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Bastaits, R. (2010). Extremely large segmented mirrors: dynamics, control and scale effects (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences appliquées – Mécanique, Bruxelles.

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ULB U

niversité

L

ibre de

B

ruxelles

Faculté des Sciences Appliquées

Extremely Large Segmented Mirrors:

Dynamics, Control and Scale Effects

Renaud Bastaits

Thesis submitted in candidature for the

degree of Doctor in Engineering Sciences June 2010

Active Structures Laboratory

Department of Méchanical Engineering and Robotics

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Jury

Supervisor : Prof. André Preumont (ULB) Members ;

Prof. Claude Jamar (AMOS, Liège) Dr Martin Dimmler (ESO, Germany) Dr Yvan Stockman (CSL, Liège) Prof. Olivier Verlinden (UPMons) Prof. Frank Dubois (ULB)

Dr Arnaud Deraemaeker (ULB)

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Remerciements

Les compétences scientifiques et pédagogiques du Professeur André Preumont, son dynamisme et sa personnalité ont constitué le ciment de ce travail. Je lui suis très reconnaissant de m’avoir accepté comme élève. Son contact restera pour moi source d’enrichissement personnel sous bien des égards, au-delà des seuls as­

pects scientifiques.

Je tiens également à remercier tout particulièrement Christophe Collette par l’intermédiaire de qui j’ai commencé mes travaux au sein du Laboratoire des Structures Actives. Ses conseils, sa bonne humeur à toute épreuve et son soutien m’ont aidé à trouver mes marques et à avancer.

De même, je souhaite remercier Gonçalo Rodrigues avec qui j’ai travaillé en étroite collaboration depuis ma réorientation dans le domaine des télescopes. Partager les sujets de recherche, le même bureau, ainsi que la plupart des repas a été source d’une stimulation constante.

Le Laboratoire des Structures Actives a constitué pour moi un environnement stimulant; par leurs qualités humaines et scientifiques, les personnes que j’y ai côtoyées m’ont aidé à avancer dans une atmosphère d’émulation toujours souri­

ante.

Je remercie également le Prof. Frank Dubois, et les Dr Yvan Stockman et Stéphane Roose pour m’avoir apporté à de nombreuses reprises leur aide précieuse pour progresser face à des questions d’optique.

Je remercie également le Fonds National de la Recherche Scientifique pour le sou­

tien financier qu’il m’a apporté, via la bourse FRIA FC76554.

Enfin, je suis heureux de partager le fruit de ce travail avec mes proches qui, chacun à leur manière, apportent ce qui fait le sel de ma vie.

V

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Abstract

Ail future Extremely Large Télescopes (ELTs) will be segmented. However, as their size grows, they become increasingly sensitive to external disturbances, such as gravity, wind and température gradients and to internai vibration sources.

Maintaining their optical quality will rely more and more on active control means.

This thesis studies active optics of segmented primary mirrors, which aims at stabilizing the shape and ensuring the continuity of the surface formed by the segments in the face of external disturbances.

The modelling and the control strategy for active optics of segmented mirrors are examined. The model has a moderate size due to the séparation of the quasi-static behavior of the mirror (primary response) from the dynamic response (secondary, or residual response). The control strategy considers explicitly the primary re­

sponse of the telescope through a singular value controller. The control-structure interaction is addressed with the general robustness theory of multivariable feed­

back Systems, where the secondary response is considered as uncertainty.

Scaling laws allowing the extrapolation of the results obtained with existing 10m

télescopes to future ELTs and even future larger télescopes are addressed and the

most relevant parameters are highlighted. The study is illustrated with a set of

examples of increasing sizes, up to 200 segments. This numerical study confirms

that scaling laws, originally developed with simple analytical models, can be used

in confidence in the preliminary design of large segmented télescopes.

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Contents

1 Introduction 1

1.1 Evolution of télescopes: Prom 1600 to 1980 ... 1

1.2 Modern-days télescopes... 3

1.2.1 A context of multiple technological breakthroughs... 3

1.2.2 The advent of active optics for monolithic mirrors... 5

1.2.3 Segmented mirrors... 8

1.2.4 Long baseline interferometry... 9

1.2.5 Space télescopes... 11

1.3 Future Extremely Large Télescopes... 13

1.4 Scale effects... 15

1.5 Outline ... 17

1.6 References... 17

2 Basics of Telescope Optics 21

2.1 Introduction... 21

2.2 Aberrations... 22

2.2.1 Définition... 22

2.2.2 Quantifying the wavefront error... 23

2.3 Common optical configurations of optical télescopes ... 26

2.3.1 Newtonian télescopes ... 26

2.3.2 Two-mirror télescopes... 27

2.3.3 Télescopes with 3 or more mirrors... 28

2.4 Wavefront error due to déviations from the design ... 28

2.4.1 Shape of optical éléments... 29

2.4.2 Relative position of optical éléments... 29

2.4.3 Linearity ... 30

2.4.4 Design trade-offs... 30

2.5 Diffraction-limited imaging ... 32

2.5.1 Définitions ... 32

2.5.2 Imaging... 32

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2.5.3 Diffraction from obscurations... 34

2.5.4 Diffraction-limited versus aberrated ...35

2.6 Télescopes with a segmented primary mirror...35

2.6.1 Conditions for optimal performances...35

2.6.2 Design trade-offs... 37

2.6.3 Diffraction in segmented télescopes... 38

2.7 Conclusions...41

2.8 Référencés... 41

3 Active control of télescopes 45

3.1 Introduction...45

3.2 External disturbances... 46

3.3 Layers of active control in modem télescopes... 50

3.3.1 Pointing and tracking... 51

3.3.2 Active optics... 52

3.3.3 Adaptive optics... 56

3.4 Active optics of the Keck telescope... 58

3.5 Conclusions...65

3.6 Référencés... 66

4 Dynamics and control 69

4.1 Introduction...69

4.2 Quasi-static approach... 70

4.3 Structural dynamics... 73

4.3.1 Model réduction... 73

4.3.2 Modal analysis... 75

4.3.3 Static response... 79

4.3.4 Dynamic response in modal coordinates... 79

4.4 Control strategy ...81

4.4.1 Dual loop controller... 82

4.4.2 Extended Jacobian SVD controller...83

4.5 Loop shaping of the SVD controller...84

4.6 Control-structure interaction ... 87

4.6.1 Multiplicative uncertainty... 88

4.6.2 Additive uncertainty... 88

4.7 Discussion...90

4.8 Conclusions...93

4.9 Référencés... 93

5 Scale effects 95

5.1 Introduction... 95

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CONTENTS XI

5.2 Static deflection under gravity... 97

5.3 First résonance frequency... 98

5.4 Control bandwidth...99

5.5 Control-structure interaction ... 101

5.6 Wind response ... 103

5.7 Summary and conclusion ... 106

5.8 References... 107

6 Structural response of large truss-supported segmented reflec- tors 111

6.1 Introduction... 111

6.2 Methodology ... 111

6.2.1 Structure... 111

6.2.2 Wind model... 115

6.2.3 Random response... 116

6.3 Results in open-loop... 117

6.4 Controlled response... 120

6.5 Effect of damping... 125

6.6 Effect of mean wind velocity... 127

6.7 Conclusions... 132

6.8 References... 132

7 Conclusions 133

7.1 Original aspects of the work... 133

7.2 Scaling laws... 134

7.3 Future perspectives... 134

A Définitions of optical design parameters 137

A. l References... 139

B Primary aberrations 141

B. l References...144

C Shack-Hartmann sensors 145 D Small-gain theorem 149

D.l General formulation...149

D.2 Stability robustness tests... 149

D.2.1 Additive uncertainty... 150

D.2.2 Multiplicative uncertainty... 150

D.3 Residual dynamics... 151

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E Mode shapes of segmented mirrors with supports 1-3 153

F Wind response of Set 3 157

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Chapter 1

Introduction

1.1 Evolution of télescopes: Prom 1600 to 1980

Astronomy distinguishes itself from other scientific disciplines by the fact that its developments are mainly based on observations, as most experiments are unprac- tical or impossible by nature. Throughout the âges, astronomers hâve relentlessly developed instruments to exploit the full potential of the sky that their naked eyes were not able to catch. Amongst the most notable inventions, it led to the development of elaborate calendars and early positioning Systems, based on the patterns formed by the astral objects.

The invention of the first refracting telescope at the end of the XVT^ century, and its improvement and use by Gahleo to observe the sky, revealed the poten­

tial of such instruments to push back the limits of the observation of objects in the skies, by focusing more light than what the naked eye is capable of, and by magnifying the image. The technological and mathematical developments to improve the refracting telescope led Isaac Newton to the construction of the first refiecting telescope around 1670, based on the use of a parabolic mirror instead of a lens as the light collecter. The refiecting telescope exhibits some advantages with respect to the refracting one, in particular the fact that they are exempt of chromatic aberrations, as refiection laws do not dépend on the wavelength of the light, while refraction laws do.

The brightness of the faintest objet that a given telescope can observe is limited by the effective area of its primary mirror (Ml) (Enard et al., 1996). Furthermore, the diameter of Ml, D, also affects the resolution and contrast characteristics of the images formed by the telescope in idéal conditions (see section 2.5). Conse- quently, improving the performance of the télescopes has called for a constant

1

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increase of D along history. Fig. 1.1 shows the évolution of the aperture diameter of optical and infrared télescopes throughout the years. Until the beginning of the century, reflecting and refracting télescopes were competing against each other, benefiting from respective technological developments leading to innova- tive designs. For apertures larger than Im, the reflecting télescopes hâve proved the most efficient.

Figure 1.1: Telescope aperture diameter in time [adapted from (Bely, 2003), p.2].

The rôle of the telescope structure is to maintain the optical performances of

the telescope during observations, by preserving the shape and alignment of the

éléments in the optical train. As those optical éléments were built larger and

thicker, and consequently heavier, so were their supporting structures. But their

sensitivity to the effects of changing gravity and température grew accordingly.

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1.2 Modern-days télescopes 3

Innovative solutions were developed, commonly referred to as passive, combin- ing, e.g., new design approaches for the sub-structures, the generalized use of kinematic mountings and an optimized choice of materials.

1.2 Modern-days télescopes

Not only should the next générations of télescopes collect more light. To be really effective, they should also ensure that the collected light is always focused on the smallest area, otherwise faint stars and slight details of extended objects are lost in a blurry luminous background. Consequently, the challenge is to build larger télescopes with an improved optical accuracy, maintained along time in spite of external disturbances, such as gravity, wind and thermal gradients.

1.2.1 A context of multiple technological breakthroughs

Before 1980, the mounts of the largest télescopes were of the équatorial type (Fig. 1.2.a), in which rotations around the polar axis (parallel to the Earth’s ro­

tation axis) and around the déclination axis (perpendicular to the polar axis) allow the initial pointing towards an object. The tracking was then simply per- formed by rotating the telescope around its polar axis, at a constant speed, to compensate for the Earth’s rotation. This simple principle could be performed in open-loop by the use of dock mechanisms. However, while élégant in its princi­

ple, the structural constraints induced by that configuration revealed unpractical for their implémentation in ever-growing télescopes, mostly because of their in- trinsic heaviness [(Bely, 2003), p.234], and cannot correct errors due to external disturbances.

The orientation of télescopes of the altitude-azimuth (alt-az) type is based on a vertical (azimuth) axis and on a horizontal (altitude or élévation) axis. Tracking is intrinsically more difhcult in this case, because it requires the axes to be rotated at variable speeds depending non-linearly on the orientation of the telescope.

However, computer control has almost cancelled that drawback. Moreover, their structures are more compact, much simpler and lighter than those of équivalent équatorial télescopes (cfr Fig.1.3), implying so significant cost savings that alt-az mounts hâve become the standard ^ . Finally, as the orientation of the altitude and azimuth axes do not change with respect to the orientation of the gravity field, the implémentation of feedforward corrections (based on lookup tables) in active optics is easier and more efficient (Enard et al., 1996).

'Other particular configurations are used in some projects such as the S ALT and HET (see section 1.2.3), but they are out of scope for this thesis, as they are not envisioned for any future ELT.

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Figure 1.2: (a) Equatorial mount - (b) Altitude-Azimuth mount.

The construction of télescopes with primary mirrors of diameters in the range of 3.5m showed the practical limits of passive techniques with respect to the severe optical tolérances required to attain the best performances, calling for complex periodical readjustments (on a timescale of weeks) (Wilson, 2003). The goal of active optics is to automate that optical maintenance procedure, during observa­

tions, on much shorter time scales (from a few tens of seconds to a few minutes).

Consequently, active optics both increases the optical performance and lengthens the timescales over which they can be maintained, making télescopes more effi­

cient.

The implémentation of active optics in modem télescopes has had a considérable impact on their overall design. First, it permitted the use of thinner mirrors (meniscus and segmented mirrors), while passive mirrors were relying solely on their thickness to minimize the sag under gravity. This consequently alleviated the requirements on the overall structure, and therefore the overall cost of the telescope. It also allowed a significant relaxation of the requirements on the low spatial frequency quality of the meniscus mirrors made active, letting the man­

ufacturer focus on mid- and high spatial frequencies that also are of practical importance (Noethe, 2009). Fig.1.3 shows the concurrent effects of the resort to active optics and alt-az mounts on the mass of télescopes.

In parallel, adaptive optics has permitted the correction of the optical aberrations

induced by the continuons local changes in the index of refraction of the atmo-

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1.2 Modern-days télescopes 5

Figure 1.3: Total mass versus Ml diameter [(Bely, 2003), p.235].

sphere, giving access to unprecedented optical performances (cfr section 3.3.3).

Finally, the development of Charged-Coupled Devices (CCD) caméras and their use in replacement of photographie plates has allowed a much higher efficiency in the use of the collected light. It also permitted the development of wavefront sensors exhibiting the required performances for their implémentation in active and adaptive optics control loops.

1.2.2 The advent of active optics for monolithic mirrors

Active optics was first implemented in the New Technology Telescope (NTT), a 3.5m telescope completed by the European Southern Observatory (ESO) in 1989 (Wilson et ah, 1987); the implémentation has two aspects, as can be seen from Fig. 1.4. The shape of the primary mirror (Ml) is controlled by actuators pushing against its back, "while the alignment of the secondary mirror (M2) with respect to Ml is maintained through the control of its rigid-body degrees of freedom.

An optical sensor, located downwards M2, measures the aberrations induced in the output wavefront and transmits the information to a controller. The lat- ter détermines the changes in shape and alignment that are responsible of those aberrations, and calculâtes the signais to apply to the actuators to compensate for them and thus obtain the best images.

Compared to similar passive primary mirrors of that time, NTT Ml was twice

thinner (Noethe, 2009). This represented a major improvement as the require-

ments on the structural design were substantially softened, and as the decrease

of the thermal inertia of the mirror also has a direct impact on the image quality

through the phenomenon of mirror seeing (see section 3.2). Fig.1.5 compares

the images produced by the NTT to those produced by other state-or-the-art

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Defocus M2

Decenter

Science instrument

Wavefront sensor

Controiler

M1 shape actuators

b)

Figure 1.4: Active optics at the New Technology Telescope (NTT) - (a) Funda- mental principles [adapted from (Wilson et al., 1987)] - (b) Back of the primary mirror: Each square corresponds to the cell of an actuator (ESO, 2010).

Figure 1.5: (a) ESO Im Schmidt; (b) ESO 3.6m (passive); (c) ESO 3.5m NTT (raw image); (d) ESO 3.5m NTT (after post-processing) (Wilson, 2003).

télescopes in 1989^.

The successful results of the technology developed for the NTT served as the basis for the design of the Unit Télescopes of ESO’s Very Large Telescope (VLT) (Fig. 1.6): Four active télescopes with a Ml of 8.2m (completed successively be- tween 1998 and 2001). It is worth noting that the thickness of VLT Ml (0.17m) is actually smaller than that of NTT Ml (0.24m), in order to fully exploit its

^The primary mirror of NTT sufîered from spherical aberration resulting from an error in polishing. Fortunately, the active optics System of NTT was able to correct it, a fact which, although it was consuming 80% of the control authority, can be seen as its first success.

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1.2 Modern-days télescopes 7

Figure 1.6: Very Large Telescope: (a) 8.2m Unit Telescope after completion - (b) Detail of the back-structure and actuators of its primary mirror (ESO, 2010).

potential in terms of light weight, thermal inertia and control authority

The success of NTT gave rise to two projects very similar to VLT: The Subaru telescope in Hawaii with a primary mirror of 8.2m diameter that was completed in 1999 by Japan (lye et al, 2004) and the Gemini observatory, consisting of two 8.1m télescopes at different sites in Hawaii and Chile completed in 2000 by an international consortium (USA, UK, Canada, Chile, Brazil, Argentina, and Aus- tralia) (Mountain et al., 1994).

An other approach to manufacturing lightweight mirrors was developed in par- allel, based on the mechanical properties of honeycomb-like structures, allowing to reduce the mass without affecting significantly the stiffness, thus minimizing the deformation under gravity. This can be achieved either by direct casting (Angel and Hill, 1982) or by machining the exceeding material. It has been used to produce mirrors such as the 8.4m mirrors of the Large Binocular Telescope (University of Arizona, 2010) and the 6.5m mirrors of the two Magellan Téle­

scopes (AURA, 2010). However, mirrors of those dimensions still require active corrections to be used at their full potential [see (Noethe, 2009) e.g.].

®This fact cornes from a requirement to the design of NTT that it could be used for astro- nomical observations even if the active optics fails (Noethe, 2009), while the images produced by the VLT are not usable without active optics (Wilson, 2003)

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1.2.3 Segmented mirrors

The idea of segmentation consists in replacing a monolithic mirror by an assem- bly of contiguous segments, constituting the tessellation of an optical surface, supported by a single mechanical structure. Segments in the 1- to 2-m-diameter range can be designed to exhibit individual deformation under gravity lower than optical tolérances, while still providing a mass per surface unit much lower than that of équivalent monolithic mirrors. However, active control is required to maintain the overall shape and continuity of the surface formed by the segments due to the deformations of the supporting structure. This is particularly critical if an optical or near infrared (IR) telescope is to be used close to its diffraction limit (see section 2.6).

The most sophisticated form of segmentation has been first implemented success- fully in the optical/near IR Keck I &: II télescopes, that saw first light respectively in 1993 and 1996 (Fig. 1.7.a). Their respective primary mirrors are made of 36 hexagonal 1.8m-diameter segments, for an effective aperture of approximately 10m. Fig. 1.7.b shows a picture of the primary mirror of the Keck télescopes:

Each segment is equipped with a set of sensors that measure the relative normal displacements between two adjacent segments and with 3 actuators that correct their positions (piston and tilts).

Figure 1.7: Keck I & II 10m télescopes: Left, the télescopes inside their enclo- sures; right, front view of the segmented Ml (Keck Observatory, 2010).

Again, the success of Keck gave rise to other projects. Inaugurated in 2009,

the Gran Telescopio Canarias (GTC - Spain) is based on a design very similar

to that of Keck, with a slightly larger segmented primary mirror (10.4m) made

up of 36 segments (Alvarez and Rodriguez-Espinosa, 2004). The Hobby-Eberly

Telescope (HET - USA) (University of Texas, 2008) and the Southern African

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1.2 Modern-days télescopes 9

Large Telescope (SALT - South Africa, Germany, Poland, USA, UK and New Zealand)(Blanco et al., 2003) also both use rectangular segmented primary mir- rors of 11x9.8 meters, made up of 91 hexagonal segments and were completed respectively in 1997 and 2005^.

1.2.4 Long baseline interferometry

Fig. 1.8 shows the basic principles of long baseline interferometry; it consists of two or more independent télescopes separated by a distance called the baseline, B, that point at the same object. Instead of being driven to their respective instruments, the image they produce are combined in a single beam illuminating a caméra. Because of the wave nature of light, instead of an image, the combi­

nation produces interférence fringes containing information about the image that can be accessed through post-processing. However, as shown by the figure, the wavefront enters the optical train of each telescope with a certain delay time.

Obtaining the best fringes requires that delay to be reduced to a portion of the wavelength; this is done through so-called delay Unes. Once phased, the aper- tures composing the interferometer can be seen as éléments of a single collecting optical surface (Enard et al., 1996).

Figure 1.8; General principles of interferometry [adapted from (ESO, 2010)]

The benefit is that the resolution of such an interferometer is proportional to (Bsin61)“^ instead of D~^. Consequently, for a given total collecting area, an in-

■*Their very spécifie design and their use without cophasing, mainly for spectroscopy, aimed at lower construction costs, making them difficult to compare to Keck or the GTC.

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terferometer can hâve a resolution several orders of magnitude higher than that of a single telescope^. Moreover, the use of interferometry at its full potential requires that active and adaptive optics should be very efficient to ensure a good phasing of the beams. Finally, the post-processing of the fringes requires exten­

sive computer power and an important observing time. Therefore, in the near future it will most likely be complementary to conventional observing techniques involving télescopes with large apertures.

Interferometric techniques are implemented in modem optical and infrared téle­

scopes. The Keck Observatory uses the 85m baseline between Keck I and II (Colavita et al., 2004). In the VLT-Interferometer, up to three of the eight téle­

scopes can be combined: The four 8.2m Unit télescopes hâve fixed locations while the four 1.8m Auxiliary télescopes can adopt different configurations to modify the length and the orientation of the baseline (Glindemann et al, 2004).

One could also mention other particular projects such as the Large Binocular Telescope (University of Arizona, 2010) or the Giant Magellan Telescope (Johns et al., 2004) that are based on respectively two and seven 8.4m lightweight pri- mary mirrors assembled on a single back-structure (Fig.1.9). The goal is to be able to operate them either with or without interferometry mode, in which the optical trains must be phased to attain the best resolution permitted by their optical designs.

Figure 1.9: Giant Magellan Telescope project (AURA, 2010).

®But the sensitivity remains a function of the sum of the areas of the mirrors composing the interferometer.

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1.2 Modern-days télescopes 11

1.2.5 Space télescopes

The atmosphère sets strong boundaries to ground-based astronomy. First, it is transparent only to a small portion of the electromagnetic spectrum, namely the visible and the near-infrared and it blocks or absorbs the rest (ultraviolet, gamma- and X-rays,... Purthermore, the quality of the wavefront emitted by celestial objects is continuously degraded by turbulence in the successive layers of the atmosphère.

Those reasons led to the launch of space télescopes programs from early 1980.

The Hubble Space Telescope (HST), launched in 1990 , is probably one of the most emblematic projects. The HST is depicted in Fig.1.10, its primary mirror is a 2.4m diameter monolithic lightweight mirror; it produces diffraction-limited images in the ultraviolet, visible and near-IR and is also used for spectrometry.

Its initial results were poor due to an error in the fabrication of its Ml: The installation of optical éléments to compensate for that error required the launch of a dedicated space mission three years later®. Thanks to this correction, the telescope was able to reach its full potential and the data it produced led to countless scientific publications.

Figure 1.10: Hubble Space Telescope (NASA, 2010a).

®The error was quite similar to that aflFecting the Ml of the NTT but the active devices in HST could not correct it.

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Following the same trend as ground-based télescopes, space télescopes with larger primary mirrors are planned for the future. Fig.1.11 depicts the James Webb Space Telescope (JWST), to be launched in 2014. Its 6.5m primary mirror will consist of 18 hexagonal segments. During the launch, JWST is folded configura­

tion in order to comply to the limited available volume in the cap; once in orbit, its active structure deploys itself and then maintains its optical configuration, to provide diffraction-limited imaging in the IR (Gardner et al, 2006).

Figure 1.11; James Webb Space Telescope: Left, folded configuration of the JWST, during the launch - Right, after deployment, once in orbit (NASA, 2010b).

However, space télescopes suffer from a limited lifetime and few or no possibilities

of maintenance, from long development times and from costs that are far above

those of ground-based Systems. The advent of adaptive optics allows ground-

based télescopes to compensate for most of the optical aberrations induced by

atmospheric turbulence. On the other hand, space télescopes can observe wave-

lengths unattainable by ground-based ones, their operation does not dépend on

the weather, they do not suffer from luminous backgrounds,...

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1.3 Future Extremely Large Télescopes 13

1.3 Future Extremely Large Télescopes

Monolithic mirrors larger than the current 8m génération are difficult to produce, and would set severe constraints on the design of their support structures, to maintain their shape and alignment to severe optical tolérances. As a resuit, segmentation seems the only promising solution to reach diameters of 20m and beyond (Strom et al., 2003), to form the class of the so-called Extremely Large Télescopes (ELTs) on which the remainder of this text will focus.

Eigure 1.12: Thirty Meter Telescope project (TMT, 2010) Several projects of future ELTs hâve been proposed since the end of the century. Three different télescopes were investigated in North America: The Cal­

ifornia Extremely Large Telescope (involving many of the persons that worked on the Keck télescopes) (California Institute of Technology, 2002), the Giant Segmented Mirror Telescope (National Optical Astronomy Observatory, 2002), and the Very Large Optical Telescope (Roberts et al, 2003), with segmented pri- mary mirrors of resp. 30m and 20m for the latter. In 2003, those projects were abandoned; their respective consortia joined their efforts into a new common project, namely the Thirty Meter Telescope (TMT), depicted in Fig.1.12. Its 30m segmented primary mirror will be tesselated with approximately 500 hexag­

onal segments (TMT Obs. Corp., 2007). Its construction officially started in

2009; TMT is expected to see first light around 2018.

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In Europe, ESO studied the concept of the Overwhelmingly Large Telescope OWL (ESO, 2004), a very ambitions project combining six reflectors, amongst which both the primary and the secondary mirrors would be segmented, the first being a 100m spherical reflector made up of more than 3000 segments, and the second a 20m reflector made up of more than 200 segments. In parallel, a consor­

tium led by the Lund Observatory in Sweden proposed a concept for the Euro50 (Lund Observatory, 2003), a telescope with a 50m primary mirror composed of 618 segments.

Eventually, some aspects of OWL were judged too risky, especially with respect to its high projected cost. A new project was developed, involving both ESO and the team working on the Euro50 (that was abandoned too): The European Extremely Large Telescope (E-ELT), with a segmented 42m primary mirror tes- selated by approximately 1000 hexagonal segments, that is depicted in Fig.1.13.

As a compromise between ambition and timeliness, certain high-risk items of OWL were avoided, such a the spherical Ml and the segmentation of M2; it is scheduled to see first light in 2017 (Gilmozzi and Spyromilio, 2008).

Japan has also started the conceptual study of a 30m telescope with a segmented primary mirror, called the Japan Extremely Large Telescope (JELT), that should be made up of approximately 1080 segments (lye et al, 2004).

Figure 1.13: European Extremely Large Telescope (ESO, 2010)

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1.4 Scale effects 15

1.4 Scale effects

The implémentation of different layers of active control hâve allowed télescopes to reach an unprecedented higher level of optical performances. In particular, active optics has allowed a much more efficient use of the telescope structure and has made segmented optics possible in the visible and in the near infrared.

The success of Keck is the promise to attain a significantly larger size of primary mirrors in the near future.

Fig.1.14 compares the Ml of some of the most celebrated télescopes, the existing ones (HST, VLT and Keck) and the future ones due to be built within the next decade (JWST, TMT and E-ELT), that will ail be segmented. Note that the size of the earth-based télescopes is one order of magnitude larger than that of space télescopes. The gap between the largest existing segmented telescope in use today (Keck) and the future ones is large and appears even larger in Table 1.1, that compares some key aspects of Keck and E-ELT.

Space Télescopes Ground-based Télescopes

Figure 1.14; Primary mirrors of current and future optical and infrared télescopes.

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Keck E-ELT

Ml diameter: D 10 m 42 m

Segment size 1.8 m 1.4 m

Collecting Area 76 m2 1250 m2

^ Segments: N 36 984

# Actuators 108 2952

^ Edge Sensors 168 5604

/segment (“1“ W^hiffle T

t

6

g

) 25 Hz ~ 60 Hz

fl {Ml) ~ 10 Hz ~ 2.5 Hz

/2 (M

2

) ~ 5 Hz ~ 1-2 Hz

Adaptive Optics d.o.f.) ~ 350 ~ 8000 Tube and mount mass ~ 110 t ~ 2000 t

Table 1.1: Keck vs. E-ELT

Moreover, as the size of the télescopes increases, they become increasingly sensi­

tive to external disturbances such as thermal gradients, gravity and wind, and to internai disturbances from support equipments such as pumps, cryocoolers, fans, etc. These disturbances can deteriorate significantly the image quality. As a resuit, the shape stability of ELTs relies more and more on active control means:

The control System involves larger loop gains, and therefore a larger bandwidth.

At the same time, the natural frequency of future ELTs is expected to be sub- stantially lower than any operating télescopés. Those conditions, combined to the very low inhérent damping of welded steel structures, increase the risk of control-structure interaction. Therefore, one can reasonably wonder if the past expérience with Keck is sufficient to warrant a sound design and optimum oper­

ation of the future ELTs, and this point alone deserves a careful attention.

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1.5 Outline 17

1.5 Outline

This text is concerned with the extrapolation of the active optics of current 10- meter class télescopes (Keck, GTC, VLT) to the next génération of 30m to 40m ELTs, and future, even larger ones. It studies how the varions factors affecting the structural response and the control-structure interaction are infiuenced by the size of the telescope.

Chapter 2 présents the basics of telescope optics. It is focused on the optome- chanical parameters that affect the optical quality.

Chapter 3 describes the varions layers of control of large télescopes, with an em- phasis on the active optics of the Keck télescopes.

The first part of chapter 4 is devoted to the numerical modelling of active optics in large segmented mirrors. The second part studies the problem of control- structure interaction in future ELTs. A parametric study is conducted, based on the numerical model developed previously.

Chapter 5 is concerned with the extrapolation of active optics of current téle­

scopes to the future ELTs. Scaling laws are proposed to evaluate the optome- chanical performances of a telescope without resorting to complicated analysis.

Chapter 6 is dedicated to the comparison of those scaling laws with numerical parametric studies involving représentative models based on the approach de- scribed in the first part of chapter 4.

1.6 Référencés

Alvarez, P. and Rodriguez-Espinosa, J. M. The GTC project: in the midst of intégration. In Oschmann, J. M., editor, Ground-based Télescopes - SPIE 5489, pages 583-591, 2004.

Angel, J. R. P. and Hill, J. M. Manufacture of large glass honeycomb mirrors. In Burbidge, G. and Barr, L. D., editors, International Conférence on Advanced

Technology Optical Télescopes - SPIE 332, pages 298-306, 1982.

AURA. Giant Magellan Telescope Observatory website, 2010. URL

http://www.gmto.org/.

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

Blanco, D. R., Pentland, G., Winrow, E. G., Rebeske, K., Swiegers, J., and Meiring, K. G. SALT mirror mount: a high performance, low cost mount for segmented mirrors. In Angel, J., R. P. and Gilmozzi, R., editors. Future Giant

Télescopés - SPIE 4^4-0-, pages 527-532, 2003.

California Instituts of Technology. California Extremely Large Telescope : con- ceptual design for a thirty-meter telescope. Technical report, 2002. URL http://celt.ucolick.org/reports/greenbook.pdf.

Colavita, M. M., Wizinowich, P. L., and Akeson, R. L. Keck Interferometer status and plans. In Traub, W. A., editor, New Frontiers in Stellar Interferometry - SPIE

5491

, October 2004.

Enard, D., Maréchal, A., and Espiard, J. Progress in ground-based optical téle­

scopes. Reports on Progress in Physics, 59:601-656, 1996.

ESO. European Southern Observatory website, 2010. URL http : //www. eso. org.

ESO. OWL Concept Design Report - Phase A design report. European SOuthern Observatory, 2004.

Gardner et al. The James Webb Space Telescope. Space Science Reviews, 123 (4):485-606, April 2006.

Gilmozzi, R. and Spyromilio, J. The 42m European ELT: status. In Stepp, L. M.

and Gilmozzi, R., editors, Ground-based and Airborne Télescopes II - SPIE 7012, 2008.

Glindemann et al. VLTI technical advances: présent and future. In Traub, W. A., editor, New Erontiers in Stellar Interferometry - SPIE 5491, 2004.

lye et al. Current Performance and On-Going Improvements of the 8.2 m Subaru Telescope. Publications of the Astronomical Society of Japan, 56(2):381-397, April 2004.

Johns, M., Angel, J. R. P., Sheetman, S., Bernstein, R., Fabricant, D. G., Mc­

Carthy, P., and Phillips, M. Status of the Giant Magellan Telescope (GMT) Project. In Oschmann, J. M., editor, Ground-based Télescopes - SPIE 5489, pages 441-453, 2004.

Keck Observatory. Keck Observatory website, 2010. URL

http://www.keckobservatory.org/.

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References 19

Lund Observatory. EuroSO - A 50m Adaptive Optics Telescope. Andersen, T., Ardeberg, A. and Owner-Petersen, M., 2003.

Mountain, C. M., Kurz, R., and Oschmann, J. Gemini 8-m télescopes project.

In M., S. L., editor, Advanced Technology Optical Télescopes V - SPIE 2199, pages 41-55, June 1994.

NASA. The Hubble Space Telescope website, 2010a. URL http://hubblesite.org/.

NASA. The James Webb Space Telescope website, 2010b. URL http://www.j wst.nasa.gov/index.html/.

National Optical Astronomy Observatory. The Giant Segmented Mirror Tele­

scope Book, 2002. URL http://www.gsmt.noao.edu/book/.

Noethe, L. History of mirror casting, figuring, segmentation and active optics.

Experimental Astronomy, 26(1-3):1-18, August 2009.

Roberts et al. Canadian very large optical telescope technical studies. In Angel, J., R. P. and Gilmozzi, R., editors. Future Giant Télescopes - SPIE

484

O, pages 104-115, January 2003.

Strom, S. E., Stepp, L., and Brooke, G. Giant Segmented Mirror Telescope: a point design based on science drivers. In Angel, J., R. P. and Gilmozzi, R., editors. Future Giant Télescopes - SPIE 4840, pages 116-128, 2003.

TMT. The Thirty Meter Telescope website, 2010. URL http://www.tmt.org/.

TMT Obs. Corp. Thirty Meter Telescope - Construction Proposai, 2007. URL http://www.tmt.org/docs/0AD-CCR21.pdf.

University of Arizona. Lare Binocular Telescope Observatory website, 2010. URL http : //médusa. as. arizona. edu/lbto/.

University of Texas. The Hobby Eberly Telescope website, 2008. URL http : //www. as. utexas. edu/mcdonald/het/het. html.

Wilson, R. N. The History and Development of the ESO Active Optics System.

The Messenger, 113:2-9, September 2003.

Wilson, R. N., Pranza, F., and Noethe, L. Active Optics I. A System for optimizing the optical quality and reducing the costs of large télescopes. Journal of Modem

Optics, 34(4):485-509, 1987.

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Chapter 2

Basics of Telescope Optics

2.1 Introduction

focal surface

Figure 2.1: Principles of imaging with a telescope.

A telescope is an instrument désignée! to image objects located at large distances from the observer. Those objects can be either point-like (e.g. stars) or extended objects (e.g. nearby planets) that can be seen as an ensemble of points. The spherical wavefront emitted by such point sources located at the infinité can be considered as plane at the level of the telescope (Fig.2.1). The rôle of the tele­

scope aperture, namely its primary mirror (Ml), is to collect the light energy;

as the radiated energy is distributed over the area of the wavefront, the larger the aperture, the more energy collected and the fainter the objects that can be observed by that telescope.

The focusing of the light is performed by the optical éléments composing the optical path of the telescope, including Ml. The designs vary greatly depending on the use and on the cost of the telescope, and can include from 1 to 6 mirrors, for the most complex design published in the literature (OWL). The quality of a telescope can be summarized by its ability to focus the energy emitted by a

21

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Figure 2.2: (a) Point object; (b) Image spot subject to diffraction; c) Image spot subject to aberrations (and diffraction).

point object into the smallest area possible on the focal surface. Diffraction as well as déviations from the initial design (aberrations) cause a spreading of the energy away from the nominal focus, setting physical and practical boundaries to the performances of the telescope in terms of resolution and contrast (Fig.2.2).

This chapter summarizes the basic concepts of optics that govern the optome- chanical performances of a telescope.

2.2 Aberrations

2.2.1 Définition

Figure 2.3: (a) Rays emerging from a spherical wavefront converge towards a single point in the image plane; (b) Rays emerging from an aberrated wavefront hit the image plane over an extended area, spreading the light energy [adapted from (Geary, 2002), p.79].

On a strictly géométrie point of view, a perfect telescope (like in Fig.2.1) should focus the light of a distant (dimensionless) point source into a (dimensionless) image point on the focal surface, to establish a point-to-point correspondence be- tween an object and its image (Fig.2.3.a). In other words, it should transform an incoming diverging spherical wavefront into a spherical wavefront converging to­

wards a point on the focal surface [(Schroeder, 2000), p.45]. However, déviations

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2.2 Aberrations 23

from the initial design in the shape or in the position of the optical éléments, or the observation of off-axis objects will induce distortions of the output wave- front, causing the light energy to be spread on the image surface as depicted in Fig.2.3.b. Those déviations are called aberrations.

Accordingly, the images generated by aberrated wavefronts resuit from a super­

position of light spots of finite size rather than points. It croates a blur in the image, the amplitude and shape of which is roughly determined by those of the aberrations présent. Historically, a classification of five primary aberrations has been established, namely spherical aberration, coma, astigmatism, field curva- ture and distortion^. They were classified according to analytical developments made by Seidel and they correspond to the optical signatures of some of the most typical déviations from the initial design. Combined with the dérivation of their analytical expressions, they were the basis of the measurements of optical quality since the XIX*^ century. A deeper discussion is provided in appendix B.

2.2.2 Quantifying the wavefront error

The wavefront error with respect to its reference sphere can be expressed as a function of space coordinates lT(r). Its root mean square (RMS) value, com- puted over its whole surface provides an effective indicator of the quality of a wavefront (or of the surface of a mirror)^. It is mostly expressed either as an absolute measurement (in units of microns e.g.) or as a relative measurement, a fraction of the wavelength of observation, A. Conventionally, a System is consid- ered as nearly perfect if the RMS wavefront error of the output beam is less than A/14 (cfr section 2.5).

In many applications, it is not required to know point by point the shape of the error. By extending the principles behind the use of the primary aberrations, it can be more convenient and efficient to express the wavefront error as a linear combination of a set of orthogonal functions defined over the whole aperture.

One of the most common analytic représentation uses the Zernike polynomials, in the form

n

W{r,d) = J2aiZi(r,9), (2.1)

1=1

where W{r,d) and

Zi{r,6)

are respectively the wavefront and the i*^ Zernike

^By nature, reflecting télescopes are not affected by chromatic aberrations, the reader can refer e.g. to [(Walker, 1998), p.l38] for more information.

^Other indicators, such as the peak-to-valley wavefront error (P-V) can be misleading as they give no information about the the area over which the error is occurring.

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polynomial expressed in polar coordinates. The coefficient a, results from the projection of W on Zi, and may be computed either by direct intégration over the unit circle or by least square fitting. The Zernike polynomials hâve élégant analytical expressions, the formulation of which can be automated easily [see e.g.

(Malacara, 1992), p.464]^. They are given up to number 11 in Table 2.1 and depicted in Fig.2.4.

Polynomial Dénomination

1

Piston

y/Âp

COS

9 Tilt y/Âp sin 9 Tilt

\/3 (2p2 - 1)

Defocus

y/6 (p^

sin

29)

Astigmatism

\/6 (p^ COS 20)

Astigmatism

\/8

(3p^

2p)

sin

9

Coma

\/8

(3p^ — 2p)

cos

9

Coma y/Sp^ sin 30 Trifoil y/8p^ cos 30 Trifoil

y/E (6p^ - 6p^ + l)

Spherical aberration

Table 2.1: Zernike polynomials [convention from (Zemax Corp., 2005)].

The piston and the tilt terms correspond respectively to a constant and to a linear phase shift ail over the wavefront; the latter only change the location of the focus on the image surface. Accordingly, none of those terms hâve an impact on the image quality. Defocus corresponds to a change of the overall radius of curvature of the wavefront, changing the position of the focus either upstream or downstream the initial image surface. The shapes of astigmatism, coma and spherical aberrations as Zernike polynomials are close (but not identical) to those of the corresponding primary aberrations.

The Zernike polynomials are usually classified with respect to their radial and azimuthal orders: The higher the orders of the polynomial, the higher its spatial frequency and, usually, the lower its amplitude in the wavefront error [this is re- ferred to as the principle of St Venant in (Wilson et al., 1987)]. In general, most of the wavefront errors due to misalignment, mechanical and thermal distortions

®Care should be taken, however, that the ordering and the normalizing of the Zernike poly­

nomials currently admit no single standard.

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2.2 Aberrations 25

Azimuthal Order

-4-3-2-10 1 2 3 4

% #

Tilt Tilt

A ▼ ^

Astigmatism Defocus Astigmatism

■ ^ ^

Trefoil Coma Coma Trefoil

rv ^ w ^ nr

Tetrafoil Spherical Aberration Tetrafoil

Figure 2.4: Zernike polynomials ranked according to their azimuthal and radial orders.

and misliguring can be described by combining the first 20 polynomials. On the contrary, they are not best fitted for the description of errors at very high spatial frequencies, such as surface roughness of mirrors, point defects, or the highest frequencies of air turbulence, that would require an unpractically high number of terms.

The Zernike polynomials hâve a zéro mean and are orthogonal over the unit circle. The mean square error of the total aberration is the weighted sum of the mean square errors of each Zernike term'* [(Schroeder, 2000), p.264]. With the normalization used in Table 2.1, the total RMS error of the wavefront is simply

^The weights dépend on the normalization.

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n

RMS=

(2.2)

The sum of Eq.(2.2) does not include piston {i = 1) and tilt (i = 2,3) as they do not affect the image quality.

The analytical expressions above are defined on unobstructed circular pupils.

Other polynomials hâve been proposed for other pupil shapes on which the wave- front error is analyzed: Circular aperture with a central circular obstruction (Mahajan, 1981), hexagonal or rectangular aperture (Mahajan and Dai, 2007), etc.

2.3 Common optical configurations of optical télescopés

The notions of f-number (//#) and field of view (FOV), that are used extensively in the following, are defined in appendix A.

2.3.1 Newtonian télescopes

Figure 2.5: Newtonian telescope: (a) Configuration giving access to the prime focus Fl - (b) A folding mirror gives an easier access to focus F

2

.

According to the fundamental property of conics, the simplest telescope could be

built with a single parabolic refiector, taking advantage of the fact that one of

its foci is at infinity. This design was implemented by Isaac Newton in his first

telescope, commonly referred to as Newtonian telescope (Fig.2.5). However, as

illustrated in appendix B, wavefront errors may arise as well from errors in the

shape of the mirror, or by observing off-axis objects. Table 2.2 synthesizes the

dependence of the primary aberrations with respect to the field angle 6, and the

//# of the overall telescope. Those relations set severe constraints on the design

and use of such a telescope, and limit its implémentation to télescopes with rather

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2.3 Common optical configurations of optical télescopés 27

small diameters of Ml, large //^ and small FOV. The most limiting aberration in this case is coma. Finally, the focus of such télescopes is difficult to access, and would complicate the design of the supporting structure for large mirrors.

Spherical (//#)“^

Coma 0(//#)"2 Astigmatism

Table 2.2: Scaling laws of primary aberrations affecting a Newtonian telescope [(Bely, 2003), p.lll].

2.3.2 Two-mirror télescopes

Figure 2.6: (a) Cassegrain telescope - (b) Gregorian telescope.

The limitations of the Newtonian telescope can be (at least partially) overcome by increasing the complexity of the optical design, consisting in a secondary conic mirror, with one of its foci collocated with that of the paraboloidal Ml. There are two important classes of two-mirror télescopes differing in the shape of the secondary mirror: The Cassegrain uses a convex h

5

^erboloid and the Gregorian a concave ellipsoid (Fig.2.6).

However, the relations of Table 2.2 still apply to the Cassegrain and Gregorian

télescopes because of their paraboloidal Ml. The dominant off-axis aberration is

still coma. The différence lies in the overall //# of the telescope, that are larger

than that of a Newtonian telescope with the same diameter and tube length, thus

allowing for larger fields. However, the need for still larger fields has called for

a more efficient use of the geometrical parameters of the conics, based on two

considérations. First, the requirement for sphericity only applies to the output

wavefront. Second, it is possible, by a proper choice of geometrical constants of

downstream mirrors, to compensate fully or partially for the aberrations induced

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by upstream mirrors^.

This led to design variations of the classical Gregorian and Cassegrain. In those variations, the paraboloid constituting Ml is replaced respectively by an ellipsoid and a hyperboloid. It can be shown from the équations [see e.g. (Schroeder, 2000), p.ll5] that the departure of Ml from a paraboloid causes the reflected wavefront to be different from a sphere, but that it can be corrected by a proper choice of M2 as équations show that there is an infinité number of combinations of the conic constants of Ml and M2 that ensure the correction of spherical aberration of the output wavefront. Amongst them, some particular combinations allow to compensate for off-axis aberrations as well, of which coma is the prévalent one. Designs that compensate for both coma and spherical aberrations are called aplanatic and the aplanatic Cassegrain is better known as the Ritchey-Chrétien]

the field of such télescopes is larger than that of their classical versions, and is limited by astigmatism according to Table 2.2.

2.3.3 Télescopes with 3 or more mirrors

The principles of the generalized Schwarzschild theorem hâve been put in practice both in analytic studies of theoretical designs, and in actual projects of future ELTs. The use of 3 or more mirrors allows for the compensation for aberrations such as the off-axis astigmatism and the distortion and field curvature on the image plane. It also opens the way to using a segmented spherical Ml that would exhibit significant advantages in terms of the manufacturing, testing and maintenance of the segments, balanced by the need for at least two corrector mirrors to compensate for the significant on-axis aberrations induced by such fast primaries®. Those designs also permit a better intégration of beam steering and déformable mirrors, respectively for image motion and wavefront correction.

2.4 Wavefront error due to déviations from the design

During observations, a telescope is subjected to external disturbances that can modify the shape of its mirrors and their relative positions in the optical train.

This section présents basic relations on the sensitivity of the wavefront with respect to those déviations in the case of a two-mirror telescope (they are equally

®This is a general principle stated in the generalized Schwarzschild theorem: "For any geom- etry with reasonable séparations between the optical éléments, it is possible to correct n primary aberrations with n powered éléments.” [(Bely, 2003), p.l23]. In this context, the term ”pow- ered éléments” refers to conics; surfaces of higher degrees could compensate for more than 1 aberration but are difRcult to produce and test.

®A mirror is said to be fast if it has a small //#, cfr appendix A

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2.4 Wavefront error due to déviations from the design 29

valid for a Cassegrain, a Gregorian and their aplanatic variations and can be extended to télescopes with more mirrors).

2.4.1 Shape of optical éléments

To a first approximation, after reflection on an aberrated mirror, the error af- fecting the wavefront is twice that of the mirrors; it can be expressed in terms of Zernike coefficients, according to Eq.(2.3). Therefore, the optical tolérances, when referring to a mirror, are twice as severe as when referring to a reflected wavefront. For curved mirrors, simulations show that the value of the coefficient is slightly smaller than 2, and that the différence grows with the amplitude of the input aberrations and when the mirror is faster (smaller //#)•

®i,output wavefront ~ 2.mirror misfigure

(2-3)

2.4.2 Relative position of optical éléments

Figure 2.7: Despace d, tilt a and decenter l.

If the mirrors are ail made up of surfaces of révolution, the influence of their rel­

ative positions is essentially governed by three relative parameters (for each pair of mirrors) that describe the déviations with respect to the initial design; they are defined on Fig.2.7, namely (axial) despace, (latéral) decenter and tilt. Table 2.3 summarizes the scaling laws of the primary aberrations induced when such déviations are présent [(Schroeder, 2000), p.l32]. A dependence in 6 indicates the variation of the induced aberration with the field angle. Those aberrations must be added to those induced when observing off-axis objects (cfr Table 2.2), either due to the observation of extended objects or due to errors in pointing.

In addition to the aberrations, the position of the image on the focal surface is

shifted of an amount proportional to / and to a. A general conclusion regarding

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Aberration Despace d [m] Decenter l [m] Tilt a [rad]

Sp / /

C ddif/#)-^ i (//#)-" «(//#)“'

A 1 /

Table 2.3: Scaling laws of primary aberrations affecting a two-mirror telescope under the effect of déviations in the relative position of the mirrors. The spher- ical aberration (Sp), coma (C) and astigmatism (A) refer to the corresponding primary aberrations.

structural aspects that can be drawn from Table 2.3 is that a telescope with a faster Ml (smaller //#) is more sensitive to position errors of any kind.

2.4.3 Linearity

The output wavefront error in terms of primary aberrations can be computed by adding those induced along the optical train of a telescope [(Schroeder, 2000), p.93]. The analytical expressions relating the low-order Zernike polynomials to the primary aberrations are developed in (Wyant and Creath, 1992). Although those relations are non-linear, simulations show that, over a quite extended régime of aberrations, the wavefront errors induced by the various éléments can be added in terms of their Zernike coefficients without generating significant déviations with respect to the actual output wavefront error (Angeli and Gregory, 2004; Noethe, 2002; Whorton and Angeli, 2003). Therefore, linear optomechanical models can approximate the Zernike coefficients of the output wavefront by

O'i, output ~ input 4“ ^ ai + üi+ Y (2-4) shape alignment off—axis

2.4.4 Design trade-ofFs

The trends in the design of télescopes can be summarized roughly by three re- quirements: A wide unaberrated field of view, a good resolution and a good light gathering power. The trade-offs consist of balancing between contradictory ad- vantages from optical and structural points of view.

The combination of good resolution and light gathering power call for large and

fast primary mirrors. The evolntion of the //# of Ml in time is illustrated

in Fig.2.8. A small //# of Ml brings several other advantages: The structure

and the enclosure are comparably smaller, having a significant impact on their

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2.4 Wavefront error due to déviations from the design 31

Figure 2.8: Evolution in time of the f-number of the primary mirrors of optical and infrared télescopes [adapted from (Bely, 2003), p.l36].

total costs. The compactness of the structure also ensures a comparably higher stiffness, a lower overall mass (and thus less thermal inertia) and a smaller M2^, leading to higher eigen frequencies. On the other hand, as shown in section 2.3, a smaller //# induces tighter tolérances on the alignment of the optical éléments, mirror shapes that are more difficult to produce and test and is responsible for higher off-axis aberrations [efr Table 2.3 and(Strom et al., 2003)].

Télescopes in the range 2-lOm largely rely on two-mirror configurations, amongst which the Cassegrain type (mostly in its Ritchey-Chrétien variation to improve the FOV) are the most common: For a given //# , a Cassegrain is more compact and has a smaller secondary. Those advantages overcome their slightly worse off- axis aberrations and difficulty to test convex M2.

More elaborate designs are considered for some future ELT projects: In the three-mirror design of JELT (Nariai and lye, 2005) and the five-mirror design of E-ELT (ELT Telescope Design Working Group, 2006; Spyromilio et al., 2008), the motivations are essentially to extend the usable field of view. In the six- mirror concept of OWL, it is also constrained by the envisioned spherical Ml.

smaller secondary offers advantages in terms of mass, thermal inertia, optical testing, obscuration of Ml (more light gathering power), area exposed to the wind and diffraction (see section 2.5.3).

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However, mirrors are the most critical éléments with respect to optical quality.

Moreover they, and their supporting structures, represent massive éléments, the position and shape of which must be maintained with tight tolérances. Therefore, the smaller and the less numerous they are, the simpler and more effective the structure. It is worth noting that the designers of the 30m TMT hâve chosen a Ritchey-Chrétien configuration (TMT Obs. Corp., 2007), building on their successful expérience with Keck.

2.5 Diffraction-limited imaging

2.5.1 Définitions

Because of the wave nature of light, even a perfect (unaberrated) optical

Sys­

tem will not image a point source as a true point, but rather as a bright core surrounded by a halo. This spreading of the light energy is called diffraction.

Light is diffracted at the edges of any opaque body that it crosses on the path between the object and the image plane: Diaphragms, mirrors, lenses, structural éléments,... Those edges modify the interférences of the light waves as they travel through space, which in turn spreads the light energy in deterministic patterns defined by the shape of the opaque body.

An optical System is said to be diffraction-limited when the aberrations are suffi- ciently small so that the size of the image point is only limited by the diffraction.

It is a lower physical boundary to the size of the image spot that a perfect imag­

ing System can produce. Therefore, it sets a limit under which the aberrations hâve little impact on the image quality: Roughly speaking, an imaging System can be considered as perfect if the area over which the rays hit the image plane is encompassed by the central bright spot produced by diffraction.

2.5.2 Imaging

The image of a point object formed by an imaging System is called its Point Spread Function (PSF). The PSF takes the diffraction and aberrations into ac- count. The image formation consists in a convolution of each point of the object by the PSF. Therefore, the narrower the PSF, the sharper the image. The PSF of a diffraction-limited imaging System with an unobstructed circular aperture, Fig.2.9, is called the Airy disk (e.g. the on-axis image formed by a parabolic mirror). It consists in a bright core surrounded by concentric rings. The central spot contains approximately 84% of the total light energy on the image surface;

its diameter is proportional to A, the wavelength of the light, and to the f /ff of

the System.

Figure

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