Single-Element Dual-Interferometer for Precision Inertial Sensing: Sub-Picometer Structural Stability and Performance as a Reference for Laser Frequency Stabilization.

Future GRACE-like geodesy missions could benefit from adopting accelerometer technology akin to that of the LISA Pathfinder, which employed laser interferometric readout at the sub-picometer level in addition to the conventional capacitive sensing, which is at best at the level of 100 pm. Improving accelerometer performance holds great potential to enhance the scientific output of forthcoming missions, carrying invaluable implications for research in climate, water resource management, and disaster risk reduction. To reach sub-picometer displacement sensing precision in the millihertz range, laser interferometers rely on suppression of laser-frequency noise by several orders of magnitude. Many optical frequency stabilization methods are available with varying levels of complexity, size, and performance. In this paper, we describe the performance of a Mach-Zehnder interferometer based on a compact monolithic optic. The setup consists of a commercial fiber injector, a custom-designed pentaprism used to split and recombine the laser beam, and two photoreceivers placed at the complementary output ports of the interferometer. The structural stability of the prism is transferred to the laser frequency via amplification, integration, and feedback of the balanced-detection signal, achieving a fractional frequency instability better than 6 parts in 1013, corresponding to an interferometer pathlength stability better than 1pm/Hz. The prism was designed to host a second interferometer to interrogate the position of a test mass. This optical scheme has been dubbed "single-element dual-interferometer" or SEDI.

On-Axis Optical Bench for Laser Ranging Instruments in Future Gravity Missions.

The laser ranging interferometer onboard the Gravity Recovery and Climate Experiment Follow-On mission proved the feasibility of an interferometric sensor for inter-satellite length tracking with sub-nanometer precision, establishing an important milestone for space laser interferometry and the general expectation that future gravity missions will employ heterodyne laser interferometry for satellite-to-satellite ranging. In this paper, we present the design of an on-axis optical bench for next-generation laser ranging which enhances the received optical power and the transmit beam divergence, enabling longer interferometer arms and relaxing the optical power requirement of the laser assembly. All design functionalities and requirements are verified by means of computer simulations. A thermal analysis is carried out to investigate the robustness of the proposed optical bench to the temperature fluctuations found in orbit.

Fiber backscatter under increasing exposure to ionizing radiation.

The Laser Interferometer Space Antenna (LISA) will measure gravitational waves by utilizing inter-satellite laser links between three triangularly-arranged spacecraft in heliocentric orbits. Each spacecraft will house two separate optical benches and needs to establish a phase reference between the two optical benches which requires a bidirectional optical connection, e.g. a fiber connection. The sensitivity of the reference interferometers, and thus of the gravitational wave measurement, could be hampered by backscattering of laser light within optical fibers. It is not yet clear if the backscatter within the fibers will remain constant during the mission duration, or if it will increase due to ionizing radiation in the space environment. Here we report the results of tests on two different fiber types under increasing intensities of ionizing radiation: SM98-PS-U40D by Fujikura, a polarization maintaining fiber, and HB1060Z by Fibercore, a polarizing fiber. We found that both types react differently to the ionizing radiation: The polarization maintaining fibers show a backscatter of about 7 ppm·m-1 which remains constant over increasing exposure. The polarizing fibers show about three times as much backscatter, which also remains constant over increasing exposure. However, the polarizing fibers show a significant degradation in transmission, which is reduced to about one third.

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 (±0.11 mrad), whereas the measurement of a wide rotation range (±5 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.

Single-Element Dual-Interferometer for Precision Inertial Sensing.

Tracking moving masses in several degrees of freedom with high precision and large dynamic range is a central aspect in many current and future gravitational physics experiments. Laser interferometers have been established as one of the tools of choice for such measurement schemes. Using sinusoidal phase modulation homodyne interferometry allows a drastic reduction of the complexity of the optical setup, a key limitation of multi-channel interferometry. By shifting the complexity of the setup to the signal processing stage, these methods enable devices with a size and weight not feasible using conventional techniques. In this paper we present the design of a novel sensor topology based on deep frequency modulation interferometry: the self-referenced single-element dual-interferometer (SEDI) inertial sensor, which takes simplification one step further by accommodating two interferometers in one optic. Using a combination of computer models and analytical methods we show that an inertial sensor with sub-picometer precision for frequencies above 10 mHz, in a package of a few cubic inches, seems feasible with our approach. Moreover we show that by combining two of these devices it is possible to reach sub-picometer precision down to 2 mHz. In combination with the given compactness, this makes the SEDI sensor a promising approach for applications in high precision inertial sensing for both next-generation space-based gravity missions employing drag-free control, and ground-based experiments employing inertial isolation systems with optical readout.

Picometer-Stable Hexagonal Optical Bench to Verify LISA Phase Extraction Linearity and Precision.

The Laser Interferometer Space Antenna (LISA) and its metrology chain have to fulfill stringent performance requirements to enable the space-based detection of gravitational waves. This implies the necessity of performance verification methods. In particular, the extraction of the interferometric phase, implemented by a phasemeter, needs to be probed for linearity and phase noise contributions. This Letter reports on a hexagonal quasimonolithic optical bench implementing a three-signal test for this purpose. Its characterization as sufficiently stable down to picometer levels is presented as well as its usage for a benchmark phasemeter performance measurement under LISA conditions. These results make it a candidate for the core of a LISA metrology verification facility.

In-Orbit Performance of the GRACE Follow-on Laser Ranging Interferometer.

The Laser Ranging Interferometer (LRI) instrument on the Gravity Recovery and Climate Experiment (GRACE) Follow-On mission has provided the first laser interferometric range measurements between remote spacecraft, separated by approximately 220 km. Autonomous controls that lock the laser frequency to a cavity reference and establish the 5 degrees of freedom two-way laser link between remote spacecraft succeeded on the first attempt. Active beam pointing based on differential wave front sensing compensates spacecraft attitude fluctuations. The LRI has operated continuously without breaks in phase tracking for more than 50 days, and has shown biased range measurements similar to the primary ranging instrument based on microwaves, but with much less noise at a level of 1 nm/sqrt[Hz] at Fourier frequencies above 100 mHz.

Line of sight calibration for the laser ranging interferometer on-board the GRACE Follow-On mission: on-ground experimental validation.

The laser ranging interferometer (LRI) on board of the GRACE follow-on spacecraft, launched in May 2018, is the first laser interferometer to perform an inter-satellite range measurement. It is designed for ranging noise levels of 80 nm Hz-1/2 for frequencies above 20 mHz, i.e., about a ten-fold improvement with respect to the GRACE follow-on main microwave ranging instrument. One of the most critical steps during the commissioning phase of the instrument is the so-called initial line of sight calibration procedure (or initial acquisition). This process is required to quantify large uncertainties with respect to laser beam pointing angles and laser frequency, which must be known to establish the interferometer link. It is a nine hour scan of five degrees of freedom, which all need to match simultaneously at least once. Here we report on laboratory tests to further validate the calibration procedure using a mock-up LRI and a set-up, the so-called laser link simulator, that creates conditions similar to those with ~220 km distance between the SC. The experiments presented here made use of LRI-like hardware and software and were carried out recreating critical conditions such as received laser powers on the pico-Watt level and their dependence on the SC misalignments, flat-top beams as receiving beams and Doppler frequency shifts. Several configurations were tested, including a full line of sight calibration with angular scans in both mock-up SC and frequency scan in one of the lasers. Results are well in agreement with the expectations and confirm, well before the LRI commissioning phase, the robustness of the procedure under realistic conditions, which had not yet been fully tested experimentally.

Phase stability of photoreceivers in intersatellite laser interferometers.

A photoreceiver (PR) is required for the opto-electrical conversion of signals in intersatellite laser interferometers. Noise sources that originate or couple in the PR reduce the system carrier-to-noise-density, which is often represented by its phase noise density. In this work, we analyze the common noise sources in a PR used for space-based interferometry. Additionally, we present the results from the characterization of the PRs in GRACE-FO, a mission which will pioneer intersatellite laser interferometry. The estimated phase noise is shot-noise limited at 10-4 rad/Hz1/2 down to 10-2 Hz, almost 4 orders of magnitude below the instrument top level requirement (0.5 rad/Hz1/2). Below 10-2 Hz, the PR finite phase response noise dominates but the levels still comply with the instrument requirement. The sub-mHz noise levels and the PR electronic noise have been identified as key design factors for the LISA PR.

Experimental demonstration of reduced tilt-to-length coupling by a two-lens imaging system.

The coupling between beam tilt and longitudinal path length readout in a setup representing a LISA test mass interferometer was reduced to below 2 µm/rad using a two lens imaging system. This was achieved by the use of a homodyne equal arm-length Mach-Zehnder interferometer and suppression of the dominating effects of higher order Gaussian modes and longitudinal actuator movement. The latter was subtracted using the phase signal of a large single element photo diode.

Experimental demonstration of deep frequency modulation interferometry.

Experiments for space and ground-based gravitational wave detectors often require a large dynamic range interferometric position readout of test masses with 1 pm/√Hz precision over long time scales. Heterodyne interferometer schemes that achieve such precisions are available, but they require complex optical set-ups, limiting their scalability for multiple channels. This article presents the first experimental results on deep frequency modulation interferometry, a new technique that combines sinusoidal laser frequency modulation in unequal arm length interferometers with a non-linear fit algorithm. We have tested the technique in a Michelson and a Mach-Zehnder Interferometer topology, respectively, demonstrated continuous phase tracking of a moving mirror and achieved a performance equivalent to a displacement sensitivity of 250 pm/Hz at 1 mHz between the phase measurements of two photodetectors monitoring the same optical signal. By performing time series fitting of the extracted interference signals, we measured that the linearity of the laser frequency modulation is on the order of 2% for the laser source used.

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.

Vanishing tilt-to-length coupling for a singular case in two-beam laser interferometers with Gaussian beams.

The omnipresent tilt-to-length coupling in two-beam laser interferometers, frequently a nuisance in precision measurements, vanishes for the singular case of two beams with identical parameters and complete detection of both beams without clipping. This effect has been observed numerically and is explained in this manuscript by the cancellation of two very different effects of equal magnitude and opposite sign.

Interspacecraft link simulator for the laser ranging interferometer onboard GRACE Follow-On.

Link acquisition strategies are key aspects for interspacecraft laser interferometers. We present an optical fiber-based setup able to simulate the interspacecraft link for the laser ranging interferometer (LRI) on gravity recovery and climate experiment Follow-On. It allows one to accurately recreate the far-field intensity profile depending on the mispointing between the spacecraft, Doppler shifts, and spacecraft attitude jitter. Furthermore, it can be used in late integration stages of the mission, since no physical contact with the spacecraft is required. The setup can also be easily adapted to other similar missions and different acquisition algorithms.

Advanced phasemeter for deep phase modulation interferometry.

We present the development of an advanced phasemeter for the deep phase modulation interferometry technique. This technique aims for precise length measurements with a high dynamic range using little optical hardware. The advanced phasemeter uses fast ADCs and an FPGA to implement a design of multiple single-bin Fourier transforms running at high sampling rates. Non-linear noise sources in the design were analyzed and suppressed. A null measurement with an optical beatnote signal using λ = 1064nm was conducted. It showed a sensitivity of 0.8µrad/√Hz below 10Hz and 13.3µrad/√Hz above, with a large dynamic range. The shown performance could enable the measuring of optical path lengths with sensitivities down to 0.14pm/√Hz and 2.3pm/√Hz, respectively, over several fringes in an interferometric setup.

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.

Highspeed multiplexed heterodyne interferometry.

Digitally enhanced heterodyne interferometry is a metrology technique that uses pseudo-random noise codes for modulating the phase of the laser light. Multiple interferometric signals from the same beam path can thereby be isolated based on their propagation delay, allowing one to use advantageous optical layouts in comparison to classic laser interferometers. We present here a high speed version of this technique for measuring multiple targets spatially separated by only a few centimetres. This allows measurements of multiplexed signals using free beams, making the technique attractive for several applications requiring compact optical set-ups like for example space-based interferometers. In an experiment using a modulation and sampling rate of 1.25 GHz we are able to demonstrate multiplexing between targets only separated by 36 cm and we achieve a displacement measurement noise floor of <3 pm/√Hz at 10 Hz between them. We identify a limiting excess noise at low frequencies which is unique to this technique and is probably caused by the finite bandwidth in our measurement set-up. Utilising an active clock jitter correction scheme we are also able to reduce this noise in a null measurement configuration by one order of magnitude.

Retroreflector for GRACE follow-on: Vertex vs. point of minimal coupling.

The GRACE Follow-On mission will monitor fluctuations in Earth's geoid using, for the first time, a Laser Ranging Interferometer to measure intersatellite distance changes. We have investigated the coupling between spacecraft rotation and the intersatellite range measurement that is incurred due to manufacturing and assembly tolerances of the Triple Mirror Assembly (TMA), a precision retroreflector to ensure alignment between in- and outgoing laser beams. The three TMA mirror planes intersect in a virtual vertex to which satellite displacements are referenced. TMA manufacturing tolerances degrade this ideal vertex, however, a Point of Minimal Coupling (PMC) between spacecraft rotation and displacement exists. This paper presents the experimental location of the PMC under pitch and yaw rotations for a prototype TMA. Rotations are performed using a hexapod, while displacements are monitored with heterodyne laser interferometry to verify the PMC position. Additionally, the vertex of the three TMA mirror planes is measured using a Coordinate Measuring Machine and compared to the PMC position. In the pitch and yaw axes, the biggest deviation between TMA vertex and PMC was 50 ± 64 µm. Thus, within the measurement uncertainties, no difference between TMA vertex and PMC could be observed. This is a key piece of information for integration of the TMA into the spacecraft: It is sufficient to use the readily-available TMA vertex location to ensure minimal rotation-to-displacement coupling during the mission.

Laser link acquisition demonstration for the GRACE Follow-On mission.

We experimentally demonstrate an inter-satellite laser link acquisition scheme for GRACE Follow-On. In this strategy, dedicated acquisition sensors are not required-instead we use the photodetectors and signal processing hardware already required for science operation. To establish the laser link, a search over five degrees of freedom must be conducted (± 3 mrad in pitch/yaw for each laser beam, and ± 1 GHz for the frequency difference between the two lasers). This search is combined with a FFT-based peak detection algorithm run on each satellite to find the heterodyne beat note resulting when the two beams are interfered. We experimentally demonstrate the two stages of our acquisition strategy: a ± 3 mrad commissioning scan and a ± 300 µrad reacquisition scan. The commissioning scan enables each beam to be pointed at the other satellite to within 142 µrad of its best alignment point with a frequency difference between lasers of less than 20 MHz. Scanning over the 4 alignment degrees of freedom in our commissioning scan takes 214 seconds, and when combined with sweeping the laser frequency difference at a rate of 88 kHz/s, the entire commissioning sequence completes within 6.3 hours. The reacquisition sequence takes 7 seconds to complete, and optimizes the alignment between beams to allow a smooth transition to differential wavefront sensing-based auto-alignment.

Analytical description of interference between two misaligned and mismatched complete Gaussian beams.

A typical application for laser interferometers is a precision measurement of length changes that results in interferometric phase shifts. Such phase changes are typically predicted numerically, due to the complexity of the overlap integral that needs to be solved. In this paper we will derive analytical representations of the interferometric phase and contrast (aka fringe visibility) for two beam interferometers, both homodyne and heterodyne. The fundamental Gaussian beams can be arbitrarily misaligned and mismatched to each other. A limitation of the analytical result is that both beams must be detected completely, which can experimentally be realized by a sufficiently large single-element photodetector.