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.

Figures (28)

• Figure 1: Optical layout of the testbed. Left: The ts provides the ob (right) with laser beams (rxgb and rxft, both in green) to simulate lisa's tmi and isi. The first beam is a TEM00 Gaussian beam provided by a fios with a width of $\sim$2 mm. The second is a flat-top beam generated by clipping a 9 mm radius beam Gaussian beam on an apodized aperture; this generates a beam flat in both intensity and phase with a diameter of $\sim$4 mm Chwalla_2016. The ts sits on top of the ob and is held by three Zerodur$^\circledR$ feet, which are indicated by the dashed lines. An optical link is established via a vertical interface (ts/ob I/F). A pair of actuated steering mirrors, named "act1" and "act2" on the left, on the ts tilt the Rx beams around the Rx-clip (this is an aperture, which is physically located on the lob) to simulate either the spacecraft jitter in the isi of the tm jitter on the tmi. An additional fios generated TEM00 Gaussian beam, the lo (blue), is used as an alignment aid for the ts with respect to the ob and as a phase reference. The ts hosts four optical copies of the Rx-clip (located on the ob). In these optical copies, one refsepd, two refqpd and a phase camera are placed. Right: The ob is a simplified version of the ob in lisa. It features a local fios generated TEM00 Gaussian beam, the Tx beam, and only one interferometer, where all three the Tx, Rx, and the lo beams interfere. At the interferometer's two output ports, the two sciqpd are placed. These output ports feature imaging systems to image the point of rotation of the Rx beam onto the center of the qpd and reduce ttl coupling. In this Figure we show the two types of imaging systems that were developed for tdobs, one featuring 2 lenses, and one featuring 4 lenses. During the work reported in this article, two 4-lense imaging systems were used. The ts's nominal position on top of the ob is indicated by the dashed lines. Such position is defined with the aid of a cqp; this is a pair of large silicon qpd positioned on the lob. When the lo beam impinges on the center of both qpd of the cqp, the Tx beam ts also propagates to the center of the pd on the ts. Figure and caption from alvise_2024.
• Figure 2: Right: Picture of a large commercial qpd. Left: definition of the used segment-nomenclature throughout the paper. The area between the segments is called slit. Depending on the specific qpd model, the slit can be either insensitive, partially sensitive, or even more sensitive than the active area itself. Note that the qpd in this experiment use a diameter of 1 mm.
• Figure 3: Measurement with three different light intensities, SPICE simulation, and fit of the electronic noise from the tia used for the pd in tdobs. The fit function is $f(x) = \sqrt{p_0^2 + (p_1x)^2}$, reproducing \ref{['eq::tia-eq-in-current-noise']}.
• Figure 4: Description of the lever arm effect between two beams, which dominates the geometrical ttl. Two beams, the reference beam and the measurement beam, interfere. The resulting intensity is measured by a pr located on the $z=0$ plane. The measurement beam can rotate about the pivot point $(0, 0, -d)^T$, representing either the Rx-clip or the tm's surface in lisa. Such a tilt causes the measurement beam to propagate over a longer path; we denote the additional path-length as $\Delta s$. The extra path length is interferometrically measured at the pr; if $\Delta s$ is not constant in time, it generates a noise term, and such noise is called ttl. Imaging systems reduce the lever arm effect between two beams and beam walk by imaging the point of rotation of the beam into the center of the pr. This means the optical path length is unchanged by the tilt, as light always takes the shortest path. As a consequence of this design, beam tilts in lisa have the center of the qpd as a rotation pivot and reduced undesired beam walk on the qpd. Figure from alvise_2024.
• Figure 5: Principle of dws. A reference beam (red) and measurement beam (blue) with a MHz-scale frequency difference produce a heterodyne beat tone on each qpd segment. When aligned (left), all beat notes share the same phase. When the measurement beam is tilted (right), the wave fronts reach the top segments earlier than the bottom ones, producing a phase difference proportional to the tilt angle. Figure from alvise_2024.