Research on the tilt-to-length coupling noise suppression method inside the gravitational wave detection telescope.

As an integral component of the laser interferometry measurement system, the tilt-to-length (TTL) coupling noise inside the telescope stands out as a critical noise factor that requires meticulous consideration. In the TianQin project, the non-geometric TTL-coupled noise inside the telescope should be less than 0.22 pm/Hz1/2. Additionally, the wavefront aberration RMS at the small pupil of the telescope needs to be better than 0.0065 λ. These requirements set for the telescope are exceptionally stringent. To address this challenge, this study aims to relax the wavefront aberration requirements by mitigating non-geometric TTL coupling noise, while ensuring the non-geometric TTL coupling noise remains below 0.22 pm/Hz1/2. By controlling the coupling aberration proportion, the wavefront aberration RMS at the small pupil of the telescope can be relaxed to 0.014 λ. Alternatively, optimizing the Gaussian beam waist radius can relax the wavefront aberration RMS to 0.016 λ. By simultaneously utilizing two optimization methods, the wavefront aberration at the small pupil of the telescope can be reduced to 0.033 λ, resulting in an impressive success rate of 91.15% in meeting the noise requirements.

Analysis of thermal instability for fiber backscatter phase and intensity.

In this Letter, we utilize the speckle model to measure the average random scattering rate of fiber backscatter and analyze its dependence on length, yielding a linear fitting coefficient of 0.23 ppm/m for a PM980-XP fiber. We incorporate the temperature coupling effect into the model and validate the model's accuracy by examining the distribution of the change rate of the backscattering rate relative to the temperature and the amplitude spectral density of the backscattered power. Our findings demonstrate that the typical shoulder-shaped noise in interferometer experiments is limited by stray light, and the dependence of shoulder-shaped noise on the fiber length and temperature noise level is analyzed.

Suppression of frequency-mixing effect for pm-level heterodyne interferometers based on "zero coupling" optical path length control.

Optical path length (OPL) noise resulting from stray light significantly constrains interferometry displacement measurements in the low-frequency band. This paper presents an analytical model considering the presence of stray light in heterodyne laser interferometers. Due to the cyclic nonlinear coupling effect, there will be some special OPLs of stray light, minimizing the frequency-mixing impact to zero. Consequently, we propose a noise suppression scheme that locks the OPL of stray light at the zero coupling point. Therefore, we significantly enhanced the interference displacement measurement noise within the low-frequency band. Experimental results show that the interferometer achieves a displacement noise level lower than 6 pm/Hz1/2 covering 1 mHz.

Study on TPD Phasemeter to Suppress Low-Frequency Amplitude Fluctuation and Improve Fast-Acquiring Range for GW Detection.

A phasemeter as a readout system for the inter-satellite laser interferometer in a space-borne gravitational wave detector requires not only high accuracy but also insensitivity to amplitude fluctuations and a large fast-acquiring range. The traditional sinusoidal characteristic phase detector (SPD) phasemeter has the advantages of a simple structure and easy realization. However, the output of an SPD is coupled to the amplitude of the input signal and has only a limited phase-detection range due to the boundedness of the sinusoidal function. This leads to the performance deterioration of amplitude noise suppression, fast-acquiring range, and loop stability. To overcome the above shortcomings, we propose a phasemeter based on a tangent phase detector (TPD). The characteristics of the SPD and TPD phasemeters are theoretically analyzed, and a fixed-point simulation is further carried out for verification. The simulation results show that the TPD phasemeter tracks the phase information well and, at the same time, suppresses the amplitude fluctuation to the noise floor of 1 µrad/Hz1/2, which meets the requirements of GW detection. In addition, the maximum lockable step frequency of the TPD phasemeter is almost three times larger than the SPD phasemeter, indicating a greater fast-acquiring range.

Experimental demonstration of weak-light inter-spacecraft clock jitter readout for TianQin.

The space-based gravitational wave detection mission, TianQin, requires high-level synchronization between independent clocks of all spacecrafts to extract the gravitational wave signals. It is necessary to measure the inter-spacecraft relative clock jitter based on laser phase-sideband clock transfer. The main challenge is the tracking and locking of clock sideband beatnote signals with low signal-to-noise ratio and frequency variation. In this paper, a systematic scheme of inter-spacecraft clock jitter readout is reported. The requirement of the clock transfer link for TianQin based on the time-delay interferometry algorithm is derived. A bi-directional laser interferometer system with a transmission optical power below 1 nW and a time delay of ∼50 µs is built up to demonstrate the weak-light clock transfer. In this scheme, frequency modulation is performed on the laser to simulate the inter-spacecraft Doppler frequency shift and its variation. Based on electrical and optical clock transfer comparison experiments, it is demonstrated that the GHz frequency synthesizer is the main noise source below the 50 mHz frequency range. The residual clock jitter noise introduced by the optical transfer link is below 40 fs/Hz1/2 above the 6 mHz frequency range, and the fractional frequency instability is less than 6.7 × 10-17 at 1000 s, which meets the requirement of the TianQin mission. Ultimately, The carrier phase measurement accuracy reaches 1 × 10-4 cycles/Hz1/2 above 6 mHz after differential clock noise correction using measured clock jitter.

Dual-frequency fundamental-mode NPRO laser for low-noise microwave generation.

Monolithic nonplanar ring oscillators (NPROs) have achieved great success in industry, scientific applications and space missions due to their excellent narrow-linewidth, low-noise, high beam-quality, lightweight and compact performances. Here, we show that stable dual-frequency or multi-frequency fundamental-mode (DFFM or MFFM) laser can be stimulated directly by tunning pump divergence-angle and beam-waist injected to NPRO. The DFFM laser has a frequency deviation of one free spectral range of the resonator and thus can be utilized for pure microwave generation by common-mode-rejection. To demonstrate the purity of the microwave signal, a theoretical phase noise model is established, and the phase noise and the frequency tunability of the microwave signal are experimentally studied. Single sideband phase noise for a 5.7 GHz carrier is measured as low as -112 dBc/Hz at 10 kHz offset, and -150 dBc/Hz at 10 MHz offset in the free running condition of the laser, which outperforms its counterparts from dual-frequency Laguerre-Gaussian (LG) modes. The frequency of the microwave signal can be efficiently tunned through two channels, with frequency tunning coefficients of 15 Hz/V by piezo, and -60.5 kHz/K by temperature, respectively. We expect that such compact, tunable, low-cost and low-noise microwave sources can facilitate multiple applications including miniaturized atomic clocks, communication and radar, etc.

Unidirectional operation criterion in monolithic nonplanar ring oscillators.

Monolithic nonplanar ring oscillators (NPROs) under an applied magnetic field can operate unidirectional single-frequency lasing due to the loss differences among its four eigenpolarizations, where the minimum was empirically estimated to be 0.01%. However, this value has never been verified because the applied magnetic field is not uniformly distributed, making it hard to resolve both theoretically and experimentally. Here, we propose a method to resolve the applied magnetic field through an NPRO by combining finite-element analysis and experimental verification. By introducing the non-uniform magnetic field information to the eigenpolarization theory, the loss differences can be calculated by path integration along the optical path in the NPRO. The critical point, where the bidirectional lasing is emerging, is identified by the relative amplitude noise (RAN) of the laser and by the beating signal between the clockwise (CW) and counterclockwise (CCW) lasing. With this method, we determine that unidirectional operation is possible with loss differences as low as 0.0001% and 0.0003%, corresponding to two different NPRO designs with out-of-plane angles of 90° and 45°, respectively, which increases the precision of the loss differences for unidirectional single-frequency lasing by more than one order of magnitude. Our findings will greatly facilitate NPRO laser design with lowered magnetic field intensity requirements.

Shot-noise-limit performance of a weak-light phase readout system for intersatellite heterodyne interferometry.

A laser interferometer will be used in the spaceborne gravitational-wave detection missions to measure the inter-satellite optical pathlength variations. The phase readout system of the interferometer needs to be carefully designed and tested to accomplish a shot-noise-limited detection performance under the situation of pico-Watt level received lights. In this work, a scheme based on dual-tone acousto-optic diffraction is presented to verify the performance of the weak-light phase readout system. By optimizing the parameters of the photoreceiver and the local strong-light power, the signal-to-noise ratio of the beat-note signal is enhanced. Thanks to the scheme's common-mode noise rejections for the laser frequency noise, and the optical-path noise, etc., the differential phase noise has achieved a performance of 2×10-4 rad/Hz1/2, which is dominated by the weak-light (∼13 pW) shot noise above the frequencies of 2 mHz.

Prototype of a monolithic cavity-based ultrastable optical reference for space applications.

We present a compact, monolithic optical reference for the frequency stabilized laser of future inter-satellite laser interferometer missions. A prototype based on the integration of a high-finesse cavity and associate optics has been designed to be space compatible while maintaining sufficient stability. The prototype has then been developed with a space-qualified bonding technique, and an in situ multi-degree-of-freedom alignment method. The performances of the optical reference have been studied by beat note analysis with another frequency stabilized laser, and the preliminary results are in agreement with the potential requirements of future space missions.

Highly linear sub-nanoradian tilt measurement based on dual-beam interferometry.

The tilt measurement method based on dual-beam interferometry is presented in this Letter. Due to symmetric property of the nonlinear errors of two displacement measurement arms, the composite nonlinearity of the tilt measurement is counterbalanced, so that a small nonlinearity over a large dynamic range can be obtained. According to the theoretical analysis with a Gaussian beam model, the second-order nonlinear error is dominant. The experimental results show that a measurement noise of 0.4nrad/Hz at 1 Hz with a nonlinearity of less than 60 nrad for a measurement range of ±500µrad has been achieved.

External right-angle measurement using a two-autocollimator system.

We report on a non-contact method for external right-angle measurement using two autocollimators. A precise mathematical model is deduced to evaluate and deduct the measuring error. The values measured with our method are very coincident compared with the results measured by a ZYGO interferometer. The measuring accuracy is superior to 0.1 arcsec for the right-angle errors within 3.0 arcsec and becomes 0.4 arcsec for the extended right-angle errors within 8.0 arcsec. This method can be widely used in situations for external right-angle measurement, such as angle measurement for the body of torsion balance and test mass in spaceborne laser interferometry.

Analysis of non-linearity in differential wavefront sensing technique.

An analytical model of a differential wavefront sensing (DWS) technique based on Gaussian Beam propagation has been derived. Compared with the result of the interference signals detected by quadrant photodiode, which is calculated by using the numerical method, the analytical model has been verified. Both the analytical model and numerical simulation show milli-radians level non-linearity effect of DWS detection. In addition, the beam clipping has strong influence on the non-linearity of DWS. The larger the beam clipping is, the smaller the non-linearity is. However, the beam walking effect hardly has influence on DWS. Thus, it can be ignored in laser interferometer.

Inter-satellite laser link acquisition with dual-way scanning for Space Advanced Gravity Measurements mission.

Laser link acquisition is a key technology for inter-satellite laser ranging and laser communication. In this paper, we present an acquisition scheme based on the differential power sensing method with dual-way scanning, which will be used in the next-generation gravity measurement mission proposed in China, called Space Advanced Gravity Measurements (SAGM). In this scheme, the laser beams emitted from two satellites are power-modulated at different frequencies to enable the signals of the two beams to be measured distinguishably, and their corresponding pointing angles are determined by using the differential power sensing method. As the master laser beam and the slave laser beam are decoupled, the dual-way scanning method, in which the laser beams of both the master and the slave satellites scan uncertainty cones simultaneously and independently, can be used, instead of the commonly used single-way scanning method, in which the laser beam of one satellite scans and that of the other one stares. Therefore, the acquisition time is reduced significantly. Numerical simulation and experiments of the acquisition process are performed using the design parameters of the SAGM mission. The results show that the average acquisition time is less than 10 s for a scanning range of 1-mrad radius with a success rate of more than 99%.

Note: Digital laser frequency auto-locking for inter-satellite laser ranging.

We present a prototype of a laser frequency auto-locking and re-locking control system designed for laser frequency stabilization in inter-satellite laser ranging system. The controller has been implemented on field programmable gate arrays and programmed with LabVIEW software. The controller allows initial frequency calibrating and lock-in of a free-running laser to a Fabry-Pérot cavity. Since it allows automatic recovery from unlocked conditions, benefit derives to automated in-orbit operations. Program design and experimental results are demonstrated.

A self-analyzing double-loop digital controller in laser frequency stabilization for inter-satellite laser ranging.

We present a digital controller specially designed for laser frequency stabilization in the application of inter-satellite laser ranging. The prototype of controller is developed using field programmable gate arrays programmed with National Instruments LabVIEW software. The controller is flexible, self-analyzing, and easily optimized with build-in system analysis. Application and performance of the controller to a laser frequency stabilization system designed for spaceborne scientific missions are demonstrated.

A dual-heterodyne laser interferometer for simultaneous measurement of linear and angular displacements.

Picometer laser interferometry is an essential tool for ultra-precision measurements in frontier scientific research and advanced manufacturing. In this paper, we present a dual-heterodyne laser interferometer for simultaneously measuring linear and angular displacements with resolutions of picometer and nanoradian, respectively. The phase measurement method is based on cross-correlation analysis and realized by a PXI-bus data acquisition system. By implementing a dual-heterodyne interferometer with a highly symmetric optical configuration, low frequency noises caused by the environmental fluctuations can be suppressed to very low levels via common-mode noise rejection. Experimental results for the dual-heterodyne interferometer configuration presented demonstrate that the noise levels of the linear and angular displacement measurements are approximately 1 pm/Hz(1/2) and 0.5 nrad/Hz(1/2) at 1 Hz.

Note: Inter-satellite laser range-rate measurement by using digital phase locked loop.

This note presents an improved high-resolution frequency measurement system dedicated for the inter-satellite range-rate monitoring that could be used in the future's gravity recovery mission. We set up a simplified common signal test instead of the three frequencies test. The experimental results show that the dominant noises are the sampling time jitter and the thermal drift of electronic components, which can be reduced by using the pilot-tone correction and passive thermal control. The improved noise level is about 10(-8) Hz/Hz(1/2)@0.01Hz, limited by the signal-to-noise ratio of the sampling circuit.

Fundamental limits on the digital phase measurement method based on cross-correlation analysis.

Ultra-precision phase measurement is a key technology for state-of-the-art laser interferometry. In this paper we present a fully digital phase measurement method based on cross-correlation analysis, and analyze the measurement errors caused by sampling quantization, intrinsic white noise and non-integral-cycle sampling. The last error source results in a cyclic error that has not been reported ever. We used a high-performance data acquisition system to carry out the cross-correlation-based phase measurement, and obtained a noise level of 1.2 × 10(-6) rad/Hz(1/2)[commercial at]1 Hz. Moreover, the cyclic phase error of about 10(-2) rad/Hz(1/2), caused by non-integral-cycle sampling, had been observed. In order to demonstrate the application of this precision phase measurement method, an ultra-precision heterodyne laser interferometer, consisting of digital phase measurement system and ultra-stable optical bench, was constructed for displacement measurement. The experimental results showed that a measurement resolution of 63 pm had been achieved.

Intersatellite laser ranging with homodyne optical phase locking for Space Advanced Gravity Measurements mission.

In this paper, we present the scheme and the preliminary results of an intersatellite laser ranging system that is designed for the Earth's gravity recovery mission proposed in China, called Space Advanced Gravity Measurements (SAGM). The proposed intersatellite distance is about 100 km and the precision of inter-satellite range monitoring is 10 nm/Hz(1/2) at 0.1 Hz. To meet the needs, we designed a transponder-type intersatellite laser ranging system by using a homodyne optical phase locking technique, which is different from the heterodyne optical phase-locked loop used in GRACE follow-on mission. Since an ultrastable oscillator is unnecessary in the homodyne phase-locked loop, the measurement error caused by the frequency instability of the ultrastable oscillator need not be taken into account. In the preliminary study, a heterodyne interferometer with 10-m baseline (measurement arm-length) was built up to demonstrate the validity of the measurement scheme. The measurement results show that a resolution of displacement measurement of about 3.2 nm had been achieved.