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coronagraphs : Application to ASPIICS
Raphaël Rougeot
To cite this version:
Raphaël Rougeot. Modeling of diffraction effects for formation flying solar coronagraphs : Application to ASPIICS. Instrumentation and Methods for Astrophysic [astro-ph.IM]. Université Côte d’Azur, 2020. English. �NNT : 2020COAZ4051�. �tel-03177643�
Calculs de diffraction
pour les coronographes solaires
en vol en formation
Raphaël ROUGEOT
Laboratoire LAGRANGE
Présentée en vue de l’obtention du grade de docteur en Physique d’Université Côte d’Azur
Dirigée par : David Mary, Rémi Flamary Co-encadrée par : Claude Aime
Soutenue le : 08/10/2020
Devant le jury, composé de : Claude Aime, Pr., OCA
Silvano Fineschi, Sen. Ass. Astron., INAF Rémi Flamary, MCF, Ecole Polytechnique Jeremy Kasdin, Pr., Princeton
Maud Langlois, Dr. CNRS, Univ. de Lyon David Mary, Pr., OCA
Agnès Mestreau-Garreau, Ing., ESA Céline Theys, MCF, OCA
Calculs de diffraction
pour les coronographes solaires
en vol en formation
Application au coronographe ASPIICS
Jury:
Rapporteurs
Silvano Fineschi, Sen. Ass. Astron., INAF
Jeremy Kasdin, Pr., Princeton
Examinateurs
Claude Aime, Pr., OCA (Président du jury)
Rémi Flamary, MCF, Ecole Polytechique
Maud Langlois, Dr. CNRS, Univ. de Lyon
David Mary, Pr., OCA
Céline Theys, MCF, OCA
Invités
Agnès Mestreau-Garreau, Ing., ESA
Thierry Viard, Ing., Thalès
Résumé
La future mission PROBA-3 de l’ESA démontrera le vol en formation de satel-lites. Elle emportera à son bord le coronographe solaire géant ASPIICS. Un satellite portera un disque de 1,42m de diamètre afin d’occulter le Soleil. Un second satellite embarquera un coronographe de Lyot et se positionnera dans l’ombre du premier à 150m, avec une précision millimétrique. ASPIICS
ob-servera la région interne de la couronne du Soleil entre 1,1 et 3,0 rayons so-laires, qui reste relativement inexplorée. La brillance de la couronne y est très peu intense, de six à dix ordres de grandeur plus faible que celle du disque solaire. Pour un tel instrument, la diffraction de la lumière du Soleil apparait donc comme un facteur clé. De plus, les contraintes associées au vol en for-mation sont nouvelles et doivent être étudiées.
Cette thèse souhaite répondre à cette problématique par une approche numérique. Dans un premier temps, la diffraction par l’occulteur est calculée par des modèles dimensionnés pour le cas d’étude. L’intensité de l’ombre résulte d’une somme incohérente sur le disque solaire. Ensuite, la propagation de Fresnel de l’onde solaire diffractée dans le coronagraphe est modélisée suiv-ant le formalisme de l’optique de Fourier. La simulation utilise des algo-rithmes FFT avec des tableaux de grandes tailles. Enfin, l’erreur de front d’onde due aux défauts de surface du télescope, et le désalignement et dé-pointage de la paire de satellites sont implémentés. Le résultat final est donné par la distribution spatiale de l’intensité de la diffraction au niveau du dé-tecteur. L’impact de la taille du masque et du stop de Lyot ainsi que des effets de vignettage sont également analysés.
Cette étude démontre que les occulteurs en dent de scie sont meilleurs que le disque simple quant à la profondeur de l’ombre, et presque aussi bons que le disque apodisé. Dans le coronagraphe, l’intensité de la diffraction reste com-parable à la brillance de la couronne solaire à 1,1 rayon solaire, mais elle est fortement réduite à partir de 1,3 rayons solaires. Ceci permet l’observation de la couronne solaire. Tandis que les erreurs de vol en formation ont un impact limité, les effets de diffusion dégradent grandement la performance. Les résultats de cette thèse ont participé au dimensionnement de ASPIICS.
Abstract
The future formation flying PROBA-3 ESA mission will fly the giant solar
coronagraph ASPIICS. One spacecraft will carry an occulter disc of 1.42m
di-ameter in front of the Sun. It will cast its shadow onto the 5cm aperture of a Lyot-style coronagraph on-board a second spacecraft that will be positioned 150m behind with millimeters accuracy. ASPIICS aims to observe the solar corona in the rather unexplored region from 1.1 to 3.0 solar radii, where the coronal brightness is six to ten orders of magnitude lower than the solar disc. For such high-contrast instrument, straylight from sunlight diffraction is a key driver for the performance. Dedicated and accurate modeling of these diffraction effects are thus required. Additionally, the novel concept of for-mation flying brings new constraints to be investigated.
This thesis aims to meet these needs. The method is numerical. First, the diffraction from the occulter is calculated by models designed for the study case. The umbra is computed as an incoherent summation over the solar disc. Second, the Fresnel propagation of the diffracted wave front through the coronagraph is built upon Fourier optics, and uses 2D FFT with large ar-rays. Perturbations are finally added to the model, like roughness scattering from the telescope, or misalignment and off-pointing of the spacecraft flying formation. The end result is the spatial distribution of the diffracted sunlight intensity at detector level. Sizing the Lyot mask and stop and vignetting ef-fects are also analyzed.
Regarding the umbra intensity, the study shows that serrated occulters are better than the simple disc and can almost reach the straylight performance of the apodized disc. At detector level, the brightness of the diffraction at 1.1 solar radius remains similar to the corona. But the coronagraph manages to reduce the straylight to the required level beyond 1.3 solar radius. While the formation flying errors have a limited impact, the scattering significantly increases the diffraction in the outer field-of-view. These results have been used to support the design of ASPIICS.
Foreword
This thesis was initiated from a preliminary theoretical study on diffraction for the externally occulted solar coronagraph ASPIICS on-board the ESA for-mation flying mission PROBA-3.
The activity got funded under a Networking and Partnering Initiative (NPI) contract scheme between the European Space Agency, the Observatory of Côte d’Azur and Thalès Alenia Space in Cannes. It started in September 2017.
The Ph.D. degree is affiliated to the doctorate school of Applied and Funda-mental Sciences of the University of Côte d’Azur, in Nice, France.
I did my research being co-located at the Lagrange Laboratory of the Obser-vatory of Côte d’Azur in France, alongside my co-directors and supervisors, and at the European Space Research and Technology Centre in the Nether-lands.
Concurrently, a position as space system engineer at ESA was offered to me. I chose to accept it, with both excitement and regret. The NPI contract thus formally ended in February 2018. I nonetheless pursued my Ph.D. in parallel to this new position. I kept on working remotely with my supervisors.
Today, I am pleased to present the result of these three years of research throughout this manuscript.
Acknowledgements
Before getting to the heart of the subject, I would like to acknowledge and thank all the people who provided me with their help and contribution, di-rectly or indidi-rectly, in one way or another.
First of all, I would like to thank Claude Aime, who I have been closely work-ing with for four years. This thesis would never be in your hand without him. Thank you Claude for all the teaching, the time, and all the passionate dis-cussions and brainstorming. I am deeply grateful to have worked along with you, and I wish to pursue our research together.
I would like to warmly thank my two co-directors Pr.David Mary and Rémi Flamary, for having supervised me through these three years. More particu-larly, I thank them for their trust, as I was working remotely.
I thank Pr.Jérémy Kasdin and Silvano Fineschi for their review and experitse. I also thank all the members of my thesis jury for their availability.
I thank the Observatory of Côtes d’Azur, in particular the Department Signal Processing and Image of the Lagrange Laboratory, for welcoming me during several months. It was a valuable and enriching period for me.
I also would like to acknowledge the contribution from Thalès Alenia Space, in particular Thierry Viard, and the European Space Agency, which enabled to kick-off this activity and provided financial support. I thank the ESA PROBA-3 team, in particular Damien Galano and Agnès Mestreau-Garreau,
who initiated this research opportunity.
Finally, I thank all my friends and family for encouraging or discouraging me to keep on. Special thank to the Bargelaan 74 team, Till, Gaspar, Anaïs and especially Bruno. Thank you Alice, Javier, Mariya, Sergio. Granolax thanks Olivier.
Contents
Abstract ii
Foreword vii
Acknowledgements ix
List of Figures xv
List of Tables xix
List of abbreviations xxi
List of physical constants and symbols xxiii
1 An introduction to solar coronagraphy 1
1.1 The Sun’s corona . . . 3
1.1.1 What is the solar corona? . . . 3
1.1.2 Why studying the Sun’s corona . . . 6
1.2 The advent of solar coronagraphy . . . 7
1.2.1 The pioneer design of the Lyot coronagraph . . . 8
1.2.2 The development of external occultation . . . 9
1.2.3 Review of past studies . . . 13
1.2.4 Past and future solar coronagraphs . . . 18
1.2.5 Comparison of solar and stellar coronagraphy . . . 19
1.3 Presentation of the thesis . . . 21
1.3.1 Motivation and objectives . . . 21
1.3.2 The ESA PROBA-3 mission . . . 22
1.3.3 ASPIICS solar coronagraph . . . 24
1.3.4 Organisation of the manuscript . . . 26
2 Theory of diffraction 31 2.1 Position of the diffraction problem . . . 32
2.1.2 The scalar theory of light . . . 35
2.1.3 Spherical and planar wave front . . . 37
2.1.4 The Kirchhoff’s integral theorem . . . 38
2.2 The Huygens-Fresnel principle . . . 39
2.2.1 The Kirchhoff boundary conditions . . . 40
2.2.2 The diffraction integral formula . . . 40
2.2.3 Babinet’s principle . . . 43
2.3 The Maggi-Rubinowicz representation . . . 43
2.4 The Fresnel and Fraunhoffer diffraction . . . 45
2.4.1 The Fresnel regime . . . 45
2.4.2 The Fraunhoffer regime . . . 49
2.5 Application to ASPIICS . . . 50
3 Diffraction from an external occulter 53 3.1 The sharp-edged occulter . . . 54
3.1.1 The Hankel transformation . . . 56
3.1.2 The 2D fast Fourier transformation . . . 59
3.1.3 Maggi-Rubinowicz representation . . . 65
3.2 The serrated occulter . . . 69
3.2.1 An old prediction . . . 70
3.2.2 Confrontation of numerical methods . . . 71
3.2.3 A two-level diffraction pattern . . . 75
3.3 The apodized and petalized occulters . . . 81
3.3.1 The apodized occulter . . . 81
3.3.2 The Vanderbei et al. approach . . . 82
3.3.3 Petalized occulters for solar coronagraphy . . . 84
3.3.4 Summary about the diffraction . . . 87
3.4 Umbra and penumbra . . . 87
3.4.1 Convolution with the Sun . . . 88
3.4.2 Umbra from serrated occulters . . . 90
3.4.3 Comparison to the literature . . . 95
4 Wave propagation inside the coronagraph 97 4.1 Presentation of the model . . . 98
4.2 Computation of the light propagation inside the coronagraph 102 4.2.1 Diffracted wave front . . . 102
4.2.2 Integrated solar intensity . . . 108
4.2.3 Numerical considerations . . . 110
4.3.1 Plane B and plane O’ . . . 113
4.3.2 Plane C . . . 119
4.3.3 Plane D . . . 120
4.4 Analysis on the externally occulted Lyot coronagraph . . . 122
4.4.1 Sizing the internal occulter and the Lyot stop . . . 122
4.4.2 Overview on vignetting effects . . . 128
4.5 Conclusion . . . 131
5 An imperfect coronagraph 133 5.1 Formation flying errors . . . 134
5.1.1 Theoretical considerations on the diffraction . . . 135
5.1.2 Asymmetric diffraction from lateral errors . . . 137
5.1.3 Insignificant longitudinal displacement . . . 145
5.2 Surface roughness of the telescope . . . 149
5.2.1 Modeling the roughness height . . . 151
5.2.2 Determining the theoretical PSDS⋆ h . . . 153
5.2.3 Application to ASPIICS . . . 155
5.2.4 A loss of diffraction filtering . . . 161
5.2.5 Sensitivity analysis on the A, B, C parameters . . . 168
5.3 Conclusion . . . 174
6 Conclusion 175 6.1 Performance of the solar coronograph ASPIICS . . . 175
6.1.1 Umbra from the external occulter . . . 175
6.1.2 End-to-end performance of the coronagraph . . . 178
6.2 Future work and perspectives . . . 180
6.2.1 Exploring other wave fronts . . . 180
6.2.2 Occulter’s edge effects . . . 182
6.2.3 Petalized occulters for solar coronagraphy . . . 184
6.2.4 Experimental approach . . . 185
A The solar brightness 187
B Maggi-Rubinowicz with Fresnel parabolic approximation 191
C Two-dimensional adaptive sampling 193
D Two-dimensional intensity in case of formation flying errors 201
F Demonstrations 213
F.1 Chapter 1 . . . 213
F.1.1 Equality of Fort et al. (1978) and Lenskii (1981) formu-las for the single disc . . . 213
F.2 Chapter 2 . . . 215
F.2.1 Demonstration of Eq.(2.21) . . . 215
F.2.2 Demonstration of Eq.(2.39) . . . 215
F.2.3 Demonstration of Eq.(2.51) . . . 217
F.3 Chapter 3 . . . 221
F.3.1 Demonstration of Eq.(3.3) and Eq.(3.24) . . . 221
F.3.2 Demonstration of Eq.(3.11) . . . 222 F.3.3 Demonstration of Eq.(3.17) . . . 224 F.3.4 Demonstration of Eq.(3.18) . . . 225 F.3.5 Demonstration of Eq.(3.20) . . . 225 F.3.6 Demonstration of Eq.(3.31) . . . 226 Bibliography 229
List of Figures
1.1 Coronal brightness . . . 6
1.2 Lyot coronagraph . . . 10
1.3 Penumbra from an occulter . . . 12
1.4 Vignetting effect . . . 13
1.5 Comparison of straylight from the literature . . . 17
1.6 Serrated and petalized occulter . . . 21
1.7 Artistic view of PROBA-3 . . . 24
2.1 Geometry of the diffraction problem . . . 34
3.1 Diffraction from the sharp-edged occulter - Hankel . . . 57
3.2 Arago spot . . . 58
3.3 Transition from shadow to light . . . 58
3.4 Deviation of sampling for the fast Fourier transformation . . . 63
3.5 Diffraction from the sharp-edged occulter - 2D FFT . . . 64
3.6 Diffraction from the sharp-edged occulter - Rubinowicz . . . . 68
3.7 Radius of Boivin . . . 71
3.8 2D FFT vs. Rubinowicz . . . 75
3.9 Two-dimensional diffraction . . . 77
3.10 Diffraction from the 30mm 64-teeth occulter . . . 78
3.11 Diffraction from serrated occulters . . . 79
3.12 Diffraction along peak and trough . . . 80
3.13 Diffraction along peak and trough - Zoom . . . 80
3.14 Diffraction from the apodized occulter . . . 82
3.15 Shape of petals . . . 85
3.16 Diffraction from petalized occulters . . . 86
3.17 Summary of diffraction intensity . . . 88
3.18 Penumbra for a fixed number of teeth . . . 92
3.19 Penumbra for a fixed size of tooth . . . 93
3.20 Integrated umbra . . . 94
4.2 Two-dimensional diffraction patterns . . . 114
4.3 Two-dimensional diffraction . . . 115
4.4 Intensity in plane B . . . 116
4.5 Intensity in plane O’ . . . 117
4.6 Ratio of straylight rejection . . . 119
4.7 Intensity in plane C . . . 120
4.8 Intensity in plane D . . . 122
4.9 Sizing the internal occulter . . . 124
4.10 Sizing the Lyot stop . . . 125
4.11 Performance map at 1.2R⊙ . . . 126
4.12 Performance map at 2.5R⊙ . . . 127
4.13 Vignetting effects . . . 130
4.14 Two-dimensional images . . . 132
5.1 Intensity in plane O’ in case of off-pointing . . . 140
5.2 Intensity in plane O’ in case of lateral misalignment . . . 141
5.3 Intensity in plane D in case of off-pointing . . . 143
5.4 Intensity in plane D in case of lateral misalignment . . . 144
5.5 Diffraction in case of longitudinal displacement . . . 147
5.6 Intensity in plane D in case of longitudinal displacement . . . 148
5.7 BTDF curves . . . 156
5.8 PSD of the roughness . . . 159
5.9 Roughness in plane A . . . 160
5.10 Intensity in plane O’ . . . 162
5.11 Relative difference in plane O’ . . . 162
5.12 Intensity in plane C . . . 163
5.13 Relative difference in plane C . . . 163
5.14 Intensity in plane D . . . 164
5.15 Relative difference in plane D . . . 164
5.16 Sizing the internal occulter with a WFE . . . 166
5.17 Sizing the Lyot stop with a WFE . . . 167
5.18 Cases A1 to A4 . . . 170
5.19 Cases B1 to B4 . . . 171
5.20 Cases C1 to C4 . . . 172
5.21 Cases Σ1 and Σ2 . . . 173
6.1 Comparison of straylight for ASPIICS. . . 177
6.2 End-to-end performance of the coronagraph . . . 179
6.4 Intensity in plane O’ for the 64-teeth occulter . . . 182
6.5 Effects of occulter roughness . . . 183
A.1 Limb-darkening . . . 189
B.1 Rubinowicz with Fresnel approximation . . . 192
C.1 Cumulative distribution function . . . 195
C.2 Radial adaptive sampling . . . 197
C.3 Two-dimensional adaptive sampling . . . 200
D.1 Two-dimensional intensity in case of off-pointing . . . 202
D.2 Two-dimensional intensity in case of lateral misalignment . . 203
E.1 Optical Aberrations for the on-axis point . . . 208
E.2 Optical Aberrations for the middle point . . . 209
E.3 Optical Aberrations for the edge point . . . 210
E.4 Leaking diffraction intensity for the on-axis point . . . 211
E.5 Leaking diffraction intensity for the middle point . . . 211
E.6 Leaking diffraction intensity for the edge point . . . 212
E.7 Optical Aberrations for the reference pattern . . . 212
List of Tables
1.1 Physical characteristics of the solar atmosphere . . . 3
1.2 Solar and stellar coronagraphy . . . 20
1.3 Proba-3 orbit . . . 24
1.4 ASPIICS characteristics . . . 27
1.5 Parameters of the manuscript . . . 29
2.1 Hypothesis about ASPIICS . . . 51
3.1 Comparison of sampling σoptand size N . . . 61
3.2 Comparison of sampling σs . . . 73
3.3 Comparison to the literature . . . 96
4.1 Coronagraphic systems . . . 100
4.2 Wave front computation . . . 107
4.3 Comparison of sampling σAand size N . . . 111
5.1 Formation flying errors . . . 138
5.2 Study cases for PSD parameters . . . 169
List of abbreviations
AAE Absolute Angular ErrorASPIICS Association de Satellites Pour l’Imagerie
et l’Interférométrie de la Couronne Solaire (French)
Association of Spacecraft for Polarimetric and Interferometric Investigation of the Corona of the Sun (English)
DARA Davos Absolute RAdiometer
BTDF Bidirectional Transmission Distribution Function CDF Cumulative Distribution Function
CME Coronal Mass Ejections CSC Coronagraph Spacecraft ESA European Space Agency FFT Fast Fourier Transformation FWHM Full Width Half Maximum
GNC Guidance, Navigation and Control ISD Inter Satellite Distance
LASCO Large Angle and Spectrometric COronagraph
METIS Multi Element Telescope for Imaging and Spectroscopy MSB Mean Solar Brightness
OPSE Occulter Position Sensor OSC Occulter Spacecraft
PROBA PRoject for On-Board Autonomy PSD Power Spectral Density
RAAN Right Ascension of the Ascending Node RDE Relative Displacement Error
RMS Root Mean Square
SECCHI Sun-Earth Connection Coronal and Heliospheric Investigation SOHO SOlar and Heliospheric Observatory
SPS Shadow Position Sensor WFE Wave Front Error
List of physical constants and
symbols
i Imaginary number√−1 π Number pi ∗ Convolution product −→ ∇ Differential operator ∆ Laplacian operator F [·] Fourier transformationJ0(·) Bessel function of the first kind Γ(·) Gamma function
δ(·) Dirac function k · k Euclidian norm
h·i Expectation
R⊙ Angular radius of the Sun
B⊙ limb-darkening function of the Sun
R⊙ Projected radius of the Sun B⊙ Stenope image of the Sun
BK,BF Brightness functions of the K- and F-corona
x, y Transverse Cartesian coordinates z Longitudinal Cartesian coordinate
α, β Cartesian angular coordinates on the sky r⊙, θ⊙ Polar coordinates on the solar disc
t Time
u, v Frequencies in the Fourier domain fx, fy Spatial frequencies of plane A
γx, γy Scatter angular directions
Ω Plane of the diffracting object
∂Ω Edge of the diffracting object dz Distance to an edge point
−→
E ,−→B Electric and magnetic fields
c Speed of Light
ε0 Dielectric permittivity of vacuum
µ0 Magnetic permeability of vacuum
̺0 electric charge density
−→
j0 electric current density
λ Wavelength
k Wave number
Ψ Scalar complex amplitude of the wave front I Sunlight intensity
U Integrated intensity
K Obliquity factor
ϕz Fresnel quadratic phase factor
NF Fresnel number
RO Radius of the external occulter
z0 Distance between the occulter and the pupil
z1 Distance between the pupil and the internal occulter
f1 Focal length of the telescope
f2 Focal length of the secondary objective
f3 Focal length of the third objective
RP Radius of the entrance aperture
RM Radius of the internal mask or Lyot mask RL Radius of the Lyot stop
HM Size of the inner hole of the internal occulter O Transmission function of the occulter
P Transmission function of the pupil
M Transmission function of the internal mask L Transmission function of the Lyot stop
Nt Number of teeth of a serrated occulter
∆t Size of a toot of a serrated occulter RB Radius of Boivin
Λ Half-apex angle of the teeth
ω1
2 Half of the angular separation
N Size of the arrays for the 2D FFT method
σ Spatial sampling for the 2D FFT method L Spatial extent of the arrays
σopt Optimal sampling for the 2D FFT method
Ns Number of sampling points for the boundary wave integral
σs Spatial sampling for the boundary wave integral
NP Number of points padding the entrance aperture
σA Numerical sampling in plane A
GK Normalization factor for the integrated intensity
Nr⊙ Number of point sources sampling the solar radius
Nθ⊙ Number of step for circular integration C Cumulative distribution function
ρ−, ρ+ Limits of the vignetted zone I Final image on the detector
δα, δβ Inertial pointing error
δx, δy Lateral displacement
δz0 Longitudinal displacement
W Wave front error in plane A h Algebraic height of the roughness huc Uncorrelated white noise
g Spatial correlation function of the roughness G Transfer function for the micro-structures Sh Power spectral density of the micro-structures
S⋆
h Theoretical model for the Power spectral density
nr Refractive index
A, B, C Parameters of the ABC-function
Chapter 1
An introduction to solar
coronagraphy
The first astronomical observation of the Sun took place at the end of the 16th century by Scheiner (1612) and Galilée (1613), revealing the dark sunspots on its surface. Centuries later, on the 10th of February 2020, the Solar Orbiter mission of the European Space Agency (ESA) is launched, carrying on-board ten scientific instruments to perform in-situ measurements and remote obser-vation of the Sun and its surroundings.
This space mission emphasizes the importance and the needs of further in-vestigating the Sun. Nowadays, our star still raises a strong scientific interest and keeps its batch of unsolved questions and mysteries. The solar corona is one of them.
The solar corona is the atmosphere of the Sun which expands millions of kilometers from its edge. But observing the Sun’s surroundings is far from being an easy task, because the nearby sunlight is too bright. Only a total solar eclipse could reveal the faint corona to a regular telescope.
In the 1930s, Bernard Lyot managed to overcome this challenge thanks to a clever invention: the Lyot coronagraph (Lyot, 1939). In particular, the French astronomer well addressed the problem of diffraction, which inevitably oc-curs and produces troublesome straylight. He managed to filter out the bright sunlight and diffraction, while letting the faint coronal light propa-gate up to the detector. It was the first time that an astronomer observed the solar corona without the help of a solar eclipse. A new instrumental science
was born: the coronagraphy.
In the following decades, the solar coronagraphy technique has further de-veloped. The goal was to observe the corona even closer to the solar edge, with even better accuracy. In this respect, straylight remained the main chal-lenge to face. More advanced coronagraphs were then imagined, improv-ing the initial concept from Lyot. Scientists and instrumentalists explored both experimental set-ups and theoretical studies in order to better estimate diffraction, and both approaches brought their respective advantages and limitations. Thanks to this built-up expertise, the solar coronagraph LASCO
-C21 on-board the SOHO2 mission (Brueckner and et al., 1995) proved to be an actual success.
But the future of solar coronagraphy is already calling for performance im-provement, targeting unexplored regions of the solar corona. The next gener-ation of coronagraphs will most likely rely on the new concept of formgener-ation flying space mission. As a pioneer, the 150m-long solar coronagraph ASPI -ICS3 (Galano et al., 2018) will fly on the ESA PROBA-3 mission (Galano et
al., 2019). This novel instrument will take advantage of such large geometry to achieve unprecedented observation of the corona of the Sun.
Today, our understanding about diffraction is much more advanced com-pared to Lyot’s time - especially in the domain of stellar coronagraphy. But that straylight problem is also brought to another level. Finer and better modeling is thus now required. My thesis falls within this framework.
This introduction chapter recalls the background.
1LASCO-C2 stands for Large Angle and Spectrometric Coronagraph. 2SOHO stands for Solar and Heliospheric Observatory.
3ASPIICS stands for Association of spacecraft for polarimetric and Interferometric and Investigation of the Corona of the Sun, or Association de Satellites Pour l’Imagerie and l’Interférométrie de la Couronne Solaire, in French.
1.1
The Sun’s corona
1.1.1
What is the solar corona?
The outer shell of the Sun is constituted by the photosphere and the chromo-sphere. From there, the solar atmosphere extents upward from a few thou-sands up to several millions of kilometers: it thus sizes more than ten times the Sun. It is a fully ionized and magnetized plasma, in a dynamic state (Koutchmy, 1988). This defines what is called the solar corona.
Table 1.1 provides a quantitative idea of the physical properties of the so-lar corona and the photosphere - the values are extracted from Aschwan-den (2005) and Eddy and Ise (1979). As can be seen, the corona greatly dif-fers from the Sun. The density is about twelve orders of magnitude lower than the photosphere itself, whereas the temperature reaches a magnitude of 106K. In comparison, the temperature of the Sun is only about 6000K. These two features, a very low density and a very high temperature, seem quite contradictory. Regarding the magnetic field, it is bigger than the Earth’s one by four orders of magnitude in the inner corona.
TABLE 1.1: Physical parameters describing the solar
atmo-sphere, extracted from Aschwanden (2005) and Eddy and Ise (1979).
Parameters Photosphere Inner corona Outer corona
Density (g.cm−3) 10−6 10−17 10−18
Temperature (K) 6.103 3.106 1.106
Pressure (P) 1.4.105 0.9 0.02
Magnetic field (T) 5.10−2 1.10−1 1.10−5 Electron density (cm−3) 2.1017 1.109 1.107
The structure of the solar corona is governed by the Sun’s dynamics. More particularly, the magnetic field plays a key role in shaping the coronal plasma. Three zones are usually defined:
The active regions which are local areas of concentrated magnetic fields. These regions are noticeable by the dark sunspots lying on the solar surface.
The quiet Sun which covers almost all the surface of the Sun, and com-monly defined as opposed to active regions. It corresponds to areas with closed magnetic field, large-scale coronal loops and arches.
The coronal holes which are dominated by open magnetic field lines. The plasma gets heated by the solar surface and is transported into the solar wind.
Several physical phenomena take place in the corona, such as magnetic field reconnections, spicules, which are direct injections of solar matter in the corona, and solar flares, giant bursts of X-rays travelling outward the Sun - to name a few. The most spectacular and interesting phenomena is the Coronal Mass Ejection (CME), an enormous loop of particle clouds ejected outward in the solar system.
These various physical processes shape the solar corona and consequently its light spectrum. This latter ranges over fourteen orders of magnitude, from radio to X-rays (Aschwanden, 2005). Every spectral band can be linked to some specific physical phenomena. For what concerns solar coronag-raphy, the visible part of the spectrum is of interest, usually in the range [500nm, 700nm]. It is sometimes called white light.
The coronal light in the visible, and by extension the corona, is usually split into three components: the K-, the F- and the E-corona.
The K-corona refers to the continuous component of the spectrum - the K stands for kontinuierliche, i.e. continuous in English. It is made of di-rect sunlight that is scattered by the free electrons of the plasma, by Thomson’s scattering process. This light is fairly polarized due to some anisotropy of the scattering.
The F-corona corresponds to the part of the spectrum containing Fraunhof-fer’s lines. It results from the scattering of sunlight by interplanetary dust particles in the surroundings of the Sun.
The E-corona is constituted by the spectral emission lines from the ions. These lines, like green FeXIV or orange FeX, provide some evidence of the plasma’s chemical composition.
As opposed to the temperature, the brightness of the solar corona in white light is much lower than the one of the Sun. It is also not uniform: there is an asymmetry between the solar pole line and the equatorial plane. Variations exist along the solar 11-years cycle: the K-corona appears less bright during a solar minimum than a solar maximum.
Cox (2000) provides photometric data of the K- and F-corona in his Table 14-19. I fitted these data points to model the coronal brightness, and I derived the following functions:
BK(r⊙) =109.609×exp(−0.809×r⊙)−10 (1.1)
for the K-corona at solar maximum, and BF(r⊙) =103.285×r
−0.717
⊙ −10 (1.2)
for the F-corona, where r⊙ ≥ 1 is the altitude above the Sun’s surface, ex-pressed in unit of solar radius. These functions are given in unit of Mean So-lar Brightness (MSB). In other words, they are relative with respect to the av-eraged radiance over the solar disc4. Regarding the E-corona, no data points of the same type has been found in the literature.
Figure 1.1 plots both functions BK(r⊙) and BF(r⊙), together with the fitted
data points. Other brightness curves can be found in Cox (2000). The coro-nal brightness ranges from 10−6(in MSB) close to the solar surface, down to 10−10−10−9 (in MSB) at an altitude of 5 solar radii. This undeniably shows the extreme faintness of the solar corona, which makes it be very hard to observe, also because of its close proximity from the Sun. In daylight, one cannot suspect its existence.
4The absolute spectral radianceB(λ, T), or spectral brightness, is commonly expressed in Watt/m2/sr/nm. It can be computed with Planck’s black-body formula:
Bλ(λ, T) = 2hc 2 λ5 × 1 exp hc λkBT −1
where λ is the wavelength, h the Planck’s constant, c the light speed, and kB Boltzmann’s constant. This formula depends on the temperature T of the celestial body. For the Sun, the surface temperature is about 5750K.
FIGURE1.1: Brightness of the solar corona in unit of MSB, as
a function of the altitude r⊙ in unit of solar radii. The red curve plots the brightness BK(⊙)of the K-corona, from Eq.(1.1).
The blue curve plots the brightness BF(r⊙) of the F-corona,
from Eq.(1.2). The dots represent the fitted data points from Cox (2000).
1.1.2
Why studying the Sun’s corona
As already mentioned, the solar corona and its properties are still not well understood by the solar physicists. It motivates the current research that re-quires new scientific observation.
The fact that the corona has such a low density and a high temperature with respect to the Sun is purely counter-intuitive. The heating process which would take place are unexplained. One hypothesis is that the phenomena of magnetic reconnection would release a lot of energy in the form of heat when it occurs (Zhukov et al., 2000).
interest. These structures would play a role in the solar wind acceleration, as well in the heating process of the corona (Zhukov et al., 2000). The cur-rent knowledge about the dynamics of the corona mostly relies on theoretical magneto-hydrodynamic modeling of the plasma. But these models need to be better corroborated and ascertained.
Finally, the solar corona consists of a key element for the understanding of the space weather in our solar system. Beyond the mere scientific interest, it has a direct impact on human activities, especially about safety. Indeed, the Sun’s surrounding is where the solar wind acceleration and the CMEs take place. These high speed clouds of protons reach the Earth and strongly inter-act with its magnetosphere, resulting in potentially harmful effects. For ex-ample, they can provoke a steep increase of the protons and electrons density in the Earth’s radiation belts. Satellites in low Earth orbit may consequently be endangered, as well as the astronauts on-board the International Space Station. These phenomena can also interfere with radio telecommunication, and even on-ground electric systems.
The study of the solar corona may be summarized but not reduced to the following questions. What processes contribute to the heating of the corona? What is the nature of the plasma’s structures at different scales? What pro-cesses drive the solar wind acceleration? What is the exact nature of the struc-ture and the dynamics of CMEs?
Today, new observation data with finer spatial and temporal resolution are required to address these questions. The exploration of the K-corona is here of crucial importance, because it is directly related to these phenomena. Solar coronagraphy appears thus necessary.
1.2
The advent of solar coronagraphy
Observing the solar corona in white light has always been a tough challenge for astronomers and scientists, because of its close proximity to the Sun and its relatively faint brightness, as shown in Figure 1.1. Any classical telescope would immediately get blinded by the direct sunlight. To overcome that
problem, one can take advantage of solar eclipses by the Moon. It is in fact a great chance that the natural apparent diameter of the Moon,
3474km
384400km =9.04.10
−3rad=31.1arcmin (1.3)
is similar to the Sun’s one R⊙ = 1391400km
149597870km =9.30.10
−3rad=32.0arcmin (1.4)
When the Moon occults the Sun, it casts its shadow on the Earth. The dark condition during a total eclipse enables to reveal the dim coronal light. Un-fortunately, such celestial event is not frequent and is very short - a few minutes. Observations of the solar corona cannot solely rely on that natu-ral event.
Solar coronagraphy is the answer to that problem. This technique aroused in the 1930s from the genial idea of Bernard Lyot. This French astronomer in-vented an astronomical instrument that recreates the eclipse’s dark condition on the detector. It is thus able to image the surrounding of the Sun without getting blinded. The Lyot coronagraph was born, taking its inventor’s name.
1.2.1
The pioneer design of the Lyot coronagraph
The concept of the coronagraph from Lyot (1939) is depicted in Figure 1.2, extracted from D’Azambuja (1952). When the instrument points to the Sun, the Sun’s image is formed on the focal plane of the telescope. The scientist set at this particular location a small tilted mirror, named the Lyot mask, whose size is slightly bigger than the Sun’s image. The sunlight is then reflected outside the instrument by this mirror. On the other hand, the light from the solar corona incomes at bigger angles of incidence. It is focused on the focal plane as well, but next to the Lyot mask, so that it can propagate further up to the detector.
However, Bernard Lyot observed a residual bright halo of sunlight, despite the presence of the mask. That straylight was very low with respect to the direct sunlight, but bright enough to perturb the observation of the solar
corona. The scientist understood that it was diffraction5 - or instrumentally scattered light in his own words. His stroke of genius was to intuit how diffraction would behave inside the instrument. The fact is that the entrance aperture diffracts, or scatters the incoming sunlight, and acts as a virtual light source. That diffracted light focuses at the location where the aperture’s im-age is formed by the optics. Bernard Lyot thus put a diaphragm there, named the Lyot stop, in order to cut-out that straylight, while letting the coronal light pass through.
The Lyot mask constitutes the first key element of the solar coronagraph. It can be either reflective, like in the original design of Lyot, or absorptive. Its radius defines the inner field-of-view of the imaging system, i.e. how much close to the Sun’s edge the corona can be observed. The Lyot stop forms the second key element. The smaller the size of this diaphragm is, the more straylight it cuts out, even through at the cost of the degradation of the an-gular resolution and the overall throughput.
Finally, Bernard Lyot put a small opaque screen, named the Lyot spot, in order to block the ghost images of the Sun. These ghosts are generated by multiple reflection in the first lens of the coronagraph.
The Lyot coronagraph proved to be a real success by taking advantage of the combined effects of the Lyot mask and stop. It managed to reveal the faint corona in the absence of a solar eclipse for the first time. Coronagraphy and the concept of high-contrast instrumentation was born and consisted of an actual breakthrough for astronomy.
1.2.2
The development of external occultation
After Lyot’s novelty, Evans (1948) introduced another technique to observe the Sun’s surroundings: the external occultation. For his sky photometer, the astronomer created an artificial eclipse by placing a disc in front of the Sun.
5Diffraction intensity is usually very low compared to the direct light from the source. It may thus be neglected in first approximation. However, in coronagraphy, the object under study, i.e. the corona, is by six orders of magnitude fainter than the bright source, i.e. the Sun. Diffraction must definitely be accounted for.
FIGURE1.2: Scheme of the original concept of the Lyot corona-graph, extracted from D’Azambuja (1952) (his Figure 113). The entrance aperture and the telescope are located in A. The Lyot mask is in B, where the image of the Sun is formed. The second objective is in C. It and makes an image of the aperture in D, where the Lyot stop is set. The Lyot spot is the small central spot in D. The final objective and detector are in E.
This disc directly casts a shadow on the aperture of the optical instrument, like the Moon during a solar eclipse.
Such an occulter disc is said external, because it is located outside the imag-ing instrument. Its angular size, as viewed from the entrance aperture, or oc-cultation angle, must be large enough to fully mask the solar disc, but small enough not to hinder too much the corona. Because of the Sun’s angular ex-tent, the shadow produced by the occulter is split into regions: the umbra and the penumbra. The umbra is the dark projection of the disc6. Geometri-cally, no sunlight reaches this area: it is perfectly dark and the instrument’s aperture shall be located there. The penumbra starts after the umbra7, where the sunlight intensity progressively increases until full Sun illumination. Fig-ure 1.3 illustrates the geometry.
Like for the Lyot coronagraph, diffraction effects inevitably occur. The sim-ple geometrical point of view depicted in Figure 1.3 is not adequate to the
6The radius of the umbra is given by R
O−z0×tan R⊙, where R⊙is the angular radius of the Sun, ROthe radius of the occulter and z0the distance between the instrument and the occulter.
7The penumbra extends from R
O−z0×tan R⊙ up to RO+z0×tan R⊙, with the same notations as above.
exact physical situation. The external occulter actually diffracts and scatters the direct sunlight, such that the umbra region is not perfectly dark in reality. That straylight irredeemably enters into the instrument aperture and hinders the solar corona.
Solar physicists and instrumentalists had understood this issue very early. In the following decades, their goal was thus to find a way to reduce as much as possible the level of diffraction. The main idea consisted of playing with the occulter’s shape in order to directly act on the diffraction. Complex and advanced occulters were thought of, leading to many theoretical or exper-imental studies and publications. I will review that literature in the next section.
First, Purcell and Koomen (1962) imagined the use of a saw-toothed disc, also called serrated occulter. The occulter edge is made of hundreds of teeth, shaped as mere triangles, and very small compared to the size the disc. These authors intuited that such a shape would provide a much deeper umbra than the simple disc. Their reasoning was based on the following geometrical considerations. Diffraction supposedly occurs perpendicularly to the edge. According to that point of view, a simple disc would diffract the light to-wards the centre, i.e. the umbra region. Conversely, the serrated mask would reject the sunlight outside - as illustrated by Figure 7 of Koutchmy (1988). Boivin (1978) developed these geometrical considerations and calculated the dark inner region that would be produced by the occulter.
A second idea was proposed by Newkirk and Bohlin (1963): the multiple disc. In that case, the occulter is made of a series of successive discs co-aligned and close to each other. Their successive diameters decrease, such that the cone angle they form is about the angular size of the Sun, c.a. 16arcmin. Double disc, triple disc and even multi-disc systems, with several tens of lay-ers, were then investigated and built in the 1970s and 1980s.
More recently, the concept of apodized external occulter was proposed by Aime (2013), based on the apodization technique developed by Jacquinot and Roizen-Dossier (1964). The principle of apodization is that the occulter has a smoothly varying optical transmission along its radius, meaning that it is not
fully opaque. It manages to drastically reduce the sunlight diffraction. Nev-ertheless, these occulters remain purely theoretical, in the sense that they cannot actually be manufactured with the required accuracy. Aime (2007) analyzed as well the use of apodized apertures for solar coronagraphy, fol-lowing the same concept.
Occulter Occultation angle
Aperture
Toward the Sun
FIGURE1.3: Illustration of the geometrical umbra and penum-bra from the external occulter. The occultation angle denotes the angular size of the external occulter as viewed from the aperture. The yellow beam represents the incoming sunlight. The dark grey is the umbra cone casted by the external occulter. The light grey shows the penumbra region.
There is a specific issue encountered with external occultation that must be pointed out: the vignetting. As the occulter is located in front the entrance aperture, it delimits the inner field-of-view of the imaging system. It may be seen as a dark spot in the centre of the detector. When observing a small area of the corona close to the occulter’s edge, the view of the instrument is partially obstructed. Only a small fraction of the entrance aperture gets illu-minated by the incoming light from that region. The effective aperture looks like a small crescent instead of a disc, as illustrated by the top drawing of Fig-ure 1.4. The direct consequence is that the overall throughput is diminished and the resolution is degraded. In addition to the straylight, this vignetting effect also contributes to the difficulty to image the inner solar corona close to the Sun’s edge.
Occulter Occulter Aperture Occultation angle Occultation angle Aperture
FIGURE1.4: Illustration of the vignetting effect by an external
occulter. The yellow pencil comes from a small region of the solar corona, e.g. a point source. In the top figure, an occul-ter of small size partially masks that light from the corona, and the entrance aperture looks like a crescent. In the bottom panel, the distance between the external occulter and the aperture is enlarged. The vignetting is consequently diminished. The oc-cultation angle of the external occulter is the same for both con-figurations.
1.2.3
Review of past studies
The investigation of the diffraction in solar coronagraphy has led to numer-ous publications. Reviewing that literature allows to know better what meth-ods were used in the past and what results were inferred, but also what is missing and what remains unaddressed.
Fort et al. (1978) did one of the first study by analytically calculating the um-bra intensity behind a single disc. They derived a residual sunlight of 10−4,
in unit of Mean Solar Brightness8 (MSB), for a disc of 15mm diameter at a distance of 450mm - the geometry used by Dollfus et al. (1968). I however notify that the analytical diffraction formula they used is not justified nor demonstrated, and may be discussed. I will come back on their expression in Chapter 3.
Five years later, Lenskii (1981) performed a mathematical analysis of the diffraction from several types of external occulters. Lenskii’s calculations relied on the Maggi-Rubinowicz diffraction theory9, also called diffraction wave boundary. The Russian scientist derived analytical expressions of the sunlight intensity in the umbra for the single disc, the double disc, the toothed disc and the scanning diaphragm - another system proposed by Koutchmy in 1977. About the single disc, I verified that the two expressions from Fort et al. (1978) (their Eq.(3)) and Lenskii (1981) (his Eq.(6)) are indeed identical -see demonstration in Annex F.1. Note that Koutchmy (1988) reported a dis-crepancy between these two expressions, that is actually erroneous. Lenskii reported an umbra intensity of the order of 10−4−10−3 for the single disc, 10−5−10−4for the double disc system, and 10−8−10−6for the serrated. In the case of the triple disc, he simply concluded on a reduction by a factor 4 with respect to the double disc system obtained by removing the middle disc. Out of this study, the serrated occulter was proved to be the most efficient to reduce straylight. Finally, this author concluded that the more distant the disc is, the darker the umbra is, for a given occultation angle.
Regarding experimental studies, Fort et al. (1978) measured the diffraction behind a single disc and a toothed disc, for a point-like source. Although they did not report quantitative results, they estimated a gain factor of at least 15 for the toothed disc. Koutchmy and Belmahdi (1987) supported this improvement by stating a factor 13: the measured intensity was 1.01.10−4 for the serrated, with respect to 1.33.10−3 for the single disc. Later, Bout et al. (2000) conducted a large experimental campaign on new types of external occulters to support the development of the solar coronagraph LASCO C2. They additionally considered the multi-threaded cone, with 160 threads, or the polished cone for example. Bout et al. (2000) made a specific comment about the difficulty of achieving a good experimental set-up and an accurate
8From now on, the intensity of the diffracted sunlight will be always given in unit of MSB, without further notifying it.
solar simulator.
Thernisien et al. (2005) investigated the triple disc occulting system in the
frame of the solar coronagraph COR-2 on-board the mission SECCHI10(Howard, 2008). They numerically computed the diffraction at a few positions using an optics
software, that they complemented by an experimental set-up. In particular, the authors investigated the optimization of the occulter configuration, and its sensitivity to a tilt or a misalignment. They concluded that an optimum in straylight rejection can be obtained when the angle between the edges of the two first discs is smaller than the angle between the two last discs.
All these results can be put into perspective, whether being theoretical or experimental. It is unfortunately difficult to provide one single and consis-tent comparison, because dimensions and wavelengths may vary across the studies. Moreover, experimental measurements are also not always explicit about the exact occulter’s geometry. I provide a comparative picture in Fig-ure 1.5, which should be interpreted with care for the aforementioned rea-sons. I chose to use the parameters from Bout et al. (2000). The description of the occulters and methods is recalled in the caption of the figure. As can be seen, the serrated occulter is theoretically predicted to provide the best performance, but this was however not fully confirmed by the experimenta-tion. The most efficient shapes appear to be the advanced three-dimensional shapes, like the polished or multithreaded cone.
More recent studies have been published in the frame of the future long-baseline solar coronagraph ASPIICS. First, Verroi et al. (2018) developed
a new analytical method based on the Fresnel-Kirchhoff theory11 to com-pute the diffraction. They considered a saw-toothed disc with only a few teeth, about 20, which seems in contradiction to what was known for solar coronagraphy. Aime (2013) was the first to implement an accurate and ex-act computation of the Fresnel diffrex-action. He found in particular that an apodized occulter theoretically manages to provide a very low umbra of 10−12−10−8. Both studies from Verroi et al. (2018) and Aime (2013) also confirmed Lenskii’s conclusion: a large distance to the occulter improves the
10SECCHI stands for Sun-Earth Connection Coronal and Heliospheric Investigation. 11It will be introduced in Chapter 2.
umbra intensity.
At last, Landini et al. (Landini et al., 2011; Landini et al., 2016; Landini et al., 2017) proposed a series of experimentation by scanning the umbra level behind various occulter edges. This campaign showed that a geometrical optimization of the shape of occulter edge, like a conical or a toroidal shape, could improve the straylight rejection performance.
FIGURE 1.5: Umbra intensity from the diffraction of sunlight behind different external occulters. The configuration is the one from Bout et al. (2000). The diameter of the occulter is 16.2mm and the distance behind the occulter is 820mm. The wavelength used for the analytical expression is 750nm. Red and black in-dicate theoretical prediction and experimental measurements, respectively. (a): single disc (Fort et al., 1978; Lenskii, 1981). (b): single disc (Bout et al., 2000). (c): single disc, a factor 2 is ap-plied with respect to (a) (Thernisien et al., 2005). (d): double disc (Lenskii, 1981). The second disc is 1mm further than the first one, and 0.18mm larger. (e): double disc (Lenskii, 1981). The second disc is 2mm further, and 0.20mm larger. (f): triple disc, obtained from the combination of the two previous double discs. A factor 4 is applied (Lenskii, 1981). (g): triple disc (Bout et al., 2000). (h): triple disc (Thernisien et al., 2005). (i): 500-teeth serrated occulter (Lenskii, 1981). (j): 1000-teeth serrated occul-ter (Lenskii, 1981). (k) serrated occuloccul-ter, where a factor 13 is applied with respect to (a) (Koutchmy and Belmahdi, 1987). (l) polished cone (Bout et al., 2000). (m): 160-threaded cone (Bout et al., 2000). (n) Electroeroded cone (Bout et al., 2000).
1.2.4
Past and future solar coronagraphs
Since the beginning of solar coronagraphy, several instruments have been built and flown. They however were unequally successful. Koutchmy (1988) provides a very complete review of the early development of past space-borne solar coronagraphs.
On the 2nd of December 1995, the SOHO mission was launched. It em-barked the Large Angle and Spectrometric Coronagraph, made of three coro-nagraphs, LASCO-C1, -C2 and -C3 (Brueckner and et al., 1995). The respec-tive field-of-views of the three LASCO overlap such that they would cover the observation of the corona from 1.1R⊙ up to 30R⊙. However, due to vi-gnetting and diffraction, clean observation of the solar corona started only around 2.2R⊙. LASCO-C1 and -C3 were designed according to the classical
concept of the Lyot coronagraph. But LASCO-C2 was more ingenious. It
com-bined together an external occulter and a Lyot-style coronagraph. The role of the internal Lyot mask was to cut-off the diffraction halo produced from the external occulter upstream, instead of the direct sunlight. The straylight re-jection is thus enhanced by this double occultation. LASCO-C2 proved to be a real success, and the scientific literature has been prolific in terms of lessons learned, with the work from Lamy, Llebaria and Koutchmy et al. for instance.
LASCOwas followed by the twin coronagraphs COR-1 and COR-2 on-board
the SECCHI mission (Howard, 2008), launched on the 26th of October 2006. They went one step closer by observing the corona from 1.5R⊙ up to 15R⊙. Today, the last solar coronagraph METIS12 (Fineschi and et al., 2012) on-board the ESA Solar Orbiter mission, will observe the inner corona from 1.5R⊙with unprecedented temporal coverage and spatial resolution.
The next generation of coronagraphs shall go even further in performance and target the region 1R⊙−2R⊙, which still remains rather unexplored. The future of coronagraphy will most likely rely on formation flying - actually for both solar and stellar coronagraphy. This new space mission concept allows to enlarge the embarked space instrument up to an unprecedented dimen-sion. The coronagraph ASPIICS on-board the ESA PROBA-3 mission will combine a 1.5m external occulter and a Lyot-style coronagraph located 144m
behind. The advantage of such a large geometry is twofold. By placing a bigger occulter at a larger distance, the entrance aperture of the coronagraph is less obstructed. This allows to mitigate the vignetting effect and improve the resolution in the inner field-of-view, as illustrated by the bottom drawing of Figure 1.4. Second, the aforementioned past studies conjecture that the umbra would be darker with such a long distance.
1.2.5
Comparison of solar and stellar coronagraphy
Stellar coronagraphy is like the twin domain of solar coronagraphy. It aims at detecting exoplanetary systems around stars. If the principle remains the same as solar coronagraphy, i.e. a faint target close to a bright source, the scales and orders of magnitude involved are actually very different. The an-gular separation between the star and the planet is usually less than 1arcsec (Traub and Oppenheimer, 2010). The level of contrast to be achieved is about 10−12−10−7 in visible or near infrared (N’Diaye et al., 2012), so more ambi-tious than the 10−10−10−6for the solar corona.
Like in solar coronagraphy, rejecting the starlight and its diffraction is one of the main challenges. But that problem is quite different between the stellar and solar case. In fact, the diffraction from a single stellar point source can be theoretically quite well manipulated in a very local area. One drawback is that it is much more sensitive to the wavelength or small perturbations. On the contrary, the Sun is an extended source: it is like made of a collection of incoherent point sources. The diffracted sunlight results from the sum of all the individual contributions over a large spatial extent. Because of that, what can work for stellar coronagraphy does not necessarily apply to solar coron-agraphy, and vice-versa. The two domains are thus quite different. Table 1.2 summarizes their respective characteristics.
In the last decades, stellar coronagraphy has been developed in parallel to solar coronagraphy. The original Lyot’s concept was adapted and upgraded to comply to exoplanet detection needs, as recalled by Ferrari et al. (2007). The development and study of shaped-pupil coronagraphs and the applica-tion of the apodizaapplica-tion technique appeared in the 2000s. The goal was to create zones of deep darkness at the location where an exoplanet is expected,
TABLE 1.2: Main differences between solar and stellar coron-agraphy, regarding the diffraction. (†): the contrast can be de-fined as the required straylight level with respect to the bright source brightness. (††): the Fresnel number is a characteristic number of diffraction problematic, associated to the geometry. It will be explained in Chapter 2, Eq.(2.58).
Features Solar coronagraphy Stellar coronagraphy
Bright parasitic source the Sun a star
extended disc of 32arcmin point source-like
Field-of-view a few degrees a few arcsec
Wavelength dependency weak strong
Required contrast(†) 10−6−10−10 10−7−10−12 External occulter a few cm to a meter a few tens of meter
Fresnel number(††) 102−104 100−101
by reshaping the point spread function of the stellar light. In particular, the apodized external occulter and its optimisation has led to numerous and rec-ognized works, alike the ones from the Princeton group that can be cited as example (Spergel, 2000; Kasdin et al., 2003; Vanderbei et al., 2003), and also more recent research like the study of Flamary and Aime (2014). I also refer the interested reader to the introduction of N’Diaye et al. (2015), which pro-vides a good overview of different types of stellar coronagraphs.
Nonetheless, the concept of optimized occulter remains purely theoretical: it cannot actually be manufactured to the required accuracy. Then, Cash (2006) and Vanderbei et al. (2007) proposed the petal-shaped occulter as an alter-native. It corresponds to the binary substitute, i.e. 0-or-1 mask, of the ideal apodized occulter. The shape of the petal that was derived is more elabo-rated than the mere triangles of the serelabo-rated edge, as depicted in Figure 1.6. Only ten to twenty petals are used, instead of hundreds. Such occulter has been named starshade.
FIGURE 1.6: The left picture illustrates the serrated occulter used in solar coronagraphy. The size of the teeth is small, and in high numbers. The right picture (Cady, 2012) shows the petal-ized occulter, or starshade, for stellar coronagraphy. It has typi-cally 20 petals.
1.3
Presentation of the thesis
1.3.1
Motivation and objectives
This introduction has explained that the capability of a solar coronagraph to successfully observe the inner solar corona is fundamentally linked to the diffraction. Therefore, understanding, computing and estimating the resid-ual diffracted sunlight turns out to be essential.
For the past solar coronagraphs, the diffraction from the external occulter was mostly estimated by experimentation, like the work from Bout et al. (2000) for LASCO-C2. But it is notoriously hard to reproduce a representative solar
source. The complementary analysis, either analytical or numerical, remains today very limited - at least in solar coronagraphy. The work of Aime (2013) can be cited as a pioneer in the domain by achieving the required precision. Furthermore, the context of formation flying brings new challenges, with the large geometry it involves. No representative full-scale experimentation in a laboratory can be performed.
It must be mentioned that classic optics software packages used in the in-dustry to design space instrumentation are not made to accurately compute
and propagate the diffraction with sufficient precision (Perrin, 1999). Dedi-cated modeling based on exact calculations of light wave propagation is thus needed.
The fact that the Sun is an extended source also complicates the problem. It cannot be treated as one single point source (Ferrari et al., 2010). In that respect, the prolific work from the scientific community in stellar coronagra-phy may actually not be applied to solar coronagracoronagra-phy. The two diffraction problems are rather different, even if they are the same in nature.
My thesis falls within this framework. It humbly intends to contribute to a better understanding of diffraction effects for externally occulted solar coron-agraphs in the context of formation flying space mission. The methodology is numerical. Firstly, it aims to pursue the work of Aime (2013). The diffraction and the umbra from large serrated external occulter is explored in particu-lar. Secondly, it provides a computational model for the propagation of the diffracted sunlight inside a Lyot-style coronagraph. My research is carried in the frame of the giant externally occulted Lyot solar coronagraph ASPIICS on-board the PROBA-3 mission .
Beside the results, the thesis aims to provide recipes in order to compute the diffraction, in a hopefully pedagogical way.
1.3.2
The ESA P
ROBA-3 mission
The performance of any astronomical instrument is closely linked to its size. Nowadays, space-borne payloads tend to be designed larger and bigger. How-ever, the dimension of a spacecraft is obviously bounded by the launcher. Formation flying provides the alternative to such a hard limit. This space mission concept distributes the payload over two (or more) spacecraft of re-duced dimension. In other words, a set of spacecraft flying as a controlled formation substitutes to a single large structure that could not be launched into space. The embarked instrument is virtually enlarged.
Today, formation flying is still not mature enough and the required precision has not been demonstrated yet. As a pioneer, the Swedish Prisma mission (Persson et al., 2006) successfully achieved an in-orbit relative position accu-racy of about 10cm between the two spacecraft Mango and Tango.
The PROBA-3 mission will pave the way to the future of formation flying. It is part of the PROBA13 family (Bernaerts et al., 2000) of the European Space
Agency dedicated to in-orbit demonstration. It will demonstrate formation flying up to unprecedented performance, including the on-board guidance, control and navigation algorithms, the associated metrology sensors, and the in-orbit and on-ground concepts of operations. The mission is developed by the ESA Directorate of Technology, Engineering and Quality.
The PROBA-3 mission is made of the Coronagraph Spacecraft (CSC) and
the Occulter Spacecraft (OSC), launched together on a highly elliptical or-bit (HEO) - the oror-bital parameters are given in Table 1.3. They will maintain a fixed configuration during six hours along the apogee arc - see Figure 1.7. The Inter Satellite Distance (ISD), i.e. the distance between the CSC and the OSC, can range from 50m to 250m, depending on what is commanded. Ded-icated impulsive manoeuvres will be executed before and after the apogee arc to acquire and break the formation in a safe way. The two spacecraft will then freely drift without any risk of collision during the perigee pass, but they will remain close enough to acquire back the formation before the next apogee.
The relative position accuracy to be achieved by PROBA-3 is 4.9mm in lateral,
and 14.8mm in longitudinal, at 150m ISD (Galano et al., 2019). To achieve this performance, a set of sensors will be used sequentially. As first, a rela-tive GPS algorithm provides a coarse estimate of the relarela-tive position, i.e. a few tens of centimeters. Then, a vision based sensor on the OSC detects and tracks a pattern of emitting photodiodes on the CSC, in order to acquire the formation. Finally, a laser metrology sensor, combined with a corner cube retro-reflector, provides the final high accuracy measurements of the lateral and longitudinal relative displacements.
TABLE 1.3: Proba-3 highly elliptical orbit parameters. The given values are the initial ones at injection. The orbit will vary over the two-years mission. The apogee and the perigee are given as altitude. (†)the initial right ascension of the ascend-ing node (RAAN) varies with the launch date.
Orbital parameter Value
Apogee 60530km Perigee 600km Eccentricity 0.811 Inclination 59◦ RAAN(†) 150−155◦ Period 19h38min
In addition to demonstrating formation flying, PROBA-3 is a science mission.
It will carry two scientific payloads: the externally occulted solar corona-graph ASPIICS (Lamy et al., 2010; Galano et al., 2018), and the Davos Abso-lute Radiometer DARA, which will finely measure the total solar irradiance.
For further details and explanations about PROBA-3, I refer the interested
reader to the paper from Galano et al. (2019), which provides the latest status of the mission at the time of the writing.
FIGURE1.7: Artistic view of the two spacecraft of the PROBA-3 mission. The images are from the ESA official website.
1.3.3
ASPIICS solar coronagraph
ASPIICS stands for Association of Spacecraft for Polarimetric and Interfero-metric and Investigation of the Corona of the Sun, or Association de Satellites
