E-TEST: a compact low-frequency isolator for a large cryogenic mirror

The AP, shown in figure 3, provides inertial control in the six degrees of freedom. It is a completely redesigned and up-scaled version of the LIGO HAM-ISI (Horizontal Access Modules- Internal Seismic Isolator) platform [37] with a new diameter of 2.5 m (to host the large cryostat) and 0.75 m in height. The natural frequency of the platform AP is designed to be about 2 Hz for the vertical, pitch and roll degrees of freedom, and about 2 Hz in the horizontal and yaw (rotation about the vertical axis) degrees of freedom. This is achieved by suspending the platform using three cantilever blade springs and three flexure elements.

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The sandwich structure of the payload, made of vertical panels, is reinforced for maintaining the first flexible mode above 300 Hz. This is to guarantee high control performance in the low frequencies. The shape of this mode is further shown in figure 4 which appears at 320 Hz.

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Particular attention has also been paid to: (1) mass and stiffness repartition, and (2) positions of the horizontal sensors and horizontal actuators. This is to facilitate plant decoupling in view of its control.

The absolute motion of the platform is reconstructed using the signals of six inertial sensors that are embedded inside of its structure: three vertical sensors for the vertical motion (developed in the PML (Precision Mechatronics Lab) [34]) and three horizontal sensors based on Watt's Linkage (developed at KU Leuven [38–40]) for the horizontal motion. They are positioned symmetrically in a circular configuration, allowing the reconstruction of the platform's degrees of freedom. Each sensor aims for a sub-picometer resolution at frequencies below 10 Hz. This is achieved by reading the proof-mass motion of the sensor mechanics using a high resolution, homodyne, quadrature, Michelson interferometer [41, 42], which has already demonstrated 1 pm resolution above 1 Hz until the Nyquist frequency [43]. The sensor's mechanics are designed to comply with the geometrical constraints of the design and to be compatible with the ultra-high vacuum environment of the platform. The sensor's performance is expected to break through the 10⁻¹³ m/$\sqrt{\text{Hz}}$ at 1 Hz.

For inertial control, AP uses six voice coil actuators (NCCO9-39-190-1 V from H2W Technologies), three for the horizontal direction and three for the vertical direction. In addition, AP uses six BOSEMs [44] to measure the relative displacement of the platform with respect to the ground. The error signals for the controller are obtained by blending, thanks to a complementary filter the information from the displacement sensors at low frequency with the signal from the inertial sensors at high frequency.

As shown in the schematic diagram of the controller in figure 5, E-TEST applies three control loops: a damping control loop (AP damping) and an isolation control loop (AP isolation) for AP, and a damping control loop at the IPP level for the horizontal and vertical modes of the passive isolation chain. The damping control loop AP is applied to AP in the local frame, then the isolation control loop AP is applied to the same stage but in the Cartesian frame, providing inertial isolation. More details on this control strategy will be published in a follow-up article whereas a brief description can be found in [45, 46]

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The isolation system consists of three inverted pendulum legs (IPLs) mounted on AP that support the IPP. Each IPL, 2.7 m high, is housed in a safety tube as shown on the label No. 4 (red) in figure 2. In addition, each upper end of the IPL is attached to a 12 kg counterweight (CW) to improve the performance of the IPP. The IPLs are connected to AP and the IPP via bends of maraging steel approximately 17 mm in diameter. The diameter of the IPP is the same as that of AP and is 2.5 m. The resonant frequency of the IPP is designed to be maintained at about 70 mHz. The IPP also has three horizontal sensors and three horizontal voice coil actuators. These sensors and actuators are used for the damping loop (to damp the horizontal resonances of the IPP and the masses of the multicascaded pendulum). In addition, the IPP contains a large GAS filter.

The GAS filter, shown in figure 6, provides additional isolation in the vertical direction, where its suspended mode is about 250 mHz. It consists of 12 triangular blades made of steel. Each blade is 10.4 kg. The GAS filter includes a linear variable differential transformer displacement sensor and a voice coil actuator for vertical direction feedback control. Additional static positioning actuators are provided consisting of a stepper motor driven vertical correction spring and a stepper motor driven Harmonic Drive bearing for adjusting the orientation of the downstream passive filtering chain in the yaw degree of freedom. Composed of twelve springs.

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A mar, shown in figure 7 (blue), hangs from a wire (0.7 m high) at the GAS filter. It contains two stepper motors with two CWs to position the payload in angular degrees of freedom. The Mar is made of aluminum with a diameter of 0.75 m and instrumented with eight coil-magnet actuators to control the pitch, roll, and yaw degrees of freedom of the cryogenic payload (CP); error signals are provided by two built-in in-vacuum optical levers measuring the orientation of the marionette with respect to the IPP platform.

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The CP, operated at cryogenic temperature, is suspended from the marionette via three wires (1.46 m high). As shown in figure 8, the payload contains on the CP and the large mirror (Mir). The CP is a cryogenic test bed for the cryogenic sensors used to detect motion at this stage (see 5 for more details on cryogenic sensors). The Mir is suspended from the CP with four silicon rods (0.4 m high) parallel to four cantilever springs (made of CuCrZr). These cantilever springs are part of the payload, not the seismic platform. Since the silicon rods are very stiff vertically, the metal springs are used to provide some vertical compliance to suspend the mirror. This results in a 13 Hz bounce mode of the final suspension stage. The silicon mirror has a diameter of 45 cm and a total weight of 100 kg. For safety reasons, the mirror is surrounded by a cage.

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As shown in figure 2 and mentioned in section 3, the cryostat is composed of two parts: the inner cryostat, suspended with the CP and the outer cryostat, including the thermal shrouds and supported by a structure placed on the optical bench of the vacuum chamber. Both cryostats are framed by interwined stack of fins, which are spaced in order to leave 2 cm gap above and below each fin when the inner and outer cryostats are in place (interlaced); this prevents any contact between fins and, consequently, the cryostat structure. In order to maximize the emissivity and the cooling efficiency, the fins must be coated with thick layer of black paint on both faces.

In order to predict the cooling efficiency of the setup, the whole cryostat was modelled with ESATAN-TMS software. The geometrical mathematical model is based on the CAD to compute the view-factors between the surfaces. One of the most critical parameter in this model is the emissivity of the black paint of the radiator fins and a worst-case approach was preferred to be conservative.

The thermal mathematical model is solved by taking into account both thermal conductivities and specific heat versus temperature. With a worst-case approach, the external cryostat assumes:

• The LN₂ panels (orange in figure 9) is supporting the GHe panels, where the external fins are connected.
• The three apertures in front of the mirror are assumed to be with 30 mm diameter through the LN₂ shield and GHe panels and are aligned such that only one internal fin is removed (keeping two consecutive external fins). In addition, these three holes are equipped with LN₂ shield and GHe tubes in order to minimize the incoming heat load in the system (figure 9).
• The three suspension apertures are designed with 50 mm diameter on the top of the cryostat for the titanium wired and cold harness, also equipped with LN₂ shield and GHe tubes protruding inside the cryostat to radiative enhance thermalization of the wires and harnesses.

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For the inner and suspended cryostat, the design includes the following features:

• Three titanium wires coming from the marionette are assumed in Ti-6Al-4 V, an alpha-beta Titanium alloy of 4 mm of diameter.
• 20 AWG24 phosphore bronze wires for sensors.
• 10 AWG32 phosphore bronze wires for temperature sensors (cooling down monitoring).
• 4 mono-crystalline silicon rods 13 mm in diameter with thinner 3 mm diameter end sections, supporting the mirror from the CP.
• The interfaces between these sapphire rods and the CuCrZr blades of the CP are improved with indium sheets.

The biggest uncertainty in this model is the emissivity of the black paint on the radiator fins. Regarding the schedule of the project and also the available budget, we have decide to use Aeroglaze Z307 as absorptive cryogenic black paint [47, 48]. Reduction uncertainties on the black paint emissivity requires specific cryogenic test not possible in this project. Consequently, the best-case/worst-case approach based on emissivity values of the black paint gives a range of possible temperature profile for the cooling of the mirror. Figure 10 shows the results of the thermal simulation assuming that the average temperature of the outer cryostat is at 20 K and for 70 m² of heat exchange area.

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Referring to the simulation, we can predict that the mirror temperature should be between 25 K and 40 K after 20 days, under vacuum conditions. The E-TEST project will be an experimental concept validation for the pure radiative cooling down method.

It is important to note that the cooling down concept presented here can be improved in order to reach lower temperatures to meet the requirements of the experiment or application. Rooms for improvements are offered by the following:

• The He temperature at the entrance of the shrouds could be as low as 4 K improving the cooling power.
• The size of the cryostat can be adapted and consequently the heat exchange surface.
• The emissivity of the black paint can be improved (by using another paint) improving the efficiency of the cryostat as presented in this section.

For the current generation GW detectors, a monolithic Watt's linkage with an interferometric readout was developed at Nikhef between 2014 and 2018 [40]. It achieved a displacement sensitivity of 8$\times10^{-15}$  m/$\surd$Hz from 30 Hz onwards and a modelled shot noise limited 2$\times10^{-15}$  m/$\surd$Hz above 10 Hz. ET will extend the detection band down to 3 Hz by, among other technologies, applying cryogenics. The current ET design features a physical link from cryocooler to the last stages of the mirror suspensions, possibly introducing unwanted vibrations near the mirror bypassing the isolation systems. The penultimate mass of the ET mirror suspensions extracts heat conductively from the 18  K mirror via its suspensions and is around 5 K.

To monitor the cryogenic penultimate stage down to 10⁻¹⁵ m/$\surd$Hz below 10 Hz, a new class of inertial sensors is needed. CSIS [49, 50]—an evolved design from the Nikhef accelerometer—aims to reduce its shot noise limited 10⁻¹⁵ m/$\surd$Hz sensitivity down to 1 Hz by lowering the thermal noise. All other readout and actuation noise can be designed to be below the thermal noise, which itself can be lowered by the reduced temperature and increased suspension quality factor Q. The sensor self-noise features a 50-fold reduction of thermal suspension noise over the previous design by increasing the mechanical Q factor. This is achieved by using niobium instead of high-strength aluminum as mechanics material and changing the actuator from coil-magnet to superconducting coil. The permanent magnet resulted in significant eddy current damping which was a dominant (viscous) damping mechanism in the previous design, reducing the Q from about 4000 to 40 [51]. The thermal noise, which determines the sensitivity at low frequency, is proportional to $1/ \surd Q$ [52] and thus, in this case, increased by a factor of 10. A schematic overview of the sensor is shown in figure 11. For a 1 kg-proof mass, the niobium Watt's linkage is $7.5\times7.5\times5$ cm³ in size.

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CSIS exploits the cryogenic temperatures by using superconducting thin film spiral coil actuators. The low-voltage superconducting coils exert magnetic pressure on the proof mass surface because of the Meissner effect. Because they are push actuators only, two actuators are needed per inertial sensor. For ET, the sensor mechanics and coils will be fabricated out of niobium. The temperature necessary for niobium, the sensor mechanics material, to be in a full Meissner state is about 5 K. The penultimate mass at around 5 K is therefore a suitable place to use the sensor for monitoring the effect of the low-vibration cryogenic systems and possibly contribute to the suspension control. The absence of magnets ensures that the suspension is structurally damped and we assume a Q factor of 10⁴. The noise budget in figure 12 shows the expected performance of the CSIS compared to the state-of-the-art, defined here as the most sensitive inertial sensor found in the GW field that is used in suspensions and on the market. At 1 Hz, the sensitivity is more than three orders of magnitude higher.

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Progress on the superconducting actuators is summarized in [55], where x-ray diffraction (XRD) measurements show that the right crystallographic phase for superconducting coils can be deposited. If fitted with higher $T_\textrm{c}$ superconducting coil and proof mass surface, the inertial sensor can operate at a temperature higher than 5 K. The CP temperature of the E-TEST prototype operates at 25 K and therefore we aim to use MgB₂ as superconductor. For E-TEST, we plan to install 3 CSISs on the CP exploiting the low-vibration cryogenic environment to perform a self-noise measurement using a huddle test [56].

In order to take full advantage of the inertial sensor performance and ensure its reliability at cryogenic temperature, the electronic part of CSIS will include a fully-custom chip fabricated in nanoscale CMOS technology, as shown in the schematic in figure 11. Many commercially available components, in fact, exhibit abnormal behavior at cryogenic temperature [57, 58] and, in some cases, turn out to be completely nonfunctional [59]. In contrast, modern nanoscale CMOS technologies are found to be fully functional and free of the anomalies that plague more mature technologies (such as kink effect and hysteresis) [60]. These modern technologies can therefore take full advantage of the benefits offered by the cryogenic environment, such as increased current driving capability [60, 61] and reduced leakage [62]. In addition to this, among all technologies operating at extremely low temperatures, nanoscale CMOS is the only one that can offer a very high level of integration and rely on over 60 years of industrial advances and optimization [63].

A first version of the chip will include the analog front-end, signal processing circuitry and the superconducting actuators current drivers. In its final version, analog-to-digital conversion will also be integrated so as to minimize connections between the cryogenic chamber and the external environment as much as possible. This leads immediately to a twofold advantage. On the one hand, the integrity of the signal is maintained and not compromised by the use of long cables often designed to prioritize thermal specifications over electrical ones. On the other hand, reducing the number of connections from the cryogenic chamber to the outside world allows to maintain a better thermal insulation and avoids the injection of unwanted thermal noise. Moreover, the full custom design allows us to tailor the design to the specific application while minimizing the sensor size and power consumption and, consequently, the heat to be dissipated. A final key advantage of a custom chip over commercial components, for applications operating in vacuum such as the one we are targeting, is the possibility to enclose the silicon die in packages with reduced outgassing [64].

However, the design of custom cryogenic CMOS circuits also presents significant challenges, first of all the absence of cryogenic models that can be used to simulate circuits in the design phase [60, 62, 65]. The design and subsequent fabrication of an integrated circuit is in fact an extremely complex process that takes considerable time (normally several months) and, therefore, it is necessary to ensure a high level of confidence in the design phase to guarantee the proper functionality of the fabricated circuit. For this reason, semiconductor foundries like TSMC and GlobalFoundries release design tools called 'Process Design Kits' that include highly reliable compact electrical models to be used in complex circuit simulation software, able to reproduce also the statistical variations intrinsic to the manufacturing process and temperature behavior. Unfortunately, the models provided by foundries are only reliable within a relatively small temperature range and cannot reproduce the behavior of devices at cryogenic temperatures [60, 62, 65, 66], which is considerably different from that at room temperature (see figure 13). For this reason, we chose to edit the foundry models in order to account for the cryogenic variations and make them usable for cryogenic design too [60, 67]. Test chips in different nanoscale technologies were fabricated to characterize active devices and integrated passive components at cryogenic temperature. A parameter extraction procedure using state-of-the-art tools and custom algorithms has also been developed to allow direct modification of the foundry models, so as to considerably reduce the difficulties inherent in this type of process, which normally require years even with the resources of a foundry. Preliminary cryogenic models are currently being used to design the first prototype of the chip that will be integrated in the sensor. The cryogenic modeling workflow developed as described here provides a fundamental step not only for CSIS, but for any cryogenic application using CMOS technology, such as quantum computing.

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Since the main prototype of the E-TEST will take several days to reach the targeted vacuum and temperature, a cryogenic test bench is used to test the cryogenic components. This will allow the components to be quickly tested, adjusted or repaired outside the chamber. The minimum temperature achieved was $T_{\textrm {min}} = 6.7 \,\text{K}$. The setup was capable of maintaining a minimum pressure level of $P_{\textrm {min}} = 5~\times~10^{-10} \,\text{mbar}$ during cooling. A typical cooling of the cryostat takes about $95 \,\text{min}$. After cooling, the temperature is stable within $\pm 0.2 \,\text{K}$. The fluctuations are caused by the cyclic nature of the cooling process. The setup is capable of achieving a temperature of $10 \,\text{K}$ with an additional heat load of $3.66 \,\text{W}$ which is well above the required design value of $1 \,\text{W}$ design value. More information on the design, cooling process, and operation, see this article [68].

To achieve the desired parameters, we plan to set up a MOPA (Master Oscillator, Power Amplifier) system, containing a crystal-based seed laser and fiber-based amplifiers. For wavelengths $\lt 2\, {\mu}$m, for example Tm-doped crystals can be used for the generation of the radiation. However, the drawbacks of these crystals are the inherent amplitude instabilities, which limit the output power stability [69, 70]. As an alternative to fulfill the desired power stability requirements, we plan to use Ho-doped crystals, which can be less sensitive to mechanical and atmospheric disturbances due to their higher gain cross section [71]. By using a Ho-doped crystal, for example a wavelength of 2090 nm can be realized.

For further power scaling, we aim to use holmium-doped fiber amplifiers, which can for example be pumped by thulium-doped fiber amplifiers at a wavelength of app. 1950 nm. This way, the quantum defect can be as low as app. $7\%$, which can result in an overall higher output power stability due to a lower impact of thermal effects within the fiber amplifier.

In the E-TEST project, a multi-stage MOPA will be developed to fulfill the laser requirements.

In most GW detectors, lasers at a wavelengths of 1064 nm are commonly used as beam sources. However, at a wavelength of 2090 nm, the development of lasers is not that advanced, which for example results in fiber components with typically worse performance parameters in comparison to 1064 nm fiber components. Within the E-TEST project, we aim to develop an innovative laser system with the following parameters:

• Wavelength: 2090 nm
• Power: 5–10 W
• Spectral linewidth: 1 kHz
• Relative intensity noise (RIN): ${\lt}$10⁻⁶ Hz$\sqrt{}$Hz $@$ 100 Hz
• Frequency noise: 100 Hz$\sqrt{}$Hz $@$ 100 Hz (assuming 1/f spectrum)
• Degree of linear polarization: ${\gt}$98%.

6.1.1. Laser design.

To achieve a wavelength of 2090 nm, holmium can be used as an active dopant in crystals and fibers. For the pumping of the holmium-doped crystals and fibers, thulium-doped lasers at a wavelength of 1950 nm. This can result in a very low quantum defect, providing a potentially low thermal-noise laser.

Therefore, we develop a solid-state laser based on Ho:YAG crystals as a seed laser, which defines the spectral properties. For the power amplification, a two-stage fiber amplifier based on holmium-doped fiber is developed. Figure 14 shows the scheme of the full laser concept. In addition to the mentioned seed laser (Seeder) and the two amplifier stages (Ho1 and Ho2), pump lasers for Ho1 and Ho2 have to be used and developed (Tm1 and Tm2). These pump lasers are thulium-doped fiber lasers at a wavelength of approximately 1950 nm. For the pumping of the crystalline Ho:YAG seeder, an earlier developed Tm:YLF laser will be used. The benefit of the multi-stage MOPA setup is its high adaptability. Depending on the output power of the seeder, one or two amplifier stages can be used. Within the development, different concepts are tested and compared to find the most practical and satisfying solution.

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To fulfill the high requirements on linewidth and stability, we aim to develop a non-planar ring oscillator (NPRO) seed laser. To achieve this, the development is divided in two phases. While state of the art NPROs at 1064 nm show satisfying results with respect to stability and linewidth, development at 2090 nm based on Ho:YAG crystals is still an ambitious approach. Therefore, in the first phase, a ring oscillator will be developed. Such a concept has a significantly higher success probability and the power amplification within a fiber resonator can be gathered. In the second phase, a compact and thermally stable NPRO will be setup and investigated to achieve the higher stability and lower linewidth.

Thanks to their compact design and achievable output powers, monolithic fiber amplifiers are a good technology to achieve a high stability at high power levels. The multi-stage fiber amplifier itself is designed to amplify low power signals of 1 mW, e.g. delivered by the NPRO, to more than 500 mW in the first stage, and to more than 10 W output power in the second stage. The fiber diameters are adapted to enable high amplification and mitigate stimulated Brillouin scattering. Since the required RIN is very ambitious, the output power of the NPRO seeder and all fiber amplifier stages will be actively controlled.

In order to provide the best performance, the optical coating must assure the lowest possible contribution to noise due to absorption. Many crystalline materials show lower optical absorption and mechanical loss than amorphous materials. However, the low and high reflective-index materials have to be lattice matched, and this limits the number of available options. Crystalline coatings made of AlGaAs and GaAs are under investigation and show promising performance [72]. The requirement of a lattice matched substrate—GaAs wafers in case of AlGaAs coatings—for the growth procedure requires a transfer of the coating to a suitable substrate after production and also can limit the maximum coating size (to about 20 cm in case of GaAs). Defects in the bond for attaching the coating to the new substrate can lead to the coating detaching during the test-mass cooling procedure or to thermal noise from a change in bulk and shear loss.

Part of the approved E-Test budget proposal is dedicated to the construction of a novel molecular beam epitaxy (MBE) coating facility to investigate new crystalline coatings for application in future GW detectors [73]. Crystalline oxide coatings will be developed with the MBE coating facility. The layers are likely to be more sensitive to the differential thermal lattice expansion between the mirror body and the coating in comparison to amorphous coatings. Therefore it will be important to test whether the crystalline coatings survive well to repeated heating and cooling cycles without the appearance of cracks. Furthermore, in all likelihood, the crystalline oxide coating will need to be bonded to the mirror body and this bond might be a fragile place where the wafer could detach with repeated temperature cycling. Another topic of investigation is whether the optical properties drift as a function of time with continued long-term exposure to ultrahigh vacuum and low temperatures and whether a protection layer will be needed.

6.2.1. MBE coating.

An important active part of the interferometer is the light detector. The photo detector detects photons exiting the interferometer at the dark fringe and thus it is susceptible to the photon number variations caused by the interaction of the GW with the interferometer. The infrared wavelength of 2 µm is one of the main innovation of the E-TEST development as it couples to requirements imposed by the low temperature operated Si mirrors and auxiliary optics. This allows for a significant reduction of the vibration noises at temperatures 5–20 K. To match the laser wavelength, a detector with an optimal spectral dependence and quantum efficiency needs to be chosen.

6.3.1. Proposed procedure.

The induced vibrations and wavefront or topology change are two issues that may appear consequence of cooling down the suspended mirror. In the framework of the ET, a mirror characterization is essential to ensure the accuracy of the results in the detection of GWs. We develop a metrology instrument for the E-TEST project to characterize the cryogenic mirrors for a GW detector in operation. The metrology instrument measures the local vibration as well as the topology change of the suspended silicon mirror at cryogenic temperatures. The instrument is a low-coherence interferometer that does not use movable components to avoid additional unwanted vibrations. It includes an innovative optical phase mask to capture on a single camera frame a set of interferogram maps of the same area observed on the test mass at different optical path differences. The optical phase mask works in transmission. It has a stair shape made of steps of different optical thicknesses of the same material. Each step on the optical phase mask simulates a different position of a conventional piezoelectric (PZT) device for surface scanning [76]. The envisaged first optical phase mask has 49 steps in a 7-step by 7-step square shape, equivalent to 49 interferogram maps of the area observed on the test mass. The spot size of the beam incident on the test mass has a diameter of 1 cm. A schematic representation of the metrology instrument, based on the optical layout of a Michelson Interferometer, and the phase mask, is included at the left of figure 15. Another critical component of the instrument is a microlens array of 49 lenses in front of the detector to image the information of each step.

We use a low-coherence source to determine the mirror location unambiguously and, therefore, the local vibration and topology. The interference pattern of a low-coherent white light source has a well-differentiated central fringe visible only when the optical paths of the two arms of the interferometer are almost exact. This region of fringe visibility, in our case around 1 µm width, is determined by the coherence length of the source. The difficulty in obtaining the interference fringes with a low-coherence source is an advantage to avoiding spurious unwanted interferometry signals from scattered rays, speckle, dust, and imperfections [77].

Each step of the optical phase mask we design will capture a point of the interference pattern. Then we reconstruct the full interference pattern. We do this at multiple local points inside the area observed at the test mass. The interferogram map of each step is 30 px by 30 px. Therefore, we have the information on local vibration and topography of 900 different local points inside the area observed of the test mass, which is around 1 cm in diameter. Due to the complexity of the optical phase mask, we expect defects and dissimilarities between steps besides test mass deformations. Due to these two factors, we measure unequally spaced sample points on the white light interferograms. To solve this, we develop new processing algorithms for the interference patterns to reach a vertical resolution of a few nanometers. [78] includes a detailed explanation of measuring vibration and topology change, besides an introduction to the analysis algorithms we develop.

We use the metrology instrument to measure low-frequency vibrations around 10 Hz. We develop subtraction methods and use signal modulation to minimize the effect of the noise from the optical table and other components on the measurements. For the deformation measurements, the lateral resolution will be a few micrometers, according to the pixel size of the detector. A prototype of the metrology instrument with an optical phase mask of 25 steps is currently under development.

Due to its high refractive index of around 3.5 at a wavelength of 2090 nm, the boundary between silicon and air reflects around 30$\%$ of light. The two surfaces of the silicon test mass form a Fabry–Pérot etalon, where light bounces back and forth. The individual light fields interfere constructively or destructively, depending on the optical path for one round-trip. As the silicon mirror is cooled down, this optical path changes due to two effects: an expansion of the material due to temperature, described by the thermo-elastic coefficient α = dL/dT, and a change in refractive index, described by the thermo-refractive coefficient β = dn/dT. Together, they form the thermo-optic coefficient of the silicon material. This thermo-optic coefficient can be measured by monitoring the interference fringes of the etalon, together with the substrate temperature [79]. Tracking the fringe pattern to measure the thermo-optic coefficients, was performed in [79]. We plan to do it in a similar way: we will start with the reconstruction and fitting of the full fringe data at room temperature and analyze the measured fringes as they pass through using the temperature-dependent coefficients. These can be compared to and correlated with conventional temperature sensor data, to further calibrate and validate the measurement, as the temperatures measured conventionally at the edges of the substrate and the middle of the substrate will not be vastly different, and especially will converge after long cooling times. Knowing the thermo-optic coefficient, the observed interference fringes can be used to obtain the temperature evolution of the silicon substrate, which allows for a contactless measurement and monitoring of the silicon temperature at the location of the laser beam. This is especially relevant for E-TEST, where we are interested in the temperature gradient during cool-down, but at the same time we cannot place conventional temperature sensors all over the test mass without impacting the performance of the seismic isolation, and more importantly, destroying the optical and mechanical quality of the mirror itself. This is why we adopted a non-contact methodology, using a µm laser source orthogonally shining the mirror surface: the beam will interfere with itself when it is reflected back and recorder via a photodetector.

Another way of probing the quality of the mirror substrate is by exciting its mechanical internal modes and see how much energy they dissipate per cycle of the kHz mode. A tiny hammer, i.e. a small mass with a spring element that can be made with e.g. a coil magnet actuator to tap the test mass ever so slightly, can excite all modes that do not have a node at the position of hammer impact. Once the modes are excited and ringing down it is up to an optical measurement to determine the envelop of the exponentially decaying ring-down.

The optical setup to do perform such measurement is depicted in figure 15(c). An interferometer uses polarizing optics to divert the light of both arms to two photodiodes [80]. The error signal is generated by subtracting the two out-of-phase photodiode output signals. This ensures all common mode noises—typically laser noises such as amplitude fluctuation—are decreased to shot noise level. The error signal is used to make the mirror attached to a piezo actuator follow to probed mirror point, such that the signal sent to the piezo actuator will show a ring down indicative of the mirror mechanical loss. Such interferometric readout systems can reach 4 fm/$\surd$Hz sensitivity [51] which is sufficient as the hammer can induce large enough internal-mode-associated mirror surface motion.

A fiducial loop design is shown in figure 16(a) with a bandwidth of 13 kHz allowing for a proper measurement of the first ten internal modes of the 100 kg, 45 cm diameter silicon mirror. The signal sent to the piezo actuator holds a superposition of all the different modes, which can be isolated using a bandpass filter. By finite element analysis, such as shown in figure 16(b), the frequency of each mode can be compared and the mode shape identified. This will also show which part of the mirror substrate (face) will move with a large enough amplitude for our sensitivity and an estimation of the expected signal strength can be made. The laser probe position on the mirror front surface can be varied by using the three different access tubes through the cryostat.

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