From Adaptive Optics systems to Point Spread Function Reconstruction and Blind Deconvolution for Extremely Large Telescopes / eingereicht von Dipl.-Ing. Roland Wagner, Bakk.techn.

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Eingereicht von Dipl.-Ing. Roland Wagner, Bakk.techn. Angefertigt am Institut für Industriemathematik Erstbeurteiler Univ.-Prof. Dr. Ronny Ramlau Zweitbeurteiler Univ.-Prof. Dr. João Alves August 2017 JOHANNES KEPLER UNIVERSITÄT LINZ Altenbergerstraße 69 4040 Linz, Österreich www.jku.at DVR 0093696

From Adaptive Optics systems

to Point Spread Function

Reconstruction and Blind

Deconvolution for Extremely

Large Telescopes

Dissertation

zur Erlangung des akademischen Grades

Doktor der Technischen Wissenschaften

im Doktoratsstudium der

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Abstract

Modern ground-based telescopes rely on Adaptive Optics (AO) systems, which com-pensate for atmospheric turbulences in order to enhance the image quality. The hardware-based technology AO uses measurements of incoming wavefronts from bright light sources and artificially created laser guide stars to reconstruct the turbulence above the telescope and adjusts quickly moving deformable mirrors accordingly. For the up-coming generation of Extremely Large Telescopes (ELTs) with diameters up to 40 m, the computational effort of this technology is strongly increasing as real-time correction at a frequency of around 500 Hertz has to be performed. This results from the fact that the atmospheric turbulences are constant for approximately only 2 ms. Thus the main challenge is having a fast enough algorithm for deriving the shape of the deformable mirror from the measurements.

Even though AO correction is used, the quality of astronomical images still is degraded due to the time delay stemming from the wavefront sensor integration time and adjust-ment of the deformable mirror(s). This results in a blur which can be mathematically described by a convolution of the original image with the point spread function (PSF). The PSF of an astronomical image varies with the position in the observed field, which is a crucial aspect on ELTs.

In this thesis, we focus on two challenges in AO based observations: First, we present new algorithms for the control of modern AO systems, in particular Single Conjugate Adaptive Optics and Ground Layer Adaptive Optics. Our focus is on matrix-free-approaches, taking into account the specific geometry of the system and known statis-tical properties. Real-life effects, such as spot elongation for laser guide stars and the statistics of the turbulent atmosphere are included in our model and, where possible, also in our reconstruction algorithms. We present results obtained in the end-to-end simulation environment from the European Southern Observatory, OCTOPUS, and show that our methods obtain comparable quality while reducing the computational time significantly compared to established methods.

Second, we turn to post-processing and present approaches for PSF reconstruction from wavefront sensor measurements for Single Conjugate Adaptive Optics and Multi-Conjugate Adaptive Optics combining atmospheric tomography and techniques for PSF reconstruction. Existing, verified techniques are fused together in a novel way to

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deliver accurate field dependent PSFs in very short time. Even though simulation re-sults in OCTOPUS suggest a good agreement between the true and the reconstructed PSF, the reconstructed PSF will never be completely accurate due to the coarse res-olution of the wavefront sensor and especially telescope specific effects, such as non common path aberrations. However, the quality of the reconstructed PSF as well as of the observed image can be further improved by using blind deconvolution methods. We opt for a Lanczos-based blind deconvolution scheme to get a fast deconvolution algorithm based on a sparse system. This method is tested to recover information from blurred star images.

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Zusammenfassung

Moderne erdgebundene Teleskope sind mit Systemen der Adaptiven Optik (AO) aus-gestattet, um atmosphärische Turbulenzen ausgleichen und damit die Bildqualität verbessern zu können. Diese Technologie verwendet Messungen der einfallenden Wel-lenfronten von hellen Lichtquellen und künstlich erzeugten Laserleitsternen, um die Turbulenzen über dem Teleskop zu rekonstruieren und schnell verformbare Spiegel entsprechend einzustellen. In der kommenden Generation von Großteleskopen (Extre-mely Large Telescopes, ELTs) mit Durchmessern von bis zu 40 m steigt der Rechen-aufwand sehr stark an, da eine Korrektur in Echtzeit mit einer Frequenz von 500 Hertz notwendig ist. Diese Freuqenz ist notwendig, da die atmosphärischen Turbulenzen nur etwa 2 ms konstant bleiben. Die größte Herausforderung ist daher die Verfügbarkeit schneller Algorithmen zur Berechnung der Form des verformbaren Spiegels aus den Messdaten.

Trotz der Korrektur durch ein AO System ist die Qualität der astronomischen Bilder immer noch vermindert, da die Wellenfrontsensoren zur Messung eine Integrationszeit benötigen und der verformbare Spiegel erst danach entsprechend eingestellt werden kann. Die resultierende Unschärfe des Bildes kann mathematisch durch eine Faltung des originalen Bildes mit der Punktspreizfunktion (point spread function, PSF) be-schrieben werden. Die PSF eines astronomischen Bildes ändert sich mit der Position im beobachteten Bereich, was insbesondere bei ELTs große Auswirkungen haben wird.

Die vorliegende Dissertation behandelt zwei Herausforderungen AO-basierter Beobach-tungen: Zuerst werden neue Algorithmen zu Regelung moderner AO System präsentiert, im Speziellen für Single Conjugate Adaptive Optics und Ground Layer Adaptive Op-tics. Der Fokus liegt dabei auf Matrix-freien Verfahren, die die spezielle Geometrie des Systems und bekannte statistische Eigenschaften berücksichtigen. Auch praktische Ef-fekte, wie Streckung des Spots auf dem Sensor bei Laserleitsternen und die Statistik der turbulenten Atmosphähre, werden im Model einbezogen und, wenn möglich, in den Re-konstruktionsalgorithmen verwendet. Wir präsentieren Simulationsresultate, die mit der end-to-end Simulationsumgebung der Europäischen Südsternwarte, OCTOPUS, erzielt wurden, und zeigen, dass unsere Methoden im Vergleich mit existierenden Al-gorithmen die gleiche Qualität erreichen, jedoch die Laufzeit signifikant reduziert wird.

Der zweite Teil der Arbeit behandelt die Bildnachverarbeitung und Methoden zur Rekonstruktion der PSF von Wellenfrontsenordaten f¨ur Single Conjugate Adaptive

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Optics und Multi-Conjugate Adaptive Optics. Diese Verfahren kombinieren existie-rende, verifizierte Algorithmen der atmosphärische Tomographie und Techniken zur Rekonstruktion der PSF auf neue Art, um rasch eine positionsabhängige PSF zu be-rechnen. Obwohl die Simulationsresultate in OCTOPUS eine gute Übereinstimmung der echten und der rekonstruierten PSF zeigen, wird die rekonstrukierte PSF nie völlig korrekt sein, da die Auflösung der Wellenfrontsensoren zu grob ist und weitere Effekte im Teleskop, wie etwa Störungen im nicht gemeinsamen optischen Pfad, auftreten. Um die Qualität der rekonstruierten PSF und auch des aufgenommenen Bildes zu ver-bessern, werden Methoden zur so-genannten blinden Entfaltung (blind deconvolution) angewandt. Wir präsentieren Verfahren basierend auf dem Lanczos-Prozess, um ein dünnbesetztes System und damit einen schnellen Entfaltungsalgorithmus zu erhalten. Als Test wird diese Methode angewandt, um Information aus verzerrten Sternbildern wiederherzustellen.

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Acknowledgments

First and foremost, I would like to express my deep thanks to my supervisor Prof. Ronny Ramlau, for giving me this unique opportunity to be involved in the work for E-ELT instrument MICADO and for his guidance, support and kindness. I generously appreciate the freedom and trust he offered me to accomplish my work. Together with his wife, Dr. Jenny Niebsch, they welcomed me with warmth and friendship.

I would like to thank my bachelor and master thesis supervisor Prof. Andreas Neubauer for his continuous support and fruitful discussions. It was an honor to have Prof. Jo˜ao Alves as second referee.

Furthermore, my thanks go to my collaboration partners and co-workers from the Aus-trian Adaptive Optics Team (Dr. Daniela Saxenhuber, Dr. Andreas Obereder, Dr. Kirk Soodhalter, Dr. Iuliia Shatokhina, Dr. Misha Yudytskiy, DI Victoria Hutterer, DI G¨unter Auzinger, DI Markus P¨ottinger, DI Stefan Raffetseder), Dr. Tapio Helin, DI Christoph Hofer, Prof. Lothar Reichel, Laura Dykes, and Dr. Miska Le Louarn for his untiring support from ESO and Dr. Gijs Verdoes Kleijn for fruitful discussions as dataflow lead within MICADO. Special thanks go also to all colleagues from the Industrial Mathematics Institute and MathConsult GmbH for their hospitality, friend-ship and cheerful (after) work hours.

My gratitude also goes to my colleagues and friends from studies, especially to DI Nora Engleitner, DI Michael Hauer, Dr. Peter Gangl and Dr. Benjamin Eichinger, and close friends, DI Christoph Göweil, DI Daniel Wöckinger, Jürgen Jungbauer MA, Paul Kralik, Mag. Florian Kronschläger and Ing. Martin Steinbach.

Finally, I would like to express my kindest thanks to my family, especially to my parents Helga and Gerhard, for their enormous support throughout my studies and without the love and care of whom I would not be where I am now. Last, but not least, my deepest gratefulness goes to Lorraine Dettmer MSc for her support and sharing her life with me.

Roland Wagner Linz, August 2017

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Chapter 1 Introduction

Since many centuries, humans try to learn more about the universe by observing ce-lestial objects which can be seen on the sky. Apart from developing a powerful natural science called astronomy, this curiosity led to the development of optical imaging sys-tems to further explore space.

Nowadays, most observations are performed through ground-based telescopes, but as-tronomers are equipped with high-tech tools. A fundamental limit for imaging systems is diffraction, which lead to the construction of telescopes with bigger diameter in order to increase the resolution. In recent years, telescopes with diameters around 10 me-ters, such as the Very Large Telescope (VLT) or the Gran Telescopio Canarias were built and the next generation with diameters of 30 meters and more is currently under design or construction.

Apart from diffraction also the occurrence of atmospheric turbulences limits the imag-ing quality by causimag-ing distortions in the light wavefronts – which can be seen as “twin-kling” of the stars and results in blurred images. As remedy for this phenomenon the field of Adaptive Optics (AO) emerged where one or more deformable mirrors are inserted in the optical path inside the telescope and reflection on these mirrors com-pensates for atmospheric distortions before the light reaches the scientific camera.

The atmospheric turbulences change rapidly and thus also the shape of the deformable mirror has to be adapted in real time, i.e., within one or two milliseconds. Due to this short time frame the shape of the mirror has to be updated with a certain time lag leading to residual turbulences blurring the science image. For current systems, the use of matrix vector multiplication (MVM) methods was sufficiently fast, even though they rely on inverting a big matrix every millisecond. However, for the in-creased amount of data in future systems such methods will not be suitable anymore, requiring the development of new methods with reduced computational cost and thus fulfilling also the restrictive time constraints.

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2 CHAPTER 1. INTRODUCTION

caused by time delay and imperfections of the systems, such as the discrete spacing of actuators on the deformable mirror. This degradation can be described by the so-called point spread function (PSF), which relates the true, unblurred image and the observed image through a convolution step. Unfortunately, no direct information on the PSF is available.

In this thesis we consider algorithms for wavefront reconstruction from measurements coming from wavefront sensors, which avoid computations of huge matrices and scale linearly with the number of unknowns. The focus of our work in this direction lies on the simplest AO system and a system correcting mainly for a big amount of tur-bulence close to the ground using a stochastic approach to inverse problems. Both methods are tested in a state-of-the-art AO simulation environment against existing techniques and show a significant speed up while maintaining comparable quality. We also develop methods for recovering information on the PSF from the wavefront sen-sor data and use this information for further deconvolving the observed images. For reconstructing the PSF, we update an existing algorithm to be suitable for future tele-scopes and enhance it to overcome the direction dependent nature of the PSF. Our blind deconvolution approach relies on the fact that the reconstructed PSF is close to the unknown true PSF and improves the PSF estimate and the image at the same time.

Figure 1.1: Illustration of the Extremely Large Telescope compared to the Very Large Telescope and St. Stephen’s Cathedral, from [43].

The background for this work is the Extremely Large Telescope (ELT), see Figure 1.1, which is currently under construction on Cerro Armazones, a 3060 m high mountain in the Atacama desert in Chile. The ELT is built and will be operated by the Eu-ropean Southern Observatory (ESO). Its first light is currently planned for 2024 and

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CHAPTER 1. INTRODUCTION 3

it is going to be the biggest ground-based telescope with a primary mirror of almost 40 meters in diameter. Attached to the ELT several instruments, relying on AO, will be available for astronomical observations to bring spectacular new insights on star and planet formation, the development of nearby galaxies, the expansion history of the Universe as well as the evolution and uniqueness of the Solar System and the Earth [43]. Among the first light instruments are MICADO (Multi-AO Imaging Camera for Deep Obserations) and METIS (Mid-infrared ELT Imager and Spectrograph).

After Austria joined ESO in 2008, as an in-kind contribution, the project “Mathemat-ical Algorithms and Software for E-ELT Adaptive Optics” was successfully carried out in Linz from 2009 until October 2013. As a follow up of this project, the Jo-hann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences and the Industrial Mathematics Institute of JKU Linz became part of the A˚_{-Consortium, funded through the Hochschulraumstrukturmittel}

by the Federal Ministry of Science, Research and Economy (BMWFW), which belongs to the instrument consortia for MICADO and METIS. Within this project the focus is on the one hand on designing the control algorithms for the AO systems and on the other hand on obtaining information on the PSF and improving the images further in a post-processing step.

This thesis is organized as follows, based on and using parts from our work in [174, 172, 173, 175, 35]:

Chapter 2 gives an introduction to AO. We present an overview of the principles of im-age formation in an optical system in the presence of atmospheric turbulence and the connection between atmospheric distortions and the point spread function. Further-more, we introduce a layered model of the atmosphere and the different components, configurations and measures of AO systems used throughout this thesis. Next, our work is related to the to-be-built ELT and its instruments and the software tools used for development are described.

In Chapter 3, we recall the mathematical theory of ill-posed inverse problems and regularization, focusing on Tikhonov regularization, the Bayesian approach to inverse problems and iterative methods. From the field of numerical linear algebra, we review the Lanczos iteration, which will be the basis for our approach to image improvement.

Chapter 4 addresses the problem of wavefront reconstruction in a Single Conjugate Adaptive Optics (SCAO) system. We revisit a cumulative reconstructor based on finite elements and its properties, before discussing an enhancement via domain de-composition and numerical simulation results.

In Chapter 5, we introduce a Bayesian approach for exploiting the statistical properties of the wavefront sensor measurements in a Ground Layer Adaptive Optics (GLAO)

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4 CHAPTER 1. INTRODUCTION

system. Then, together with an existing fast method for wavefront reconstruction, a new algorithm is developed and results from numerical simulation are presented.

Chapter 6 contains an introduction to the optical transfer function in the case of aber-rations and a review of existing methods for point spread function reconstruction from SCAO data. One of the existing methods is updated to fulfill the needs of ELTs. An extensive discussion on implementation details and numerical results, also considering the reduction of input data, concludes this chapter.

In Chapter 7, we extend the ideas for point spread function reconstruction to a Multi-Conjugate Adaptive Optics (MCAO) system. To this end the existing methods for MCAO systems are recalled and a three-step approach for atmospheric tomography is explained. Based on these ideas, a novel approach for direction dependent point spread function reconstruction is developed.

Chapter 8 considers the problem of image improvement by deconvolution of the ob-served image after an observation. After the problem description and an overview of existing methods, a fast blind deconvolution approach is introduced. Numerical results are obtained in the framework of observations through ground based telescopes, using the reconstructed point spread functions from the previous chapters.

Finally, in Chapter 9, we give a conclusion as well as an outlook on possible future work.

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CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS 5

Chapter 2 Astronomical Adaptive Optics

Light traveling from a distant astronomical object, e.g., a star, through the atmo-sphere gets distorted. Observed on a ground-based telescope, this results in a blurred image. Adaptive Optics (AO) offers a technique to compensate for these atmospheric aberrations. Especially in large ground-based telescopes, the aim is to mechanically correct the effects of the atmosphere by the means of a deformable mirror [71, 138]. This is obtained by measuring the incoming wavefronts from a bright astronomical object or an object created artificially by a strong laser and calculating and adjusting the shape of the deformable mirror accordingly.

Figure 2.1: The principle idea of Adaptive Optics (AO): correcting a perturbed in-coming wavefront by means of a deformable mirror, source [176].

The fundamental idea of astronomical adaptive optics, illustrated in Figure 2.1, is to use the reflection on the deformable mirror to propagate a then corrected wavefront to the science camera, resulting in better image quality. As atmospheric aberrations are varying fast, the deformable mirror has to be adjusted around 500 times per second.

In this chapter, we recall some basics of image formation in an optical system related to Fourier optics, based on [68, 67, 139], and additional effects arising in ground based

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6 CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS

astronomy [139, 110]. After a short introduction on modeling the atmosphere, we present the basic components of AO systems and different AO configurations. We conclude this chapter with a discussion of relevant measures in AO systems and relat-ing this work to the ELT and its instruments.

2.1 Image formation

Figure 2.2: Diffraction geometry, source [139].

We start with a short introduction to geometrical and Fourier optics. The goal is to present the basic formulae which relate the optical field at an observation point and the optical field in the diffracting aperture, having a spatial separation. Figure 2.2 presents an illustration of the geometry of the diffraction calculation, where the x0y0

-plane is the aperture -plane, the x1y1-plane is the observation plane, z indicates the

distance between these two, and ~r01 is the vector between points in the aperture and

observation plane, locating the source point at ~x0 and the observation point at ~x1.

The problem to be solved is the calculation of the optical field at the observation point under the knowledge of the field in the aperture plane. In this section, we follow the lines of [139, 68, 67].

One solution to the problem described above is given by the Rayleigh-Sommerfeld diffraction formula, being the most general solution for the situation presented above, and stems from scalar diffraction theory, where the vector nature of optical field quan-tities is ignored and they are simply treated as scalars. It has been shown that this simplification is accurate if the diffracting aperture is much larger than the optical wavelength and the separation between aperture and observation plane is many wave-lengths. Both conditions are easily satisfied for the situation of our interest.

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Further-CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS 7

more, it is only valid for monochromatic optical fields.

The mathematical starting point to obtain the Rayleigh-Sommerfeld diffraction for-mula are Maxwell’s equations. Using scalar diffraction theory, these equations can be reduced to a Helmholtz equation. Using clever Green’s functions gives a representation of the optical field in the image plane ui by the optical field uo diffracted in the pupil

plane P as uipx1, y1q “ ż P hpx1, y1, x0, y0quopx0, y0q dpx0, y0q, (2.1) hpx1, y1, x0, y0q “ 1 iλ exppikrpx1, y1, x0, y0qq rpx1, y1, x0, y0q cospθpx1, y1, x0, y0qq, (2.2)

where px1, y1q is a point in the image plane, px0, y0q in the aperture plane with distance

z between these two planes, as in Figure 2.2, and i the complex unit. The quantity rpx1, y1, x0, y0q is the distance between point px1, y1q in the image plane and point

px0, y0q in the pupil plane, thus given by

rpx1, y1, x0, y0q “ a z2_{` px} 1´ x0q2` py1´ y0q2, (2.3) and cospθpx1, y1, x0, y0qq “ z rpx1, y1, x0, y0q . (2.4)

One should note that cospθq and r only depend on the distances px1´ x0q and py1´ y0q.

Therfore, (2.1) can be written as a convolution

uipx1, y1q “

ż

P

¯

hpx1´ x0, y1´ y0quopx0, y0q dpx0, y0q,

with ¯hpx1 ´ x0, y1´ y0q “ hpx1, y1, x0, y0q, or, alternatively, by exploiting (2.4), as

uipx1, y1q “ z iλ ż P uopx0, y0q exppikrpx1, y1, x0, y0qq rpx1, y1, x0, y0q2 dpx0, y0q. (2.5)

As in (2.3) a square root appears, the above formulae are difficult to handle. The so-called Fresnel approximation offers a possibility to overcome these difficulties. One performs a binomial expansion of the square root in (2.3) and reduces it to the constant and linear terms, which yields

rpx1, y1, x0, y0q « z ˆ 1 ` 1 2 ´x 1´ x0 z ¯2 ` 1 2 ´y 1´ y0 z ¯2˙ .

This approximation is valid for distances z much larger than the extent of the image and the pupil plane.

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With the above notation, we get the Fresnel approximation for the Rayleigh-Sommerfeld theory as uipx1, y1q “ eikz iλz ż P uopx0, y0q exp " ik 2z “ px1´ x0q2` py1´ y0q2 ‰ * dpx0, y0q, (2.6)

or, equivalently expanding the squares, by

uipx1, y1q “ eikz iλze ik 2zpx 2 1`y21q ż P ! uopx0, y0qe ik 2zpx 2 0`y20q ) e´2iπλzpx1x0`y0y1qdpx₀, y₀_q. (2.7)

Figure 2.3: Lens geometry, source [139].

Before we turn to a more generalized imaging geometry, we model the effect of a lens on an incident optical field. This means how a field is changed when propagating from the incident side of a lens to the transmitting one, as shown in Figure 2.3. Mathematically the relationship between incident and transmitted field is given by

utpxq “ uipxqtlpxq,

where tlpxq is the transparency function describing the effect of the lens, and ut and

ui are the incident and transmitted field respectively. Under the so-called “thin lens”

assumptions (cf, e.g., [68]), the transparency function can be written as

tlpxq “ expp´i

k 2f|x|

2

q,

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Figure 2.4: Generalized imaging geometry, source [139].

2.1.1 Coherent image formation

To describe the limits of an imaging system arising from the wave nature of light, we need some ideas from Fourier optics (cf, e.g., [68, 139]). Let us consider a general-ized imaging system as shown in Figure 2.4 in coherent imaging conditions. Coherent refers to spatial coherence of the illuminating optical field and implies a perfectly corre-lated, i.e., deterministic, optical field. Using the linearity of the wave propagation and the Fresnel approximation conditions being satisfied, gives the following relationship between the object field uo and the image field ui as

uipxq “ u0pxq ˚ hpxq, (2.8)

where the imaging system impulse response h is given by

hpxq “ F pP pf λdiqq, (2.9)

with P pxq the pupil aperture function, λ the optical wavelength, and di the distance

between exit pupil and imaging pupil as in Figure 2.4.

The pupil aperture function is the indicator function on the pupil aperture, i.e., con-stant within the pupil and zero else. We omit the extensive discussion why quadratic terms must be eliminated from the diffraction result and refer to [68]. For the moment the pupil aperture function just describes the entrance pupil of the system. We will discuss a more generalized version later.

2.1.2 Incoherent image formation

In contrast to the previous subsection, we now consider incoherent optical fields. This means that all field points are completely independent of all other field points. In

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reality, there exists always some relationship for two points, if they are close enough (e.g., smaller than the wavelength). However, most imaging systems are unable to resolve such small distances, so we can keep this simplification. In case of incoherent image formation the quantity of interest is the intensity of the image I, also called irradiance, and defined through the following relation

Ipx, yq “ |uopx, yq ˚ hpx, yq|2 (2.10) “ ´ uopx, yq ˚ hpx, yq ¯´ uopx, yq ˚ hpx, yq ¯ “ ż R2 ż R2 uopξ1, η1quopξ2, η2qhpx ´ ξ1, y ´ η1qhpx ´ ξ2, y ´ η2qdpξ1, η1qdpξ2, η2q,

where ¯z represents the complex conjugate.

In this setting, uo is according to [139], a random quantity, and thus inherits this

randomness onto Ipx, yq as well. The system response h is depending on the point in the plane which is considered. Especially due to the longer illumination time of the sensor, the average intensity becomes the quantity of interest. Average in this case might be an ensemble or time average and is denoted by x¨y. Taking the average on both sides of (2.10) gives

xIpx, yqy “ ż R2 ż R2 xuopξ1, η1qugpξ2, η2qyhpx ´ ξ1, y ´ η1qhpx ´ ξ2, y ´ η2qdpξ1, η1qdpξ2, η2q. (2.11) for an incoherent field. Using the properties of an incoherent field, i.e., spatial inde-pendence, gives a contribution only for ξ1 “ ξ2 and η1 “ η2 as

xuopξ1, η1quopξ2, η2qy “ κxIopξ1, η1qyδpξ1´ ξ2, η1´ η2q, κ P R, (2.12)

where δ is the usual Dirac distribution (cf., e.g., [67]).

Inserting (2.12) into the formula for incoherent image formation, the average intensity of the observed image xIpx, yqy is given by

xIpx, yqy “ κxIopx, yqy ˚ |hpx, yq| 2

, (2.13)

where xIopx, yqy is the average light distribution of the object, hpx, yq the imaging

system impulse response given by (2.9) and κ P R`_.

The term |hpx, yq|2 in (2.13) is called point spread function (PSF). From the knowledge on h from coherent image formation (2.9), we can rewrite this as

PSF px, yq “ |hpx, yq|2

“ |F pP pf λdiqq |2px, yq, (2.14)

where P is the pupil function, λ the wavelength and di the distance between the lens

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Note that PSF depends on the distance di, but it can be made independent of this

distance using a transformation into angular coordinates by α “ x

di and β “ f di. We

omit the complete discussion to rewrite PSF in angular coordinates, resulting in the same expressions except for a constant.

The relation in (2.13) can also be viewed in the frequency domain, by simply applying the Fourier transform on both sides and exploiting its rules, as

F pIqpf1, f2q “ F pIgqpf1, f2q ¨ F pP SF qpf1, f2q.

The expression F pPSF q is called the Optical Transfer Function (OTF), and |F pPSF q| is denoted as the Modulation Transfer Function (MTF) (cf, [68]). Equivalent up to a constant, the OTF can be defined as normalized function through

OT F pf1, f2q “ ş R2|hpx, yq| 2 e´2πipf1x`f2yq_{dpx, yq} ş R2|hpx, yq| 2 dpx, yq . (2.15)

2.1.3 Diffraction limited PSF

Diffraction limited PSF with no central obstruction

We want to calculate the PSF of a diffraction limited telescope without a central ob-struction. This is a very simple geometry and diffraction limited observation can, e.g., be obtained when a telescope is operating above the atmosphere of the Earth.

For this case the PSF can be calculated analytically as in [139] by

PSFTprq “ |F pχΩDqprq| 2 “ πD 2 4λ2 ˆ 2J1pπDr{λq πDr{λ ˙ ,

where D is the diameter of the telescope aperture and λ the wavelength. The func-tion J1 is a Bessel function of first kind of order one and thus solution to the Bessel

differential equation x2d 2_y dx2 ` x dy dx ` px 2 ´ 1qy “ 0.

This Bessel function is finite at the origin and can be defined as series expansion around zero by J1pxq “ 8 ÿ m“0 p´1qm m!Γpm ` 2q ´x 2 ¯2m`1 ,

with Γ the gamma function Γpnq “ pn ´ 1q! for positive integers. The intensity distribution PSFT is often also referred to as Airy pattern. For increasing diameter

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Diffraction limited PSF with central obstruction

Due to the optical layout, most modern telescopes have circular apertures. As dis-cussed in [110, Section 4.10], for a diffraction limited telescope with a circular aperture of diameter D and a circular central obstruction of diameter d, the PSF is given by

PSFTprq “ 4 ¨ ˜« J1pπDr_λ q πDr λ ´ˆ d D ˙2 J1pπdr_λ q πdr λ ff ¨ 1 1 ´ p_Ddq2 ¸2 .

Usually the central obstruction is small, but one could think also about increasing it. For d Ñ D, meaning a central obstruction covering almost the whole telescope, the central part of PSFT will become narrower and thus improve the image quality.

However, this improvement is small and additionally, as a drawback, a higher fraction of the incoming light is transferred into the outer rings of the Airy pattern [110].

2.1.4 Seeing limited PSF

In most observations, the incoming wavefront is not planar, but has some aberrations, as discussed in [68]. Different sources for these aberrations are possible: There might be some focusing error, the system might suffer from inherent properties of perfectly spherical lenses, such as spherical aberrations, or the aberrations come from atmo-spheric turbulence (see following Section 2.2). Aberrations cause the PSF to be no longer limited by diffraction but by the seeing conditions and, therefore, such situa-tions are referred to as seeing limited.

When such aberrations exist, one could also view this as if the wavefront is deformed when leaving the pupil. Thus the deformation of the wavefront would be part of the optical system, which motivates the definition of a generalized pupil function. If ϕpx, yq is the wavefront aberration at point px, yq, then the a generalized pupil function is given by Ppx, yq “ P px, yq exp ˆ i2π λ ϕpx, yq ˙ , (2.16)

where λ is the corresponding wavelength.

One could also relate the generalized pupil function Ppx, yq to the phase φpx, yq, which is related to the wavefront aberration via

φ “ 2π λ ϕ.

From this one can clearly see that the wavefront aberration is wavelength dependent. In addition to that, phase and wavefront aberration have different units: while a phase is given in radians, a wavefront is described in optical path distance, i.e., in meters.

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Now, let us replace P px, yq in (2.14) by Ppx, yq from (2.16). This translates into the following formula PSFφpx, yq “ |hpx, yq|2 “ ˇ ˇ ˇF ´ P pλdix1, λdiy1qeiφpλdix 1_,λd iy1q ¯ˇ ˇ ˇ 2 px, yq, (2.17) and in angular coordinates

PSFφpα, βq “ |hpα, βq| 2 “ ˇ ˇ ˇF ´ P pλα1_{, λβ}1 qeiφpλα1,λβ1q ¯ˇ ˇ ˇ 2 pα, βq.

For the ideal case of having no aberrations, i.e., φ “ 0, the PSF is exactly the diffrac-tion limited case of Secdiffrac-tion 2.1.3.

Example 2.1. One of the easiest special cases of aberrations are so called tip-tilt aberrations of a wavefront. This terminology is widely used in the astronomical com-munity and comes from the fact that the first two, non-constant Zernike polynomials are called tip and tilt. Such an aberrations has the form φθpx, yq “ 2πpθ1x ` θ2yq,

where θ “ pθ1, θ2q represents tip and tilt, respectively. For this aberration, it can be

shown (cf, e.g., [40] that the resulting PSF is just a shift of the diffraction limited PSF by the vector θ, i.e.,

PSFφθpx, yq “ P SFTpx ´ θ1, y ´ θ2q. (2.18)

2.2 Atmospheric turbulence

In ground based astronomy, one of the major obstacles is turbulence in the Earth’s atmosphere. Being heated by the sun during daytime and cooling down during night time together with wind shears causes turbulent motion of air. In this motion, turbu-lent eddies occur which are random regions of specific temperatures [139, 127]. The refractive index of air strongly depends on the temperature and therefore varies in the Earth’s atmosphere. Changes in the refractive index cause a distortion of initially planar wavefronts when traveling through the atmosphere.

Two characteristic parameters for atmospheric turbulence are L0, called outer scale,

and l0, called inner scale. The first one describes the largest appearing turbulent

ed-dies and usually ranges from tens to hundreds of meters. The latter one is the size of the smallest appearing eddies and only on the order of few millimeters (cf, e.g., [127]).

In the following, we will present two models for atmospheric turbulence stemming from [88] and [171], respectively.

2.2.1 Kolmogorov turbulence model

Kolomogorov [88] introduced a model for atmospheric turbulence, based on the as-sumption that in the inertial regime, lying between l0 and L0, a universal description

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of the turbulence spectrum exists. This assumption leads to a homogeneous and isotropic turbulence between l0 and L0, i.e., the turbulence statistics can be modeled

by a stationary, isotropic Gaussian random field. To exploit this further, the statistics of turbulence is independent of the position and higher order statistical moments de-pend only on the radial distance between two points.

The structure function Dxpr, sq of a random variable x measured at positions r, s gives

means to describe the spatial structure of a random process and is defined by Dxpr, sq “ E

`

|xprq ´ xpsq|2˘ .

Assuming homogeneous and isotropic turbulence, Dxpr, sq only depeneds on |r ´ s|

and is independent of the direction [127]. Therefore, according to [139, 127], it can be rewritten as

Dnpr, sq “ c2n|r ´ s|

2{3_, _(2.19)

where the index structure coefficient c2

n is the local strength of the index of refraction

fluctuation, depending on the height. From this formula, we get that the turbulence spectrum is described only by the turbulence strength. Note, that c2_n has units m´2{3_.

In practice, c2

n often is given relative to the overall turbulence in the atmosphere, i.e.,

as a fraction of one. The total amount of turbulence along a line of sight, also called seeing, can be obtained by integration over c2_n along this line. The values of c2_n show a clear dependence on altitude and weather. In practice, c2

n values are measured

em-pirically and therefore we present typical values in Section 2.2.4.

The power spectral density (PSD) Φnpκq is used for describing the statistical

distri-bution of the number and size of the turbulent eddies in the atmosphere and can be derived from (2.19). According to [139], the PSD of the Kolmogorov model is given by:

ΦK_npκq “ 0.033 ¨ c2_n|κ|´11{3

for the inertial range 2π{L0 ă |κ| ă 2π{l0, κ “ pκ1, κ2, κ3q the spatial frequency of

the turbulence, |κ| “ aκ2

1` κ22` κ23 Euclidean norm, and superscript K indicates

Kolmogorov. For |κ| Ñ 0 problems due to singularity occur. This relates to the fact that for eddies larger than L0, the assumption of homogeneity is not fulfilled. To

overcome these problems, the von Karman was introduced.

2.2.2 Von Karman turbulence model

The singularities of the Kolmogorov statistics at κ “ 0 can be overcome by a modifica-tion known as von Karman statistics [171, 139]. The PSD of the von Karman statistic is given by ΦV_npκq “ 0.033 ¨ c 2 n pκ2` κ2₀q11{6 exp ˆ ´κ 2 κ2 m ˙ ,

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where V indicates von Karman. Using the values κ0 “ 2π{L0 and κm “ 5.92{l0, ΦVn

has a finite value at the origin and shows a fast decay towards zero for large frequencies.

2.2.3 Atmospheric parameters

The description of the atmosphere is related to few parameters, among them the Fried parameter r0 and the isoplanatic angle θ0.

Fried parameter r0

Following the lines of [139], the structure function D can be related to the Fried parameter r0, which is given in meters by

r0 “ 0.185 ¨ ˜ λ2 ş8 0 c 2 nphqdh ¸3₅ ,

at the wavelength λ. It was originally introduced in [48] and has a close relation to seeing conditions. The Fried parameter gives the limiting aperture size, i.e., for telescope apertures larger than r0 the resolution will not improve. In other words,

telescopes with aperture diameters smaller than r0 are limited by diffraction, whereas

for apertures bigger than r0 observations are limited by the seeing. This can be seen

in the following example.

Example 2.2. In Figure 2.5, two subsequent observations from the 4.2 m William Herschel telescope on the Canary Islands in Spain are shown which were presented in [146]. As the observed object was a single star, the image represents essentially the PSF. In the left image, the PSF obtained by a seeing limited observation (r0 „ 0.15 m,

without AO) has no explicit peak. In contrast to that, the right image, being the result of an observation with AO correction, shows a clear peak. For bigger, to-be-built telescopes, such as the ELT, this effect even increases. As these telescopes do not exist right now, we can only present PSFs as simulation results from [153] for the 42 m ELT in Figure 2.6. Again, for the PSF without AO correction, no explicit peak shows up (top), whereas an AO correction clearly improves the quality (bottom). The refractive index structure constant c2_n is not zero up to the maximum turbulence height, which is usually around 20 to 30 km. The ratio between wavelength λ and the Fried parameter r0 is called atmospheric seeing.

An alternative definition of the Fried parameter can be found, e.g., in [163] and is given as r0 “ ˆ 0.423 ¨ˆ 2π λ ˙ psec zq żHmax 0 c2_nphqdh ˙´3₅ ,

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Figure 2.5: Comparison of an uncompensated image (left) and an AO corrected image (right), source [146].

where z is the angle to center direction (also known as zenith angle). From the above formulae, it follows that the Fried parameter, which is also known as coherence radius, depends on the specified wavelength λ by r09λ6{5.

Isoplanatic angle θ0

In [110], an isoplanatic patch is described as an area over which the PSF remains invariant. Isoplanaticity in ground-based astronomy depends on both the path of light through the atmosphere and the telescope optics. For the telescope optics, [110] suggests to assume the existance of an isoplanatic patch in the image plane. The iso-planatic angle θ0 describes the separation at which two speckle images begin to look

increasingly different, see [110] for details. Thus, seeing limited instantaneous PSFs differ, if they are separated by more than θ0. The effect arising when two objects

are separated by an angle bigger than θ0 is called angular anisoplanatism and clearly

limits ground-based astronomical observations.

The AO system gives the best possible correction of the wavefronts in direction of the guide star. If an observation is made at an angle θ of the guide star direction, the phase variance can be expressed as

Epσ2φq “

ˆ θ θ0

˙5{3

,

where θ0 is the isoplanatic angle, as defined in [127, 163, 49] by

θ0 “ ˜ 2.914ˆ 2π λ ˙2 psecpzqq8{3 ż c2_nphqh5{3dh ¸´3{5 ,

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Figure 2.6: Averaged PSFs of a 42 m telescope with annular aperture: without AO correction and with AO correction, source [153].

or, using the definition of the Fried parameter r0

θ0 “ 0.314 cospzq r0 H, with H “ ´_{ş c}2 nphqh5{3dh ş c2 nphq dh ¯3{5

, called the mean effective turbulence height, and z the zenith angle of the observation. The isoplanatic angle is usually given in arcmin, arcsec or radian. Nowadays, separate instruments such as DIMM (Differential Image Motion Monitors, cf, e.g., [148]) are used to measure θ0 additionally to other seeing

parameters.

2.2.4 Layered model

Taylor’s hypothesis of frozen flow, introduced in [162], claims that the atmosphere consists of layers of turbulence that are distinguishable and traveling parallel to the Earth’s surface at a certain velocity. This originates from observing that the wind speed is faster than the time frame of changes in the turbulence pattern of such an atmospheric layer. In other words, a ”frozen“ pattern is blown over the telescope aperture at a typical wind speed around 20 m/s (cf, e.g., [127]). Therefore, one can

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assume that the c2_n-variations are constant over a short time frame.

Following the standard assumptions, i.e., the turbulence in subregions has approxi-mately a homogeneous statistics and statistical independence of each layer, the atmo-sphere can be modeled by a finite number of layers, L. Each layer is infinitely thin and lies at an altitude hl, l “ 1, . . . , L [136]. The amount of turbulence in layer l is

defined as γl:“ c2nphlq, where in numerical simulations the relative turbulence strength

γ “ pγ1, . . . , γLq is normalized such that

řL

l“1γl “ 1.

As the refractive index structure constant, also known as c2_n-profile, is a crucial part of all layered atmosphere models, extensive studies have been carried out in order to ob-tain good results. However, the turbulence structure is varying with altitude, location and time of day. From experimental measurements, e.g., made by balloons, several models where developed for night time observations [139], such as the Hufnagel-Valley profile [121]. Due to its variation in time, the c2

n-profile needs to be measured

fre-quently by external instruments. Several instruments have been developed such as a Multi Aperture Scintillation Sensor (MASS) [90] or the Scintillation Detection and Range Finding (SCIDAR) [168, 5, 102]. In recent years, extensions of the latter such as Stereo-SCIDAR [120] have been developed. Nowadays such instruments are suc-cessfully installed at observation sites such as Paranal (VLT-site). The profile can also be estimated by a method called Slope Detection and Ranging (SLODAR) from Shack-Hartmann wavefront sensor data [62, 26, 73].

Even though modern atmospheric profiles consist of up to 40 layers, it has been shown that it is sufficient to consider a much smaller number of layers [150, 4]. Therefore it is reasonable to use only a 9- or 10-layer atmosphere in simulations.

ESO Standard 9-layer atmosphere

The simulations in Chapter 4, 5 and 7 are performed with a 9-layer atmospheric model from ESO, introduced in [101]. This model is based on measurements at ESO’s site Paranal in the Atacama desert with a Fried parameter r0 “ 12.9 cm. In Table 2.1, the

values for the 9-layer medium seeing atmosphere are given.

Layer 1 2 3 4 5 6 7 8 9

Height(m) 47 140 281 562 1125 2250 4500 9000 18000 c2_n-profile 0.522 0.026 0.044 0.116 0.098 0.029 0.059 0.043 0.06

Table 2.1: 9-layer median atmosphere.

ESO 10-layer atmosphere

As preparation for the ELT from recent measurements at Cerro Armazones a propri-etary atmospheric model with 10 layers was derived by ESO. We use this model with

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a Fried parameter r0 “ 12.1 cm in Chapter 6. In Table 2.2, the values for the 10-layer

atmosphere are given.

Layer 1 2 3 4 5 6 7 8 9 10

Height(m) 123.8 251.4 415 740.9 1394 2699 5309 9079 12849 16329 c2

n-profile 0.59 0.02 0.04 0.06 0.01 0.05 0.09 0.04 0.05 0.05

Table 2.2: 10-layer atmosphere.

2.3 Components of AO systems

AO systems were developed from the need to compensate for atmospheric distortions, i.e., reducing the image degradation due to turbulence. A wavefront sensor (WFS) measures incoming wavefronts from a guide star (GS). The real-time control (RTC) unit computes from these measurements the optimal correcting shape of a deformable mirror (DM). The reflection on the DM is compensating then for the atmospheric tur-bulence. The update of the DM shape needs to be done in less than a millisecond, thus the measurements of the WFS and the calculations of the RTC have to be performed at the same frequency.

First ideas for AO systems were developed in [6]. A sketch of an AO system is presented in Figure 2.7. The beamsplitter is necessary as not all incoming light should go to the WFS, but the major part should be left to observe a corrected high resolution image with the scientific camera.

2.3.1 Guide star

A guide star (GS) is a bright astronomical object, e.g., a star, which is used for getting information on the turbulence in the atmosphere. The measurements from this guide star are the input for the reconstruction process of the RTC. Guide stars can be ei-ther natural astronomical objects that are point sources, so-called natural guide stars (NGS), or artificial ones, so-called laser guide stars (LGS). The reason for creating artificial guide stars is the limited sky-coverage with NGS.

An LGS is formed by shooting a strong laser up into the night sky, which stimulates a sodium layer at approximately 90 km height in the atmosphere and thus creates an artificial astronomical object.

Effects of LGS

From the creation of LGS by stimulation of a sodium layer three major effects arise that need to be taken into account: Spot elongation, cone effect and tip-tilt indetermination.

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Figure 2.7: A general AO system, source [29].

The thickness of this sodium layer is modeled by a Gaussian random variable via its mean hLGS and a full width half maximum parameter FWHMN a. Usually, in models

hLGS “ 90 km and FWHMN a“ 11.4 km is assumed. Note that FWHM for a Gaussian

random variable relates to the standard deviation σ as FWHM “ 2?2 ln 2 ¨ σ.

Due to the “thickness” of the sodium layer, its vertical width, the scattering of the laser beam is not a single point as for an NGS, but rather a small stripe on the night sky. The spot, registered by a detector on the telescope, appears elongated, see Figure 2.8. This effect is called spot elongation. This elongation degrades the measurement accu-racy and the error increases linearly with the elongation of the spot in the direction of the centroid [23]. Furthermore, spot elongation introduces correlation between the X and Y measurements in the subaperture [161]. We follow the lines of [23], when discussing the compensation of spot elongation in Chapter 5.

Additionally to the vertical width of the sodium layer, its finite height causes the light to travel through a cone-like volume in the atmosphere. This effect is referred to as

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Figure 2.8: Graphical representation of the photon distribution of an NGS (left) and the spot elongation generated by an LGS (right) on the detector, from [178].

cone effect, illustrated in Figure 2.9. In atmospheric tomography, as in Section 7.2, the cone effect has to be taken into account as a scaling factor cl relating hLGS and

the height of an atmospheric layer hl.

Figure 2.9: Illustration of the cone effect of an LGS, from [178].

A third complication arising from LGS is the so-called tip-tilt indetermination, meaning that the planar part of the incoming wavefront is wrongly observed at the telescope. This effect stems from the fact that the laser beam passes through the same atmosphere

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twice, a first time when traveling up and a second one when being back scattered from the sodium layer. Thus, a tip or tilt of the incoming wavefront due to, e.g., a layer of differently tempered air, as shown in Figure 2.10, cannot be observed and the “correct” position of the LGS remains unknown. As remedy for this phenomenon additional NGS are used, from which just the tip-tilt information is obtained, as their exact positions are known. Detailed discussions on this effect and how it can be overcome can be found, e.g., in [149, 178, 165].

Figure 2.10: Illustration of the tip-tilt indetermination, from [3].

2.3.2 Wavefront sensor

The device to measure light coming from a guide star is called wavefront sensor (WFS). A high enough spatial resolution and measurement speed are important requirements for a successful real time compensation by the AO system. The techniques developed vary from focal plane techniques to pupil plane techniques [138]. Most current WFS, such as the Shack-Hartmann WFS and Pyramid WFS (cf, e.g., [129, 131, 130]), provide indirect measurements of the atmospheric distortions. These are two pupil plane WFS, which are commonly used nowadays. Apart from these also other sensors, such as the curvature WFS (cf [137]), exist.

Shack-Hartmann WFS

The Shack-Hartmann WFS (SH-WFS) [40, 152, 123] consists of an array of small lense-lets and a photo detector lying behind. These sensors measure the average gradient of the incoming wavefront in each subaperture. More specifically, the detector mea-sures the x- and y-coordinates of the points where the light of each lenslet is focused. These measurements are related to the slope of the incoming wavefront via the center

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of gravity of the focal spots. To compute the center of gravity, a variety of methods exists, such as (weighted-)centroiding or matched filtering, e.g., [61]. The principle of a SH-WFS is illustrated in Figure 2.11.

Usually having one Shack-Hartmann measurement over the whole aperture is not enough, so Shack-Hartmann WFS consist of a n ˆ n grid of apertures as described above, now called subapertures. Such a sensor covers the whole aperture of the tele-scope and Ω “ Yn

i,j“1Ωij, where Ω is the telescope aperture and Ωij is one aperture of

the SH-WFS. Not all subapertures are illuminated if the sensor is square. One thus has to distinguish between active and inactive subapertures.

The action of one subaperture Ωij of the SH-WFS on the incoming wavefront can be

formalized by the operator Γ : Hs Ñ R2n2 as

sxri, js “ pΓxϕqri, js :“ 1 |Ωij| ż Ωij Bϕ Bxpx, yq dpx, yq, syri, js “ pΓyϕqri, js :“ 1 |Ωij| ż Ωij Bϕ Bypx, yq dpx, yq, for i, j P t1, . . . , nu.

Note that Γ is well-defined for s ą 1₂ according to [116]. The waffle mode (checkerboard pattern) and the piston mode (constant function) lie in the nullspace N pΓq. However, the piston mode can be neglected for the wavefront reconstruction. The additional condition

ż

Ω

ϕpx, yq dpx, yq “ 0 (2.20)

enforces a piston mode zero for ϕ on Ω in the reconstruction process.

We combine the operators Γx, Γy to a system of equations on the whole telescope

aperture, defined by

Γϕ “ s, Γ “ pΓx, Γyq, s “ psx, syq. (2.21)

In a discretized version, the action of Γ on ϕ can be rewritten as

sxri, js “

pϕri, j ` 1s ´ ϕri, jsq ` pϕri ` 1, j ` 1s ´ ϕri ` 1, jsq

2 ,

sxri, js “

pϕri ` 1, js ´ ϕri, jsq ` pϕri ` 1, j ` 1s ´ ϕri, j ` 1sq

2 ,

where ϕri, js denotes the wavefront evaluated in the subaperture with index pair pi, jq [149].

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Figure 2.11: Principle of a Shack-Hartmann WFS, source [163].

2.3.3 Deformable mirror

The deformable mirror (DM) is a thin, flexible and highly reflective mirror that can be moved and its shape can be controlled by actuators attached from below, as il-lustrated in Figure 2.12. While in many current telescopes the DM consists of a continuous faceplate, the DM of the ELT will be different as it will consist of several hexagonal segments. Around 5200 actuators are planned to allow the surface to be readjusted in less than a millisecond. Each segment will be made of ceramic glass at a thickness of approximately 2 millimeters. The actual size of the DM is around 2.5 meters in diameter, while still optically scaled to the aperture size of the telescope [167, 18]. The actuators are controlled by applying voltages.

2.3.4 Real-time control

The shape of the DM, and moreover the commands for the actuators of the DM, are derived from the WFS measurements by the real-time control (RTC) unit. For this action specific fast algorithms are required. We will develop algorithms for different AO systems in Chapter 4 and 5.

2.4 Adaptive Optics configurations

As several AO systems exist, specific algorithms for each setting have to be developed. While the first AO systems were capable of only handling measurements from one NGS and one DM, nowadays more complicated ones have several WFS and also multiple

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Figure 2.12: Principle of a deformable mirror, source [54].

DMs.

Before presenting the different flavors of AO systems, we want to introduce the notion of closed loop and open loop. These two concepts are completely different. In an AO system running in closed loop, the WFS is placed in the optical path behind the DM, i.e., the WFS only sees residual aberrations arising, e.g., from time delay. On the opposite, in an open loop configuration the WFS is placed before the DM, i.e., the WFS measures the full, uncorrected atmospheric aberrations. In Figure 2.7, a closed loop system is shown.

For some AO systems relying on atmospheric tomography, it might make sense to consider open loop data even though the system is running in closed loop. This can be obtained by transforming closed loop data and the corresponding DM shape and is often referred to as pseudo-open loop, see Section 2.5.2 for details.

2.4.1 Single Conjugate AO

The most simple, and first, AO system is called Single Conjugate AO (SCAO). It uses only one NGS with a corresponding WFS and one DM. The RTC basically recon-structs the incoming wavefront of the GS and maps it to the corresponding DM shape.

Due to the properties of this system a good correction is only possible in the direction of the GS. This results in the drawback that for good astronomical observations one has to find a bright enough star close to the science object in order to get a good correction in the desired part of the science image. As the sky is barely covered by bright enough point sources this technique clearly has its limits. Using LGS in SCAO

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is tricky, due to certain effects from the laser that might be impossible to correct for without having an additional NGS (e.g., for the so-called tip-tilt).

In Chapter 4, an algorithm to reconstruct wavefronts in an SCAO system will be presented, and in Chapter 6, a reconstruction method for the PSF of the AO system will be developed.

2.4.2 Ground-Layer AO

Having still only one DM, but multiple GS and corresponding WFS, maybe also LGS, one could think of a more complex AO system. In a Ground-Layer AO (GLAO) sys-tem the main aim is to correct for the amount of turbulence being located close to the ground. This is related to the fact that standard models for the atmosphere, as presented in Section 2.2.4, have a strong ground layer. Thus, most atmospheric dis-tortions affecting the image quality are located in the ground layer. If one can correct for these distortions over the whole field of view, a huge increase in image quality is the immediate consequence.

The DM is positioned in the optical path such that it is conjugated to the ground layer. Considering only the Ground Conjugated DM in Figure 2.13, gives a sketch of a GLAO system.

In Chapter 5, a wavefront reconstruction algorithm for a GLAO system will be intro-duced.

2.4.3 Multi-Conjugate AO

Increasing the number of DMs additionally to increasing the number of WFS, one can set up a Multi-Conjugate AO (MCAO) system, as introduced in [10]. MCAO has been introduced to extend the area of wavefront correction compared to SCAO. The deformable mirrors are conjugated to different heights in the atmosphere in order to compensate atmospheric turbulence in a large field of view. Both, NGS and LGS in several directions, are used to reduce anisoplanatism and obtain a uniform correction over the field of view. For atmospheric tomography, such systems were the starting point as it is needed for reconstructing a turbulent atmosphere from multiple guide stars [135].

A sketch of the system is shown in Figure 2.13. In Chapter 7, a reconstruction algo-rithm for an MCAO system from [149] will be presented and then used to obtain a reconstruction of the PSF in multiple directions.

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Figure 2.13: MCAO system, from [43].

2.5 Control in AO systems

2.5.1 Integrator control

The deformable mirror needs some control, in order to combine the reconstructions from the previous and the current time step. In a standard method, the so called integrator control, the new mirror shape pΦDMqt is related to the old mirror shape

pΦDMqt´1 and the new reconstruction pΦDMqt via

pΦDMqt“ p1 ´ γqpΦDMqt´1` γpΦDMqt, (2.22)

where γ is called the loop gain and .

The same strategy can also be applied directly as output gain in the atmospheric to-mography step, i.e., instead of pΦDMqt one takes the reconstructed layers Φt.

In a closed loop system, only an update for the DM shape pΦDM updateqt is computed

and as a control only a damped version of this update is added to the current DM shape, therefore (2.22) changes to

pΦDMqt“ pΦDMqt´1` γclpΦDM updateqt, (2.23)

where the subscript cl indicates that the gain, also called loop gain, is clearly different to the one in (2.22).

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2.5.2 Pseudo-open loop control

The alignment of WFS and DM differs for different AO systems. As introduced in Section 2.4, systems run in closed or open loop. In order to enforce stability and ro-bustness of the reconstruction algorithm, pseudo-open loop control (POLC) has been introduced in AO, e.g., in [39]. It offers an alternative to integrator control in closed loop system and provides stabilization over time for the reconstruction algorithm. To use POLC, open loop data have to be created from the original closed loop data. We use an approach introduced in [143, 145] for MCAO systems.

In modern reconstruction methods for MCAO systems, such as a three-step approach (cf, e.g., [149, 151]), an intermediate result are incoming wavefronts ϕ in closed loop. Subtracting the mirror updates of the last time step gives a transformation into open-loop like data, see (2.24). The following steps can then be performed on this artificial open-loop data. Due to this behaviour, statistical information of the atmsophere can be directly used, and additionally to the usual integrator gain, another gain, referred to as input gain in [149], can be used,

ϕol “ ϕcl´ ˜AΦDM, (2.24)

ϕol _{“ p1 ´ inputgainq ¨ ϕ}ol,old_{` inputgain ¨ ϕ}ol_,

where ˜A projects DM shapes from their conjugated height to the ground, ΦDM are

the DM shapes and superscript ol and cl indicate open and closed loop, respectively.

2.6 Measures in AO systems

2.6.1 Quantities and sizes in AO systems

Within AO systems certain system specific quantities appear that are related either to the observation or that are inherited by the technical set up. We discuss the quantities that are needed throughout this work.

Minute and second of arc

Positions of stars on the sky are often given in angular coordinates and also we will often refer to arcmin (1’) and arcsec (1”). Both of them are angular measurements, being 1{21600 and 1{1296000 of a circle, respectively, and thus given by

11 “ π 10800rad „ 2.9 ¨ 10 ´4_, 12 “ π 648000rad „ 4.8581 ¨ 10 ´6_.

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Field of view (FoV)

The field of view (FoV) is usually described by the diameter of a circle in arcmin or arcsec. Two different FoV have to be distinquished: the science FoV and corrected FoV. The first one can be much smaller than the latter one, especially for complex AO systems. We will most of the time refer to the latter one which is determined by the guide star asterism.

Frame rate

The duration of sensing at the CCD of the WFS is called the frame rate of a specific AO system. For many AO system, it lies around 500 Hz, meaning that the time frame for calculating the DM shape(s) is approximately one millisecond. By the frame rate, the length of one time step in simulations is determined.

Wavelength λ

Observations can be performed in different wavelengths and, as already discussed, many parameters are wavelength dependent. In this work, we focus on observations mostly in K-band, i.e., λ P r2.0, 2.4sµm, being a near infrared wavelength. One has to distinguish between sensing and evaluation wavelength, which do not need to coincide. In general, the AO performance can change drastically with changing λ.

Photon flux nph

The intensity of light reaching the aperture (and consequently the WFS) is measured in photons. Typically the photon flux denotes the number of photons that reach one subaperture of the WFS per frame. In our tests, the photon flux ranges from low (1 to 500 photons per subaperture per frame) to high (10000 and more) flux.

The photon flux is directly related to the magnitude m, the astronomical measure for brightness of an object, at a certain wavelength λ, through several formulae described in [156], which are recalled in the following. We denote with F0 the flux in Jy (Jansky)

(i.e., [10´26_{W m}´2_Hz´1_{s) at magnitude zero for a given wavelength, being a fixed}

quantity known from measurements. Fλ is the spectral flux density at wavelength λ

at magnitude m and related to F0 by

Fλ “ 10´0.4¨m¨ F0¨ β{λ2,

where β is a constant to convert between different units, in our case β “ 3 ¨ 10´12_.

The photon flux Fph is then computed as

(42)

with the photon engery Eph “ h ¨ c{λ, ∆λ being the width of the wavelength band, h

the Planck constant and c the speed of light.

Finally, nph can be calculated as

nph“ Fph¨ QER¨ Tatm¨ Toptics¨ 10´3¨ ∆t,

where QERis the number of electrons per photon, Tatmand Topticsare the transmission

factors of the atmosphere and the telescope optics, respectively, and ∆t is the time per frame, calculated from the frame rate. Note that some of the used quantities are constant, while others depend on the used observing system and wavelength.

2.6.2 Quality measures

In order to evaluate the quality of the run of an AO system, some quality measures are introduced. Various quality measures are possible and their application is also dependent on the used AO system. All quantities depend on the observing wavelength λ.

Strehl ratio

The Strehl ratio relates the PSF of an AO run (and its corresponding residual wave-front) to the perfect telescope PSF, i.e., a diffraction limited PSF as presented in Section 2.1.3. While the PSF itself is the most global way to represent the quality of the observed image, the Strehl ratio gives just one number for evaluating the image quality.

The Strehl ratio (SR) is defined through

SR “ PSFφp0q PSFtelp0q

,

where PSFφ is the point spread function related to the (residual) atmospheric

aber-rations resulting in a wave phase φ and PSFtel is the diffraction limited PSF of the

telescope, cf., e.g., [139, 110]. One should note that the evaluation at the origin of PSFφ and PSFtel gives a relation between their respective peaks, if they are

per-fectly centered. However, due to aberrations, the peak of PSFφ might be slightly of

the center leading to a much lower Strehl ratio. Note furthermore that 0 ď SR ď 1, where the equality on the right is obtained only with a perfect atmospheric correction.

The above formula is rather complicated to evaluate as one needs the full PSF of an AO run. For good corrections, related to high Strehl ratios, an approximation can be made, known as Mar´echal criterion which holds for SR ą 0.1 (cf, e.g., [138, 139]). For small residual wave front phase σ2 “ _|Ω1

D|} ¯φ ´ φ} 2 L2_pΩ Dq, where ¯φ “ 1 |ΩD| ş ΩDφprq dr, i.e.

From Adaptive Optics systems to Point Spread Function Reconstruction and Blind Deconvolution for Extremely Large Telescopes / eingereicht von Dipl.-Ing. Roland Wagner, Bakk.techn.