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Model-Based Wavefront Reconstruction Approaches For Submitted by Dipl.-Ing. Victoria Hutterer, BSc. Submitted at Industrial Mathematics Institute Supervisor and First Examiner Univ.-Prof. Dr. Model-based wavefront Ronny Ramlau reconstruction approaches Second Examiner Eric Todd Quinto, Ph.D., Robinson Profes- for pyramid wavefront sensors sor of Mathematics in Adaptive Optics October 2018 Doctoral Thesis to obtain the academic degree of Doktorin der Technischen Wissenschaften in the Doctoral Program Technische Wissenschaften JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, Osterreich¨ www.jku.at DVR 0093696 Abstract Atmospheric turbulence and diffraction of light result in the blurring of images of celestial objects when they are observed by ground based telescopes. To correct for the distortions caused by wind flow or small varying temperature regions in the at- mosphere, the new generation of Extremely Large Telescopes (ELTs) uses Adaptive Optics (AO) techniques. An AO system consists of wavefront sensors, control algo- rithms and deformable mirrors. A wavefront sensor measures incoming distorted wave- fronts, the control algorithm links the wavefront sensor measurements to the mirror actuator commands, and deformable mirrors mechanically correct for the atmospheric aberrations in real-time. Reconstruction of the unknown wavefront from given sensor measurements is an Inverse Problem. Many instruments currently under development for ELT-sized telescopes have pyramid wavefront sensors included as the primary option. For this sensor type, the relation between the intensity of the incoming light and sensor data is non-linear. The high number of correcting elements to be controlled in real-time or the segmented primary mirrors of the ELTs lead to unprecedented challenges when designing the control al- gorithms. Therefore, a precise mathematical investigation of the pyramid sensor and its forward models is essential for accurate and fast wavefront reconstruction. In this Thesis, we present new approaches for wavefront reconstruction from pyramid wavefront sensor data for which our detailed analytical study of the pyramid sensor lays the mathematical foundation. Therefore, we derive the pyramid sensor models in a distributional sense and provide a thorough analysis of the underlying operators, e.g., we linearize the models or calculate Fréchet derivatives and adjoint operators. Using further approximations that are suggested by the physical setting of the sensor itself, we can describe the sensor’s forward models as roof sensor configurations or as a variant of the finite Hilbert transform of the incoming wavefront. The analysis is extended to these operators as well. Based on various approximate models, several algorithms have been developed in re- cent years for a stable, high-quality, and high-speed wavefront correction. Among those, we emphasize interaction-matrix-based approaches, Fourier domain methods, iterative algorithms, and methods based on the inversion of the finite Hilbert trans- form. We briefly present the core ideas of already existing algorithms and explicitly provide the theoretical background of new approaches for wavefront reconstruction from pyramid and roof wavefront sensor data. i We divide the new algorithms into two groups: linear and non-linear wavefront re- construction methods. With linear approaches, we use a singular value expansion for wavefront reconstruction or apply well-known iterative methods such as the conjugate gradient method, the steepest descent algorithm, Landweber iteration or Kaczmarz versions of the previously mentioned methods. With non-linear approaches, we employ non-linear Landweber iteration and Landweber-Kaczmarz iteration to the problem of wavefront reconstruction from pyramid sensor data. An additional complication arises due to the structure of the ELTs. So called telescope spiders create areas where the information of the phase is isolated on the wavefront sensor detector, leading to pupil fragmentation and a break in the spatial continuity of the data, and further to extremely poor wavefront reconstruction. These unwanted er- rors make several existing control algorithms unfeasible for telescope systems with wide spiders. To overcome the effects induced by telescope spider obstruction we propose a hybrid scheme, the so called Split Approach, which combines model-based algorithms with methods that provide the lost information directly from isolated reconstructors resulting in accurate wavefront reconstruction for segmented pupils. The effectiveness of the proposed algorithms is demonstrated in the context of the 39 m Extremely Large Telescope. Numerical evaluations using European Southern Obser- vatory’s end-to-end simulation tool Octopus include detailed comparisons of speed and reconstruction quality of the proposed new methods to previously developed al- gorithms. ii Zusammenfassung Aufgrund von atmosphärischen Turbulenzen und Lichtbrechung sind Bilder von Him- melskörpern, die mit erdgebundenen Teleskopen aufgenommen werden, verschwom- men. Um die Störungen, hervorgerufen von Windströmungen oder kleinen, veränder- lichen Temperaturzonen in der Atmosphäre, auszugleichen, verwendet man für die neue Generation von Extremely Large Telescopes (ELTs) die Methode der Adaptiven Optik (AO). Ein AO System besteht aus Wellenfrontsensoren, Kontrollalgorithmen und verstellbaren Spiegeln. Ein Wellenfrontsensor misst einfallende, verzerrte Wellen- fronten, der Kontrollalgorithmus verbindet die Wellenfrontsensormessungen mit den Kommandos für die Spiegelaktuatoren und die verstellbaren Spiegeln korrigieren die atmosphärischen Bildfehler mechanisch in Echtzeit. Die Rekonstruktion der unbekann- ten Wellenfront aus den gegebenen Sensormessungen ist ein Inverses Problem. In vielen der Instrumente, die derzeit für Teleskope der Größenordnung von ELTs eint- wickelt werden, ist der Pyramid Wellenfrontsensor als erste Option vorgesehen. Für diese Art von Sensor ist die Verbindung zwischen der Intensität des einfallenden Lich- tes und der Sensordaten nichtlinear. Die hohe Anzahl der Korrekturelemente, die in Echtzeit kontrolliert werden müssen, oder die segmentierten Hauptspiegel der ELTs führen zu noch nie dagewesenen Herausforderungen bei der Umsetzung von Kontroll- algorithmen. Deshalb ist für eine genaue und schnelle Wellenfrontrekonstruktion eine präzise mathematische Untersuchung des Pyramid Wellenfrontsensors and seiner Vor- wärtsmodelle essentiell. In dieser Dissertation präsentieren wir neue Ansätze zur Wellenfrontrekonstruktion aus Pyramid Wellenfrontsensordaten, für welche unsere detaillierte Untersuchung des Pyramid Sensors die mathematische Grundlage bildet. Für diesen Zweck leiten wir die Pyramid Sensormodelle im Distributionensinn her und geben eine ausführliche Analy- se der zu Grunde liegenden Operatoren an; zum Beispiel linearisieren wir die Modelle oder berechnen Fréchet Ableitungen und adjungierte Operatoren. Unter Verwendung weiterer Approximationen, die vom physikalischen Aufbau des Sensors abgeleitet wer- den, können wir die Vorwärtsmodelle des Sensors als eine Roof Sensor–Konfigurationen oder als eine Variante der endlichen Hilbert Transformation der einfallenden Wellen- front beschreiben. Die Analyse wird auch auf diese Operatoren ausgedehnt. Basierend auf verschiedenen Vorwärtsmodellen sind in den letzten Jahren zahlreiche Algorithmen für eine stabile und hochqualitative Hochgeschwindigkeits–Wellenfront- korrektur entwickelt worden. Wir beschreiben unter diesen Ansätze solche, die auf iii einer Interaktions–Matrix basieren, Methoden im Fourierraum, iterative Ansätze und Verfahren, die auf der Inversion der endlichen Hilbert Transformation beruhen. Wir präsentieren kurz die Kernideen von bereits existierenden Algorithmen and stellen den theoretischen Hintergrund von neuen Herangehensweisen zur Wellenfrontrekonstruk- tion aus Pyramid und Roof Wellenfrontsensordaten im Detail dar. Die neuen Algorithmen teilen wir in zwei Gruppen auf: lineare und nichtlineare Wellen- frontrekonstruktionsmethoden. Als lineare Ansätze verwenden wir eine Singulärwert– Erweiterung für die Wellenfrontrekonstruktion oder bekannte iterative Methoden wie das CG–Verfahren, den Algorithmus des steilsten Abstiegs, Landweber Iteration oder Kaczmarz Versionen der zuvor genannten Methoden. Als nichtlineare Ansätze benut- zen wir nichtlineare Landweber Iteration und Landweber–Kaczmarz Iteration für das Problem der Wellenfrontrekonstruktion aus Pyramid Sensordaten. Eine zusätzliche Komplikation tritt aufgrund des Aufbaus der ELTs auf. Sogenann- te Teleskop–Spiders erzeugen Bereiche, in welchen die Information über die Phase am Detektor des Wellenfrontsensors isoliert ist, was zu einer Fragmentierung der Teleskop- apertur und einem Verlust der gebietlichen Verbundenheit der Daten und weiters zu äußerst schlechter Wellenfrontrekonstruktion führt. Diese unerwünschten Fehlerquel- len machen existierende Kontrollalgorithmen unbrauchbar für Teleskopsysteme mit breiten Spider–Strukturen. Um die Effekte, die durch Teleskop–Spider–Beschattung entstehen, zu korrigieren, entwickeln wir ein zweiteiliges Schema, den sogenannten Split Ansatz, welcher Modell–basierende Algorithmen mit Methoden, die die verlorene Information direkt aus unabhägigen Rekonstruktoren gewinnen, kombinert und so in genaue Wellenfrontrekonstruktion für segmentierte Teleskopöffnungen resultiert. Die Effektivität der eingeführten Algorithmen wird im Kontext des 39 m großen Extre- mely Large Telescopes demonstriert. Numerische Evaluierungen aus der “end–to–end“ Simulationsumgebung Octopus
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