AdaptiveOpticsWithSegmentedDeformableBimorphMirrors Universit´eLibredeBruxelles

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Universit´ e Libre de Bruxelles

F a c u l t ´e d e s S c i e n c e s A p p l i q u ´e e s

Adaptive Optics With

Segmented Deformable Bimorph Mirrors

Gon¸calo Mendes da Costa Rodrigues

Thesis submitted in candidature for the degree of Doctor in Engineering Sciences

Active Structures Laboratory

Department of Mechanical Engineering and Robotics

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Jury

President : Prof. Frank Dubois (ULB) Supervisor : Prof. Andr´e Preumont (ULB)

Members :

Dr. Fr´ed´eric Falzon (Thales Alenia Space, France) Prof. Enrico Filippi (UPMons)

Prof. Marc Haelterman (ULB) Dr. Norbert Hubin (ESO, Germany) Prof. Claude Jamar (AMOS, Li`ege) Prof. Carlos Mota Soares (IST, Portugal) Dr. Yvan Stockman (CSL, Li`ege)

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Acknowledgments

First of all, I would like to thank Professor Andr´e Preumont, for accepting me at the Active Structures Laboratory (ASL) and for permanently challenging and guiding me through the extremely exciting and relevant topic that constitutes this thesis.

I am also grateful to all my colleagues at ASL. Very particularly to Renaud Bastaits who implemented the optical measurement system used in the experi- mental optical bench, and by doing his thesis in the related field of Active Optics, provided a permanent source of interaction throughout these four years. I am equally grateful both to Dr. Mihaita Horodinca, who co-designed with me and machined the mechanical supports of the two prototypes developed, and fully assembled them and to Dr. Ioan Burda who designed and assembled the elec- tronic board of the second prototype. Samuel Veillerette is also acknowledged by having generated the CAD model illustrating the concept proposed in this thesis and which forms its cover. Even if not directly involved in this research, I want to thank the permanent availability of Dr. Bruno de Marneffe for sharing some of his extensive knowledge of the finite element code SAMCEF, as well as that of Pierre Letier and Dr. Arnaud Deraemaeker in helping to solve many IT problems.

From the ULB Technology Transfer Office, I am very indebted to Alain Weymeer- sch who provided an enormous support to the preparation of the applied patent and I am also grateful to Pierre Galland who took his role since January 2009.

I am equally thankful to Prof. Frank Dubois and Dr. Yvan Stockman, who formed myCommit´e d’Accompagnement, for their feedback and encouragement.

Due to its multidisciplinary character, the research that lead to this thesis in- cluded many partners in industry, universities and research institutes to whom I am very grateful. Dr. Steve Roose, from theCentre Spatiale de Li`ege (CSL), gave extensive support in the implementation of the optical measurement sys- tem. Dr. Andreas Schoenecker, Dr. Sylvia Gebhardt and Stefan Uhlig, from Fraunhofer Institute - IKTS in Dresden, developed an extensive effort in tuning the process of Screen Printing to the needs of bimorph deformable mirrors. Prof.

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Pierre Villon, from the Universit´e Technique de Compi`egne (UTC), was available for several fruitful discussions that helped in the establishment of the control method used. Jean-Philipe Verschueren, from Micromega Dynamics, checked the design of the first electronic board and suggested very useful improvements. Dr.

Arnaud Liotard, from Thales Alenia Space, kindly provided the routines used for simulating the atmospheric turbulent screens. Finally, Kris de Coninck, Dr.

Xavier Hutsebaut, Dr. Luc Joannes and Dr. Olivier Dupont from Lambda-X for allowing us to freely use their optical sensor NIMO and providing considerable support.

I am also very thankful to those suppliers who by their curiosity and availabil- ity in exploring new solutions exceeded their expected role. These were Bruno Vilters from Electronic Apparatus NV, Hein Schellens from Heinmade BV and Ralph Dorn and Dr. Johannes Pfund from Optocraft GmbH.

I am extremely grateful to the PortugueseFunda¸c˜ao para a Ciˆencia e a Tecnolo- gia(FCT) for funding my PhD grant SFRH/BD/21732/2005 as well as the bench fees. Equally, to the BelgianFonds de la Recherche Scientifique (FNRS) for fund- ing almost entirely the hardware purchased and developed within this research.

The European Science Foundation (ESF) is also acknowledged for organizing and funding several scientific meetings in the framework of the EUROCORES S3T Programme and for covering the expenses of our participation.

I am also very thankful to Dr. Juli´an Santiago from the European Space Agency (ESA) who is in the origin of my coming to ASL.

I want to thank my friends, Carlos Sim˜ao Ferreira, Herdis Heinemann, Fausto Heinemann Ferreira, Ana Catarino and Rui S´a whose presence nearby was ex- tremely supportive. Also to my farther away friends Miguel, Ana, Maria Jos´e and Juviano Bacalhau as well as to Nuno Silva, for having come to Brussels on purpose to visit me.

And last, but certainly not the least, to my family for more than can be expressed by words. To the parents and grandparents of Ver´onica. To my aunts, uncles and cousins. To my grandparents. To my mother and my father. To Cl´ovis, for the inspiring sleep deprived nights of the last 3 months. And of course, to Ver´onica.

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Abstract

The degradation of astronomical images caused by atmospheric turbulence will be much more severe in the next generation of terrestrial telescopes and its compen- sation will require deformable mirrors with up to tens-of-thousands of actuators.

Current designs for these correctors consist of scaling up the proven technologies of flexible optical plates deformed under the out-of-plane action of linear actua- tors. This approach will lead to an exponential growth of cost with the number of actuators, and in very complex mechanisms.

This thesis proposes a new concept of optical correction which is modular, robust, lightweight and low-cost and is based on the bimorph in-plane actuation.

The adaptive mirror consists of segmented identical hexagonal bimorph mirrors allowing to indefinitely increase the degree of correction while maintaining the first mechanical resonance at the level of a single segment and showing an in- crease in price only proportional to the number of segments.

Each bimorph segment can be mass-produced by simply screen-printing an array of thin piezoelectric patches onto a silicon wafer resulting in very compact and lightweight modules and at a price essentially independent from the number of actuators.

The controlled deformation of a screen-printed bimorph mirror was experimen- tally achieved with meaningful optical shapes and appropriate amplitudes; its ca- pability for compensating turbulence was evaluated numerically. The generation of continuous surfaces by an assembly of these mirrors was numerically simulated and a demonstrator of concept consisting of 3 segments was constructed.

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Contents

Jury iii

Acknowledgments v

Abstract vii

1 Introduction 1

1.1 Adaptive Optics and Astronomy . . . 1

1.2 The Atmosphere . . . 4

1.2.1 Refraction . . . 5

1.2.2 Absorption . . . 5

1.2.3 Emission . . . 6

1.2.4 Turbulence . . . 6

1.2.5 Sites for Optical Astronomy . . . 8

1.3 Adaptive Optics Systems . . . 10

1.3.1 The State-Of-The-Art of Adaptive Optics Systems . . . 12

1.3.2 Adaptive Optics for Extremely Large Telescopes . . . 13

1.4 Outline . . . 14

1.5 References . . . 16

2 Atmospheric Turbulence 19 2.1 Introduction . . . 19

2.2 Diffraction Limited Imaging . . . 20

2.3 Imaging Through Atmospheric Turbulence . . . 23

2.3.1 Effect On The Image . . . 23

2.3.2 Effect On The Wavefront . . . 25

2.4 Modal Decomposition . . . 25

2.4.1 Zernike Polynomials . . . 26

2.4.2 Karhunen-Lo`eve functions . . . 28

2.5 Zonal Correction . . . 30

2.6 Influence Of Primary Mirror . . . 30 ix

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2.7 Temporal Bandwidth . . . 32

2.8 Conclusions . . . 34

2.9 References . . . 36

3 Deformable Mirrors 37 3.1 Introduction . . . 37

3.2 Actuation Principles . . . 40

3.2.1 Segmented Mirrors . . . 40

3.2.2 Bimorph Mirrors . . . 41

3.2.2.1 Monomorph Mirrors . . . 43

3.2.3 Piston Actuated Mirrors . . . 45

3.2.3.1 Micro Electro Mechanical Systems - MEMS . . . . 46

3.3 Existing Technologies . . . 47

3.3.1 Segmented Mirrors . . . 47

3.3.2 Bimorph Mirrors . . . 48

3.3.3 Piezo-Stacked Actuated Mirrors . . . 49

3.3.4 Voice-Coil Deformable Mirrors . . . 51

3.3.5 Electrostatic Membranes . . . 54

3.3.6 Micro Electro Mechanical Systems - MEMS . . . 54

3.4 Conclusions . . . 55

3.4.1 Challenge . . . 55

3.4.2 Scalable Bimorph Mirrors . . . 55

3.5 References . . . 58

4 Control of Bimorph Mirrors 61 4.1 Control of Adaptive Optics Systems . . . 61

4.2 Control Architectures: Zonal Control . . . 63

4.3 Control Architectures: Modal Control . . . 64

4.3.1 Minimizing the Mean Square Error . . . 64

4.3.1.1 Real-Time Matrix Operations . . . 67

4.3.2 Discrete Control . . . 68

4.4 Numerical Simulations Of Shape Control . . . 69

4.4.1 Finite Element Analysis Of Mirror Deformation . . . 69

4.4.2 Generating The Zernike Polynomials . . . 71

4.4.3 Simulating The Compensation of Turbulence . . . 76

4.5 Conclusions . . . 79

4.6 References . . . 81

5 Screen Printed Silicon Bimorph Mirrors 83 5.1 Introduction . . . 83

5.2 Screen-Printed Monomorph mirrors . . . 84

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CONTENTS xi

5.3 Experimental Results . . . 86

5.3.1 Adaptive Optics Bench . . . 86

5.3.2 Closed-Loop Control . . . 89

5.3.3 Open-Loop Control . . . 91

5.4 Design Sensitivity . . . 93

5.4.1 Increased Size . . . 94

5.4.2 Increased Thickness Of The Active Layer . . . 94

5.5 Conclusions . . . 95

5.6 References . . . 96

6 Scalable Bimorph Mirrors 97 6.1 Introduction . . . 97

6.2 Modular Bimorph Mirrors . . . 98

6.2.1 Segment Co-Phasing . . . 99

6.2.2 Dynamic Behavior . . . 99

6.2.3 Performance Scale-Up . . . 100

6.3 Assembly Shape Control . . . 100

6.3.1 Mixed Zonal-Modal Control . . . 100

6.3.2 Simulations . . . 102

6.3.3 Real-Time Advantages . . . 104

6.3.3.1 Segment Parallel Control . . . 104

6.4 3-Segment Prototype . . . 106

6.5 Permanent Curvature . . . 111

6.6 References . . . 115

7 Conclusions 117 7.1 Original Aspects Of The Work . . . 118

7.2 Future Perspectives . . . 118

A Astronomical Sites 123 B Generation of Turbulent Screens 125 C Global Deformations of Bimorph Mirrors 127 C.1 Bimorph Mirror . . . 127

C.2 Monomorph Mirror . . . 128

C.3 References . . . 130

D Simulation of Distributed Shape Correction 131 D.1 Finite Element Analysis . . . 131

D.2 Least-Squares Projection . . . 132

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E Shack-Hartman Sensors 135

F Patent Application 141

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Chapter 1

Introduction

1.1 Adaptive Optics and Astronomy

Observations of the sky with the naked eye led to the establishment of calendars in many civilizations throughout the world. The invention, in the early 17th century by Hans Lippershey, of the refractive telescope constituted of two lenses allowed Galileo to observe the moons of Jupiter and provided flagrant evidence of the inadequacy of the Geocentric model of the Universe. Soon after, reflective telescopes enabled the construction of larger observatories without the chromatic aberration of refractive optics, yielding observations with greater detail and the detection of fainter celestial objects. However, it soon became clear that at- mospheric turbulence introduced a plateau on the level of detail that could be resolved for astronomical objects, despite further enhancements of telescope tech- nologies. In 1704, Isaac Newton proposed the first mitigating actions: ”The only remedy is a most serene and quiet air, such may perhaps be found on the tops of the highest mountains...”. Constructed in the beginning of the 20th century, the altitude observatory at Mount Wilson, near Los Angeles, combined a more stable atmosphere and a state-of-the-art telescope with 2.5 m diameter. In the 1920’s, it allowed Edwin Hubble to estimate the scale of the Universe and to observe its expansion.

The first form of active compensation of the atmospheric effects in astronomy can be traced back to the fast automatic guiding of the telescope, introduced in the 1930’s (Whitford and Kron, 1993). It enabled the stabilization of the position of the image and thereby rejected the disturbances introduced by the deformation of the telescope structure as well as by the atmospheric turbulence.

The elimination of the blurring in astronomical images introduced by the turbu- lent atmosphere was first achieved in the early 1980’s at the American Air Force

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Maui Optical Site, at Mount Haleakala in the Hawaii islands. The objective was to capture clearer images of russian satellites by employing deformable mirrors operated in real-time in a closed-loop shape control system. As a by-product, it provided the first application of the field of Adaptive Optics to astronomical observations (Hardy, 1998). The first full compensation of atmospheric turbu- lence effects on astronomical imaging was achieved at the Observatoire de Haute Provence. It was the result of a common effort of a consortium constituted by the French National Laboratory of Aerospace Research (ONERA), the Observatoire de Paris - Meudon, the University of Paris 7, the French company Laserdot (now CILAS) and the European Southern Observatory (ESO) (Rousset et al., 1990).

While the field of Adaptive Optics focused on the capacity for correcting at- mospherically induced aberrations with greater spatial complexity, the field of Active Optics evolved in parallel, aiming at compensating the response of the telescope structure to the Earth environment which produces optical aberrations with higher amplitude, though slower and more spatially uniform. Active Optics eliminated the need for constructing massive, rigid and inert telescopes which would become extremely costly and difficult to orientate. Instead, it allowed the construction of telescopes with primary mirrors with more than 8mof diameter formed of meniscus glass plates or multiple segments. In these flexible structures, the distortions induced by gravity, wind and thermal loads are compensated by hundreds of actuators. Also, lighter structures supporting secondary mirrors and exposed to wind buffeting could be constructed despite the requirements of shape accuracy being in the order of the tens of nanometers. Additional improvements were attained with refined dome design for reducing wind excitation and adequate ventilation to prevent the formation of turbulence in the vicinity of the telescope.

These innovations were incorporated in the state-of-the-art north-american Keck telescope (Waldrop, 1990) and the European Very Large Telescope (VLT) (Eu- ropean Southern Observatory, 1998). These are illustrated in Figs. 1.1 and 1.2, respectively. An additional feature of these telescopes is the fact that they are constituted of multiple units that can operate in an interferometric mode forming a sparse aperture with enhanced resolution.

Global awareness about space astronomy increased in 1993 after a servicing mis- sion by the Space Shuttle corrected a fabrication error of the Hubble Space Tele- scope. Space astronomy allows for access to the bands of the electromagnetic spectrum which are blocked by the Earth atmosphere, it avoids the aberrating effects of atmospheric turbulence and it is made in an environment with very low background radiation. However, the price to pay for making astronomy in space is very high. To the launch cost, it has to be added the cost associated to the

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1.1 Adaptive Optics and Astronomy 3

Figure 1.1: The Keck observatory is constituted of 2 telescopes. Each telescope has a primary mirror 10m wide and formed of 36 hexagonal segments arranged as a honeycomb. The observations of the two units can be combined by interfer- ometry providing images with a higher resolution (such as could be obtained by a telescope with a diameter equal to their distance of 90m).

development and fabrication with very tight tolerances and the extensive verifi- cation campaigns. Due to the short-range and extremely high operating-cost of the Space-Shuttle, space telescopes need to operate autonomously and with very low perspectives of maintenance, reparation or up-grading. Moreover, the current launch vehicles provide a diameter of the payload compartment of the order of 5 m, and thus, larger apertures will have to be folded during launch and deploy once in orbit with a precision in the order of the tens of nanometers (Gardner et al., 2006). This poses not only additional technological challenges but also greater development, construction and operating costs.

So far, Active and Adaptive Optics enabled ground-astronomy to overtake the Hubble Space Telescope. The next 15 years will see the construction of tele- scopes with primary mirrors in the order of 3050m. The European Extremely Large Telescope (E-ELT) (Evans, 2008) and the North-American Thirty-Meter- Telescope (TMT) (Nelson and Sanders, 2008) aim at observing back in time when galaxies were being formed and at providing direct images of planets orbit- ing other stars. In addition to the challenge posed by constructing and accurately maintaining the figure of such large optical components, atmospheric turbulence will be much more uncorrelated over such large areas. It will thus be much more difficult to correct and it will call for deformable mirrors with up to tens-of- thousands of degrees of freedom, creating a new challenge for the Adaptive Optics

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Figure 1.2: The VLT observatory. Left: the 4 main telescopes and 3 of the auxiliary telescopes. The auxiliary units ensure a permanent utilization of the interferometric mode while the main telescopes operate independently. Top right:

VLT on the top of Cerro Paranal and the permanent cloud cover of the Pacific Ocean. Bottom right: monolithic glass mirror of the main telescopes with 8m of diameter.

community. This thesis proposes a paradigm-shift in the design and construction of deformable mirrors having such a complex capacity of correction while offering robustness, simplicity of control and low-cost fabrication.

This introductory chapter proceeds with a brief description of the effects of the atmospheric turbulence on the light emitted from outer-space, leading to the cri- teria for selecting the best sites for astronomical observations. Follows a descrip- tion of the main components of a typical Adaptive Optics system with reference to the state-of-the-art of existing systems as well as the systems foreseen for the future 3050m class telescopes.

1.2 The Atmosphere

The atmosphere of the Earth constitutes an obstacle to light incoming from astro- nomical objects. Its non-uniformity, which induces the bending of the light rays, the molecular excitation which absorbs most of the electromagnetic spectrum, the de-excitation that creates background radiation and its unsteady turbulent motion, affect light propagating through it and consequently degrade the quality of astronomical images.

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1.2 The Atmosphere 5

1.2.1 Refraction

The density of the atmosphere, and thus the index of refraction, decrease with the height. This causes the continuous bending of light incoming from outside the atmosphere due to refraction, by the same phenomenon occurring in any change of media like that producing the apparent localized bending of objects semi- immersed in liquids. In astronomy, refraction apparently places objects higher in the sky, the deviation increasing when the observation becomes more horizontal.

1.2.2 Absorption

The molecules of the atmosphere absorb most of the electromagnetic radiation incoming from outer space. This filtering shields life on earth from dangerous radiations but also blocks a big portion of the electromagnetic spectrum from reaching the surface of the Earth, as is illustrated in Fig. 1.3.

Observations on the surface of the Earth can therefore only be made for two re- gions: the optical window which encompasses the visible and part of the infrared and the radio window. In the optical window, the atmosphere is essentially trans-

Figure 1.3: Filtering of the electromagnetic spectrum by the atmosphere of the Earth. Only the optical and radio windows reach the surface of the Earth.

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parent in the range of 0.30.8 µm which includes the visible, the source of all visual information for humans. Some bands of the infrared, between 0.825µm, can also be observed. Above the 25 µm all the radiation is filtered up to the millimeter range.

In radio astronomy, the wavelengths are much longer, which relaxes the required surface accuracy of the reflectors, allowing for larger baselines of observation.

The millimetric range, 1100 mm constitutes the microwaves domain which are highly absorbed by water vapor in the atmosphere in the same phenomenon that heats food inside a microwave oven. Microwave observatories must thus be located at high altitude with a low mass of water vapor above. The ALMA obser- vatory (Atacama Large Millimeter Array) is currently being constructed in the Atacama desert in Chile at an altitude of 5000m, and should be operational by 2012. On the other hand, the radio telescope of Arecibo, Puerto Rico, operates at wavelengths of tens of centimeters, to which the atmospheric molecules and even the clouds are completely transparent. It is located at only 300m altitude and the 300 m diameter dish is suspended just a few meters above a luxuriant tropical forest.

1.2.3 Emission

Emission and dispersion of light by atmosphere molecules or particles in suspen- sion creates a background which complicates the isolation of astronomical objects due to a lower sensitivity in the observations. Scattering of the sun and moon lights makes observations in the visible and near-infrared impossible during day- time and significantly reduces the sensitivity of observations during the full moon.

At night, the main source of emission at the optical wavelengths is the ”airglow”

which consists of the de-excitation of molecules and ions in the atmosphere. Ther- mal emission of the atmosphere between 230 and 300 K is specially important for wavelengths above 2.3µm and needs to be added to the thermal emission of the telescope itself. The background emissions can be partially eliminated by a technique calledchopping. It consists of tilting the mirror for imaging at a high frequency in order to produce slightly deviated images. The subtraction between the two images allows the significant elimination of the background radiation and consequently increases sensitivity.

1.2.4 Turbulence

The energy received from the sun heats up the air in the atmosphere inducing convection flow. The transition of this flow to turbulent causes the breaking and

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1.2 The Atmosphere 7

Figure 1.4: The random temporal and spatial distribution of the index of refrac- tion of the turbulent atmosphere non-uniformly delays the wavefront initially flat before entering the atmosphere.

mixing of layers at different temperatures and thus having different index of re- fraction. In this way, light incoming from outer space will propagate through a non-uniform medium. The non-uniform refraction suffered by the rays of light, illustrated in Fig. 1.4, does not significantly bend them in their propagation through the atmosphere and has no significant impact on imaging, but the non- uniform delay it causes to an initially flat wavefront does. For radio astronomy, the spatial delay introduced is very small compared to the wavelengths so that the changes in phase and the image degradation due to atmospheric turbulence are negligible. At the optical wavelengths, however, the change in phase is sig- nificant, resulting in image motion and blur. In the visible, telescopes with a diameter larger than about 15 cm do not provide any improvement in terms of resolution due to this blur known asseeing.

Seeing can be characterized in terms of a few parameters. The Fried length r0 (Fried, 1965) corresponds to the diameter of a bundle or rays travelling to- gether and in phase. It can be regarded as the maximum diameter of a telescope imaging without significant atmospherically induced degradation. The Fried pa- rameter depends on the observation site, time of the day and angle of observation.

Better images are obtained with conditions corresponding to a longer Fried length.

Typical values in a good observation site are about 0.2min the visible and 1.4m in the near infra-red (Bely, 2003), obeying the tendency (Hardy, 1998):

r0 λ6/5 (1.1)

The intensity of turbulence can also be described by the angular parameter called seeing disk (Dierickx, 1992) which corresponds to the minimum angle that can be

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resolved in the sky considering that the position of the image has been stabilized but no other active means of correction are employed. It is the ratio between the wavelength of observationλand the Fried lengthr0:

seeing = λ

r0 (1.2)

Better resolution corresponds to smaller seeing angles which in the good obser- vation sites are about 0.5” in the visible and 0.33” in the near infra-red.

Since the main source of turbulence above high altitude sites consists essentially of layers of wind shear, the temporal phase fluctuation caused by turbulence can be realistically attributed to a permanent horizontal random pattern of index of refraction being blown by the wind (this is called the frozen turbulence hypoth- esis). A characteristic time of atmospheric turbulence can then be estimated by considering the transit of a Fried length over the line of sight with a given wind velocityV (Fried, 1990):

τ0= r0

V (1.3)

τ0 is typically 10ms in the visible and 50ms in the near infrared.

Since the turbulence pattern is horizontally non-uniform, rays of light originating from two sources with a given angular deviation will undergo different random delays. If the light from one of these sources is used for characterizing the ex- isting turbulence and specifying the correction to be applied, there is a limit to the angular deviation to which this correction will be effective. This limit is the isoplanatic angle. Following the approach of one main layer of atmospheric tur- bulence, it can be approximated in terms of the ratio between the Fried length and the height of the layer of most intense turbulence (Fried, 1982):

θ0 = 0.3r0

h (1.4)

In a good observation site, the isoplanatic angle is typically 2” in the visible and 10” in the near infrared.

1.2.5 Sites for Optical Astronomy

The selection of the observatory site plays a crucial role on the intensity of the en- vironmental disturbances witnessed and on the construction and operation costs.

The characteristics of the ideal site for astronomy and a list of the current main observatories are presented in Appendix A.

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1.2 The Atmosphere 9

The site selection of the future E-ELT (Vernin et al., 2008) and TMT (Sch¨ock et al., 2008) telescopes was initiated with the analysis of satellite data to identify the locations with low cloud cover and low precipitable water. This allowed a pre-selection of sites where small autonomous stations were installed for moni- toring the local seeing properties.

Seeing originates from three main sources of turbulence:

Turbulence in the ground winds triggered by ground rugosity and topogra- phy;

Turbulence from the planetary and atmospheric boundary layers originating from the vertical air convection of the thermal diurnal cycle;

Turbulence due to shear between wind layers at high altitude in the free atmosphere.

Observatories located on the top of single peak islands benefit from the fact that the ground airflow predominantly diverges when reaching the island instead of climbing up mountain. It is even better if the islands are located away from large land masses since the airflow will be undisturbed by the ocean. This is the case of the Mauna Kea observatory in Hawaii and the Canary Islands. Cold seas that originate lower convection and reduce the height of the planetary boundary layer are even more beneficial. Single mountain sites located near flat-plains or deserts enjoy a similar protection to ground flows as single peak islands. Coastal sites with prevailing undisturbed winds from the sea are also attractive. This is the case of the Paranal observatory in Chile. Both on isolated peaks or islands or coastal sites, telescopes located near ridges will benefit from a more undisturbed airflow than if they are located further inland in the plateaus.

Tropical locations, which lie within ±23 latitude, present weaker high altitude winds which constitute the main source of high atmosphere seeing.

Despite being a remote location and having a harsh environment, the antarc- tic continent is privileged from the point of view of atmospheric seeing. It has the thinnest atmosphere, which is stable and stratified during the long winter nights. It is the coldest place on earth with very low atmospheric or optical- system thermal emission and is dust-free. It is clearly far-away from cities and the light pollution they create. In particular, on the Antarctic plateau it is ob- served that the majority of the optical turbulence is confined to a stable 30 to 40 m thick boundary layer. Therefore, observations will greatly benefit from being performed on the top of a sufficiently high tower (Travouillon et al., 2009).

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1.3 Adaptive Optics Systems

An astronomical observatory is schematized in Fig. 1.5. The large primary mir- ror (M1) collects the light incoming from the sky, which was disturbed by the atmosphere, and reflects it to the secondary mirror (M2). The secondary mirror, in turn, reflects the light back to the telescope infrastructure for image formation, scientific analysis and adaptive correction of blur and position.

The Adaptive Optics system is schematized inside the dashed box in Fig. 1.5.

The aberrations introduced to the beam of light by the atmosphere or the im- perfect optics of the telescope are cancelled by deformable mirrors. Right before the formation of the image, a portion of the light beam is extracted through a beam-splitter and is analyzed by the wavefront sensor. The wavefront sensor characterizes the aberrations still remaining, and feeds these measurements in real-time to a computer that determines the control signals to be applied to the deformable mirrors. The Adaptive Optics correction is beneficial to astronomy by enabling the formation of images with higher resolution and sensitivity, thus allowing the observation of finer details and fainter objects. It is also beneficial for spectrography by increasing both spatial and spectral resolution.

The optical correction is typically performed by a rigid tip-tilt mirror for stabi- lizing the image position and a deformable mirror (DM) for correcting the blur of the image. For correcting this blur, the shape of the deformable mirror is changed with the negative of the non-uniform delay introduced by the atmosphere. Since in the optical window, the speed of light in the air is essentially the same for all the wavelengths, the delays introduced by the fluctuating temperature are also reasonably independent of the wavelength and a given adjustment of the shape of the deformable mirror provides effective correction in a broad spectral band.

Alternatively, the optical correction can be made at the secondary mirror of the telescope,M2. In this case, the same mirror will provide blur correction and image stabilization. This configuration enormously simplifies the optical setup of the telescope and by reducing the number of mirrors, it increases the light throughput to the wavefront sensor and scientific instruments, and also reduces the scattering of light and reduces the noisy infrared background caused by ther- mal emission. This configuration has been successfully implemented on several telescopes (Brusa et al., 2003; Riccardi et al., 2003) and is now under develop- ment for the VLT (Arsenault et al., 2006).

Wavefront sensors (WFS) characterize the aberrations present in a beam of light

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1.3 Adaptive Optics Systems 11

mirror M2

mirror M1

Figure 1.5: Telescope and Adaptive Optics system (Egner, 2006). The light collected by the large primary mirrorM1is reflected back by the secondary mirror M2 allowing for wavefront correction, image formation and spectrography.

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by providing the topography of the points with the same phase, the wavefront.

This constitutes a three dimensional representation of the non-uniform delay in- troduced by the atmosphere and the imperfect telescope optics and corresponds to the symmetric of the shape to be imposed to the deformable mirror that would yield a perfect correction. The most common wavefront sensor used is of the Shack-Hartmann type. It discretizes the wavefront through an array of lenslets and determines the average slope of the wavefront within each lenslet in the x and y directions. Alternatively, there are sensors that measure the wavefront curvature and that can be coupled to a particular type of deformable mirrors (bimorph mirrors) significantly simplifying the Adaptive Optics control problem.

Most astronomical objects of interest are not bright enough to enable a cor- rect measurement by the wavefront sensor. Only in some cases, a brighter object exists in a nearby angular position whose light will traverse the same isoplanatic angle. Wavefront sensors with higher sensitivity (Ragazzoni, 1996) will mitigate this difficulty. Another alternative consists in creating artificial guide stars in the sky by means of lasers that excite the atoms in the atmospheric sodium layer situated at a height of 90 km or that are scattered back to the ground by the atmosphere.

1.3.1 The State-Of-The-Art of Adaptive Optics Systems

The main characteristics of the Adaptive Optics systems currently operating in the VLT and the Keck telescopes are presented in table 1.1. The sampling rate of the WFS,FS, is typically the most limiting element on the temporal rate of cor- rection, and the bandwidth of the Adaptive Optics system can be approximately estimated as one tenth of the sampling rate of the WFS.

NAOS, VLT MACAO, VLT Keck, AO System

DM, φaperture 115mm 100mm 146mm

DM, #actuators 185 60 349

DM, stroke p-v 10µm 48 µm 5µm

WFS, # sub-apertures 14 x 14 60 20 x 20

WFS, FS 500 Hz 350 Hz 700 Hz

Table 1.1: Main characteristics of current Adaptive Optics Systems.

NAOS was the first Adaptive Optics system to be installed in VLT (Fusco et al., 2004; Amico and Lidman, 2008; Feautrier et al., 2003). It comprises a tip-tilt mirror for providing image stabilization and a deformable mirror based on linear

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1.3 Adaptive Optics Systems 13

piezoelectric (PZT) actuators perpendicular to a flexible reflecting surface. The wavefront sensors are of the Shack-Hartmann type and can be used with 14 x 14 or 7 x 7 lenslet arrays, allowing a trade-off between spatial resolution and sensitivity (with more photons collected per sub-aperture).

The MACAO system (Donaldson et al., 2000), which stands for Multi-Application Curvature Adaptive Optics System, couples a bimorph deformable mirror to a curvature wavefront sensor. The deformable mirror is mounted on a tip-tilt plat- form avoiding the need for a separate tip-tilt mirror. In the bimorph deformable mirror, the actuators consist of active patches or continuous active layers sol- idary to the flexible reflecting surface. Each bimorph actuator induces a curva- ture which is predominantly localized within its boundaries and null elsewhere.

In this way, each actuator can be coupled to a single sub-aperture of the cur- vature sensor fully decoupling the control problem. The in-plane character of the bimorph actuators provide a much simple and robust configuration making bimorph deformable mirrors lightweight and low-cost than the more widespread technology of linear actuators. The main drawback is a lower frequency of cor- rection.

Like NAOS, the Adaptive Optics system of the Keck telescopes (van Dam et al., 2004; Oppenheimer et al., 1997) is based on a deformable mirror with linear actuators and a Shack-Hartmann wavefront sensor.

1.3.2 Adaptive Optics for Extremely Large Telescopes

Within the much larger apertures of the future extremely large telescopes the atmosphere will introduce aberrations with much more complex shapes and with globally higher amplitude. Deformable mirrors will thus demand a much higher number of control channels and will have as well a higher stroke. Wavefront sen- sors will require much denser lenslet arrays. Concerning image motion, the main contributor will be wind-induced vibrations of the telescope structure which will be more flexible and more exposed to wind rather than the effects of atmospheric turbulence on the propagation of light.

The baseline Adaptive Optics system for the TMT, NFIRAOS (Ellerbroek et al., 2008), will possess two deformable mirrors with linear actuators of about 10µm stroke. The first mirror will have 300mmof diameter and will be mounted on a tip-tilt platform for correcting high amplitude image motion. It will have an array of 63 x 63 linear actuators to correct atmospheric turbulence of the ground-layer.

A second mirror with 360mmdiameter will have an array of 75 x 75 actuators for the correction of the turbulence at high altitude. The wavefront sensors will have an array of about 60 x 60 lenslets. This method of employing several deformable

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mirrors each dedicated to a specific region of turbulence and utilizing a dedicated wavefront sensor is called Multi-Conjugate-Adaptive-Optics (MCAO) and aims at increasing the isoplanatic angle of an observation.

The E-ELT telescope will follow a different approach, by making the aberra- tion correction with a large 2.5 m diameter deformable mirror in an approach similar to the current secondary deformable mirrors and the stabilization of the image position also with rigid mirror of similar size (Vernet et al., 2008). Two detailed designs are currently under development for the main Adaptive Optics mirror, each based on a different technology. These concepts aim at achieving al- most 100µmstroke with 5000 to 8000 actuators. Additional deformable mirrors are foreseen for MCAO and other niche Adaptive Optics applications.

Multipurpose Adaptive Optics systems like NFIRAOS feed multiple astronomical experiments therefore avoiding the duplication of infrastructures. If the correc- tion is done at the level of the main mirrors of the telescope like theM2 of VLT or theM4 of E-ELT, it becomes even more advantageous by avoiding additional optics and mechanisms. However, some experiments have specific requirements such as the experiment SPHERE of VLT which aims at discovering and studying new Extra-Solar planets. This particular research requires a very high contrast but can be performed in very narrow Fields of View and therefore utilizing MEMS deformable mirrors only a fewcmwide and with several thousands of actuators.

1.4 Outline

This thesis proposes a new concept of deformable mirrors having the very high degree of correction demanded by future ELTs. The concept proposed is mod- ular, robust, lightweight and low-cost and provides a rupture with the current design paradigm.

Chapter 2 briefly characterizes turbulent atmospheric optics and how it drives the requirements imposed on deformable mirrors.

Chapter 3 reviews the state-of-the-art of deformable mirrors highlighting how scaling up proven technologies for compensating turbulence in the future ELTs will lead to very complex mechanisms and an explosion in price.

Chapter 4 details the architecture of the control loop of Adaptive Optics sys- tems and shows the simulation of the shape control of a bimorph mirror and its compensation of atmospheric turbulence.

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1.4 Outline 15

Chapter 5 shows the successful experimental results of the shape control of a sili- con bimorph mirror. This mirror was fabricated by automatically screen-printing the piezoelectric actuating patches, being lightweight and very compact.

Finally, chapter 6 details the design and predicts the scalability of performance of a modular assembly of bimorph mirrors. The construction of a demonstrator of concept consisting of 3 segments is also presented, along with the challenges posed by the permanent convex curvature generated during the sintering following the screen-printing process.

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1.5 References

Amico, P., A. N. and Lidman, C. VERY LARGE TELESCOPE NaCo User Manual. User manual, ESO, June 2008.

Arsenault, R., Biasi, R., Gallieni, D., Riccardi, A., et al. A deformable secondary mirror for the VLT. In Bonaccini Calia, D. and Ellerbroek, editors, Advances in Adaptive Optics II, Proc. SPIE 6272, 2006.

Bely, P. Y. The Design and Construction of Large Optical Telescopes. Springer, 2003.

Brusa, G., Riccardi, A., Wildi, F. P., Lloyd-Hart, M., et al. MMT adaptive secondary: first AO closed-loop results. In Tyson, R. K. and Lloyd-Hart, M., editors, Astronomical Adaptive Optics Systems and Applications, Proc. SPIE 5169, pages 26–36, 2003.

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

Donaldson, R., Bonaccini, D., Brynnel, J., Buzzoni, B., Close, L. M., Delabre, B., DuPuy, C., Farinato, J., Fedrigo, E., Hubin, N. N., Marchetti, E., Stroebele, S., and Tordo, S. MACAO and its application for the VLT interferometer. In Wizinowich, P. L., editor, Adaptive Optical Systems Technology, Proc. SPIE 4007, pages 82–93, 2000.

Egner, S. E. Multi-Conjugate Adaptive Optics for LINC-NIRVANA. PhD thesis, Max-Planck Institute for Astronomy, University of Heidelberg, November 2006.

Ellerbroek, B., Adkins, S., Andersen, D., Atwood, J., et al. Progress toward developing the TMT adaptive optical systems and their components. In Hubin, N. Max, C. E. W. P. L., editor, Adaptive Optics Systems, Proc. SPIE 7015, 2008.

European Southern Observatory. The VLT White Book. European Southern Observatory (ESO), Garching near Munich, 1998.

Evans, C. The European Extremely Large Telescope.Astronomy and Geophysics, 49, August 2008.

Feautrier, P., Rousset, G., Dorn, R. J., Cavadore, C., et al. Performance and results of the NAOS visible wavefront sensor. In Wizinowich, P. L. and Bonac- cini, D., editors, Adaptive Optical System Technologies II, Proc. SPIE 4839, pages 250–258, 2003.

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References 17

Fried, D. L. Statistics of a Geometric Representation of Wavefront Distortion.

Journal of the Optical Society of America, 55:1427–1435, November 1965.

Fried, D. L. Anisoplanatism in adaptive optics. Journal of the Optical Society of America (1917-1983), 72, 1982.

Fried, D. L. Time-delay-induced mean-square error in adaptive optics. Journal of the Optical Society of America A, 7, July 1990.

Fusco, T., Rousset, G., Rabaud, D., Gendron, E., et al. NAOS on-line character- ization of turbulence parameters and adaptive optics performance. Technical Report 6, 2004.

Gardner, J. P., Mather, J. C., Clampin, M., Doyon, R., et al. Space Science Reviews, 123:485–606, April 2006. doi: 10.1007/s11214-006-8315-7.

Hardy, J. W. Adaptive Optics for Astronomical Telescopes. Oxford University Press, Inc., New York, US, 1998.

Nelson, J. and Sanders, G. H. The status of the Thirty Meter Telescope project.

In Stepp, L. M., G. R., editor,Ground-based and Airborne Telescopes II, Proc.

SPIE 7012, 2008.

Oppenheimer, B. R., Palmer, D., Dekany, R. G., Sivaramakrishnan, A., et al.

Investigating a Xinξtics Inc. deformable mirror. In Tyson, R. K. and Fugate, R. Q., editors, Adaptive Optics and Applications, Proc. SPIE 3126, 1997.

Ragazzoni, R. Pupil plane wavefront sensing with an oscillating prism. Journal of Modern Optics, 43:289–293, February 1996. doi: 10.1080/095003496156165.

Riccardi, A., Brusa, G., Salinari, P., Busoni, S., et al. Adaptive secondary mir- rors for the Large binocular telescope. In Tyson, R. K. and Lloyd-Hart, M., editors, Astronomical Adaptive Optics Systems and Applications, Proc. SPIE 5169, pages 159–168, 2003.

Rousset, G., Fontanella, J. C., Kern, P., Gigan, P., et al. First diffraction-limited astronomical images with adaptive optics. Astronomy and Astrophysics, 230:

L29–L32, 1990.

Sch¨ock, M., Els, S., Riddle, R., Skidmore, W., et al. Status of the Thirty Meter Telescope site selection program. In Stepp, L. M., G. R., editor, Ground-based and Airborne Telescopes II, Proc. SPIE 7012, 2008.

Travouillon, T., Jolissaint, L., Ashley, M. C. B., Lawrence, J. S., et al. Over- coming the Boundary Layer Turbulence at Dome C: Ground-Layer Adaptive

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Optics versus Tower. Publications of the Astronomical Society of the Pacific, 121:668–679, 2009.

van Dam, M. A., Le Mignant, D., and Macintosh, B. A. Performance of the Keck Observatory Adaptive-Optics System. Applied Optics, 43:5458–5467, October 2004.

Vernet, E., Jochum, L., La Penna, P., Hubin, N., et al. The field stabilization and adaptive optics mirrors for the European Extremely Large Telescope. In Hubin, N. Max, C. E. W. P. L., editor, Adaptive Optics Systems, Proc. SPIE 7015, 2008.

Vernin, J., Mu˜noz-Tu˜non, C., and Sarazin, M. E-ELT site characterization status.

In Stepp, L. M., G. R., editor,Ground-based and Airborne Telescopes II, Proc.

SPIE 7012, 2008.

Waldrop, M. M. Keck’s First-Light. Science, 1990.

Whitford, A. E. and Kron, G. E. Photoelectric guiding of astronomical telescopes (Review of Scientific Instruments 1937). Selected Papers on Instrumentation in Astronomy, 1993.

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Chapter 2

Atmospheric Turbulence

2.1 Introduction

The atmosphere of the Earth has permanent thermal gradients resulting from the diurnal cycle and which originate convection currents and the turbulent mix- ing of layers with different temperature. This turbulent mixing is characterized by a random spatial distribution of vortices with a broad spectrum of sizes and evolving randomly in time. More globally, the distribution of these vortices can be regarded as homogeneous and the temporal changes as stationary. The differ- ences of temperature between these vortices imply different indices of refraction and therefore, light propagating through different vortices will be delayed in a different manner. As a result, a wavefront initially plane when entering the at- mosphere will be randomly delayed spatially and temporally while propagating through it, becoming distorted and originating a loss of resolution and sensitivity of astronomical images.

The degree of distortion introduced by the atmosphere is quantified by analyzing the light emitted by objects in the sky and which propagates through the same atmospherical path as the light from the astronomical objects being observed.

Natural stars which are bright enough to enable accurate measurements by the wavefront sensors can only be found in a small portion of the sky. In this way, the great majority of observations will require the creation of artificial guide stars by exciting the sodium layer of the atmosphere, at an altitude of 90km, with laser beams. The light emitted by these laser guide stars has a wavelength of 589nm and will allow to quantify the disturbances introduced by the atmosphere at this wavelength alone. However, light of different wavelengths will be affected in a different manner, and this mismatch will create an additional source of error to the Adaptive Optics correction. It is expected that in the future ELTs, mea-

19

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surements of the wavefront at 589nm will enable effective correction at all the wavelengths above 500nm. Atmospheric aberrating effects are very dependent on the wavelength below the 500nm range, completely compromising Adaptive Optics correction at the lower wavelengths of the visible (Devaney et al., 2008;

Owner-Petersen and Goncharov, 2004).

This chapter compactly characterizes atmospheric turbulence in a sense useful to the design of a deformable mirror. It starts with a brief description of the pro- cess of imaging and how this is affected by atmospheric turbulence at different wavelengths of observation. It then focuses on the observation site seeing char- acteristics, and the diameter of the primary mirror of the telescope. It is shown that the statistical description of the fluctuations of the optical path greatly ben- efits from resorting to a modal expansion in terms of polynomials of Zernike and Karhunen-Lo`eve functions, in particular for identifying the spatial spectral re- quirements imposed by the size of the primary mirror. The required temporal bandwidth of correction and stroke of the system are also addressed.

2.2 Diffraction Limited Imaging

The process of imaging in a telescope consists of the diffraction of the collected light through the telescope aperture, i.e. the primary mirror in a reflective tele- scope, and its focalization on the image plane.

The light-wave E(x) can be compactly described at each point of coordinates xwithin the aperture plane by its amplitude A(x) and phase ϕ(x) according to the phasor notation:

E(x) =A(x)e−jϕ(x) (2.1)

The image will be formed according to the distribution of the intensity of light focalized on the image plane. This consists of the squared amplitude of the Fourier Transform of the phasor of the light-wave at the aperture. For a point-source located at infinity, this intensity distribution is called the Point-Spread-Function (PSF) and its is given by:

P SF =||F{E(x)}||2 (2.2) The light emitted by a far away point-source arrives at the Earth’s atmosphere as a plane wave. In a non-aberrated system (i.e. without atmospheric turbu- lence, structural deformations and having perfect optics) the light arrives at the telescope aperture (the primary mirror) still as a plane wave (Fig. 2.1). It has a constant phase and an amplitude which is constant within the boundaries of the

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2.2 Diffraction Limited Imaging 21

Point Source Aperture Plane Image Plane

Figure 2.1: Imaging in a diffraction-limited optical system.

aperture and null elsewhere. This constant value of phase can then be taken as a reference and the phasor written as:

E(x) =

½ A0 within the aperture

0 elsewhere (2.3)

The image produced by diffracting a plane wave through a circular aperture is called the Airy disk and is shown on the left of Fig. 2.2 a). The radial profile of intensity is depicted as the solid line on the right.

If the point source is deviated from the optical axis of the imaging system by an angle αo, the plane wave will impinge obliquely on the aperture plane and the corresponding PSF will be shifted proportionally to αo. The resolution of an optical system resides on its capability for distinguishing the images of two distinct point sources, i.e. on distinguishing their PSF. It can be quantified, with sufficient accuracy for the purpose of this thesis, as the angular deviation at which the 2 PSF from 2 different point sources intersect at their half-maxima, as illustrated on the right of Fig. 2.2 a). Geometrically, it can be observed that this minimum angular deviation coincides with the full width at the half maximum (F W HM) of the PSF. For the diffraction through the circular aperture of an ideal un-aberrated system, theF W HM shows the following dependence:

F W HM ' λ

D rad (2.4)

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I

Aligned Point Source Deviated

Point Source

Strehl S =

Idiffraction limited

Iaberrated

Iaberrated

Idiffraction limited

Speckles

seeing angle

a)

b)

c)

FWHM

o

Figure 2.2: Images produced by a point source at infinity, i.e. PSF, (left) and corresponding intensity profiles (right). a) Airy disk produced by a diffraction- limited system: the resolution corresponds to the full width at half maximum (F W HM) of the PSF. b) Speckle pattern of an aberrated short-exposure image:

Definition of the number of Strehl,S, in terms of the intensity ratio between aber- rated and diffraction limited PSF. c) Long-exposure aberrated image: Definition of seeing as theF W HM of an averaged aberrated PSF.

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2.3 Imaging Through Atmospheric Turbulence 23

In Eq. 2.4, λ stands for the wavelength of observation and D stands for the diameter of the aperture. As can be seen, better resolution can be achieved with a largerD (as theF W HM decreases). This is consistent with the fact that the PSF is the squared magnitude of the Fourier Transform of the aperture function.

In fact, if the aperture increases indefinitely, the PSF will converge to a Dirac deltaδ(x), making the intersection of the PSF from two different sources less and less likely and thus allowing to distinguish closer sources.

2.3 Imaging Through Atmospheric Turbulence

The introduction of aberrations by the atmospheric turbulence distorts the ini- tially plane wavefronts and produces a non-constant distribution of phase at the aperture plane, as is shown Fig. 2.3. In the image plane, this results on the spread- ing of the PSF and therefore on the degradation of resolution and sensitivity of the astronomical images.

Point Source Aperture Plane Image Plane

Atmosphere

Figure 2.3: Imaging in an optical system with aberrations.

2.3.1 Effect On The Image

A short-exposure image affected by the atmospheric turbulence shows an irreg- ular spreading of the PSF in a pattern of speckles as depicted in Fig. 2.2 b).

Atmospheric turbulence can thus be seen affecting the image in two ways:

The loss of resolution (blurring), since the spreading of the PSF makes it harder to distinguish the images of two close point-sources.

The loss of sensitivity, as the maximum intensity of a point source dimin- ishes with relation to the background.

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The degree of aberration of an image can be quantified by the ratio between the maximum intensity of the aberrated PSF and that of the diffraction limited PSF, as illustrated in Fig. 2.2 b). This is the so-called number of Strehl:

S = max(Iaberrated)

max(Idif f raction limited) (2.5) In long-exposure images, the effects of the atmospheric turbulence tend to average- out and so the PSF recovers its axi-symmetric character as illustrated in Fig. 2.2 c).

The resolution of an image aberrated by the atmospheric turbulence is called the seeing angle, and corresponds once again to the F W HM of the PSF, now flat- tened with relation to the diffraction limited one and averaged. It is defined in terms of the wavelength, λ, and the Fried Length, ro, which characterizes the intensity of turbulence:

seeing = λ

r0 (2.6)

The seeing angle corresponds to the resolution of an aberrated image but whose position is assumed stabilized. It is therefore the F W HM of the surface aver- aging a PSF whose shape changed along time but which remained in the same position of the image plane.

A parallel can be established between the definition of seeing angle given just above and the definition of resolution from Eq. 2.4 for a diffraction limited sys- tem. For the seeing angle, the diameterDof the aperture being replaced by the coherence length r0 that corresponds to the diameter of the bundle of rays that propagate through the atmosphere without significant relative delays.

An average number of Strehl can also be established for long exposure images.

It is interesting to observe that because of the Parseval’s theorem of the Fourier Transform (Goodman, 1996) the volume of the PSF, i.e. the energy content in the images, remains constant for all the speckles and therefore for all the 3 Figs. 2.2 a), b) and c). The theorem states that:

Z

−∞

||E(x)||2dx= Z

−∞

P SF dα (2.7)

and it is applicable because it was assumed that the only effect of the atmospheric turbulence is a de-phasing produced by changes in the optical path which do not affect the integrand on the left side.

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2.4 Modal Decomposition 25

2.3.2 Effect On The Wavefront

The effects of the atmospheric turbulence can intuitively be described by the spatial distribution of the phase delayϕ(x) it introduces to an initially plane wave.

Specially meaningful is the average in time of the Mean-Square-Error (MSE) of the phase ϕ(x) with relation to a plane wave. For a circular aperture, and considering a distribution of the sizes of the vortices in the atmosphere according to the spectrum of Kolmogorov, it has been shown to depend only on the diameter of the telescope,D, and the Fried lengthr0(Roddier, 1999; Roggeman and Welsh, 1996):

σ2 =

¿ R ϕ(x)2dx A

À

= 1.03 µD

r0

5

3 (2.8)

Equation 2.8 shows that a diameter of the aperture equal to the Fried length, D=r0, yields an error root-mean-square (RMS) of the wavefront of σ ' 1 rad which is considered a threshold of acceptable image quality. This, in turn, sets a threshold for the telescope aperture which requires the use of Adaptive Optics:

For telescope apertures shorter than the Fried length r0, the resolution is limited by diffraction, no Adaptive Optics is required and the resolution of the system can be computed by Eq. 2.4.

Increasing the aperture abover0 will not improve resolution without Adap- tive Optics correction and, considering that the image stabilization is per- fectly achieved, resolution is given by Eq. 2.6.

A relation between wavefront error and image quality can be established by means of the ”Mar´echal Approximation” which is valid for a number of StrehlS greater than 0.1. In this case, the following conversion betweenSandσ2 holds (Roddier, 1999; Roggeman and Welsh, 1996):

S=e−σ2, S >0.1 (2.9) The threshold of image acceptability, i.e. σ = 1 rad, corresponds then to S w 0.37.

2.4 Modal Decomposition

Greater insight into the effects of atmospheric turbulence on imaging can be achieved by the decomposition of the phase distribution within the aperture in terms of a set of orthonormal base functions. The phase within the aperture

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varying randomly in time is expanded by means of randomly varying coefficients ai(t) and deterministic base functionsφi(x) in the form:

ϕ(x, t) = X

i=1

ai(t)φi(x) (2.10)

The base functions respect the orthonormality condition over the aperture:

Z

φi(x)φj(x)dx=δij (2.11)

In this framework, the statistics of the atmospheric turbulence can now be studied in a discrete manner, in terms of the covariance matrix of the expansion coeffi- cients< ai, aj >. Due to the orthonormality of the base functions, σ2 depends solely on the diagonal terms of the covariance matrix (Roddier, 1999; Roggeman and Welsh, 1996; Goodman, 1985). From Eqs. 2.8 and 2.10, it can be shown to be:

σ2 = X

i=1

< a2i > (2.12) The main advantage of performing the modal expansion of the phase becomes now clear. It provides a quick assessment of the performance of an adaptive optics system, if the degrees of freedom of correction are made to coincide with the orthonormal modes of the phase decomposition of Eq. 2.10. In this way, if the adaptive optics system is capable of correcting the first N modes of the expansion, then, the residual mean square phase averaged over the aperture that remains to be corrected can be written in the form:

σ2res N =σ2 XN

i=1

< a2i > (2.13) Moreover, as it will become clear later on, the modal decomposition provides a means for simulating turbulent screens.

2.4.1 Zernike Polynomials

The decrease ofσ2resin Eq. 2.13 with the number of corrected modes depends on the choice of the base functions. The shapes of some of the Zernike polynomials are specially attractive due to both their simple analytical representation and that the low order terms correspond to the common optical aberrations of piston (a constant change in phase), tilt (which corresponds to shifts in image posi- tion), defocus, decentering coma, astigmatism... The shapes of the polynomials

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2.4 Modal Decomposition 27

of Zernike, φi(x) =Zi(x), are illustrated in Fig. 2.4. Their analytical simplicity enables a closed form analytical solution to the components of the covariance matrix (Noll, 1976).

It is observed that the components of the covariance matrix associated to pis- ton are infinite. This stems from the fact that the Kolmogorov model is not applicable to the very low spatial frequencies. However, since the piston term of phase has no impact on imaging, this problem is overcome by simply removing the piston term from the modal expansion. To understand why the image is unaffected by the piston term in the wavefront, it suffices to observe Eq. 2.2 and that the magnitude of the Fourier Transform will remain the same regardless of any term of constant phase which is multiplied by the phasor of the light-wave

Piston

Tilt Tilt

Astigmatism Defocus Astigmatism

Trefoil Coma Coma Trefoil

Spherical Aberration

Astigmatism Astigmatism

Tetrafoil Tetrafoil

Coma Coma

Pentafoil Pentafoil

Spherical Aberration

Hexafoil Tetrafoil Astigmatism Astigmatism Tetrafoil Hexafoil

Azimuthal Order Radial

Order

0 1 2 3 4 5 6

-1 -2 -3 -4 -5 -6

0

1

2

3

4

5

6

Trefoil Trefoil

Figure 2.4: Polynomials of Zernike classified according to their radial (top to bottom) and azimuthal (left to right) orders.

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Zernike Modes

Karhunen-Loève Modes Zonal Correction

Number of corrected modes, N

100 101 102

100

10-1

10-2

10-3 DN,rad2

Figure 2.5: Coefficients ∆N of the residual σres2 in terms of the number of cor- rected modesN.

at the aperture in Eq. 2.1.

The residualσ2res can also be factored in terms of

³D ro

´5/3

in the form:

σres N2 =σ2 XN

i=1

< a2i >= ∆N µD

ro

5/3

(2.14) In this manner, the coefficients ∆N define the decrease of the residualσres2 with the number of corrected modes normalized by

³D ro

´5

3. These coefficients were computed by several authors (Noll, 1976; Roddier, 1990; Dai, 1995, 1996), and are illustrated in the solid line in Fig. 2.5. After the correction (discarding) of piston, the coefficients ∆N become finite, then, it is the correction of modes 1 and 2, i.e. tip-tilt, that produce the biggest decrease inσ2res.

2.4.2 Karhunen-Lo`eve functions

The base of Karhunen-Lo`eve functions shows a covariance matrix < ai, aj >

which is diagonal and has the highest diagonal components than any other base

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2.4 Modal Decomposition 29

of functions. It provides therefore a faster decay of the residual σ2res with the number of retained functions than for the base the Zernike polynomials or any other base of functions (from Eq. 2.13). However, Karhunen-Lo`eve functions have no analytical representation and must be expressed as a linear combination of other functions, like for example the Zernike polynomials, the coefficients being obtained by diagonalizing their covariance matrix.

Let us assume the expansion in terms of the optimal base in the form:

ϕ(x, t) = X

j=1

bj(t)ej(x) (2.15)

The first step for determining this optimal base consists of computing< ai, aj >

of a certain base, let us say the base of polynomials of Zernike, and to diagonalize it by means of a Singular Value Decomposition (SVD) (Roggeman and Welsh, 1996):

< ai, aj >=UΛUT (2.16) where Λ is the diagonal covariance matrix of the coefficients of the intended optimal expansion:

Λ=< bj, bj > (2.17)

and the matricesU contain the eigenvectors that transform the coefficients from one base to the other:

ai= X

j=1

Uijbj (2.18)

bj = X

i=1

Uijai (2.19)

Since< ai, aj >is Hermitian,U is a rotation matrix.

Finally, by replacing Eq. 2.18 in Eq. 2.10, it is possible to obtain the modes of the optimal base, ej(x), expressed in terms of the modes of the original base φi(x), in this case the Zernike polynomialsφi(x) =Zi(x) :

ej(x) = X

i=1

UijZi(x) (2.20)

Figure

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