Capacitive sensing of test mass motion with nanometer precision over millimeter-wide sensing gaps for space-borne gravitational reference sensors

We report on the performance of the capacitive gap-sensing system of the Gravitational Reference Sensor on board the LISA Pathfinder spacecraft. From in-flight measurements, the system has demonstrated a performance, down to 1 mHz, that is ranging between 0.7 and $1.8\mathrm{aF}{\mathrm{Hz}}^{-1/2}$. That translates into a sensing noise of the test mass motion within 1.2 and $2.4\mathrm{nm}{\mathrm{Hz}}^{-1/2}$ in displacement and within 83 and $170\mathrm{nrad}{\mathrm{Hz}}^{-1/2}$ in rotation. This matches the performance goals for LISA Pathfinder, and it allows the successful implementation of the gravitational waves observatory LISA. A $1/f$ tail has been observed for frequencies below 1 mHz, the tail has been investigated in detail with dedicated in-flight measurements, and a model is presented in the paper. A projection of such noise to frequencies below 0.1 mHz shows that an improvement of performance at those frequencies is desirable for the next generation of gravitational reference sensors for space-borne gravitational waves observation.

Figure 1

A block diagram of the single TM sensing channel electronics along the $x$ axis. Two pairs of electrodes ($\mathrm{A}+/\mathrm{A}-$, $\mathrm{B}+/\mathrm{B}-$) allow simultaneous measurement of the TM displacement and rotation, which is achieved by measurement of the gaps between the TM and the electrodes at opposing sides of the TM. For small TM displacements, the capacitance is proportional to the respective TM—electrode gap. The actuation circuit (one for each electrode) electrostatically applies forces and torques on the TM using the same electrodes. An ac injection bias (100 kHz) is injected on the TM. Differential currents are measured, amplified, and converted into an ac sensing voltage proportional to the TM motion. Sensing voltage is filtered at the ac injection frequency, and its amplitude is demodulated and finally converted into a digital value.

Figure 2

Average square root of PSD of the GRS capacitive sensing channels measured during the LPF mission. As the data sample rate is 1 Hz and the sensing noise is flat down to 1 mHz, the values presented are representative of the precision of the instrument in the frequency range $[{10}^{-3},0.5]\mathrm{Hz}$. Sensing precision is reported in $\mathrm{aF}{\mathrm{Hz}}^{-1/2}$ as this is the physical observable sensed by the instrument (differential capacitance). In order to convert such values in nm ${\mathrm{Hz}}^{-1/2}$, the conversion factor $1.7\mathrm{nm}/\mathrm{aF}$ applies.

Figure 3

Fractional change of the average sensing noise with respect to the first in-flight measurement performed in February 2016.

Figure 4

TM1 and TM2 time series for the different GRS sensing channels. Data were acquired during LPF commissioning from February 9th, 2016, at $14:30$ UTC until February 11th, 2016, $9:05$ UTC. Test masses were grabbed by the plungers touching them on the z axis surfaces. The sampling frequency is 10 Hz. Despite the different absolute values, the y ranges for the plots are spanning just $10/20\mathrm{aF}$. We also observe the differences in the high-frequency noise between $1x$, $1y$, $1z$ and $2x$, $2y$, $2z$ channels as already reported in Sec. 2.

Figure 5

Square root of the power spectral density of three selected channels, together with the corresponding models provided by Eq. (1) with the fit parameters reported in Table 1. In all figures, we report the three available data sets (two measured in February 2016 and one measured in April 2017). (a) Measurements and models corresponding to channel TM1 $1x$. (b) Measurements and models corresponding to channel TM2 $1x$. (c) Measurements and models corresponding to channel TM1 $1z$. The difference observed between the measurements in February 2016 and in April 2017 is entirely due to the different position along $z$ of the test mass during the two measurements. The average readout during February 2016 was $-3.4\mathrm{fF}$, while during April 2017, it was $-21.3\mathrm{fF}$. The increase of TM displacement corresponds to an enhancement of the effect of the position dependent noise (noise coefficient B).

Figure 6

Projection of the LPF GRS capacitive sensing noise within the extended LISA measurement band. The dark green line represents a possible design target for LISA. The dashed green, yellow, and red lines are the contribution of B noise [(position dependent; see Eq. (1)] for a TM displacement of 1, 10, and $100\mu \mathrm{m}$ respectively. The light blue shaded region represents the contribution from C noise [position independent; see Eq. (1)]. The lower and upper boundaries of the C noise area are set by the minimum and maximum values for C reported in Table 1.