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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 10 13 , corresponding to an interferometer pathlength stability better than 1 pm / 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. [ABSTRACT FROM AUTHOR]

The GRACE follow-on satellites carry the very first interspacecraft Laser Ranging Interferometer (LRI). After more than four years in orbit, the LRI outperforms the sensitivity of the conventional Microwave Instrument (MWI). However, in the current data processing scheme, the LRI product still needs the MWI data to determine the unknown absolute laser frequency, representing the "ruler" for converting the raw phase measurements into a physical displacement in meters. In this paper, we derive formulas for precisely performing that conversion from the phase measurement into a range, accounting for a varying carrier frequency. Furthermore, the dominant errors due to knowledge uncertainty of the carrier frequency as well as uncorrected time biases are derived. In the second part, we address the dependency of the LRI on the MWI in the currently employed cross-calibration scheme and present three different models for the LRI laser frequency, two of which are largely independent of the MWI. Furthermore, we analyze the contribution of thermal variations on the scale factor estimates and the LRI-MWI residuals. A linear model called Thermal Coupling (TC) is derived, which significantly reduces the differences between LRI and MWI to a level where the MWI observations limit the comparison. [ABSTRACT FROM AUTHOR]

The GRACE Follow-On satellite mission measures distance variations between its two satellites in order to derive monthly gravity field maps, indicating mass variability on Earth on a scale of a few 100 km originating from hydrology, seismology, climatology and other sources. This mission hosts two ranging instruments, a conventional microwave system based on K(a)-band ranging (KBR) and a novel laser ranging instrument (LRI), both relying on interferometric phase readout. In this paper, we show how the phase measurements can be converted into range data using a time-dependent carrier frequency (or wavelength) that takes into account potential intraday variability in the microwave or laser frequency. Moreover, we analyze the KBR-LRI residuals and discuss which error and noise contributors limit the residuals at high and low Fourier frequencies. It turns out that the agreement between KBR and LRI biased range observations can be slightly improved by considering intraday carrier frequency variations in the processing. Although the effect is probably small enough to have little relevance for gravity field determination at the current precision level, this analysis is of relevance for detailed instrument characterization and potentially for future more precise missions. [ABSTRACT FROM AUTHOR]

The precise tracking of distance variations between two satellites in low Earth orbit can provide key data for the understanding of the Earth's system, specifically on seasonal and sub-seasonal water cycles and their impact on water levels. Measured distance variations, caused by local variations in gravitational field, serve as inputs to complex gravity models with which the movement of water on the globe can be identified. Satellite missions GOCE (2009–2013) and GRACE (2002–2017) delivered a significant improvement to our understanding of spatial and temporal gravity variations. Since 2018, GRACE Follow-On has been providing data continuity and features for the first time through the use of a laser interferometer as the technology demonstrator, in addition to a microwave ranging system as the main instrument. The laser interferometer provides an orders-of-magnitude lower measurement noise, and thereby could enable a significant improvement in the measurement of geoids if combined with suitable improvements in auxiliary instrumentation and Earth system modelling. In order to exploit the improved ranging performance, the ESA is investigating the design of a 'Next Generation Gravity Mission', consisting of two pairs of satellites with laser interferometers, improved accelerometers and improved platform performance. In this paper, we present the current design of the laser interferometer developed by us, the development status of the individual instrument units and the options available. [ABSTRACT FROM AUTHOR]

The quadrant photoreceivers (QPRs) in the Laser Interferometer Space Antenna (LISA) mission must feature an input-referred current noise below 2 pAHz $^\frac{-1}{2}$ at 25 MHz to avoid degradation of the signal-to-noise ratio. We present a QPR with a dc-coupled transimpedance amplifier (TIA) based on heterojunction bipolar transistors (HBTs). The QPR features an input-referred current noise density below 1.5 pAHz $^\frac{-1}{2}$ at 25 MHz, a 37 MHz bandwidth, and a 56.2 mW power consumption. Additionally, the dc-coupled, HBT-based TIA allows the QPR to fulfill the size and interface requirements of the LISA mission. [ABSTRACT FROM AUTHOR]

The gravity field maps of the satellite gravimetry missions Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On are derived by means of precise orbit determination. The key observation is the biased inter-satellite range, which is measured primarily by a K-Band Ranging system (KBR) in GRACE and GRACE Follow-On. The GRACE Follow-On satellites are additionally equipped with a Laser Ranging Interferometer (LRI), which provides measurements with lower noise compared to the KBR. The biased range of KBR and LRI needs to be converted for gravity field recovery into an instantaneous range, i.e. the biased Euclidean distance between the satellites' center-of-mass at the same time. One contributor to the difference between measured and instantaneous range arises due to the nonzero travel time of electro-magnetic waves between the spacecraft. We revisit the calculation of the light time correction (LTC) from first principles considering general relativistic effects and state-of-the-art models of Earth's potential field. The novel analytical expressions for the LTC of KBR and LRI can circumvent numerical limitations of the classical approach. The dependency of the LTC on geopotential models and on the parameterization is studied, and afterwards the results are compared against the LTC provided in the official datasets of GRACE and GRACE Follow-On. It is shown that the new approach has a significantly lower noise, well below the instrument noise of current instruments, especially relevant for the LRI, and even if used with kinematic orbit products. This allows calculating the LTC accurate enough even for the next generation of gravimetric missions. [ABSTRACT FROM AUTHOR]

Intersatellite laser interferometers feature quadrant photoreceivers to produce electrical signals from the interfered optical beams. In the particular case of Laser Interferometer Space Antenna, the expected optical ac beat note has an amplitude of the order of nanowatts. This requires photoreceivers with an input current noise density of a few ${\mathrm {pA}} \cdot {\mathrm {Hz}}^{-1/2}$ in each channel up to 25 MHz. In addition, the significant number of photoreceivers in a single spacecraft imposes tight constraints on the power consumption per device. We present the experimental characterization of a quadrant photoreceiver based on discrete heterojunction bipolar transistors and an off-the-shelf 0.5-mm-diameter InGaAs quadrant photodiode, showing an input current noise density of 1.9 ${\mathrm {pA}} \cdot {\mathrm {Hz}}^{-1/2}$ at 25 MHz, a 3-dB bandwidth of 37 MHz, and a total power consumption of 178 mW. [ABSTRACT FROM AUTHOR]

The Gravity Recovery and Climate Experiment (GRACE) mission has yielded data on the Earth's gravity field to monitor temporal changes for more than fifteen years now. The GRACE twin satellites use microwave ranging with micrometer precision to measure distance variations between two satellites caused by the Earth's global gravitational field. GRACE Follow-on (GRACE-FO) will be the first satellite mission to use inter-satellite laser interferometry in space. The laser ranging instrument (LRI) will provide two additional measurements compared to the GRACE mission: interferometric inter-satellite ranging with nanometer precision and inter-satellite pointing information. We have designed a set of simulated GRACE-FO data, which include LRI measurements, apart from all other GRACE instrument data needed for the Earth's gravity field recovery. The simulated data files are publicly available via https://doi.org/10.22027/AMDC2 and can be used to derive gravity field solutions like from GRACE data. This paper describes the scientific basis and technical approaches used to simulate the GRACE-FO instrument data. [ABSTRACT FROM AUTHOR]

We report on the progress in the realization of an electronic / optical simulator for space based, long arm interferometry and its application to the eLISA mission. The goal of this work is to generate realistic optics and electronics signals, especially simulating realistic propagation delays. The first measurements to characterize the simulator are also presented. With the present configuration, noise reduction factors of 5×10 for optical beat notes and 10 for RF beat notes have been achieved using the Time Delay Interferometry algorithm. The principle of the experiment has been validated and further work is ongoing to identify the residual noise sources and optimize the apparatus. [ABSTRACT FROM AUTHOR]

A method based on phase-shifting Fizeau interferometry is presented with which retroreflectors with large incoming-outgoing beam separations can be tested. The method relies on a flat Reference Bar that is used to align two auxiliary mirrors parallel to each other to extend the aperture of the interferometer. The method is applied to measure the beam coalignment of a prototype Triple Mirror Assembly of the GRACE Follow-On Laser Ranging Interferometer, a future satellite-to-satellite tracking device for Earth gravimetry. The Triple Mirror Assembly features a lateral beam offset of incoming and outgoing beam of 600 mm, whereas the acceptance angle for the incoming beam is only about ±2 mrad. With the developed method, the beam coalignment of the prototype Triple Mirror Assembly was measured to be 9 μrad with a repeatability of below 1 µrad. [ABSTRACT FROM AUTHOR]

Intersatellite laser interferometry is a central component of future space-borne gravity instruments like Laser Interferometer Space Antenna (LISA), evolved LISA, NGO and future geodesy missions. The inherently small laser wavelength allows us to measure distance variations with extremely high precision by interfering a reference beam with a measurement beam. The readout of such interferometers is often based on tracking phasemeters, which are able to measure the phase of an incoming beatnote with high precision over a wide range of frequencies. The implementation of such phasemeters is based on all digital phase-locked loops (ADPLL), hosted in FPGAs. Here, we present a precise model of an ADPLL that allows us to design such a readout algorithm and we support our analysis by numerical performance measurements and experiments with analogue signals. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Laser phase noise is the dominant noise source in the on-board measurements of the space-based gravitational wave detector LISA (Laser Interferometer Space Antenna). A well-known data analysis technique, the so-called timedelay interferometry (TDI), provides synthesized data streams free of laser phase noise. At the same time, TDI also removes the next largest noise source: phase fluctuations of the on-board clocks which distort the sampling process. TDI needs precise information about the spacecraft separations, sampling times and differential clock noise between the three spacecrafts. These are measured using auxiliary modulations on the laser light. Hence, there is a need for algorithms that account for clock noise removal schemes combined with TDI while preserving the gravitational wave signal. In this paper, we will present the mathematical formulation of the LISA-like data streams and discuss a compliant algorithm that corrects for both clock and laser noise in the case of a rotating, non-breathing LISA constellation. In contrast to previous papers, we consider the current optical bench design (split interferometry configuration), i.e. the test mass readout is done by the local oscillators only, instead of reflecting the weak inter-spacecraft light off the test mass. Furthermore, the absolute order of laser frequencies is taken into account and it can be shown that the TDI equations remain invariant. This is a crucial issue and was, up to now, completely neglected in the analysis. [ABSTRACT FROM AUTHOR]

We report on the design, implementation, and characterization of fully digital control loops for laser frequency stabilization, differential phase-locking, and performance optimization of the optical metrology system on-board the LISA Pathfinder space mission. The optical metrology system consists of a laser with modulator, four Mach–Zehnder interferometers, a phase-meter and a digital processing unit for data analysis. The digital loop design has the advantage of easy and flexible controller implementation and loop calibration, automated and flexible locking and resetting, and improved performance over analog circuitry. Using the practical ability of our system to modulate the laser frequency allows us to accurately determine the open-loop transfer function and other system properties. Various noise sources and their impact on system performance are investigated in detail. [ABSTRACT FROM PUBLISHER]

Based on large quantities of co-orbital phenomena in the motion of natural bodies and spacecraft, a model of the co-orbital restricted three-body problem is put forward. The fundamental results for the planar co-orbital circular restricted three-body problem are given, which include the selection of variables and equations of motion, a set of approximation formulas, and an approximate semi-analytical solution. They are applied to the motion of the barycenter of the planned gravitational observatory LISA constellation, which agrees very well with the solution of precise numerical integration. [ABSTRACT FROM AUTHOR]

The Laser Interferometer Space Antenna (LISA) is a joint ESA-NASA mission for detecting low-frequency gravitational waves in the frequency range from 0.1 mHz to 1 Hz, by using accurate laser interferometry between three spacecrafts, which will be launched around 2018 and one year later reach their operational orbits around the Sun. In order to operate successfully, it is crucial for the constellation of the three spacecrafts to have extremely high stability. Based on the study of operational orbits for a 2015 launch, we design the operational orbits of beginning epoch on 2019-03-01, and introduce the method of orbit design and optimization. We design the orbits of the transfer from Earth to the operational orbits, including launch phase and separation phase; furthermore, the relationship between energy requirement and flight time of these two orbit phases is investigated. Finally, an example of the whole orbit design is presented. [ABSTRACT FROM AUTHOR]

We require voltage references with a relative output voltage noise of at most 10-6°/√Hz in the frequency range of 0.1 mHz-1 Hz. We have investigated the output voltage noise and the external noise sources of a selection of references. Two external noise sources, i.e., supply voltage and temperature, and their coupling to the output voltage of the references were studied. The effect of supply voltage noise was negligible, whereas temperature noise had to be suppressed by stabilization. We found that samples of ADS87LN- and MAX6126AASA50-type references fulfilled the requirements if the ambient temperature was stabilized to a level of approximately 50 mK/√Hz. [ABSTRACT FROM AUTHOR]

The Laser Interferometer Space Antenna (LISA) mission is a joint ESA-NASA mission for detecting low-frequency gravitational waves in the frequency range from 0.1 mHz to 1 Hz, by using accurate distance measurements with laser interferometry between three spacecraft, which will be launched around 2015 and one year later reach their orbits around the Sun. In order to operate successfully, it is crucial for the constellation of the three spacecraft to have extremely high stability. In this paper, several problems of the orbit optimization of the LISA constellation are discussed by using numerical and analytical methods for satisfying the requirements of accuracy. On the basis of the coorbital restricted problem, analytical expressions of the heliocentric distance and the trailing angle to the Earth of the constellation's barycenter are deduced, with the result that the approximate analytical solution of first order will meet the accuracy requirement of the spacecraft orbit design. It is proved that there is a value of the inclination of the constellation plane that will make the variation of the arm-length a minimum. The principle for selecting the optimum starting elements of orbits at any epoch is proposed. The method and programming principles of finding the optimized orbits are also presented together with examples of the optimization design. [ABSTRACT FROM AUTHOR]

On the basis of many coorbital phenomena in astronomy and spacecraft motion, a dynamics model is proposed in this paper — treating the coorbital restricted problem together with method for obtaining a general approximate solution. The design of the LISA spacecraft orbits is a special 2+3 coorbital restricted problem. The problem is analyzed in two steps. First, the motion of the barycenter of the three spacecraft is analyzed, which is a planar coorbital restricted three-body problem. And an approximate analytical solution of the radius and the argument of the center is obtained consequently. Secondly, the configuration of the three spacecraft with minimum arm-length variation is analyzed. The motion of a single spacecraft is a near-planar coorbital restricted three-body problem, allowing approximate analytical solutions for the orbit radius and the argument of a spacecraft. Thus approximative expressions for the arm-length are given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]