A method for modifying occulter shapes

Transcription

1 A method for modifying occulter shapes Eric Cady a, Stuart B. Shaklan b, N. Jeremy Kasdin a,anddavidspergel a a Princeton University, Princeton, NJ, USA b Jet Propulsion Laboratory, Pasadena, CA, USA ABSTRACT An occulter is an instrument designed to suppress starlight by diffraction from its edges; most are designed to be circular, with a set of identical petals running around the outside. Proposed space-based occulters are lightweight, deployed screens tens of meters in diameter with challenging accuracy requirements. We describe a general method for modifying the shape of an occulter to accommodate engineering considerations and show how to calculate the resulting wavefront. This method can be used to place hinges and tensioning elements between petals, to reduce tolerancing requirements by allowing gaps between petals to be moved elsewhere, and to potentially reduce the number of petals required on an occulter. 1. INTRODUCTION One of the most promising technologies being developed for exoplanet science is direct imaging recording the photons emitted directly for the planet and filtering them from the starlight by optical means. Direct imaging would allow spectra of planets to be taken directly, outer planets to be found without requiring them to undergo a full orbit, and rocky Earth-like planets to be found in habitable zones. One method proposed to do direct imaging of planets from space is with the use of an occulter, a large spacecraft that flies between a telescope and its target star to block the starlight and leave light from nearby planets unaffected. Current designsarebasedonansmoothapodizationar) which changes with radius and is constant in angle; a number of different apodizations have been proposed to do this. 1 3 The shadow from the occulter can then be calculated in a straightforward manner. Unfortunately these smooth apodizations are virtually impossible to manufacture with real materials, and so we approximate them with binary occulters, which either allow all or none of the light through at any point. The resulting shape has a series of structures along the edge, called petals, which vary their width with radius so that if a circle is drawn at a radius r, the fraction of the that circle which is blocked by petals is Ar). It can be shown 4 that the resulting electric field is then the same as that of a smooth apodization, with a series of additional terms from the scattering. Previous designs have used a single function Ar) to define the apodization; however, it turns out that multiple profiles can be used, as long as the total area blocked remains equal to Ar) ateachr. It also requires that these structures be repeated about the edge, as with a single profile, to keep too much light from being scattered into the telescope aperture.) In this paper, we show how the fields from these branched occulters may be calculated, and give some potential applications.. PROPAGATION AROUND BRANCHED OCCULTERS Consider a planar occulter with N symmetric petals, whose width is determined by a single function Ar). The set of points which are on the occulter is defined as Ω in polar coordinates 3 : 4 where Ω={r, θ) : r R, θ Θr)} Θr) = N 1 n= [ N π N Ar), N + π N Ar)] 1) Techniques and Instrumentation for Detection of Exoplanets IV, edited by Stuart B. Shaklan, Proc. of SPIE Vol. 744, SPIE CCC code: X/9/$18 doi: / Proc. of SPIE Vol

2 Suppose we instead want a series of functions A 1 r),a r),... to define the occulter, creating a symmetric petal with multiple branches. With m profiles, Ω b can be written: Ω b = {r, θ) : r R, θ Θ b r)} where Θ b r) = N 1 [ n= N π N A 1r), N π N A r)... N π N A mr), N + π N A mr)... N + π N A r), N + π N A 1r) ] ) We can use this to calculate the electric field at the telescope aperture by using Babinet s principle: the field will be the difference between a propagated plane wave and the Fresnel propagation of the electric field in a hole with the shape of Ω b : ) E b ρ, φ) = E e πiz = E e πiz 1 1 iz 1 1 R iz Ω b r,θ) Θ b r) z r +ρ ) e πirρ z cos θ φ) rdrdθ z r +ρ ) e πirρ z cos θ φ) rdrdθ where ρ, φ) are polar coordinates in the plane of the telescope, and is the wavelength. We can use the Jacobi-Anger expansion to rewrite Eq. 4 as a sum of integrals over a simpler integrand in θ: E β ρ, φ) =E e πiz 1 1 R [ ) ] ) z r +ρ ) πrρ i m J m e imθ φ) rdrdθ 5) iz z Θ b r) m= ) 3) 4) which can be rewritten as: E β ρ, φ) =E e πiz 1 m= i m e imφ iz R ) [ z r +ρ ) πrρ J m z e imθ dθ Θ b r) ] ) rdr 6) The integrals over θ can be evaluated explicitly: N 1 e imθ N π βr) N Ar)+ N dθ = e imθ dθ... Θ b r) which gives: = n= + + E b ρ, φ) = E e πiz E e πiz N 1 n= N 1 n= N π βr) N A1r)+ N N + π βr) N Amr)+ N N π βr) N Amr)+ N N + π βr) N A1r)+ N N + π βr) N Ar)+ N e imθ dθ... e imθ dθ 7) π [A 1 r) A r)+a 3 r)...], m =, j [sin jπa 1r) sin jπa r)+sinjπa 3 r)...], m = ±N,±N,±3N,..., 8), else, 1 π iz j=1 R 1) j π iz ) ) z r +ρ ) πrρ J [A 1 r) A r)+...] rdr z R ) ) z r +ρ ) πrρ sin jπa1 r) sin jπa r)+... J jn rdr z jπ cosjnφ π/))) 9) Proc. of SPIE Vol

3 If we choose our profiles so that: Ar) =A 1 r) A r)+... 1) then we end up with the same φ-independent part as the shadow produced by an occulter without branching: Eρ, φ) = E e πiz 1 π R ) ) z r +ρ ) πrρ J Ar)rdr iz z E e πiz 1) j π R ) ) z r +ρ ) πrρ sin jπa1 r) sin jπa r)+... J jn rdr iz z jπ j=1 cosjnφ π/))) 11) and the only difference show up in the integrals in the series. 3. APPLICATIONS The similarity between Eq. 9 and Eq. 11 means that branches can be easily added as long as they maintain the condition in Eq. 1. The only differences will come in the scattering terms, which can be made small in the central region by increasing N. Often, if the terms in the series were small in Eq. 11, they will be small in Eq. 9, especially if the changes made by branching are relatively small. Some potential applications of this are listed below. All simulations are done for the THEIA occulter, 5 which is 4m tip-to-tip in diameter and is designed to operate at two distances: 55km and 35km. It images from 5nm to 7nm at 55km, and 7nm to 1nm at 35km. All shadows are shown at 5nm and 55km, as the magnitude of the terms in the series in Eq. 9 are the largest at this wavelength. 3.1 Tensioning elements Occulter designs generally have a solid central disk out to a radial distance a, atwhichpointthepetals begin. Beyond this, the petals are floppy and unconnected to each other. To mitigate this, we can introduce a structure between the petals further out than the disk to hold the petals together. As long as holes can be added elsewhere to ensure A 1 r) A r) +... = Ar), this will not affect the main shadow, only the perturbation terms. One thing to note is that this rules out adding concentric rings, as these always correspond to Ar) = 1. However, elements between petals that are not perpendicular to lines of constant θ, including most straight-line segments, are acceptable, though they cannot be so thick that holes cannot be added in the center. One example is the creation of a truss structure between petals, such as the one in Fig. 1. We define the distance between the outer edge of the truss and the nominal profile to be T r), and the width of the truss members to be W r). If the truss falls between radial points r 1 and r,thenwecanwritetheshapewith the branching formulation: A 1 r) = A r) = A 3 r) = Ar) r<r 1 OR r>r Ar)+T r) r 1 r r Ar) r<r 1 OR r>r Ar)+T r) W r) r 1 r r Ar) r<r 1 OR r>r Ar) W r) r 1 r r The corresponding shadow is in Fig.. For thin truss members, the first few perturbation terms remain virtually unchanged, as: sin jπa 1 r) sin jπa r)+sinjπa 3 r) = sinjπar), r < r 1 OR r>r sin jπa 1 r) sin jπa r)+sinjπa 3 r) = sinjπar)[cosjπtr)+cosjπwr) cos jπt r) W r))] +cosjπar)[sinjπtr) sin jπwr) sin jπt r) W r))], r 1 r r 13) 1) Proc. of SPIE Vol

4 and the second part is sin jπar) for small W r). 3. Gaps One of the difficult parts of occulter deployment is the presence of long gaps between petals with small widths. These are extremely long sometimes on the order of meters and thin, usually millimeter-scale or less. These are expected to be quite difficult to manufacture, deploy, and maintain to the necessary accuracies. An option when creating optimized occulters is to explicitly include a lower limit on the size of the gap, though this constraint may narrow the parameter space available. Another option is to use the branching formulation to move these gaps to a different part of the petal, where they can be manufactured as a hole in a single piece of material, rather than the space between two floppy, unattached petals. We can do this with the branching notation by letting the shifted section be between the radial positions r 1 and r, and choosing: A 1 r) = A r) = Ar) r<r 1 OR r>r 1 r 1 r r r<r 1 OR r>r 1 Ar) r 1 r r 14) Choosing r 1 and r appropriately allows the minimum width of the gap to be nearly as large as desired. One example of this is shown in Fig. 3, and corresponding shadows in Fig. 4. A similar method can be used to compensate for structural devices, such as hinges between petals, which would be placed in the gaps between petals This would effectively change Ar); the area blocked by the hinge may be added to the center to compensate. 3.3 Petals Adding additional petals to an occulter increases the risk and complexity of the system, as structures are required to unfold each petal, and the entire occulter will fail if any of them fails to work. Designs try to use as few petals as they can, although the series in Eq. 9 sets a lower limit. If we wish to get around this, we can partially combine two or more) petals into a single one, so that they can be treated as a single structure for deployment purposes. We can then reshape the tips of these combined petals to minimize the first one or more terms in the series. Strictly speaking, breaking up the petals in this manner will only reduce the number of parts that need to unfold; there will usually remain the same number of tips at the end, unless the profiles are constrained to bring them back together.) While this can be done using any number of combined petals, it is easiest to demonstrate with just two. We can write these in the branching formulation as: A 1 r) = A r) = Ar) r<r 1 Ar)+Δr) r r 1 r<r 1 Δr) r r 1 15) We wish to minimize the first perturbation term in the series, which we can do by choosing Δr) to minimize the magnitude of: R ) ) z r +ρ ) πrρ sin πar) Δr)) sin πδr) J N rdr 16) z π for appropriate ρ,, and z. This can simplified considerably by noting that: sin πar) Δr)) sin πδr) =sinπar)sec πar) cos πδr)+ πar) ) 17) Proc. of SPIE Vol

5 If we let: then Eq. 16 becomes: R Δ r) = cos πδr)+ πar) ) Δr) = arccos Δ r) π Ar) ) ) z r +ρ ) πrρ sin πar)sec πar) Δ r) J N rdr z π 18) 19) ) which is a linear optimization in Δ r), and straightforward to solve globally. One example of this is shown in Fig. 5, and the corresponding shadow in Fig. 6. Here, this methods reduces the number of petals from to 16. Generally, we place constraints on Δ r) so that the petal doesn t separate into two parts until near the tip. It is worth noting that, without these constraints, Δ r) = is a global minimizer; the result breaks each petal into two identical, symmetric petals, equivalent to doubling the petal number. 4. CONCLUSIONS We have shown a number of modifications that can be applied to an occulter to attempt to improve its performance, stability, and manufacturability. These are all variations on a central concept: keeping the fraction of the circle with radius r blocked by the petals constant. We expect to extend this work to a more general class of designs by removing the requirement of symmetric petals. Acknowledgements This work was performed under NASA contract NNX8AL58G, as part of the Astrophysics Strategic Missions Concept Studies ASMCS) series of exoplanet concept studies. REFERENCES 1. C.J. Copi and G.D. Starkman. The Big Occulting Steerable Satellite [BOSS]. Astrophysical Journal, 53:581 59,.. W. Cash. Detection of earth-like planets around nearby stars using a petal-shaped occulter. Nature, 44:51 53, R.J. Vanderbei, E.J. Cady, and N.J. Kasdin. Optimal occulter design for finding extrasolar planets. Astrophysical Journal, 665: , R.J. Vanderbei, D. Spergel, and N.J. Kasdin. Circularly symmetric apodization via star-shaped masks. Astrophysical Journal, 599: , N. Jeremy Kasdin, Paul Atcheson, Matt Beasley, Rus Belikov, Morley Blouke, Eric Cady, Daniela Calzetti, Craig Copi, Steve Desch, Phil Dumont, Dennis Ebbets, Rob Egerman, Alex Fullerton, Jay Gallagher, Jim Green, Olivier Guyon, Sally Heap, Rolf Jansen, Ed Jenkins, Jim Kasting, Ritva Keski- Kuha, Marc Kuchner, Roger Lee, Don J. Lindler, Roger Linfield, Doug Lisman, Rick Lyon, John MacKenty, Sangeeta Malhotra, Mark McCaughrean, Gary Mathews, Matt Mountain, Shouleh Nikzad, Bob OConnell, William Oegerle, Sally Oey, Debbie Padgett, Behzad A Parvin, Xavier Prochaska, James Rhoads, Aki Roberge, Babak Saif, Dmitry Savransky, Paul Scowen, Sara Seager, Bernie Seery, Kenneth Sembach, Stuart Shaklan, Mike Shull, Oswald Siegmund, Nathan Smith, Remi Soummer, David Spergel, Phil Stahl, Glenn Starkman, Daniel K Stern, Domenick Tenerelli, Wesley A. Traub, John Trauger, Jason Tumlinson, Ed Turner, Bob Vanderbei, Roger Windhorst, Bruce Woodgate, and Bob Woodruff. THEIA: Telescope for habitable exoplanets and interstellar/intergalactic astronomy. dns/theia/nas theia v14.pdf. Proc. of SPIE Vol

6 Figure 1. An occulter modified to include tensioning elements between its petals. Figure. Left. The shadow of the occulter with no truss structure. Right. The shadow of the occulter with a structure between the petals. Proc. of SPIE Vol

7 Figure 3. An occulter modified to move the gaps between its petals to the centers of the petals. Figure 4. Left. The shadow of the occulter with the gaps untouched. Right. The shadow of the occulter with the gaps moved to the center. Proc. of SPIE Vol

8 Figure 5. An occulter modified to reduce the petal number by adding structure at the petal tips. Figure 6. Left. The shadow of the occulter with standard petals. Right. The shadow of the occulter with two branches on the petals. Proc. of SPIE Vol