ExtremelyLargeSegmentedMirrors:Dynamics,ControlandScaleEffects Universit´eLibredeBruxelles

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Universit´ e Libre de Bruxelles

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

Extremely Large Segmented Mirrors:

Dynamics, Control and Scale Effects

Renaud Bastaits

First vibration mode(l/2) Flat mirror l/2

Thesis submitted in candidature for the

degree of Doctor in Engineering Sciences June 2010

Active Structures Laboratory

Department of Mechanical Engineering and Robotics

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Jury

Supervisor : Prof. Andr´e Preumont (ULB) Members :

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

Dr Arnaud Deraemaeker (ULB)

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Remerciements

Les comp´etences scientifiques et p´edagogiques du Professeur Andr´e Preumont, son dynamisme et sa personnalit´e ont constitu´e le ciment de ce travail. Je lui suis tr`es reconnaissant de m’avoir accept´e comme ´el`eve. Son contact restera pour moi source d’enrichissement personnel sous bien des ´egards, au-del`a des seuls as- pects scientifiques.

Je tiens ´egalement `a remercier tout particuli`erement Christophe Collette par l’interm´ediaire de qui j’ai commenc´e mes travaux au sein du Laboratoire des Structures Actives. Ses conseils, sa bonne humeur `a toute ´epreuve et son soutien m’ont aid´e `a trouver mes marques et `a avancer.

De mˆeme, je souhaite remercier Gon¸calo Rodrigues avec qui j’ai travaill´e en ´etroite collaboration depuis ma r´eorientation dans le domaine des t´elescopes. Partager les sujets de recherche, le mˆeme bureau, ainsi que la plupart des repas a ´et´e source d’une stimulation constante.

Le Laboratoire des Structures Actives a constitu´e pour moi un environnement stimulant; par leurs qualit´es humaines et scientifiques, les personnes que j’y ai cˆotoy´ees m’ont aid´e `a avancer dans une atmosph`ere d’´emulation toujours souri- ante.

Je remercie ´egalement le Prof. Frank Dubois, et les Dr Yvan Stockman et St´ephane Roose pour m’avoir apport´e `a de nombreuses reprises leur aide pr´ecieuse pour progresser face `a des questions d’optique.

Je remercie ´egalement le Fonds National de la Recherche Scientifique pour le sou- tien financier qu’il m’a apport´e, via la bourse FRIA FC76554.

Enfin, je suis heureux de partager le fruit de ce travail avec mes proches qui, chacun `a leur mani`ere, apportent ce qui fait le sel de ma vie.

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Abstract

All future Extremely Large Telescopes (ELTs) will be segmented. However, as their size grows, they become increasingly sensitive to external disturbances, such as gravity, wind and temperature gradients and to internal 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 separation 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 telescopes to future ELTs and even future larger telescopes 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 telescopes.

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Contents

1 Introduction 1

1.1 Evolution of telescopes: From 1600 to 1980 . . . 1

1.2 Modern-days telescopes . . . 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 telescopes . . . 11

1.3 Future Extremely Large Telescopes . . . 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 Definition . . . 22

2.2.2 Quantifying the wavefront error . . . 23

2.3 Common optical configurations of optical telescopes . . . 26

2.3.1 Newtonian telescopes . . . 26

2.3.2 Two-mirror telescopes . . . 27

2.3.3 Telescopes with 3 or more mirrors . . . 28

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

2.4.1 Shape of optical elements . . . 29

2.4.2 Relative position of optical elements . . . 29

2.4.3 Linearity . . . 30

2.4.4 Design trade-offs . . . 30

2.5 Diffraction-limited imaging . . . 32

2.5.1 Definitions . . . 32

2.5.2 Imaging . . . 32 ix

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2.5.3 Diffraction from obscurations . . . 34

2.5.4 Diffraction-limited versus aberrated . . . 35

2.6 Telescopes 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 telescopes . . . 38

2.7 Conclusions . . . 41

2.8 References . . . 41

3 Active control of telescopes 45 3.1 Introduction . . . 45

3.2 External disturbances . . . 46

3.3 Layers of active control in modern telescopes . . . 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 References . . . 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 reduction . . . 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 References . . . 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 resonance 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 Definitions of optical design parameters 137 A.1 References . . . 139

B Primary aberrations 141 B.1 References . . . 144

C Shack-Hartmann sensors 145 D Small-gain theorem 149 D.1 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 telescopes: From 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 ages, astronomers have 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 XVIth century, and its improvement and use by Galileo 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 reflecting telescope around 1670, based on the use of a parabolic mirror instead of a lens as the light collector. The reflecting telescope exhibits some advantages with respect to the refracting one, in particular the fact that they are exempt of chromatic aberrations, as reflection laws do not depend 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 (M1) (Enard et al., 1996). Furthermore, the diameter of M1,D, also affects the resolution and contrast characteristics of the images formed by the telescope in ideal conditions (see section 2.5). Conse- quently, improving the performance of the telescopes has called for a constant

1

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increase ofDalong history. Fig.1.1 shows the evolution of the aperture diameter of optical and infrared telescopes throughout the years. Until the beginning of the XXth century, reflecting and refracting telescopes were competing against each other, benefiting from respective technological developments leading to innova- tive designs. For apertures larger than 1m, the reflecting telescopes have proved the most efficient.

1600 1700 1800 1900 2000

Galileo

Aperturediameter[m]

0.1 1 10 100

Newton

VLT, Gemini, Subaru Keck

TMT, JELT E-ELT

OWL?

Year

Monolithic reflecting Refractive

Segmented reflecting Hubble

JWST NTT Active optics

GTC

2100

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

The role of the telescope structure is to maintain the optical performances of the telescope during observations, by preserving the shape and alignment of the elements in the optical train. As those optical elements were built larger and thicker, and consequently heavier, so were their supporting structures. But their sensitivity to the effects of changing gravity and temperature grew accordingly.

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1.2 Modern-days telescopes 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 telescopes

Not only should the next generations of telescopes 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 telescopes 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 telescopes were of the equatorial type (Fig.1.2.a), in which rotations around the polar axis (parallel to the Earth’s ro- tation axis) and around the declination 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 clock mechanisms. However, while elegant in its princi- ple, the structural constraints induced by that configuration revealed unpractical for their implementation in ever-growing telescopes, mostly because of their in- trinsic heaviness [(Bely, 2003), p.234], and cannot correct errors due to external disturbances.

The orientation of telescopes of the altitude-azimuth (alt-az) type is based on a vertical (azimuth) axis and on a horizontal (altitude or elevation) axis. Tracking is intrinsically more difficult 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 equivalent equatorial telescopes (cfr Fig.1.3), implying so significant cost savings that alt-az mounts have become the standard 1 . Finally, as the orientation of the altitude and azimuth axes do not change with respect to the orientation of the gravity field, the implementation of feedforward corrections (based on lookup tables) in active optics is easier and more efficient (Enard et al., 1996).

1Other particular configurations are used in some projects such as the SALT 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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Altitude axis

Azimuth axis

Earth

Declination axis

Polar axis Earth

’srot ation

axis

a) b)

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

The construction of telescopes with primary mirrors of diameters in the range of 3.5m showed the practical limits of passive techniques with respect to the severe optical tolerances 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 telescopes more effi- cient.

The implementation of active optics in modern telescopes has had a considerable 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 telescopes.

In parallel, adaptive optics has permitted the correction of the optical aberrations induced by the continuous local changes in the index of refraction of the atmo-

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1.2 Modern-days telescopes 5

M1 diameter [m]

Totalmass[tons]

VLT Keck

D2.5

0 5 10

102 103

101 Equatorial

Alt-az 1

Figure 1.3: Total mass versus M1 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) cameras and their use in replacement of photographic 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 implementation 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 al., 1987); the implementation has two aspects, as can be seen from Fig. 1.4. The shape of the primary mirror (M1) is controlled by actuators pushing against its back, while the alignment of the secondary mirror (M2) with respect to M1 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 determines the changes in shape and alignment that are responsible of those aberrations, and calculates the signals 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 M1 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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M1 shape actuators M1

M2 Defocus

Science instrument

Wavefront sensor

Controller

Decenter

a) 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 1m Schmidt; (b) ESO 3.6m (passive); (c) ESO 3.5m NTT (raw image); (d) ESO 3.5m NTT (after post-processing) (Wilson, 2003).

telescopes in 19892.

The successful results of the technology developed for the NTT served as the basis for the design of the Unit Telescopes of ESO’s Very Large Telescope (VLT) (Fig.1.6): Four active telescopes with a M1 of 8.2m (completed successively be- tween 1998 and 2001). It is worth noting that the thickness of VLT M1 (0.17m) is actually smaller than that of NTT M1 (0.24m), in order to fully exploit its

2The primary mirror of NTT suffered 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 telescopes 7

a) b)

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 authority3. 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 (Iye et al, 2004) and the Gemini observatory, consisting of two 8.1m telescopes 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 Tele- scopes (AURA, 2010). However, mirrors of those dimensions still require active corrections to be used at their full potential [see (Noethe, 2009) e.g.].

3This fact comes 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 tolerances, while still providing a mass per surface unit much lower than that of equivalent 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 telescopes, 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 telescopes:

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 telescopes: Left, the telescopes inside their enclo- sures; right, front view of the segmented M1 (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 telescopes 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 11×9.8 meters, made up of 91 hexagonal segments and were completed respectively in 1997 and 20054.

1.2.4 Long baseline interferometry

Fig.1.8 shows the basic principles of long baseline interferometry; it consists of two or more independent telescopes separated by a distance called thebaseline, 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 camera. Because of the wave nature of light, instead of an image, the combi- nation produces interference 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 lines. Once phased, the aper- tures composing the interferometer can be seen as elements of a single collecting optical surface (Enard et al., 1996).

B delay

delay line beams combination

baseline D

q

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

The benefit is that the resolution of such an interferometer is proportional to (Bsinθ)−1 instead ofD−1. Consequently, for a given total collecting area, an in-

4Their very specific 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 have a resolution several orders of magnitude higher than that of a single telescope5. 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 telescopes with large apertures.

Interferometric techniques are implemented in modern optical and infrared tele- 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 tele- scopes can be combined: The four 8.2m Unit telescopes have fixed locations while the four 1.8m Auxiliary telescopes 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).

5But 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 telescopes 11

1.2.5 Space telescopes

The atmosphere 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,. . . ). Furthermore, the quality of the wavefront emitted by celestial objects is continuously degraded by turbulence in the successive layers of the atmosphere.

Those reasons led to the launch of space telescopes 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 M1: The installation of optical elements to compensate for that error required the launch of a dedicated space mission three years later6. 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).

6The error was quite similar to that affecting the M1 of the NTT but the active devices in HST could not correct it.

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Following the same trend as ground-based telescopes, space telescopes 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 telescopes 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 telescopes to compensate for most of the optical aberrations induced by atmospheric turbulence. On the other hand, space telescopes can observe wave- lengths unattainable by ground-based ones, their operation does not depend on the weather, they do not suffer from luminous backgrounds,. . .

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1.3 Future Extremely Large Telescopes 13

1.3 Future Extremely Large Telescopes

Monolithic mirrors larger than the current 8m generation 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 tolerances. As a result, 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 Telescopes (ELTs) on which the remainder of this text will focus.

Figure 1.12: Thirty Meter Telescope project (TMT, 2010)

Several projects of future ELTs have been proposed since the end of the XXth century. Three different telescopes were investigated in North America: The Cal- ifornia Extremely Large Telescope (involving many of the persons that worked on the Keck telescopes) (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 ambitious 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 as the spherical M1 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 (Iye 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 implementation of different layers of active control have allowed telescopes 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 M1 of some of the most celebrated telescopes, 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 all be segmented. Note that the size of the earth-based telescopes is one order of magnitude larger than that of space telescopes. 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.

VLT - 8.4 m

JWST - 6.5 m

TMT - 30 m

E-ELT - 42 m

Keck - 11m HST - 2.6 m

Space Telescopes Ground-based Telescopes

Figure 1.14: Primary mirrors of current and future optical and infrared telescopes.

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Keck E-ELT

M1 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

fsegment (+ Whiffle Tree) 25 Hz 60 Hz

f1 (M1) 10 Hz 2.5 Hz

f2 (M2) 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 telescopes increases, they become increasingly sensi- tive to external disturbances such as thermal gradients, gravity and wind, and to internal disturbances from support equipments such as pumps, cryocoolers, fans, etc. These disturbances can deteriorate significantly the image quality. As a result, 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 telescopes. Those conditions, combined to the very low inherent damping of welded steel structures, increase the risk of control-structure interaction. Therefore, one can reasonably wonder if the past experience 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 telescopes (Keck, GTC, VLT) to the next generation of 30m to 40m ELTs, and future, even larger ones. It studies how the various factors affecting the structural response and the control-structure interaction are influenced by the size of the telescope.

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

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

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 tele- 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 representative models based on the approach de- scribed in the first part of chapter 4.

1.6 References

Alvarez, P. and Rodriguez-Espinosa, J. M. The GTC project: in the midst of integration. In Oschmann, J. M., editor,Ground-based Telescopes - 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 Conference on Advanced Technology Optical Telescopes - 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 Telescopes. 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 Telescopes - SPIE 4840, pages 527–532, 2003.

California Institute 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., Mar´echal, A., and Espiard, J. Progress in ground-based optical tele- scopes. Reports on Progress in Physics, 59:601–656, 1996.

ESO. European Southern Observatory website, 2010. URLhttp://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 Telescopes II - SPIE 7012, 2008.

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

Iye 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., Shectman, 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 Telescopes - 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. Euro50 - 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 telescopes project.

In M., S. L., editor, Advanced Technology Optical Telescopes 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.jwst.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 Telescopes - SPIE 4840, 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 Telescopes - SPIE 4840, pages 116–128, 2003.

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

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

University of Arizona. Lare Binocular Telescope Observatory website, 2010. URL http://medusa.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., Franza, F., and Noethe, L. Active Optics I. A system for optimizing the optical quality and reducing the costs of large telescopes.Journal of Modern Optics, 34(4):485–509, 1987.

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Chapter 2

Basics of Telescope Optics

2.1 Introduction

M1

M2

... 8

focal

surface

object wavefront

Figure 2.1: Principles of imaging with a telescope.

A telescope is an instrument designed 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 infinite can be considered as plane at the level of the telescope (Fig.2.1). The role of the tele- scope aperture, namely its primary mirror (M1), 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 elements composing the optical path of the telescope, including M1. 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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a) b) c)

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 deviations 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 Definition

a) image plane b) image plane

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 geometric 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, deviations

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2.2 Aberrations 23

from the initial design in the shape or in the position of the optical elements, 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 deviations are calledaberrations.

Accordingly, the images generated by aberrated wavefronts result from a super- position of light spots of finite size rather than points. It creates a blur in the image, the amplitude and shape of which is roughly determined by those of the aberrations present. Historically, a classification of five primary aberrations has been established, namely spherical aberration, coma, astigmatism, field curva- ture and distortion1. They were classified according to analytical developments made by Seidel and they correspond to the optical signatures of some of the most typical deviations from the initial design. Combined with the derivation of their analytical expressions, they were the basis of the measurements of optical quality since the XIXth 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 W(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)2. 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,λ. Conventionally, a system is consid- ered asnearly perfect if the RMS wavefront error of the output beam is less than λ/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 representation uses the Zernike polynomials, in the form

W(r, θ) = Xn

i=1

aiZi(r, θ), (2.1)

where W(r, θ) and Zi(r, θ) are respectively the wavefront and the ith Zernike

1By nature, reflecting telescopes are not affected by chromatic aberrations, the reader can refer e.g. to [(Walker, 1998), p.138] for more information.

2Other 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 ai results from the projection of W on Zi, and may be computed either by direct integration over the unit circle or by least square fitting. The Zernike polynomials have elegant analytical expressions, the formulation of which can be automated easily [see e.g.

(Malacara, 1992), p.464]3. They are given up to number 11 in Table 2.1 and depicted in Fig.2.4.

Polynomial Denomination

1 Piston

4rcosθ Tilt

4rsinθ Tilt

2r2

Defocus

r2sin 2θ¢

Astigmatism

r2cos 2θ¢

Astigmatism

3r32r¢

sinθ Coma

3r32r¢

cosθ Coma

8r3sin 3θ Trifoil

8r3cos 3θ Trifoil

6r46r2+ 1¢

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 all over the wavefront; the latter only change the location of the focus on the image surface. Accordingly, none of those terms have 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 theprinciple of St Venant in (Wilson et al., 1987)]. In general, most of the wavefront errors due to misalignment, mechanical and thermal distortions

3Care 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

Tilt

Astigmatism Astigmatism

Trefoil Coma Trefoil

Tetrafoil Tetrafoil

Radialorder

Piston

Defocus

Spherical Aberration Azimuthal Order

0

Tilt

Coma

1 2 3 4

-1 -2

-3 -4

0

1

2

3

4

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

and misfiguring 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 have a zero 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 term4 [(Schroeder, 2000), p.264]. With the normalization used in Table 2.1, the total RMS error of the wavefront is simply

4The weights depend on the normalization.

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RM S = vu utXn

i=4

a2i . (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 have 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 telescopes

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

2.3.1 Newtonian telescopes

F1

F2

M1 M1

a) b)

Figure 2.5: Newtonian telescope: (a) Configuration giving access to the prime focusF1 - (b) A folding mirror gives an easier access to focusF2.

According to the fundamental property of conics, the simplest telescope could be built with a single parabolic reflector, 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 asNewtonian 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 angleθ, and the f /# of the overall telescope. Those relations set severe constraints on the design and use of such a telescope, and limit its implementation to telescopes with rather

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2.3 Common optical configurations of optical telescopes 27

small diameters of M1, largef /# and small FOV. The most limiting aberration in this case is coma. Finally, the focus of such telescopes is difficult to access, and would complicate the design of the supporting structure for large mirrors.

Spherical (f /#)−3 Coma θ(f /#)−2 Astigmatism θ2(f /#)−1

Table 2.2: Scaling laws of primary aberrations affecting a Newtonian telescope [(Bely, 2003), p.111].

2.3.2 Two-mirror telescopes

M1 M2

M1 M2

a) b)

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 M1. There are two important classes of two-mirror telescopes differing in the shape of the secondary mirror: The Cassegrain uses a convex hyperboloid and the Gregorian a concave ellipsoid (Fig.2.6).

However, the relations of Table 2.2 still apply to the Cassegrain and Gregorian telescopes because of their paraboloidal M1. The dominant off-axis aberration is still coma. The difference lies in the overallf /# 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 considerations. 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 mirrors5.

This led to design variations of the classical Gregorian and Cassegrain. In those variations, the paraboloid constituting M1 is replaced respectively by an ellipsoid and a hyperboloid. It can be shown from the equations [see e.g. (Schroeder, 2000), p.115] that the departure of M1 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 equations show that there is an infinite number of combinations of the conic constants of M1 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 prevalent one. Designs that compensate for both coma and spherical aberrations are called aplanatic and the aplanatic Cassegrain is better known as theRitchey-Chr´etien;

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

2.3.3 Telescopes with 3 or more mirrors

The principles of the generalized Schwarzschild theorem have 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 M1 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 primaries6. Those designs also permit a better integration of beam steering and deformable mirrors, respectively for image motion and wavefront correction.

2.4 Wavefront error due to deviations 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 presents basic relations on the sensitivity of the wavefront with respect to those deviations in the case of a two-mirror telescope (they are equally

5This is a general principle stated in the generalized Schwarzschild theorem: ”For any geom- etry with reasonable separations between the optical elements, it is possible to correctn primary aberrations with n powered elements.” [(Bely, 2003), p.123]. In this context, the term ”pow- ered elements” refers to conics; surfaces of higher degrees could compensate for more than 1 aberration but are difficult to produce and test.

6A mirror is said to befastif it has a smallf /#, cfr appendix A

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2.4 Wavefront error due to deviations from the design 29

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

2.4.1 Shape of optical elements

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 tolerances, 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 difference grows with the amplitude of the input aberrations and when the mirror is faster (smallerf /#).

ai,output wavefront = 2.ai,mirror misfigure (2.3) 2.4.2 Relative position of optical elements

M1 M2

d l

optical axis a

Figure 2.7: Despace d, tiltα and decenter l.

If the mirrors are all made up of surfaces of revolution, the influence of their rel- ative positions is essentially governed by three relative parameters (for each pair of mirrors) that describe the deviations with respect to the initial design; they are defined on Fig.2.7, namely (axial) despace, (lateral) decenter and tilt. Table 2.3 summarizes the scaling laws of the primary aberrations induced when such deviations are present [(Schroeder, 2000), p.132]. A dependence in θ 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 tol and to α. A general conclusion regarding

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

Sp d(f /#)−3 / /

C θd(f /#)−2 l(f /#)−2 α(f /#)−2

A / l(f /#)−1 /

Table 2.3: Scaling laws of primary aberrations affecting a two-mirror telescope under the effect of deviations 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 M1 (smallerf /#) 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 regime of aberrations, the wavefront errors induced by the various elements can be added in terms of their Zernike coefficients without generating significant deviations 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

ai, output =ai, input+ X

shape

ai+ X

alignment

ai+ X

of f−axis

ai . (2.4)

2.4.4 Design trade-offs

The trends in the design of telescopes 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 evolution of the f /# of M1 in time is illustrated in Fig.2.8. A smallf /# of M1 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 deviations from the design 31

Keck I & II

E-ELT, TMT GTC VLT

1950 1970 Year 1990 2010

0 1 2 3 4 5

Gemini, Subaru NTT

Primarymirrorf/#

Figure 2.8: Evolution in time of the f-number of the primary mirrors of optical and infrared telescopes [adapted from (Bely, 2003), p.136].

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 M27, leading to higher eigen frequencies. On the other hand, as shown in section 2.3, a smallerf /# induces tighter tolerances on the alignment of the optical elements, mirror shapes that are more difficult to produce and test and is responsible for higher off-axis aberrations [cfr Table 2.3 and(Strom et al., 2003)].

Telescopes in the range 2-10m largely rely on two-mirror configurations, amongst which the Cassegrain type (mostly in its Ritchey-Chr´etien variation to improve the FOV) is the most common: For a givenf /# , 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 Iye, 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 M1.

7A smaller secondary offers advantages in terms of mass, thermal inertia, optical testing, obscuration of M1 (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 elements with respect to optical quality.

Moreover they, and their supporting structures, represent massive elements, the position and shape of which must be maintained with tight tolerances. 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 have chosen a Ritchey-Chr´etien configuration (TMT Obs. Corp., 2007), building on their successful experience with Keck.

2.5 Diffraction-limited imaging

2.5.1 Definitions

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 elements,. . . Those edges modify the interferences 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 bediffraction-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 have 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 toλ, the wavelength of the light, and to thef /# of the system.

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2.5 Diffraction-limited imaging 33

d1 d2 d1= 2.44lf/#

d2= 4.48lf/#

(EE = 84%) (EE = 91%) image plane

intensity profile

Figure 2.9: Airy disk, diffraction pattern produced by a perfect imaging system with a circular pupil (EE refers to the percentage of encircled energy) [adapted from (Born and Wolf, 1997), p.416 and (Walker, 1998), p.51].

Theresolutionof a telescope,δθ, is the minimum angular separation between two point objects (of the same brightness) to appear as two separate images. Because of diffraction, there is an overlap between the energies associated with each image spot; therefore, if the images are too close to each other, they might be seen as a single spot by the detector. By convention, the limit of resolution is expressed to a first order in terms of the equivalent Airy disk produced by a telescope, and corresponds to

δθ= 1.22λ/D. (2.5)

This corresponds to a situation where the peak of one Airy disk falls in the first dark ring of the other Airy Disk8. Therefore, increasing D improves the resolu- tion (to the limit of a constant level of aberrations RMS).

An other important practical aspect of that definition is thatλ actually defines the order of magnitude of the optical precision required to consider an optical system as perfect. A diffraction-limited image requires that the phase differences over the wavefront be inferior to a fraction of λ. Conventionally, a system is considered as diffraction limited if the RMS wavefront error of the output beam is less than λ/149. Therefore, it is much easier to build large and fast radio- telescopes that operate at significantly longer wavelengths (λ >0.5m).

8This convention is known as the Rayleigh criterion. There are other criteria, amongst which a less conservative definition that simply usesδθ=λ/D.

9Therefore, a mirror is considered as perfect if its RMS surface error is inferior to λ/28.

Practically, the requirements change in function of the spatial frequencies of the defects. The disturbances and means of compensation that we consider in this work are concerned only with low spatial frequencies, i.e. larger than approx. D/10.

Figure

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