Differential Wavefront Sensing Research Articles

Measuring the impact of laser relative intensity noise on heterodyne interferometers using differential wavefront sensing

Laser-intensity fluctuations cause undesired phase noise even in balanced heterodyne interferometers. However, in space missions in particular, such as LISA Pathfinder or LISA, direct measurements of these fluctuations at the relevant frequencies are often not available. Hence, it can be challenging to estimate their impact on the interference phase. To address this, we propose a new method for characterizing laser relative intensity noise (RIN) using differential wavefront sensing (DWS), with the latter being a well-established technique typically used for angular sensing and control. Unlike other methods, this approach does not require an additional reference interferometer and instead takes advantage of the inherent phase subtraction of DWS. This allows us to estimate the RIN value at the heterodyne frequency and its harmonic, relative to the sensor noise floor of the total measurement system. Moreover, it provides a strategy to identify the ideal set point for minimizing RIN couplings in DWS. Published by the American Physical Society 2024

Analysisof Beam Walk in Inter-Satellite Laser Link: Implications for Differential Wavefront Sensing in Gravitational Wave Detection

Achieving space-based gravitational wave detection requires the establishment of an interferometer constellation. It is necessary to establish and maintain stable laser interferometric links using the differential wavefront sensing (DWS) technnique. When the distant measurement beam experiences pointing jitter, it causes beam walk on the surface of the local detector. The reduced overlap between the local reference spot and the distant spot increases the nonlinear errors in the DWS technique, which need to be suppressed. Numerical analysis was conducted on the spatial beam interference signals of the DWS technique when the distant measurement beam experienced pointing jitter. An experimental measurement system was designed, and the beam walk was suppressed using a conjugate imaging system. The results show that within a range of 300 μrad, the optical path with the imaging system can reduce measurement errors by at least 83%. This way also helps to reduce pointing jitter noise in inter-satellite links, thereby improving laser pointing control accuracy.This method would provide a valuable reference for future DWS measurement systems.

High-precision laser beam lateral displacement measurement based on differential wavefront sensing.

Accurately lateral displacement measurement is essential for a vast of non-contact sensing technologies. Here, we introduce a high-precision lateral displacement measurement method based on differential wavefront sensing (DWS). Compared to the conventional differential power sensing (DPS) method, the DWS method based on phase readout has the potential to achieve a higher resolution. The beam lateral displacement can be obtained by the curvature distribution of the wavefront on the surface of the detector. According to the theoretical model of the DWS method, the sensitivity of the lateral displacement can be greatly improved by increasing the wavefront curvature of the measured laser beam by means of lenses. An optical system for measuring the lateral displacement of the laser beam is built and calibrated by a high-precision hexapod. The experimental results show that the DWS-based lateral displacement measurement achieves a resolution of 40 pm/Hz1/2 (at 1-10 Hz) with a linear range of about 40 µm, which is consistent with the theoretical model. This technique can be applied to high-precision multi-degree-of-freedom interferometers.

Using DWS Optical Readout to Improve the Sensitivity of Torsion Pendulum.

In space gravitational wave detection missions, a drag-free system is used to keep the test mass (TM) free-falling in an ultralow-noise environment. Ground verification experiments should be carried out to clarify the shielding and compensating capabilities of the system for multiple stray force noises. A hybrid apparatus was designed and analyzed based on the traditional torsion pendulum, and a technique for enhancing the sensitivity of the torsion pendulum system by employing the differential wavefront sensing (DWS) optical readout was proposed. The readout resolution experiment was then carried out on an optical bench that was designed and established. The results indicate that the angular resolution of the DWS signal in optical readout mode can reach the level of 10 nrad/Hz1/2 over the full measurement band. Compared with the autocollimator, the sensitivity of the torsional pendulum is noticeably improved, and the background noise is expected to reach 4.5 × 10-15 Nm/Hz1/2@10 mHz. This method could also be applied to future upgrades of similar systems.

Tilt-to-Length Coupling Analysis of an Off-Axis Optical Bench Design for NGGM

A new off-axis optical design alternative to that of the GRACE Follow-on mission for future NGGM missions is considered. In place of the triple-mirror assembly of the GRACE Follow-on mission, a laser retro-reflector is instead generated by means of lens systems. The receiving (RX) beam and transmitting (TX) beam are enforced to be anti-parallel by a control loop with differential wavefront sensing (DWS) signals as readout, and a fast-steering mirror is employed to actuate the pointing of the local beam. The tilt-to-length (TTL) coupling noise of the new off-axis optical bench layout is carefully studied in the present work. Local TTL originated from piston noise as well as assembly and alignment errors of optical components are studied. Effort is also made to have an in depth understanding of global TTL due to relative attitude jitter between spacecraft. The margin of TTL noise in the position noise budget for laser ranging is examined. With an open loop control of the offset between the reference point of the optical bench and the centre of mass of a satellite, the TTL noise of the new off-axis optical bench design may be suppressed efficiently.

Postprocessing subtraction of tilt-to-length noise in LISA

The coupling of an angular jitter into the interferometric phase readout is summarized under the term tilt-to-length (TTL) coupling. This noise is expected to be a major noise source in the intersatellite interferometry for the Laser Interferometer Space Antenna (LISA) space mission. Despite efforts to reduce it by satellite construction, some remaining TTL noise will need to be removed in postprocessing on Earth. Therefore, such a procedure needs to be developed and validated to ensure the success of the LISA mission. This paper shows a method to calibrate and subtract TTL noise that has no impact on LISA science operations. This solution relies on noise minimization and uses the differential wavefront sensing (DWS) measurements to estimate the TTL contribution. Our technique is applied after the laser frequency noise is suppressed via the time-delay interferometry (TDI) postprocessing algorithm. We use a simulation to show as a proof-of-principle that we can estimate the TTL coefficients to the required accuracy level based on the current design configuration of LISA. We then use these estimates to subtract the TTL noise, ensuring that any remaining TTL noise is below the current estimate of the other noise sources. We validate the procedure on simulated data for different operating scenarios. Our work shows that it is indeed possible to estimate the effect of TTL coupling and subtract it a posteriori from the TDI data streams.

A Differential Phase-Modulated Interferometer with Rotational Error Compensation for Precision Displacement Measurement

In this paper, a differential phase-modulated interferometer (DPMI) is proposed to compensate for the rotational error for precision displacement measurement. In DPMI, a reference interferometer sharing the same reference arm with the measurement interferometer is constructed. Using the two interferometers to differentially measure the displacement, the unbalanced environmental disturbance on the measurement can be minimized. An integrated 2 × 2 array photodetector (APD) is adopted in DPMI. Based on APD with differential wavefront sensing (DWS) technology, the rotation angle can be detected and compensated. Therefore, precision displacement without rotational error and unbalanced environmental disturbance can be achieved. Three confirmatory experiments were performed, and the experimental results show that the maximum displacement drift is reduced from 902.9 nm to 16.3 nm in 100 min stability test, the standard deviations between the pitch and yaw angles obtained by DPMI and Renishaw interferometer are 1.68 × 10−5° and 1.86 × 10−5°, respectively, and the maximum deviation between the measurement result of DPMI and the stage positioning before and after angle compensation is reduced from 5.207 μm to about 0.719 μm.

Analysis of GRACE Follow-On Laser Ranging Interferometer Derived Inter-Satellite Pointing Angles

Gravity Recovery and Climate Experiment Follow-On (GRACE-FO) was launched on May 22, 2018. It carries the Laser Ranging Interferometer (LRI) as a technology demonstrator that measures the inter-satellite range with nanometer precision using a laser-link between satellites. To maintain the laser-link between satellites, the LRI uses the beam steering method: a Fast Steering Mirror (FSM) is actuated to correct for misalignment between the incoming and outgoing laser beams. From the FSM commands, we can compute the inter-satellite pitch and yaw angles. These angles provide information about the spacecraft's relative orientation with respect to line-of-sight (LOS). We analyze LRI derived inter-satellite pointing angles for 2019 and 2020. Further, we present its comparison with the pointing angles derived from GRACE-FO SCA1B data, which represents the spacecraft attitude computed from star cameras and Inertial Measurement Unit (IMU) data using a Kalman filter. We discuss the correlations seen between the laser based attitude data and the spacecraft temperature variations. This analysis serves as the basis to explore the potential of this new attitude product obtained from the Differential Wavefront Sensing (DWS) control of a FSM.

Laser acquisition experimental demonstration for space gravitational wave detection missions

The laser acquisition-pointing technique is one of the most important techniques for space gravitational wave detection missions, like the Taiji program and the LISA (Laser Interferometer Space Antenna) program. The laser acquisition system suppresses the laser deviation angle to 1 µrad at the receiving aperture. Corresponding to 80 times of telescope magnification, the acquisition accuracy should reach 80 µrad at the acquisition camera. In order to verify the feasibility of the laser acquisition scheme, a laser acquisition ground simulation experimental system is designed and constructed. The experimental system simulates the actual acquisition process of the Taiji from three aspects: optical path, acquisition accuracy and acquisition scanning process. In the experiment, the coupling between the laser acquisition system and the laser pointing system is considered by introducing the DWS (Differential Wave-front Sensing) technique to calibrate the reference position of the acquisition camera and read out the acquisition precision. Due to limited beam propagation distance in the ground experiment, the in-flat top properties of the transmitting beam will greatly affect the acquisition precision. Based on the analysis of the influence, an improved acquisition ground simulation scheme is introduced. The experimental results indicate that the experimental system can achieve the acquisition accuracy of sub-10 µrad magnitude at the acquisition camera. The experimental system realizes methodological demonstration of the acquisition scheme. The results offer the experimental foundation and theoretical basis for the acquisition system of the Taiji/LISA program.

Applying Differential Wave-Front Sensing and Differential Power Sensing for Simultaneous Precise and Wide-Range Test-Mass Rotation Measurements

We propose to combine differential wave-front sensing (DWS) and differential power sensing (DPS) in a Mach-Zehnder type interferometer for measuring the rotational dynamics of a test-mass. Using the DWS method, a high sensitive measurement of 6 nrad Hz−1/2 in sub-Hz frequencies can be provided around the test-mass nominal position ( mrad), whereas the measurement of a wide rotation range ( mrad) is realized by the DPS method. The interferometer can be combined with deep frequency modulation (DFM) interferometry for measurement of the test-mass translational dynamics. The setup and the resulting interferometric signals are verified by simulations. An optimization algorithm is applied to find suitable positions of the lenses and the waist size of the input laser in order to determine the best trade of between the slope of DWS, dynamic range of DPS, and the interferometric contrast. Our simulation further allows to investigate the layout for robustness and design tolerances. We compare our device with a recent experimental realization of a DFM interferometer and find that a practical implementation of the interferometer proposed here has the potential to provide translational and rotational test-mass tracking with state-of-the-art sensitivity. The simple and compact design, and especially the capability of sensing the test-mass rotation in a wide range and simultaneously providing a high-precision measurement close to the test-mass nominal position makes the design especially suitable for example for employment in torsion pendulum setups.

Modulated Differential Wavefront Sensing: Alignment Scheme for Beams with Large Higher Order Mode Content

Modulated differential wavefront sensing (MDWS) is an alignment control scheme in the regime of beams with strong higher order transversal modes (HOMs). It is based on the differential wavefront sensing (DWS) technique. MDWS represents a significant upgrade over conventional techniques used in the presence of high HOM content as it allows for higher control bandwidths while eliminating the need of auxiliary alignment modulations, that otherwise cause loss of applied squeezing. The output port of gravitational wave (GW) interferometers (IFO) is one such place where a lot of HOMs are present. These are filtered out by a cavity called the output mode cleaner (OMC), whose alignment gets challenging due to the presence of HOMs. In this paper, we present the first demonstration of the MDWS scheme for aligning the fundamental mode from the IFO to the OMC at the gravitational wave detector-GEO 600.

Tracking Length and Differential-Wavefront-Sensing Signals from Quadrant Photodiodes in Heterodyne Interferometers with Digital Phase-Locked-Loop Readout

We propose a method to track signals from quadrant photodiodes (QPD) in heterodyne laser interferometers that employ digital phase-locked loops for phase readout. Instead of separately tracking the four segments from the QPD and then combining the results into length and Differential Wavefront Sensing (DWS) signals, this method employs a set of coupled tracking loops that operate directly on the combined length and angular signals. Benefits are increased signal-to-noise ratio in the loops and the possibility to adapt the loop bandwidths to the different dynamical behavior of the signals being tracked, which now correspond to physically meaningful observables. We demonstrate an improvement of up to 6 dB over single-segment tracking, which makes this scheme an attractive solution for applications in precision inter-satellite laser interferometry in ultra-low-light conditions.

Linearity performance analysis of the differential wavefront sensing for the Taiji programme

ABSTRACTThe Taiji programme is a Chinese space-based laser interferometer gravitational wave detection mission and the differential wavefront sensing (DWS) technique is introduced to reduce the effect of the laser pointing noise. As the distance between two adjacent satellites is about , the wavefront of the receiving beam is a flat top beam. In this paper, we construct an analytical model of the DWS signal under the circumstances of Gaussian beam-flat top beam interference and study the linearity performance. The position offset of the beams is found to greatly affect the linearity range. A numerical method is used to simulate the linearity range of the DWS signal under various operating conditions. The linearity range decrease with the increase of the offset angle of the Gaussian beam and the offset length of the flat top beam. The offset range of the beams based on the requirement of the Taiji is given at last.

LISA Pathfinder: Understanding DWS noise performance for the LISA mission

ESA’s L3 Laser Interferometer Space Antenna (LISA) mission contains a mechanism to compensate for out-of-plane angles between the received and emitted beams of the three satellites. Depending on the configuration of this Point-Ahead Angle Mechanism (PAAM) it is expected to contribute readout noise through Differential Wavefront Sensing (DWS). This was investigated with LISA Pathfinder (LPF) through a dedicated investigation. One of the two free-falling test masses was rotated via the on-board electrostatic actuators while the resulting angular noise in the differential interferometer between the two test masses was measured. For angles between −250 μrad to 250 μrad and corresponding contrast in the range of 59.4 % to 97.9 % an increased spectral density was found. The differential displacement noise remains almost unchanged for these misalignments.

Spatially resolved photodiode response for simulating precise interferometers

Quadrant photodiodes (QPDs) are used in laser interferometry systems to simultaneously detect longitudinal displacement of test masses and angular misalignment between the two interfering beams. The latter is achieved by means of the differential wavefront sensing (DWS) technique, which provides ultra-high precision for measuring angular displacements. We have developed a setup to obtain the spatially resolved response of QPDs that, together with an extension of the simulation software IfoCAD, allows us to use the measured response in simulations and accurately predict the desired longitudinal and DWS phase observables. Three different commercial off-the-shelf QPD candidates for space-based interferometry were characterized. The measured response of one QPD was used in optical simulations. Nonuniformities in the response of the device and crosstalk between segments do not introduce significant variations in the longitudinal and DWS measurands with respect to the standard case when a uniform QPD without crosstalk is used.

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.

Beam geometry, alignment, and wavefront aberration effects on interferometric differential wavefront sensing

Heterodyne interferometry is a widely accepted methodology with high resolution in many metrology applications. As a functionality enhancement, differential wavefront sensing (DWS) enables simultaneous measurement of displacement, pitch, and yaw using a displacement interferometry system and a single beam incident on a plane mirror target. The angular change is measured using a weighted phase average between symmetrically adjacent quadrant photodiode pairs. In this paper, we present an analytical model to predict the scaling of differential phase signals based on fundamental Gaussian beams. Several numerical models are presented to discuss the effects of physical beam parameters, detector size, system alignment errors, and beam wavefront aberrations on the DWS technique. The results of our modeling predict rotational scaling factors and a usable linear range. Furthermore, experimental results show the analytically predicted scaling factor is in good agreement with empirical calibration. Our three degree-of-freedom interferometer can achieve a resolution of 0.4 nm in displacement and 0.2 μrad in pitch and yaw simultaneously.

Development of a Micro-Thruster Test Facility which fulfils the LISA requirements

In the context of investigations for a sufficient attitude control thruster for LISA, we have developed a thruster test facility which consists of a highly precise thrust balance coupled with plasma diagnostics. In parallel to the test facility development, investigations to downscale a High Efficiency Multistage Plasma Thruster (HEMP-T) are also being carried out. The thruster has been used to demonstrate the measurement capabilities of the facility. The setup allows a parallel operation of all instruments and can also be used for other types of μN propulsion systems including cold gas thrusters. The thrust balance consists of two pendulums. As read out a heterodyne laser interferometer is used. Differential wave front sensing (DWS) enables the measurement of the pendulum tilt which, via suitable calibration using an electrostatic comb, can be converted to a thrust. The whole setup is a symmetric configuration enabling a common-mode rejection of the dominant noise sources (e.g. seismic noise etc.). The thrust balance has a demonstrated precision of 0.1 μN. Based on our unique design, this precision can be attained down to 10-3 Hz. Thus, the measurement setup is especially suitable for characterising the thrust noise of potential eLISA propulsion candidates. We give an overview of the design, the present performance and the future plans.

Laser beam steering for GRACE Follow-On intersatellite interferometry.

The GRACE Follow-On satellites will use, for the first time, a Laser Ranging Interferometer to measure intersatellite distance changes from which fluctuations in Earth's geoid can be inferred. We have investigated the beam steering method that is required to maintain the laser link between the satellites. Although developed for the specific needs of the GRACE Follow-On mission, the beam steering method could also be applied to other intersatellite laser ranging applications where major difficulties are common: large spacecraft separation and large spacecraft attitude jitter. The beam steering method simultaneously coaligns local oscillator beam and transmitted beam with the laser beam received from the distant spacecraft using Differential Wavefront Sensing. We demonstrate the operation of the beam steering method on breadboard level using GRACE satellite attitude jitter data to command a hexapod, a six-degree-of-freedom rotation and translation stage. We verify coalignment of local oscillator beam/ transmitted beam and received beam of better than 10 μrad with a stability of 10 μrad/ √Hz in the GRACE Follow-On measurement band of 0.002...0.1 Hz. Additionally, important characteristics of the beam steering setup such as Differential Wavefront Sensing signals, heterodyne efficiency, and suppression of rotation-to-pathlength coupling are investigated and compared with analysis results.

Methodological demonstration of laser beam pointing control for space gravitational wave detection missions

In space laser interferometer gravitational wave (G.W.) detection missions, the stability of the laser beam pointing direction has to be kept at 10 nrad/√Hz. Otherwise, the beam pointing jitter noise will dominate the noise budget and make the detection of G.W. impossible. Disturbed by the residue non-conservative forces, the fluctuation of the laser beam pointing direction could be a few μrad/√Hz at frequencies from 0.1 mHz to 10 Hz. Therefore, the laser beam pointing control system is an essential requirement for those space G.W. detection missions. An on-ground test of such beam pointing control system is performed, where the Differential Wave-front Sensing technique is used to sense the beams pointing jitter. An active controlled steering mirror is employed to adjust the beam pointing direction to compensate the jitter. The experimental result shows that the pointing control system can be used for very large dynamic range up to 5 μrad. At the interested frequencies of space G.W. detection missions, between 1 mHz and 1 Hz, beam pointing stability of 6 nrad/√Hz is achieved.