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%%\title{CH.6 Laser stabilization for 6D isolation injection}
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\chapter{Laser frequency stabilization for 6D isolation system device}
In this chapter I will introduce the 6-Dimension (6D) device, a new technology for inertial isolation. This project was presented to the scientific community at the 10th ET Symposium in 2019 \cite{poster}. My contribution to the development of this technique focused on the sensing side: a laser will be injected into the device and will need to be stabilized in frequency for a low-noise readout of the sensing system at lower frequencies. To do it, we propose a new technique based on compact interferometry.\\
The experiment was built and tested in-depth: the laser stabilisation results were limited by excess noise in the sensors and many tests were made to identify and reduce noise. During these tests, it was determined that one of the devices was intrinsically noisier than the other.\\
This work was done entirely at UoB: the design of the project was conducted in 2020, while the experiment was built and tested from September 2020, when the University allowed the return to the laboratory, to July 2021.

\section{6D inertial isolation system overview}
The 6D inertial isolation system is a device based on a new technology under development at University of Birmingham and at Vrije Univestiteit in Amsterdam, which could enable detection of gravitational waves below 10 Hz \cite{6d}. We have already seen the importance for this frequency window to be opened (chap 2): this sensor could be installed on 2nd generation Earth-based interferometers, with major upgrades, on or under ground, allowing the different instruments to easily use the same device.\\
As the name reminds, the 6D system investigates the motion of a reference mass in all 6 degrees of freedom, using 6 interferometers. In Fig. \ref{6d} it is shown a sketch of the design of the facility. \\

\begin{figure}[h!]
\centering
\includegraphics[scale=1.2]{images/6d.png}
\caption[6D design.]{Sketch of the 6D device (Figure taken from \cite{6d}). The working principle is based on an isolated, suspended reference mass which is monitored by compact interferometers, detecting the relative motion between the mass and the platform; actuators apply corrections to the platform and the whole apparatus is in vacuum.}
\label{6d}
\end{figure}
\noindent
All six degrees of freedom are simultaneously low-noise, reducing the cross-coupling affecting low force-noise measurements.\\
The reference mass, suspended from a single, thin, fused-silica fibre, provides supports in the vertical (Z) degree of freedom. An interferometric readout and control are used in all 6 degrees of freedom.\\
The major advantages is that this system can improve sensitivity, thermal noise, and tilt-to-translation coupling, providing isolation in all the degrees of freedom with the use of only one device. Currently aLIGO is seismically isolated by three seismometers and twelve geophones \cite{lisa}: the use of the 6D device would replace three seismometers and six geophones on Stage 1 of the chambers.\\
\noindent
What we expect from the 6D system is isolation at low frequencies and reduction of fundamental noises: the thermal noise of the suspension is suppressed by the quasi-monolithic, fused-silica fibre; temperature gradients are kept under control thanks to the vacuum enclosure.\\
The expected performance is shown in Fig. \ref{6dsens}: the 6D isolator provides an improvement of the performance of more than two orders of magnitude with respect to what is possible with state of the art seismometers \cite{6d}.\\
The key point is to reduce the motion in order to limit the control noise and allow the bandwidth of control loops to be lowered. This is a goal set for a detector sensitive to low frequencies, and for which the 6D device can contribute \cite{yu}.

%The control noise will become negligible above 5Hz because the bandwidth for the control loops will be reduced to 0.5 Hz.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/6dsens.PNG}
\caption[6D sensitivity curve]{A comparison of the expected performance of the 6D isolator and that of current seismometers (Figure taken from \cite{6d}).}
\label{6dsens}
\end{figure}

\subsection{HoQI technology}
The 6 interferometers dedicated to the sensing role of the facility are the HoQI devices (Homodyne Quadrature Interferometer) developed at UoB \cite{hoqi}.\\
HoQI is a compact, fibre-coupled interferometer with high sensitivity and large working range. In Fig. \ref{hoqi1} you can see the optical layout of the device:

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/hoqi1.png}
\caption[HoQI optical layout]{HoQI optical layout (Figure taken from \cite{hoqi}). The working principle is based on a Mach-Zender interferometer which uses two different beam splitters to recombine the beam independently. Moreover, the required differential phase shift is generated by a polarization scheme conveniently designed.}
\label{hoqi1}
\end{figure}
\noindent
This device has been designed to sense motion at low frequency, with a sensitivity of 2 $\times$ 10$^{-14}$ m/ $\sqrt(Hz)$ at 70 Hz and 7 $\times$ 10$^{-11}$ m/ $\sqrt(Hz)$ at 10 mHz \cite{hoqi}. In the frame of compact devices, HoQIs are designed to be very small in size, so they can easily be attached to sensors: good results were obtained when combining HoQI devices to inertial sensors to create an "interferometric inertial sensor" \cite{sam}. They are then ideal for 6D purposes, not only for their high sensitivity, but also for their small size.

\noindent
The six HoQIs used for the 6D device need to be fed by a laser source that is sent into the vacuum chamber: my project focused on this source, specifically how to stabilize it in frequency.

\section{Laser stabilization: requirements and key technology}
The laser chosen as source for 6D is a 1064 nm RIO ORION Laser Module (see Fig. \ref{rio}). This has been chosen for its low frequency noise, inexpensiveness and small size, relatively to other options. The key point in the stabilization of the frequency noise of this source is that the technology will be based on HoQIs: the same devices used by the 6D, but with a longer arm-length mismatch, are sensitive enough to be installed also to stabilize the laser source. This solution is very convenient in terms of costs and presents practical advantages: the HoQIs are known devices, compact in size and, as we will see, they allow the setup to be moved easily (in vacuum or in air), according to the main 6D requirements.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/rio.jpg}
\caption[Rio Orion laser modules]{Picture of the RIO Orion laser mounted on a breadboard.}
\label{rio}
\end{figure}
\noindent
What we want from this source is a low-noise readout for the HoQIs inside the 6D tank, and thus the laser source needs to be as low noise in frequency fluctuations as possible at frequencies below 0.5 Hz, because this is the range of frequencies where the 6D isolator is aimed to detect and control seismic noise: we are going to use two Rio Orion laser modules to obtain a frequency stabilization suitable for 6D requirements. Constraints to these requirements are mainly given by the HoQIs. For 6D readout, HoQIs are built in such a way that the arm length mismatch is as small as practically possible, e.g. L$_{6D}$ $<$ 3 mm. Limitations to this number are given by BOSEM size ($\pm$ 2 mm) and the ability to adjust it, once the devices are in vacuum. Another parameter to take into account is the noise of HoQIs, which is H = 6 $\times$ 10$^{-14}$ m/$\surd{Hz}$ at about 1 Hz \cite{hoqi}. Frequency fluctuations depend on both these parameters and we want it to meet the following requirement:

\begin{equation}
\centering
\delta f_{6D} \ll f \times \frac{H}{L_{6D}} \simeq 5000 \frac{Hz}{\surd{Hz}},
\end{equation}\\
\noindent
where f is the laser frequency.\\
The technique we are going to adopt to stabilize the laser in frequency, as anticipated, is to use HoQIs, because we can associate frequency fluctuations to fluctuations of arm length:\\
\begin{equation}
\centering
\delta f = \frac{\delta L}{L}\cdot f,
\label{df}
\end{equation}\\
and this arm length can belong to a HoQI placed on the optical bench. The use of compact interferometers to stabilize the frequency of solid-state lasers without the use of cavity locking is new and allows the whole set up to be small in size. This technique, in combination with cheap laser sources, makes the set up competitive with other more expensive products.\\
We can then apply the same relation of. eq. \ref{df} to the arm length of the HoQI used for the laser stabilization, remembering that the requirement of $\delta$f $\ll$ 5000 Hz/$\surd{Hz}$ must remain valid. So, constraints to the arm length in this case are due also to the size of the bench and the whole set up.\\ We said we want a compact setup, but the arm length of this HoQI (say L$_{stab}$) can have a wider range of sizes to fit the requirement. For example, for L$_{stab}$ = 1 m we have:\\

\begin{equation}
\centering
\delta f_{stab} = f \times \frac{H}{L_{stab}} \simeq 16 \frac{Hz}{\surd{Hz}},
\end{equation}\\
which is still much lower than the threshold of 5000 Hz/$\surd{Hz}$.\\
Since we want the setup to be as much compact as possible, we need to find the lowest possible L$_{stab}$ which gives an interesting $\delta f_{stab}$, compared to the current performance of RIO Orion and the best products available.\\
In the plot in Fig. \ref{perf} there is the analysis and comparison with two of the best products available.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/perf.png}
\caption[Analysis and comparison of RIO Orion laser with other products]{Analysis and comparison of RIO Orion laser with other products and with the configuration involving HoQIs. For an arm mismatch of 10 cm, $\delta f_{stab}$ is still below the threshold, being 168 Hz/$\sqrt{Hz}$, still fitting the 6D requirements.}
\label{perf}
\end{figure}
\noindent
It is evident that we cannot build a HoQI with L=10 m. If we want our device to be competitive even with the best product (ADJUSTIK X15, shown in orange line), we will need a L=30 cm. However, our purpose is to make the set up \textit{compact} and \textit{competitive} with most of the available products, so the best compromise is choosing L=10 cm. With this configuration, the device will still be competitive with ADJUSTIK X15 in terms of price.\\
Fig. \ref{free} shows a plot of the measured frequency noise of the Rio Orion laser modules and the level of stabilization required by the 6D with the chosen L$_{stab}$.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/free.png}
\caption[Free running noise and requirements]{Measured free running frequency noise of the Rio Orion laser modules compared with the frequency noise of the Mephisto Light laser module. The dotted line is the threshold of the 6D requirements for stabilization of its laser source, which is the goal we aim to achieve with the proposed setup. This threshold has been built using L$_{stab}$=10 cm and the HoQI noise and ADC floors as in \cite{hoqi}.}
\label{free}
\end{figure}

\section{Experiment design}
The optical set up for the stabilization of the laser source is built on a 800 mm $\times$ 650 mm breadboard: this choice allows to adjust the position of the laser source easier when the light is sent to the 6D vacuum chamber. The setup includes two RIO Orion lasers with output power 12 mW each, and a double-check of the light signal through an optical heterodyne detection: the beat frequency is monitored to assure that the two light frequencies are as much similar as possible. The frequency noise from the heterodyne detection is measured by a frequency counter: we chose this device because it is the least ambiguous, lowest noise, lowest systematic-error, best calibrated measure of the relative fluctuations of the two lasers available for our purposes.\\
The frequency of the lasers is tunable via temperature and input modulation: in the first case, the Thermo-Electric-Controller (TEC) is driven by a software provided by the manufacturer, while in the second case the module can be integrated to any software code to produce an output voltage between +4 V and - 4 V, that is sent to the lasers: the frequency tuning efficiency with this method is around 80 MHz/V, when a sinusoidal modulation is applied at 10 kHz. From specifications, the range of tuning spans between 50 and 100 MHz/V.\\
To minimise airflows, the optical setup has been enclosed into a box made of foam.

\paragraph*{Opto-mechanical design}
The optical layout is shown in Fig. \ref{las}: the two lasers have a twin optical layout. There is a Faraday Isolator (FI) at each output and then a 1 to 4 fibre beam splitter (BS) which separates the beam into 4 outputs of equal power: 3 outputs go into the vacuum chamber (for a total of 6 laser inputs, one for each 6D HoQI into the vacuum chamber). The remaining output is sent through a fibre coupler to a Schafter-Kirchhoff collimator and gives an output of about 1.2 mW for each laser; this proceeds freely on the breadboard towards a 1 inch, 10/90 (R/T) beam splitter: 10$\%$ of the light is sent to a fast DC coupled 125-MHz photoreceiver (PD) acted to sense the beat-note of the two lasers; two 1 inch mirrors deviate one of the two laser beams towards the transmitting surface of another 1 inch beam splitter, which combines the light from both lasers towards the photoreceiver; the other 90$\%$ of it is sent to the HoQIs, one for each laser. The optical path lengths (OPL) have been set to be equal, to assure the same beam size from both lasers at the photoreceiver.\\
The photoreceiver has strict constraints about the beam size and the input power: a focussing lens in front of the active area assures that the beam size is suitable to fit the 0.3 mm active area. Neutral density damping filters are added along the OPL, because the maximum input power of the device is 55 $\mu$W.\\
The whole optical setup lies on the breadboard and it is relatively easy to align because all the optomechanical components have been manufactured to make the beams out of the collimator to travel at the same height as HoQI components and the photoreceiver, so that there is no need of pitch tuning.

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{images/layout.png}
\caption[Optical layout]{Optical layout of the laser stabilization experiment.}
\label{las}
\end{figure}

\paragraph*{HoQI design}
HoQIs for 6D laser stabilization have been built to fit the requirements, as shown previously: the adjustable arm-length mismatch is of 10 cm; the photodiodes have a bigger active area with respect to the one of the HoQIs inside the 6D vacuum chamber, because the laser spot size is larger than the one travelling into the 6D device. Moreover, this type of HoQI is independent from any inertial sensor, so both the arms end with mirrors on steering mounts, and instead of corner cubes used to for the A+ devices. Table \ref{hoqi} shows other small details that have been adapted for this experiment.\\

\begin{table}[h!]
\centering
\begin{tabular}{|c|| c| c| }
\hline
 & \textbf{A+ HoQIs} & \textbf{Laserstab HoQIs}\\
\hline
Platform thickness & 6 mm & 1 cm\\
\hline
Size &65$\times$75mm &65$\times$200mm\\
\hline
Corner Cubes & 2 & 0 \\
\hline
PD active area &3.6$\times$3.6mm &5.8 $\times$5.8mm\\
\hline
Steering mirrors & 1 & 2\\
\hline
\end{tabular}
\caption{Main differences between the original HoQI and the one used for 6D laser stabilization.}
\label{hoqi}
\end{table}

\noindent
In Fig. \ref{chia} there is a photo of the HoQI built for this experiment.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/HoQIChia.jpg}
\caption[Photo of a HoQI]{Photo of one of the HoQIs built for the laser stabilization experiment.}
\label{chia}
\end{figure}
\noindent
The photodiodes are Hamamatsu S2386-8K and the optical layout is similar to the one shown in the figure \ref{hoqi1}: there are commercial 0.5 inch cubic beam splitters (three polarizing and 1 non polarising) mounted on custom cubic bases, two 0.5 inch mirrors on steering mounts, one 0.5 inch $\lambda$/2 waveplate and one 0.5 inch $\lambda$/4 waveplate, both mounted on a custom base allowing them to rotate for fine tuning. The whole optical set up is placed on a 75 mm $\times$ 200 mm $\times$ 10 mm baseplate.\\
The two HoQIs have been tuned to obtain the best fringe visibility, which is 0.8 for HoQI1 and 0.6 for HoQI2. The technique used to measure the fringe visibility is based on measured power on a sin vs cos plot, once the beams have been aligned to overlap in far field and produce interference. The plot show a \textit{lissajous} figure, which can be tuned by improving the alignment with the steering mirrors and rotating the waveplates for power adjustments. When the adjustments are optimal (same power amplitude and in quadrature phase offset), the \textit{lissajous} is a circumference\footnote{For details about the characteristic equations of HoQIs and the working principle, refer to \cite{hoqi}.}. Fig. \ref{fringes} shows the sinusoids of the HoQI photodiodes after fringe visibility optimization.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/fringes.png}
\caption[HoQI fringes alignment]{Example of how we set the fringes of the HoQIs to obtain the desired alignment. Every photodiode detects the fringes independently from the others: to obtain the same response, we adjusted the offsets and the gains of each diode on the pre-amplifier and via software.}
\label{fringes}
\end{figure}

\section{AC-coupled control loop}
To acquire our data, we need to connect the HoQIs and the beat-note receiver to a data acquisition system. At UoB we have 3 CDS racks and one of them is dedicated to 6D. As shown in Fig. \ref{signals}, HoQIs will need a pre-amplifier, an Anti-Aliasing (AA) device and an Analog to Digital Converter (ADC) before connecting to CDS (Fig. \ref{signals}).

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/signals.png}
\caption[Scheme of the signals]{Basic scheme of the row signals for laser frequency control: the power detected by the photodiodes is converted into $\mu$A and sent to a pre-amp, one for each HoQI; the pre-amps convert the signal into double-ended voltage to be sent to an ADC. The output of the CDS is a $\pm$ 10 V double-ended signal out of the DAC: since the lasers require a $\pm$ 4 V single-ended input, the double-ended signal is converted into single with a custom differential- to single-ended amplifier of gain 2.5.}
\label{signals}
\end{figure}
\noindent
The beat-note receiver is 15 V powered and connected to a frequency counter, and then to the CDS. A detailed scheme of the electronics is shown in Fig. \ref{cables}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/cables.png}
\caption[Scheme of the electronics]{Detailed scheme of the electronics designed for this experiment. The different colors of the arrows represents different types of cables. Bacardi and Peapsy are the front-end and the workstation controlling it, as we named them at UoB. Differential to single ended converters (Diff2SE) are needed because the CDS supports differential outputs while the pre-amps are single ended. The Mokulab is the device used as an oscilloscope and/or as a spectrum analyser, connected to the beat-note fast photoreceiver. Green arrows indicate power supplies. The frequency counter can be connected to a computer to acquire data or a USB drive can be inserted to save data directly from the device. The temperature modulation requires the use of a software provided by the manufacturer and installed on computers. Each laser needs its own software connection. }
\label{cables}
\end{figure}
\noindent
The sensing and control system of the experiment is based on the HoQIs: the software code manipulating the HOQIs signal and driving the input modulation is written with Matlab Simulink and controlled by the CDS. The controller filter has been built taking into account all the features of the loop and implemented into the CDS: the filters are shown in Fig. \ref{filter}: their performance has been tested looking at the stability of the beat-note peak when each filter is switched on, and when they are on together as the full controller filter.\\
The lasers can then be controlled via input modulation of the current feeding the lasers, through the feedback control loop built via the CDS, where HoQIs act as the sensors. 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/filter.png}
\caption[Controller filter]{Plot of the controller filter installed into the CDS (green) and of the closed-loop expected gain when this filter is applied (magenta). This design should push the gain from below 0.1 Hz, assuring stability when applied at lower frequencies.}
\label{filter}
\end{figure}

\section{Noise hunting}
The performance of the setup depend strongly on the HoQIs because they are the sensing and feedback devices of the setup: the noise budget in Fig. \ref{noiseb} shows that the measured HoQI readut follows the free running frequency noise of the lasers detected by the frequency counter at low frequencies, and then it sits on the ADC noise (estimated as in \cite{hoqi}) at frequencies above 100 Hz. What we are interested in is getting the lowest possible frequency noise from the lasers, reducing the noises affecting the HoQIs. The improvement of HoQIs sensitivity is crucial to obtain the best performance in sensing frequency fluctuations, in order to provide the correct and stable control and feedback to the setup.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/noisebudget.png}
\caption[Noise budget]{Noise budget of the laser stabilization setup, in free air with the controller turned off. The paper which provided the HoQI noise and the ADC noise is given by \cite{hoqi}.}
\label{noiseb}
\end{figure}

\subsection{Tested noise sources}
There are several noise sources to take into account and that we tested and minimized: air currents and vibrations from electronics and cables have been reduced placing the optical setup into a foam box and moving the electronic devices suitably. The lasers have been left outside the box to avoid overheating inside, due to their heat dissipation. Cables have been isolated from the table and the breadboard by rubber feet.\\

\newpage
\paragraph{Offset/gain parameter matching}
When aligning HoQIs, a significant contribution to the results was given by the fact that the fringes needed to be adjusted to match in gain and offsets: this was done via CDS and it provided one of the most important issues (and noise sources) during the tests. Imperfect fringe-visibility and parameter matching dominate the coupling from intensity to measured-phase, and we saw substantial improvements when these values were optimised. We had insufficient diagnostics to fix the issue, mainly because, to create fringes, the HoQIs needed to be mechanically shaken and tuned, and this could be done only manually and in dark room (to avoid room light to affect the offsets), providing imprecise results. We had no proper way to adjust the laser frequency without changing the intensity, and we had no way to adjust the intensity without changing the frequency. This made the offsets and gains change and pollute the measurements. The plots in Fig. \ref{parameters} show the difference between the row fringes generate while shaking HoQI1 and the ones generated after optimizing the parameters. We observed variations in the settings, which needed to be double-checked and readjusted before every measurement. This was an important issue that slowed the tests down and that it is worthy of further tests and data analysis.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{images/row.jpg}\\
\includegraphics[scale=0.4]{images/opt.jpg}
\caption[Row and optimized fringe parameters for HoQI1]{Row and optimized fringe parameters for HoQI1. This fringes has been generated by changing the temperature of laser1: this induced a frequency variation, detected by HoQI. The plots show the row fringes from each photodiode (upper) and the ones after the optimization of the gains and offsets for each photodiode, with respect to a reference one (lower).}
\label{parameters}
\end{figure}

\newpage
\paragraph{Acoustic noise}
The test in Fig. \ref{sound} shows that the setup is sensitive to acoustic noise: we injected a sound at 75 Hz and both HoQIs clearly detected it. Moreover, we found out that HoQI1 is detecting some noise around 22 Hz that HoQI2 is not able to sense: the two peaks in the figure are present in every condition of the laboratory and time of the day. The source of this noise is still under investigation: it could be a permanent sound in the lab non audible by humans. The fact that only HoQI1 can detect it could be due to its position with respect to the noise source: it might be closer to it than HoQI2. Imperfections in the optics and general setup of the HoQIs are also taken into account.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=0.33]{images/sound75hz.png}
\caption[Test of the setup to sound stimulation.]{Test of the setup to sound stimulation. The peaks at 75 Hz shows that both HoQIs are sensitive to acoustic stimulation; HoQI1 is also detecting another noise around 22 Hz, which is instead non-visible by HoQI2.}
\label{sound}
\end{figure}


\paragraph{The role of the temperature}
Temperature changes affected dramatically the measurements. The two lasers can be driven also via temperature modulation: this method has been used to move the beat-note peak along the frequencies and set it around 60 MHz, being this the setpoint we decided for it. However, both laser modules are sensitive to changes of the room temperature, which make the peak move out from the setpoint on large time scales (~hours): this affects long time measurements. The stabilization of the room temperature requires the use of the air conditioning, which in turn creates air currents visible by the setup below 10 Hz (Fig. \ref{ACtest} shows the difference between two tests taken with and without AC).\\
Temperature changes are also responsible for deformations of metals; this induces noises into the HoQI platforms because of the different materials they are built of: temperature change induces differential expansion. In a rigid structure this creates stress and some of the bolted connections slide, introducing noise. This issue has been reduced by inserting rubber rings between the connections: this allows the baseplate to expand differentially without creating stress.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/AConoff.png}
\caption[Test of the impact of AC on the frequency stability]{Test of the impact of air conditioning (AC) on the frequency stability. From this plot the free running frequency measured by the beat-note is compared to the frequency measured when the setup is in loop in different AC conditions: the red trace shows a measurement taken when the AC was on, during the night: the AC creates air currents, and it is also responsible for changes in the temperature of the room and of the lasers, and it can induce dust in the OPL. All these contributions are affecting the setup below 10 Hz; the black trace shows the same test with no AC: below 10 Hz the trace is much quieter. The higher noise above 10 Hz is due to the fact that this test has been taken in daylight time, and HoQIs suffered the vibrations of the building. After this test we reduced the free space OPL between the optics were possible, filled the empty spaces of the box and reduced the free space between the last optic and the beat-note photoreceiver, to reduce air flows. All the following tests have been taken with AC off.}
\label{ACtest}
\end{figure}

\paragraph{HoQIs performance}
When monitoring the output of the HoQIs, we noticed that HoQI2 is much noisier than HoQI1: Fig. \ref{hoqi2} shows an out of loop measurement of the output of both HoQIs. This discrepancy has been investigated: possible reasons for that could arise from the laser source of HoQI2, alignment and clipping on the optics, fringe visibility, spurious light, mechanical defects in HoQI2 setup. The laser source has been changed to be the same as HoQI1 and further tests showed that HoQI2 is performing the same way. This relieved the laser of any responsibility, since now the same source is feeding the two HoQIs in the same way. The alignment and the clipping on the optics have been carefully checked and possible sources of stray lights have been meticulously covered. The fringe visibility has been double-checked: the test is still showing more noise from HoQI2 output. What remains to inspect is the possibility of mechanical defects in the optics or in the setup of HoQI2, the latter being a concrete possibility due to errors in manufacturing.\\
Another reason of concern about HoQI2 behaviour is that it is not consistent with tests in loop (see Fig. \ref{looptest} later): this might be due to intensity noise coupling, which effect might be more evident than in HoQI1 due to internal defects, giving a noisier output when the setup is in loop. 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/hoqi2_issue.png}
\caption[Difference between HoQI1 and HoQI2 outputs]{Difference between HoQI1 and HoQI2 outputs in an open-loop test.}
\label{hoqi2}
\end{figure}

\paragraph{Power variations}
The datasheets of the laser modules put in evidence that the power modulation is different for the two lasers: in particular, for laser2 it is higher than laser1 (compare the datasheets reports in \cite{rio} and \cite{riologbook}). This might in part explain why HoQI2 is in general noisier than HoQI1. We monitored the power of both in-loop lasers: according to the datasheet, the power modulation of laser 2 is about 1.56 times higher than for laser 1. In Fig. \ref{power}, the power variation measured at HoQIs input during frequency modulation for the two lasers is consistent with the datasheet. The higher power variations of laser 2 might induce higher intensity fluctuations, which could affect HoQI2.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.45]{images/power.jpg}
\caption[Power variations of RIO Orion lasers]{Power measured at the input of the two HoQIs, after the polarizing beam splitter (PBS) over 1 hour of data. Laser 2 shows a higher power variation over the same time with respect to laser 1, for a change about 1.56 higher. This is consistent with the datasheet statements, and might also be a reason for the higher noise into HoQI2. However, for practical reasons the power photodiode has been placed after the PBS, a contribution of this optic in this effect is not excluded: there might be internal imperfections which could be accounted for possible polarization variations there on in the fibres.}
\label{power}
\end{figure}

\subsection{Loop performance}
The behaviour of the two HoQIs has been tested in loop and out of loop, to check if they are detecting and responding correctly to the injection of the controller filters through the input modulation of the laser modules. The expectation is that the HoQIs output in out-of-loop mode should show the injection of the gain: Fig. \ref{looptest} shows that the expectations are satisfied.\\
This test confirms that HoQI2 is in general noisier than HoQI1, especially above 1 Hz: this affects laser stabilization measurement and loop stability, thus it has been deeply investigated. The higher intensity fluctuations of laser2 can partially explain the reason of HoQI2 noise.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/hoqisOLCL.png}
\caption[In-loop test of HoQIs performance]{In-loop test of HoQIs performance. The out-of-loop traces (cyan and purple) are following the free running frequency noise trace (blue) as expected, while when the loop is closed the HoQI outputs (green and red) show that the controllers are pushing the expected gain (orange). There is an evident un-match with the orange trace below 0.4 Hz and this is likely due to spectral leakage.}
\label{looptest}
\end{figure}

\section{Laser stabilization: tests and results}
The tests have been performed measuring the stability of the beat-note peak around the 60 Hz setpoint: the frequency counter used for this measurements is a Keysight 53230A 350 MHz - 20 ps. The output of the fast photoreceiver is DC-coupled and can be directly connected to the counter. The measurements has been recorded on a USB drive: the data provided by the counter are in frequency (Hz).\\
Several tests have been taken in different conditions for noise hunting along the frequency range of interest, the best measurements are shown in Fig. \ref{test}. The tests with the heterodyne detection revealed that the system is very sensitive to the external noise sources described above and that the two HoQIs are not robust and stable enough to assure the stability of the loop, despite the robustness of the controller filter. The solution for reducing these noises might be placing the HoQIs in vacuum.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/result.png}
\caption[Results of frequency stabilization tests]{Results of frequency stabilization with respect to the free running frequency noise: the in-loop red trace shows the frequency stabilized lasers as detected by the frequency counter, monitoring the beat-note between the two lasers in the lower frequency range. This trace is the best measurement we obtained below 1 Hz, where we reached 1.67 $\times$ 10$^4$ Hz/$\sqrt{Hz}$ at 0.05 Hz; The green trace is a test taken with the counter set to a higher frequency range: this test shows a result of 3.6 $\times$ 10$^3$ Hz/$\sqrt{Hz}$ at 1 Hz. This is also the test which showed the quietest results above 10 Hz, demonstrating that the HoQIs can reach a good level of stability in air. The black trace is the expected gain activated by the controllers, which is set to maximise the stabilization below 1 Hz: when it is lower than the green and red curves, we are not limited by loop gain. When it is below the dashed-black curve, there is sufficient loop gain to meet our noise target.}
\label{test}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/experiment.jpg}
\caption[Photo of the experiment setup]{Photo of the experiment setup for frequency stabilization of the 6D laser source. Only the optical setup has been placed inside the foam box, which suitable apertures to let the beams go to the power photodiodes and the photoreceiver outside the box. The gaps inside the box have been filled in order to reduce the air currents along the OPL in free air and a lid covers the whole box.}
\label{expsetup}
\end{figure}

\section{An alternative test}
An alternative test has been made to make sure that the input modulation is effectively reducing the frequency noise of the lasers. Since we found that HoQI2 is noisier than HoQI1 and that laser2 has larger power fluctuations, we decided to use laser1 to feed both HoQIs. The new concept is to let only HoQI1 be the in-loop sensor, while HoQI2 will act as the out-of-loop sensor.\\
Results are shown in Fig. \ref{alt} and are encouraging: the frequency noise of the out-of-loop sensor is lowered by about one order of magnitude when the controller on laser1 is active. This means that the control loop works well and that there are still more external noise sources that are reducing the performance of the HoQIs, impacting also on the measurement through the heterodyne detection.\\
This test confirmed also that HoQI2 is still noisier than HoQI1, especially in closed loop, despite the use of the laser with less power fluctuations: this clarifies that HoQI2 noise arises from other external sources that HoQI1 is non-sensitive to or imperfections of the setup. A test in vacuum could solve the doubts about the external sources and HoQI2 assembly.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/CLOLtest2907.png}
\caption[Alternative stabilization test]{Alternative test for frequency stabilization: the out-of-loop sensor (HoQI2) is less noisy by an order of magnitude when the in-loop sensor is actuated by the controller filter. This is a proof that the frequency noise of the laser is reduced by the control loop.}
\label{alt}
\end{figure}

\newpage
\section*{Conclusions}
\addcontentsline{toc}{section}{Conclusions}
The results of this experiment showed that it is possible to stabilize the frequency of the laser source of the 
6D device using the technology presented: a compact, easy to handle setup which makes use of small
interferometers of the same type that are used inside the 6D sensor. With this technology, we managed to reach a
frequency stabilization of 3.6 $\times$ 10$^3$ Hz/$\sqrt{Hz}$ at 1 Hz, without the need of installing the
prototype in vacuum.\\
This is already a promising result, but not yet sufficient for the requirements of 6D, especially below 1 Hz. The results showed that we are not limited by loop gain, that acoustic noise and vibrations are important noise sources, that intensity noise and frequency-to-intensity coupling limit performance, and that both HoQIs show different coupling to these effects. The alternative test showed an indirect measurement of laser stabilization, because the out-of-loop HoQI improved its output signal of about one order of magnitude when the laser was modulated by the controller filter. This test highlighted that the frequency stabilization through the heterodyne detection depends on the stability and robustness of the HoQIs. A concrete plan for next tests is to place the setup in vacuum: this will suppress all the external noises and will possibly highlight the intrinsic issues of HoQIs.\\
This test has already been considered as a possibility during the experiment design: the HoQI setups can be placed inside the 6D vacuum tank independently from the rest of the components of the stabilization breadboard and electronics.





%\begin{thebibliography}{9}
%
%\bibitem{poster} \textit{https://dcc.ligo.org/LIGO-G1900741}
%
%\bibitem{6d} C.M. Mow-Lowry, D. Martinov, \textit{A 6D interferometric inertial isolation system}, 2019 Class. Quantum Grav. 36, 24 
%
%\bibitem{hoqi} S. Cooper et. al, \textit{A compact, large-range interferometer for precision measurement and inertial sensing}, 2018 Class. Quantum Grav. 35 095007
%
%\bibitem{sam} S. Cooper, \textit{Breaking the Seismic Wall: How to Improve Gravitational Wave Detectors at Low Frequency}, PhD thesis, University of Birmingham, 2019
%
%\bibitem{rio} $https://rio-lasers.com/1064-nm-laser-module/$
%
%\bibitem{adj} $https://www.photonicsolutions.co.uk/upfiles/ADJUSTIKFiberLaserLG13Dec17.pdf$
%
%\bibitem{mephi} $https://edge.coherent.com/assets/pdf/COHR\_ MephistoNPRO\_ WP\_ 9\_ 24\_ 19.pdf$
%
%
%\end{thebibliography}
%
%\end{document}

