Tilt-to-length coupling in LISA Pathfinder: A data analysis

We present a study of the tilt-to-length coupling noise during the LISA Pathfinder mission and how it depended on the system’s alignment. Tilt-to-length coupling noise is the unwanted coupling of angular and lateral spacecraft or test mass motion into the primary interferometric displacement readout. It was one of the major noise sources in the LISA Pathfinder mission and is likewise expected to be a primary noise source in LISA. We demonstrate here that a recently derived and published analytical model describes the dependency of the LISA Pathfinder tilt-to-length coupling noise on the alignment of the two freely falling test masses. This was verified with the data taken before and after the realignments performed in March (engineering days) and June 2016, and during a two-day experiment in February 2017 (long cross-talk experiment). The latter was performed with the explicit goal of testing the tilt-to-length coupling noise dependency on the test mass alignment. Using the analytical model, we show that all realignments performed during the mission were only partially successful and explain the reasons why. In addition to the analytical model, we computed another physical tilt-to-length coupling model via a minimizing routine making use of the long cross-talk experiment data. A similar approach could prove useful for the LISA mission.

Figure 1

Simplified sketch of the LPF setup. The yellow squares represent the two TMs which are surrounded by the electrodes from the electrode housing. The light-blue square in the center is the optical bench. The optical setup is reduced to the components relevant for the TM distance changes and control loop measurements. The same holds for the shown beam paths. The red beam represents the measurement beam, i.e., the beam affected by the motion of the $\mathrm{S}/\mathrm{C}$ or the TMs. The blue beam is not affected by this motion and holds as a reference. Further, the principle of the nominal control mechanisms is shown. The drag-free loop utilized the measurement of the $\mathrm{S}/\mathrm{C}$ relative to TM1 to define $\mathrm{S}/\mathrm{C}$ accelerations via the thrusters. The suspension loop is fed in by the readout of the $\mathrm{S}/\mathrm{C}$-TM2 alignment used to control the TM2 alignment via electrostatic forces. Figure reproduced from [12].

Figure 2

Histogram of the coupling coefficients fitted for 120 2-hour timespans during the noise run which started on 13th Feb. 2017. The coefficients are not constant but spread about a mean. The dashed red lines are offset from the mean by the root-mean-square. With twice this deviation, we would cover all coefficient estimates but the outliers.

Figure 3

Changes of the $\mathrm{\Delta }g$ observable due to realignments of the TMs. Dark green: $\mathrm{\Delta }g$ before the realignments in June (three day noise run ending on the 18th June 2016). Light green: $\mathrm{\Delta }g$ in between the realignments in June (noise run from the 20th to 24th June 2016). Yellow: $\mathrm{\Delta }g$ after the realignments in June (one hour data with low noises at the 26th June 2016). These noise runs are defined in further detail in Sec. 7b1. The gray curves show the $\mathrm{\Delta }g$ observable before and after the realignment during the ED for comparison. Black dashed curve: LPF performance model for the time of the ED.

Figure 4

Performance of the fit model for the twelve subexperiments of the LXE. The red curve shows the ASD of the observable $\mathrm{\Delta }g$. The yellow curve shows the respective density of the linear model fitted to the data. The blue curve shows the residual remaining after the subtraction of the fit from $\mathrm{\Delta }g$. The noise peaks at the injection frequencies of the $y$-injections (10, 12 and 17 mHz) are well subtracted, and the noise at the injection frequencies of the $z$-injections (5 and 10 mHz) is significantly reduced. The noise above 20 mHz is suppressed except for the multiples of the injections frequencies not covered by the linear fit model.

Figure 5

Performance of the analytical model for the twelve subexperiments of the LXE. The red curve shows the ASD of the observable $\mathrm{\Delta }g$. The yellow curve shows the ASD of the linear analytical model [Eq. (2)]. The dashed green line pictures the ASD of the stiffness terms and the cross-coupling of residual $\mathrm{S}/\mathrm{C}$ motion along the optical axis fitted to the difference between $\mathrm{\Delta }g$ and the analytical model. The blue curve shows the residual remaining after the subtraction of the analytical model and the fitted terms from the measurement. The subtraction analytical model significantly reduces the noise at the full frequency range dominated by the TTL coupling, with the exception of the multiples of the injections frequencies, which are not covered by this linear model.

Figure 6

Comparison of the fitted (blue) and analytically derived (red) coupling coefficients for the 12 subexperiments of the LXE. They are representatively plotted in the mid of the timespan of the experiments each. The fit was conducted in the frequency range from 2 to 70 mHz. The error bars of the fit are plotted. The error bars of the analytical coefficients correspond to their uncertainties given in Eq. (3). Moreover, we show the respective additional double root-mean-square of the fit deviations within a noise run to cover unmodeled instabilities (see Sec. 4b) in light red. Times are given in UTC. Most coefficients match when considering the error bars and expected variations of the fit results. Only in the case of the $C_{\eta }$ coefficient we find few mismatches.

Figure 7

Same as Fig. 6 but with the fit using a frequency range restricted to the bump only (i.e., 20 to 70 mHz). The results of this fit is shown in blue. The distribution of the analytical coefficients is again plotted in red. Their offset was adjusted to the new mean of the fitted coefficients in experiments in which they are not expected to (significantly) change. Compared to Fig. 6, the fitted $C_{\eta }$ coefficients do not split into two levels. Therefore, the fitted and analytical coefficients match better.

Figure 8

Comparison of the fitted (injections only, green) and analytically derived (red) coupling coefficients for the 12 subexperiments of the LXE. The time domain fit for the injection frequencies is further described in Sec. 6b1. Compared to Fig. 7, the resulting $C_{\eta }$ coefficients distribute about a significantly lower level, while the $C_{\phi }$ coefficients did not significantly change. The distribution of the analytical coefficients is plotted in red. Their offset was adjusted to the mean of the fitted coefficients (green) in experiments in which they are not expected to (significantly) change. The blue coefficients correspond to the original fit result shown in Fig. 6.

Figure 9

The ASDs of the noise contributors during the subexperiments of the LXE. The observable $\mathrm{\Delta }g$ is given in gray. The colored curves show the measurements in the respective DoF scaled by their fitted coupling coefficients. In all experiments, the acceleration noise in the $xz$-plane is dominant at TTL noise bump. The most dominant is the $\ddot{\overline{z}}$ noise contribution, which almost only explains the bulge between 55 and 70 mHz.

Figure 10

Residuals after subtracting the second-order models for the twelve subexperiments of the LXE. The continuous gray curve shows the ASD of the observable $\mathrm{\Delta }g$. The colored curves show the ASD of the residuals after subtracting the second-order models from $\mathrm{\Delta }g$. Red: second-order analytical model plus the fitted stiffness and $o_1$ terms. Dark blue: fit model with one additional angular second-order coefficient in each plane. Blue: fit model with two additional angular second-order coefficients in each plane. For comparison, we show the ASD of the residual after subtracting the linear fit model (dashed gray). All three models subtract the peaks at the multiples of the injections frequencies that were not subtracted by the linear models. However, the extensions change very little at the injection frequencies. The fit model with two second-order coefficients in each plane subtracts the noise at the differential injection frequencies better (see 7 mHz peaks).

Figure 11

Comparison of the coefficients derived via the minimizing routine with the fitted (fit range 20 to 70 mHz) and analytically derived coupling coefficients. The minimizer and fit coefficients match well. The coefficient changes due to TM realignments show the same characteristic behavior as analytically predicted with the deviations discussed in Sec. 6b. The time is given in UTC.

Figure 12

Noise contributors in the pre- and postengineering phase of the ED. The noise contributions due to accelerations in $\phi$, $y$, $z$ and $o_1$ were reduced due to the realignments. The largest noise contributors after the realignments were the accelerations in $\eta$ and $y$.

Figure 13

Analytical modeling of the expected noise after realigning the TMs in accordance to the analytically derived angles (right hand side of Table 7). The results validate the analytical model [Eq. (2)]. Red curves: measured $\mathrm{\Delta }g$ before the realignment (preengineering data). Yellow curve: residual after the subtraction of the fit model [Eq. (1)] from the measured $\mathrm{\Delta }g$ before the realignment. Dashed black line: LPF performance model. Left figure: The analytical model [Eq. (2)] for the analytically derived realignment angles (Table 7) added to the preengineering noise (dashed purple line), and the sum of the fitted stiffness and $o_1$-noise contributions (dashed gray line). Right figure: The analytical model added to the preengineering noise minus the fitted stiffness and $o_1$-noise contributors (purple line). The overlap of the purple and the yellow curve indicates that a full TTL noise mitigation via realignment could have been achieved via the analytically derived TM angles. The noise added above 200 mHz originates from the measurement noise of the GRS scaled by the TTL coefficients. It would not be present if performing the realignment.

Figure 14

Analytical modeling of the expected noise after realigning the TMs to the angles defined during the mission (left hand side of Table 7). The results further validate the analytical model [Eq. (2)]. Red curves: measured $\mathrm{\Delta }g$ before realignment. Green curve: measured $\mathrm{\Delta }g$ after the realignments. Dashed black line: LPF performance model. Left figure: The analytical model (2) evaluated for the realignments applied during the engineering days added to the preengineering noise (dashed purple line), and the sum of the fitted stiffness and $o_1$-noise contributions (dashed gray line). Right figure: Analytical model added to the preengineering noise minus the fitted stiffness and $o_1$-noise contributors (purple line). The overlap of the purple and the green curve indicates that the observed noise decrease due to the commanded realignment could have been predicted with the analytical model. In comparison to Fig. 13 we see that not the realignment angles defined during the mission but the new analytically derived ones would have yielded a suppression of TTL noise.

Figure 15

Noise contributors during the noise run prior to the TM realignments in June 2016. The noise run started on 15th June and lasted almost three days until 18th June 2016. The coupling of the acceleration in $y$ and $\eta$ was the main cause of the TTL bump between 20 and 200 mHz.

Figure 16

Noise contributors during the noise run in between the two TM realignments in June 2016. It started a day after the realignment performed on 19th June 2016 and lasted for four days. In comparison to Fig. 15, we see a decrease of the $\eta$- and $z$-acceleration coupling but an increase of the $\phi$- and $y$-acceleration noise. The overall TTL noise increased between 20 and 200 mHz.

Figure 17

Noise contributors after the realignment on 25th June 2016. Considered are data from 26th June 2016 from 7∶00 to 8:00 AM UTC. Due to this realignment, the $\phi$- and $y$-acceleration noise decreased, but the $\eta$- and $z$-acceleration coupling increased again. In sum, the TTL bump is smaller than before this realignment.