Optimal design of petal-shaped occulters for extra-solar planet detection Eric J. Cadya , N. Jeremy Kasdina , Robert Vanderbeib , and Ruslan Belikova a Department b Department

of Mechanical and Aerospace Engineering, Princeton University of Operations Research and Financial Engineering, Princeton University ABSTRACT

One of the proposed methods for directly imaging extrasolar planets is via a free-flying occulter for blocking the starlight. The occulter would fly between a conventional telescope and the target star. It has long been known that a solid occulting disk does not produce a deep shadow; diffraction effects result in a bright spot in the center that would mask a planet. However, utilizing recent results in shaped pupil optimization, we have developed designs for an occulter with a shape that does effectively block the light from the star, allowing the planet light to be seen even at a small angular separation. The shadow created by the occulter is wavelength-dependent and quite sensitive to the shape of the outer edge. We present an optimization approach for producing these occulter designs to meet contrast requirements over multiple wavelengths and also discuss tolerancing requirements on the shade manufacture and control. Keywords: occulter, exoplanets, optimization

1. INTRODUCTION One area currently of great interest in astronomy is the discovery and characterization of planetary systems. Currently, there are 248 known extrasolar planets in 212 systems, with more discoveries being announced every year. [1] Few of these planets have been found by direct imaging, however–most were found by radial velocity measurements, an indirect method that infers the presence of the planet from the motion of the star. Direct imaging of a rocky extrasolar planet in the optical regime is a difficult task, due to the approximately 1010 difference in intensity between the planet and the star, and the small angular separation between the two. The combination causes the PSF of the imaged star to overwhelm the faint signal from the planet. Even if the PSF can be shaped to provide sufficient contrast at the location of the planet, aberrations in the optics will scatter starlight onto the planet’s location. Despite these difficulties, direct imaging has a number of properties that make it attractive. Direct imaging allows spectroscopy, which is essential to characterizing the atmosphere and surface of any planets it finds. While transit observations can potentially, and do, collect spectra, they only work for edge-on disks, while direct imaging could examine face-on systems as well. Direct imaging also allows large swaths of the region about a star to be examined at once, and would be able to examine circumstellar disks. One candidate solution to the problem of direct imaging, which has been proposed in various incarnations over the last decade [2] [3] [4] [5] , is to remove the starlight before it ever reaches the telescope using a separate spacecraft called an occulter. The occulter would be maneuvered between the telescope and the target, and the body of the spacecraft itself would block the incoming starlight. This would not only allow a 360◦ region about the star to be imaged, it would eliminate the wavefront control problem, as the starlight would be too dim to obscure the planet. It would also allow arbitrarily small inner working angles to be achieved, as long as the spacecraft is maintained at a sufficient distance away. Unfortunately, not just any shape can be used, as diffractive effects from the occulter’s edges will tend to throw light into the dark region directly behind the occulter. Rather, the edge needs to be carefully shaped to ensure the scattered starlight tends to destructively interfere with itself in the center of the shadow, creating a dark region where the telescope is located. Cash[4] proposed doing this using a set of N identical structures around the edge of a disk, termed petals, whose radial profile was a shifted hypergaussian function. Vanderbei et al.[6] extended this design to optimize its performance over a large region of the spectrum. Techniques and Instrumentation for Detection of Exoplanets III, edited by Daniel R. Coulter Proc. of SPIE Vol. 6693, 669304, (2007) · 0277-786X/07/$18 · doi: 10.1117/12.734465

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In this paper, we discuss methods of designing the shape of the outside of the occulter to ensure 1010 contrast over a wide spectral band in the shadow with two optimization techniques. We will also discuss how the shadow is altered under errors in position, orientation, and edge shape, and how these affect both the optimization of the occulter shape and the control of the occulter’s location and attitude.

2. OPTIMAL OCCULTER DESIGN The first procedure for creating occulter designs by optimization is covered in depth in here.

[6]

–we review it briefly

To begin, we consider the scalar form of Babinet’s principle, which states that the sum of an electric field due to a hole in an infinite conducting sheet, and the electric field due to its complement, gives the field as if there had been no obstruction at all. This is written: Ehole + Eocc = Eunobs

(1)

where Ehole is the field in the case of a hole, Eocc is the field in the case of an occulter, and Eunobs is the field from unobstructed propagation. In general, we treat the incident field as a plane wave. From this, we can calculate the electric field downstream of an occulter by combining the fields of a propagating plane wave and the due to a hole. This provides significant mathematical advantages for occulters that are small with respect to the distance to the telescope, as the propagation from the hole can be approximated by Fresnel diffraction, and the plane wave can be solved from the Helmholtz equation directly. Before we consider designing an occulter with petals, it is useful to look first at a simpler design–a radiallysymmetric occulter, or hole, whose attenuation can vary smoothly between 0 and 1. For this, we will introduce the function A(r), which represents the light attenuation due to the occulter. (If the occulter is not radiallysymmetric, we generalize this to A(r, θ).) When A = 1, no light can pass through, while when A = 0, the occulter poses no barrier at all. Consider an incident plane wave with electric field E0 . We can then write the electric field in the case of a hole as: (2) Ehole (ρ, 0) = E0 A(r) at the hole, and as:

ikρ2

keikz e 2z E0 Ehole (ρ, z) = iz


0



R

J0

krρ z

 A(r)e

ikr 2 2z

rdr

(3)

at a distance z downstream, after integration over θ due to the azimuthal symmetry. This is combined with the plane wave, which is simply E0 eikz , to give:     ikρ2
R 2z ikr 2 krρ ke Eocc (ρ, z) = E0 eikz 1 − J0 (4) A(r)e 2z rdr iz z 0 A(r) can be chosen analytically; examples include the shifted hypergaussian in [4] and the polynomial representation in [2] . These can provide the desired contrast and inner working angle within certain wavelength bands for appropriately-sized occulters. However, we would like to do better–we want to be able to: • Use the smallest feasible occulter. • Place the occulter at the closest possible distance. • Have the occulter provide the deepest shadow possible. • Have the occulter provide the largest shadow possible. • Have the occulter provide the desired contrast over the largest spectral band.

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To best meet these goals, we optimize A(r) numerically. The first step is to choose how these goals are weighted. For our designs, we note that the science requirements, specifically biomarker spectroscopy and intensity of scattered light from the planet, drive the selection of the wavelength band. Following the recommendations in [7] , we chose 400nm-1100nm for the wavelengths, and 1010 for the depth of the shadow. These values are set as constraints for all of our optimizations; we run multiple optimizations to explore the parameter space for different distances, occulter sizes, and shadow sizes. We solve the following optimization problem: R minimize A(r)rdr 0 subject to

|Eocc (ρ, λ)|2 ≤ 10−10 |E0 |2 , 0 ≤ A(r) ≤ 1,

0 ≤ ρ ≤ ρmax 0≤r≤R

400nm ≤ λ ≤ 1100nm

(5)

where Eocc is as defined in equation Eq. 4, R is the radius of the occulter, and ρmax gives the radius of the dark hole. Here we’ve chosen the cost function to be an approximation of the area–if A(r) was 0/1-valued, it would be the area, but for 0 ≤ A(r) ≤ 1 we call it the pseudoarea [8] . The advantage to this cost function is its linearity; it is by no means the only possible choice. This is a quadratic program. Since linear programs are computationally more efficient, √ we could separate the real and imaginary parts of Eocc , and constrain them separately to be less than 10−5 / 2. Note this linear representation is slightly more conservative than the quadratic one. Also, when the optimization is set up numerically, ρ, r, λ, and the integrals need to be discretized. Other modifications can be added, such as monotonicity constraints, curvature constraints, and alternative cost functions; see [6] for more detail.

2.1 Petalization The optimization produces occulter shapes for smooth, radially-symmetric apodization functions, but not for occulters with petals. We can, however, relate these shapes to occulters with petals by considering a circle of radius r centered on the occulter. Traveling along the perimeter, we find alternating bands of empty space and occulter petal. If we choose the fraction of this circle that is covered by occulter petals to be A(r), then as shown in [8] , the electric field will be: Eocc,petal (ρ, φ)

= Eocc (ρ) −E0 eikz

∞  (−1)j k j=1

iz


0

R

e

2 2 ik 2z (r +ρ )

 JjN

× (2 cos (jN (φ − π/2)))

krρ z



sin (jπA(r)) rdr jπ



(6)

where Eocc is again the electric field due to a smoothly varying occulter with radial profile A(r). The new field is the sum of the smooth field and an infinite number of perturbation terms. Since the jth term in the sum of the perturbation terms goes as the integral of JjN , however, we can truncate the sum with negligible error near the center, and we can truncate this sum sooner for large N . Petalization, the first way of making occulters with petals, is then done by simply computing a smooth apodization by linear programming as above, and then adding petals, in the form of perturbation terms. This is done for decreasing numbers of petals until the contrast in the shadow at some wavelength is worse than 1010 . See Fig. 1 and Fig. 2 for examples.

2.2 Optimal Petalization Instead of optimizing the attenuation of a continuous profile and adding on perturbation terms, we could instead optimize the profile and perturbations together. This is equivalent to optimizing the shape of the petal directly, and we call it optimal petalization. This method directly accounts for the degradation at the edges of the shadow due to petalization, theoretically enabling fewer petals to be used, or deeper nulls achieved with existing numbers of petals. The penalty is a nonlinear optimization, and one that extends over two variables (ρ and φ).

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The nonlinearity arises because of the sin (jπA(r)) terms; this makes convergence to a global maximum not guaranteed. We can improve the situation by choosing the initial conditions carefully; one technique that may be of use is to run a linear optimization first, and use the output as the starting condition for the optimal petalization. The second problem can be alleviated by noting that for each value of ρ, there exist a φmax and φmin which maximize and minimize the value of the real or imaginary part of the electric field at that radius. If the real and imaginary parts of the electric field meet the contrast constraints using φ = φmax and φ = φmin , then the perturbed field will still give 1010 contrast at any angle. Consider the following n-term truncated Fourier series: P =

n 

aj (ρ) cos (jN φ)

(7)

j=1

This term corresponds to the perturbation term in Eq. 6. We can let the aj be either the real or the imaginary part of the ρ-dependent perturbation integral, and assume it includes all of the negative signs and constants. It n will be real but not necessarily positive. For a given ρ, this sum is conservatively bounded by ± j=0 |aj |; we can get a more precise bound by noting that cos nθ = Tn (cos θ)—that is, the Chebyshev polynomial of the first kind. We rewrite Eq. 7 as: n  aj (ρ)Tj (cosN φ) (8) P = j=1

We can find the location of maxima and minima by examining the first derivative: ∂P ∂φ

= =

∂P ∂ cos N φ ∂ cos N φ ∂φ n  aj (ρ)jUj−1 (cos N φ) (−N sin N φ)

(9) (10)

j=1

where Uj (x) is the Chebyshev polynomial of the second kind. The result is that ∂P ∂φ = 0 at φ = 0, π/N and at up to n − 1 other points between 0 and π/N , which are the zeros of the (n − 1)th degree polynomial n a (ρ)jU (x) in x = [−1, 1]. (We use the interval [0, π/N ], as going beyond this will just cause the shape j j−1 j=1 of the petals to repeat, due to symmetry.) From these, we could test to see which are the global maxima and minima, and these would be φmax and φmin for that ρ. Note that aj depends on A(r), and thus will have to be recalculated at each iteration. One useful detail is that since U0 = 1, and |Uj | ≤ j + 1 in [−1, 1], we can say that if: |a1 | >

∞ 

j 2 |aj |

(11)

j=2

then the only zeros of Eq. 10 on φ = [0, π/N ] will be 0 and π/N . Since aj goes as the integral of JjN , the series can fall off quickly for sufficient numbers of petals. One practical way to compromise on the need to reevaluate the aj after each iteration is to do an optimization using 0 and π/N as the maximum and minimum, and verify afterwards, using the final A(r), that the above inequality is satisfied; only if it is not would the full optimization with recalculations need to be done. In practice, we have found the best strategy is to produce a petalized occulter, and search for an optimallypetalized one only if the petal count needs to be reduced. The reason for this is that petalized designs take on the order of minutes to produce, while the optimal designs can require tweaking of the initial conditions and take hours or days. See Fig. 1 and Fig. 2 for examples.

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3. CONSIDERATIONS FOR DESIGN In our optimizations, we can set ρmax , R, z, λmax , λmin , the contrast level, and number of petals. The question then arises how best to choose these. As discussed previously, some of these choices are driven by scientific requirements: to provide a shadow over a requested spectral band, support a telescope with diameter D, provide some set inner working angle (IWA), and provide some desired contrast in the shadow. The remainder of the choices are dependent on limitations in engineering and optimization. We can further divide these and say that some are driven by manufacturing, some by estimation and control, and some by spacecraft engineering. 1. ρmax : radius of shadow at telescope • Bounded below by the diameter of the telescope aperture, D–the telescope must fit in the shadow. • Bounded above by the state estimation and control of the occulter. The occulter must be able hold its position and orientation well enough to keep the telescope from leaving the shadow. • Also bounded above by the optimization, as larger shadows are more difficult to ensure a consistently deep shadow across. 2. R: radius of occulter, center to tip • Bounded below by the optimization. In practice, it has proven difficult to get small shade designs to produce feasible solutions, particularly for large ρmax . The boundary will vary depending on the other constants selected, but must be strictly greater than ρmax . • Bounded above by spacecraft engineering considerations–larger shades require more fuel to move and orient, reducing the number of scientific targets that can be examined. Thus we would desire smaller shades, but it doesn’t place a hard upper limit on size. • Bounded above by the selection of z and the choice of minimum IWA, since the IWA is ≈ R/z. (The actual inner working angle may be smaller, as the tips of the petals don’t block much planet light.) • Bounded above and below by the practicalities of designing a spacecraft. It must be large enough to hold a spacecraft bus on which the craft’s systems are placed, and it must be small enough to fold up into a launch vehicle. 3. z: distance from occulter to telescope • Bounded below by the selection of R and the choice of minimum IWA, since the IWA is ≈ R/z. • Bounded below by the optimization. In practice, much as with R, it has proved sometimes difficult to produce feasible designs close in to the telescope. • Bounded above by spacecraft engineering considerations. The farther away the occulter rests, the faster it must move to remain between the star and the telescope, and the more fuel it must use. A note on error in z: in Eq. 4 and Eq. 6, z appears with λ as λz every time it appears (in k/z), except in the eikz terms from the propagation. If a change from z to z + ∆z is accompanied by a change in λ∆z , then the λz term is unchanged, and the shadow will be the same at the wavelength from λ to λ − z+∆z ikz telescope. (The e terms disappear when intensity is calculated, and so we can neglect the effect on it.) Thus, if we want the shade to give a shadow a tolerance of ∆z in each direction, we expand the range of ∆z ∆z ), λmax (1 + z+∆z )]. This wavelengths over which to optimize in Sec. 2 from [λmin , λmax ] to [λmin (1 − z+∆z will guarantee the deep shadow for the entire spectral band. For this reason, λmax will not necessarily be the upper wavelength requested by science, and similarly for λmin . 4. λmax : maximum wavelength used in the optimization • Bounded below by the maximum wavelength in the band over which spectra are taken.

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• Bounded above by the limitations of the estimation and control system–as mentioned above, the shadow due to error in the z-direction is identical (to within a phase factor) to the shadow at a different wavelength. The upper bound needs to be high enough that the residual z-error after estimation and correction will not be equivalent to a shadow from a wavelength above λmax . 5. λmin : minimum wavelength used in the optimization • Bounded below by the limitations of the estimation and control system, for reasons identical to those for λmax . • Bounded above by the minimum wavelength in the band over which spectra are taken. 6. Contrast level • Bounded above by the scientific requirements of the mission. Effectively, it varies with the type of planet being investigated; we expect rocky planets need ≈ 1010 , and gas giants ≈ 108 . See [7] for more discussion. It also can be affected by the wavelength band, as at mid-infrared wavelengths contrast receives a boost from the planet’s blackbody emission. • Bounded below by the optimization; for sufficiently high contrasts, it is difficult to make the optimization converge. 7. Number of petals • Bounded above by manufacturing considerations–the more petals that are added to an occulter, the finer the structure at the tips have to be, and the more accurate the edges have to be. (Errors in the shape of the manufactured edge manifest themselves as changes in A(r).) Increased numbers of petals also increase the perimeter, giving more edge length for errors to occur. This will be limited by our ability to construct the shape accurately. • Bounded above by spacecraft engineering considerations, as each petal must be extended once it is released from the launch housing, and adding additional booms to extend these adds mass and risk. • Bounded below by optimization considerations–it is quite difficult to find designs that produce broadband high contrast over a large shadow with small numbers of petals. Given the complex interaction of the parameters above, it is difficult to select a set of ideal parameters from first principles. Instead, our design process is an iterative approach. We find a shade we can feasibly design, and attempt to make a control system that can accommodate it. If we can produce a suitable control system and set of control laws, then we are finished; if not, we find the best we can do, and use that to iterate on another shade design. The designs used in Fig. 1 and 2 use ρmax = 3m, R = 30m, z = 100000km, λmax = 1100nm, λmin = 400nm, and a contrast level of 1010 . Fig. 1 uses 12 petals, while Fig. 2 uses 16.

4. SUMMARY AND FUTURE WORK We have given some optimization techniques to design petal-shaped occulters, and discussed how the parameters of the optimization interact with the manufacturing, launching, and control of an occulter. The control system itself will be discussed by the authors in a future paper.

ACKNOWLEDGMENTS The third author would like to acknowledge support from ONR grant #N00014-98-1-0036.

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REFERENCES 1. J. Schneider, “The extrasolar planets encyclopaedia.” http://exoplanet.eu, August 2007. 2. C. Copi and G. Starkman, “The Big Occulting Steerable Satellite [BOSS],” Astrophysical Journal 532, pp. 581–592, 2000. 3. A. Schultz, I. Jordan, M. Kochte, D. Fraquelli, F. Bruhweiler, J. Hollis, K. Carpenter, R. Lyon, M. DiSanti, C. Miskey, J. Leitner, R. Burns, S. Starin, M. Rodrigue, M. Fadali, D. Skelton, H. Hart, F. Hamilton, and K.-P. Cheng, “UMBRAS: A matched occulter and telescope for imaging extrasolar planets,” in Proceedings of SPIE–High-Contrast Imaging for Exo-Planet Detection, 4860, 2003. 4. W. Cash, “Detection of earth-like planets around nearby stars using a petal-shaped occulter,” Nature 442, pp. 51–53, 2006. 5. M. Janson, “Celestial exoplanet survey occulter: A concept for direct imaging of extrasolar earth-like planets from the ground,” Publications of the Astronomical Society of the Pacific 119, pp. 214–227, 2007. 6. R. Vanderbei, E. Cady, and N. Kasdin, “Optimal occulter design for finding extrasolar planets,” Astrophysical Journal 665, pp. 794–798, 2007. 7. D. D. Marais, M. Harwit, K. Jucks, J. Kasting, D. Lin, J. Lunine, J. Schneider, S. Seager, W. Traub, and N. Woolf, “Remote sensing of planetary properties and biosignatures on extrasolar terrestrial planets,” Astrobiology 2(2), pp. 153–181, 2002. 8. R. Vanderbei, D. Spergel, and N. Kasdin, “Circularly symmetric apodization via star-shaped masks,” Astrophysical Journal 599, pp. 686–694, 2003.

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Broadband shadow at φ = 0 0

−1

−1

−2

−2

−3

−3

−4

−4 log(contrast)

log(contrast)

Broadband shadow at φ = 0 0

−5 −6

400nm 500nm 600nm 700nm 800nm 900nm 1000nm 1100nm 3m shadow

−7 −8 −9 −10 −11

0

5

10 15 Distance [meters]

20

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−7 −8 −9 −10 −11

25

0

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−1

−1

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400nm 500nm 600nm 700nm 800nm 900nm 1000nm 1100nm 3m shadow

−7 −8 −9 −10 −11

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10 15 Distance [meters]

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Broadband shadow at φ = π/N

0

log(contrast)

log(contrast)

Broadband shadow at φ = π/N

10 15 Distance [meters]

20

−5 −6

400nm 500nm 600nm 700nm 800nm 900nm 1000nm 1100nm 3m shadow

−7 −8 −9 −10 25

−11

0

5

10 15 Distance [meters]

20

25

Figure 1. This figure shows the performance over a 400nm-1100nm band of two profiles, both designed to try and produce a 3m-radius shadow, being applied to an occulter with 12 petals. Left top and bottom. These profiles were produced by petalizing a smooth profile, as described in Sec. 2.1. They are shown at two angles, one aligned with the petal tip, and one aligned with the trough between petals. Right top and bottom. These profiles were produced by optimal petalization, as described in Sec. 2.2. They are shown at the same angles as the petalized profile. Note the improvement in contrast–there is a marked advantage to using optimal petalization for 12 petals.

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Broadband shadow at φ = 0 0

−1

−1

−2

−2

−3

−3

−4

−4 log(contrast)

log(contrast)

Broadband shadow at φ = 0 0

−5 −6

400nm 500nm 600nm 700nm 800nm 900nm 1000nm 1100nm 3m shadow

−7 −8 −9 −10 −11

0

5

10 15 Distance [meters]

20

−5 −6

400nm 500nm 600nm 700nm 800nm 900nm 1000nm 1100nm 3m shadow

−7 −8 −9 −10 −11

25

0

5

0

−1

−1

−2

−2

−3

−3

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400nm 500nm 600nm 700nm 800nm 900nm 1000nm 1100nm 3m shadow

−7 −8 −9 −10 −11

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Broadband shadow at φ = π/N

0

log(contrast)

log(contrast)

Broadband shadow at φ = π/N

10 15 Distance [meters]

20

−5 −6

400nm 500nm 600nm 700nm 800nm 900nm 1000nm 1100nm 3m shadow

−7 −8 −9 −10 25

−11

0

5

10 15 Distance [meters]

20

25

Figure 2. This figure shows the performance over a 400nm-1100nm band of two profiles, both designed to try and produce a 3m-radius shadow, being applied to an occulter with 16 petals. Left top and bottom. These profiles were produced by petalizing a smooth profile, as described in Sec. 2.1. They are shown at two angles, one aligned with the petal tip, and one aligned with the trough between petals. Right top and bottom. These profiles were produced by optimal petalization, as described in Sec. 2.2. They are shown at the same angles as the petalized profile. Note that both profiles give 1010 contrast within 3m–there is no advantage to using optimal petalization for a 16-petal design.

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