Mitigating effects of nonlinearities in homodyne quadrature interferometers

TL;DR

This work shows that Homodyne Quadrature Interferometers (HoQIs) suffer nonlinear readout when sensing displacement, especially across interferometer fringes. By employing a linear-motion fused-silica resonator as a test mass, the authors quantify nonlinearities via ellipse-like distortions in the HoQI quadrature plane and demonstrate powerful real-time correction using a ringdown-derived ellipse, implemented in the data acquisition system. They further develop post-processing strategies including hysteresis compensation and an hour-scale, two-parameter ellipse fit to suppress residual nonlinearity, aided by coherent subtraction of witness sensors. The results indicate significant noise reduction across motion regimes, and the compatibility of whitening filters with HoQIs, suggesting HoQIs can meet the stringent isolation needs of current and next-generation gravitational-wave detectors. Overall, the work extends the applicability and robustness of HoQIs for displacement sensing in seismic isolation and other precision measurement contexts.

Abstract

Homodyne Quadrature interferometers (HoQI) are an interferometric displacement sensing scheme proven to have excellent noise performance, making them a strong candidate for sensing and control schemes in gravitational wave detector seismic isolation. Like many interferometric schemes, HoQIs are prone to nonlinear effects when measuring displacements. These nonlinearities, if left unsuppressed, would substantially limit the use cases of HoQIs. This paper first shows a means of measuring and quantifying nonlinearities using a working HoQI and a mechanical resonator. We then demonstrate a method for real-time correction of these nonlinearities and several approaches for accurately calibrating the correction technique. By correcting in real time, we remove one of the biggest obstacles to including HoQIs in upgrades to future gravitational wave detectors. Finally, we discuss how to post correct data from HoQIs, suppressing even further the nonlinearity-induced errors, broadening the appeal of such sensors to other applications where measurement data can be reconstructed after the fact. We demonstrate all of this on a working HoQI system and show the measured suppression of nonlinear effects from each of these methods. Our work makes HoQIs a more broadly applicable tool for displacement sensing.

Figures (13)

• Figure 1: Beam paths in a HoQI. The beam is split by a polarising beam splitter (PBS) and the parts are sent to a reference mirror and measurement mirror. The beams containing phase information of the two different arms are shown with separate arrows despite being overlapped in reality. The returning light is split by a non-polarising beam splitter (NPBS) before being measured by 3 different PDs. The delay of one of the beams due to the quarter wave plate (QWP) is shown by a shift of the arrows.
• Figure 2: Mode shape of the mechanical resonator used to induce arm length changes in the HoQI. The axis indicates the local displacement as a fraction of maximum motion. The resonator was set up such that the z-axis was aligned to the incident beam from the HoQI, such that motion in this direction was measured. The resonator is a monolithic fused silica piece, realised using direct bonding, to achieve the high Q factor. Its test mass weighs 3 g. A gold coating has been applied to the centre of the piece to reflect incident laser light.
• Figure 3: Trace drawn by the data on $Q_1 Q_2$-plane during an hour of near one FSR motion (blue scatter), ellipse fit performed with parameters from ringdown measurement ("ringdown par.", black dotted line), and ellipse fit performed with scipy.curvefit routine ("fit scipy", orange dashed line) on only one minute of these data in post-processing.
• Figure 4: ASD of HoQI self-noise (after coherently subtracting motion registered by the witness sensors) for one hour of near one FSR motion data. Ellipse correction with parameters from ringdown measurement (blue line) is compared against a fit with scipy.curve_fit on roughly the first minute of data, with correction applied then to the full hour of data (orange line) and no ellipse correction (gray line).
• Figure 5: Measurement of the transfer function of the whitening filters for each PD channel multiplied with the previously calculated digital de-whitening filter.