Mathematical derivation and verification of the amplitude of LISA's interferometric signals on an ultra-stable interferometer testbed

Abstract: The Laser Interferometer Space Antenna (LISA) mission aims to detect gravitational waves by interferometrically measuring the change of separation between free-falling test masses (TMs). LISA's interferometers must deliver pm/rtHz sensitivity while accommodating beam tilts up to 1 mrad at the photodiodes, which degrade the interferometric amplitude and increase the induced readout noise coupling. This paper uses an analytical framework developed by the authors in a previous work, based on minimal and justified approximations, that relates beam tilt to the resulting heterodyne signal amplitude in a generic two-beam interferometer with circular-area photodiodes (PDs). A set of interferometric topologies is analyzed, all of high relevance for LISA. We derive the exact amplitude response for an infinite detector and a closed-form approximation for finite detectors, and we validate both against numerical simulations and experimental measurements on an ultra-stable LISA-representative testbed. We then use this model to quantify the phase-noise amplification arising from reduced signal-to-noise ratio (SNR) under tilt, showing that curvature mismatches between the interfering beams substantially enhance this effect. Finally, we introduce a compact function that captures the angular dependence of correlated and uncorrelated phase noises in quadrant photodiode (QPD)-based readouts. Here, a new noise feature, caused by wavefront curvature mismatch, is predicted and measured for the first time. These results indicate that controlling wavefront curvature mismatch in the test mass interferometer (TMI) is essential to limit excess phase noise. The models and results derived in this paper, although originating in the context of LISA, are general and can be applied to any interferometric topology undergoing tilts with pivot on the detector plane.