HAM Relay Triple Suspension (HRTS)

• HRTS is a triple pendulum suspension with 6-DOF control, using three stages (total 390 mm length) to isolate test-mass motion in LIGO/ET applications.
• It serves as a testbed to compare conventional BOSEM shadow sensors with COBRI interferometric sensors by isolating their noise contributions in a suspended Fabry–Pérot cavity.
• Detailed mechanical modeling and state-space analysis are used to evaluate noise budgets and predict performance improvements, especially around the 5 Hz regime.

The HAM Relay Triple Suspension (HRTS) is a triple pendulum suspension intended for LIGO and ET Pathfinder applications and, in the experimental concept reported in "Reducing suspension control noise with interferometric sensors -- an experimental concept" (Weickhardt et al., 24 Jul 2025), it serves as the central mechanical plant for quantifying how local displacement sensing limits suspension control noise. The proposed experiment uses two HRTSs facing one another, each suspending a mirror so that the pair forms a short suspended Fabry–Pérot cavity. By comparing cavity length stability when local damping is performed with conventional BOSEM shadow sensors versus COBRI interferometric sensors, the setup isolates the contribution of local sensing noise to control-induced displacement noise. In this role, the HRTS is not merely a support structure for a test optic; its residual motion, actuation paths, and transfer functions determine the observable cavity length noise and therefore the measurable benefit of interferometric local sensing (Weickhardt et al., 24 Jul 2025).

1. Mechanical configuration and experimental role

In the reported experiment, two HRTSs are arranged facing one another on a passively isolated optical table inside the actively isolated vacuum chamber VATIGrav. Each HRTS suspends one mirror, and the two mirrors form the suspended optical cavity. The paper states that the HRTS has three pendulum stages, control of all six degrees of freedom, and a total pendulum length of 390 mm (Weickhardt et al., 24 Jul 2025).

The cavity formed by the two suspended mirrors is described with preliminary parameters:

Quantity	Symbol	Value
Suspended cavity length	$l_\textrm{sus}$	$\qty{40}{\cm}$
Suspended cavity finesse	$\mathcal{F}_\textrm{sus}$	3140
Laser wavelength	$\lambda$	$\qty{1064}{\nm}$
Laser power	$P_0$	$\qty{1}{\W}$

The suspended resonator is read out with Pound–Drever–Hall (PDH) sensing. The laser is first stabilized to a separate rigid reference cavity, then passed through a mode cleaner, and then injected into the suspended cavity. The reference optics are given as a reference cavity length $l_\textrm{ref} = \qty{20}{\cm}$ with reference cavity finesse $\mathcal{F}_\textrm{ref} = 3140$, and a mode-cleaner round-trip length $L_\textrm{MC} = \qty{53}{\cm}$ with mode-cleaner finesse $\qty{40}{\cm}$0 (Weickhardt et al., 24 Jul 2025).

This geometry defines the HRTS as the mechanical subsystem whose test-mass motion is converted directly into cavity length fluctuations. The cavity is primarily considered between the test masses of the two HRTSs. The authors also note that a cavity between penultimate masses was considered, but that configuration would increase both sensor-noise coupling and seismic coupling. A plausible implication is that the test-mass cavity was retained to preserve a more observatory-relevant balance between sensing sensitivity and isolation.

2. Suspension architecture, coordinates, and dynamic significance

The HRTS is treated as a 6-DOF suspension, with the paper focusing mainly on cavity-axis translation $\qty{40}{\cm}$1 and pitch because these dominate coupling into cavity length in the present analysis (Weickhardt et al., 24 Jul 2025). The suspension notation distinguishes the support point and successive suspended stages:

• suspension-point motion: $\qty{40}{\cm}$2
• top-mass motion: $\qty{40}{\cm}$3
• lower stages ending in test mass: $\qty{40}{\cm}$4

This notation is central because the experiment probes how disturbances injected or sensed at the upper stages propagate to the optic that forms the cavity. The paper uses transfer-function notation in which $\qty{40}{\cm}$5 denotes a transfer function; for example,

$\qty{40}{\cm}$6

denotes displacement transfer from suspension-point motion $\qty{40}{\cm}$7 to test-mass motion $\qty{40}{\cm}$8. For force-to-displacement response, the susceptibility is defined as

$\qty{40}{\cm}$9

where $\mathcal{F}_\textrm{sus}$0 is a force on stage $\mathcal{F}_\textrm{sus}$1, and $\mathcal{F}_\textrm{sus}$2 is the displacement response of that stage.

As a triple pendulum, the HRTS provides isolation above resonance roughly as

$\mathcal{F}_\textrm{sus}$3

with $\mathcal{F}_\textrm{sus}$4, yielding the usual $\mathcal{F}_\textrm{sus}$5 pendulum isolation trend at high frequency. At the mechanical resonances, however, motion is amplified. The paper does not tabulate all HRTS resonance frequencies numerically, but it states that the relevant excess motion and damping action occur in the few-Hz region, especially around 3–10 Hz, with the clearest expected sensing advantage around 5 Hz (Weickhardt et al., 24 Jul 2025).

This operating regime explains why the HRTS is suitable for evaluating local sensing technologies. The local damping loop suppresses resonant RMS motion so that the suspended optics remain within a range where interferometric cavity control remains linear and robust. This suggests that the HRTS is being used not only as a representative observatory suspension but also as a calibrated transducer from local sensing noise to cavity-observable displacement noise.

3. Six-degree-of-freedom sensing and control

The planned final experiment implements sensing and control in all six DOFs using both BOSEMs and COBRIs. A crucial implementation detail is that BOSEMs remain in place for actuation, COBRIs are mounted opposite each BOSEM, and the experiment can switch sensing from BOSEMs to COBRIs while keeping the same BOSEM coil actuators (Weickhardt et al., 24 Jul 2025). This isolates the effect of the sensor without changing the actuator chain: the HRTS mechanical plant and coil-magnet actuation remain fixed while only the local readout is changed.

The control architecture is a standard local damping loop acting at the top mass, comprising:

• a local displacement sensor measuring top-stage motion
• a digital damping filter
• force applied through BOSEM coils

The authors state explicitly that the HRTS uses 6 DOF sensing and control at the top mass with both BOSEMs and COBRIs (Weickhardt et al., 24 Jul 2025). The damping filters used in the analysis are taken from Jeffrey Kissel’s LIGO SWG filters, described as filters “designed to work for all DOFs on most suspensions.” The paper stresses that these filters are not yet optimized for interferometric sensors, so the reported performance is conservative with respect to what COBRI-based sensing might ultimately enable.

The sensor comparison is the core of the experiment. The conventional HRTS local sensor is the BOSEM shadow sensor with integrated coil-magnet actuator. The paper quotes the BOSEM peak sensitivity as roughly

$\mathcal{F}_\textrm{sus}$6

depending on variant and frequency. The alternative local sensor is the Compact Balanced Readout Interferometer (COBRI), based on deep frequency modulation, and intended to provide multi-fringe sensing, absolute ranging, and theoretical peak sensitivity around

$\mathcal{F}_\textrm{sus}$7

The COBRI noise model used in the paper is an estimate based on the Cramér–Rao lower bound, with low-frequency rise proportional to $\mathcal{F}_\textrm{sus}$8, attributed to residual laser frequency noise coupling, and high-frequency noise dominated by electronic readout noise. The paper explicitly notes that the COBRI noise is dominated at low frequency by laser frequency noise and at high frequency by electronic readout noise (Weickhardt et al., 24 Jul 2025).

These properties are operationally important for the HRTS. Because the sensor must function around the suspension resonances in the few-Hz band, COBRI’s lower self-noise is expected to reduce control-induced displacement noise directly. The paper also identifies angular control as a particularly strong use case: in LIGO, BOSEMs are not used to control rotations, because their self-noise is too large relative to rotational seismic motion; in the HRTS analysis, BOSEMs would inject additional rotational motion above 3 Hz, exceeding the existing measured rotational motion of the optical table, whereas COBRI self-noise lies well below the measured rotational motion. Accordingly, BOSEMs are considered infeasible for damping rotations in this setup, not only pitch but any rotation (Weickhardt et al., 24 Jul 2025).

4. Mathematical modeling and noise-coupling structure

The paper uses a mechanical model of the HRTS derived from the Advanced LIGO suspension Mathematica model, exported into state-space form and analyzed in Python using spicypy, Python Control Systems Library, and GWpy. This model underlies all simulated HRTS transfer functions, noise projections, and control-loop analyses (Weickhardt et al., 24 Jul 2025).

The open-loop transfer function for longitudinal damping is given as

$\mathcal{F}_\textrm{sus}$9

with $\lambda$0 the damping filter and $\lambda$1 the top-mass force-to-top-mass displacement susceptibility. For local sensing noise injected into the controlled HRTS, the manuscript prints a truncated equation, but the intended formula is stated as

$\lambda$2

where $\lambda$3 is the sensor-noise ASD and $\lambda$4 is the susceptibility from top-mass force to test-mass displacement. This relationship expresses the central mechanism of control-induced displacement noise: sensor noise is filtered by the controller, injected through the actuator, and shaped by the mechanical coupling from the actuated stage to the optic.

The main longitudinal coupling chain is described as:

1. local sensor measures top-mass $\lambda$5-motion
2. digital filter $\lambda$6 produces damping signal
3. BOSEM coils actuate on top mass
4. HRTS mechanical susceptibility $\lambda$7 transmits force to test-mass $\lambda$8-motion
5. test masses of the two HRTSs form cavity-length noise

The paper also emphasizes actuator noise. From the undamped top-mass RMS motion under current seismic conditions, the authors infer a required DAC range for control of

$\lambda$9

including a safety factor of two. The DAC noise assumption is

$\qty{1064}{\nm}$0

for a $\qty{1064}{\nm}$1 V range, corresponding to coil current noise

$\qty{1064}{\nm}$2

The actuator-induced displacement noise is then

$\qty{1064}{\nm}$3

with force constant

$\qty{1064}{\nm}$4

After reducing the effective DAC range based on expected motion, the paper says the noise after the coil driver is attenuated to $\qty{1064}{\nm}$5 of the initial estimate. Under the improved seismic target, the RMS drops by roughly two orders of magnitude, but the practical DAC-range reduction is limited by ADC noise in BOSEM readout to about one order of magnitude (Weickhardt et al., 24 Jul 2025).

Because the experiment uses two HRTSs, the figure of merit is the differential cavity motion between the suspended mirrors. The authors assume that the two suspensions reject 99% of common seismic motion, leaving a 1% mismatch coupling between the two HRTSs. Using time-series outputs from the mechanical models, they compute the ASD of differential motion and add all noise terms. This differential readout is the quantity used to determine whether COBRI-based local damping yields lower cavity length noise than BOSEM-based damping.

5. Noise budget, facility limits, and predicted performance

The HRTS cavity noise budget combines control noise with seismic, thermal, quantum, and frequency-noise terms (Weickhardt et al., 24 Jul 2025). Measured table motion was obtained with the Trillium Horizon 120 translational seismometer and the Blueseis-3A plus a triangular TH120 configuration for rotational sensing. For the HRTS simulations, however, only translation in $\qty{1064}{\nm}$6 and pitch rotation were injected; all other DOFs were set to zero because their coupling to cavity-axis motion was considered negligible.

A key conclusion is that with the current table motion, cavity noise remains heavily dominated by seismic coupling below about 7 Hz, masking much of the BOSEM-versus-COBRI difference. The paper therefore develops a target seismic spectrum requiring roughly one to two orders of magnitude improvement in isolation, depending on frequency. The target isolation shape is described as an $\qty{1064}{\nm}$7 slope at low frequency and flat at high frequency, limited by TH120 self-noise.

The present and target performance regimes are summarized concisely:

Condition	Main outcome	Frequency range
Current facility isolation	only a slight difference between BOSEM and COBRI; BOSEM influence largely masked by seismic coupling	4–7 Hz
Current facility isolation, conclusion summary	up to a factor of two in length stability along the cavity axis	3.5–7 Hz
Improved pre-isolation target	COBRI-damped suspensions significantly lower than BOSEM-damped suspensions; improvement expected to be at least an order of magnitude	especially around 5 Hz; generally below 30 Hz

This distinction is crucial for interpreting HRTS performance. In the current facility configuration, the HRTS itself is not the sole limitation; seismic motion of the support environment obscures the full benefit of the interferometric sensor. The paper is explicit that the current seismic pre-isolation in VATIGrav is not yet low enough, that translational ground coupling dominates cavity motion below about 7 Hz, that rotational measurements using the TH120 triangular method are not used in simulation due to calibration uncertainty, and that only $\qty{1064}{\nm}$8 and pitch were simulated (Weickhardt et al., 24 Jul 2025).

Once local control noise is reduced, other terms become relevant. The paper gives the mirror coating thermal-noise PSD as

$\qty{1064}{\nm}$9

with

$P_0$0

The cavity displacement noise from shot noise is written, though truncated in the text, as

$P_0$1

with

$P_0$2

Radiation pressure noise is written as

$P_0$3

with

$P_0$4

Laser frequency noise is related to reference-cavity length fluctuation by

$P_0$5

and mapped to suspended cavity displacement noise by

$P_0$6

Temperature coupling in the ULE reference cavity is modeled as

$P_0$7

with

$P_0$8

and simulation using the conservative value

$P_0$9

These terms are not HRTS mechanical noises per se, but they set the floor against which HRTS cavity stability is interpreted.

6. Interpretation, limitations, and prospective modifications

The reported work does not introduce a new HRTS design; rather, it provides an experimental and modeling framework for understanding how local sensing quality limits a triple suspension such as the HRTS (Weickhardt et al., 24 Jul 2025). The paper’s central conclusions are that the HRTS is a realistic testbed for comparing conventional shadow sensing and interferometric local sensing in a 6-DOF, observatory-relevant suspension, that local sensor noise enters the HRTS control loop through the top-stage actuation path and suspension susceptibility, and that the practical BOSEM-versus-COBRI tradeoff depends strongly on facility pre-isolation.

Several limitations are stated explicitly. The damping filters are generic LIGO filters and are not optimized for COBRI. BOSEM readout electronics impose an ADC-noise-related limit on how far the DAC range can be reduced. Only $\qty{1}{\W}$0 and pitch were simulated. Rotational measurements from the TH120 triangular method were excluded from simulation because of calibration uncertainty. The present facility is therefore intentionally conservative with respect to the performance that interferometric sensing might permit.

The paper also considers whether the suspension mechanics could be altered to strengthen tests of sensor-noise coupling. It cites the following suspension-design principles:

• use four wires from the actuated mass to the test mass
• match masses and moments of inertia between stages
• tune wire attachment angles

In simulations, matching masses and moments of inertia increased top-to-test-mass coupling by about a factor of two over

$\qty{1}{\W}$1

and increasing wire attachment angle increased it further. This suggests that future HRTS-like suspensions could be engineered specifically to magnify the measurable impact of local sensor noise. However, the present work deliberately uses the real HRTS because it is a well-tested, relevant observatory-style suspension (Weickhardt et al., 24 Jul 2025).

A common misconception would be to treat the reported improvement as a universal property of the suspension alone. The paper argues otherwise: with current facility isolation, the expected longitudinal improvement is only modest, whereas with improved pre-isolation the simulations indicate a much clearer advantage from interferometric sensing. The HRTS therefore emerges as a platform in which the measurable superiority of COBRI depends jointly on suspension mechanics, sensing noise, controller shape, actuator noise, common-mode rejection, and support-platform seismic motion.

In that sense, the HRTS is significant less as an isolated mechanical artifact than as a controlled testbed for low-frequency suspension control in gravitational-wave detectors. Within the regime emphasized by the paper—especially around 5 Hz and more generally below 30 Hz—the move from BOSEM shadow sensing to interferometric sensing like COBRI is expected to reduce active-control-induced displacement noise directly, while also enabling practical low-noise angular damping that BOSEMs cannot provide in the analyzed setup (Weickhardt et al., 24 Jul 2025).