Mark Chun

Christopher Baranec

Olivier Lai

Jessica R Lu

Ruihan Zhang

Stefan Kuiper

Wouter Jonker

Matthew Maniscalco

A new adaptive secondary mirror for astronomy on the University of Hawaii 2.2-meter Telescope

Keywords:

We are developing a new adaptive secondary mirror (ASM) for the University of Hawaii 2.2-meter telescope based on a novel and very efficient hybrid variable reluctance actuator developed by TNO. The actuator technology has broad implications on the ASM design and results in an ASM with a thicker facesheet, lower power dissipation, and simple controls. We report here preparations and plans for lab testing as well as on-sky demonstration of the ASM. The lab calibrations of the ASM influence functions will use a phase measuring deflectometry setup. The on-sky tests will include the evaluation of the use of the ASM for narrow field AO observations at visible through near infrared wavelengths, for very wide fields of view ground-layer adaptive optics, and for seeing limited non-adaptive optics observations.

INTRODUCTION

Adaptive secondary mirrors (ASM) greatly enhance two key observational capabilities for ground-based adaptive optic systems: low-thermal emissivity background observations and wide-field of view seeing-improved observations. Telescopes currently with ASMs include the MMT [START_REF] Lloyd-Hart | Lessons learned from the first adaptive secondary mirror[END_REF] , LBT [START_REF] Christou | Living with adaptive secondary mirrors 365/7/24[END_REF] , Magellan [START_REF] Close | Status of MagAO and review of astronomical science with visible light adaptive optics[END_REF] , and recently VLT [START_REF] Madec | Adaptive Optics Facility: from an amazing present to a brilliant future[END_REF] . The current ASMs have relatively large actuator densities so high levels of wavefront correction can be obtained. The principal downside of the current ASMs is that their use of a voice-coil actuator adds complexity to the systems making it difficult for them to meet the very stringent reliability requirements demanded of the telescope optics.

Here, we present the status and plans for testing the new adaptive secondary mirror we're building for the University of Hawaii 2.2-meter telescope (UH88). The ASM project is an on-sky demonstrator for an actuator technology developed by TNO and that we believe is an ideal match to adaptive secondaries. The goal of the project is to demonstrate the technology's suitability for astronomical adaptive optics in various AO modes (single-conjugate adaptive optics and wide-field ground-layer adaptive optics), and as a fixed secondary for seeing limited observations. The project is a collaboration between TNO, the University of Hawaii's Institute for Astronomy, and our industrial partners VDL and Hyperion.

The project is centered about TNO's hybrid variable reluctance actuators [START_REF] Kuiper | Advances of TNO's electromagnetic deformable mirror development[END_REF] . The key features of these actuators are that they produce a large force output for a given current making them capable of producing large strokes in very efficient packaging volumes. TNO has demonstrated the actuators in a 160mm-diameter deformable mirror with an 18mm actuator pitch. Within the linear regime of these actuators (+/-150mA), the force output of the actuators is about +/-8N. This is about an order of magnitude more force typical of voice-coil actuators used in current adaptive secondaries.

The efficiency and output force of these actuators make them well suited to ASMs and lead to important improvements in the optical facesheet and the controls of the ASM compared to current adaptive secondaries. The optical facesheets in current ASM are thin (~1.6mm) due to the lower force output of voice-coil actuators. This makes them fragile to handle and difficult to fabricate. The higher force output of the TNO actuators allow the optical face sheet to be thicker. In the case of the UH88 ASM the facesheet will be ~3.5mm thick, similar to a car windshield thickness, and still provide many tens of microns of stroke. This makes for an optical facesheet that is easier and safer to handle. The large stroke provided *markchun@hawaii.edu; phone 1 808 932-2317; by the actuators also eases the fabrication requirements on the optical facesheet. Rather than grind a large piece of glass down to the final thickness and shape, the facesheet for the UH88 ASM will be slumped on a master mold and MHD polished to remove high-frequency residual errors. We believe this process has a significant cost savings and is scalable to large aperture ASMs. The higher force output is expected to lead to a simpler control system. The TNO actuator+facesheet combination in the UH88 ASM design is mechanically stiff due to the thickness of the facesheet, the inherent stiffness of the actuators, and the fact that the actuators are mechanically connected to the facesheet. This is in contrast with current adaptive secondaries where the thin facesheet magnetically floats over the reference body and the mechanical stiffness is achieved by adding an additional capacitive position sensors/servos that control the position of the actuator+facesheet. This position servo loop is needed to translate the lower force output of the voice coil actuators into a position displacement but produces a system that goes-to and maintains a commanded shape. This makes it effectively a stiff system. The cost in doing this is added system complexity with the sensors, servo electronics, thin gap between the facesheet and optical reference body, and a considerable heat load at the top end of the telescope due to the servo electronics. The TNO actuator/ASM avoids the use of this extra control system and its complexity and power dissipation. For comparison, the ASM and drive electronics for the UH88 ASM will dissipate about 20-30W. This is more than an order of magnitude less than that for a similarly sized ASM based on voice coil actuators. In addition most of this dissipation is in the drive electronics which does not need to be colocated with the ASM. Finally, the greater force output of the TNO actuators provides flexibility in the design of the ASM. In particular the force output is large enough that the spacing of the actuators can be set by the requirements of the desired adaptive optics correction rather than the need to be spaced to mechanically support a thin optical facesheet from deformations due to gravity. The goal of the UH88 ASM project is to build an ASM around the TNO technology and demonstrate its performance and operational readiness in a number of AO modes and for seeing limited observations. The ASM will be deployed on the University of Hawaii 2.2-meter telescope (Figure 1). While this telescope just celebrated its 50th year of operation, it has a number of features that make this a nice match for the development of an ASM. The telescope is on Mauna Kea, Hawaii next to 4-meter and 8-/10-meter class telescopes. The UH 2.2-meter telescope is operated by the Institute for Astronomy and we can dedicate, and historically have dedicated, on-sky time to astronomical technology development.

BASIC REQUIREMENTS OF THE UH88 ASM

The telescope is home to two adaptive optics systems: Robo-AO [START_REF] Baranec | High-efficiency Autonomous Laser Adaptive Optics[END_REF] and soon Robo-AO-2 [START_REF] Baranec | The Robo-AO-2 facility for rapid visible/near-infrared AO imaging and the demonstration of hybrid techniques[END_REF] , a Rayleigh laser guide star AO system with both visible and near-infrared science cameras, and `imaka 8 , a ground-layer testbed/demonstrator with 3-5 NGSs and a visible-light CCD imager covering 11×11 arcminute FOV in the center. In addition the telescope has two seeing limited instruments -an integral field visible light spectrograph and a wide-field imager (0.25 °×0.25°). The telescope's equatorial mount simplifies the ASM control since the orientation of the ASM is fixed in the WFS(s).

In order to feed all of these instruments the ASM will replicate the currently used f/10 secondary. The resulting ASM is relatively large for a telescope of this size (620mm in diameter) but this provides a large field of view at the Cassegrain focal plane that allows us to test very wide field GLAO with an unvignetted field of view of 0.5 degrees.

The ASM will interface with the existing secondary interface plate. The telescope has two secondaries mounted at all times and the top end can "flip" to put either into the beam (Figure 1). For most of the first 50 years, the two secondaries were (1) for the Cassegrain focus (f/10) and ( 2) for the Coude feed. The Coude feed is no longer used so it will be replaced by an ASM that optically replicates the f/10 secondary. We can then "flip" between the fixed and adaptive secondaries in approximately 30 minutes. This gives us flexibility in when/how we use the ASM. It also keeps the ASM continuously exposed to the observatory environment allowing us to test the long-term operability and reliability in such conditions. While this approach has obvious advantages, it does have an important design constraint. The volume, location, and mass of the ASM must be similar to the replaced secondary to maintain the balance of the top end and to put the secondary within the range of the existing focusing mechanism. The volume requirement has an important impact on the dynamic behavior of the ASM that we will discuss in the next section.

The AO specifications of the ASM are driven by the desire to integrate it with the Robo-AO-2 wavefront sensors (WFS) and control system and to provide correction at visible and near-infrared wavelengths. The ASM's 210 actuators in a circular arrangement is well matched to the 16×16 subaperture Shack-Hartmann WFSs in the Robo-AO-2 system and the dynamic response of the ASM will be consistent with a 0-dB closed loop bandwidth of about 100Hz. This should give good performance into the red end of the visible on a 2.2-meter telescope. The stroke requirements for the ASM are driven more by the desire to have sufficient range to accommodate any residual loworder aberrations in the fabrication of its slumped facesheet. Nominally we specified this to be +/-15 microns and +/-4 microns inter-actuator stroke.

DESIGN AND STATUS

Several other contributions in these proceedings cover the design and current fabrication status [START_REF] Wouter | Design and manufacturing status of the UH-88 adaptive secondary mirror[END_REF] , and performance estimates [START_REF] Kuiper | Performance Analysis of an adaptive secondary mirror for the University of Hawaii 2.2-meter Telescope[END_REF] . Here we highlight a few of items as they apply to the suite of tests we plan for the system. For more details we refer you to Jonker et al. [START_REF] Wouter | Design and manufacturing status of the UH-88 adaptive secondary mirror[END_REF] and Kuiper et al. [START_REF] Kuiper | Performance Analysis of an adaptive secondary mirror for the University of Hawaii 2.2-meter Telescope[END_REF] The key high-level design features of the UH88 ASM are:  A slumped facesheet 3.5mm thick  Struts that connect the face sheet to the actuators  An aluminum-alloy support structure that holds the actuators and provides the mechanical base.

 210 actuators in a radial arrangement with a 39 mm pitch  An update rate of 2kHz and a full assembly response consistent with an AO system 0dB bandwidth of 100Hz.

 An interface to the existing telescope interface plate.

TNO has modeled the performance of the system [START_REF] Kuiper | Performance Analysis of an adaptive secondary mirror for the University of Hawaii 2.2-meter Telescope[END_REF] . Some key aspects of the modeling include a detailed analysis of the surface figure errors due to changes in the telescope orientation, changes in temperature, epoxy print-thru, and the actuator stroke/stiffness. TNO has done breadboarding/prototyping has been done to ground the results [START_REF] Wouter | Design and manufacturing status of the UH-88 adaptive secondary mirror[END_REF] . The dynamic behavior of the system has been modeled both for the ASM and for the telescope top-end. The lowest resonant mode of the system is due to flexure of the ASM structure/body and is a tip/tilt mode at about 160Hz. This is a relatively low frequency but (1) its well above the bandwidth we need for tip/tilt correction on the 88" telescope and (2) we believe that we can suppress the resonance of this particular mode with relatively simple control algorithms. In particular we plan to control the tip/tilt modes with a separate control gain than the higher order modes [START_REF] Kuiper | Performance Analysis of an adaptive secondary mirror for the University of Hawaii 2.2-meter Telescope[END_REF] . We note that this resonant frequency is largely set by the design of the ASM support structure and this is constrained by the requirement that the UH88 ASM fit within the existing volume of the fixed secondary it replaces. Relaxing this requirement and making the ASM structure/body thicker or from a stiffer material would move this resonance to higher frequencies so the low resonance seen in the UH88 ASM is not fundamental to the approach. The first resonance of the actuators themselves is around a 1kHz.

LAB TESTING

In this section we summarize the suite of tests we plan to do with the ASM. We will focus here on the tests with the fully integrated ASM in the lab and on the telescope and without the AO systems. Other contributions in this conference discuss tests characterizing the TNO actuators in a 60-actuator flat deformable mirror [START_REF] Bowens-Rubin | Performance of Large Format-Deformable Mirror Constructed with TNO Variable-Reluctance Actuators[END_REF] and plans to test a new lab prototype DM based with 19 of the TNO UH88-style actuators [START_REF] Wouter | Design and manufacturing status of the UH-88 adaptive secondary mirror[END_REF][START_REF] Bowens-Rubin | Performance of Large Format-Deformable Mirror Constructed with TNO Variable-Reluctance Actuators[END_REF] with a 39 mm pitch and a flat facesheet 3.5mm thick. These DMs serve as prototypes for the technology and importantly test the fabrication processes.

The first tests of the UH88 ASM assembly will be done at TNO using a set of six capacitive sensors to test basic functionality and provide a first look at the performance of the assembly. The sensors measure the change in position of the facesheet with respect to the ASM body/structure. In this way they are similar to the capacitive sensors in current ASMs except that we are using the sensors to sense differential motion and not as an absolute position reference. To provide flexibility on what region of the ASM is tested, the ASM body will have a series of machined openings into which we can place one of the sensors.

The capacitive sensors provide good temporal resolution (~10kHz), sub-nm vertical resolution and sparse but sufficient spatial resolution across the surface. Measurements from the sensors can be combined with our finite-element models of the ASM to provide a good first test of the system. In addition to basic functional tests, we can use these to test the basic "static" performance of the system: actuator stroke, repeatability tests, linearity, hysteresis, and inter-actuator coupling (via models). In addition, in combination with our FEA models and accelerometers, we may be able measure the resonant mode frequencies of the ASM. Importantly, the capacitive sensors provide an independent calibration for later tests. In particular the they ground the measurements of the zonal and modal influence functions albeit at a limited number of points about the mirror.

The full optical tests of the ASM normally require a large complex optical setup. These setups serve two functions. They calibrate the static (and possibly dynamic) response of the ASM and they allow the team to integrate and test the ASM with the rest of the AO components (e.g. wavefront sensor(s) and real-time controller). Each of these are important steps that we need to perform with the UH88 ASM. However, such test setups are complex and expensive [START_REF] Stalcup | A test stand for the MMT Observatory adaptive secondary[END_REF] . For this project we have decided to split the measurement of the static performance of the system (e.g. the zonal/modal influence functions) and the dynamic/AO system integration tests into two separate test setups.

The ASM influence functions (both zonal and modal) require high-spatial resolution measurements that preferably cover the entire surface in each measurement. Both the zonal and modal influence functions are needed in the control of the ASM (e.g. pseudo synthetic interaction matrices and modal control/gains). However, testing the large convex surface of the ASM in one metrology setup presents a challenge. Typical approaches such as a Hindle sphere test or a transmissive variant using a lens [START_REF] Stalcup | A test stand for the MMT Observatory adaptive secondary[END_REF] require a large optical element of at least the size of the ASM to test the full surface of the mirror. In our case the Hindle sphere test requires a spherical mirror several meters in diameter. For this project such an optical setup is not possible. Instead to measure the influence functions we'll use an approach called phase measuring deflectometry (PMD). Others have used this for optical metrology [START_REF] Kolb | Laboratory results from the AOF system testing[END_REF] and the principle is commonly used in commercial "structured light" 3D scanners. The basic idea is shown in Figure 2 below. A camera looks at the reflection of a screen off the mirror being tested. The approach effectively traces rays from the screen to the mirror and to the camera. These ray paths depend on the geometry of the setup and the local gradients across the surface of the mirror. We know or can calibrate the geometry of the setup so the gradient of the surface can be derived. These slopes can then be integrated to deduce the surface shape.

The bookkeeping of the raytracing is done via a judicious choice of the pattern(s) displayed on the screen. A series of two-dimensional sinusoidal patterns with known phase shifts between images in the sequence effectively encode the positions on the screen so they can be tracked to positions on the mirror in the distorted images captured by the camera. This is analogous to the reduction of fringe patterns from interferometers and techniques to unwrap 2-pi ambiguities in the analysis are commonly applied. However, unlike an interferometric fringe pattern, we can display any set of fringes we like (and even different colors) on the screen. Doing so allows us to measure both the x-and y-slopes with a single pattern. This makes the data acquisition fast though it results in a very high resolution gradient map (e.g. thousands of elements across the mirror). Turning these into a surface shape requires a large matrix multiplication. The beauty of the PMD approach is that the required hardware elements are very simple and inexpensive. For the UH88 ASM to cover the entire surface of the mirror in one measurement we need a camera and two ~ 70-inch 4k LCD monitors. This is 2-3 order of magnitude reduction in the cost of the hardware compared to traditional approaches and scales to larger ASMs. The approach was used an LBT ASM [START_REF] Kolb | Laboratory results from the AOF system testing[END_REF] with results that compare favorably to interferometric approaches.

The challenge with deflectometry is that the position of the camera, mirror, and screens must be known to high precision. Errors in these positions can not be distinguished with low-order aberrations on the surface of the mirror. However, the measurement of the zonal and modal influence functions is inherently differential so as long as the calibration of the setup is stable during the measurements (e.g. poke+ and poke-) then these errors are do not contribute to the estimate of the influence functions.

The deflectometry setup is a lab only test but importantly provides high-resolution full aperture measurements. With it we will repeat many of the basic functionality tests done with the capacitive sensors and also obtain the influence functions. Here the capacitive sensors provide independent point measures of the influence to which we can compare the deflectometry measurements.

Note that we can not perform any measurements of the ASM resonant modes as these are at too a high a temporal frequency for the deflectometry measurements. To do these (and to integrate the ASM with the AO components) we need a second setup. For these tests we will deploy a transmissive lens variant of the Hindle sphere test similar to that done by Stalcup et al for the MMT ASM [START_REF] Stalcup | A test stand for the MMT Observatory adaptive secondary[END_REF] . For budget reasons we only make these measurements over a portion of the mirror at one time. Figure 3 above shows the approach. A meniscus lens (MHS) smaller than the diameter of the ASM is placed in front of the ASM and acts to divert the beam from an illuminator, placed at the focus of the ASM's hyperboloid, onto the ASM. The back surface the meniscus lens acts as the Hindle sphere and reflects rays bouncing off the ASM back to the front focus. At this location we place our wavefront sensor. With this setup we can not test all actuators at once but we can close the loop on a portion of the mirror in the lab. The MHS lens can be moved to other locations to test other actuators/portions of the ASM. The setup has the additional advantage that it can be used in the lab and when the ASM is installed on the telescope. The illuminator and wavefront sensor will be mounted at the telescope focus and the MHS lens can be installed when the ASM is on the telescope.

ON-SKY TESTING

The rapid reconfiguring of the UH 2.2-meter telescope to put the ASM into the optical beam and the fact that we can obtain significant time on sky to test the ASM allows us to step thru the on-sky tests in a simple phased approach. In the first phase we test the functionality of the system during the day using the MHS lens setup and then transition to SCAO on a bright natural guide star. The instrument setup is shown on the left in Figure 4. For the daytime tests the Hindle sphere test lens is put in place and at the focus of the telescope we mount the integration WFS, a small-pixel CMOS camera, and the illuminator. This allows us to do simple environmental tests with the ASM (gravity, temperature). We'll also go to the sky for initial tests with this same basic setup, without the Hindle sphere lens to do SCAO on a bright natural guide star. We will start the on-sky tests with this setup since it does not require integration with the RoboAO or `imaka systems. However, the imaka RTC will be used for the real-time control of the ASM.

The integration WFS and camera also allow us to do seeing limited tests of the ASM. In particular we will quantify how the image quality of the telescope is maintained via a look up table or a simple low-order wavefront sensor (for example a modified telescope guider) that drives the ASM. These tests will mostly be done with the integration wavefront sensor/ camera but we can also feed the optical beam to one of our seeing limited instruments to measure image quality over a wider field of view (albeit with less precision on the image quality).

More complete SCAO tests begins when we integrate the ASM with the Robo-AO-2 system. The ASM is sized to be compatible with the Robo-AO-2 natural guide star and laser guide star wavefront sensors and we will modify the control system to send the wavefront correction to either its internal DM or the ASM. This allows us to do a side-by-side comparison of the AO correction by the ASM and by the DM in Robo-AO-2 at visible and near-infrared wavelengths. This will be the first test of the system at longer wavelengths -but note the configuration does not test the thermal emissivity gains of the ASM since RoboAO's optical relay is warm.

Finally, we will use the ASM in a wide-field GLAO configuration. The simplest implementation is to modify the existing `imaka GLAO system to drive the ASM but still use its `imaka optical relay. This provides an up to 0.3×0.4 deg field of view for the five NGS WFSs. However, for these tests we plan to remove the imaka optical relay and place the imaka WFSs and science camera at the f/10 telescope focus. This gives a 0.5-deg (unvignetted) FOV for the WFS GSs and a science camera that subtends a 0.25×0.25 degrees.

CONCLUSIONS

In the coming year we will complete the fabrication of the UH88 ASM and begin testing of the system in Hawaii. We have planned a long road of activities with the system from classical single-conjugate narrow field AO observations in the visible thru near infrared wavelengths, to half-degree field of view ground-layer adaptive optics, and for seeing limited non-adaptive optics observations.

Figure 1 .

 1 Figure 1. (LEFT) The University of Hawaii 2.2-meter telescope (LEFT) shown with the Robo-AO system mounted to the North bent-Cassegrain port. The Robo-AO Rayleigh laser is the long black rectangular box seen midway up the side of the telescope tube. (RIGHT) The top-end of the telescope showing its two secondary mirrors. The secondary currently "in the beam" is the f/ 10 fixed secondary. The secondary out of the beam (pointed away from the telescope tube) is the no longer used Coude secondary. The UH88 ASM will replace this Coude secondary. The top-end can rotate (~30min) to place either secondary in the beam.

Figure 2 .

 2 Figure 2. The geometry of the deflectometry setup with the camera positioned along the optical axis of the mirror.

Figure 3 .

 3 Figure 3. A transmissive Hindle-sphere test setup for the UH88 ASM.

Figure 4 .

 4 Figure 4. Various configurations of the UH88 ASM from Left to Right: Phase 1 -with the integration WFS and camera, Phase 2 -in seeing-limited mode, Phase 3 -with Robo-AO-2 in SCAO mode with NGS and LGS, and Phase 4 -with imaka in GLAO mode.

ACKNOWLEDGMENTS

The UH88 ASM work is funded by the National Science Foundation (AST-1910552).


S Kuiper

W A Jonker

M Maniscalco

E Van De Ven

Robbert Voorhoeve

M Chun

O Lai

Performance analysis of the adaptive secondary mirror for the UH2.2 telescope

Keywords:

TNO and industrial partners are developing a new type of adaptive secondary mirrors (ASM) for the University of Hawaii 2.2-meter telescope, consisting of 210 actuators, in an overall volume of ø63cm by 15cm height, and having an aspherical convex mirror-shell of 3,5mm thickness. The novel actuator technology enables a compact system without active cooling that can be retro-fitted within the same mass and volume of an existing passive secondary mirror. This development enables affordable and reliable ASM systems for the world's larger telescopes as well as the many telescopes in the 2-4 meter class. This paper presents the overall design of this ASM and focusses on the performance analysis regarding its figure quality, its dynamical behavior and the related closed loop performances.

INTRODUCTION

Several observatories are exploring the use of Adaptive Secondary Mirrors (ASM) to enable effective aberration correction over wide fields of view via Ground Layer Adaptive Optics (GLAO). Such ASM's are highly complex systems due to the size, high number of actuators (hundreds to thousands), and the (aspherical) convex or concave optical surfaces. Furthermore, as these ASM's are an integral part of the telescope located atop of the spider structure, they need to be highly reliable, compact and able to cope with exposure to the changing environment.

TNO and industrial partners VDL ETG and Hyperion are developing a new type of Adaptive Secondary Mirror (ASM), based on an in-house developed variable reluctance actuator [START_REF] Kuiper | Advances of TNO's electromagnetic deformable mirror development[END_REF][START_REF] Bowens-Rubin | Performance of Large Format-Deformable Mirror Constructed with TNO Variable-Reluctance Actuators[END_REF]. These actuators have a high efficiency, enabling a relatively compact overall system without active cooling that can be retro-fitted within the same mass and volume of existing passive secondary mirrors (M2). Due to the overall low complexity of this technology it is inherently reliable, which is vital to guarantee the required uptime of the telescope.

As a first project to demonstrate this technology in an observatory setting, an ASM is being developed for the University of Hawaii's 2.2-meter (UH2.2) telescope in close collaboration with the University of Hawaii. The overall goal of this project is to demonstrate this ASM technology for GLAO on Mauna Kea. The UH2.2 telescope is considered an almost ideal test-site for this new technology, given its moderate telescope size, its location close to other big telescopes, and the fact that it already has substantial AO-infrastructure into which this ASM can be integrated. Furthermore, the UH2.2 has a rotatable spider structure with two secondary mirrors, which allows the ASM to be installed on one of the positions (replacing the obsolete Coudé-focus secondary) while maintaining the availability of the original passive secondary mirror. It is expected that once successful, this development will open up new opportunities for affordable and reliable ASM's for the world's larger telescopes, as well as many telescopes in the 2-4-meter class. This paper focusses on the performance analysis that has been performed during the critical design phase of the ASM development, with regard to the wavefront error performance, the dynamical performance, and the impact of an actuator failure. For more in-depth discussion about the mechanical design and MAI-status the reader is referred to [START_REF] Jonker | Design and MAIT Status of the UH2.2 Adaptive secondary Mirror[END_REF], while [START_REF] Chun | A new adaptive secondary for astronomy on the University of Hawaii 2.2-meter telescope[END_REF] discusses the test-strategy and test-preparations for this ASM.

ASM FINAL DESIGN

Figure 1 shows a CAD rendering of the final design of this novel ASM as mounted on the rotatable spider structure of the UH2.2 telescope. This ASM contains 210 actuators in a circular grid with a radial spacing of 39mm. These actuators are based on the variable reluctance principle and have a high efficiency in terms of force per unit volume, and force per unit power. The overall volume of the ASM is Ø63x13,5cm which fits within the volume of the original passive M2. The overall mass of the ASM is 55kg, which is around 15kg lower than the original passive M2.

Figure 2 shows a cross-sectional view of this ASM, with the actuators fitted at the bottom of a support structure and mechanical struts protruding through the support structure which transmit the actuator forces to the face sheet (or thin mirror shell). This mirror shell as a mechanical diameter of Ø62cm, and a thickness of 3,5mm. These actuators provide ±8N within its linear range (99.5%), which via an internal mechanical lever gets amplified to ±22N on the strut that interfaces with the face sheet. Given the stiffness of the actuators and the facesheet this enables a linear displacement range of around 35µm PV when all the actuators are moving together, and 4µm PV inter-actuator. Recent test results on these actuators on an earlier build-standard DM are discussed in [START_REF] Bowens-Rubin | Performance of Large Format-Deformable Mirror Constructed with TNO Variable-Reluctance Actuators[END_REF].

Next to these actuators' provisions are made for capacitive sensors to measure the relative displacement of the face sheet with respect to the support frame during operation. These capacitive sensors are foreseen to be used in early stages of the ASM commissioning to verify the overall actuator performances. The actuators are driven by high efficiency Pulse-Width-Modulated (PWM) current amplifiers that are stowed in a 15x15x50cm electronics box in the shadow of the ASM.

Due to the high efficiency of the actuators, the power dissipation at the ASM itself is estimated to be below 3Watts, while the electronics drivers will dissipate around 20Watts. This low overall power dissipation allows the ASM to be operated without additional cooling which significantly simplifies the overall system and adds to the robustness.

SURFACE FORM ERROR PERFORMANCE ANALYSIS

The main performance specification of the ASM is the achievable wave front correction performance. In full Adaptive Optics (AO) operation the 'best-flat' performance target is set to λ/15 Wave Front Error (WFE @630nm), which corresponds to ~21nm rms Surface Form Error (SFE). This performance needs to be achieved over the entire operational temperature range from +20°C down to -10°C, and with a changing gravity vector when the whole telescope is moving in elevation. Besides the full AO-mode, it is also important that the ASM can be operated in a 'Static-mode' where it maintains a static figure with reasonable accuracy. This is needed during first initialization, where the initial aberrations must fall well within the detection range of the wave-front sensor. Furthermore, the ASM is also required to maintain a high-quality static figure during seeing-limited observations without high-bandwidth AO-corrections. For such static mode operation, a target WFE requirement is set at λ/4 (corresponding to 80nm rms SFE) over its entire operating range.

To achieve this the ASM will be flattened with pre-calibrated 'flattening-commands' to compensate for the initial shape errors of the mirror. Furthermore, a look-up table will be used to compensate the SFE-induced by the gravity sag at the different elevation angles and at different temperature ranges. To further improve the figure accuracy for seeing-limited observations, it is foreseen that slow wavefront corrections are applied via low-order wavefront sensors. These slow figure corrections will particularly target the characteristic mode shapes corresponding to the gravity sag and thermal elastic behavior of the structure.

The UH2.2 telescope is equipped with a refocusing stage that allows correction for focus variations during observations. As a result, spherical (r 2 ) aberrations of the ASM induced by for instance the thermal behavior can be compensated by this refocusing stage. This significantly reduces the stroke requirements for the ASM actuators and enables the use of a support structure with a relatively high coefficient of thermal expansion (CTE). Table 2 shows the SFE budget of the ASM, where the SFE errors are subdivided in (i) focus aberrations which can be compensated by the refocusing stage, (ii) low and mid-spatial frequency aberrations that can be compensated by the ASM actuators, and (iii) high spatial frequency aberrations that cannot be compensated by the ASM actuators. The blue column lists the low/mid spatial frequency errors before compensation, while the orange column lists the same contributors when compensated with pre-calibrated actuator commands with an accuracy of 90%. This 90% is considered a conservative assumption given the low hysteresis of the actuators (~2%). The bottom rows list the total SFE in the different modes of operation (i) initial errors, (ii) after pre-calibrated flattening, and (iii) in full AO-mode.

The main figure error source is expected to be the initial figure error stemming from the manufacturing of the face-sheet. The values presented in the row named 'Initial figure errors' of Table 2 are the target specifications for manufacturing of the face sheet as agreed upon with the manufacturer. It is expected that for future ASM's higher polishing qualities can be achieved at the expense of higher polishing efforts.

The thermal behavior of the ASM structure is extensively analyzed via Finite Element Modeling (FEM), as shown in Figure 3. As can be seen in Figure 3, the kinematic strut interface between the actuators and the face sheet ensure that the face sheet can thermally 'breathe' with respect to the support frame, ensuring minimal stresses in the face sheet. As a result, the overall temperature variation yields mostly a spherical aberration of 3450nm PV which can be well compensated by the refocusing stage of the telescope. The remaining aberrations are around 78nm rms in the correctable range and 3nm in the uncorrectable range at the maximum temperature difference of 30ºC. Notice that this is really considered worst case and better performances are expected during nominal conditions. Another SFE-contributor is the gravity sag of the ASM structure, which has been analyzed with FEM in the two extreme cases: (i) with the gravity vector perpendicular to the mirror surface, and (ii) parallel to the mirror surface, as shown in Figure 4. Notice that a global, low spatial-frequency effect can be distinguished in which the entire support structure shows a sag with respect to its telescope interface, and a more local high spatial frequency effect of the face sheet sagging in between its actuator support points. The latter is not compensable by the actuators and is considered the main driver for the 3.5mm thickness of the face sheet yielding less than 1nm rms sag in the when the ASM is in the horizontal position. It is foreseen that in seeing-limited mode of operation the low spatial frequency aberrations induced by gravitysag will be compensated to a large extent via a calibrated look-up table which stores the compensation commands for the different elevation positions.

To summarize the total SFE error performance; the second-to-last row of Table 2 shows an expected SFE performance of 72nm rms in static operation, and 21nm rms in full AO-mode which are both within the target requirements. The latter is dominated by the polishing quality of the face sheet which, if needed, can be improved by additional polishing steps. As mentioned above, it is expected that in seeing-limited mode the static figure can be further improved by (low spatial and temporal frequency) wave-front-sensors targeted at the characteristic deformation modes.

Print-through analysis

Print-through resulting from the bond-connections of the actuators to the face sheet may lead to specific diffraction effects due to its regular pattern. Based on optical analysis, a target budget of 10nm rms is defined for this pattern. Various sources of print-through are identified and analyzed with FEM. Table 3 summarizes these results in an overall print-through budget, adding up to a worst case of 6.9nm rms at dT=30°C and with gravity perpendicular to the surface which is well within the requirement. Note that for future systems print-through can be reduced by better thermal matching of the CTE at the local bonds and by CTE matching between the support structure and the mirror material.

Impact of actuator failure

While the actuators are inherently reliable given their low complexity and absence of wear and aging, it is of interest to understand the impact of an actuator failure on the wavefront correction performance. This impact is highly dependent on the applied actuator commands. The largest source of SFE is the initial static figure error of the face sheet (see Table 2) targeted at 5µm PV and mostly within the lower spatial frequency range. The impact of failed actuators is analyzed by generating 250 different sample shapes via a superposition of the first 21 Zernike polynomials, normalized such that the PV in the optical aperture is 5µm. Given these sample shapes, the achievable SFE is analyzed with up to 4 randomly chosen 'failed' actuators. This analysis is done by fitting and subtracting the actuator influence functions which are generated based on FEM. The results are summarized in Figure 5. For the 250 different realizations, the top-left graph shows how often the SFE falls in a certain RMS range, with up to four randomly chosen failed actuators. The table on the bottom of Figure 5 summarizes these results, showing an average SFE of 8,77 nm rms with four failed actuators.

It is concluded that the SFE performance degrades gracefully with an increased number of failed actuators and the average degradation with four failed actuators is relatively small with respect to the overall SFE budget. This graceful failure is due to the relatively actuator coupling of around 35%, which allow neighboring actuators to partially compensate for a failure.

Notice that the actuator design enables replacing each individual actuator during a daytime servicing operation. This is done by disconnecting the bond between the actuator and the strut via a dedicated removal tool, after this a new actuator can be placed. This procedure is further explained and shown in [START_REF] Jonker | Design and MAIT Status of the UH2.2 Adaptive secondary Mirror[END_REF].

DYNAMIC ANALYSIS

Understanding the dynamical behavior of the ASM is important to achieve the required overall AO-control performance. The target disturbance rejection bandwidth is set at 100Hz (-3dB) and is limited by the overall dynamic behavior of the ASM and the surrounding structure. This chapter discusses the dynamical FEM modeling results and the control strategy developed to control to cope with the ASM dynamics with the aim of meeting the 100Hz bandwidth requirements.

Dynamical FEM modeling

Figure 6 shows the dominant resonance modes of the ASM obtained by dynamic FEM modeling. The first global resonance frequencies are relatively low, which requires special attention in the design of the mechanics and the feedback controllers. Since with this ASM concept the face sheet is rigidly supported via stiff actuators, the resonances of the mirror occur at relatively high frequencies (>5kHz).

For analysis of the achievable control performance and robustness, the main concerns are (i) the resonance modes that occur at a frequency close to the target control bandwidth, and (ii) mode shapes which show a significant displacement in the vertical direction as these might be excited by the actuator commands. As can be seen from Figure 6, these are mostly the tip and tilt resonances (#7 and #8) at 164Hz, and the astigmatism modes (#43 and #44) at 634 Hz.

Dynamic behavior of the spider structure is also analyzed, showing that the low-frequency modes of the spider structure have little impact on the overall control performance of the ASM. This is explained by the fact that the modal mass corresponding to these resonance modes is significantly higher that the moving mass of the actuators.

Control strategy

The resonances at frequencies near the target control bandwidth can be excited when the actuators are driven in a pattern that matches these dynamic mode shapes of the structure. Of particular concern are the tip/ tilt modes at 164Hz and the astigmatism modes at 634 Hz, as these have a significant deformation in the vertical direction, in-line with the actuation direction. The piston mode at 205Hz is not considered, as the corresponding global piston commands are not of use in controlling the aberrations and thus will not be excited.

To handle these dynamics in the design of the feedback controller for the ASM, TNO is developing a control strategy based on modal-decoupling [START_REF] Langedijk | Decentralized Control for Large Deformable Mirrors[END_REF]. To study and explain the dynamic behavior of the ASM, a simplified dynamic model is initially used, of which the dynamic responses are shown in the bottom-left graph of Figure 7. To enable modal decoupling, the interaction matrices between the actuators and the wave-front sensors are decomposed to specific orthogonal patterns via the input and output matrices Tu and Ty in Figure 7. Commonly such a decomposition is based on Singular Value Decomposition (SVD) based on the set of measured interaction matrices between the actuators and wave-front sensor responses. However, in this case the first rows/columns of the input/output matrices are based on the specific mode shapes of the relevant low frequency modes (e.g. tip/tilt modes, and astigmatism modes). The remainder of the input/output matrices are based on SVD of the interaction matrices after subtraction of these dynamic mode shapes.

As a result, the excitation of the first resonance modes will only occur via those corresponding rows of the input matric Tu, and are therefore decoupled from the rest of the system responses (see bottom-right graph in Figure 7). This enables robust control of the ASM via specifically tuned Single Input/Single Output (SISO) feedback controllers for each particular mode. As depicted in the bottom-right graph of Figure 7, the first rows of the decoupled dynamic system will show a typical fourth order response with a resonance and anti-resonance pair at the frequency of the corresponding resonance mode. Due to the decoupling, those resonances will not appear in the other system-responses which are thus not affected nor limited by these low frequency dynamics. The SISO-loops corresponding to the low frequency dynamic modes may be tuned to a lower feedback bandwidth to prevent instability from these modes, or more advanced control techniques may be applied such as notch-filters to push the bandwidth close to or beyond the frequency of these corresponding modes.

This effect of this control strategy has verified based on a dynamical FEM model of the UH 2.2 ASM. Figure 8 shows the achievable control bandwidth per mode in case of using standard Zernike decomposition, and when using the proposed combination of Modal-shapes and SVD decomposition. In the latter case the 100Hz bandwidth requirement is met for all the mode apart from the first tip-tilt modes which are limited by the low tip-tilt resonances of 164 Hz (see Figure 6). Notice that this analysis assumes a standard decoupled Proportional-Integral (PI) feedback controller for each mode. It is expected that the bandwidth of the tip/tilt modes can be further improved utilizing more advanced feedback controller design for these modes, with for instance notch-filters to minimize the excitation of those specific modes.

OUTLOOK

Based on the analysis discussed above, it is expected that the UH2.2 ASM will achieve is main performance goals in terms of Surface Form Error, and control bandwidths within its operational range. At the time of writing of this paper the main components of this ASM are being manufactured, and integration is scheduled for the beginning of 2021. After integration, its performance will be verified with initial functional tests, off-telescope tests at the summit, and finally full on sky-testing to demonstrate the expected imaging improvements. These tests are further described in [START_REF] Chun | A new adaptive secondary for astronomy on the University of Hawaii 2.2-meter telescope[END_REF].

In parallel to these activities, conceptual design studies are ongoing for application of this technology for an ASM for the APF-telescope, and the Keck-telescope [7].

Figure 1 :

 1 Figure 1: CAD rendering of the ASM mounted on the rotatable telescope spider structure.

Figure 2 :

 2 Figure 2: UH2.2 ASM cross-sectional view

Figure 3 :Figure 4 :

 34 Figure 3: Thermomechanical analysis results

Figure 5 :

 5 Figure 5: Actuator failure impact study results

Figure 6 :

 6 Figure 6: List of dominant dynamic modes (left), and a FEM image of their specific Zernike mode shapes

Figure 7 :Figure 8 :

 78 Figure 7: Principle of modal decoupling demonstrated via the simplified dynamical model. Input and output matrices are defined to diagonalize the ASM dynamic response

Table 1 :

 1 UH 2.2 ASM main features

	Basic Properties	Value
	Actuator count	210, spacing 39mm on a circular grid
	Facesheet	Ø620mm, 3.5mm thickness, convex aspherical (4.2m RoC, conic constant -3.604)
	Optical Coating	protected aluminum
	Total Mass	~55kg (current passive M2: ~72kg)
	Performance Specifications	Value	Comment
	Actuator stroke (free)	35µm
	Actuator stroke (inter-actuator)	4µm	Tunable via actuator and facesheet stiffness
	Linearity	99.5%	Measured in existing prototype mirrors
	Resolution (surface)	0.5nm	16-bit drive electronics current resolution
	Actuator coupling with neighbors	35%	Tunable via actuator and face sheet stiffness
	Actuator dissipation for flattening <3 Watts	Total over 210 actuators
	Drive electronics power dissipation <20Watts
	First actuator resonance	~1 kHz	Tunable via actuator stiffness
	Achievable correction bandwidth	~100 to 150Hz	factor ~10 below actuator resonance mode

Table 2 :

 2 ASM surface form error budget Surface

form error budget (nm)

		Focus	Low spatial frequencies SFE (correctable) High spatial SFE	Totals
			Initial (uncorrected)	Calibrated	(uncorrectable)	(RSS)
					correction (90%)
		PV	PV	RMS	RMS	RMS
	Initial figure error	3000	6000	630	63	20
	Thermal effect at dT=30°C	3450	490	78	8	3
	Gravity effect	631	1230	280	28	3
	Unpowered -target 633nm SFE (2λ WFE)		689		21	690
	Static mode -target 80nm SFE (ʎ/4 WFE)			69	21	72
	AO mode -target 21nm SFE (ʎ/15 WFE)				21

Table 3 :

 3 print-through budget

	Error source	SFE (rms) Comments
	1. Gravity sag in Z-axis	1	FEM analysis
	2. Bond shrinkage due to epoxy curing	0.25	Breadboard testing
	3. Bond CTE mismatch (dT=30°C)	2.4	FEM analysis
	4. Strut bending due to global thermal deformation (dT=30°C)	3.25	FEM analysis
	Total	6.9	Requirement 10nm rms