Designing asymmetric and branched petals for planet-finding occulters

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 telescope largely undisturbed everywhere outside the occulter. The shape of an occulter for finding planets is carefully prescribed to destructively interfere the starlight at the telescope, while leaving the 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 the resulting occulter. One example of these is shown in Fig. 1.

These petals are approximations to radial apodization functions, and the diffraction from the petals can be modeled as the propagation integral for a smooth apodization, plus a series of additional terms which represent the effect of using binary petals. These functions can be defined analytically [12, 14] or by optimization [15]; either method can produce occulters that remove 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 is located. Adding more petals makes these terms also become small at points further and further radially outward from the center. In general, we select the number of petals for an occulter by adding petals until the additional terms in the pupil plane of the telescope are small enough to be 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 would be catastrophic to an occulter mission.

However, the petals are not required to be symmetric about their center axis, nor are they limited to being defined by a single smooth function. These are additional degrees of freedom that have not been investigated previously. In this paper, we demonstrate that by introducing an optimized asymmetry into the petals, the additional terms can be minimized across the telescope aperture, 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 useful modifications 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 as in polar coordinates[15, 16]:

Ω = {(r, θ) : 0 \leq r \leq R, θ ∊ Θ (r)}

where Θ (r) = ⋃_{n = 0}^{N - 1} [\frac{2 π n}{N} - \frac{π}{N} A (r), \frac{2 π n}{N} + \frac{π}{N} A (r)] .

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

Ω_{β} = (r, θ) : 0 \leq r \leq R, θ ∊ Θ_{β} (r)

where Θ_{β} (r) = ⋃_{n = 0}^{N - 1} [\frac{2 π n}{N} - \frac{π}{N} A (r) + \frac{β (r)}{N}, \frac{2 π n}{N} + \frac{π}{N} A (r) + \frac{β (r)}{N}]

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

\begin{matrix} E_{β} (ρ, ϕ) = E_{0} \exp (\frac{2 πiz}{λ}) (1 - \frac{1}{iλz} \\ \times \int_{0}^{R} \int_{Θ_{β} (r)} \exp [\frac{πi}{λz} (r^{2} + ρ^{2})] \exp [\frac{2 πirρ}{λz} \cos (θ - ϕ)] r d r d θ) . \end{matrix}

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)/(λz)) gives:

\begin{matrix} E_{β} (ρ, ϕ) = E_{0} \exp (\frac{2 πiz}{λ}) (1 - \frac{1}{iλz} \\ \times \int_{0}^{R} \int_{Θ_{β} (r)} \exp [\frac{πi}{λz} (r^{2} + ρ^{2})] [\sum_{m = - \infty}^{\infty} i^{m} J_{m} (\frac{2 πrρ}{λz}) \exp [im (θ - ϕ)]] r d r d θ), \end{matrix}

which can be rewritten as:

\begin{matrix} E_{β} (ρ, ϕ) = E_{0} \exp (\frac{2 πiz}{λ}) (1 - \sum_{m = - \infty}^{\infty} \frac{i^{m} \exp (- imϕ)}{iλz} \\ \times \int_{0}^{R} \exp [\frac{πi}{λz} (r^{2} + ρ^{2})] J_{m} (\frac{2 πrρ}{λz}) [\int_{Θ_{β} (r)} \exp (imθ) d θ] r d r) . \end{matrix}

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

\int_{Θ_{β} (r)} \exp (imθ) d θ = \sum_{n = 0}^{N - 1} \int_{\frac{2 πn}{N} - \frac{π}{N} A (r) + \frac{β (r)}{N}}^{\frac{2 πn}{N} + \frac{π}{N} A (r) + \frac{β (r)}{N}} \exp (imθ) d θ

which gives:

E_{β} (ρ, ϕ) = E_{0} \exp (\frac{2 πiz}{λ}) (1 - \frac{2 π}{iλz} \int_{0}^{R} \exp [\frac{πi}{λz} (r^{2} + ρ^{2})] J_{0} (\frac{2 πrρ}{λz}) A (r) r d r)

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) = E_{0} \exp (\frac{2 πiz}{λ}) (1 - \frac{2 π}{iλz} \int_{0}^{R} \exp [\frac{πi}{λz} (r^{2} + ρ^{2})] J_{0} (\frac{2 πrρ}{λz}) A (r) r d r)

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 shadow across the aperture of a telescope.

2.1. Optimization

The number of petals required for an occulter is determined by the wavelength band the occulter must 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 the series 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 the scattered light comes from the j = 1 term in the infinite series. To reduce the number of petals required 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). To deal with this, we separate the real and imaginary parts of the j = 1 term, ignoring the constant phase term – exp(2πiz/λ)i ^N-1:

E_{1} (ρ, ϕ) ≔ (R_{c} + i I_{c}) \cos (Nϕ) + (R_{s} + i I_{s}) \sin (Nϕ), where

R_{c} (β (r), ρ, λ) = E_{0} \frac{4}{λz} (\int_{0}^{R} \cos (\frac{π}{λz} (r^{2} + ρ^{2})) J_{N} (\frac{2 πrρ}{λz}) \sin (πA (r)) \cos β (r) r d r),

I_{c} (β (r), ρ, λ) = E_{0} \frac{4}{λz} (\int_{0}^{R} \sin (\frac{π}{λz} (r^{2} + ρ^{2})) J_{N} (\frac{2 πrρ}{λz}) \sin (πA (r)) \cos β (r) r d r),

R_{s} (β (r), ρ, λ) = E_{0} \frac{4}{λz} (\int_{0}^{R} \cos (\frac{π}{λz} (r^{2} + ρ^{2})) J_{N} (\frac{2 πρ}{λz}) \sin (πA (r)) \sin β (r) r d r),

I_{s} (β (r), ρ, λ) = E_{0} \frac{4}{λz} (\int_{0}^{R} \sin (\frac{π}{λz} (r^{2} + ρ^{2})) J_{N} (\frac{2 πrρ}{λz}) \sin (πA (r)) \sin β (r) r d r) .

Note: for most occulters, A(r) = 1 and sin(jπA(r)) = 0 from r = 0 out to some r = a, and so the integral is generally done as ∫^R _a . We can then discretize ρ and λ, and write a conservative vector of the maximum intensities at each (ρ_i, λ_i):

\overset{}{f} (β) = [\begin{matrix} R_{c} {(β (r), ρ_{1}, λ_{1})}^{2} + I_{c} {(β (r), ρ_{1}, λ_{1})}^{2} + R_{s} {(β (r), ρ_{1}, λ_{1})}^{2} + I_{s} {(β (r), ρ_{1}, λ_{1})}^{2} \\ R_{c} {(β (r), ρ_{2}, λ_{2})}^{2} + I_{c} {(β (r), ρ_{2}, λ_{2})}^{2} + R_{s} {(β (r), ρ_{2}, λ_{2})}^{2} + I_{s} {(β (r), ρ_{2}, λ_{2})}^{2} \\ ⋮ \end{matrix}] .

Each of these terms are a conservative upper bound on the intensity ∣E _j=1∣², as:

E_{j = 1} = (R_{c} + i I_{c}) \cos (Nϕ) + (R_{s} + i I_{s}) \sin (Nϕ)

{∣ E_{j = 1} ∣}^{2} = ((R_{c} + i I_{c}) \cos (Nϕ) + (R_{s} + i I_{s}) \sin (Nϕ)) ((R_{c} - i I_{c}) \cos (Nϕ) + (R_{s} - i I_{s}) \sin (Nϕ))

_

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

\frac{1}{2} {[{(R_{c}^{2} + I_{c}^{2} - R_{s}^{2} - I_{s}^{2})}^{2} + {(2 R_{s} R_{c} + {2 I}_{s} I_{c})}^{2}]}^{\frac{1}{2}} = \frac{1}{2} (I_{c}^{4} + 2 I_{c}^{2} I_{s}^{2} + I_{s}^{4}

Since

8 I_{c} I_{s} R_{c} R_{s} - 2 I_{s}^{2} R_{c}^{2} - 2 I_{c}^{2} R_{s}^{2} \leq 2 I_{s}^{2} R_{c}^{2} + 2 I_{c}^{2} R_{s}^{2},

we have

\frac{1}{2} {[{(R_{c}^{2} + I_{c}^{2} - R_{s}^{2} - I_{s}^{2})}^{2} + {(2 R_{s} R_{c} + {2 I}_{s} I_{c})}^{2}]}^{\frac{1}{2}} \leq \frac{1}{2} (I_{c}^{4} + 2 I_{c}^{2} I_{s}^{2} + I_{s}^{4}

\Rightarrow {∣ E_{j = 1} ∣}^{2} \leq (R_{c}^{2} + I_{c}^{2} + R_{s}^{2} + I_{s}^{2}) .

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

max_{ϕ} ∣ E_{j = 1} ∣ = \frac{1}{2} (R_{c}^{2} + I_{c}^{2} + R_{s}^{2} + I_{s}^{2}) + \frac{1}{2} {[{(R_{c}^{2} + I_{c}^{2} - R_{s}^{2} - I_{s}^{2})}^{2} + (2 R_{s} R_{c} + 2 I_{s} I_{c})^{2}]}^{\frac{1}{2}} .

This will be equal to R ² _c + I ² _c + R ² _s + I ² _s when R_sI_c = R_cI_s, and smaller otherwise.

To optimize, we write an unconstrained optimization to minimize the norm of f̄:

Minimize : {∥ \bar{f} (β) ∥}_{p},

where ∥ · ∥_p is the p-norm of f̄. In the following, we use the 1-norm of f̄, 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 to β(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 the same locations, but the choice is arbitrary.)

Fig. 2. β(r) for the THEIA occulter.

2.3. Results

There are number of ways to do unconstrained optimization; for our optimization, we used a conjugate gradient algorithm with Polak-Ribière update step to find local minima of the cost function. The conjugate gradient method is a optimization algorithm for unconstrained functions which uses a set of search directions which are successively orthogonal to one another to converge quickly to a minimum [17]. We use a nonlinear variant of this; other nonlinear optimization 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 be optimized by taking advantage of their properties. There are a few ways for calculating the update step for nonlinear conjugate gradient methods, but Polak-Ribie`re update tends to converge faster than others.

This algorithm was applied to the design of the THEIA occulter [18]. The THEIA occulter is a 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, and from 700 – 1000nm when the occulter is moved in to 35000km. (See Table 2.2 and [18] for more details.) The nominal occulter is shown in Fig. 1.

For the THEIA occulter, the optimization was only run from 250nm to 450nm, as at longer wavelengths the Bessel function in the integral is small, and so is the perturbation. (J_n(x) reaches its first peak at x ≈ n [19].) The close-in distance was not involved, as the scaling relations make the field identical to a corresponding field at the further distance. Figure 2 shows one example 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 improved at other wavelengths as well, but the performance gain at 250nm is the best. The occulter itself is 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 method can 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 at 250nm. Center. The shadow at the THEIA telescope aperture with 12 petals and β(r) = 0 at 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.

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 they obey 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 use A ₁(r), A ₂(r), … to create a shape with a more complicated (but still symmetric) structure. (As the simplest application would be to make the ends of a petal branch out like a tree, we will refer to these as branched occulters.) In a similar manner to Eq. (2), we can define the set of points that are covered by a branched occulter as:

Ω_{b} = {(r, θ) : 0 \leq r \leq R, θ \in Θ_{b} (r)}

where M is the the number of profiles, and 0 ≤ A _2m ≤ A _2m-1 ≤ … A ₁ ≤ 1. (If the petal uses an odd number of profiles, A _2m = 0.) The electric field following this occulter is:

E_{b} (ρ, ϕ) = E_{0} \exp (\frac{2 πiz}{λ}) (1 - \frac{1}{iλz}

By choosing:

A (r) = A_{1} (r) - A_{2} (r) + A_{3} (r) - \dots,

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

3.1. Structural elements

Under dynamic and thermal loading, the petals of an occulter can move independently of one another, rotating in the occulter plane and bending out of it. These motions change the optical properties of the occulter; depending on the design, these motions may change the shape significantly enough that the occulter can no longer suppress the starlight sufficiently to detect a planet. (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 help support and fix them with respect to their neighbors. These can be created using multiple profiles to define the edges of the elements. Figure 7 shows an example of a simple cross introduced between petals; a close-up view is shown in Fig. 8. This design may be created by between r = r ₁ and r = r ₂ by choosing A ₁(r), A ₂(r) and A ₃(r) as shown in Table 3.1. Here T(r) defines the outer edge of the profile, and W(r) defines the width of the crossbars.

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

Fig. 6. Image plane performance at 250nm for the modifications described in this paper. A planet 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 rather than longer ones. (For THEIA, for example, petal length was constrained by the size of the launch vehicle fairing.) This can be accomplished by using multiple profiles to move the small gaps between petals to inner regions of the petal. This may have the additional benefit of reducing the need to make sure that two independent petals are kept at a consistent spacing. An example of this is shown in Fig. 9, with a close-up view in Fig. 10. The profiles to create this are defined in Table 3.2, where the shifted region falls between r ₁ and r ₂.

Table 3. The profile definitions for shortening petals.

3.3. Isolating regions

This may also be used to change the shape of a region of the outer edge of a petal, moving a more complex shape to the inside and replacing it with something simpler. As shapes on the interior of the petal could have additional shielding placed all the way around them to mitigate sun glint, they may be more readily manufacturable than the outer edge. The higher tolerance edge could then be chosen as a shape less difficult to manufacture (e.g. a straight line.) One example is shown in Fig. 11, and a close-up view in Fig. 12. The equations for this are shown in 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. A ₁(r) is in green, A ₂(r) is in blue, and A ₃(r) is in red. The three profiles are identical outside the truss region.

Table 4. The profile definitions for isolating regions.

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

\int_{r_{1}}^{r_{2}} \exp [\frac{πi}{λz} (r^{2} + ρ^{2})] J_{N} (\frac{2 πrρ}{λz}) \frac{\sin [πS (r)] - \sin [πS (r) - πA (r)] - \sin [πA (r)]}{π} r d r

We note that using larger r ₁ and r ₂ can make the contribution from the Bessel term significantly larger, especially when 2’03C0;r’03C1;/(λz) is close to N. If the region being isolated is near the petal tip, this can cause more light to be diffracted into the aperture than other modifications might cause, 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 of profiles in Table 3.4:

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

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

Fig. 10. A close-up of the shifted gaps between petals on the occulter. A ₁(r) is in red, A ₂ (r) is in blue.

Here Δ(r) is the new shape of the outer edge of the petal; this procedure pulls a single petal outward into two subpetals after r = r ₁. If we choose r ₁ to be sufficiently larger than a so that the two profiles are joined for most of the petal, the two reflected petals can be in practice deployed as a single petal with a complicated tip, and don’t require separate deployment mechanisms. In particular, 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, we can define a Δ^* (r):

Δ^{*} (r) = \cos (π Δ (r) + \frac{πA (r)}{2}),

and note that

\sin π (A (r) + Δ (r)) - \sin π Δ (r) = \sin πA (r) \sec \frac{πA (r)}{2} \cos (π Δ (r) + \frac{πA (r)}{2}),

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

(\int_{0}^{R} \exp [\frac{πi}{λz} (r^{2} + ρ^{2})] J_{N} (\frac{2 πrρ}{λz}) \frac{\sin πA (r) \sec \frac{πA (r)}{2} Δ^{*} (r)}{π} r d r) .

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

Δ (r) = \frac{\arccos Δ^{*} (r)}{π} - \frac{A (r)}{2} .

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

Fig. 12. A close-up of an isolated region at the end of a petal. A ₁(r) is in red, A ₂(r) is in blue.

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

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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:

E_{β} (ρ, ϕ) = E_{0} \exp (\frac{2 πiz}{λ}) (1 - \frac{2 π}{iλz}

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 most general form of an occulter, which uses a set of arbitrary profiles f ₁(r), f ₂(r), …, f _2m(r) to create its shape. As with Eq. (2) and Eq. (26), we can write the set of points defining the occulter:

Ω_{f} = {(r, θ) : 0 \leq r \leq R, θ \in Θ_{f} (r)}

- 1 \leq f_{1} (r) \leq f_{2} (r) \leq \dots \leq f_{2 m} (r) \leq 1,

which produce the following electric field:

E_{β} (ρ, ϕ) = E_{0} \exp (\frac{2 πiz}{λ}) (1 - \frac{2 π}{iλz}

where F(r, ϕ) is:

F (r, ϕ) = \frac{1}{2} \sum_{ℓ = 1}^{m} [\frac{\sin (jπ f_{2 ℓ} (r)) - \sin (jπ f_{2 ℓ - 1} (r))}{jπ} \cos (jNϕ)

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. We note additionally that by choosing f ₁(r), f ₂(r), etc. appropriately, Eq. (37) will reduce to a standard, asymmetric, or branched occulter. (For example, letting f ₁(r) = -A(r) and f ₂(r) = A(r) reduces Eq. (37) to Eq. (10).) However, modifying an occulter via this process will be more difficult than any of the modifications shown above, as both the matching condition in the ϕ-independent term and the quadratic equalities in the series will have to be met, and so we have 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 delve into two specific cases of this — asymmetric and branched occulters — with potential engineering applications. 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 a petal profile that can be produced with this optimization. We have also demonstrated a number of possible branched profiles, and described practical applications. We hope to improve the quadratic optimization procedure to be able to make more conclusive statements about the global minima of the cost functions.

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

We also note a couple additional directions that could be investigated with nonlinear optimization: rather than creating A(r) and β(r) separately, they could be created simultaneously in a single optimization. Additionally, while using only the j = 1 term keeps the optimization more simple, higher terms can be included to potentially allow even fewer petals. Both of these would increase the complexity of the optimization considerably, however, and this is a tradeoff worth 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 Astrophysics Strategic Missions Concept Studies (ASMCS) series of exoplanet concept studies.

References and links

1. J. Schneider, “The extrasolar planets encyclopaedia,” http://exoplanet.eu.

2. S. Udry, D. Fischer, and D. Queloz, “A Decade of Radial-Velocity Discoveries in the Exoplanet Domain,” in Protostars and Planets V, B. Reipurth, D. Jewitt, and K. Keil, ed. (University of Arizona Press, 2006), pp. 685 – 699.

3. D. Charbonneau, T. M. Brown, A. Burrows, and G. Laughlin, “When Extrasolar Planets Transit Their Parent Stars,” in Protostars and Planets V, B. Reipurth, D. Jewitt, and K. Keil, ed. (University of Arizona Press, 2006), pp. 701–716.

4. O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical limits on extrasolar terrestrial planet detection with coronagraphs,” Astrophys. J. Suppl. Ser. 167, 81–99 (2006). [CrossRef]

5. D. J. Des Marais, M. O. Harwit, K. W. Jucks, J. F. Kasting, D. N. Lin, J. I. Lunine, J. Schneider, S. Seager, W. A. Traub, and N. J. Woolf,. “Remote sensing of planetary properties and biosignatures on extrasolar terrestrial planets,” Astrobiology 2, 153–181 (2002). [CrossRef] [PubMed]

6. J. W. Evans, “A photometer for the measurement of sky brightness near the sun,” J. Opt. Soc. Am 38, 1083–1085 (1948). [CrossRef] [PubMed]

7. G. E. Brueckner, R. A. Howard, M. J. Koomen, C. M. Korendyke, D. J. Michels, J. D. Moses, D. G. Socker, K. P. Dere, P. L. Lamy, A. Llebaria, M. V. Bout, R. Schwenn, G. M. Simnett, D. K. Bedford, and C. J. Eyles, “The large angle spectroscopic coronagraph (LASCO),” Sol. Phys. 162, 357–402 (1995). [CrossRef]

8. R. A. Howard, J. D. Moses, and D. G. Socker, “Sun earth connection coronal and heliospheric investigation (SECCHI),” Proc. SPIE 4139,259–283 (2000). [CrossRef]

9. S. Vivès, P. Lamy, M. Venet, P. Levacher, and J. L. Boit, “The giant, externally-occulted-coronagraph ASPIICS for the PROBA-3 formation flying mission,” Proc. SPIE 6689, 66890F (2007). [CrossRef]

10. E. Verroi, F. Frassetto, and G. Naletto, “Analysis of diffraction from the occulter edges of a giant externally occulted solar coronagraph,” J. Opt. Soc. Am. A 25, 182–189 (2008). [CrossRef]

11. L. Spitzer, “The beginnings and future of space astronomy,” Am. Sci. 50, 473–484 (1962).

12. C. J. Copi and G. D. Starkman, “The Big Occulting Steerable Satellite [BOSS],” Astrophys. J. 532, 581–592 (2000). [CrossRef]

13. A. B. Schultz, I. J. E. Jordan, M. Kochte, D. Fraquelli, F. Bruhweiler, J. M. Hollis, K. G. Carpenter, R. G. Lyon, M. DiSanti, C. Miskey, J. Leitner, R. D. Burns, S. R. Starin, M. Rodrigue, M. S. Fadali, D. Skelton, H. M. Hart, F. Hamilton, and K.-P. Cheng, “UMBRAS: A matched occulter and telescope for imaging extrasolar planets,” Proc. SPIE 4860, 54–61 (2003). [CrossRef]

14. W. Cash, “Detection of earth-like planets around nearby stars using a petal-shaped occulter,” Nature 442, 51–53 (2006). [CrossRef] [PubMed]

15. R. J. Vanderbei, E. J. Cady, and N. J. Kasdin, “Optimal occulter design for finding extrasolar planets,” Astrophys. J. 665, 794–798 (2007). [CrossRef]

16. R. J. Vanderbei, D. Spergel, and N. J. Kasdin, “Circularly symmetric apodization via star-shaped masks,” Astro-phys. J. 599, 686–694 (2003). [CrossRef]

17. J. R. Shewchuk, “An Introduction to the Conjugate Gradient Method Without the Agonizing Pain,” http://www.cs.cmu.edu/͠quake-papers/painless-conjugate-gradient.pdf.

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 astronomy,” http://www.astro.princeton.edu/͠dns/Theia/nas theia v14.pdf.

19. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1944).

20. E. Cady, S. Shaklan, N. J. Kasdin, and D. Spergel, “A method for modifying occulter shapes,” Proc. SPIE 7440, 744007 (2009). [CrossRef]

21. P. Dumont, S. Shaklan, E. Cady, J. Kasdin, and R. Vanderbei, “Analysis of external occulters in the presence of defects,” Proc. SPIE 7440, 744008 (2009). [CrossRef]

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Fig. 1. A 20-petal occulter.

Fig. 2. β(r) for the THEIA occulter.

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

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

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

Fig. 8. A close view of the trusses, with the profiles outlined. A ₁(r) is in green, A ₂(r) is in blue, and A ₃(r) is in red. The three profiles are identical outside the truss region.

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

Fig. 10. A close-up of the shifted gaps between petals on the occulter. A ₁(r) is in red, A ₂ (r) is in blue.

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

Fig. 12. A close-up of an isolated region at the end of a petal. A ₁(r) is in red, A ₂(r) is in blue.

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

Tables (5)

Table 1. THEIA occulter specifications.

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

Table 3. The profile definitions for shortening petals.

Table 4. The profile definitions for isolating regions.

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