Ranging measurement principle

The LRI is an optical interferometer, which is operated with a near-IR wavelength of 1064nm and where the optical layout is arranged in a race-track configuration as shown in fig. 1. The direction of laser beams is reversed on each S/C by an approx. 60 cm long triple-mirror assembly (TMA), utilizing three mutual perpendicular mirrors in a corner-cube arrangement. The vertex of the TMA is co-located with the accelerometer and S/C center-of-mass (CoM) in order to minimize to a large extent the tilt-to-length coupling. The lateral separation of 60 cm between in-coming and out-going beam was required due to space constraints on the S/C, i.e. the central axis between the CoM is occupied by the KBR instrument and by cold-gas tanks. This spatial separation of beams has the advantage that one can relinquish polarization multiplexing as usually present in single-axis interferometers, which reduces the number of optical components and the complexity of the interferometer.

Figure 1.

Sketch of the LRI measurement principle. Image credit: Vitali Müller / AEI Hannover

The LRI is a heterodyne interferometer designed for phase-tracking of beatnote frequencies between 4.16 MHz, which are produced by the interference of optical fields with the MHz difference in optical frequency. Both S/C accommodate identical optical bench assemblies (OBA), which are connected by optical fibers to the NPRO laser with 25 mW output power. The laser light is guided over a two-axis fast steering mirror (FSM) onto the recombination beamsplitter (RBS, fig. 1). A small fraction of the light is transmitted at the RBS towards some imaging optics and the hot-redundant quadrant phtotodiodes (QPD, fig. 1), while most of the laser light leaves the OBA towards the TMA. The TMA retro-reflects the beam with approx. 2.5 mm 1/e² radius and 20 mW optical power in the direction of the distant S/C. The laser light expands to a diameter of over 50 meter and gains a Doppler shift (w.r.t. the receiver craft) while traveling the 220 km distance. This Doppler frequency shift contains the ranging information, which is read out by the phase measurements.

Only a small fraction of the expanded laser beam is cut out by the 8 mm diameter receiving aperture on the OBA, yielding an optical power of less than 2 nW on the OBA to be interfered with the local laser beam. In principle, both OBAs obtain Doppler shifted laser light and measure the phase of the received light w.r.t. to the local laser. However, the LRI is operated in a transponder scheme, where one satellite is designated as slave with an active frequency-offset phase-locked loop. This control loop actuates with high gain and high bandwidth the NPRO laser phase so that a constant phase ramp with a slope of 10 MHz is measured by the phasemeter on the slave S/C. This means that the phase of the local laser is locked to the incoming phase and the phase information is transported on the laser light to the master S/C. The master side measures twice, or the round-trip, Doppler shift in addition to the frequency-offset imprinted by the slave, while the slave satellite measures a trivial constant phase ramp due to the frequency-offset.

The master side exhibits an active laser frequency stabilization by means of an ULE optical cavity⁵ in order to minimize phase fluctuations, which couple via the round-trip propagation into the final phase measurement on the master side. Since the frequency stabilization and the optical cavity are present on both S/C, the master and slave role is interchangeable between the S/C yielding additional redundancy.

Laser beam steering

The LRI requires accurate pointing of the transmit laser beam, which has a 1/e² half-angle divergence of approx. 140 μrad, in order to transmit a sufficient amount of light power to the other S/C. Since the S/C pointing uncertainty and attitude variations are much larger than the LRI pointing requirements, active beam steering is employed by means of control loops and fast steering mirrors (FSM). Therefore, quadrant photodiodes (QPD) measure the tip and tilt of the local beam with respect to the incoming beam, using the differential wavefront sensing (DWS) technique^6,7. This principle is depicted in fig. 2. This DWS signal is fed back to the control loop, commanding the steering mirror in such a way that the DWS signal is driven towards a programmed set point. In an ideal scenario (with perfect component placement and alignment), this set point is zero and both phasefronts are parallel in front of the QPDs as long as the local S/C misalignment is within the range of the optics and steering mirror. The FSM control loop compensates for misalignments of the local S/C and ensures that the heterodyne efficency and corresponding signal-to-noise ratio of the phase readout is maximized. With a closed control loop and zero DWS set point, the incoming phasefront is parallel to the outgoing phasefronts on the OBA, which means that after retro-reflection at the TMA, the emitted light is propagating in the direction of the received light and towards the distant S/C.

Figure 2.

Working principle of DWS and FSM. Image credit: Vitali Müller / AEI Hannover

Pointing angle measurement

Since the FSM is permanently commanded to compensate for the misalignment of the received light field and the local laser beam, the FSM positions provide very accurate information about the alignment of the spacecraft with respect to the line of sight (LoS), and hence about the spacecraft LoS pointing angles. This method directly measures the pointing angles, i.e. the deviation of the local frame from the line-of-sight frame. As opposed to that, other attitude sensors (e.g. Star Cameras) measure the satellites’ attitudes compared to inertial space, which can be combined with orbit data in order to obtain pointing angles. However, the LRI does not measure the spacecraft roll angle (which is the angle around the satellites’ line-of-sight), only the vertical and horizontal angles which correspond to the so-called pitch and yaw angles.

Initial acquisition strategy

The LRI has a field-of-view of plus-minus a few milliradians, which is designed to be larger than the pointing variations of the satellites and initial uncertainties, e.g. due to alignment tolerances during instrument integration. It is however required to perform a search within this field-of-view, in order to close the interferometric link. This pointing offset, which is unknown initially, has to be determined in-flight. Furthermore, the laser frequency difference between master and slave is required to be in the measurable range of 4.16 MHz. To this end, an initial acquisition scan is performed on both spacecraft simultaneously (“initial line of sight calibration procedure”⁴). Afterwards, the pointing offset is determined from the steering mirror positions recorded at the instances when the phasemeter detects flashes, i.e. when the beatnote amplitude exceeds a threshold. These split-second flashes are seen only if all of the 4 angles (pitch and yaw for both spacecraft) and laser frequency offset are very close to their optimal values at the same time. Thus, a complicated five-dimensional scan pattern is needed, which is depicted in fig. 3 for the angular domain. The total duration of the scan is approximately nine hours. During this initial acquisition scan there is no transition into science mode. The data is merely recorded and analyzed afterwards. When the optimal set point is determined, a parameter file is uploaded to the spacecraft.

Figure 3.

Schematic of the LRI initial acquisition scan. Image credit: Alexander Koch / AEI Hannover

Once the initial pointing offsets are uploaded, the LRI is commanded into the re-acquisition mode. In this mode, the instrument performs only a short scan with smaller range, starting with the values that were previously determined by the initial line of sight calibration procedure. These values are already very close to the optimum and the instrument automatically attempts to transition from the re-acquisition mode into the science mode, once a re-acquisition flash is seen.

Result

This initial acquisition scan was performed on 13 June 2018. The data recorded during the scan was sent to ground and analyzed the following night. The analysis showed that enough flashes were recorded in order to determine the initial offsets for the re-acquisition scan, which include the major misalignment of the LRI OBA w.r.t. the satellite frame (SF) and the minor TMA co-alignment error and estimation errors in the pointing towards the distant spacecraft. The parameter files containing this information were uploaded shortly afterwards. The same day, the instrument successfully established the interferometer laser link. The following ground-station pass of the satellites confirmed that the LRI was in science mode and taking continuous measurements.

TMA vertex point offset

One of the expected error sources in the LRI ranging measurements is the TTL coupling, i.e. the ranging error which is caused by spacecraft attitude variations. A main reason for this is an offset between the satellite CoM and the so-called vertex point (VP) of the TMA, which is the virtual intersection point of the three mirrors⁸. One of the advantages of the LRI concept is that the interferometer round-trip length is exactly two times the distance from VP to VP. This implies that the measured range is independent of rotations around those points. However, rotations around a different point change the position of the VP which then causes undesired variations in the ranging measurement.

TTL calibration

A mathematical derivation of the known TTL coupling effects indicate that a model which is linear in the pointing angles can be used to explain the main part of the TTL error. That is, we assume that each of the 6 pointing angles (roll, pitch, yaw for each of the two spacecraft) linearly adds to the ranging measurement, weighted by a coupling factor. Since the attitudes of the satellites are known quite accurately, this error can in theory be computed and subtracted from the measured range, if the 6 coupling factors can be estimated well enough.

One possible way to estimate the TTL coupling factors is to use a dedicated satellite rotation maneuver, i.e. commanding the satellite AOCS to produce periodic pointing angle oscillations. Similar approaches are used for the calibration of the KBR instrument and for the accelerometer calibration. The KBR calibration maneuvers cannot be used for this purpose, since it uses angle amplitudes which are too large for the LRI to cope with. The goal of the accelerometer calibration or “center-of-mass calibration” (CMC) maneuver is to determine the offset between the CoM of the satellite and the CoM of the accelerometer proof mass⁹. A CMC maneuver consists of small periodic pointing angle excitations for a duration of 180 seconds. The oscillation period is 12 seconds, which corresponds to 83.3 mHz. These calibration maneuvers, which are planned to be performed regularly, could be useful for the LRI as well. If needed, the maneuver parameters can also be adjusted to values that are optimal for the purpose of LRI TTL calibration.

Another possible strategy is to use the naturally occuring pointing variations of the satellites. The advantage of this approach is that it requires merely the regular data stream, which is typically available for long durations without large gaps. On the other hand, the data analysis in this case is more complex. The ranging signal in lower frequencies is dominated by the gravity signal, which has to be carefully subtracted beforehand. In higher frequencies, the pointing variations are very small, so that the TTL error signal is expected to be buried by other noise sources and especially hard to detect.

In either case, the first step is to prepare a data set of 6 pointing angles and a quantity that contains the TTL error, which is either directly the measured range or derived from it (e.g. post-fit residuals). The data is synchronized and bandpass filtered. TTL coupling factors can be estimated by using linear least-squares, i.e. by minimizing the expression

where x_i, i=1,…,6, are the pointing angles, c_i are the coupling factors, and y is the range or range-derived quantity. This procedure can be improved by using advanced formalisms, where additional information (e.g. knowledge about instrument noise) can be taken into account. The analysis is ongoing and detailed results will be published when they are available.

Proof-of-concept

After the LRI had started taking range data on 14 June 2018, it has proven that it can take measurements for many orbits without any interruptions. A good opportunity for a first test of the ranging data came up when the GFO satellites were flying over the Himalayas. After highpass filtering, the ranging data clearly shows a signal that is caused by change in the inter-spacecraft separation due to the presence of the large mass of the Himalayas. This proof-of-concept visualization shown in fig. 4 is hence called “Himalaya Plot”. Fig. 4 was originally published on the AEI Hannover web page¹⁰, a similar plot was published at the same time on the JPL web page¹¹. Likewise, a Himalaya plot for the GFO KBR instrument¹² can be found on the NASA web page¹².

Figure 4.

“Himalaya Plot”. Top: LRI ranging data measured as the satellites flew over the Himalayas. Middle and bottom: Topography of the ground track (source: SRTM). Image credit: Gerhard Heinzel / AEI Hannover