Thesis submitted in candidature for the

Université libre de Bruxelles

É c o l e P o l y t e c h n i q u e d e B r u x e l l e s

Piezoelectric Mirrors for Adaptive Optics in Space Telescopes

David ALALUF

Thesis submitted in candidature for the

degree of Doctor in Engineering Sciences September 2016

Active Structures Laboratory

Department of Mechanical Engineering and Robotics

Jury

Supervisor: Prof. André Preumont (ULB) President: Prof. Frank Dubois (ULB) Secretary: Prof. Michel Kinnaert (ULB)

Membres:

Prof. Emanuele Garone (ULB)

Dr. Gonçalo Rodrigues (ESA - ATG Europe B.V) Dr. Vincent Moreau (AMOS)

iii

Remerciements

Tout d’abord, je tiens à remercier mon superviseur, le Professeur André Preumont, directeur du Laboratoire des Structures Actives de l’ULB, pour tout ce que j’ai ap- pris à ses côtés et pour son encadrement exceptionnel pendant ces 4 années. Je n’oublierai jamais ma première rencontre avec lui et tous les moments que l’on a passés par la suite.

Je remercie également Renaud Bastaits, auprès de qui j’ai travaillé pendant deux ans et demi et qui a contribué aux résultats présentés dans cette thèse. Je n’oublie pas non plus Geoffrey Warniez, qui a tant travaillé sur nos prototypes.

Je remercie chaleureusement mes collègues et amis: prof. Mihaita Horodinca, prof.

Iulian Romanescu et prof. Ioan Burda, pour leur aide indispensable à la réussite de nos projets, ainsi que pour tous les instants inoubliables que l’on a pu partager.

Je remercie bien sûr Eric Wille et Gonçalo Rodrigues, les Technical Officer auprès de l’Agence Spatiale Européenne (ESA) des projets BIALOM et MATS, pour leurs conseils, ainsi que pour les discussions fructueuses que nous avons eues tout au long de ces projets. Gonçalo tient également une place importante pour moi puisque sa thèse de doctorat sur les miroirs déformables a précédé la mienne, et je suis très heureux de travailler avec lui aujourd’hui.

Le Laboratoire des Structures Actives a été pour moi comme une seconde maison pendant presque 4 ans, j’y ai passé toutes les heures d’une journée et y ai rencon- tré des personnes que je n’oublierai jamais. Je remercie chaleuresement tous les membres du laboratoire qui ont contribué à ces moments passés et à cette atmo- sphère multiculturelle si enrichissante pour moi: mon grand ami Matisek, Bilal et Momo avec qui j’ai passé tant de temps à rire, Thibaut Grasset qui saura pourquoi, Gecko qui m’a tant inspiré, Georges, Carlos, Mylène et Francesco pour toutes les soirées, Romain avec qui j’ai couru des kilomètres, la dream team Andrea et Matteo, mes amis Yves, Gabriele, Luigi, Tristan, Anis, Hussein, Ali, Kainan, Zhui, Yanghai, Mathieu D., Paul, les 2 Alexandres, Romain, Goska et Neda. La liste serait trop

v

longue, mais je remercie bien sûr également tous mes amis en dehors du laboratoire pour tout ce que l’on vit tous ensemble au quotidien (bon, sauf durant les moments de rédaction de thèse...). Merci les amis.

Je tiens également à remercier Timo Scholehwar (PI Ceramic) qui nous a très gen- tillement aidé à améliorer la qualité de nos PZT; Grégory Martic (BCRC), qui a consacré beaucoup de temps à mettre au point la technique de l’ablation laser;

Pierre Taquet (Sonaca) pour nous avoir fourni la plaque support en CFRP de notre prototype de miroir segmenté; le personnel de la SABCA et de Thales Alenia Space à qui l’on a fait appel pour les soudures électriques de nos 2 demonstrateurs. Je remer- cie également vivement les partenaires de nos projets BIALOM et MATS: AMOS, le Centre Spatial de Liège, Materia Nova, le Certech et Samtech, avec qui j’ai eu l’occasion de travailler au cours de ces projets.

Merci également à tous ceux qui m’ont appris quelque chose, un jour.

Finalement, je remercie de tout mon coeur mes parents, ma soeur jumelle et mes grands parents...

Le travail réalisé dans cette thèse s’inscrit dans le cadre des projets BIALOM et MATS financés par l’Agence Spatiale Européenne (ESA).

vi

Abstract

Future generations of space-based telescopes will require increasingly large primary reflectors, with very tight optical-quality tolerances. However, as their size grow, it becomes more and more difficult to meet the requirements, due to the manufactur- ing complexity and the associated costs.

Chapters 2 and 3 propose two concepts of Adaptive Optics deformable mirrors, intended to be used as secondary corrector to compensate for manufacturing errors, gravity release and thermal distortion of large lightweight primary mirrors of space telescopes: (i) A scalable segmented bimorph mirror, based on independent PZT patches glued on Silicon wafers, providing a large number of degrees of freedom, a low mass while overcoming the problem of a low resonance mode; and (ii) A mono- lithic bimorph mirror, controlled by an array of independent electrodes, done by laser ablation on a single PZT patch. The modelling, the control strategy and the technological aspects are described. The performances of the manufactured proto- types are demonstrated experimentally. These prototypes have been developed in the framework of the ESA project, Bimorph Adaptive Large Optical Mirror Demon- strator (BIALOM).

Chapter 4 introduces alternative designs, allowing to face the thermal distortion inherent to the bimorph architecture. They are compared in terms of stroke, volt- age budget and first resonance frequency. These designs are required to be controlled in both directions using only positive voltages.

Finally, the last chapter explores the feasibility of the shape control of a small size active thin shell reflector (with double curvature). The prototype is intended to be a technology demonstrator of a future large and very light active primary reflector.

The behavior of the shell is studied through numerical simulations, and a prelimi- nary design is proposed. This investigation is carried out in the framework of the ESA project: Multilayer Adaptive Thin Shell Reflectors (MATS).

vii

ix

"Ayez des rêves fous et poursuivez les, l’important n’est pas tant d’accomplir ses rêves, mais de profiter du chemin qui vous y mène..."

David Saint-Jacques, Astronaute de l’Agence Spatiale Canadienne

Glossary

List of acronyms

AO Adaptive Optics

ASL Active Structures Laboratory

BIALOM Bimorph Adaptive Large Optical Mirror Demonstrator CAD Computer Aided Design

CTE Coefficient of Thermal Expansion DLS Damped Least Squares

DM Deformable mirror

EELT European Extremely Large Telescope ELT Extremely Large Telescope

ESA European Space Agency

ESO European Southern Observatory HST Hubble Space Telescope

IF Influence Function

JWST James Webb Space Telescope

LS Least Squares

LBT Large Binocular Telescope M1 Primary mirror

M2 Secondary mirror

MATS Multilayer Adaptive Thin Shell Reflectors for Future Space Telescopes MEMS Microelectromechanical systems

NASA National Aeronautics and Space Administration NIMO Phase-shift Schlieren NIMO RE2507 wavefront sensor NTT New Technology Telescope

PMN Lead Magnesium Nioboate PV Peak to Valley

PVDF Polyvinylidene fluoride PZT Lead-Zirconate-Titanate RMS Root Mean Square SAR Sensor-to-actuator ratio

xi

SH Shack-Hartmann

SVD Singular Value Decomposition ULB Université Libre de Bruxelles VLT Very Large Telescope

WFS Wavefront sensor

List of symbols

α Damping factor (used for the Damped Least Squares method) c(.) Condition number of (.)

D Diameter of a circular deformable mirror or of a shell (chapter 5)

D

Primary mirror diameter D

Diameter of the optical pupil

D

Radial size of the electrode (chapter 5) D

Segment external diameter

d

, d

, d

Piezoelectric constants

∆α Mismatch of CTE between the substrate and the Patch/Ring

∆T Temperature change

∆V Voltage range

δθ Angular resolution

E Electric field

f

First resonance frequency

g Gap between adjacent electrodes (Chapter 2) J Jacobian: (m × n) matrix

J

Pseudo-inverse of the Jacobian: (n × m) matrix κ, κ

, κ

Total curvature of the mirror,

curvature induced by the patch and by the ring L

Characteristic length

λ Wavelength of the light

M , M

Biaxial Young’s modulus of the substrate and of the patch m Number of signals provided by a wavefront sensor

(that is two times the number of micro lenses in the case of a SH sensor) n Number of actuators of a deformable mirror

ν Poisson’s ratio of the substrate

R Radius of a circular deformable mirror or of a shell (chapter 5)

R

Radius of curvature

R

Radius of the optical pupil R

, R

Patch radius and ring width

xii

r Rank of J (Chapter 1)

r Radial coordinate (Chapters 2 to 5); or Electrode size (Chapter 2)

r

Fried length

ρ Transition region

% Density of the substrate

S Strehl ratio

s Threshold used for the Singular Values Decomposition s (m × 1) vector containing the slopes of a deformable reflector σ RMS Surface figure error

σ

i

singular value of J

t, t

, t

Thickness of respectively the substrate, the PZT patch and the Passive/Active ring

v (n × 1) vector containing the voltages applied to the actuators W Stroke of a deformable mirror

w (m × 1) vector containing the displacements of a deformable reflector ξ Mechanical Damping Coefficient

Y , Y

Young’s modulus of the substrate and of the patch

xiii

Contents

Jury iii

Remerciements v

Abstract vii

Glossary xi

1 Introduction 1

1.1 Astronomy and telescopes . . . . 1

1.2 Modern-day telescopes . . . . 2

1.2.1 Active Optics . . . . 2

1.2.2 Adaptive Optics . . . . 3

1.2.3 Ground-based telescopes . . . . 5

1.2.4 Space telescopes . . . . 7

1.3 Deformable mirrors . . . . 11

1.3.1 Stacked Array Mirrors . . . . 11

1.3.2 Voice-Coil Actuator Mirrors . . . . 12

1.3.3 Microelectromechanical Systems . . . . 13

1.3.4 Bimorph/Unimorph Mirrors . . . . 15

1.4 Feedforward control . . . . 20

1.4.1 Singular Value Truncation . . . . 21

1.4.2 Tikhonov regularization . . . . 22

1.5 Outline . . . . 25

1.6 References . . . . 27

2 Segmented Bimorph Mirrors 31 2.1 Introduction . . . . 31

2.2 Segment Design . . . . 32

2.2.1 Bimorph mirror . . . . 32

2.2.2 Isostatic support . . . . 37

xv

xvi CONTENTS

2.2.3 Initial curvature of the segment . . . . 39

2.3 Control strategy . . . . 43

2.3.1 Singular Value Decomposition . . . . 45

2.3.2 Damped least squares . . . . 48

2.3.3 Wavefront sensor . . . . 48

2.4 Comparison of various segmented configurations . . . . 51

2.5 Single segment experiment . . . . 55

2.5.1 Influence functions . . . . 55

2.5.2 Turbulent screen . . . . 55

2.6 Dimples . . . . 58

2.7 Conclusion . . . . 65

2.8 References . . . . 66

3 Keystone Electrode Pattern 69 3.1 Introduction . . . . 69

3.2 Design and manufacturing . . . . 71

3.2.1 Deformable mirror . . . . 71

3.2.2 Gluing under voltage . . . . 73

3.2.3 Electrical insulator . . . . 76

3.2.4 Electrical connections . . . . 77

3.2.5 Isostatic mount . . . . 78

3.2.6 Finite Element model . . . . 80

3.2.7 Experimental set-up . . . . 82

3.3 Experimental results . . . . 86

3.3.1 Feedforward control . . . . 86

3.3.2 Static shape . . . . 88

3.3.3 Zernike modes . . . . 89

3.3.4 Rigid-body modes . . . . 94

3.3.5 Long term stability in open-loop . . . . 95

3.3.6 Eigenfrequencies . . . . 96

3.4 New design . . . . 97

3.5 Conclusion . . . . 99

3.6 References . . . . 100

4 Additional considerations 101 4.1 Introduction . . . . 101

4.2 Current design . . . . 101

4.2.1 Critical analysis . . . . 101

4.2.2 Thermally balanced design . . . . 103

4.3 Active ring configuration . . . . 108

4.4 Clamped configuration . . . . 111

CONTENTS xvii

4.5 Conclusion . . . . 114

4.6 Summary . . . . 115

4.7 References . . . . 116

5 Adaptive Thin Shell Reflector 117 5.1 Introduction . . . . 117

5.2 Shell behaviour . . . . 118

5.3 Auxetic materials . . . . 122

5.4 Demonstrator design . . . . 126

5.4.1 Electrode pattern . . . . 129

5.5 Performances of the clamped design . . . . 131

5.6 Conclusion . . . . 134

5.7 References . . . . 135

6 Conclusions 137 6.1 Original aspects of the work . . . . 138

6.2 Future work . . . . 139

A Zernike modes 143

A.1 References . . . . 144

Chapter 1

Introduction

1.1 Astronomy and telescopes

The first telescope saw the light at the end of the 16th century, thanks to Hans Lippershey, a German-Dutch gentleman who is generally credited for the invention of the refracting telescope. Galileo improved the invention and used it to make the first observations of the sky with an instrument allowing to focus more light than the naked eye and to magnify the images. This was a major breakthrough for the astronomy.

Around 1668, the idea that the light-gathering element could be a mirror instead of a lens led Isaac Newton to build the first practical reflecting telescope as an alterna- tive to refracting one. Reflecting telescopes do not suffer from chromatic aberration since the reflection law does not depend on the wavelength of the light.

A telescope can be seen as a set of mirrors (or lenses) positioned extremely ac- curately relative to each other, collecting a part of the light emitted from a point source and imaging it at a point. Therefore, the larger the diameter of the aperture, the more light gathered, allowing to improve the resolution of the images and to observe fainter objects. Both refracting and reflecting telescopes have called for a constant increase of the size of the objectives. Nevertheless, for apertures larger than 1m, reflecting telescopes have proven more effective. Obviously, since the aperture of the telescopes is finite, it is not possible to collect all the light emitted by a point source, hence the image will never result in a point but rather in a blurred spot called Airy disk, because of diffraction. A perfect (unaberrated) optical system which is only limited by the diameter of the primary mirror is said to be diffraction-limited.

Conventionally, a system is considered as diffraction-limited if the RMS wavefront

1

2 1 Introduction

error is smaller than λ/14 (Maréchal criterion)

.

1.2 Modern-day telescopes

As their size increases, modern telescopes are increasingly sensitive to external dis- turbances (e.g gravity, wind etc.) To cope with this, the mirrors and the structural components have been made thicker and stiffer. However, such passive methods started to show practical limitations for telescopes with primary mirror diameter larger than 4m. This was due to the extremely high costs and to the difficulty to manufacture and to orientate such heavy structures. At the end of the 20th century, the advent of active and adaptive optics has been a considerable step forward for the astronomy as explained hereafter.

1.2.1 Active Optics

Active optics refers to all the control systems aimed at compensating for manufac- turing errors, deformation induced by gravity, thermal effects, wind excitation and vibration propagation. Its bandwidth is typically below 10 Hz and the disturbance amplitude can be up to few millimeters.

A schematic drawing illustrating the active optics principle is depicted in Figure 1.1. When the light enters in the telescope, it is reflected by the mirrors and is directed to the beam splitters which separate the beam in two parts. The first part directly goes to the science instrument to form an image of the object of interest.

The second part is directed to a wavefront sensor (WFS) providing information about the aberrations to compensate. Based on the measurements of the WFS, a controller determines the input of the force actuators, pushing against the back of the primary mirror, M1, to deform it so as to compensate for the aberrations intro- duced by the external perturbations. The remaining aberrations are compensated by the rigid-body motion of the secondary mirror, M2, and the orientation of the whole telescope itself (altitude-azimuth).

The implementation of active optics in telescopes allows to alleviate considerably the requirements on the mirrors as well as on the overall structure. Indeed, thinner mirrors (monolithic or segmented mirrors) can be manufactured and actively con- trolled instead of using rigid and heavy mirrors which are extremely costly, difficult to manufacture and to manipulate.

1Therefore, the deformable mirror requires a RMS error smaller thanλ/28.

1.2 Modern-day telescopes 3

M1 M2

Science instrument

Shack-Hartmann (WFS)

cameraCCD

Controller

Controller

Alt-az axes

Tilt

Force actuators

Rigid-body motion

Beam splitter

Figure 1.1: General principles of active optics (adapted fromESO-NTT).

1.2.2 Adaptive Optics

Until the advent of active optics, it was not possible to build and maintain telescopes for a long time with a good optical accuracy. Therefore, the relative impact of the aberrations produced by the atmosphere had only a second order impact on the optical performance. Nevertheless, thanks to active optics, telescopes have been able to work closer to their diffraction limit and have quickly been limited by atmospheric turbulence. Indeed, when the light incoming from the celestial objects reaches the outer layers of the atmosphere, it can be considered as a perfectly plane wavefront.

However, thermal gradients and convection currents of the atmosphere locally change the refraction index of the air and distort the wavefront. Instead of being focalized at a point by the telescope (which is supposed to be perfect in this case), the light is spread in the focal plane in spots 10 to 100 times larger than the diffraction limit, which results in a loss of resolution and sensitivity (Figure 1.2). The limit of resolution is defined as the smallest angular separation between two objects that can be distinguished by an instrument, that is:

δθ = 1.22 λ/D

(1.1)

where λ is the wavelength of the light and D

is the diameter of the optical aperture of the instrument. However, due to atmospheric turbulence, the resolution is limited to (Dierickx, 1992):

δθ = λ/r

(1.2)

where r

is the Fried length (Fried, 1965), corresponding to the aperture on which

4 1 Introduction

a) image plane b) image plane

Figure 1.2: a) Rays emanating from a perfect spherical wavefront, focused in a point in the image plane; b) Rays emanating from an aberrated wavefront, focused over an extended area (adapted fromGeary,2002).

the rays are in phase. The Fried length depends on the location of the telescope and it can be shown that (Hardy):

r

∝ λ

(1.3)

For visible wavelengths, r

is close to 15 cm. Therefore, without making any cor- rection to cope with atmospheric turbulence, a telescope with a large primary mir- ror diameter will not provide better resolution than a 15 cm mirror diameter. A telescope limited by the atmospheric turbulence is said to be seeing-limited. Atmo- spheric turbulence causes aberrations at high temporal frequencies, ranging from 0.1 Hz to several hundred Hz, and with an amplitude around 1 µm.

To cope with this, a deformable mirror, generally much smaller than the primary mirror, is introduced in the optical train of the telescopes and is deformed in real- time to produce a wavefront correction compensating most of the wavefront error introduced by the atmosphere. A typical adaptive optics system is schematized in Figure 1.3. As for active optics, a beam splitter allows to direct part of the beam to a wavefront sensor characterizing the aberrations and feeding a controller which determines the deformation to apply to the deformable mirror. Besides, the DM can be mounted on a rigid-body mount allowing to control the orientation of the DM so as to control the stabilization of the image. It is worth noting that in some telescopes, the secondary mirror plays the role of the DM which simplifies the setup of the telescope by reducing the number of optical elements and limits the risk to contaminate the wavefront.

Adaptive optics also aims at correcting the residual error from active optics. Both

are implemented in all state-of-the-art telescopes and give access to unprecedented

performances allowing to reach near diffraction-limited images.

1.2 Modern-day telescopes 5

8

Deformable Mirror

Controller Wavefront sensor

Science camera

AO OFF

AO ON Atmospheric

turbulences

Disturbed wavefront Plane wavefront

Telescope

Figure 1.3: Schematic drawing illustrating adaptive optics and images with and with- out adaptive optics (adapted from Hickson,2008).

1.2.3 Ground-based telescopes

The concept of Adaptive Optics has been introduced for the first time in 1953 by Ho- race Babcock (Babcock, 1953), an American astronomer. It is only in the seventies that the American Air Force built the first AO system for observing Soviet satellites.

Around 1990, Adaptive and Active Optics systems were for the first time success-

fully installed in telescopes, respectively in the Observatoire de haute Provence in

France and in the New Technology Telescope (NTT) in Chile. These successes en-

couraged the development of larger telescopes such as the Very Large Telescope

(VLT) built soon after by the European Southern Observatory (ESO) in the Ata-

cama desert in Chile (Figure 1.4). The VLT consists of four Unit Telescopes (UT),

each with a 8.2 m monolithic deformable primary mirror with a thickness of only 175

mm and its shape is actively controlled by means 150 axial force actuators (active

optics). More recently, the secondary mirror of the VLT has been equipped with a

6 1 Introduction

deformable mirror operated by 1170 magnetic actuators, to perform adaptive optics.

Above 8 m, the primary mirror becomes extremely difficult to manufacture and to transport to the site, and a segmented configuration becomes preferable. The primary mirror is made of a set of segments of hexagonal shape about 2 m arranged in honeycomb; active control is needed to ensure the continuity of the surface formed by the segments. The rigid body motion of the segments is controlled by a set of 3 actuators, and the relative position of the various segments is measured by edge sensors. The twin Keck telescopes, built in 1993, located near the summit of Mauna Kea in Hawaii constitute the most representative example, they have a primary mir- ror made of 36 hexagonal segments of 1.8 m-diameter for a total aperture of ≈ 10 m (Figure 1.5). They are equipped with active and adaptive optics technologies to perform near diffraction-limited observations.

The future European Extremely Large Telescope (EELT) will be the world biggest eye on the sky, with its 39m-primary mirror composed of a tessellation of 798 seg- ments. It will include two adaptive optics deformable mirrors with more than 5000 actuators to compensate for atmospheric turbulence and to stabilize the image. It will be built in Cerro Armazones in Chile and should see the first light in 2024.

Figure 1.4: Top: The four Units of the Very Large Telescope. Left: 8.2m-monolithic primary mirror, made in Zerodur. Right: 0.94m-secondary deformable mirror, made in Beryllium (European Southern Observatory,2010).

1.2 Modern-day telescopes 7

Figure 1.5: Keck telescope with its 10m-segmented primary mirror (Keck Observatory, 2010).

1.2.4 Space telescopes

Space astronomy made a giant step in the 1990’s with the stunning success of NASA’s Hubble Space Telescope (HST) providing the first astronomical observations un- blurred by the Earth’s atmosphere (Figure 1.6 a). HST has a monolithic mirror of 2.4 m of diameter (surface area of 4.5m

), with an areal density of 180 kg/m

. More recently, the Herschel telescope was launched by ESA in 2009, with a primary mirror of 3.5m of diameter (surface area of 9.6 m

) constructed by brazing 12 petals of SiC with an areal density of 22 kg/m

. Unlike HST which operates in low orbit and could be repaired by astronauts in several instances, Herschel operated until 2013

at the distant Lagrange point L2, at millions of kilometers from the Earth.

This monolithic telescope was the largest telescope ever launched; larger telescopes will need to be folded in order to be stowed inside the fairing of launch vehicles. The James Webb Space Telescope (JWST) (Gardner, 2006) is currently in its final stage of design; its primary mirror consists of 18 hexagonal segments (made of Beryllium

3

), which are folded during launch and deployed in orbit to provide a primary mirror

2The lifespan of Herschel was governed by the amount of coolant available for its instruments.

At the end of the mission, Herschel has been placed into a Heliocentric orbit.

3A gold optical coating has been deposited to reach a high reflectivity.

8 1 Introduction

of 6.5m of diameter (surface area of 33 m

) with an areal density of 20 kg/m

(Fig- ure 1.6 b and Figure 1.7). The primary mirror is actively controlled for cancelling deployment errors as well as orbital disturbances. Unlike the HST which is mainly sensitive in the ultraviolet, JWST will operate mostly in infrared (to search for light from the first stars and galaxies that formed in the Universe after the Big Bang).

Figure 1.6: a) Hubble Space Telescope launched in 1990 (NASA, 2010a); b) James Webb Space Telescope, folded in the launcher and after deployment. The JWST is expected to be launched in 2018 from French Guiana on an Ariane 5 (NASA,2010b).

From another side, the EELT will have the capability to separate a small planet from its star, and will have a sufficient collecting area to feed spectrographs. However, even with adaptive optics, the atmospheric turbulence makes it difficult to block all the light of the mother star with coronagraphs, which still overwhelms any light from a close planet (Conti and Clampin, 2015). The state of the knowledge in the study of exoplanets suggests that a space telescope with a diameter of at least 12 m is necessary to find and directly image earth-size planets around sunlike stars.

In addition, space telescopes are needed for performing Earth observation and for astronomical observation in the wavelengths which are blocked by the atmosphere.

The concept of the JWST can, in principle, be scaled up by adding more segments (and more cost and complexity). Alternatively, the space community is looking for a change of paradigm in terms of stowability and areal densities of 3 kg/m

or less.

Such a change of paradigm is offered by the so-called gossamer spacecrafts (Jenk-

ins, 2001). Figure 1.8 shows Bekey’s vision of the evolution of the space telescopes

leading eventually to formation flying of membrane optics (Bekey, 2003).

1.2 Modern-day telescopes 9

Figure 1.7: Inside the cleanroom at NASA Goddard in April 2016. The JWST’s pri- mary mirror will soon be rotated to install the instruments on its back (NASA,2010b).

a) b) c) d)

Figure 1.8: Evolution of space telescopes: a) Monolithic primary mirror (e.g HST);

b) Segmented primary mirror (e.g JWST); c) Lenticular membrane primary reflector;

d) Thin shell primary mirror (Bekey,2003).

10 1 Introduction

There are two broad classes of candidate gossamer telescope structures: the lentic- ular, pressure stiffened membranes (Figure 1.9 a) and the doubly curved, form stiff- ened elastic shells (Figure 1.9 b). Lenticular membrane reflectors are made of two circular plane membranes glued to each other on the edge; one of them is covered with a reflecting material on the inside and will form the reflector, while the other is transparent and will form the canopy. The membrane is inflated with an inter- nal pressure which can be adjusted to control the focal length of the system. The lenticular structure is attached to a supporting torus by tie rods. Because of the limited control capability of the uniform internal pressure, the wavefront error of lenticular structures tends to be dominated by the spherical aberration (Marker and Jenkins, 1997). The tie rods may also be tensioned for control (boundary control), and the reflecting membrane may be covered by a film of a piezoelectric material such as PVDF (or any electroactive material with strain actuation properties) with segmented electrodes, to produce additional control degrees of freedom.

(a) (b)

Figure 1.9: a) Lenticular membrane reflector (NASA); b) Polyimide thin shell reflec- tor in deployed and rolled configurations (source MEVICON).

The adaptive elastic shell is an alternative option (Flint et al., 2003; Flint et al.,

2006); the reflector is molded in its final shape and rolled for stowage. Once released

in orbit, the reflector will unfold on its own strain energy. The sources of surface

figure error of such a system are the manufacturing errors, the possible creep in

rolled configuration, the thermal gradient and the gravity gradient. In the same

way as for the membranes, the final shape of the shell can be controlled by means of

active layers. Note that the required final wavefront error of a fraction of wavelength

(e.g. λ/14) may be relaxed (e.g. to the order of 10 µm) if a secondary wavefront

corrector is used (Figure 1.10); the latter acts in a way similar to Adaptive Optics,

with a large number of degrees of freedom and only a small stroke; it is intended to

remove as much as possible of the residual wave front error that will be inevitably

introduced by the manufacturing process, gravity release and thermal gradients.

1.3 Deformable mirrors 11

Deformable mirror

8

Primary reflector

Figure 1.10: Use of adaptive optics as secondary wavefront corrector (deformable mirror). When the primary reflector is a thin shell or a membrane, it may also be covered by an active layer.

1.3 Deformable mirrors

Most of the wavefront correctors are based on the deformation of a reflecting surface.

The main parameters of a DM are: the number of actuators (number of independent degrees of freedom); the stroke (typically a few microns); the size (related to the field of view); the temporal bandwidth (typically 50-100 Hz, which brings a lower limit to the first natural frequency of the mirror). They range from Microelectromechanical System (MEMS) with only a few mm of aperture, to secondary mirrors up to several meters, depending on their position in the optical train and on the application.

The actuation principle is generally piezoelectric, electromagnetic or electrostatic as explained below. The following sections give an overview of the DM technologies for adaptive optics and astronomy (Madec, 2012).

1.3.1 Stacked Array Mirrors

The shape of the mirror is controlled by an array of actuators pushing its back

and controlled through a voltage (Figure 1.11). The actuators consist of stacks

of individual plates made of Piezoelectric Lead Zirconate Titanate (PZT) or elec-

12 1 Introduction

trostrictive Lead Magnesium Nioboate (PMN). When a voltage is applied between the electrodes of the actuator, a deformation in the longitudinal direction occurs (d

mode), achieving an out of plane deformation of the mirror.

The main drawback of stacked array mirrors resides in the bulky electronic required when many actuators are used. On the other hand, their first natural frequency is often above the kHz, thanks to the individual stiffness of each actuator. Moreover, their excellent accuracy and large stroke (≈ 8 µm) make them a well-suited solution for future extremely large telescopes.

The companies producing stacked array mirrors are CILAS and AOA-Xinetics, they equipped very large telescopes such as the VLT

(Chile), Gran Telescopio Canarias (la Palma) and Keck, Subaru and Gemini in Hawaii.

Figure 1.11: Stacked array DM. The mirror is deformed by an array of linear piezo- electric actuators acting perpendicular to the flexible reflecting surface (CILAS).

1.3.2 Voice-Coil Actuator Mirrors

The deformation of the mirror is performed without any contact, by means of voice- coils fixed on a rigid reference plate (Figure 1.12). Permanent magnets are bonded to the rear face of the mirror, which is "floating" on the magnetic field created by the array of voice-coils. When a change of current is sent through a voice-coil, the mirror is locally deformed. A local capacitive sensor at the extremity of the voice-coil is used to measure the distance between the reference plate and the mirror, which is

4In the adaptive optics system NAOS.

1.3 Deformable mirrors 13

of the order of 50 to 70 µm.

The main drawbacks of this technology are the power consumption, the complexity of the manufacturing process and the cost. On the contrary, the use of voice-coils allows to get very high strokes (> 50 µm), opening the door to tip/tilt atmospheric compensation. Moreover, their excellent accuracy, long-term stability and short time response make them very attractive for many large reflectors (e.g the company AdOptics is now working on the M4 of the future EELT, with a diameter of 2.5 m and ≈ 5000 actuators).

Figure 1.12: 1170 permanent magnets are bonded to the back side of the VLT’s sec- ondary mirror. The DM is "floating" and deformed thanks to the magnetic field created by an array of voice-coils actuators fixed on the rigid base plate. Capacitive sensors are integrated to the voice-coils to measure locally the distance between the mirror and the rigid plate (European Southern Observatory, 2010). The secondary mirror of the Large Binocular Telescope (LBT) has also been equipped with 672 actuators, improving considerably the sharpness of the astronomical images.

1.3.3 Microelectromechanical Systems

Microelectromechanical systems (MEMS) started to be used for adaptive optics

around 1990. Several MEMS concepts can be found, generally relying on the con-

trol of a reflecting surface (continuous or segmented) through an electrostatic or a

magnetic field. Figure 1.13 shows a configuration where a thin reflecting membrane

is attached to an intermediate diaphragm whose local deformation is controlled by

means of an electrostatic field generated by an array of electrodes located on a sup-

port plate.

14 1 Introduction

MEMS present long-term stability, reliability, almost no hysteresis, a good tem- poral bandwidth and are fairly low-cost. However, they are too small (1.5 to 25 mm aperture) for some applications requiring a wide field-of-view and their stroke is generally limited.

Among the companies producing MEMS, one can mention OKO technologies who develops the "OKO Mirror" for low-order aberrations correction, with 37 actuators and a diameter of 15 mm. Besides, Boston Micromachines proposes DMs ranging from a few tens of actuators to more than 4000, with a stroke up to ≈ 5 µm (Figure 1.14). Finally, the French company ALPAO develops DM with up to 277 actuators and 15 µm stroke.

Electrode

Membrane Mirror

Electrostatically Actuated diaphragm

Support plate

Figure 1.13: MEMS DM concept. The membrane mirror is locally deformed thanks to an electric field created by an array of electrodes on the support plate.

Figure 1.14: Left: A 4096-actuator micro-electrical-mechanical-system deformable mirror, stroke of 3.5 µm, aperture 25 mm. Right: Electrical connections. (Boston Micromachine Corporation).

1.3 Deformable mirrors 15

1.3.4 Bimorph/Unimorph Mirrors

Bimorph mirrors (Schwartz et al., 1994) are made of two piezoelectric layers, coated with electrodes on both faces and bonded together. The most common piezoelectric materials are PZT and Polyvinylidene fluoride (PVDF). When these materials are subjected to an external electric field perpendicular to the plane, they undergo an in-plane deformation whose sign do depend on the polarization of the materials (d

mode or transverse piezoelectric effect). Therefore, by choosing opposite polarities for the piezoelectric layers, one of them expands while the other one shrinks, re- sulting in a bending moment (Figure 1.15). It is worth mentioning that generally, piezoelectric materials do not have symmetric range deformation in traction and in compression. This feature leads to various designs aiming to deform the mirror in both directions. Bimorph mirrors are rather difficult to manufacture and require a high quality surface on the optical face.

On the contrary to bimorph layout, unimorph mirrors are made of only one piezo- electric layer bonded to a passive substrate such as glass or Silicon (Figure 1.15), hence they are much easier to manufacture. Once again, when a electric field is applied through the PZT, its deformation results in the bending of the passive sub- strate.

It can be shown (Rodrigues, 2010), that for a given aperture, the maximum stroke achievable with bimorph and unimorph mirrors reaches respectively :

f or bimorph mirrors : W ∝ E

d

f

, (1.4)

and

f or unimorph mirrors : W ∝ t

t

E

d

f

(1.5)

where E

, d

, f

, t

and t stand respectively for the maximum electric field, the piezoelectric coefficient, the first natural frequency, the thickness of the PZT and the thickness of the substrate. These relations clearly show the conflict between increasing the stroke while maintaining a reasonable first natural frequency.

Since unimorph mirrors are a particular case of bimorph mirrors, the terminol-

ogy "bimorph" will be applied for both cases without any distinction. Next sections

present the main manufacturers of bimorph mirrors.

16 1 Introduction

V⁺ Ground

electrodes

Optical side

pol.

PZT PZT

Positive electrode

pol.

Bimorph mirror

V⁺ Ground

electrode

Optical side

Substrate PZT

Positive electrode

pol.

Unimorph mirror

Figure 1.15: Left: The bimorph mirror is made of two active layers (with opposite polarity) bonded together. When a voltage is applied on the positive electrode, one layer shrinks, while the other one extends resulting in the bending of the mirror. Right: The unimorph mirror is made of only one active layer bonded to a passive substrate which bends when a voltage is applied on the active layer.

Compagnie Industrielle des Lasers (CILAS)

Bimorph mirrors has been pioneered by the French company, CILAS. They supplied mirrors for the biggest telescopes in the world such as the VLT (in SINFONI and in the Very Large Telescope Interferometer). They also developed a 188-electrodes bimorph DM for the 8.2-meter Subaru telescope (Figure 1.16). Recently, they under- took the development of a DM for space telescopes aiming at observing the Earth.

Figure 1.16: 188-actuator bimorph mirror, with a ring of electrodes outside the optical pupil. The DM is installed in the Subaru telescopes in Hawaii. The active material is PZT (CILAS).

1.3 Deformable mirrors 17

Ningbo University

Nigbo University and UST China (Jianqiang et al., 2011) developed a unimorph DM by gluing a 100 µm thick PZT to a 200 µm thick Silicon wafer and clamping its edge on a glass ring. The DM has a 40 mm diameter with an optical pupil of 15 mm (Figure 1.17). The central part of the rear face is paved by 37 hexagonal actuators (honeycomb pattern) aiming at deforming the mirror in a convex way, while the external ring actuator takes advantage from the clamping to deform the mirror in a concave way. The first resonance frequency is very high but the maximal stroke is modest (Eq.1.4). Furthermore, the optical pupil of the DM is rather small with respect to the dimensions of the device. A more detailed comparison between the main configurations of bimorph DM is presented in chapter 4.

Figure 1.17: Unimorph deformable mirror using 37 hexagonal actuators (arranged as a honeycomb pattern) to achieve convex deformations and complex shapes. The outer- ring electrode is only used to achieve a bias (concave defocus) by means of the clamping of the DM on its edge.

18 1 Introduction

Münster University

The Photonics lab of Münster (Rausch et al., 2016) developed a deformable mirror for space applications, based on a 700 µm thick piezoelectric layer (PIC 155), glued on a 550 µm thick N-BK 10 glass. Thanks to the large thickness of the PZT material, the maximal voltages that can be applied on the patch is rather high, ±400 V such that the mirror can be deformed in both directions. Besides, the DM is supported by three external "arms" dedicated to achieve tip/tilt motion (isostatic mount), while the central electrodes, belonging to the keystone family, aim at deforming the mirror (Figure 1.18). The mirror has an optical pupil diameter of 50 mm, 41 electrodes and a first resonance frequency around 230 Hz. This design is rather heavy (≈ 1.8 kg) and its 3 arm mechanism make him vulnerable with respect to external loads.

Figure 1.18: Unimorph mirror using three external arms to control the orientation (tip/tilt motion) of the mirror. The electrodes pattern is the classical keystone pattern used in many deformable mirrors.

1.3 Deformable mirrors 19

UC Davis

The University of California Davis (Horsley et al.) designed a 20 mm diameter bimorph mirror, actuated by two active layers made of electrostrive materials such as PMN (Figure 1.19). Since the electrostrictive materials have a quadratic strain- electric field relation, they can deform only in one direction, no matter the sign of the applied voltage. Therefore, the front-face electrode is used to produce a concave curvature by means of a bias voltage, while the back face electrodes are used to generate a convex curvature and local deformations. The DM is supported by two rubber o-rings positioned on the edge of the DM, hence its first resonance frequency reaches 1.2 kHz. Let us note that the quadratic dependency to the electric field make the control rather complex.

Figure 1.19: Bimorph mirror manufactured by AOptix. It uses two active layers to deform the mirror in both directions. The mirror is supported by two rubber-rings.

20 1 Introduction

1.4 Feedforward control

The goal of the control is to compute the voltages to be applied to the actuators of the DM, in order to change the shape of the reflector to fit a specific target surface.

One can write

:

w = J v (1.6)

where w is a (m × 1) vector containing the displacements of the reflector, v is a (n × 1) vector containing the voltages applied to the actuators and J is the so-called Jacobian matrix which is, in general rectangular with dimensions (m × n). Let us note that within this section, the number of measurements, m, is supposed to be larger than the number of actuators, n. The Jacobian makes the link between the voltages applied to the actuators and the shape of the reflector. It is built according to:

J = [f

, f

, ..., f

, ..., f

] (1.7) where f

is a (m × 1) vector containing the i

influence function (IF), corresponding to the sensor signals obtained by applying a unit voltage to the i

actuator, that is the i

element of v. Therefore, it should be pointed out that one Jacobian is associated to one pattern of actuators.

The control problem amounts to solving the inverse problem:

v

= J

w

, (1.8)

where J

stands for the pseudo-inverse of J, also called the Moore-Penrose inverse of J and v

is the voltages to be applied to the actuators to fit the target shape, w

(Figure 1.20). It is important to note that since there are less actuators than measurements, this problem has no exact solution.

J

Target shape

w

v

J

Pseudo-inverse of the Jacobian

w

Figure 1.20: Block diagram of the feedforward control. The loop is manually closed as explained in section 3.3.1

5The system is supposed to be linear, that is, complex shapes can be achieved by applying the principle of superposition.

1.4 Feedforward control 21

The rank of J, r, is defined as the number of independent rows or columns of J (Strang, 1986)

. It is smaller or equal to the minimum value between m and n:

r ≤ min{m, n}. (1.9)

When J has full column rank (r = n), J

J is invertible; the pseudo-inverse reads

J

= (J

J)

J

, (1.10)

hence,

v

= (J

J)

J

w

, (1.11)

which is the solution to the problem: Find v

so that

min

{kw

− Jv

k

}. (1.12) However, in practice J

J may be difficult to invert. This happens basically either when actuators have a very small authority with respect to the others or to the sen- sitivity of the sensor, or when two or more influence functions are linearly dependent (they appear similar as seen by the sensor). Small electrodes or electrodes located outside the optical pupil, far enough from it are typical causes of conditioning prob- lems resulting in very high voltages as shown hereafter. The ill-conditioning of the matrix J

J can be handled as follows.

1.4.1 Singular Value Truncation

A Singular Value Decomposition (SVD) of J consists of expressing it as (Strang, 1986):

J = UΣV

=

r

X

i

σ

u

v

, (1.13)

where U and V are orthogonal matrices and Σ is a diagonal matrix containing the r singular values (SV) of J, σ

, that is the square root of the non-zero eigenvalues of JJ

and J

J. The columns u

of U represent the orthonormal sensor modes while the columns v

of V are the orthonormal actuator modes. The pseudo-inverse of J reads

J

= VΣ

U

=

r

X

i

1

σ

v

u

. (1.14)

6It can be shown that the number of independent rows of a matrix is always equal to the number of independent columns.

22 1 Introduction

We note that the various contributions to the sum are proportional to σ

, hence it is singular when σ

approaches zero, leading to diverging voltages.

The condition number of J

J is defined as:

c(J

J) = ( σ

σ

)

, (1.15)

where σ

and σ

are respectively the largest and the smallest singular values of J. A problem with a low condition number is said to be well-conditioned, while a matrix with a large condition number is closer to the singularity and is said to be ill-conditioned. To avoid the inversion of almost-singular matrix, the terms of the expression (1.14) associated with the smallest singular values can be truncated.

This reduces the voltages v

, at the expense of a slight increase of the reconstruction error. Figure 1.21 illustrates this with a DM with 25 keystone electrodes (this DM will be analyzed in detail in chapter 3). In the example, the target surface is a coma

with an amplitude of 1 µm. The SV of J

J range from 10

to 10

. If all the SV are taken in the computation of the pseudo-inverse, the voltage distribution is that shown in the bottom left of the Figure 1.21; the maximum voltage difference is ∆V = 211 V and the RMS surface figure error is 9.6 nm. Notice the large values of the voltages outside the pupil because they have little effect on the mirror shape inside the pupil. If the lowest three SV (below 10

) are removed from the expression, the voltage difference is reduced to ∆V = 51 V, at the expense of a slight increase of the RMS error, to 13.5 nm.

1.4.2 Tikhonov regularization

As an alternative, the Tikhonov regularization, also known as the Damped Least Squares method (DLS), can be used to increase the smallest singular values of the Jacobian. In fact, rather than just finding the minimum vector, v, that gives the best solution to Eq.(1.12), the problem can be replaced by (Fuhry and Reichel, 2012) Find v

so that

min

{kw

− Jv

k

+ α

kv

k

}. (1.16) where α is the damping factor and weighs the conflicting requirements of minimizing the surface figure error at the sensor and minimizing the voltage budget. In other words, the bigger α, the higher is the RMS error and the smaller are the voltages.

It is easy to show that the solution to this problem is given by:

v

= (J

J + α

I)

J

w

. (1.17)

7See appendixAfor more details on coma and other Zernike modes

1.4 Feedforward control 23

DLS:

3 modes truncated SVD:

Condition number

10 15 30

5 20 25

10^-10

10^-12

10^-14

10^-16

103 V

-107 V

= 9.6 nm RMS_Err.

D V = 211 V

= 13.5 nm RMS

D V = 51 V

= 13.3 nm RMS

D V = 49 V

SVD DLS

J

J J

J +

a I

10^-18

Index i [/]

Coma

2

: Singular value s

a =1x10

s + a

i2 2 2

s

s

s

Figure 1.21: Top: singular valuesσ²_i ofJ^TJand ofJ^TJ+α²I forα= 10⁻⁸. Bottom:

Voltage distribution leading to a coma of1µm; Left: Moore-Penrose inverse [the condi- tion number ofJ isc(J) = 10⁴]; Center: the lowest 3 singular values are truncated [re- ducing toc(J) = 10³], Right: the DLS is used [in such a way thatc(J^TJ+α²I) = 10⁶]).

In particular, when α is set to zero, Eq.(1.17) is equivalent to Eq.(1.11). In practice,

α is chosen so as to make (J

J+α

I) invertible. α should be large enough so that the

solution for the voltages, v

, are well-behaved near singularities. On the other hand,

if α is too large, unreasonable weight is attached to the voltages and it results in an

augmentation of the RMS error, in the same way as the SVD truncation decreases

the quality reconstruction of the target shape.

24 1 Introduction

The damped least squares method can be understood using the SVD as shown in (Buss, 2004):

(J

J + α

I)

J

=

r

X

i

σ

σ

+ α

v

u

. (1.18) The comparison between Eq.(1.18) and Eq.(1.14) makes clear the relationship be- tween the damped least squares method and the pseudo-inverse. When α is small with respect to the singular values, both expression are equivalent. Conversely, when α is bigger than σ

:

1 σ

6= σ

σ

+ α

f or σ

−→ 0. (1.19)

Since the singular values of (J

J + α

I) are σ

+ α

, when α

is bigger than σ

, the smallest singular values are close to α

as shown by the red curve in Figure 1.21.

Thus, the damping factor α acts as a cut-off and tends to smooth the performance of the pseudo-inverse in the neighborhood of singularities, that is when σ

−→ 0.

This gives a rule to select α: α

should be equal to the threshold below which the SV σ

should be discarded. The DLS method is also illustrated in Figure 1.21 when α

= 10

. The results are very close to those obtained by the SV truncation. A second approach to select α is explained in 2.3.2.

The voltages map of Figure 1.21 also shows that (1) the SVD and the DLS methods

lead to the same voltages field and RMS error, since the condition number is very

close in both cases and (2) the voltages of the electrodes outside the optical pupil are

much more affected than the inner-electrodes which are almost unchanged (Alaluf

et al., 2016).

1.5 Outline 25

1.5 Outline

This thesis is concerned with the shape control of piezoelectric mirrors for Adaptive Optics. The technological aspects are investigated, numerical simulations are used to predict the behavior of plate and shell reflectors, and experiments are performed.

The study has been supported by the European Space Agency (ESA), in the frame- work of the GSTP program.

The thesis is organized in the following way:

• Chapter 2 presents a concept of a light weight segmented bimorph mirror for adaptive optics. The deformable mirror is based on independent PZT patches glued in a honeycomb layout on Silicon wafers. The technological aspects of the segment design are described, the morphing strategy is discussed, and experimental results are presented.

• Chapter 3 describes the design, the manufacturing and the testing of a mono- lithic deformable mirror controlled by independent actuators with a keystone layout and done by laser ablation on a single PZT patch. The performances of the demonstrator are evaluated in terms of RMS wavefront error on the first fifteen Zernike modes, voltage budget, rigid-body actuation, reflectivity and lowest eigen-frequency.

• Chapter 4 is dedicated to the analysis of alternative designs, with a particular attention devoted to the thermal equilibrium and the capability to control the mirror with only positive voltages.

• Finally, chapter 5 investigates the feasibility of the shape control of a small size active thin shell (with double curvature). The prototype is intended to be a technology demonstrator of a future large and very light primary reflector for space telescopes.

Figure 1.22 summarizes the Adaptive Optics activities carried out at Active Struc-

tures Laboratory.

26 1 Introduction

Figure 1.22: Evolution of the Adaptive Optics activities at ASL.

1.6 References 27

1.6 References

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Dierickx, P. Optical performance of large ground-based telescopes. Journal of Mod- ern Optics, 39:569:588, 1992.

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American Institute of Aeronautics and Astronautics, 2003.

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Gardner, J. The james webb space telescope. Space Science Reviews, page 123, 2006.

Geary, J. M. Introduction to Lens Design - With Practical ZEMAX Examples.

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Hardy, J. W. Adaptive optics for astronomical telescopes. Oxford University Press.

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Horsley, D. A., Park, H., and Laut, S. P. Characterization of a bimorph deformable mirror using stroboscopic phase-shifting interferometry.

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Jianqiang, M., Ying, L., Ting, H., Baoqing, L., and Jiaru, C. Double drive modes unimorph deformable mirror for low-cost adaptive optics. Applied optics, 50(29):

5647–5654, 2011.

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keckobservatory.org/.

Madec, P.-Y. Overview of deformable mirror technologies for adaptive optics and astronomy. In SPIE Astronomical Telescopes+ Instrumentation, pages 844705–

844705. International Society for Optics and Photonics, 2012.

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References 29

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30 References