Designing asymmetric and branched petals for planet-finding occulters

Designing asymmetric and branchedpetals for planet-finding occulters

Eric Cady1,∗, N. Jeremy Kasdin1, and Stuart Shaklan2

1: Dept. of Mechanical and Aerospace Engineering, Princeton University,Olden St., Princeton, NJ, 08544

2: Jet Propulsion Lab, 4800 Oak Grove Drive,Pasadena, CA, 91109∗[email protected]

Abstract: One of the proposed methods for finding small extrasolarplanets is through use of an occulter, a spacecraft which flies in formationwith a space-based telescope to block the light from a star, while leavingnearby planets unaffected. This is accomplished by placing the occulter farenough from the telescope to give it a small angular size, and by carefullychoosing the shape to strongly suppress the starlight at the telescopeaperture. For most designs, this shape takes the form of a number ofbilaterally-symmetric structures called petals, arrayed about a circularcentral disk. In this paper, we show that the necessary number of petalsmay be reduced by the introduction of an asymmetry in the petal shape, anddescribe a a general procedure for producing such a shape by optimizationfor any occulter with petals. In addition, we show that permitting openingswithin each petal allows a number of additional modifications to be madewithout affecting the suppression.

© 2010 Optical Society of America

OCIS codes: (050.1940) Diffraction, (050.1970) Diffractive optics, (120.6085) Space instru-mentation, (350.6090) Space optics

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#118002 - $15.00 USD Received 1 Oct 2009; revised 9 Nov 2009; accepted 9 Nov 2009; published 4 Jan 2010

(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 523

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18. N. J. Kasdin, P. Atcheson, M. Beasley, R. Belikov, M. Blouke, E. Cady, D. Calzetti, C. Copi, S. Desch, P. Du-mont, D. Ebbets, R. Egerman, A. Fullerton, J. Gallagher, J. Green, O. Guyon, S. Heap, R. Jansen, E. Jenkins,J. Kasting, R. Keski-Kuha, M. Kuchner, R. Lee, D. J. Lindler, R. Linfield, D. Lisman, R. Lyon, J. MacKenty,S. Malhotra, M. McCaughrean, G. Mathews, M. Mountain, S. Nikzad, B. OConnell, W. Oegerle, S. Oey, D.Padgett, B. A. Parvin, X. Prochaska, J. Rhoads, A. Roberge, B. Saif, D. Savransky, P. Scowen, S. Seager, B.Seery, K. Sembach, S. Shaklan, M. Shull, O. Siegmund, N. Smith, R. Soummer, D. Spergel, P. Stahl, G. Stark-man, D. K. Stern, D. Tenerelli, W. A. Traub, J. Trauger, J. Tumlinson, E. Turner, R. Vanderbei, R. Windhorst, B.Woodgate, and B. Woodruff, “THEIA: Telescope for habitable exoplanets and interstellar/intergalactic astron-omy,” http://www.astro.princeton.edu/˜dns/Theia/nas theia v14.pdf.

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defects,” Proc. SPIE 7440, 744008 (2009).

1. Introduction

To date, over 400 planets have been found around other stars [1]. These planets have beendiscovered primarily using radial velocity methods [2] and transit methods [3], and they mostlyare large gas giants, often in very close orbits. However, these methods are not very sensitive tosmall, rocky planets, especially orbiting larger stars or in larger orbits.

Direct imaging from space is both capable of imaging terrestrial planets and providing spec-tra, and a large body of literature has been produced proposing ways to do this. (See for example[4] for an overview of many methods.) However, the small angular separation, e.g. 100mas fora 1AU planet at 10pc, and the large ratio between the emitted flux from the star and the planet,roughly 1010 [5] for a Sun-Earth analog, make finding these planets around nearby stars quitechallenging.

One of the proposed methods to do this is by combining a telescope in space with an occulter.An occulter is a diffractive optical element which is placed in front of the telescope to blockmost of the light before it reaches the optics inside. Occulters were first proposed for solarcoronagraphy in 1948 [6], and small occulters have been flown on solar missions such as SOHO[7] and STEREO [8]. These occulters were circular and attached to the telescope on booms;more recently, shaped solar occulters have been proposed to fly in formation with the telescopeat distances of approximately 100m [9, 10]. The potential of occulters for the observation ofextrasolar planets was realized as early as 1962 [11]. In contrast to solar occulters, however,occulters for extrasolar planet-finding [12, 13, 14, 15] need to be very precisely shaped andlocated tens of thousands of kilometers from the telescope to provide both the small angular



Fig. 1. A 20-petal occulter.

size and the deep suppression.The occulter’s size and distance are chosen so that the occulter has a smaller angular size than

the expected orbits of planets it is designed to image, so planet light will enter the telescopelargely undisturbed everywhere outside the occulter. The shape of an occulter for finding planetsis carefully prescribed to destructively interfere the starlight at the telescope, while leavingthe light from nearby planets unaffected. This diffraction is created by a series of bilaterally-symmetric structures along the outer edge, termed petals for the flower-like appearance of theresulting occulter. One example of these is shown in Fig. 1.

These petals are approximations to radial apodization functions, and the diffraction fromthe petals can be modeled as the propagation integral for a smooth apodization, plus a seriesof additional terms which represent the effect of using binary petals. These functions can bedefined analytically [12, 14] or by optimization [15]; either method can produce occulters thatremove sufficient starlight. (The derivation is given in Sec. 2.) The magnitude of the additional



terms from the petals is smallest in the center of the downstream field, where the telescope islocated. Adding more petals makes these terms also become small at points further and furtherradially outward from the center. In general, we select the number of petals for an occulter byadding petals until the additional terms in the pupil plane of the telescope are small enough tobe neglected. We use the minimum number of petals possible; from an engineering perspective,fewer petals means less cost and less risk, as the failure of any petal to deploy correctly wouldbe catastrophic to an occulter mission.

However, the petals are not required to be symmetric about their center axis, nor are theylimited to being defined by a single smooth function. These are additional degrees of freedomthat have not been investigated previously. In this paper, we demonstrate that by introducing anoptimized asymmetry into the petals, the additional terms can be minimized across the telescopeaperture, decreasing the number of petals required to keep light out of the aperture. In addition,we show that if multiple functions are used to define the occulter edge, a number of usefulmodifications can be made without losing suppression.

2. Asymmetry

Consider an occulter of N symmetric petals. We model this occulter as a binary mask, being(0,1)-valued at all locations in a plane. The set of points which are on the occulter is defined asΩ in polar coordinates[15, 16]:

Ω = {(r,θ) : 0 ≤ r ≤ R,θ ∈ Θ(r)} (1)

where Θ(r) =N−1⋃

n=0

[2πnN

− πN

A(r),2πnN

+πN

A(r)]. (2)

Here A(r) is the smooth apodization function which the petals are approximating, and (r,θ) arepolar coordinates in the plane of the occulter. In this design, each petal is symmetric about itscenter axis. However, there is no fundamental requirement that the petals be symmetric. We cangain extra degrees of freedom by exploiting asymmetry. For example, let us shift the centerlineof each petal to follow a curve β (r):

Ωβ = (r,θ) : 0 ≤ r ≤ R,θ ∈ Θβ (r) (3)

where Θβ (r) =N−1⋃

n=0

[2πnN

− πN

A(r)+β (r)

N,2πnN

+πN

A(r)+β (r)

N

]. (4)

Stars of interest are sufficiently small and far enough away to model starlight as fa plane waveincident on the occulter. The telescope will be located a distance z from the occulter, alignedwith the occulter and the target star. We can calculate the shadow produced by the occulterusing Babinet’s principle: the electric field at the telescope pupil is well-approximated by theFresnel transform of the complementary hole subtracted from a propagated plane wave:

Eβ (ρ,φ) =E0 exp

(2πiz

λ

)(1− 1

iλ z(5)

×∫ R

0

∫

Θβ (r)exp

[πiλ z

(r2 +ρ2)]

exp

[2πirρ

λ zcos(θ −φ)

]rdrdθ

).

where λ is the wavelength and (ρ,φ) are polar coordinates in the pupil plane of the telescope.



The Jacobi-Anger expansion of exp(2πirρ cos(θ −φ)/(λ z)) gives:

Eβ (ρ,φ) =E0 exp

(2πiz

λ

)(1− 1

iλ z(6)

×∫ R

0

∫

Θβ (r)exp

[πiλ z

(r2 +ρ2)][ ∞

∑m=−∞

imJm

(2πrρ

λ z

)exp [im(θ −φ)]

]rdrdθ

),

which can be rewritten as:

Eβ (ρ,φ) =E0 exp

(2πiz

λ

)(1−

∞

∑m=−∞

im exp(−imφ)iλ z

(7)

×∫ R

0exp

[πiλ z

(r2 +ρ2)]Jm

(2πrρ

λ z

)[∫

Θβ (r)exp(imθ)dθ

]rdr

).

The integral over Θβ (r) can be evaluated explicitly:

∫

Θβ (r)exp(imθ)dθ =

N−1

∑n=0

∫ 2πnN + π

N A(r)+ β (r)N

2πnN − π

N A(r)+ β (r)N

exp(imθ)dθ

=

⎧⎨

⎩

2πA(r), m = 0,2Nm sin(mπA(r)/N)exp [imβ (r)/N], m = ±N,±2N, . . . ,± jN, . . .

0, else,(8)

which gives:

Eβ (ρ,φ) =E0 exp

(2πiz

λ

)(1− 2π

iλ z

∫ R

0exp

[πiλ z

(r2 +ρ2)]J0

(2πrρ

λ z

)A(r)rdr

)

−E0 exp

(2πiz

λ

) ∞

∑j=1

i jN exp(−i jNφ)iλ z

×(∫ R

0exp

[πiλ z

(r2 +ρ2)]JjN

(2πrρ

λ z

)2sin( jπA(r))

jexp [i jβ (r)]rdr

)

−E0 exp

(2πiz

λ

) ∞

∑j=1

i− jN exp(i jNφ)iλ z

×(∫ R

0exp

[πiλ z

(r2 +ρ2)]J− jN

(2πrρ

λ z

)2sin(− jπA(r))

− jexp [−i jβ (r)]rdr

)

=E0 exp

(2πiz

λ

)(1− 2π

iλ z

∫ R

0exp

[πiλ z

(r2 +ρ2)]J0

(2πrρ

λ z

)A(r)rdr

)(9)

−E0 exp

(2πiz

λ

) ∞

∑j=1

i jN4πiλ z

×(∫ R

0exp

[πiλ z

(r2 +ρ2)]JjN

(2πrρ

λ z

)sin( jπA(r))

jπcos [ jNφ − jβ (r)]rdr

).

We note that when β (r) = 0, i.e. when the centerline of the petal is not shifted, Eq. (9) reduces



to the formula for an occulter with symmetric petals [15]:

E(ρ,φ) =E0 exp

(2πiz

λ

)(1− 2π

iλ z

∫ R

0exp

[πiλ z

(r2 +ρ2)]J0

(2πrρ

λ z

)A(r)rdr

)(10)

−E0 exp

(2πiz

λ

) ∞

∑j=1

i jN4πiλ z

cos( jNφ)

×(∫ R

0exp

[πiλ z

(r2 +ρ2)]JjN

(2πrρ

λ z

)sin( jπA(r))

jπrdr

).

We also note that, since β (r) applies equally to all petals, the first term of Eq. (9) and Eq. (10)are the same—β (r) does not affect the φ -independent term that primarily defines the shadowacross the aperture of a telescope.

2.1. Optimization

The number of petals required for an occulter is determined by the wavelength band the occultermust work over (specifically, the shortest wavelength), the occulter size, shape, and distance,and the telescope diameter. To find the appropriate number, we increase N in Eq. (10) until theseries is small enough to neglect over the spectral band we care about, for values up to ρ = D/2,where D is the telescope diameter.

For occulters with fewer petals than this, but with otherwise identical properties, most of thescattered light comes from the j = 1 term in the infinite series. To reduce the number of petalsrequired in an occulter, then, we can optimize β (r) to minimize the j = 1 term of the series.Unfortunately, the φ -dependence is not separable from the ρ-dependent integral in Eq. (9). Todeal with this, we separate the real and imaginary parts of the j = 1 term, ignoring the constantphase term −exp(2πiz/λ )iN−1:

E1(ρ,φ) := (Rc + iIc)cos(Nφ)+(Rs + iIs)sin(Nφ), where (11)

Rc(β (r),ρ,λ ) = E04

λ z

(∫ R

0cos

(πλ z

(r2 +ρ2))

JN

(2πrρ

λ z

)sin(πA(r))cosβ (r)rdr

), (12)

Ic(β (r),ρ,λ ) = E04

λ z

(∫ R

0sin

(πλ z

(r2 +ρ2))

JN

(2πrρ

λ z

)sin(πA(r))cosβ (r)rdr

), (13)

Rs(β (r),ρ,λ ) = E04

λ z

(∫ R

0cos

(πλ z

(r2 +ρ2))

JN

(2πrρ

λ z

)sin(πA(r))sinβ (r)rdr

), (14)

Is(β (r),ρ,λ ) = E04

λ z

(∫ R

0sin

(πλ z

(r2 +ρ2))

JN

(2πrρ

λ z

)sin(πA(r))sinβ (r)rdr

). (15)

Note: for most occulters, A(r) = 1 and sin( jπA(r)) = 0 from r = 0 out to some r = a, and sothe integral is generally done as

∫ Ra . We can then discretize ρ and λ , and write a conservative

vector of the maximum intensities at each (ρi,λi):

f (β ) =

⎡

⎢⎣Rc(β (r),ρ1,λ1)2 + Ic(β (r),ρ1,λ1)2 +Rs(β (r),ρ1,λ1)2 + Is(β (r),ρ1,λ1)2

Rc(β (r),ρ2,λ2)2 + Ic(β (r),ρ2,λ2)2 +Rs(β (r),ρ2,λ2)2 + Is(β (r),ρ2,λ2)2

...

⎤

⎥⎦ .

(16)



Each of these terms are a conservative upper bound on the intensity |Ej=1|2, as:

Ej=1 = (Rc + iIc)cos(Nφ)+(Rs + iIs)sin(Nφ) (17)

|Ej=1|2 = ((Rc + iIc)cos(Nφ)+(Rs + iIs)sin(Nφ))((Rc − iIc)cos(Nφ)+(Rs − iIs)sin(Nφ))

= (R2c + I2

c )cos2 (Nφ)+(R2s + I2

s )sin2 (Nφ)+(2RsRc +2IsIc)sin(Nφ)cos(Nφ)

=12(R2

c + I2c +R2

s + I2s )+

cos(2Nφ)2

(R2c + I2

c −R2s − I2

s )+sin(2Nφ)

2(2RsRc +2IsIc).

(18)

(19)

We note that the magnitude of the φ -dependent part is:

12

[(R2

c + I2c −R2

s − I2s )2 +(2RsRc +2IsIc)2]1/2

=12

(I4c +2I2

c I2s + I4

s

+2I2c R2

c −2I2s R2

c +R4c +8IcIsRcRs

−2I2c R2

s +2I2s R2

s +2R2cR2

s +R4s

)1/2. (20)

Since8IcIsRcRs −2I2

s R2c −2I2

c R2s ≤ 2I2

s R2c +2I2

c R2s , (21)

we have

12

[(R2

c + I2c −R2

s − I2s )2 +(2RsRc +2IsIc)2]1/2 ≤1

2

(I4c +2I2

c I2s + I4

s

+2I2c R2

c +2I2s R2

c +R4c +2I2

c R2s

+2I2s R2

s +2R2cR2

s +R4s

)1/2

=12(R2

c + I2c +R2

s + I2s ) (22)

=⇒ |Ej=1|2 ≤ (R2c + I2

c +R2s + I2

s ). (23)

We note that the bound is still conservative, and a more complicated but tight upper bound forthe intensity could be used by choosing

maxφ

|Ej=1| = 12(R2

c + I2c +R2

s + I2s )+

12

[(R2

c + I2c −R2

s − I2s )2 +(2RsRc +2IsIc)2]1/2

. (24)

This will be equal to R2c + I2

c +R2s + I2

s when RsIc = RcIs, and smaller otherwise.To optimize, we write an unconstrained optimization to minimize the norm of f :

Minimize : || f (β )||p, (25)

where || · ||p is the p-norm of f . In the following, we use the 1-norm of f , for simplicity.We also note for completeness that the solution to the optimization will not be unique; the

bounds on the electric field will be the same under rotation—equivalent to adding a constant toβ (r)—and reflection—equivalent to replacing β (r) with −β (r). (In general, we set β (a) to 0,where a is the radial distance at which the petals begin, so the bases of the petals remain at thesame locations, but the choice is arbitrary.)



10 12 14 16 18 20−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

meters

β (r

adia

ns)

β(r)

Fig. 2. β (r) for the THEIA occulter.

2.2. Results

There are number of ways to do unconstrained optimization; for our optimization, we used aconjugate gradient algorithm with Polak-Ribiere update step to find local minima of the costfunction. The conjugate gradient method is a optimization algorithm for unconstrained func-tions which uses a set of search directions which are successively orthogonal to one another toconverge quickly to a minimum [17]. We use a nonlinear variant of this; other nonlinear op-timization methods may be suitable, but this one proved to perform well enough. In addition,the form of the cost function chosen in Eq. (25) and of the integrals in Eq. (12) through Eq.(15) allow the derivative calculations required by the nonlinear conjugate gradient method to beoptimized by taking advantage of their properties. There are a few ways for calculating the up-date step for nonlinear conjugate gradient methods, but Polak-Ribiere update tends to convergefaster than others.

This algorithm was applied to the design of the THEIA occulter [18]. The THEIA occulter isa 40m occulter, nominally with 20 petals, designed to operate at 2 distances from the telescope.The telescope would do observations from 250−700nm when the occulter is at 55000km, andfrom 700− 1000nm when the occulter is moved in to 35000km. (See Table 2.2 and [18] formore details.) The nominal occulter is shown in Fig. 1.

For the THEIA occulter, the optimization was only run from 250nm to 450nm, as at longerwavelengths the Bessel function in the integral is small, and so is the perturbation. (Jn(x)reaches its first peak at x ≈ n [19].) The close-in distance was not involved, as the scaling rela-tions make the field identical to a corresponding field at the further distance. Figure 2 shows oneexample of a β (r), in this case optimized for the 40m THEIA occulter for use with 12 petals,rather than the expected 20. The starlight intensity is shown in Fig. 3; performance is improvedat other wavelengths as well, but the performance gain at 250nm is the best. The occulter itselfis shown in Fig. 4 and Fig. 5.

It is worth noting that, since A(r) is only used as a parameter to produce β (r), this methodcan be used regardless of whether A(r) is produced analytically or by optimization.



Fig. 3. Left. The shadow at the THEIA telescope aperture with 20 petals and β (r) = 0 at250nm. Center. The shadow at the THEIA telescope aperture with 12 petals and β (r) = 0at 250nm. Right. The shadow at the THEIA telescope aperture with 12 petals and β (r)optimized at 250nm.

Fig. 4. The 12-petal THEIA occulter, with asymmetry.



Fig. 5. A close-up of the tip of one of the petals on the occulter.

Table 1. THEIA occulter specifications.

OcculterOcculter radius (m) 20

Occulter nominal distance (km) 55000Occulter inner distance (km) 35000

Occulter nominal angular size (mas) 75Occulter inner angular size (mas) 118

Telescope diameter (m) 4Size of secondary mirror (m) 0.8

Nominal number of petals 20Spectral band (nm) 250-700

Petal length (m) 10

3. Branching

A petal asymmetry is not the only modification that can be made to a binary occulter. [20]showed that multiple profiles can be used to define the edge of the occulter, as long as theyobey a certain matching condition. We review this derivation briefly here.

Suppose instead of demarcating the edges of the binary occulter with A(r), we wish to useA1(r),A2(r), . . . to create a shape with a more complicated (but still symmetric) structure. (Asthe simplest application would be to make the ends of a petal branch out like a tree, we willrefer to these as branched occulters.) In a similar manner to Eq. (2), we can define the set ofpoints that are covered by a branched occulter as:

Ωb ={(r,θ) : 0 ≤ r ≤ R,θ ∈ Θb(r)}

where Θb(r) =N−1⋃

n=0

{M⋃

m=1

[2πnN

− πN

A2m−1(r),2πnN

− πN

A2m(r)]

⋃ M⋃

m=1

[2πnN

+πN

A2m(r),2πnN

+πN

A2m−1(r)]}

. (26)

where M is the the number of profiles, and 0 ≤ A2m ≤ A2m−1 ≤ . . .A1 ≤ 1. (If the petal uses an



odd number of profiles, A2m = 0.) The electric field following this occulter is:

Eb(ρ,φ) =E0 exp

(2πiz

λ

)(1− 1

iλ z

×∫ R

0

∫

Θb(r)exp

[πiλ z

(r2 +ρ2)]

exp

[2πirρ

λ zcos(θ −φ)

]rdrdθ

)

=E0 exp

(2πiz

λ

)(1− 2π

iλ z

×∫ R

0exp

[πiλ z

(r2 +ρ2)]J0

(2πrρ

λ z

)[A1(r)−A2(r)+ . . .]rdr

)

−E0 exp

(2πiz

λ

) ∞

∑j=1

i jN4πiλ z

cos( jNφ) (27)

×(∫ R

0exp

[πiλ z

(r2 +ρ2)]JjN

(2πrρ

λ z

)sin jπA1(r)− sin jπA2(r)+ . . .

jπrdr

)

By choosing:A(r) = A1(r)−A2(r)+A3(r)− . . . , (28)

the first term of Eq. (27) is matched to the first term in Eq. (9) and Eq. (10), ensuring that addingbranches will not affect the φ -independent part of the equation. This method can be used to addstructural elements between petals, shorten petals, isolate complex edge shapes, and reduce thenumber of petals, though to a lesser extent than asymmetry can. Examples of these applicationsfollow, again using the THEIA occulter. Their image plane performance is plotted at 250nmin Fig. 6, along with the asymmetric occulter shown in Fig. 4 and the unmodified THEIAdesign. (We use 250nm because the performance degrades the most quickly at the shortestwavelengths.)

3.1. Structural elements

Under dynamic and thermal loading, the petals of an occulter can move independently of oneanother, rotating in the occulter plane and bending out of it. These motions change the opticalproperties of the occulter; depending on the design, these motions may change the shape sig-nificantly enough that the occulter can no longer suppress the starlight sufficiently to detect aplanet. (See [21] for examples of the effects of rotated and bent petals on THEIA.)

One possible solution is to create connecting elements between the petals, in order to helpsupport and fix them with respect to their neighbors. These can be created using multiple pro-files to define the edges of the elements. Figure 7 shows an example of a simple cross introducedbetween petals; a close-up view is shown in Fig. 8. This design may be created by betweenr = r1 and r = r2 by choosing A1(r), A2(r) and A3(r) as shown in Table 3.1. Here T (r) definesthe outer edge of the profile, and W (r) defines the width of the crossbars.

Table 2. The profile definitions for a 3-profile truss.

r < r1 r1 ≤ r ≤ r2 r2 < rA1(r) A(r) A(r)+T (r) A(r)A2(r) A(r) A(r)+T (r)−W (r) A(r)A3(r) A(r) A(r)−W (r) A(r)



0 20 40 60 80 100 120 140 160 180 20010

−16

10−14

10−12

10−10

mas

250nm, intensity at image plane

No modificationsAsymmetry with 12 petals

0 20 40 60 80 100 120 140 160 180 20010

−16

10−14

10−12

10−10

mas


No modificationsStructural elements

0 20 40 60 80 100 120 140 160 180 20010

−16

10−14

10−12

10−10

mas


No modificationsShortened petals

0 20 40 60 80 100 120 140 160 180 20010

−16

10−14

10−12

10−10

mas


No modificationsIsolated regions

0 20 40 60 80 100 120 140 160 180 20010

−16

10−14

10−12

10−10

mas


No modificationsBranching with 16 petals

Fig. 6. Image plane performance at 250nm for the modifications described in this paper. Aplanet peaks at intensity of 10−10, at a 75mas separation.

3.2. Shortened petals

For engineering reasons, it may be preferable to design the occulter with shorter petals ratherthan longer ones. (For THEIA, for example, petal length was constrained by the size of thelaunch vehicle fairing.) This can be accomplished by using multiple profiles to move the smallgaps between petals to inner regions of the petal. This may have the additional benefit of re-ducing the need to make sure that two independent petals are kept at a consistent spacing. Anexample of this is shown in Fig. 9, with a close-up view in Fig. 10. The profiles to create thisare defined in Table 3.2, where the shifted region falls between r1 and r2.

Table 3. The profile definitions for shortening petals.

r < r1 r1 ≤ r ≤ r2 r2 < rA1(r) A(r) 1 A(r)A2(r) 0 1−A(r) 0

3.3. Isolating regions

This may also be used to change the shape of a region of the outer edge of a petal, moving amore complex shape to the inside and replacing it with something simpler. As shapes on theinterior of the petal could have additional shielding placed all the way around them to mitigatesun glint, they may be more readily manufacturable than the outer edge. The higher toleranceedge could then be chosen as a shape less difficult to manufacture (e.g. a straight line.) Oneexample is shown in Fig. 11, and a close-up view in Fig. 12. The equations for this are shownin Table 3.3; here S(r) is the simpler profile on the edge.



Fig. 7. An occulter with connecting elements between its petals.



Fig. 8. A close view of the trusses, with the profiles outlined. A1(r) is in green, A2(r) is inblue, and A3(r) is in red. The three profiles are identical outside the truss region.

Table 4. The profile definitions for isolating regions.

r < r1 r1 ≤ r ≤ r2 r2 < rA1(r) A(r) S(r) A(r)A2(r) 0 S(r)−A(r) 0

For an isolated region, the difference between the nominal and modified j = 1 term is:

∫ r2

r1

exp

[πiλ z

(r2 +ρ2)]JN

(2πrρ

λ z

)sin [πS(r)]− sin [πS(r)−πA(r)]− sin [πA(r)]

πrdr (29)

We note that using larger r1 and r2 can make the contribution from the Bessel term significantlylarger, especially when 2πrρ/(λ z) is close to N. If the region being isolated is near the petaltip, this can cause more light to be diffracted into the aperture than other modifications mightcause, hence the higher profile in Fig. 6.

3.4. Reduced petal number

Branching may also be used to reduce the number of petals. Consider the following pair ofprofiles in Table 3.4:

Table 5. The profile definitions for reducing petal number with branching.

r < r1 r1 ≤ rA1(r) A(r) A(r)+Δ(r)A2(r) 0 Δ(r)



Fig. 9. An occulter with the gaps between petals shifted to the centers of the petals for thefirst 2.5 meters of the petal.



Fig. 10. A close-up of the shifted gaps between petals on the occulter. A1(r) is in red, A2(r)is in blue.

Here Δ(r) is the new shape of the outer edge of the petal; this procedure pulls a single petaloutward into two subpetals after r = r1. If we choose r1 to be sufficiently larger than a so that thetwo profiles are joined for most of the petal, the two reflected petals can be in practice deployedas a single petal with a complicated tip, and don’t require separate deployment mechanisms. Inparticular, requiring Δ(r) = 0 at the tip will bring the subpetals back together.

As with asymmetry, we select Δ(r) to minimize the j = 1 term in Eq. (27). In particular, wecan define a Δ∗(r):

Δ∗(r) = cos

(πΔ(r)+

πA(r)2

), (30)

and note that

sinπ(A(r)+Δ(r))− sinπΔ(r) = sinπA(r)secπA(r)

2cos

(πΔ(r)+

πA(r)2

), (31)

such that the j = 1 term in Eq. (27) becomes:(∫ R

0exp

[πiλ z

(r2 +ρ2)]JN

(2πrρ

λ z

)sinπA(r)sec πA(r)

2 Δ∗(r)π

rdr

). (32)

In other words, we can finding a mapping Δ(r)→ Δ∗(r) which turns the problem of minimizingthe j = 1 term into a linear optimization. We then find Δ∗(r) to minimize the j = 1 term vialinear optimization, following for example [15], and invert the mapping in Eq. (30) to produceour outer edge Δ:

Δ(r) =arccosΔ∗(r)

π− A(r)

2. (33)



Fig. 11. An occulter with a 2m section of the edge moved into the center of the petal, andreplaced by a straight edge.



Fig. 12. A close-up of an isolated region at the end of a petal. A1(r) is in red, A2(r) is inblue.



Fig. 13. An occulter with the number of petals reduced by changing the tip shape.

One example of this can be seen in Fig. 13.Multiple profiles can also be used in combination with asymmetry; in this case, Eq. (9)

becomes:

Eβ (ρ,φ) =E0 exp

(2πiz

λ

)(1− 2π

iλ z∫ R

0exp

[πiλ z

(r2 +ρ2)]J0

(2πrρ

λ z

)[A1(r)−A2(r)+ . . .]rdr

)

−E0 exp

(2πiz

λ

) ∞

∑j=1

i jN4πiλ z

(∫ R

0exp

[πiλ z

(r2 +ρ2)]JjN

(2πrρ

λ z

)(34)

× sin jπA1(r)− sin jπA2(r)+ . . .

jπcos [ jNφ − jβ (r)]rdr

).

However, the optimization associated with this integral is more nonlinear than either Eq. (9) or



Eq. (32), and may prove difficult to produce satisfactory results.

4. General form

As a note, both branched and asymmetric occulters may be considered subsets of the mostgeneral form of an occulter, which uses a set of arbitrary profiles f1(r), f2(r), . . . , f2m(r) tocreate its shape. As with Eq. (2) and Eq. (26), we can write the set of points defining theocculter:

Ω f ={(r,θ) : 0 ≤ r ≤ R,θ ∈ Θ f (r)}

where Θ f (r) =N−1⋃

n=0

[2πnN

+πN

f1(r),2πnN

+πN

f2(r)]

∪[

2πnN

+πN

f3(r),2πnN

+πN

f4(r)]∪ . . .

∪[

2πnN

+πN

f2m−1(r),2πnN

+πN

f2m(r)]

(35)

−1 ≤ f1(r) ≤ f2(r) ≤... ≤ f2m(r) ≤ 1, (36)

which produce the following electric field:

Eβ (ρ,φ) =E0 exp

(2πiz

λ

)(1− 2π

iλ z

×∫ R

0exp

[πiλ z

(r2 +ρ2)]J0

(2πrρ

λ z

)12

m

∑�=1

[ f2�(r)− f2�−1(r)]rdr

)(37)

−E0 exp

(2πiz

λ

) ∞

∑j=1

4πi jN

iλ z

(∫ R

0exp

[πiλ z

(r2 +ρ2)]JjN

(2πrρ

λ z

)F(r,φ)

jπrdr

),

where F(r,φ) is:

F(r,φ) =12

m

∑�=1

[sin( jπ f2�(r))− sin( jπ f2�−1(r))

jπcos( jNφ)

−cos( jπ f2�(r))− cos( jπ f2�−1(r))jπ

sin( jNφ)]. (38)

This is the most general form for describing the electric field from a binary occulter with petals,under the condition that each of the petals is identical under a rotation by 2π/N radians. Wenote additionally that by choosing f1(r), f2(r), etc. appropriately, Eq. (37) will reduce to astandard, asymmetric, or branched occulter. (For example, letting f1(r) = −A(r) and f2(r) =A(r) reduces Eq. (37) to Eq. (10).) However, modifying an occulter via this process will bemore difficult than any of the modifications shown above, as both the matching condition inthe φ -independent term and the quadratic equalities in the series will have to be met, and so wehave not produced any designs with the general form thus far.

5. Conclusions and future work

In this paper, we describe a general form for all occulters symmetric under a rotation, and delveinto two specific cases of this—asymmetric and branched occulters—with potential engineeringapplications. We have presented a general optimization framework for taking advantage of the



additional degree of freedom introduced by petal asymmetry, and provided an example of apetal profile that can be produced with this optimization. We have also demonstrated a numberof possible branched profiles, and described practical applications. We hope to improve thequadratic optimization procedure to be able to make more conclusive statements about theglobal minima of the cost functions.

These designs are intended to illustrate possible avenues for modifying the design to meetcertain objectives, such fewer petals, broader bandwidth, or simplified engineering. Tolerancinganalysis has not been performed and is currently under study—we expect the results of thisanalysis will inform the utility of these modifications.

We also note a couple additional directions that could be investigated with nonlinear opti-mization: rather than creating A(r) and β (r) separately, they could be created simultaneouslyin a single optimization. Additionally, while using only the j = 1 term keeps the optimizationmore simple, higher terms can be included to potentially allow even fewer petals. Both of thesewould increase the complexity of the optimization considerably, however, and this is a tradeoffworth considering when examining these occulters.

Acknowledgements

The authors would like to thank Robert Vanderbei and David Spergel for useful discussions.This work was performed under NASA contract NNX08AL58G, as part of the AstrophysicsStrategic Missions Concept Studies (ASMCS) series of exoplanet concept studies.



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Designing asymmetric and branched petals for planet-finding occulters

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