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Space-based gravitational-wave detectors, such as LISA, record interferometric measurements on widely separated satellites. Their clocks are not synced actively. Instead, clock synchronization is performed in on-ground data processing. It relies on measurements of the so-called pseudoranges, which entangle the interspacecraft light travel times with the clock desynchronizations between emitting and receiving spacecraft. For interspacecraft clock synchronization, we need to isolate the differential clock desynchronizations, i.e., disentangle the pseudoranges. This further yields estimates for the interspacecraft light travel times, which are required as delays for the laser frequency noise suppression via time-delay interferometry. Previous studies on pseudorange disentanglement apply various simplifications in the pseudorange modeling and the data simulation. In contrast, this article derives an accurate pseudorange model in the barycentric celestial reference system, complemented by realistic state-of-the-art LISA data simulations. Concerning pseudorange disentanglement, this leads to an a priori under-determined system. We demonstrate how on-ground orbit determinations, as well as onboard transmission and on-ground reception time tags of the telemetry data, can be used to resolve this degeneracy. We introduce an algorithm for pseudorange disentanglement based on a nonstandard Kalman filter specially designed for clock synchronization in systems where pseudorange measurements are conducted in different time frames. This algorithm achieves interspacecraft clock synchronization and light travel time estimation with submeter accuracy, thus fulfilling the requirements of time-delay interferometry.

The Laser Interferometer Space Antenna (LISA) is poised to revolutionize astrophysics and cosmology in the late 2030's by unlocking unprecedented insights into the most energetic and elusive astrophysical phenomena. The mission envisages three spacecraft, each equipped with two lasers, on a triangular constellation with 2.5 million-kilometer arm-lengths. Six inter-spacecraft laser links are established on a laser-transponder configuration, where five of the six lasers are offset-phase-locked to another. The need to determine a suitable set of transponder offset frequencies precisely, given the constraints imposed by the onboard metrology instrument and the orbital dynamics, poses an interesting technical challenge. In this paper we describe an algorithm that solves this problem via quadratic programming. The algorithm can produce concrete frequency plans for a given orbit and transponder configuration, ensuring that all of the critical interferometric signals stay within the desired frequency range throughout the mission lifetime, and enabling LISA to operate in science mode uninterruptedly., Comment: 16 pages, 11 figures

The Laser Interferometer Space Antenna (LISA) is a gravitational wave detector in space. It relies on a post-processing technique named time-delay interferometry (TDI) to suppress the overwhelming laser frequency noise by several orders of magnitude. This algorithm requires intersatellite-ranging monitors to provide information on spacecraft separations. To fulfill this requirement, we will use on-ground observatories, optical sideband-sideband beatnotes, pseudo-random noise ranging (PRNR), and time-delay interferometric ranging (TDIR). This article reports on the experimental end-to-end demonstration of a hexagonal optical testbed used to extract absolute ranges via the optical sidebands, PRNR, and TDIR. These were applied for clock synchronization of optical beatnote signals sampled at independent phasemeters. We set up two possible PRNR processing schemes: Scheme 1 extracts pseudoranges from PRNR via a calibration relying on TDIR; Scheme 2 synchronizes all beatnote signals without TDIR calibration. The schemes rely on newly implemented monitors of local PRNR biases. After the necessary PRNR treatments (unwrapping, ambiguity resolution, bias correction, in-band jitter reduction, and/or calibration), Scheme 1 and 2 achieved ranging accuracies of 2.0 cm to 8.1 cm and 5.8 cm to 41.1 cm, respectively, below the classical 1 m mark with margins., Comment: 17 pages, 13 figures

Time-delay interferometry (TDI) is a data processing technique for space-based gravitational-wave detectors to create laser-noise-free equal-optical-path-length interferometers virtually on the ground. It relies on the interspacecraft signal propagation delays, which are delivered by intersatellite ranging monitors. Also delays due to onboard signal propagation and processing have a nonnegligible impact on the TDI combinations. However, these onboard delays were only partially considered in previous TDI-related research; onboard optical path lengths have been neglected so far. In this paper, we study onboard optical path lengths in TDI. We derive analytical models for their coupling to the second-generation TDI Michelson combinations and verify these models numerically. Furthermore, we derive a compensation scheme for onboard optical path lengths in TDI and validate its performance via numerical simulations.

Wavefront errors are a common artifact in laser light generation and imaging. They can be described as an aberration from the spherical wavefront of an ideal Gaussian beam by combinations of higher-order Hermite- or Laguerre-Gaussian terms. Here, we present an algorithm called Beamfit to estimate the mode composition from a series of CCD images taken over the Rayleigh range of a laser beam. The algorithm uses a user-defined set of Hermite- or Laguerre-Gaussian modes as the basis of its theoretical model. A novel method reduces the number of calculations needed to compute the model's intensity profiles. For a given model containing $N$ modes, the number of Hermite-Gaussian complex amplitudes needed to calculate are reduced from orders of $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and replaced by simple multiplications. Additionally, non-beam parameters are pre-calculated to further reduce the search space dimension and its resulting calculation time. It is planned to release the Beamfit software to the public under an open-source license., Comment: 13 pages, 5 figures

To reach sub-picometer sensitivity in the millihertz range, displacement sensors based on laser interferometry require suppression of laser-frequency noise by several orders of magnitude. Many optical frequency stabilization methods exist with varying levels of complexity, size, and performance. In this paper, we describe the performance of a compact Mach-Zehnder interferometer based on a 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 $10^{-12}$ m$/\sqrt{\mathrm{Hz}}$., Comment: 10 pages, 4 figures

To achieve sub-picometer sensitivities in the millihertz band, laser interferometric inertial sensors rely on some form of reduction of the laser frequency noise, typically by locking the laser to a stable frequency reference, such as the narrow-linewidth resonance of an ultra-stable optical cavity or an atomic or molecular transition. In this paper we report on a compact laser frequency stabilization technique based on an unequal-arm Mach-Zehnder interferometer that is sub-nanometer stable at $10\,\mu$Hz, sub-picometer at $0.5\,$mHz, and reaches a noise floor of $7\,\mathrm{fm}/\!\sqrt{\mathrm{Hz}}$ at 1 Hz. The interferometer is used in conjunction with a DC servo to stabilize the frequency of a laser down to a fractional instability below $4 \times 10^{-13}$ at averaging times from 0.1 to 100 seconds. The technique offers a wide operating range, does not rely on complex lock acquisition procedures, and can be readily integrated as part of the optical bench in future gravity missions., Comment: 9 pages, 7 figures

Interspacecraft ranging is crucial for the suppression of laser frequency noise via time-delay interferometry (TDI). So far, the effects of on-board delays and ambiguities on the LISA ranging observables were neglected in LISA modelling and data processing investigations. In reality, on-board delays cause offsets and timestamping delays in the LISA measurements, and pseudo-random noise (PRN) ranging is ambiguous, as it only determines the range up to an integer multiple of the PRN code length. In this article, we identify the four LISA ranging observables: PRN ranging, the sideband beatnotes at the interspacecraft interferometer, TDI ranging, and ground-based observations. We derive their observation equations in the presence of on-board delays, noise, and ambiguities. We then propose a three-stage ranging sensor fusion to combine these observables in order to gain accurate and precise ranging estimates. We propose to calibrate the on-board delays on ground and to compensate the associated offsets and timestamping delays in an initial data treatment (stage 1). We identify the ranging-related routines, which need to run continuously during operation (stage 2), and implement them numerically. Essentially, this involves the reduction of ranging noise, for which we develop a Kalman filter combining the PRN ranging and the sideband beatnotes. We further implement crosschecks for the PRN ranging ambiguities and offsets (stage 3). We show that both ground-based observations and TDI ranging can be used to resolve the PRN ranging ambiguities. Moreover, we apply TDI ranging to estimate the PRN ranging offsets.

The Gravity Recovery And Climate Experiment - Follow On (GRACE-FO) satellite mission (2018-now) hosts the novel Laser Ranging Interferometer (LRI), a technology demonstrator for proving the feasibility of laser interferometry for inter-satellite ranging measurements. The GRACE-FO mission extends the valuable climate data record of changing mass distribution in the system Earth, which was started by the original GRACE mission (2002-2017). The mass distribution can be deduced from observing changes in the distance of two low-earth orbiters employing interferometry of electromagnetic waves in the K-Band for the conventional K-Band Ranging (KBR) and in near-infrared for the novel LRI. This paper identifies possible radiation-induced Single Event Upset (SEU) events in the LRI phase measurement. We simulate the phase data processing within the Laser Ranging Processor (LRP) and use a template-based fitting approach to determine the parameters of the SEU and subtract the events from the ranging data. Over four years of LRI data, 29 of such events were identified and characterized.

LISA is an upcoming ESA mission that will detect gravitational waves in space by interferometrically measuring the separation between free-falling test masses at picometer precision. To reach the desired performance, LISA will employ the noise reduction technique time-delay interferometry (TDI), in which multiple raw interferometric readouts are time shifted and combined into the final scientific observables. Evaluating the performance in terms of these TDI variables requires careful tracking of how different noise sources propagate through TDI, as noise correlations might affect the performance in unexpected ways. One example of such potentially correlated noise is the relative intensity noise (RIN) of the six lasers aboard the three LISA satellites, which will couple into the interferometric phase measurements. In this article, we calculate the expected RIN levels based on the current mission architecture and the envisaged mitigation strategies. We find that strict requirements on the technical design reduce the effect from approximately 8.7 pm/rtHz per inter-spacecraft interferometer to that of a much lower sub-1 pm/rtHz noise, with typical characteristics of an uncorrelated readout noise after TDI. Our investigations underline the importance of sufficient balanced detection of the interferometric measurements., Comment: 15 pages, 10 figures, 2 tables

The GRACE Follow-On satellites carry the very first inter-spacecraft 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 that significantly reduces the differences between LRI and MWI to a level where the MWI observations limit the comparison., Comment: 26 pages, 15 figures

This paper is the second in a set of two investigating tilt-to-length (TTL) coupling. TTL describes the cross-coupling of angular or translational jitter into an interferometric phase signal and is an important noise source in precision interferometers, including space gravitational wave detectors like LISA. We discussed in 10.1088/2040-8986/ac675e the TTL coupling effects originating from optical path length changes, i.e. geometric TTL coupling. Within this work, we focus on the wavefront and detector geometry dependent TTL coupling, called non-geometric TTL coupling, in the case of two interfering fundamental Gaussian beams. We characterise the coupling originating from the properties of the interfering beams, i.e. their absolute and relative angle at the detector, their relative offset and the individual beam parameters. Furthermore, we discuss the dependency of the TTL coupling on the geometry of the detecting photodiode. Wherever possible, we provide analytical expressions for the expected TTL coupling effects. We investigate the non-geometric coupling effects originating from beam walk due to the angular or translational jitter of a mirror or a receiving system. These effects are directly compared with the corresponding detected optical path length changes in 10.1088/2040-8986/ac675e. Both together provide the total interferometric readout. We discuss in which cases the geometric and non-geometric TTL effects cancel one-another. Additionally, we list linear TTL contributions that can be used to counteract other TTL effects. Altogether, our results provide key knowledge to minimise the total TTL coupling noise in experiments by design or realignment.

The GRACE Follow-On satellite mission measures distance variations between the two satellites in order to derive monthly gravity field maps, indicating mass variability on Earth on a few 100 km scale due to hydrology, seismology, climatology and others. 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 potential intraday variability in the microwave or laser frequency into account. 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, the analysis is of relevance for detailed instrument characterization and potentially for future more precise 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., Comment: 18 pages, 8 figures

The Laser Interferometer Space Antenna (LISA) aims to observe gravitational waves in the mHz regime over its 10-year mission time. LISA will operate laser interferometers between three spacecrafts. Each spacecraft will utilize independent clocks which determine the sampling times of onboard phasemeters to extract the interferometric phases and, ultimately, gravitational wave signals. To suppress limiting laser frequency noise, signals sampled by each phasemeter need to be combined in postprocessing to synthesize virtual equal-arm interferometers. The synthesis in turn requires a synchronization of the independent clocks. This article reports on the experimental verification of a clock synchronization scheme down to LISA performance levels using a hexagonal optical bench. The development of the scheme includes data processing that is expected to be applicable to the real LISA data with minor modifications. Additionally, some noise coupling mechanisms are discussed., Comment: 9 pages, 4 figures

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., Comment: 18 pages, 9 figures

The gravity field maps of the satellite gravimetry missions GRACE (Gravity Recovery and Climate Experiment) 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 non-zero 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., Comment: This is a preprint of an open-access article published in Journal of Geodesy. The final authenticated version is available online at: https://doi.org/10.1007/s00190-021-01498-5

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., Comment: 11 pages, 8 figures

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 degree of freedom two-way laser link between remote spacecraft succeeded on the first attempt. Active beam pointing based on differential wavefront 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 $1\,{\rm nm}/\sqrt{\rm Hz}$ at Fourier frequencies above 100 mHz.

Since the 2017 Nobel Prize in Physics was awarded for the observation of gravitational waves, it is fair to say that the epoch of gravitational wave astronomy (GWs) has begun. However, a number of interesting sources of GWs can only be observed from space. To demonstrate the feasibility of the Laser Interferometer Space Antenna (LISA), a future gravitational wave observatory in space, the LISA Pathfinder satellite was launched on December, 3rd 2015. Measurements of the spurious forces accelerating an otherwise free-falling test mass, and detailed investigations of the individual subsystems needed to achieve the free-fall, have been conducted throughout the mission. This overview article starts with the purpose and aim of the mission, explains satellite hardware and mission operations and ends with a summary of selected important results and an outlook towards LISA. From the LISA Pathfinder experience, we can conclude that the proposed LISA mission is feasible., Comment: Preprint of an article submitted for consideration in Lepton-Photon 2017 conference proceedings \copyright 2019 copyright World Scientific Publishing Company

Deep frequency modulation interferometry combines optical minimalism with multi-fringe readout. Precision however is key for applications such as optical gradiometers for satellite geodesy or as dimensional sensor for ground-based gravity experiments. We present a single-component interferometer smaller than a cubic inch. Two of these are compared to each other to demonstrate tilt and displacement measurements with a precision of less than $20\,\mathrm{nrad}/\sqrt{\mathrm{Hz}}$ and $1\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$ at frequencies below $1\,\mathrm{Hz}$., Comment: 6 pages, 4 figures, accepted in Physical Review Applied

The science operations of the LISA Pathfinder mission has demonstrated the feasibility of sub-femto-g free-fall of macroscopic test masses necessary to build a LISA-like gravitational wave observatory in space. While the main focus of interest, i.e. the optical axis or the $x$-axis, has been extensively studied, it is also of interest to evaluate the stability of the spacecraft with respect to all the other degrees of freedom. The current paper is dedicated to such a study, with a focus set on an exhaustive and quantitative evaluation of the imperfections and dynamical effects that impact the stability with respect to its local geodesic. A model of the complete closed-loop system provides a comprehensive understanding of each part of the in-loop coordinates spectra. As will be presented, this model gives very good agreements with LISA Pathfinder flight data. It allows one to identify the physical noise source at the origin and the physical phenomena underlying the couplings. From this, the performances of the stability of the spacecraft, with respect to its geodesic, are extracted as a function of frequency. Close to $1 mHz$, the stability of the spacecraft on the $X_{SC}$, $Y_{SC}$ and $Z_{SC}$ degrees of freedom is shown to be of the order of $5.0\ 10^{-15} m\ s^{-2}/\sqrt{Hz}$ for X and $4.0 \ 10^{-14} m\ s^{-2}/\sqrt{Hz}$ for Y and Z. For the angular degrees of freedom, the values are of the order $3\ 10^{-12} rad\ s^{-2}/\sqrt{Hz}$ for $\Theta_{SC}$ and $3\ 10^{-13} rad\ s^{-2}/\sqrt{Hz}$ for $H_{SC}$ and $\Phi_{SC}$., Comment: 16 pages, 10 figures

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., Comment: 5 pages, 4 figures

LISA is a proposed space-based laser interferometer detecting gravitational waves by measuring distances between free-floating test masses housed in three satellites in a triangular constellation with laser links in-between. Each satellite contains two optical benches that are articulated by moving optical subassemblies for compensating the breathing angle in the constellation. The phase reference distribution system, also known as backlink, forms an optical bi-directional path between the intra-satellite benches. In this work we discuss phase reference implementations with a target non-reciprocity of at most $2\pi\,\mathrm{\mu rad/\sqrt{Hz}}$, equivalent to $1\,\mathrm{pm/\sqrt{Hz}}$ for a wavelength of $1064\,\mathrm{nm}$ in the frequency band from $0.1\,\mathrm{mHz}$ to $1\,\mathrm{Hz}$. One phase reference uses a steered free beam connection, the other one a fiber together with additional laser frequencies. The noise characteristics of these implementations will be compared in a single interferometric set-up with a previously successfully tested direct fiber connection. We show the design of this interferometer created by optical simulations including ghost beam analysis, component alignment and noise estimation. First experimental results of a free beam laser link between two optical set-ups that are co-rotating by $\pm 1^\circ$ are presented. This experiment demonstrates sufficient thermal stability during rotation of less than $10^{-4}\,\mathrm{K/\sqrt{Hz}}$ at $1\,\mathrm{mHz}$ and operation of the free beam steering mirror control over more than 1 week., Comment: 20 pages, 8 figures, submitted to Classical Quantum Gravity

The future space-based gravitational wave detector Laser Interferometer Space Antenna (LISA) requires bidirectional exchange of light between its two optical benches on board of each of its three satellites. The current baseline foresees a polarization-maintaining single-mode fiber for this backlink connection. Phase changes which are common in both directions do not enter the science measurement, but differential ("non-reciprocal") phase fluctuations directly do and must thus be guaranteed to be small enough. We have built a setup consisting of a Zerodur$^{\rm TM}$ baseplate with fused silica components attached to it using hydroxide-catalysis bonding and demonstrated the reciprocity of a polarization-maintaining single-mode fiber at the 1 pm/$\sqrt{\textrm{Hz}}$ level as is required for LISA. We used balanced detection to reduce the influence of parasitic optical beams on the reciprocity measurement and a fiber length stabilization to avoid nonlinear effects in our phase measurement system (phase meter). For LISA, a different phase meter is planned to be used that does not show this nonlinearity. We corrected the influence of beam angle changes and temperature changes on the reciprocity measurement in post-processing.

Following the selection of The Gravitational Universe by ESA, and the successful flight of LISA Pathfinder, the LISA Consortium now proposes a 4 year mission in response to ESA's call for missions for L3. The observatory will be based on three arms with six active laser links, between three identical spacecraft in a triangular formation separated by 2.5 million km. LISA is an all-sky monitor and will offer a wide view of a dynamic cosmos using Gravitational Waves as new and unique messengers to unveil The Gravitational Universe. It provides the closest ever view of the infant Universe at TeV energy scales, has known sources in the form of verification binaries in the Milky Way, and can probe the entire Universe, from its smallest scales near the horizons of black holes, all the way to cosmological scales. The LISA mission will scan the entire sky as it follows behind the Earth in its orbit, obtaining both polarisations of the Gravitational Waves simultaneously, and will measure source parameters with astrophysically relevant sensitivity in a band from below $10^{-4}\,$Hz to above $10^{-1}\,$Hz., Comment: Submitted to ESA on January 13th in response to the call for missions for the L3 slot in the Cosmic Vision Programme

Low frequency high precision laser interferometry is subject to excess laser-frequency-noise coupling via arm-length differences which is commonly mitigated by locking the frequency to a stable reference system. This approach is crucial to achieve picometer level sensitivities in the 0.1 mHz to 1 Hz regime, where laser frequency noise is usually high and couples into the measurement phase via arm-length mismatches in the interferometers. Here we describe the results achieved by frequency stabilising an external cavity diode laser to a quasi-monolithic unequal arm-length Mach-Zehnder interferometer read out at mid-fringe via balanced detection. We find this stabilization scheme to be an elegant solution combining a minimal number of optical components, no additional laser modulations and relatively low frequency noise levels. The Mach-Zehnder interferometer has been designed and constructed to minimize the influence of thermal couplings and to reduce undesired stray light using the optical simulation tool ifocad. We achieve frequency-noise levels below 100 Hz/$\sqrt{\textrm{Hz}}$ at 1 Hz and are able to demonstrate the LISA frequency prestabilization requirement of 300 Hz/$\sqrt{\textrm{Hz}}$ down to frequencies of 100 mHz by beating the stabilized laser with an iodine-locked reference., Comment: 6 pages, 6 figures, published in Phyiscal Review Applied

The Laser Interferometer Space Antenna (LISA) is a future space-based interferometric gravitational-wave detector consisting of three spacecraft in a triangular configuration. The interferometric measurements of path length changes between satellites will be performed on optical benches in the satellites. Angular misalignments of the interfering beams couple into the length measurement and represent a significant noise source. Imaging systems will be used to reduce this tilt-to-length coupling. We designed and constructed an optical test bed to experimentally investigate tilt-to-length coupling. It consists of two separate structures, a minimal optical bench and a telescope simulator. The minimal optical bench comprises the science interferometer where the local laser is interfered with light from a remote spacecraft. In our experiment, a simulated version of this received beam is generated on the telescope simulator. The telescope simulator provides a tilting beam, a reference interferometer and an additional static beam as a phase reference. The tilting beam can either be a flat-top beam or a Gaussian beam. We avoid tilt-to-length coupling in the reference interferometer by using a small photo diode placed at an image of the beam rotation point. We show that the test bed is operational with an initial measurement of tilt-to-length coupling without imaging systems. Furthermore, we show the design of two different imaging systems whose performance will be investigated in future experiments.

Precision phase readout of optical beat note signals is one of the core techniques required for intersatellite laser interferometry. Future space based gravitational wave detectors like eLISA require such a readout over a wide range of MHz frequencies, due to orbit induced Doppler shifts, with a precision in the order of $\mu \textrm{rad}/\sqrt{\textrm{Hz}}$ at frequencies between $0.1\,\textrm{mHz}$ and $1\,\textrm{Hz}$. In this paper, we present phase readout systems, so-called phasemeters, that are able to achieve such precisions and we discuss various means that have been employed to reduce noise in the analogue circuit domain and during digitisation. We also discuss the influence of some non-linear noise sources in the analogue domain of such phasemeters. And finally, we present the performance that was achieved during testing of the elegant breadboard model of the LISA phasemeter, that was developed in the scope of an ESA technology development activity., Comment: submitted to Review of Scientific Instruments on April 30th 2015

Space-borne gravitational wave detectors, such as (e)LISA, are designed to operate in the low-frequency band (mHz to Hz), where there is a variety of gravitational wave sources of great scientific value. To achieve the extraordinary sensitivity of these detector, the precise synchronization of the clocks on the separate spacecraft and the accurate determination of the interspacecraft distances are important ingredients. In our previous paper (Phys. Rev. D 90, 064016 [2014]), we have described a hybrid-extend Kalman filter with a full state vector to do this job. In this paper, we explore several different state vectors and their corresponding (phenomenological) dynamic models, to reduce the redundancy in the full state vector, to accelerate the algorithm, and to make the algorithm easily extendable to more complicated scenarios., Comment: 12 pages

The most promising concept for low frequency gravitational wave observatories are laser interferometric detectors in space. It is usually assumed that the noise floor for such a detector is dominated by optical shot noise in the signal readout. For this to be true, a careful balance of mission parameters is crucial to keep all other parasitic disturbances below shot noise. We developed a web application that uses over 30 input parameters and considers many important technical noise sources and noise suppression techniques. It optimizes free parameters automatically and generates a detailed report on all individual noise contributions. Thus you can easily explore the entire parameter space and design a realistic gravitational wave observatory. In this document we describe the different parameters, present all underlying calculations, and compare the final observatory's sensitivity with astrophysical sources of gravitational waves. We use as an example parameters currently assumed to be likely applied to a space mission to be launched in 2034 by the European Space Agency. The web application itself is publicly available on the Internet at http://spacegravity.org/designer., Comment: 36 pages, 20 figures

The present work reports on a feasibility study commissioned by the Chinese Academy of Sciences of China to explore various possible mission options to detect gravitational waves in space alternative to that of the eLISA/LISA mission concept. Based on the relative merits assigned to science and technological viability, a few representative mission options descoped from the ALIA mission are considered. A semi-analytic Monte Carlo simulation is carried out to understand the cosmic black hole merger histories starting from intermediate mass black holes at high redshift as well as the possible scientific merits of the mission options considered in probing the light seed black holes and their coevolution with galaxies in early Universe. The study indicates that, by choosing the armlength of the interferometer to be three million kilometers and shifting the sensitivity floor to around one-hundredth Hz, together with a very moderate improvement on the position noise budget, there are certain mission options capable of exploring light seed, intermediate mass black hole binaries at high redshift that are not readily accessible to eLISA/LISA, and yet the technological requirements seem to within reach in the next few decades for China.

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. This paper was published in Applied Optics and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: [http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-54-5-1010]. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.

During the On-Station Thermal Test campaign of the LISA Pathfinder the data and diagnostics subsystem was tested in nearly space conditions for the first time after integration in the satellite. The results showed the compliance of the temperature measurement system, obtaining temperature noise around $10^{-4}\,{\rm K}\, {\rm Hz}^{-1/2}$ in the frequency band of $1-30\;{\rm mHz}$. In addition, controlled injection of heat signals to the suspension struts anchoring the LISA Technology Package (LTP) Core Assembly to the satellite structure allowed to experimentally estimate for the first time the phase noise contribution through thermo-elastic distortion of the LTP interferometer, the satellite's main instrument. Such contribution was found to be at $10^{-12}\,{\rm m}\, {\rm Hz}^{-1/2}$, a factor of 30 below the measured noise at the lower end of the measurement bandwidth ($1\,{\rm mHz}$)., Comment: Submitted to CQG

In this paper, we describe a hybrid-extended Kalman filter algorithm to synchronize the clocks and to precisely determine the inter-spacecraft distances for space-based gravitational wave detectors, such as (e)LISA. According to the simulation, the algorithm has significantly improved the ranging accuracy and synchronized the clocks, making the phase-meter raw measurements qualified for time- delay interferometry algorithms., Comment: 14 pages, Phys. Rev. D 90, 064016 (2014)

A typical application for laser interferometers is a precision measurement of length changes that result in interferometric phase shifts. Such phase changes are typically predicted numerically, due to the com- plexity 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., Comment: 8 pages, 2 figures

Inter satellite laser interferometry is a central component of future space-borne gravity instruments like LISA, eLISA, NGO and future geodesy missions. The inherently small laser wavelength allows 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, 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, hosted in FPGAs. Here we present a precise model of an all digital phase locked loop that allows to design such a readout algorithm and we support our analysis by numerical performance measurements and experiments with analog signals., Comment: 17 pages, 6 figures, accepted for publication in CQG

An elegant breadboard model of the LISA phasemeter is currently under development by a Danish-German consortium. The breadboard is build in the frame of an ESA technology development activity to demonstrate the feasibility and readiness of the LISA metrology baseline architecture. This article gives an overview about the breadboard design and its components, including the distribution of key functionalities., Comment: 5 pages, 3 figures, published in ASP Conference Series, Vol. 467, 9th LISA Symposium (2012), pp 271-276

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 analogue 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., Comment: 10 pages, 7 figures; draft only, for edited version see journal link

LISA Pathfinder is a technology demonstration mission for the Laser Interferometer Space Antenna (LISA). The main experiment on-board LISA Pathfinder is the so-called LISA Technology Package (LTP) which has the aim to measure the differential acceleration between two free-falling test masses with an accuracy of 3x10^(-14) ms^(-2)/sqrt[Hz] between 1 mHz and 30 mHz. This measurement is performed interferometrically by the Optical Metrology System (OMS) on-board LISA Pathfinder. In this paper we present the development of an experimental end-to-end testbed of the entire OMS. It includes the interferometer and its sub-units, the interferometer back-end which is a phasemeter and the processing of the phasemeter output data. Furthermore, 3-axes piezo actuated mirrors are used instead of the free-falling test masses for the characterisation of the dynamic behaviour of the system and some parts of the Drag-free and Attitude Control System (DFACS) which controls the test masses and the satellite. The end-to-end testbed includes all parts of the LTP that can reasonably be tested on earth without free-falling test masses. At its present status it consists mainly of breadboard components. Some of those have already been replaced by Engineering Models of the LTP experiment. In the next steps, further Engineering Models and Flight Models will also be inserted in this testbed and tested against well characterised breadboard components. The presented testbed is an important reference for the unit tests and can also be used for validation of the on-board experiment during the mission.

We present a real-time differential phasefront detector sensitive to better than 3 mrad rms, which corresponds to a precision of about 500 pm. This detector performs a spatially resolving measurement of the phasefront of a heterodyne interferometer, with heterodyne frequencies up to approximately 10 kHz. This instrument was developed as part of the research for the LISA Technology Package (LTP) interferometer, and will assist in the manufacture of its flight model. Due to the advantages this instrument offers, it also has general applications in optical metrology.

We present recent sensitivity measurements of the LISA Technology Package interferometer with articulated mirrors as test masses, actuated by piezo-electric transducers. The required longitudinal displacement resolution of 9 pm/sqrt[Hz] above 3 mHz has been demonstrated with an angular noise that corresponds to the expected in on-orbit operation. The excess noise contribution of this test mass jitter onto the sensitive displacement readout was completely subtracted by fitting the angular interferometric data streams to the longitudinal displacement measurement. Thus, this cross-coupling constitutes no limitation to the required performance of the LISA Technology Package interferometry., Comment: Applied Physics B - Lasers and Optics (2008)

We have developed a method to equip homodyne interferometers with the capability to operate with constant high sensitivity over many fringes for continuous real-time tracking. The method can be considered as an extension of the "J_1...J_4" methods, and its enhancement to deliver very sensitive angular measurements through Differential Wavefront Sensing is straightforward. Beam generation requires a sinusoidal phase modulation of several radians in one interferometer arm. On a stable optical bench, we have demonstrated a long-term sensitivity over thousands of seconds of 0.1 mrad/sqrt[Hz] that correspond to 20 pm/sqrt[Hz] in length, and 10 nrad/sqrt[Hz] in angle at millihertz frequencies.

At present a 10\,m prototype interferometer facility is being set up at the AEI Hannover. One unique feature of the prototype will be the suspension platform interferometer (SPI). The purpose of the SPI is to monitor and stabilise the relative motion between three seismically isolated optical tables. The in-vacuum tables are suspended in an L-shaped configuration with an arm length of 11.65\,m. The design goal of the SPI is to stabilise longitudinal differential displacements to a level of 100\,pm/$\sqrt{\mathrm{Hz}}$ between 10\,mHz and 100\,Hz and relative angular noise of 10\,nrad/$\sqrt{\mathrm{Hz}}$ in the same frequency band. This paper covers the design aspects of the SPI, e.g. cross-coupling between the different degrees of freedom and fibre pointing noise are investigated. A simulation is presented which shows that with the chosen optical design of the SPI all degrees of table motion can be sensed in a fully decoupled way. Furthermore, a proof of principle test of the SPI sensing scheme is shown., Comment: 12 pages, 6 figures

The AEI 10 m prototype interferometer facility is currently being constructed at the Albert Einstein Institute in Hannover, Germany. It aims to perform experiments for future gravitational wave detectors using advanced techniques. Seismically isolated benches are planned to be interferometrically interconnected and stabilized, forming a low-noise testbed inside a 100 m^3 ultra-high vacuum system. A well-stabilized high power laser will perform differential position readout of 100 g test masses in a 10 m suspended arm-cavity enhanced Michelson interferometer at the crossover of measurement (shot) noise and backaction (quantum radiation pressure) noise, the so-called Standard Quantum Limit (SQL). Such a sensitivity enables experiments in the highly topical field of macroscopic quantum mechanics. In this article we introduce the experimental facility and describe the methods employed, technical details of subsystems will be covered in future papers.

The scientific objectives of the Lisa Technology Package (LTP) experiment, on board of the LISA Pathfinder mission, demand for an accurate calibration and validation of the data analysis tools in advance of the mission launch. The levels of confidence required on the mission outcomes can be reached only with an intense activity on synthetically generated data. A flexible procedure allowing the generation of cross-correlated stationary noise time series was set-up. Multi-channel time series with the desired cross correlation behavior can be generated once a model for a multichannel cross-spectral matrix is provided. The core of the procedure is the synthesis of a noise coloring multichannel filter through a frequency-by-frequency eigendecomposition of the model cross-spectral matrix and a Z-domain fit. The common problem of initial transients in noise time series is solved with a proper initialization of the filter recursive equations. The noise generator performances were tested in a two dimensional case study of the LTP dynamics along the two principal channels of the sensing interferometer., Comment: Accepted for publication in Physical Review D (http://prd.aps.org/)

We present the first experimental confirmation of the so-called "self-phaselocked delay interferometry". This laser frequency stabilization technique consists basically in comparing the prompt laser signal with a delayed version of itself that has been reflected in another LISA satellite 5 million km away. In our table-top experiment, the phase of a voltage-controlled oscillator is stabilized by means of a control loop based on this technique. In agreement with the theory, the measured unity gain frequency is not limited by the inverse of the used delay (1.6 microseconds). In the time domain, the system also behaves as predicted, including the appearance of a quasi-periodic "ringing" just after the lock acquisition, which decays exponentially. Its initial amplitude is smaller when the loop gain is slowly ramped up instead of suddenly switched on., Comment: 9 pages, 9 figures