\section{Introduction}
\label{sec:intro}

\begin{figure}[h]
   \includegraphics[scale=0.4]{./images/AssyBeamsplitterMasses.png}
   \caption{3D CAD rendering of the \gls{BBSS} triple. Top Left: the suspended masses; \acrfull{M1}, \acrfull{M2}, \acrfull{M3} that are mounted to the cage through the \gls{SUSP}. Top Right: the suspended masses together with the cage which is fixed to the \gls{ISI} platform.  Bottom: detailed view showing the proposed sensor and actuator arrangement at \gls{M2}, showing \gls{BOSEM}s (used only for actuation) in the corners and \gls{HOQI}s (only capable of sensing) along the vertical and horizontal axes. }
    \label{fig:TripleSuspensions}
\end{figure}

\begin{figure}[h]
   \includegraphics[scale=0.3]{./images/damping.png}
   \caption{Simplified side-view schematic of the triple \gls{BBSS} showing the sensing and actuation points for damping the remainder of ground motion not suppressed by the \gls{ISI}. Left: at \gls{M1}, multiple \gls{BOSEM}s are mounted on the cage, measuring the relative displacement between the cage and \gls{M1} in all six degrees of freedom. Similarly, at \gls{M2} four \gls{HOQI}s are mounted on the cage, measuring the relative displacement between the cage and \gls{M2}, but only in \acrfull{L}, \acrfull{P}, and \acrfull{Y}. Right: at both \gls{M1} and \gls{M2} there are multiple \gls{BOSEM}s to actuate on the suspension. By default, the \gls{BOSEM}s at \gls{M1} are used for local sensing and control, and the \gls{BOSEM}s at \gls{M2} for actuation from global interferometer signals. We investigate using the \gls{BOSEM}s at \gls{M2} for actuation based on both local and global signals.}
    \label{fig:TripleSuspensionsSchematic}
\end{figure}

Gravitational waves were predicted by Einstein's Theory of General Relativity, and were first observed in 2015 \cite{GW150914}.  Since then multiple events  \cite{O21,O3} have been detected by the Advanced \gls{LIGO} \cite{AdvancedLigo} and Advanced Virgo \cite{AdvancedVirgo} interferometers. These interferometers precisely measure the \gls{DARM} changes of the long (3-4\,km) arm cavities. Passing gravitational waves induce differential strain in the perpendicular arms, allowing interferometric detection.

The first detection of a binary neutron star inspiral \cite{BNS,BNSMMS} demonstrated the importance of gravitational wave detectors for multi-messenger astronomy. Gravitational wave detectors provide sky localization from triangulation with multiple detectors. Inspiral detection can provide early alerts for electromagnetic and particle observatories. 

Improvements to sensitivity at 10-20\,Hz enable earlier detections, greater signal-to-noise ratios, and further astrophysical reach into space \cite{A+}. All events observed so far have been inspirals, where the frequency increases until the objects collide and merge. However, the signals have much longer duration at lower frequencies, with the time-until-merger proportional to $f^{-8/3}$. Improvements to low frequency sensitivity are therefore especially important for earlier detections and increasing the time in the measurement band, which in turn improves sky localisation \cite{5hz}.

One of the largest noise sources in earth-bound gravitational-wave detectors is ground vibration, which is ten orders of magnitude larger than the signal \cite{GW150914} and moves the optics of the interferometer. The \gls{LIGO} observatory has seismic isolation systems consisting of cascaded passive \cite{QuadUpdate} and active isolation \cite{ISI1, ISI2} to suppress this noise, and facilitate gravitational wave detection. 

Passive isolation is achieved through the use of pendula, and mass-spring-systems. Active isolation employs a blend of relative displacement and inertial sensors for feedback control of the passive isolation systems. 

Despite the success of these isolation systems in reducing direct vibration coupling, Advanced \gls{LIGO}'s low-frequency sensitivity is limited at 10-20\,Hz by `control noise' from the global interferometer's \gls{asc} and the auxiliary \acrlong{LSC} systems \cite{O33}.

Local controls minimize the transfer of motion to an individual optic, while global control keep the optics of the interferometer correctly placed and oriented to each other.  

Improved local sensors for controlling the suspension chains will result in a quieter, simpler, and more stable plant for global control. Thereby reducing noise that is associated with non-linear, bi-linear, and non-stationary couplings that cannot currently be suppressed in post-processing \cite{O3,MachineNoise}. This, in turn, reduces the control noise in the 10-20\,Hz region, allowing detections which were previously impossible.

One of the \gls{LIGO} A+ upgrades, planned for after the upcoming fourth observing run, is the installation of a new, larger beamsplitter \cite{APDR}. This necessitates a new \acrfull{BBSS}, it contains slots for \acrlong{HOQI}s (\acrshort{HOQI}s)\cite{HoQI} to be used as relative displacement sensors. If installed, they will demonstrate the effects of better damping performance for future upgrades.

Figure \ref{fig:TripleSuspensions} shows an overview of the \gls{BBSS}, a cascade of 3 masses. The suspension system isolates the \gls{M3} from the residual ground motion of the \acrfull{ISI} platform. Every step down this chain is further, $f^2$ isolated above its pendulum frequency, from movement. The chain's resonances must be damped to limit the \gls{RMS} and resonant motion of \gls{M3}. The damping system works by measuring and actuating between the rigid cage and the suspended stages, depicted in figure \ref{fig:TripleSuspensionsSchematic}. The baseline A+ damping design uses \gls{BOSEM}s \cite{Bosem} mounted on the cage to measure and actuate on the \acrfull{M1} (we call this `\gls{M1} \gls{BOSEM} damping'). 

Interferometric sensors provide significant performance improvements over optical shadow sensors. For a review of compact interferometric sensor designs see \cite{intorev}. \gls{HOQI}s are one such sensor with a mature design, and about 3-4 orders of magnitude more sensitive than \gls{BOSEM}s, as shown in figure \ref{fig:SensorNoise}. Improved sensitivity allows them to be used for local control closer to the optic, without disturbing the sensitivity in the critical measurement band above 10\,Hz. 

In this paper we present a detailed investigation of the damping and noise performance if more sensitive sensors, \gls{HOQI}s specifically, are mounted between the \acrfull{M2} and the cage. The combination of \gls{M2} \gls{HOQI} sensing and \gls{M2} \gls{BOSEM} actuation is referred to as `\gls{M2} \gls{HOQI} damping'. We provide a noise-budget breakdown of the contributions to optic motion in both \gls{L} and \gls{P}. Three targets were identified as crucial; reducing suspension resonance peaks, meeting 10\,Hz performance requirements, and reducing \gls{RMS} motion. We also considered the impact of global actuation signals on the final optic motion, especially in the presence of cross-coupling, but we were unable to sufficiently justify quantitative predictions.

The evaluation of improved damping for the \gls{BBSS} is a case-study applicable to other triple suspensions, and with some modifications, to the quadruple suspensions used for the test mass optics. Installation of \gls{HOQI}s at the \gls{BBSS} would be an effective technology demonstration. We elaborate on the connection between local sensors and improved detector sensitivity at low frequencies, supporting the statement that improved damping is one of the elements required for breaking the `low-frequency wall' \citep{5hz}. 



\section{Simulating the \gls{HOQI} damping performance}
\label{sec:Method} %

The purpose of the damping system is to lower the quality factor of the resonances of the suspension chain at its eigenfrequencies. This reduces the total motion at \gls{M3}, provides a simpler plant for implementing global interferometer controls, and reduces the \gls{RMS} motion of the suspended masses and corresponding non-linear effects. Active damping is preferred over passive damping to have more freedom in shaping the frequency response of the dissipation and to allow for fine-tuning after installation. In addition to damping performance, this system should not introduce noise into the sensing region of the detector (10\,Hz and above). To design the damping system we need models of the damping performance and an understanding of the noise contributions.

The following method is used to show the damping performance of \gls{M2} \gls{HOQI} damping in comparison to \gls{M1} \gls{BOSEM} damping. A Matlab model for simulating the damping performance of the \gls{BOSEM}s at \gls{M1} exists \cite{bbssmatlab, BBSSM}. This model provides an open-loop (undamped) state-space for the suspension dynamics, and includes feedback paths and damping filters for sensing and actuating at \gls{M1}. These produce the closed-loop \gls{BOSEM}-damped state-space.

We expanded this implementation to include feedback paths from \gls{HOQI}s sensors to \gls{BOSEM}s actuators, all at \gls{M2}, and designed appropriate new damping filters. This study is limited to the \gls{L} and \gls{P} degrees of freedom, which are strongly cross-coupled in the underlying mechanical equations of motion. Similar damping performance is expected for \gls{Y}. 

Three primary noise sources are identified: actuator noise, sensor noise and inertial noise. Actuator noise has been left out of this investigation. In practice, the actuator noise is only relevant at $\approx$10\,Hz, and it will not affect the conclusions of this study.

Sensor noise depends on the resolution of the individual sensors and their geometrical placement. Depending on the control filters and the loop gain, the sensor noise will be injected via actuators into the suspension resulting in physical movement. 

Both \gls{HOQI}s and \gls{BOSEM}s are relative position sensors. To measure translation and rotation, multiple sensors are used at different locations of \gls{M1} and \gls{M2}. Common sensor output corresponds to translational movement, and the noise is lower noise than that of an individual sensor due to the multiplicity of sensors. Differential sensor outputs correspond to rotation and the noise is higher than for an individual sensor due to the short lever arm. Scripts were used to obtain the sensing noise in each \gls{DoF} based on the geometric placement of \gls{BOSEM}s for the \gls{M1} \gls{BOSEM} damping model  \cite{OSEMEUL} and modified for use with the \gls{M2} \gls{HOQI} damping model. 

Inertial noise consists of two parts. The first part is `\gls{ISI} noise', the transmission of motion from the \gls{ISI} (rigidly connected to \gls{SUSP}) to \gls{M3}. The second part is `cage noise'.
At high frequencies, the masses \gls{M1}-\gls{M3} move much less than the cage, but the sensors cannot distinguish the origin of the motion. To model this we projected the \gls{ISI} movement to the \gls{M1} and \gls{M2} sensor locations.
It is non-trivial to accurately determine the rotation of the \gls{ISI} at low frequencies, and we used recent predictions of expected tilt, synthesised from several sensors \cite{isiblend}.

Since the \gls{BBSS} hangs down from the \gls{ISI}, the motion of the cage due to rotation of the \gls{ISI} increases at each stage. However, damping performance increases substantially when measuring and actuating closer to the optic. With a total noise budget and the complete damped suspension model, the effect of \gls{M1} \gls{BOSEM} versus \gls{M2} \gls{HOQI} damping was compared quantitatively.



\section{\acrshort{HOQI} vs \acrshort{BOSEM} damping performance}
\label{sec:Results} %


We close the control loops on the \gls{BBSS} model with sensing and actuation at \gls{M2}. 
		
	\begin{figure}[h]
	   \includegraphics[scale=0.36]{./images/11.pdf}
	   \caption{In loop damped transfer function from \gls{ISI} translation to optic translation in the `\gls{L}' direction (parallel to the optical axis) for different input frequencies. A comparison is made between three local sensing and control configurations: only at \gls{M1}, only at \gls{M2}, and both \gls{M1} + \gls{M2}. Outside of the displayed range the responses were identical.}
	    \label{fig:LtoL}
	\end{figure}
	
	
	\begin{figure}[h]
	   \includegraphics[scale=0.36]{./images/64.pdf}
	   \caption{Noise budget of \gls{M3} translation in the \gls{L} direction, when using \gls{BOSEM} damping at \gls{M1} (black and brown), or \gls{HOQI} damping at \gls{M2} (light and dark blue).}
	    \label{fig:NtoL}
	\end{figure}
	
	
	\begin{figure}[h]
	   \includegraphics[scale=0.36]{./images/74.pdf}
	   \caption{Noise budget of \gls{M3} rotation in the \gls{P} direction, when using \gls{BOSEM} damping at \gls{M1} (black and brown), or \gls{HOQI} damping at \gls{M2} (light and dark blue). The figure is zoomed in around the resonance frequencies. A broader frequency span is shown in the appendix in Figure \ref{fig:NtoP2}.}
	    \label{fig:NtoP}
	\end{figure}
	
	Figure \ref{fig:LtoL} shows the transfer function of \gls{ISI} \gls{L} motion to \gls{M3} \gls{L} displacement with different damping configurations shown. The baseline sensing and damping at \gls{M1}, our proposed sensing and damping at \gls{M2}, and both damping methods simultaneously. It can be seen that the resonances around 0.4\,Hz, 1.1\,Hz and 1.8\,Hz are suppressed by up to a factor of eight with \gls{M2} \gls{HOQI} damping compared with (only) \gls{M1} \gls{BOSEM} damping. The results for \gls{M2} \gls{HOQI} damping, and \gls{M2} \gls{HOQI} + \gls{M1} \gls{BOSEM} damping are very similar in performance. The performance improvements in \gls{P} are slightly better than those in \gls{L} and can be found in the appendix figure \ref{fig:LtoP2}. We have only considered  configurations which are both physically possible and feasible for the \gls{BBSS}.

Figures \ref{fig:NtoL} and \ref{fig:NtoP} present the closed-loop noise budgets, showing different noise sources contributing to optic \gls{L} and \gls{P} movement with either \gls{M1} \gls{BOSEM} or \gls{M2} \gls{HOQI} damping. It can be seen that above 1\,Hz, the \gls{HOQI} damped system always has a lower total noise than the \gls{BOSEM}-damped system. Above 1\,Hz the \gls{BOSEM}-damped system is limited by \gls{BOSEM} sensor noise, while the \gls{HOQI} damped system is always limited by \gls{ISI} and cage noise. Finally, the accumulated \gls{RMS} movement in the \gls{P} direction at 1\,Hz is a factor 60 lower for the \gls{M2} \gls{HOQI} damped system when compared to the \gls{M1} \gls{BOSEM}-damped system.

The results show that adding \gls{M2} \gls{HOQI} damping obtains better damping performance without introducing more noise. Figure \ref{fig:LtoL} shows that when using \gls{M2} \gls{HOQI} damping, additional \gls{M1} \gls{BOSEM} damping does not bring big improvements. Combining this information with the lower noise injection of the \gls{HOQI} system in figures \ref{fig:NtoL} and \ref{fig:NtoP} provides strong motivation for deactivating \gls{BOSEM} damping in degrees of freedom where \gls{HOQI} damping can be used (\gls{L}, \gls{P}, \gls{Y}).

\gls{BOSEM} noise is driving the noise budget above 1\,Hz in the \gls{M1} \gls{BOSEM}-damped system. Cage noise, inertial motion of the \gls{ISI} and the cage, is dominating the \gls{M2} \gls{HOQI} damped budget, especially in \gls{L}. If the sensors were not placed on a rigid cage but on a suspended reaction chain, present in the \gls{LIGO} test mass optics quadruple suspensions \citep{AdvancedLigo}, this cage noise is strongly attenuated. This will result in even better performance for the \gls{HOQI} damped system. Cage noise can be reduced with `sensor correction', a feed-forward technique designed to subtract the inertial motion of the \gls{ISI} from the \gls{HOQI} sensor output. This is possible using the inertial sensors on the \gls{ISI}, which measure motion in the relevant band with significant signal-to-noise ratio \cite{isiblend}.



\section{Impact of suspension damping on gravitational-wave detection}
\label{sec:Interpretation} %

Current simulation models don't correctly predict the \gls{LIGO} interferometers' total noise budget for the 10-20\,Hz region. Steps are being taken to model non-linear couplings into \gls{DARM} \cite{Lightsaber} and to use machine learning for non-linear noise prediction and suppression \cite{MachineNoise,MachinesSensitivity}. 

Further efforts are being made to iterate on control filter design by closing an over-arching loop, with detailed technical noise projections \cite{closedloop}. Figure \ref{fig:ControlNoise} illustrates some of the interconnections between distinct control systems. An integrated loop over the different control systems would allow for quickly evaluating the \gls{DARM} performance gains seen by improving one element in the chain.

As these models are currently under development, we aren't able to quantify the improvements of better damping (for single or multiple optics) on \gls{DARM}. Instead we outline the mechanisms that link suspension dynamics and optic motion with \gls{DARM} sensitivity.

Reducing the \gls{RMS} motion of \gls{LIGO}'s suspended optics allows for a reduction of interferometric sensing and control bandwidths, and therefore reduced control forces and lower optical sensor noise injection. Lower resonance peaks mean lower cross-coupling into different degrees of freedom. A better damped \gls{BBSS} (evidenced by `smoother' transfer functions), enables more robust and simpler global control options, and less of a need for frequency dependent features, which affect phase. Better damping (figures \ref{fig:M2M3},\ref{fig:ISIM3}) and `smoother' phase features should improve the quality and stability of the feed-forward, used to de-couple degrees of freedom.

The effects of having multiple better damped and therefore easier to control core optics are outlined conceptually in figure \ref{fig:Darm}. From interferometric \gls{asc} studies \cite{ASC} it is known that all interferometer auxiliary degrees of freedom impact \gls{DARM}. Improvements in any part of this chain can lower the `technical noise' contributions to \gls{DARM}. Figure \ref{fig:Darm} stems from strategies which have helped resolve problems in the 20-40\,Hz range \cite{SAMSRCL2}, and investigations into a main offender for the 10-20\,Hz region \cite{SRCLMysteries}. 

The model adaptations we have made can be readily transferred to other triple suspensions in future upgrades. Initial studies showing the benefits for quadruple suspensions have been performed \cite{QuadDamp} and we expect that similar results can be obtained for all core suspended optics. The increased sensitivity of \gls{HOQI}s deeply suppresses the contribution of sensor noise and the inertial movement of the cage (cage noise) is expected to dominate the motion of the optic for \gls{BBSS} triple suspension. As such, lower total noise is anticipated if \gls{HOQI}s are placed on the suspended `reaction chain' of the quadruple suspensions that support the test masses (main arm-cavity optics). Even without a precise quantification, which is necessarily dependent on the (evolving) status of the interferometer, we have strong experimental evidence linking better damping to reduced noise in \gls{DARM} at low frequencies. 

\begin{figure}[h]
   \includegraphics[scale=0.37]{./images/ControlNoise.png}
   \caption{Simplified flow chart of the different stages of control and types of noise injections. The \gls{ISI} actively lowers the ground movement propagating to the suspensions. Measurement noises are seen as actual motion such that counteracting this perceived motion will cause actual motion. Sensor noise and ground noise result in \gls{ISI} noise. \gls{ISI} noise, cage noise and \gls{HOQI} and \gls{BOSEM} noise aggregate to create optic movement. Optic movement and global sensor noise (such as shot noise) cause movement between optics, which couples to \gls{DARM}. In the 10-20\,Hz region these effects limit the detector performance.} 
    \label{fig:ControlNoise}
\end{figure}


\begin{figure}[h]
   \includegraphics[scale=0.37]{./images/Darm2.png}
   \caption{A simplified flow chart sketching how better suspension damping can improve \gls{DARM} sensitivity. Better suspension damping reduces the \gls{RMS} motion of the optic, and (especially) reduces eigenmode resonance peaks, making the plant easier to control. 
When global sensor noise in auxiliary degrees of freedom is limiting the performance, the reduced \gls{RMS} motion allows for auxiliary control bandwidths to be lowered with the same closed-loop motion. Lower control bandwidth reduces the injection of optical sensor noise. Lower resonance peaks make a more robust plant, allowing for more accurate feed-forward and more aggressive roll-off. Finally, energy in resonance peaks is strongly cross-coupled between eigenmodes that are close in frequency. Damping therefore reduces the strength of the mechanical cross-coupling between resonances. Combined, these improvements result in improved \gls{L}-to-angle and angle-to-\gls{L} coupling, which will result in better gravitational wave sensitivity.}
   
    \label{fig:Darm}
\end{figure}

\section{Conclusions}
\label{sec:conclusions} 

The \acrlong{BBSS} design has slots for installing \gls{HOQI}s, providing the potential to test improved sensors (and as such better damping) for this suspension, and act as a technology demonstrator for future upgrades. We have simulated the damping performance if \gls{HOQI}s are fitted and used the resulting performance as a case study.

We have shown that by using \gls{HOQI} damping at \gls{M2}, instead of \gls{BOSEM} damping at \gls{M1}, we can reduce the resonance peaks of the plant by a factor of up to eight while simultaneously reducing the motion of the suspended optic. Reduced \gls{RMS} motion means lower global control bandwidth, reduced control forces, and less non-linear, bi-linear and non-stationary couplings.
A simpler plant allows for more flexible and robust global control options. This better plant stability permits improvements to the feed-forward control. These aspects will improve the sensitivity of the \gls{LIGO} interferometers in the control noise limited 10-20\,Hz region. 





\bibliographystyle{unsrt2authabbrvpp}

