Discovering Exo-Earths: optimization of an external occulter.

Christophe Bellisario

June 29, 2010

Abstract

Before looking for extra-terrestrial life, detecting exo-planets is one of the most promising quest in astronomy.However, the difficulties that science and technology encounter make tough the detection and observation of exo-planets, especially exo-Earths. In this paper, I itemize concretely what we are looking for and the various methodsused up to now in the search of exo-planets. I will be paying particular attention to the direct imaging, which enablesspectral characterization of a planet.

Suppressing the light coming from the host star by orders of magnitude to reveal the faint light coming from theexo-planet is one of the most efficient way for characterizing an exo-Earth and maybe, finding life. This can be doneby the use of internal or external occulters (coronagraphs or starshades) with numerous different properties. I discusshere how to compute the physical expression of the light intensity when combining a starshade with a telescope. Ialso explain how to build the shape of an occulter through an apodization function coming from a numerical andanalytical optimization, which I implemented. Then, I investigate all the related parameters such as the diameter,the petal length, the inner working angle, etc..., to highlight all the various behaviors of the apodization through arange of data corresponding to the science we aim to do.

Contents

1 Science of extra-solar planets 21.1 Exo-Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Gas Giants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Terrestrial exoplanets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Detection methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Radial Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Transit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.4 Puslar Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.5 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.6 Direct Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Starshade: design, optimization and properties 112.1 Design and optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Free-Space propagation from starshade to telescope . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Binary apodization, shape of the occulter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Optimization of the occulter apodization function . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Global study of the parameter space and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 First comparison between analytical and numerical methods . . . . . . . . . . . . . . . . . . . . 182.2.3 Contrast as a function of the diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.4 Distance of the occulter as a function of the diameter . . . . . . . . . . . . . . . . . . . . . . . 212.2.5 Action of the petal length on the contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.6 Action of the Shadow Oversize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.7 Another way to define the HyperGaussian function . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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Chapter 1

Science of extra-solar planets

1.1 Exo-Planets

Are we alone in the Universe? This fundamental and philosophical question could find a simple scientific yes-or-noanswer by finding life in other solar systems, other Earths around stars. Since we discovered the first exoplanets, theresearch in this field has grown tremendously and more than 450 planets have been catalogued to create a huge archiveof various planets. There are planets with masses from 2 M⊕ ([1]) to 13 MJup (limiting mass for thermonuclear

fusion of deuterium [2]), the radius varies from 2 R⊕ ([3]) to 2 RJup ([4]) and the temperature from 50◦K ([5]) to

1100◦K ([6]). The boundaries of the high values are quite difficult to establish as the limit between gas giants andbrown dwarfs/aborted stars is as of now not really clear and can be defined based on different formation scenariosinstead of the mass ([7] & [8]). Gravity is also important as it depends on the density and radius; it can hardly mouldthe shape and consistency of the surface. Moreover, if we look for life, the age of the star will be another importantfactor as young stars won’t be good candidates since it took Earth at most one billion year to appear. The next andlast step in the search of life deals with the characterization of the atmosphere. Nowadays, we count few planetswhich show evidence of carbon dioxide, methane and sodium ([9], [10] & [11]).In order to find candidate planets sheltering life, we have to differentiate the kinds of exoplanets: gas giant, ice giantsand terrestrial planets, etc. For a given terrestrial planet, the habitability is defined by the presence of liquid water(habitable zone – def HZ). If life exists, then we can search for it by using biomarkers. Many more hot Jupiter-likeplanets have been found in comparison to terrestrial-like exoplanets, and most of them have been classified in differentkinds. In the next two parts we give a general description of gas giants and terrestrial planets, more precisely wherethe search of life starts.

1.1.1 Gas Giants

Based on Jovian planets of our solar system, gas giants are also called hot/cold Jupiters. They mainly represent aclass of gaseous planets, that are almost always the mass of Jupiter or more, and are also above a vague boundaryof 10 M⊕. Their types have been classified by David Sudarsky in regards of some characteristics like temperatureand composition ([12]). The gas giants like Jupiter and Saturn are mostly composed of hydrogen and helium, whichare the most abundant elements in the Sun whereas the ice giants like Uranus and Neptune are primarily made ofheavier components such as oxygen, carbon, nitrogen, and sulfur. They also differ by the size of their respective coresaround which the gas orbits. The scientific community first attempted to observe them while they were far awayfrom their host stars, like it is the case for Jupiter, Saturn, Uranus and Neptune. However, a significant proportionof discoveries of gas giants that were very close to their host stars (which helped for their detection) and also morediscoveries of retrograde orbits made scientists rethink their ideas of star system formation.Hydrogen, helium, methane and ammonia are the main components detected in gas giants. Age and temperature areimportant properties and the latter is principally governed by the distance to the host star, e.g. closer planets meanhotter planets. Their temperature will lead to different structures as the fluid becomes either a gas or solid. Moreover,younger gas giants are significantly more luminous, which makes them easier to observe with direct imaging in thenear infrared.The big question about whether a detection is a gas giant, a brown dwarf, or a binary star is still relevant. Thereare many observations of speculative close gas giants that need confirmation before they can be recognized officiallyas new exoplanets.

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1.1.2 Terrestrial exoplanets

As previously stated, exo-Earths are the Holy Grail of exo-planet research. Up to now, the lightest exoplanetdiscovered is GJ 581e with a minimum mass of 1.9 M⊕ ([13]) and the smallest is CoRoT-Exo-7b with a radius of1.76 R⊕ ([3]). In our quest for life, we mainly look around Sun-like stars, even if it might look anthropocentric tosearch for similar biological markers. However, starting with the various forms that life can take on our planet, wemight include huge possibilities of other means of life.The key parameters to describe a possible Earth-twin are orbit, mass, radius, visible/infrared spectrum, and alsotheir variations during time. These parameters and their combinations provide us information about numerous otherproperties, like effective temperature, density (→ surface gravity with the help of the radius) and albedo (→ surfacereflectance).The first step is to define an habitable zone. In our galaxy, disruptive gravitational forces or strong emissions ofinfrared radiation and X-rays could cause the impossibility for life to grow close to the galactic center. In the outerlimit of the galaxy, the abundance of heavy elements decreases due to galactic chemical evolution. Next, as the lifeform we know could not inhabit planets like Neptune or Venus, the stellar-habitable zone has be defined to set arange of distance around the star where all these conditions are met to encourage the development of life mainlydefined by the presence of water ([14]). To help find this range, we can use our own solar system as a base. Here thehabitable zone goes from 0.7 AU to 1.5 AU (Earth being, of course, at 1 AU). We scale this frame by the square rootof the stellar luminosity, which will lead to the following equation:

Habitable Zone (AU) ∈ [0.7− 1.5]×

√L∗L�

(see figure 1.1). We can also translate the habitable zone in AU into an angular distance and through

θ(”) =a(AU)

d(pc)=

√L∗/L�d(pc)

which will lead to a frame of 70 milli-arcseconds (mas) to 120 mas (by taking the ratio L∗/L� between 0.5 and 1.5).A useful parameter which expresses the angular separation is the Inner Working Angle (IWA). It expresses the anglebetween the host star and the planet seen from the telescope and takes the previous values in the section devoted tothe starshade.

Figure 1.1: Habitable zone (HZ) in Earth radius as a function of the star masses. As we can see, for increasing mass,the luminosity will be higher and it will push away the HZ. For a Sun-like star, Earth is between the [0.7 - 1.5] AUboundaries whereas Mars is just at the limit of the HZ. Figure credit: GFDL.

In spite of the increase of discoveries thanks to transit and radial velocity methods, characterization of the atmosphererequires a higher level of planet detection. Direct imaging is the key to getting spectroscopic data. Currently, transitspectroscopy provides many spectral characterizations ([15]) but remains inefficient for terrestrial planets closer totheir host stars (except for some dwarfs), in the habitable zone. The wavelengths of interest are included in a range

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of 1 to 12 µm which reveal information about water, methane and carbon monoxide signatures. They help us tounderstand about the surface type, clouds or atmospheric retention too. Finally, O2 and O3, as biogenic traces willindicate evidences for life ([16]).

Figure 1.2: Histograms of the number of exoplanets discovered as a function of radius and mass (figure credits:exoplanet.eu).

1.2 Detection methods

1.2.1 Radial Velocity

Radial Velocity has been so far the more prolific method in detecting exo-planets. Also called the Doppler spec-troscopy, it uses gravitational laws and the Doppler effect. Historically, it has also been the first method used byastronomers ([17]) to discover a Jupiter-class planet around 51 Pegasi.The exoplanets in a system have elliptical orbits and the host star moves in a small counter-orbit around a commonbarycenter due to the attraction of planets. This movement will change the radial velocity of the star seen from Earthand the spectral lines will show small blue shifts and red shifts.At first, the errors of radial velocity measurements were too big to detect exo-planets. For example, the Sun getsan additional movement due to Jupiter of 13 m/s and the errors were of 1000 m/s. However, in 1988, the Canadianastronomers Bruce Campbell, G. A. H. Walker, and S. Yang suggested that a planet was orbiting the star GammaCephei using a method that allowed them to detect radial velocity movements to a precision of 15 m/s. Now theHigh Accuracy Radial velocity Planet Searcher (HARPS) at La Silla Observatory in Chile, can reach a precision ofalmost 0.97 m/s (in comparison, Earth causes a fluctuation of 10 cm/s for the Sun).This method provides a lower mass limit to the planet since radial velocity measure M∗ · sin(i) due to the inclinationof the orbital plane, with i being the angle of inclination. Further astrometric observations tracking the movement ofthe star may change an exo-planet discovery to a brown dwarf detection. As of now, there are 425 planets discovered,with numerous ground missions already working: AFOE, Anglo-Autralian Planet Search Program, Automated PlanetFinder, California & Carnegie Planet Search, Coralie at Leonard Euler Telescope, Elodie, Sophie, Exoplanet Tracker,HARPS, Hobby-Eberly Telescope Magellan 6.5m Telescope, Mc Donald Observatory, NK2 Consortium, TNG HighResolution Spectrograph, UVES. In the next years, the Absolute Astronomical Accelerometry, Carmenes, HARPS-N,OWL, PRVS will complete the huge panel of missions using radial velocity ([18]).

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Figure 1.3: Principle of the radial velocity method: when the star moves away, the Doppler shift will be red, andwhen the star gets closer, the shift moves to blue. Figure credit: NASA/JPL image.

1.2.2 Transit

First proposed by Otto Struve in 1951, the idea is to study the luminosity of a star with a reasonable sized-telescope.Periodical variations of the luminosity could come from a planet located between the star and the Earth. In thatcase, we need to see the system in the ecliptic plane. Otherwise, no planet can be detected. Some calculations withgeometric probabilities suggest an estimate that there are almost 5% of stars with a detectable exo-planet.

Here, detection will occur with a fall of luminosity and characterization is possible with the spectral analysis of

Figure 1.4: Principle of the transit method. Figure credit: NASA/JPL image

the light received. It is the primary transit. The composition and scale height used with the help of the absorptionof starlight passing through the planet’s atmosphere is called transit spectroscopy. Next, when the planet is almostbehind the star, the secondary eclipse happens and provides a direct detection of the planet’s spectrum. We cantherefore get information about components, reflectivity and temperature. The radius, mass and orbits follow fromtime and depth of the fall. When the planet goes behind the star, we can get the light from the planet by subtractionof the star light; this semi-direct method also allows planet characterization. For example, the Spitzer Space Telescope(NASA) managed to produce a 7.5-14.7 µm spectrum for the transiting extrasolar giant planet HD 189733b ([19]).Currently 81 planets have been discovered with the ongoing ground missions: Alsubai’s Project, ASP, BEST,E.P.R.G., HATNetwork, LCOGT and UStAPS (also doing lensing), MEarth, MONET, OGLE III, PASS, PIRATE,PISCES, STARE, Super WASP, STEPSS, UNSWEPS Project, Tenessee Automatic Photoelectric Telescope, TrES,TRESCA, Vulcain South, WHAT, XO Project. In space, CoRoT, EPOCh, Fabra-ROA Camera, Gaia, KEPLER al-ready found many planets, and in project: Plato, GEST for space telescopes and GITPO, STELLA for ground-basedmission ([18]).

1.2.3 Gravitational Lensing

In order to use gravitational lensing for the detection of exoplanets, we need first a background star. Next, whenanother star crosses the distance between Earth and the background star, the gravitational field acts like a lens andthe light of the background star is modified. If the star owns a planet, its gravitational field will contribute to thelense and be detected from the Earth (see figure 1.5). However, this event is very rare and only occurs once for eachplanet as the alignment cannot happen again, making a confirmation impossible.This method provides some advantages: the mass can be measured, we can reach Earth-like masses, and the angularseparation is also known. We can also detect planets in other galaxies. But there are drawbacks: the observationtime required is huge as well as the number of stars which need to be observed for a small number of detections.Moreover, the mass and orbit size depend on the properties of the host star, which also need to be known.With gravitational lensing, 10 planets have been observed, especially with the use of OGLE (the Optical Gravitational

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Lensing Experiment) and the others projects: University of St. Andrews Planet Search (UStAPS), Las CumbresObservatory Global Telescope Network (LCOGT), MACHO, Microlensing Planet Search Project (MPS). In thefuture, GEST will join them ([18]).

Figure 1.5: Principle of gravitational lensing: the lensing effect of the star is disturbing by the presence of anexoplanet. Figure credit: Abe et al.

1.2.4 Puslar Timing

Pulsars are neutron stars hosting a strong magnetic field and are the fastest spinning objects discovered so far. Twobeams of radiation are ejected at the pole of the star, and we get on Earth a brief pulse each time the beam crossesthe path of the Earth. This need to have an alignment between Earth and a pulsar makes the proportion of pulsarswe cannot observe from Earth rather large. However, the existence of a planet around it makes them both orbitaround their center of mass. Similar to the radial velocity method, the act of measuring the periodic changes in thetime between each pulse will be a way to estimate the semi-major axis of the planet’s orbit, and a lower limit of theplanet’s mass ([20]).Actually, planets around pulsars are not very interesting in the search for life in our galaxy. Indeed, pulsars arecreated by stellar explosions like supernovae, which may prevent life to expand in such systems. At this moment,only 8 planets in 5 different solar systems have been discovered with this method by the ongoing mission PulsarPlanet Detection.

1.2.5 Astrometry

Figure 1.6: Principle of the astrometry method. Figure credit: http://www.astro.wisc.edu/

Basically, astrometry is used for determining positions of stars. Using already well-known coordinates of nearstars, it determines the location of an unknown star in the same image by comparison. In the detection of extra-solarplanets, we measure the displacement of stars around a supposed center of mass of the system composed by a star andits planets, providing position and mass of the planet(see figure 1.6). Many white dwarfs have been discovered as theyinvolve high variations. However, the accuracy needed for exo-planets is very precise and ground-based astrometry isnot enough powerful, due to the distorting effects of the Earth’s atmosphere for example. As of now, only one planet

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has been discovered with astrometry (with HST Astrometry, [21]).There will be space missions using this method in the next few years, such as NASA’s SIM Lite (Space InterferometryMission) projected to launch in 2015, European Space Agency’s GAA, due to launch in 2012, Origins Billion StarSurvey (OBSS) in study, and on the ground: STEPS, Radio Interferometric Planet Search (RIPL), PRIMA-DDL(VLTI, under study), Keck Interferometer, ASPENS. All of them will detect terrestrial planets orbiting close to theirstars with astrometric techniques ([18]).

1.2.6 Direct Imaging

About direct imaging

Providing a direct picture of exo-planets is the most difficult challenge to take up. Their extremely faint luminosityrequires a high contrast for imaging. For young gas giants (1 MJup, 70 Myr), at a distance of 10 pc, a contrast of

10−7 with its host star will be required ([22]). In comparison, for exo-Earths, still at 10 pc, 10−10 of contrast will benecessary. These values correspond to a difference of magnitude from 17.5 to 25 mag (as m = −2.5 ∗ log10(F ) + C,with F the flux and C a constant). There are some directly imaged exoplanets (12 as of now [18]).In detection of Earth-like planets, one issue for getting high contrast is dust. Primary disks of dust and asteroidsdating from the creation of stellar systems may still remain like our Jupiter and Neptune Trojans. We also add thedust coming from the numerous collisions between asteroids and comets. All of that represents the exozodiacal dust.It is a source of background noise contaminating the spectrum. For example, the zodiacal dust in our solar system isthe most luminous component after the Sun ([23]). It is called the local zodi. Earth signature might appear in thisdisk as a clump inside it so that even if the exozodiacal disk is not helpful for a detection, its structure might revealthe presence of an unseen planet or could be sign of the system’s orbital dynamic.The exozodiacal light is measured with the ratio between the infrared luminosity and stellar luminosity: (LIR/L∗) ≈10−6 − 10−5 in most of the cases. In comparison, the local zodi represents the zodiacal disk in our asteroid belt andthe ratio (LIR/L∗) is approximately equal to 10−7 which is the referee value, 1 zodi. Even if it represents most of thesource of noise, the exozodiacal light provides us information about the elliptic inclination of the planetary systemby supposing a circularly distribution around the star. Therefore, dust may provide a clue about the planets’ orbits,which is very important for science. The field of exozodiacal disks is of great importance to understand the behaviorin planetary systems as well as planetary formation and systems evolution.One other issue lies in interferences between wavefronts. Optical aberrations cause speckles on the image that canbe confused with a planet signal. Speckles are also created by small thermal variations. These aberrations are timedependent which makes the subtraction by calibration difficult. The use of deformable mirror and adaptive opticsreduces the atmospheric distortions for ground-based telescopes and the speckle intensity. However, 100% efficiencycannot be achieved and there will still be remnants.From visible light to infrared, there are several ways to detect exoplanets. In the case of infrared light, the main methodis the use of an interferometer. It consists of a few telescopes connected together to build a larger telescope withhigh resolution power. A nulling interferometer can also be built to reduce the intensity of the host star to show thefaint light of the planet, e.g. the canceled Darwin mission and the ongoing ground-based Large Binocular Telescope.Many projects are already in use or planned for the future ([18]): Keck Interferometer ([24]), Very Large TelescopeInterferometer VLTI (ESO, Paranal), both ground-based telescopes and others are in project: The Antarctic PlateauInterferometer (API) and CARLINA Hypertelescope Project on Earth, Space Infrared Interferometric Telescope(SPIRIT, NASA) in space.The coronagraph, invented by the French astronomer Bernard Lyot in the 1930’s, is a powerful instrument usedfirstly to observe the corona around the Sun. It simulates an artificial eclipse by blocking the light with an occultingspot or mask whereas the surrounding light stays undisturbed. In search for exoplanets, the coronagraph helps toget the faint light coming from a planet around its star. There are several types of coronagraphs, from the band-limited coronagraph (present on the JWST’s Near-Infrared Cam), phase-mask coronagraph, apodized pupil lyotcoronagraph to the optical vortex coronagraph. More information about coronagraphs are listed by Guyon ([25]) andclassified by Quirrenbach ([26]). The space missions are: Pupil mapping Exoplanet Coronagraphic Observer (PECO,under study), Super-Earth Explorer (SEE-COAST, under project [27]), and the ground-based missions: Spectro-Polarimetric Imaging (SPHERE, under construction [28]), Gemini Planet Imager (GPI, under construction [22]).And the last method, subject of our interest, is the external occulter. The idea of building an occulter in orderto block the light coming from a star was emitted by Lyman Spitzer in 1962 ([29], see figure 1.7). As one wouldthink, a circular-shaped screen could block the light of a star to help observe the faint light coming from a nearbyplanet. However, strong edges cause Fresnel diffraction effects that will brighten the shadow created by the occulterand cause starlight to come through the telescope. An apodization has to remove the strong edges of the occulter.By advanced calculations, many shapes have been designed in the early 1980s ([30]) but finally, the petal-shape hasbeen adopted (a 20 point star-shaped mask was also another candidate, [31]). To ensure the best achievement, theexternal occulter has to remain accurately at its position during observation time. In our case, the telescope will be

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in an orbit around the Sun-Earth L2 point and the occulter has to follow the orbit. To avoid noise coming from theSun reflectance on the occulter, the observations have to be reduced to the ones which match an angular position tothe Sun from 45 to 85 degrees (or slightly beyond if the occulter can be tilted, [32]). Currently, there is no missionin activity but a few missions are being studied e.g. the proposed THEIA (Telescope for Habitable Exoplanets andInterstellar/Intergalactic Astronomy) and NWO (New Worlds Observer). Now, the future JWST (James Webb SpaceTelescope) is the best candidate for an occulter mission on a short time scale.

Figure 1.7: Principle of a starshade: the light coming from the star is stopped whereas the weak light from the planetis not stopped. Figure credit: Northrop Grumman Corporation

Managing direct imaging is the most important march for the search of life. However, as indirect methods likeastrometry or radial velocities provide a direct measurement of the masses, orbital parameters and coordinates of theplanets, combining these methods would complete the characterization and supply all the information we want to getabout an exoplanet ([33]).

Comparison between coronagraph and external occulter

Both occulter and coronagraph are acting in the same way: suppressing the light of the star to reveal the faintlight of the orbiting planet. However, if they do almost the same thing, we can wonder why one would send a hugespacecraft thousand of kilometers away from the telescope. An external occulter presents several advantages overinternal coronagraphs.But first, we will discuss some of the drawbacks of an occulter. As the word ’external’ says, an occulter requiresanother spacecraft to maneuver it and thus the lifetime of the occulter depends on fuel consumption and on themission design. Since micro-engineering costs a lot for high performance coronagraphs, the cost of a occulter in orbitis not that much higher but it increases with its size. The time of travel between each target is also a burden forthe occulter. Almost two weeks of traveling is necessary to move to the next target (Design Reference Mission isbuilt to optimize as much as possible all these time constraints by reducing all the impacted parameters). Next,lower contrast caused by the deformations of the occulter, and speckles may arise from manufacturing, deploymentor micro-meteorite hits in flight.The small size of a coronagraph may be source of material imperfections. One parameter only controlled by the ex-ternal occulter, as it can move backward and forward, is the inner working angle. It is not fixed, and there is no limitfor outer working angle too (due to the absence of deformable mirror correction of the coronagraph speckles). Stilldealing with the inner working angle, in the case of coronagraph, it depends on the wavelength (IWA ∝ λ/D): thehigher the wavelength, the higher the inner working angle. Because exo-Earths will be found at low separations, thecoronagraphs are usually designed to provide a spectrum between 250 and 1000 nm ([34]). Starshades are typicallynot limited in the size of the bandpass. For internal coronagraphs, starlight suppression over a broad band is morechallenging and typically limited to 10-20%. The contrast obtained is also better for external occulters and 10−10 isachieved for most of the cases (see the result part) and this can be helped by the fact that the primary mirror andsupporting optics of the telescope have less constraints with an occulter. Finally, as a Lyot-type coronagraph is muchmore complex, a simple reasoning explains that with fewer optical systems, less signal is lost.The following table sums up the characteristics we compare between a coronagraph and the external equivalent:

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Characteristics Coronagraph External OcculterCost + -

Signal - +Lifetime + -

Deployment + -Scattered light - +Position control - +

Higher suppression - +Inner Working Angle - +

Usable for any telescope instruments - +Performance in spectral characterization - +

Instead of wondering if an occulter is better than a coronagraph, the idea of doing both light suppression have beenexcogitated. Combining an occulter with a coronagraph makes the science really more complicated. The occulterhas to be designed for the coronagraph and moving the occulter will require the coronagraph to be changed. And foreach new configuration, tests need to be done to verify the ability of the system. These are called hybrid occulters.They could reduce the size of the occulter and therefore the distance, economizing time and fuel. Occulters withapodized pupil Lyot coronagraph and achromatic interfero-coronagraph have been described by Cady ([35]) showingtheir advantages and drawbacks.

JWST

JWST is a large space telescope, optimized for infrared observations, scheduled for launch in 2014. Its goals are tofind first galaxies, planetary systems formation, and evidence of the reionization ([36]) during a five-year mission butwill have enough fuel to run over 10 years. It is due to an international collaboration between NASA, the EuropeanSpace Agency (ESA), and the Canadian Space Agency (CSA). The James Webb Space Telescope was named after aformer NASA Administrator.The telescope will stand in the Sun-Earth Lagrangian 2 point, at 1.5 millions km from the Earth, after a trip of 30days. It is composed of a large mirror, 6.5 meters and offers 4 scientific instruments covering infrared wavelengths:NIRCam (Near Infrared Camera), NIRSpec (Near Infrared Spectrograph), TFI (Tunable Filter Imager) and MIRI(Mid Infrared Instrument). Here we explain most of their abilities in the case of a starshade, at low resolution.

• NIRCam: insuring observations between 0.6 and 5 microns (tow arms, one short from 0.6 µm to 2.5 µm and onelonger from 2.5 µm to 5 µm), with a high sensitivity between 2.2 and 5 microns, the NIRCam is the first cameraof the JWST. It will be used for the search of planetary companions, mainly Jupiter-sized planets, for the searchfor protoplanetary disks, stellar populations in nearby galaxies, and also for the characterization of galaxies atvery high redshift, mapping dark matter. NIRCam is composed of two modules for broad- and intermediate-band imaging where traditional focal plane coronagraphic mask plates will be used and two different wavelengthchannel outputs. Lastly, NIRCam will be used as a wave front control module to check alignment and shape ofthe 18 hexagonal-shaped mirror segments ([37]).

• NIRSpec: also covering from 0.6 µm to 5 µm with different quantum efficiency above and below 1.0 µm(respectively ≥ 80% and ≥ 70%), the Near Infrared Spectrograph uses a 0.2 arcsec slit with resolution of100, 1000 and 2700. Low resolution is advised with regards to sensitivity and exposure time whereas higherresolution could be used for giant planets.

• TFI: the Tunable Filter Imager is, as it is called, an imager. The fact that the light is not dispersed reducesconsiderably the contrast sensitivity, making this instrument inappropriate with the use of a starshade.

• MIRI: the Mid Infrared Instrument is composed of two spectrographs, one for low resolution and on for medium.Giant planets with an external occulter could be the goal of this instrument, however, similar efficiency asNIRCam and NIRSpec is reached for an habitable zone at 400 mas, and there is an impossibility to combinethe low resolution spectrograph with filters. Theses particularities jeopardize a sufficient efficiency in the searchof exo-Earths with the use of a starshade.

A starshade for JWST is one of the New Worlds Observer mission ([38] & [39]), in supplementation of THEIA ([40]),CESO ( Celestial Exoplanet Survey Occulter [41]) and O3 (Occulting Ozone Observatory [42]) which all use apodizedof binary occulters, optimized for a wide variety of wavelengths. JWST by itself will require the help of an externalocculter to be capable of directly imaging planets in the habitable zone; the Hubble successor is one of the perfectcandidates for this mission. It would be the fastest and most affordable path to the discovery of life as the resultingcost of this kind of mission have been estimated at about 1 billion dollars ([43]).

9

Figure 1.8: James Webb Space Telescope, with the on-axis primary mirror of 6.5 meters diameter composed of 18hexagonal mirrors. Figure credit: NASA

The starshade will be launched after the JWST (in the best case, 6 months after). The starshade will be in orbitaround the Sun-Earth L2 point. Due to the large distance between the occulter and the telescope, it has to covermany thousands of km for each star. In the 5 year planned mission, almost one week is required between each starobservation for the travel, 24 hours for imaging and 2 weeks for the spectroscopy science. There is a Design ReferenceMission (DRM, [34], [44] & [45]) built for optimizing the time travel and the number of discoveries based on thefrequency of Earth-like planets. Thus the DRM would maximize the number of planets discovered, their spectralcharacterizations, productions of orbital fits. At the same time it would maximize the percentage of the targetlist observed. At the end, for an occurrence rate of planets of 0.3, there would be 5 habitable Earth-mass planetsdiscovered for a small fraction of JWST observing time (say 7%) and the probability of zero discoveries would be0.004 ([45]).

Figure 1.9: Route followed by the starshade to join the Sun-Earth Lagrangian 2 point, for a case of a 50 meterocculter at 50000 km, launched 3 years after the space telescope. Figure credit: W.Cash et al.

10

Chapter 2

Starshade: design, optimization andproperties

2.1 Design and optimization

2.1.1 Free-Space propagation from starshade to telescope

As the name tells us, an external occulter, also called a starshade, obscures the light coming from a star. We placea large occulter in the path of the light between the star and our telescope. The latter has to remain in the shadowproduced by the occulter. We manipulate the design and position of the occulter to have a control over the size ofthe shadow, and also over the contrast between both stellar and planet luminosity.Our purpose is to get the expression of the light intensity, traveling in finite distances. One of the unwanted effectsdue to a circular disk mask is the Poisson’s spot, a bright spot in the center of the shadow which irradiance is nearlythe same as without any occulter ([46]). It results in a diffraction pattern dependent on size, shape and distance ofthe starshade relative to the telescope.On paper, we work with the electric field in the telescope’s pupil plane to express the light observed since its intensityis given by the squared modulus of the electric field. We start with the Babinet’s theorem ([47]). The light propagatingfrom an unobstructed star (Eu) is the same as the light coming from an on-axis hole (Eh) plus the light coming fromthe complement of that hole (Eo):

Eu = Eh + Eo.

Next we can write the plane wave equation in terms of the polar coordinates (ρ, φ) of the telescope pupil plane, (ρ=0being the center of the plane and ρmax the top)

Eu(ρ, φ) = E0e2πizλ ,

with z being the distance between the occulter and the telescope. As previously said, a circular-shaped screen cannotstop the light of a star without Fresnel diffraction effects. In this way, Spitzer ([29]) built up a way to suppressthe Poisson’s spot with the help of an apodization function A(r, θ), meaning that we consider the occulter beingpartially attenuated, with r and θ, the polar coordinates of the occulter. As we assume circular symmetry, we haveA(r, θ) = A(r) (same result for φ). This function is equal to 1 for total obscurity and 0 when all the light propagates,so that it describes the whole occulter. All these tools combined, we have ([46]):

Eo = Eu − Eh,Eu(ρ) = E0e

2πiz/λ,

Eh(ρ) = E02π

iλze2πiz/λeπiρ

2/λz ×∫ R

0

J0(2πrρ

λz)A(r)e(πi/λz)r

2

rdr,

with J0, Bessel function of the first kind, order 0, and the Fresnel integral being Eh(ρ), which gives us the total fieldfor the occulter-telescope system:

Eo(ρ) = E0e2πiz/λ

(1− 2π

iλz×∫ R

0

A(r)J0(2πrρ

λz)e

πiλz (r

2+ρ2)rdr

).

11

2.1.2 Binary apodization, shape of the occulter

A continuous graded occulter cannot be built. The idea has been proposed by Spitzer ([29]) to build a binary occulterwith a workable shape. Thus we approximate our continuous apodization by a binary occulter composed of N evenidentical petals arranged around a circular central part ([31] then [38]). We use a similar new expression of theimage-plane electric field

E(ρ, φ) =

∫ ∫S

e−2πiρ cos(θ−φ)rdrdθ

with S being the mathematical description of the petal-shape mask. Using among others the Jacobi-Anger expression([48]), we compute the following result for a propagated field with an occulter ([49] & [50]):

Eo(ρ, φ) =

Eo(ρ)− E0e2πiz/λ

∞∑j=1

2π(−1)j

iλz

(∫ R

0

eπi(r2+ρ2)/λzJjN

(2πrρ

λz

)sin(jπA(r))

jπrdr

)∗ (2 cos(jN(φ− π/2)))

with Eo(ρ) being the previous electric field for a graded apodization, JjN the jN th order Bessel function whose effectexponentially decreases for high N and j > 0 so that Eo(ρ) becomes predominant. Indeed, terms in the sum over jconverge to 0 quickly enough so that our optimization codes only require writing Eo(ρ) ([31]).Next, the translation between the apodization function and the final shape of the occulter is easy: the angular width∆θ(r) will be expressed as:

∆θ(r) =2π

NA(r)

where N , the number of petals. Hence, the width of the petals, R∆θ, is directly mapping the apodization functionwith R being the radius:

∆ = R ∗∆θ = R2π

NA(r)

If N →∞ we result in a graded occulter. However to keep a range for a feasible occulter, calculations of the averagesuppression over the shadow profile on the telescope for different wavelengths have shown that 16 petals were a goodcompromise ([51] & [38]).

2.1.3 Optimization of the occulter apodization function

Among the numerous variables, we can fit the best occulter based on the physical constraints. As previously said,we need to get a contrast (or a suppression) of about 10−10 in the focal plane. A starshade can be optimized fordiameter, suppression level, wavelength range, shadow size, petal length and inner working angle (IWA). Other linkedvariables, like the telescope aperture size itself, can also be changed in the research field of adding external occultersto general space telescopes.

Parameters

To shape an occulter, there are several parameters which come into play. Here we describe the main characteristicsof the occulter and we will see in the result section how they behave together.The diameter is one of the most important features. As the suppression increases with the size, bigger occulterswould be more favorable. However, if the contrast increases with the size for a constant geometric IWA, so does thedistance, the cost and the time for moving it. A good frame size for an occulter is 60 - 100 meters in diameters for alarge telescope of the size of JWST since the size of the occulter depends on the size of the shadow. As of now, anocculter larger than 80 meters would encounter technological issues.Then, with the addition of a binary apodization function, we establish a difference between a circular central partand the petals. Their lengths modify the suppression achieved and they are also subject to deployment concerns, i.e.longer petals would be harder to bloom and control. Decreasing the size of the petals will reduce the suppression fora reasonable number of petals, as we come closer to the circular occulter without apodization.The size of the shadow is important too. If we need to suppress the light coming from the star, the shadow provided bythe occulter needs to be large enough to cover the telescope aperture over the wavelengths studied. The shadow needsto have at least a 1 meter margin to make sure we achieve the required contrast ([52] & [43] and see section 2.2.6).Even if we first consider the contrast as a goal, it can also be considered as a parameter we get after optimization.How much contrast we get as a function of the shape will be explained in depth in the section 2.2.3.The Fresnel number is defined by

F =D2

λz

12

with z, the distance of the occulter. For a given Fresnel number, the form of the shadow, or the contrast, created bythe occulter will remain the same. Thus, for a defined wavelength, we are able to establish a proportionality betweenthe distance z and the diameter D. Moreover, since the inner working angle equals IWA = D/2z, we have:

F =2 ∗ IWA ∗D

λ

Thereby, the science we are looking for will set the range of parameters describing the starshade, and by taking aFresnel number, we can figure how IWA, diameter and wavelength are correlated. To keep the same contrast, thefresnel number needs to be identical. For a constant diameter, λ ∗ z remains constant. Therefore the starshade canbe used for observations at longer wavelengths by moving it closer to the telescope (this in turn increases the IWA).The range of wavelengths we look at matters in our case of a starshade for JWST. Spectral characterization of the life-signatures like O2, O3, H2O, CO2, or CH4 are optimally found between 0.7 µm and 2 µm. In the case of the JWST,NIRSpec and NIRCam detectors work best between 0.6 and 5 µm, with different quantum efficiency depending onthe wavelength. However outside the optimal band pass, especially close to the red above 2 µm, the starshade startsto leak starlight, reducing considerably the contrast. This light takes the form of speckles in the focal plane and thusthe use of filters with good out-of-band rejection is required to suppress enough this red leak ([43]).In summary the basic parameters for the starshade are

• occulter diameter

• petal length

• inner working angle

• distance (related to occulter & IWA)

• wavelength range

• shadow size.

We next use them as the ’x’ values of the apodization function and write the physical equation of the electric fieldso that we can easily manipulate these parameters.

Analytical optimization

A useful tool to shape the petals through the apodization function is the hypergaussian function. It has been usedand set up to be the best mathematical expression for the apodization. Developed by Cash ([38]) the function isbased on the following expression:

A(r) =

1 ∀ r ≤ a,

exp

[−(r − ab

)n]∀ r ≥ a,

where a is the radius of the central part of the occulter and therefore b is the complementary distance which gives usthe radius of the occulter, i.e. the petal length. As the Hypergaussian is a mathematical function, the exponentialis endless and the definition of the inner working angle has been provided by Cash at AIWA = 1

e , corresponding to atransmission of almost 63%. a and b are given intrinsically by the value of the occulter central part (OCP ) and theocculter diameter (D): a = OCP ∗D and b = (1−OCP ) ∗D. n is a parameter for the petal shape set to the value6 ([52]). The hypergaussian function presents the advantage of being an analytical application reducing, the timeof calculation. This is in contrary to the following optimization which requires minimization of calculation paths.Moreover, the hypergaussian is independent of the number of wavelengths since it is monochromatic. For each steprequiring the wavelength, the average value λ = 1.7 µm is taken, corresponding to the maximum of the broadbandof interest. For smaller wavelengths, the contrast will be better (see figure 2.5).Another function given by Copi & Starkman ([53]), is based on the transmission function τ(r) which equals 1−A(r).This transmission function expresses the expansion of the diffraction pattern through Chebyshev polynomials. Theywrote the transmission function as

τN (y) =

N∑n=0

cnyn

where N the order of the occulter and with

y =(r/R)2 − ε

1− ε

13

where ε is the fractional radius of the center of the occulter, which gives us, for a four-order occulter:

A4(y) = 1− τ4(y) = 1− (35y4 − 84y5 + 70y6 − 20y7).

They took ε=0.15 in their consideration. It turns out that the hypergaussian function is similar to the first order ofthe Copi & Starkman development, which makes it a harder but analytical way of optimization.I wrote a code to generate starshade profiles with the hypergaussian function which calculates the contrast for arange of parameters. I integrated the calculation with the existing functions for numerical optimization described inthe next section, in order to compare both approaches.

Numerical optimization

The main numerical way to get an optimized apodization function has been developed by Robert J. Vanderbei [51]using linear programming. This mathematical operation, also used in economics, management or engineering, appliesthis following operation:

Minimize: (c ·A)

Subject to: (m ·A)

∣∣∣∣∣∣∣≥=

≤b & A

∣∣∣∣∣∣∣≥=

≤d.

with c and A vectors, m a matrix, and b and d vectors corresponding to the constraints.In our case, the goal is to get the best occulter shape by using Fourier optics for the system telescope + starshade.To do so, we reproduce the approach used by Vanderbei et al. ([31]) and constrain the intensity ratio of the light overthe pupil plane to be less than 10−10. As the intensity is not linear, we settle the matter by expressing the constrainton the related electric field so that:

|Eo(ρ)|2 ≤ 10−10|Eo|2

with

Eo(ρ) = E0e2πiz/λ

(1− 2π

iλz×∫ R

0

A(r)J0(2πrρ

λz)e

πiλz (r

2+ρ2)rdr

).

We can already see that the first exponential e2πiz/λ disappears in the calculation of the modulus. As written above,the electric field is described by the apodization function A(r). We use linear programming to minimize the sumof the apodization function, which is described in matrix format by the scalar product c · A(r), with c, a simpleunit vector, so that c · A(r) = A(r). This objective function has little impact on the results in this case, becausethe contrast goal will be placed on the constraints (together with other constraints described below). Since Eo(ρ) iscomplex, the assumption is taken that the constraint

Re(Eo)2 + Im(Eo)

2 ≤ 10−10

will correspond to −10−5√

2≤Re(Eo) ≤

10−5√2

−10−5√2≤Im(Eo) ≤

10−5√2,

with the amplitude E0 removed and the√

2 coming from the constraint expression scaled to a circle around 10−5.Thus, we get a system of four inequalities. In order to make it linear for optimal solutions ([54]), we write in the code

the integral inside Eo(ρ) as the sum of all the area elements under the curve∑i J0( 2πriρ

λz )eπiλz (r

2i+ρ

2)ri ∗A(ri), like aRiemann sum approximation of the integral. For convenience, we write

J0(2πriρ

λz)e

πiλz (r

2i+ρ

2)ri → φi, χi

so thatRe(Eo) ∝ φi ∗A(ri) & Im(Eo) ∝ χi ∗A(ri).

Moreover, we can add constraints on the apodizer itself. Firstly, by definition, A(r) will be bound by 0 and 1.Secondly, we impose A(r) to be equal to one for the central part of the occulter (the fully opaque central disk).Thirdly, in the monochromatic case, the natural solution of such an optimization is a ”bang-bang” solution which is adiscontinuous function (like a bar-code). In order to avoid this problem we add a smoothness constraint σ bounding

14

the second derivative of A(r). We also add a constraint on the first derivative so that the petal width decreasesmonotonically (the width of the petal is directly proportional to the apodization function). We finally have for A(r):

A(r) = 1 ∀ 0 ≤ r ≤ a,0 ≤ A(r) ≤ 1 ∀ a ≤ r ≤ R,

A′(r) ≤ 0 ∀ 0 ≤ r ≤ R,|A′′(r)| ≤ σ ∀ 0 ≤ r ≤ R.

The first and second derivatives are expressed using respectively the finite difference expressions:

limh→0

A(r + 1)−A(r)

hand lim

h→0

A(r + 1)− 2A(r) +A(r − 1)

h2.

Here, h will correspond to the number of points n taken for the calculations, so that limh→0 will be limn→∞. Thelinear programming formalism allows to combine different constraints. For that, we simply combine all the constraintsin one large matrix:

Minimize: c ·A(r) = ( .. 1 .. ) ·A(r),

Suject to:

m·A︷ ︸︸ ︷.. Re(Eo) ..

.. Im(Eo) ..

.. A′(r) ..

.. |A′′(r)| ..

b︷ ︸︸ ︷∣∣∣∣∣∣∣∣∣∣≥ and ≤ ±10−5/

√2

≥ and ≤ ±10−5/√

2

≤ 0

≤ σ

&

∣∣∣∣∣A(rr<a)

A(rr≥a)

∣∣∣∣∣ = 1

∈ [0, 1].

and it returns a table of the function A(r) in the case where the data set can converge to a solution. If not, thecalculation stops and returns an error message.As Eo depends on the wavelength, the calculations have to be done over a sufficient range of λ covering the bandpasswe are looking at, meaning that all the constraints are also repeated for each wavelength.

The contrast as a parameter

I used the first optimization code described above to generate starshade designs for a large range of parameters. Thisapproach is appropriate to design a starshade for a specific goal, but for a systematic study of the parameter spacethe optimizer may or may not deliver a result; e.g. there may not be a possible starshade profile achieving 10−10

suppression for a given set of constraints (diameter, IWA, shadow size, petal length). Given an existing code for thebasic optimization of starshade described above, I wrote a new optimization code that includes the contrast in theobjection function of the optimizer. This is described below following a method described by Cady ([35]). Next, Ichanged the code to add the contrast as the result of the optimization, no longer constraint.Starting from:

Re(Eo) ≤10−5√

2& Re(Eo) ≥

−10−5√2

Im(Eo) ≤10−5√

2& Im(Eo) ≥

−10−5√2

,

the square root of the contrast 10−10 becomes k, a parameter, such as:Re(Eo) ≤

k√2

& Re(Eo) ≥k√2

Im(Eo) ≤k√2

& Im(Eo) ≥k√2,

The trick is to remove the contrast of the constraints. To do so, we write:Re(Eo)−

k√2≤ 0 & Re(Eo)−

k√2≥ 0

Im(Eo)−k√2≤ 0 & Im(Eo)−

k√2≥ 0,

Then, the vector expressing the apodization requires the addition of one term, the contrast k. We rewrite the matrixto express Re(Eo)− 1√

2so that:

(.. φi, χi ..

−1√2

)·

..

A(ri)..k

=∑i

φi, χi ∗A(ri)−k√2

15

and the last modification will be in the scalar product minimized: c · A(r). As we need to get the best contrast, wewrite:

c such as

c1..ci..cn

·A(r1)..

A(ri)..k

minimized, give us the minimum value of k, i.e. c =

0..0..1

.

Constraints dealing with the first and second derivatives of the apodization function just need to be resized to thedimension of A(r) and we burke the suppression by putting a 0 inside the new vectors. With this scheme, the sum ofthe apodization profile is no longer minimized. This does not impact the result since the optimizer is entirely drivenby the constraints in this case. With this new version of the code, the optimizer delivers a result whatever the set ofconstraints may be. The output of the optimizer is both the apodization profile and contrast, concatenated in a longvector. This code is more adapted for the parameter space study we describe in the next section.

Creation of widths at tips and gaps

Another improvement I made is creating a concrete expression of the apodization function. The numerical solutionshave limited size petals, corresponding to the size of the array used for A(r), however the width of the tip of thepetal is unconstrained and may reach non-realistic values. Analytical optimization has endless petals that need tobe truncated at some radius, which is studied in section 2.2.2. The problem is the same for the ”valleys” betweentwo petals. The purpose is to create a width characterizing both tip and gap size at the same time (they can takedifferent values). In order to do so, we add two lines of constraints ([35]): one will ask the translation of apodization

Figure 2.1: Positions of the tip and gap on an external occulter. We want to build width of the order of the millimeter,and as the petal size is more than 10 meters, gap and tip widths will not be observable.

function in width at the bottom of the petal (r1) to be 1 minus the gap, ∆gap. The second one will ask it to beequal to a thickness ∆tip at the edge (r2). We use ∆ = R∆θ(r) = R π

NA(r) (here, the factor of 2 disappears as we

deal with half of the petal) to write the following new constraints:Rπ

N∗ (1−A(r1)) ≥ ∆gap

Rπ

N∗A(r2) ≥ ∆tip

− πN∗A(r1) ≥

∆gap

R− π

N

π

N∗A(r2) ≥

∆tip

R

i.e., in our matrix:

(.. 0 ..

r1︷︸︸︷− πN.. 0 ..) ·A(r) ≥

∆gap

R− π

N

(.. 0 .. 0 .. ..

r2︷︸︸︷π

N) ·A(r) ≥

∆tip

R,

16

As we ask the function to make a leap in two points, the smoothness constraints might be relaxed to allow the op-timization convergence. Here, I wrote it by nullifying the expression of the second derivative on three points aroundthe gap so that the discrepancy is not taken into account.Of course, we can combine both improvements in order to get the best suppression for these new petals. A realisticnumber for manufacturability would be a minimum size of 2 mm for these features. This lower limit is sufficient forthe tip and gap widths, without changing the efficiency of the starshade as we can observe in figure 2.2 (this is easyto imagine when we think about the size of a petal, between 15 and 20 meters). More calculations have to be made in

0 1 2 3 4

1. ´ 10-10

2. ´ 10-10

3. ´ 10-10

4. ´ 10-10

Width HmmL

Con

tras

te

Figure 2.2: Contrast as a function of the width at tips and gaps (both the same) for an occulter of 75 meters indiameter, at 100 mas and with an occulter central part of 50%. The contrast remains under the 10−10 requirementat 3 mm and starts going up exponentially after, exceeding 10−10. Here the changes in the program made the startat 0 mm different of 1 · 10−11 between the original one, creating a small discrepancy in the results.

order to see the difference in behavior between the tip and the gap over all ranges of diameter, petal length and innerworking angle. Following similar logic, the addition of tensioning elements and some changes in the petal structurehave been used to see how the contrast varied between them ([49]). However, in general, the more constraints weadd to the problem, the larger the starshade becomes.

17

2.2 Global study of the parameter space and results

2.2.1 Goal

We now have enough tools to describe the starshade properties giving the suppression we are looking for. We runmany calculations to explore the abilities of the starshade over the range of parameters describing the stellar system.The inner working angle will take values from 80 to 120 mas, corresponding to most of the range of the habitablezone. We start with starshade diameters of 60 up to 100 meters. 100 meters would certainly be unrealistic, but weinclude a large range of diameter to understand the behavior of the starshade according to its design parameters.The distance between the occulter and the telescope is between 50,000 km and 100,000 km. Next, the petal lengthvaries between 30% and 70% of the total size. More than 70% would be unfeasible and less than 30% would makethe occulter come closer to a one without apodization. The shadow size will take values from -1 to 8 meters (for themargin in comparison to the telescope size). At first, the contrast, as a constraint, will be set at 10−10 as planned.Then, we want to optimize this value in regards to all the others.

About time of calculation

Each creation of an apodizer profile for a given set of data, in the case of numerical optimization runs between 10and 20 minutes depending on the precision (number of points, constraints). Typically we use 4000 points alongthe apodizer profile, 11 wavelengths and 11 points along the shadow profile at the telescope aperture. For figureslike figure 2.11, figure 2.9, or figure 2.6, the diameter takes almost 8 values, the inner working angle 5 values, thecentral part 9 values, which gives us almost 360 different panels. Thus we have 360*20*60/3600 = 120 hours ofcalculation. This long time can be reduced by decreasing the number of wavelengths, or the number of points acrossthe apodization profile A(r), and the number of points across the shadow profile at the telescope aperture. Howeverthis will significate less accurate values (it gets a useful practice only for testing the programs). Moreover, trying toget an higher accuracy or changing the values of the shadow oversize, gap and tip sizes, or any other parameter (likechanging the step of the diameter to 1 or 2 meters) will increase considerably the time of calculation.On the contrary, since the hypergaussian function proceeds to take an analytical optimization so that the apodizationis already known, we only calculate which contrast it returns. Because the hypergaussian is always better at shorterwavelengths, we calculate the monochromatic contrast at the longest wavelength of the band. This monochromaticcharacter also reduces the time and makes the hypergaussian easier to manipulate. Each calculation takes less than aminute mainly limited by the Fresnel propagation from occulter to the aperture. However, for the second method ofcalculating an analytical optimization, seen in 2.2.7, we need to compute a complete numerical apodization so thatit takes as much time as the numerical solution.

2.2.2 First comparison between analytical and numerical methods

For similar properties, we have a look at the behavior of the two optimization methods for the occulter’s shape.Figure 2.3 shows the differences between linear programming and hypergaussian in the apodization function. We

0 1000 2000 3000 40000.0

0.2

0.4

0.6

0.8

1.0

Radial position

Occ

ulte

rpr

ofile

Analytical

Numerical

Figure 2.3: Apodization function for numerical approximation with linear programming and analytical optimizationwith the hypergaussian function for an occulter diameter of 75m, an IWA of 100 mas, and a central part of 50% inthe case of a JWST-like telescope.

can see that there are some steps on the numerical apodization. We also note the hypergaussian function is fitted

18

to the constraints so that it goes down faster than the numerical. Moreover, because the hypergaussian function isexponential, it never reaches the 0 value, but reaches very small values quickly as the typical value for the hypergaus-sian exponent (which is characteristic of the fall) is n=6. It is difficult to define where to end the written apodizer.Indeed, in our plot, the true diameter of the analytical apodizer is not known. In order to calculate the IWA, Cash([38]) defined the diameter for a transmission to be up to 1 − 1/e. In this way, we understand why the differenceis well marked. A 50 meter diameter defined in this way would actually be a 60 meter diameter, tip to tip ([38]).However, the propagation takes into account all the apodizer profiles up until a zero value. The next plot 2.4 showsus how the contrast varies when we expand the diameter to its true value. To do so, we make a run over a coefficientwhich extends the radius for the propagation calculation. We start from 1 up to 1.6.

As we can see, the radius needs to be multiplied at least by 1.3 to satisfy the constraints and provide a good

60 65 70 75 80 85 90 95

10-6

10-4

0.01

1

True Diameter HmL

Con

tras

t

Figure 2.4: Behavior of the contrast as a function of the true diameter. We multiply the value of the radius by acoefficient which will express how much longer the true diameter is (in the case of a 60 meter diameter). Beyond acertain diameter the truncation of the hypergaussian has virtually no effect (here beyond 75m). Therefore, we use acoefficient from the 1/e diameter to include the entire tip of the hypergaussian. Since the propagation is calculatedfor the entire array, the result does not depend on the coefficient value.

suppression (here, the 60 meters occulter made with the analytical optimization provides 10−8 suppression). Theocculter diameter will proceed from 60 to 78 meters (and so on for bigger starshade).As previously said, the hypergaussian function is monochromatic. Figure 2.5 shows us how the contrast changes whenwe change the maximum wavelength. Clearly, increasing the range of the spectrum, and therefore the wavelengthof interest, will damage the contrast quickly and thus a bigger starshade will be required. Later in section 2.2.7, wewill create the hypergaussian in another way by fitting a numerical optimization to it in order to find a consistentdefinition for both approaches.

0.8 1.0 1.2 1.4 1.6

5 ´ 10-12

1 ´ 10-11

5 ´ 10-11

1 ´ 10-10

5 ´ 10-10

Maximum Wavelength HΜmL

Con

tras

t

Figure 2.5: Contrast of the hypergaussian apodization as a function of the wavelength. Smaller wavelengths willprovide a better suppression, and increasing the wavelength will deteriorate the contrast in a logarithmic way.

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2.2.3 Contrast as a function of the diameter

Here, we have a look at the general behavior of the contrast subject to the variations of diameter at different innerworking angles for a constant central part size of the occulter. We make an imposing run over the whole range ofdiameters and inner working angles.

In figure 2.6, we can see a logarithmic behavior of the contrast. As the inner working angle gets smaller, the

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Figure 2.6: Values of the contrast as a function of the occulter diameter for different inner working angles. Left:obtained with the numerical optimization. Right: obtained with the help of the hypergaussian function. For thenumerical optimization, we notice that up to 85 meters would be sufficient to get all the range of inner working angleswanted.

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Figure 2.7: Left: Shape of the apodization for a similar contrast between numerical and analytical optimization at62m. Right: their comparison in suppression at various diameter, overlaying at 62m. Very early, the numericaloptimization becomes more efficient than the other one, with a peak at around 85 meters. At 75m, the numericalsolution is nearly 9 times better (Contrast for the numerical: 4.383·10−11 and contrast for the analytical: 4.301·10−10).Similarly, the 75 m numerical solution has a contrast of 4.383·10−11; this contrast is obtained for a diameter of almost93 meters with an analytical solution.

suppression requirement of 10−10 is achieved for high values of diameter. On the contrary, high inner working angleswill easily reach the suppression for smaller diameters. This is consistent with the normal thought: bigger occulter= higher suppression = smaller inner working angle.In the second plot of figure 2.7, we have the two methods for a given inner working angle (here, 100 mas). We can seethat for deeper contrast, the difference between the hypergaussian and linear programming apodization gets higher,and for the 10−10 requirements, we have a difference of almost 12 meters in the telescope diameter. The following isthe evidence of the advantage of a numerical optimized apodization-built occulter in comparison to the hypergaussianfunction. If we increase the value of the inner working angle, the separation between the two methods will go down,for as much as 8 meters for 120 mas. However, at 100 mas, an occulter smaller than 62 meters would be preferablybuilt with the help of the hypergaussian function. The first plot is similar to figure 2.15 and expresses how the shapeof both apodizations will look like for a similar contrast (our 62 meter occulter in that case). It is interesting to seethat even for a similar contrast, the shape is actually quite different. But we have to remember that the size of thisanalytical apodization is defined up until the 1/e transmission point and will in fact correspond to a diameter of atleast 10 meters more (considering the supposed-infinite length of the function).

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2.2.4 Distance of the occulter as a function of the diameter

In this part, we change the variables. Using the fact that the inner working angle, diameter and distance betweenthe occulter and telescope are linked together through IWA = (Diameter D)/(2 ∗Distance z), we get the distancewith:

z(km) =D(m)

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For a given ratio between the central part and petal length (here it is 49%), we select the 10−10 contrast. For eachdiameter, we select the smaller distance between the occulter and the telescope. In figure 2.8, the plot describes

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Figure 2.8: Distance of the occulter for less than 10−10 as a function of its diameter. Left: by numerical optimization.Right: by analytical optimization. In the case of the analytical apodization, each time the diameter is not plotted, itmeans that a design was not generated for this set of constraints (too small of a diameter).

the distance from the occulter to the telescope required to reach 10−10 suppression. There are two similar behaviorsbetween the numerical and analytical optimizations but with different configurations of data. We can see for example,an angular resolution of 100 mas in the extra-solar system will require an occulter of about 89 m at a distance of90,000 km for the hypergaussian apodization. In comparison, the numerical apodization requires an occulter of about75 m at 75,000 km. Calculations of the distance are made in regards to the formula IWA = D/2z. By translatingthe curve, we clearly see how much we gain in the occulter size and distance: almost 15 meters in diameter and 15000 km in distance in average.

2.2.5 Action of the petal length on the contrast

Using the basic optimization scheme with the contrast as a constraint, the program stops each time the constraintscannot be satisfied. This method presents the advantage to directly show the limit of the starshade. However, wedon’t have any information about what would be the best contrast for this set of constraints. Here, we make a runover different sizes of petal and occulter in order to see how behave these two parameters together. In figure 2.9,

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Figure 2.9: Minimum of the petal length ratio (first plot) and petal length (second plot) as a function of the diameterin order to reach the suppression requirement of 10−10. Increasing the size of the diameter means decreasing thepetal length, which is more practical for the starshade’s engineering.

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we can see that the binary apodization requires smaller petals when we increase the size of the occulter. This can beunderstood when we think about the circular continuous apodization. For a perfect occulter, it would be necessaryto have an infinite number of small petals. This effect is advantageous for practical purposes. It is easier to build ahuge opaque central part and deploy small petals over it. A reasonable frame locates the petal length between 15and 25 meters, which corresponds to a ratio of 50% (the value of the occulter’s central part will be fixed on someprevious and future calculations at 50%).The next two figures show us numerically optimized starshades for diameters of 65 and 75 meters (see figure 2.10).Both are built with 16 petals.

65 m tip to tip, IWA=100mas, petals=24m D=75 m, IWA=100mas, petals=19.5m

Figure 2.10: Optimized Starshade for a diameter of 65m and 75m.

Here, with the help of the new apodization function giving us the value of contrast, we examine the behavior of thesuppression according to the full-opacity occulter size (figure 2.11). We make the run over the range of 30% - 70%,for an inner working angle of 80, 90, 100, 110, 120 mas, and diameters from 60 meters to 100 by steps of 5. There isa big difference between the two methods. Even if both are following a logarithmic scale, the hypergaussian presentsrelatively constant suppression between 40% and 55% whereas the suppression keeps on going down as the centralpart is reduced in the case of numerical optimization. As previously seen (figure 2.9), a bigger starshade will requiresmaller petals, in other words, a bigger central part. 50% seems to be a good value for the next calculation in thecase of an occulter with a 75 meter diameter.In figure 2.12, for different inner working angles, we plot the contrast as a function of the petal length. In the case

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Figure 2.11: Suppression as a function of the OCP (occulter central part) for different Inner Working Angles with adiameter occulter of 75 meters. Left: by the numerical optimization. Right: by the analytical optimization. Clearly,there is no similar behavior, and we note this especially before 40% of the central part. The hypergaussian apodizationpresents a constant part whereas the numerical is monotone.

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Figure 2.12: Contrast as a function of the petal length for an occulter diameter of 75 m. Left: by the numericaloptimization. Right: by the analytical optimization. As the central part and petals are linked together, we regainthe same behavior as previously shown and confirm the range of petal lengths we defined before.

of the hypergaussian, we recognize the plateau seen in figure 2.11 as the petal length of maximum contrast whichgives a similar efficiency for a few meters. Increasing the IWA will increase the length of the constant part, depth ofthe contrast and minimum petal length. For the required parameters at 100 mas, a petal length of more than 19 mis necessary. In these cases, a petal length of 19 m for an occulter of 75 m results in a central part of almost 50% andfor the 85 m, 56%.These plots show us the limit of the analytical optimization through the hypergaussian function. As the shape willremain pretty much the same for huge variations of parameters, the hypergaussian function will not be the best fittingof the numerical optimizations. In that case, changing the n parameter would be a way to get round the problem byreducing the speed of fall.

2.2.6 Action of the Shadow Oversize

The shadow oversize is the parameter describing how big the shadow is in comparison to the telescope diameter. Wecan wonder how the contrast will be change for different margins chosen. First, we take the numerical optimization.We run the calculation for the JWST with a diameter of 6.5 meters and we look for the contrast as a function of theocculter diameter, the inner working angle and the shadow size (from 5 meters to 15 meters, so a margin from -1.5to 8.5 meters). In these plots, the ratio between petal lengths and occulter size is set up at 50%. For each size of theshadow, we look at the smallest occulter diameter for which we get our required contrast of 10−10 (see figure 2.13).We note the contrast is better past the constant part of the graph. We also note 10−10 is reached for some values of

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Figure 2.13: Occulter diameter as a function of the margin size of the shadow for which we have 10−10 suppressionfor 2 IWA, 110 and 120 mas (chosen for their obvious behaviors as smaller IWA displays the same plateau and thengrowth, but for higher diameter the plateau is longer).

the shadow oversize before the optimization cannot get the required contrast. 3 meters of margin seems to be enoughfor an inner working angle of 120 mas, confirming our supposition of 1 meter. Increasing the shadow oversize bymore than 3 meters will make the contrast worse, so that a bigger starshade will be required. Then, the size requiredfollows a linear approximation: increasing the diameter of 1 meter would be a solution if we want to increase the

23

shadow oversize of 1 meter too.Now, with figure 2.14, we look at how the contrast behaves as a function of the margin for the case of a 75 meterocculter at 100 mas, numerically and analytically optimized. In the case of numerical optimization, the growth of the

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Figure 2.14: Contrast as a function of the margin of the shadow oversize. Left: by numerical optimization. Right: byanalytical optimization. Both are for a 75 meter diameter, at 100 mas. The contrast for the numerical optimizationfollows a logarithmic behavior all along the size of the shadow, whereas the analytical optimization displays a constantpart. Finally, the contrast grows up in a logarithmic way.

variation of contrast is logarithmic and the 10−10 limit is reached for almost 3 meters (so a shadow size of 3+6.5=9.5meters). On the contrary, with the hypergaussian, there is a constant part where the contrast doesn’t change, andafter 5 meters of margin (→ 11.5 meter shadow), the contrast gets worse. At this limit, the contrast for a 75 metersstarshade is almost the same for both methods. We can conclude that increasing the shadow size will damage thecontrast so that the performance of a numerically-built starshade will be the same as the analytically-built starshade.

2.2.7 Another way to define the HyperGaussian function

We defined previously the hypergaussian apodization function through the parameters a and b as being respectively:OCP ∗ D and (1 − OCP ) ∗ D ([55]). This was the base of all the calculations we made earlier. Now, let us lookat a different method. Instead of looking for the hypergaussian function with the same parameters as the numericalapodization, we first create a numerical apodization and search for the best analytical fit.To do so, the steps are the following:

• Creation of the numerical apodization function. We keep using the optimization which minimizes the contrast.We then separate the given values of contrast from the rest of the apodization function.

• We create a function depending on the parameters a and b which plots an hypergaussian with the same diameterand points as the numerical apodization.

• We minimize the physical distance between the two curves following

minimizing a and b such that∑i

(Apodizernum −ApodizerHG)2 is minimum.

• Now that we have the best value of a and b, we calculate the new analytical starshade to be built.

Now, for a given numerical optimized starshade, we are able to compare the contrast to its fitted analytical starshade.For small occulters, the shape of the petals looks a lot like the hypergaussian function. As we see on the top left plotin figure 2.15, for an occulter with a diameter of 60 meters, the two plots are very similar and the differences reach0.7%. However, increasing the diameter will stretch the petals so that a hypergaussian will not be able to fit properlythe apodization function. We see on the bottom left plot in figure 2.15 that there are some margins between bothoptimizations up to 5%.Since we have a hypergaussian function which takes a leaf out the numerical optimization, we calculate the corre-sponding contrast we get at different occulter diameters but with all the same parameters. The figure 2.16 shows usthe suppression we reach as a function of the occulter diameter at 80 mas, 100 mas and 120 mas for the numericaloptimization and its corresponding analytical optimization. Clearly, there is a huge gap between both methods. Thesum over the small errors for the petal shapes is enough to create a difference of 4 orders of magnitude. For 80

24

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Figure 2.15: Numerical optimized apodization functions with their analytical doppelgangers and close-up on thedifference between the two curves for each. On top, case for an IWA of 80 mas and an occulter diameter of 60 meters,whereas case for 120 mas and 100 meters below (the extremes). For a 60 m diameter and 80 mas, the fit is goodenough to make both of the numerical and analytical apodizations indistinguishable in the first plot. As we can see,the differences are below 1% in the second plot. For a 100 m diameter and 120 mas, the hypergaussian cannot fit aswell as before the numerical optimization and the differences are up to 5%.

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Figure 2.16: For IWA of 80 mas (blue), 100 mas (violet), 120 mas (green), contrast as a function of the occulterdiameter for numerical optimization (full line) and its new analytical version (dashed line). We notice the predictedimprovement of the contrast with the size of the occulter diameter except at 120 mas for the hypergaussian apodiza-tion. Further comparisons would have to be done to make sure the hypergaussian tips are not truncated in thisprocess.

and 100 mas, we recognize the same behavior: the suppression increases with the size of the occulter, however, for120 mas, the differences with the new analytical optimization are too strong and we get a contrast of almost 10−6.This comparison is not yet completely fair with the hypergaussian apodization. The next step in the comparisonwill be to match the profiles (so that distance and geometric IWA match), and to extend the tip truncation of the

25

hypergaussian function to get to the regime shown in figure 2.4. Indeed it is possible that the comparison truncatesthe function too short and that the contrast is further degraded. However the numbers are consistent with the firstcomparison where the performance is worse for the same parameters.

26

2.3 Conclusion

We described the general methods for the optimization of starshade designs which include numerical (with linearprogramming) and analytical (with the hypergaussian function) approaches. Based on the general calculations of theapodization function, I developed a new code to enable the optimization of the contrast for a numerical starshade. Ialso wrote the corresponding optimization with the hypergaussian function, describing an analytical way of buildingthe shape of an occulter. Finally I added constraints such as minimum sizes for the tip and gap.Using these optimization tools, we studied the parameter space of the starshade for a telescope of the size of JWSTin a systematic way, generally by getting the shape and the returned contrast. Several parameters are involved: theocculter diameter taking a range from 60 meters to 100 meters, the petal length from 30% to 70% of the diameter,the shadow oversize with a few meters of margin, the inner working angle related to the habitable zone, and of course,the contrast itself required to be 10−10 for exo-Earths detection.We have featured some characteristics compatible with previous thoughts (increasing the size increases the contrast,section 2.2.3) and others not expected (bad contrast with a hypergaussian based on a numerical optimization, sec-tion 2.2.7). We also discovered a useful fact in section 2.2.5 where the calculations showed us that increasing the sizeof the occulter will need smaller petals, which are easier to manufacture. However, for a given occulter diameter,increasing the size of the petals will help for a better contrast, for numerical optimization and almost similarly foranalytical optimization (see section 2.2.5). In section 2.2.6, we got a range of values allowed for the margin of theshadow oversize of a few meters and justified the value taken for all other calculations. At last, we made a concretecomparison between the numerical optimization and the corresponding analytical optimizations, not only inside thecharacteristic simulations (like in section 2.2.3) but also in the shape (see 2.2.2) and properties of the starshade (seesection 2.2.7).After all this work is done, we conclude that the analytical optimization through the hypergaussian function is notas efficient as the numerical optimization, but provides us with faster calculations and more manufacturable petals.This can be understood by the fact that linear programming optimizes the values for the given parameters. Theanalytical optimization can still be efficient for small occulters and for small contrasts (around 60 meters, like we seein section 2.2.3) but the issues about the size from tip to tip may counteract this advantage (since we would haveto add 10 meters more). As of now, more calculations are required to see how the fit of the numerical optimizationbehaves (seen in section 2.2.7), but also a complete investigation must be made of the Copi & Starkman function.Indeed, as the hypergaussian function is only a case of the Copi & Starkman function, finding another expression andanother order would be useful for combining numerical and analytical optimizations. It would add more parametersthan the 3 already describing the shape of an hypergaussian function.Using an external occulter turns out to be a promising technology which would provide an entire characterization ofan Earth-like exoplanet. However, due to money, engineering, political and scientific concerns, we cannot establishwhat the design for the perfect occulter would be for the next possible launch. As of now, this is one of the best wayof finding life on other planets. Actions and choices have to be made as soon as possible to make a starshade possiblefor a forthcoming mission, like the James Webb Space Telescope.

Acknowledgments

I would like to give special thanks to Remi Soummer for his welcome, the trust that he placed in me and all thehelp he provided me, Adrien Thormann and Brendan Hagan for making me feel at home and their help with theEnglish (or should I say American) language, Judit Szulagyi, Ignacio Mendigutıa and all the other PhD students orpostdoctoral researchers for the pleasant sharing of the dungeon. Finally I would like to thank all the people fromthe Space Telescope Science Institute of Baltimore for their cordial welcome.

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