Mathematical derivation and verification of LISA’s angular readout signal on an ultrastable interferometer testbed

The Laser Interferometer Space Antenna (LISA) aims to detect gravitational waves by interferometrically measuring the separation between free falling test masses (TMs) with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>pm</a:mi> <a:mo>/</a:mo> <a:msqrt> <a:mi>Hz</a:mi> </a:msqrt> </a:math> sensitivity down to mHz frequencies. Achieving this sensitivity requires the suppression of spurious path length variations induced by unavoidable tilts of the TMs and spacecraft, a phenomenon known as tilt-to-length coupling. LISA employs differential wave front sensing (DWS) to measure angular misalignments between the interfering beams’ wave fronts. To first-order approximation, the DWS signal is linearly dependent on the tilt angle for small angular displacements (on the order of microradians). The linear coupling coefficient is well-documented for the idealized case of a photodiode (PD) much larger than the beam spot radius. However, LISA’s setup violates this assumption, and the anticipated tilt range exceeds the linear regime. Accurately characterizing the DWS signal thus requires a more comprehensive approach. This paper presents a novel analytical framework for calculating the first two Maclaurin coefficients of the DWS signal for two fundamental-mode Gaussian beams with a small wave front curvature mismatch, incident on PDs of arbitrary size. The validity of this approach is verified by comparison with numerical simulations and experimental calibrations conducted on an ultrastable testbed representative of LISA. Moreover, we utilize this framework to determine the phase sensitivity required to meet LISA’s stringent measurement requirements. The results of this paper will facilitate the calculation of the DWS gain of an interferometric topology and give a reference target to the experiments aiming to demonstrate LISA’s performance requirement.