the current auto-correlation function. The distribution of individual ticks exhibits a tradeoff between accuracy, resolution, and dissipation, as expected from previous literature. Going beyond the distribution of individual ticks, we investigate how clock accuracy varies over different integration times by computing the Allan variance. We observe non-monotonic features in the Allan variance as a function of time and applied voltage, which can be explained by the presence of temporal correlations between ticks. These correlations are shown to yield a precision advantage for timekeeping over the timescales that the correlations persist. Our results illustrate the non-trivial features of the tick series produced by nanoscale clocks, and pave the way for experimental investigation of clock thermodynamics using nanoelectromechanical systems.

Broadband and robust Mach-Zehnder interferometer for Rydberg atomic system.

We present a broadband and robust Mach-Zehnder interferometer (MZI) with meter-scale arm length, aiming to acquire the full information of an atomic system. We utilize a pre-loading phase shifter as servo actuator, broadening the servo bandwidth to 108 kHz without sacrificing the size of the piezoelectric transducer (PZT) and mirror. An auxiliary laser at 780 nm, counter-propagating with the probe laser, is employed to achieve arbitrary phase locking of the MZI, boosting a phase accuracy of 0.45 degrees and an Allan deviation of 0.015 degrees, which breaks the current record. By utilizing our robust MZI, the measurement accuracy of atomic system can be theoretically predicted to improve by 2.3 times compared to the most stable MZI in other literatures. In addition, we also demonstrate the sensitivity improvement in imaginary part and real part of the susceptibility in virtue of the completed interferometer, which exhibits tremendous potential in atom-based measurement system.

Near-Infrared Off-Axis Cavity-Enhanced Optical Frequency Comb Spectroscopy for CO2/CO Dual-Gas Detection Assisted by Machine Learning.

Cavity-enhanced direct frequency comb spectroscopy (CE-DFCS) is widely used as a highly sensitive gas sensing technology in various gas detection fields. For the on-axis coupling incidence scheme, the detection accuracy and stability are seriously affected by the cavity-mode noise, and therefore, stable operation inevitably requires external electronic mode-locking and sweeping devices, substantially increasing system complexity. To address this issue, we propose off-axis cavity-enhanced optical frequency comb spectroscopy from both theoretical and experimental aspects, which is applied to the detection of single- and dual-gas of carbon monoxide (CO) and carbon dioxide (CO2) in the near-infrared. An erbium-doped fiber frequency comb with a repetition frequency of ∼41.709 MHz is coupled into a resonant cavity with a length of ∼360 mm in an off-axis manner, exciting numerous high-order modes to effectively suppress cavity-mode noise. The performance of multiple machine learning models is compared for the inversion of a single/dual gas concentration. A few absorbance spectra are collected to build a sample data set, which is then utilized for model training and learning. The results demonstrate that the Particle Swarm Optimization Support Vector Machine (PSO-SVM) model achieves the highest predictive accuracy for gas concentration and is ultimately applied to the detection system. Based on Allan deviation, the detection limit for CO in single-gas detection can reach 8.247 parts per million by volume (ppmv) by averaging 87 spectra. Meanwhile, for simultaneous CO2/CO measurement with highly overlapping absorbance spectra, the LoD can be reduced to 13.196 and 4.658 ppmv, respectively. The proposed optical gas sensing technique indicates the potential for the development of a field-deployable and intelligent sensor system capable of simultaneous detection of multiple gases.

Transponder-type laser interferometer prototype for spaceborne gravitational wave detectors.

Transponder-type laser interferometry is essential in spaceborne gravitational wave detection missions. This paper presents a transponder-type laser interferometer prototype for potential noise calibration of spaceborne gravitational wave detectors. Using a digital optical phase-locked loop, we successfully locked the phase of the slave laser to the master laser (∼200p W). Once the link between the master laser and the slave laser is established, the two satellites (essentially two lasers) form a transponder-type laser interferometer. We carefully analyze the measurement stability and noise characteristics of the interferometer, and the results show that the Allan deviation of the zero drift can reach 243.2pm at t=0.429s, while the noise spectral density has a typical 1/f line shape with a floor of 21p m/H z 1/2 at 1Hz. The coherence analysis shows that the temperature drift is an important factor limiting the performance of the interferometer below 2mHz, while the frequency noise of the master laser is not dominant in the experiment. Transponder-type laser interferometers have a wide range of applications in intersatellite communication and measurement. Our design can serve as a valuable reference for gravitational wave detection missions such as LISA.

Evaluation of timing and time transfer with PPP using Galileo High Accuracy Service

The Galileo High Accuracy Service (HAS) provides free corrections for real-time precise point positioning (PPP) worldwide. PPP can provide high-stability time dissemination and remote clock comparisons, and consequently is widely adopted in time metrology. We collected HAS data of 134 days to present a comprehensive performance evaluation in timing and time transfer. Concerning the signal-in-space range error (SISRE), HAS corrections reach orbital/total root mean square errors (RMSE) of 3.0/6.0 cm and 4.2/13.7 cm for Galileo and GPS, respectively. The timing results show overlapping Allan deviation (OADEV) values 1 · 10-12–5 · 10-15 for Galileo and 4 · 10-12–1 · 10-14 for GPS, for intervals 30–100,000 s, and RMSE of PPP timing 1.9 ns and 1.4 ns for Galileo and GPS, respectively. For remote clock comparisons, the OADEV values are 5 · 10-13–5 · 10-15 for Galileo and 7 · 10-13–4 · 10-15 for GPS.

Dual closed-loop control method for resonant integrated optic gyroscopes with combination differential modulation

In order to optimize the detection accuracy and output stability of Resonant Integrated Optic Gyroscopes (RIOG), a dual closed-loop control method for combined differential modulation (DCM Control) has been proposed. Two optical signals are differentially modulated in sync, and their differential-mode output results are compensated in real-time. Considering the distinct characteristics of the two frequency servo loops, they are respectively utilized to tune the laser frequency to track the resonant frequency's wide fluctuations and continuously changing resonant frequency due to random rotation. Through this novel structure, we have enhanced reciprocity while reducing optical components, suppressing RIM noise and laser frequency noise in the gyroscope system, and effectively reducing control errors. We analyze the principles affecting angular velocity detection accuracy and the system's output model. Experimental results indicate that this method can effectively achieve high reciprocity and high-precision frequency control, offering a larger dynamic detection range and better linearity. The long-term bias instability (BI) based on Allan deviation is measured at 0.95°/h, and the angle random walk (ARW) is 0.119°/√h, providing a simple and efficient approach for RIOG closed-loop signal processing.

Simultaneous dual-harmonic detection for LITES with a multi-pass configuration based on combined flexural modes of a quartz tuning fork

The simultaneous dual-harmonic detection for light-induced thermoelastic spectroscopy (LITES) with a multi-pass configuration based on combined flexural modes of a quartz tuning fork (QTF) is demonstrated. To enhance the photothermal energy absorbed by a custom QTF, the laser beam is designed to pass through the QTF multiple times. With dual-frequency modulation methodology, the second harmonic (2f_fun) signal and the first harmonic (1f_ove) signal are simultaneously inspired at resonant frequencies of fundamental and overtone flexural modes of the QTF, respectively. Thus, precise light intensity and gas concentration can be retrieved concurrently with corresponding 1f and 1f-normalized 2f signals using frequency division multiplexing technology. In comparison to traditional LITES under the same condition sensor, our triple-pass configuration performs 1.84 times better in signal-to-noise ratio (SNR). According to Allan variance analysis, 1f-normalized 2f signal enables a longer integration time of 1100 s and a lower detection limit of 0.07 ppm. The corresponding normalized noise equivalent absorption coefficient (NNEA) is 0.8 × 10−9 cm−1W/Hz1/2.

Quantum cascade laser absorption sensor for in-situ, real-time and sensitive measurement of high-temperature SO2 and SO3

We report a mid-infrared quantum cascade laser absorption sensor capable of measuring SO2 and SO3 simultaneously and sensitively at elevated temperatures. In the sensor development, the intense transitions of SO2 and SO3 in the mid-infrared region of 1129 cm−1 and 1398 cm−1 were exploited by two quantum cascade lasers. A high-temperature multipass cell was adopted to increase the absorption path length to 10 m. The quantitative concentrations of SOx were directly obtained from the calibration-free wavelength modulation spectroscopic method, which was validated at varied temperature and pressure conditions. From Allan deviation analysis, we achieved a minimum detection limit of 8 parts per billion (ppb) for SO2 and 3 ppb for SO3, with an average time of 100 s. Lastly, we successfully demonstrated the real-time and sensitive measurement of SO2 and SO3 during the oxidation reaction of SO2 by O3 at 460 K. Our laser sensor shows great potential for in-situ and real-time monitoring of SOx from combustion emissions.

Triple-range adjustable plane-concave multipass gas cell for high-precision trace gas detection based on TDLAS

A triple-range adjustable plane-concave multipass gas cell (PC-MPGC) was developed for high-precision trace gas detection based on tunable diode laser absorption spectroscopy. Effective optical path lengths (EOPLs) of 64.5, 42.5, and 20.5 m could be easily achieved by adjusting the incident beam angle. The trace gas sensor system employed a 1.653 µm distributed feedback diode laser to target a strong methane (CH4) absorption line, and direct absorption spectroscopy (DAS) was used to evaluate the performance of the system with a 64.5 m EOPL. A Savitzky–Golay filtering algorithm was employed to denoise the measured signal, and a third harmonic signal was used for laser frequency locking. A signal-to-noise ratio (SNR) of 191 was achieved, which corresponds to a detection sensitivity of 0.78 ppmv. The system response time was obtained through measurement of CH4 at concentration of 0 ppmv and 20 ppmv at 298 K and 1 atm, and it was found to be 11 s. Kalman filtering was applied to improve the detection sensitivity of the gas sensor. The stability of the system was analyzed using Allan deviation, and a minimum detectable limit of 21 ppbv was obtained for an integration time of 580 s. In situ continuous monitoring of atmospheric CH4 characteristics was performed for 72-h. The development of a new type of PC-MPGC paves the way for the detection of multicomponent gases at different absorption path lengths.

A laterally sensitive quartz vibrating beam accelerometer for inertial navigation application

To improve the sensitivity and stability of a quartz vibrating beam accelerometer (QVBA), a laterally sensitive structure was developed that utilizes the force-frequency characteristics of a quartz double-ended tuning fork (DETF) to realize quasi-digital measurement of acceleration. Two DETFs were purposely designed for the sidewalls of the proof mass, and the hinge of the spring-mass structure was along the proof-mass thickness, which improved the sensitivity. Laterally sensitive structures can also solve the fabrication problems of hinges, including accuracy, deformation, and residual stress. A femtosecond laser was used to release the anchor between the out-of-frame and the proof mass to protect the hinge. The isolation structure was designed to float both the DETFs and hinge to reduce the interference of external stress. Electrodes were deposited on the four surfaces of the vibrating beam in a three-segment configuration to excite the DETF into the antiphase in-plane vibration model. As a result, an accelerometer prototype is fabricated, and the performance characterization results exhibit the sensitivity and stability of 188.61 Hz/g and 7.81 ppm, respectively, and the bias instability (Allan deviation) of 1.26 μg, which proves the design rationality of laterally sensitive QVBA.

Repetition Frequency Control of a Mid-Infrared Ultrashort Pulse Laser

In this study, a method for controlling the repetition frequency of a mid-infrared ultrashort pulse laser with a central wavelength of 2.8 µm is developed. A ring cavity that is insensitive to the polarization state of the laser light emitted from the fiber end was constructed to stabilize the oscillation of the mid-infrared ultrashort pulse laser. More oscillation conditions for the ultrashort pulse laser based on nonlinear polarization rotation are found than the conventional method. To confirm that the pulse oscillation is mode-locked, ultrashort pulse oscillation was confirmed by an autocorrelator. The pulse repetition frequency of this robust ultrashort pulse laser was controlled. The control method was based on the phase-locked loop (PLL) control. A wedge window was inserted into the cavity and mounted on a linear stage driven by a piezoelectric transducer. By driving the piezoelectric transducer, the position of the wedge window changed, and the resulting optical path length also changed. The repetition frequency was controlled based on this principle. Optical path length control by the wedge window and temperature control provides an Allan deviation of approximately 1 mHz.

Re-injection off-axis integrated cavity output spectroscopy for the simultaneous detection of N2O, H2O and CO with a mid-infrared QCL laser.

Off-axis integrated cavity output spectroscopy (OA-ICOS) has attracted much interest because it potentially allows highly sensitive field measurements with robust optical alignment. In this paper, a novel instrument that employs a high-finesse optical cavity as an absorption cell has been developed for sensitive measurements of multi-component gases N2O, H2O and CO in the atmosphere based on a mid-infrared quantum cascade laser (QCL) and OA-ICOS. In order to improve the energy utilization and increase the signal-to-noise ratio (SNR) of the signal, a new type of optical path structure of the laser re-injection method is adopted. Furthermore, the system performance can be effectively improved by using a new intervention method of injecting radio frequency (RF) white noise into a laser driver to suppress cavity mode noise and combining the wavelength modulation method (WMS). We compared the sensitivity of the second harmonic signal demodulation between the re-injection method and the standard OA-ICOS, and the SNR increased by 2.68 times compared to the latter. Analysis of the spectral measurements with Allan variance indicates that within an integration time of 1 s, the measurement accuracy of N2O, H2O, and CO is 6.71 ppb, 13.945 ppm, and 1.81 ppb, respectively, and within an integration time of 820 s, the measurement accuracy of N2O, H2O, and CO can be further improved to 1.26 ppb, 2.089 ppm, and 172 ppt, respectively. Our approach represents an underlying analytical method that provides guidelines for monitoring of representative gases in the atmosphere, industrial processes, emergency safety, etc.

Near infrared ammonia sensing system based on on‐axis cavity‐enhanced absorption spectroscopy using board‐level mode‐locked circuit

AbstractWe proposed a near‐infrared gas sensing system based on mode‐locked cavity‐enhanced absorption spectroscopy. A distributed feedback fiber laser (DFB‐FL) with a central wavelength of 1550 nm was employed as the light source, coupling with a Fabry–Perot (F‐P) cavity with an effective optical path length of 42.7 m served as the gas cell. By utilizing a homemade mode‐locked electric circuit and Pound–Drever–Hall (PDH) technique, the mode‐locking between the DFB‐FL and F‐P cavity was achieved with long‐term fluctuation of 3.2%. System performance was verified using ammonia as the analyte gas with a target line at 1550.17 nm. Allan variance analysis revealed that the minimum detection limit was 15.8 ppm with an averaging time of 216 s, corresponding to an absorption coefficient of 2.3 × 10−7 cm−1.

Measurement of Ammonia Leakage by TDLAS in Mid-Infrared Combined with an EMD-SG Filter Method

Ammonia (NH3) has been used as the main refrigerant in various types of cold storages, exceeding 90% of cooling applications. However, NH3 leakage occurs frequently due to damage of the refrigeration compressor, resulting in environmental pollution and intoxication accidents. Therefore, reliable and highly sensitive NH3 detection is a critical requirement in cold storage. In this paper, tunable diode laser absorption spectroscopy (TDLAS) is used to measure NH3 leakage, where a new denoising method combining empirical mode decomposition with the Savitzky-Golay smoothing algorithm (EMD-SG) is proposed to improve the signal-to-noise ratio (SNR) of absorbance signals. The results show that a 5.5 times improvement of SNR and 86% reduction of the linear fitting error in calibration is obtained; furthermore, this system achieves a detection limit of 0.2 ppm at an average time of 69 s by Allan variance analysis. Hence, it can be concluded that the EMD-SG algorithm can effectively improve the performance of TDLAS systems to meet the requirements for NH3 leakage detection in cold storages.

Double-channel sensors for high precision measurement of methane based on a dual-path Herriott cell.

A double-channel methane (CH4) sensor was developed using a dual-pass multipass cell (DP-MPC) and a novel method that combines averaging dual-channel concentration signals with optimized detector gain configuration. This DP-MPC features two input/output coupling holes, resulting in absorption path lengths of approximately 95.8 m and 35.8 m, respectively. By optimizing the photodetector gain configuration and averaging the dual-channel concentration signals, the detection performance of the sensor was further enhanced. Allan deviation analysis indicated that after optimizing the detector gain, the measurement precision after dual-channel averaging reaches 21 ppb with an integration time of 1 s at a concentration of 2 ppm CH4, which is approximately 1.4 times higher than the measurement precision of the long-path channel (31 ppb) and short-path channel (30 ppb). The time required to achieve a measurement precision of 21 ppb is 2.4 s for the long-path channel and 2.1 s for the short-path channel. The response speed of the dual-channel averaging is approximately 2 times that of any single channel. Meanwhile, the sensor demonstrated its stability and reliability through continuous outdoor atmospheric CH4 measurements.

Application of quantum-limited optical time transfer to space-based optical clock comparisons and coherent networks

With the demonstration of quantum-limited optical time transfer capable of tolerating the losses associated with long ground-to-space links, two future applications of free-space time transfer have emerged: intercontinental clock comparisons for time dissemination and coherence transfer for future distributed sensing in the mm-wave region. In this paper, we estimated the projected performance of these two applications using quantum-limited optical time transfer and assuming existing low-size, low-weight, and low-power hardware. In both cases, we limit the discussion to the simplest case of a single geosynchronous satellite linked to either one or two ground stations. One important consideration for such future space-based operations is the choice of reference oscillator onboard the satellite. We find that with a modestly performing optical reference oscillator and low-power fiber-based frequency combs, quantum-limited time transfer could support intercontinental clock comparisons through a common-view node in geostationary orbit with a modified Allan deviation at the 10−16 level at 10-s averaging time, limited primarily by residual turbulence piston noise. In the second application of coherence transfer from ground-to-geosynchronous orbit, we find the system should support high short-term coherence with ∼10 millirad phase noise on a 300 GHz carrier at essentially unlimited integration times.

Highly sensitive detection of oxygen based on light-induced thermoelastic spectroscopy with a high power diode laser

Oxygen (O2) is a fundamental element in the atmosphere. It is important to detect it in various fields. In this paper, a sensitive O2 sensor based on light-induced thermoelastic spectroscopy (LITES) with a high power diode laser is reported. A distributed feedback (DFB) diode laser emitting at 760 nm with output power of 16.8 mW was employed as the light source. The O2 absorption line at 13144.54 cm-1 was selected for the investigation. Wavelength modulation spectroscopy (WMS) technique and second harmonic signal demodulation (2f) were adopted to improve the sensing performance. For the concentration dynamic range from 20% to 80%, a linear relationship between the amplitude of 2f signal and O2 concentration was demonstrated. Allan deviation analysis was conducted to evaluate the long-term stability of the sensor system and the minimum detection limit (MDL) was measured to be 583 ppm at the optimum integration time of 100 s. Further improvement of this O2-LITES sensor system was also discussed.

Simultaneous atmospheric CH4, CO2 and O2 detection using fiber-optic switch (FOS)-based time-division multiplexed NIR laser long optical path absorption spectroscopy

A near-infrared (NIR) distributed feedback (DFB) laser sensor based on long optical path absorption spectroscopy technique was developed for simultaneous measurement of atmospheric methane (CH4), carbon dioxide (CO2) and oxygen (O2). A fiber-optic switch (FOS) in conjunction with the time-division multiplexing (TDM) technique was adopted to enable different lasers scan over the target gas absorption lines in turn. The long optical path wavelength modulation spectroscopy (WMS) technique with second harmonic detection as well as a laser power fluctuation correction method was used to improve the sensor’s precision and accuracy. For the selected gas absorption lines, located at 6046.945 cm-1, 6361.245 cm-1 and 7877.647 cm-1, the detection limit (1σ) of 0.034 parts per million (ppm) for CH4, 11.921 ppm for CO2 and 0.14% for O2 was achieved according to the concentration and noise level. Allan deviation analysis indicates that the 1-s measurement precision was 0.030 ppm for CH4, 11.518 ppm for CO2 and 0.14% for O2, which could be improved to 6.5E-4 ppm, 0.255 ppm and 0.005% at an optimal average time of 800 s. A two-day continuous measurement of the CH4, CO2, and O2 concentration from ambient air was carried out, which verifies the stability and robustness of the developed multi-gas sensor.

Trace gas sensor based on a multi-pass-retro-reflection-enhanced differential Helmholtz photoacoustic cell and a power amplified diode laser.

A high-sensitive photoacoustic spectroscopy (PAS) sensor, which is based on a multi-pass-retro-reflection-enhanced differential Helmholtz photoacoustic cell (DHPAC) and a high power diode laser amplified by erbium-doped fiber amplifier (EDFA), is presented in this work for the first time. In order to improve the interaction length between the light and target gas, the incident light was reflected four times through a multi-pass-retro-reflection-cell constructed by two right-angle prisms. A 1.53 µm distributed feedback (DFB) diode laser was selected to excite photoacoustic signal. Moreover, its power was amplified by an EDFA to 1000 mW to improve the amplitude of photoacoustic signal. Acetylene (C2H2) was chosen as the target analysis to verify the reported sensor performance. Compared to double channel without multiple reflections, the 2f signal of double channel with four reflections was improved by 3.71 times. In addition, when the output optical power of EDFA was 1000 mW, the 2f signal has a 70.57-fold improvement compared with the multi-pass-retro-reflection-cell without EDFA. An Allan deviation analysis was carried out to evaluate the long-term stability of such PAS sensor. When the averaging time was 400 s, the minimum detection limit (MDL) of such PAS sensor was 14 ppb.

Higher-order Allan variance for atomic clocks of arbitrary order: mathematical foundation

In this paper, we perform a time-domain analysis of the higher-order Allan variance for atomic clock models of arbitrary order. Adopting a standard atomic clock model where the time series of the clock reading deviation is expressed as a Wiener or integrated Wiener process, we define the higher-order Allan variance as the mean squared higher-order difference of the clock reading deviation. The main results of this paper are threefold. First, we prove that the higher-order difference operation of the clock reading deviation, which can be interpreted as a linear aggregation with binomial coefficients, is not only sufficient but also necessary for a resulting aggregated time series to be an independent and identically distributed Gaussian process. Second, we derive a complete analytical expression of the higher-order Allan variance, which consists of both time-dependent and time-independent terms. Third, we prove that the higher-order Allan variance is time-independent if and only if the order of difference operation is greater than or equal to the order of the atomic clock model.