Abstract

One of the proposed methods for finding small extrasolar planets is through use of an occulter, a spacecraft which flies in formation with a space-based telescope to block the light from a star, while leaving nearby planets unaffected. This is accomplished by placing the occulter far enough from the telescope to give it a small angular size, and by carefully choosing the shape to strongly suppress the starlight at the telescope aperture. For most designs, this shape takes the form of a number of bilaterally-symmetric structures called petals, arrayed about a circular central disk. In this paper, we show that the necessary number of petals may be reduced by the introduction of an asymmetry in the petal shape, and describe a a general procedure for producing such a shape by optimization for any occulter with petals. In addition, we show that permitting openings within each petal allows a number of additional modifications to be made without affecting the suppression.

© 2010 Optical Society of America

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    [CrossRef]

2009 (2)

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

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]

2008 (1)

2007 (2)

S. Vivés, 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]

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

2006 (2)

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]

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

2003 (2)

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]

R. J. Vanderbei, D. Spergel, and N. J. Kasdin, "Circularly symmetric apodization via star-shaped masks," Astrophys. J. 599, 686-694 (2003).
[CrossRef]

2002 (1)

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]

2000 (2)

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

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]

1995 (1)

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]

1962 (1)

L. Spitzer, "The beginnings and future of space astronomy," Am. Sci. 50, 473-484 (1962).

1948 (1)

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

Bedford, D. K.

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]

Boit, J. L.

S. Vivés, 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]

Bout, M. V.

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]

Brueckner, G. E.

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]

Bruhweiler, F.

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]

Burns, R. D.

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]

Cady, E.

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

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]

Cady, E. J.

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

Carpenter, K. G.

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]

Cash, W.

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

Cheng, K.-P.

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]

Collins, B.

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]

Copi, C. J.

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

Dere, K. P.

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]

Des Marais, D. J.

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]

DiSanti, M.

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]

Dumont, P.

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]

Evans, J.W.

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

Eyles, C. J.

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]

Fadali, M. S.

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]

Fraquelli, D.

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]

Frassetto, F.

Guyon, O.

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]

Hamilton, F.

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]

Hart, H. M.

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]

Harwit, M. O.

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]

Hollis, J. M.

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]

Howard, R. A.

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]

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]

Jordan, I. J. E.

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]

Jucks, K. W.

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]

Kasdin, J.

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]

Kasdin, N. J.

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

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

R. J. Vanderbei, D. Spergel, and N. J. Kasdin, "Circularly symmetric apodization via star-shaped masks," Astrophys. J. 599, 686-694 (2003).
[CrossRef]

Kasting, J. F.

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]

Kochte, M.

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]

Koomen, M. J.

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]

Korendyke, C. M.

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]

Kuchner, M. J.

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]

Lamy, P.

S. Vivés, 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]

Lamy, P. L.

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]

Leitner, J.

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]

Levacher, P.

S. Vivés, 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]

Lin, D. N.

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]

Llebaria, A.

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]

Lunine, J. I.

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]

Lyon, R. G.

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]

Michels, D. J.

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]

Miskey, C.

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]

Moses, J. D.

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]

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]

Naletto, G.

Pluzhnik, E. A.

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]

Ridgway, S. T.

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]

Rodrigue, M.

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]

Schneider, J.

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]

Schultz, A. B.

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]

Schwenn, R.

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]

Seager, S.

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]

Shaklan, S.

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

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]

Simnett, G. M.

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]

Skelton, D.

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]

Socker, D. G.

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]

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]

Spergel, D.

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

R. J. Vanderbei, D. Spergel, and N. J. Kasdin, "Circularly symmetric apodization via star-shaped masks," Astrophys. J. 599, 686-694 (2003).
[CrossRef]

Spitzer, L.

L. Spitzer, "The beginnings and future of space astronomy," Am. Sci. 50, 473-484 (1962).

Starin, S. R.

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]

Starkman, G. D.

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

Traub, W. A.

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]

Vanderbei, R.

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]

Vanderbei, R. J.

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

R. J. Vanderbei, D. Spergel, and N. J. Kasdin, "Circularly symmetric apodization via star-shaped masks," Astrophys. J. 599, 686-694 (2003).
[CrossRef]

Venet, M.

S. Vivés, 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]

Verroi, E.

Vivés, S.

S. Vivés, 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]

Woolf, N. J.

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]

Am. Sci. (1)

L. Spitzer, "The beginnings and future of space astronomy," Am. Sci. 50, 473-484 (1962).

Astrobiology (1)

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]

Astrophys. J. (3)

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

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

R. J. Vanderbei, D. Spergel, and N. J. Kasdin, "Circularly symmetric apodization via star-shaped masks," Astrophys. J. 599, 686-694 (2003).
[CrossRef]

Astrophys. J. Suppl. Ser. (1)

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]

J. Opt. Soc. Am (1)

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

J. Opt. Soc. Am. A (1)

Nature (1)

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

Proc. SPIE (5)

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

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]

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]

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]

S. Vivés, 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]

Sol. Phys. (1)

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]

Other (6)

J. Schneider, "The extrasolar planets encyclopaedia," http://exoplanet.eu.

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.

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.

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.

N. J. Kasdin, P. Atcheson, M. Beasley, R. Belikov, M. Blouke, E. Cady, D. Calzetti, C. Copi, S. Desch, P. Dumont, 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. Starkman, 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.

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

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Figures (13)

Fig. 1.
Fig. 1.

A 20-petal occulter.

Fig. 2.
Fig. 2.

β(r) for the THEIA occulter.

Fig. 3.
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.
Fig. 4.

The 12-petal THEIA occulter, with asymmetry.

Fig. 5.
Fig. 5.

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

Fig. 6.
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.
Fig. 7.

An occulter with connecting elements between its petals.

Fig. 8.
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.
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.
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.
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.
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.
Fig. 13.

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

Tables (5)

Tables Icon

Table 1. THEIA occulter specifications.

Tables Icon

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

Tables Icon

Table 3. The profile definitions for shortening petals.

Tables Icon

Table 4. The profile definitions for isolating regions.

Tables Icon

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

Equations (71)

Equations on this page are rendered with MathJax. Learn more.

Ω = { ( r , θ ) : 0 r R , θ Θ ( r ) }
whereΘ ( r ) = n = 0 N 1 [ 2 π n N π N A ( r ) , 2 π n N + π N A ( r ) ] .
Ω β = ( r , θ ) : 0 r R , θ Θ β ( r )
where Θ β ( r ) = n = 0 N 1 [ 2 π n N π N A ( r ) + β ( r ) N , 2 π n N + π N A ( r ) + β ( r ) N ]
E β ( ρ , ϕ ) = E 0 exp ( 2 πiz λ ) ( 1 1 iλz × 0 R Θ β ( r ) exp [ πi λz ( r 2 + ρ 2 ) ] exp [ 2 πirρ λz cos ( θ ϕ ) ] r d r d θ ) .
E β ( ρ , ϕ ) = E 0 exp ( 2 πiz λ ) ( 1 1 iλz × 0 R Θ β ( r ) exp [ πi λz ( r 2 + ρ 2 ) ] [ m = i m J m ( 2 πrρ λz ) exp [ im ( θ ϕ ) ] ] r d r d θ ) ,
E β ( ρ , ϕ ) = E 0 exp ( 2 πiz λ ) ( 1 m = i m exp ( imϕ ) iλz × 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J m ( 2 πrρ λz ) [ Θ β ( r ) exp ( imθ ) d θ ] r d r ) .
Θ β ( r ) exp ( imθ ) d θ = n = 0 N 1 2 πn N π N A ( r ) + β ( r ) N 2 πn N + π N A ( r ) + β ( r ) N exp ( imθ ) d θ
= { 2 π A ( r ) m = 0 , 2 N m sin ( A ( r ) / N ) exp [ imβ ( r ) / N ] , m = ± N , ± 2 N , , ± jN , 0 , else ,
E β ( ρ , ϕ ) = E 0 exp ( 2 πiz λ ) ( 1 2 π iλz 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J 0 ( 2 πrρ λz ) A ( r ) r d r )
E 0 exp ( 2 πiz λ ) j = 1 i jN exp ( ijN ϕ ) iλz
× ( 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J jN ( 2 πrρ λz ) 2 sin ( jπA ( r ) ) j exp [ ijβ ( r ) ] r d r )
E 0 exp ( 2 πiz λ ) j = 1 i j N exp ( ijN ϕ ) iλz
× ( 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J jN ( 2 πrρ λz ) 2 sin ( jπA ( r ) ) j exp [ ijβ ( r ) ] r d r )
= E 0 exp ( 2 πiz λ ) ( 1 2 π iλz 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J 0 ( 2 πrρ λz ) A ( r ) r d r )
E 0 exp ( 2 πiz λ ) j = 1 i j N 4 π iλz
× ( 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J jN ( 2 πrρ λz ) sin ( jπA ( r ) ) cos [ jNϕ ( r ) ] r d r ) .
E ( ρ , ϕ ) = E 0 exp ( 2 πiz λ ) ( 1 2 π iλz 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J 0 ( 2 πrρ λz ) A ( r ) r d r )
E 0 exp ( 2 πiz λ ) j = 1 j jN 4 π iλz cos ( jNϕ )
× ( 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J jN ( 2 πrρ λz ) sin ( jπA ( r ) ) r d r ) .
E 1 ( ρ , ϕ ) ( R c + i I c ) cos ( ) + ( R s + i I s ) sin ( ) , where
R c ( β ( r ) , ρ , λ ) = E 0 4 λz ( 0 R cos ( π λz ( r 2 + ρ 2 ) ) J N ( 2 πrρ λz ) sin ( πA ( r ) ) cos β ( r ) r d r ) ,
I c ( β ( r ) , ρ , λ ) = E 0 4 λz ( 0 R sin ( π λz ( r 2 + ρ 2 ) ) J N ( 2 πrρ λz ) sin ( πA ( r ) ) cos β ( r ) r d r ) ,
R s ( β ( r ) , ρ , λ ) = E 0 4 λz ( 0 R cos ( π λz ( r 2 + ρ 2 ) ) J N ( 2 πρ λz ) sin ( πA ( r ) ) sin β ( r ) r d r ) ,
I s ( β ( r ) , ρ , λ ) = E 0 4 λz ( 0 R sin ( π λz ( r 2 + ρ 2 ) ) J N ( 2 πrρ λz ) sin ( πA ( r ) ) sin β ( r ) r d r ) .
f ( β ) = [ 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 ] .
E j = 1 = ( R c + i I c ) cos ( ) + ( R s + i I s ) sin ( )
E j = 1 2 = ( ( R c + i I c ) cos ( ) + ( R s + i I s ) sin ( ) ) ( ( R c i I c ) cos ( ) + ( R s i I s ) sin ( ) )
= ( R c 2 + I c 2 ) cos 2 ( ) + ( R s 2 + I s 2 ) sin 2 ( ) + ( 2 R s R c + 2 I s I c ) sin ( ) cos ( )
= 1 2 ( R c 2 + I c 2 + R s 2 + I s 2 ) + cos ( 2 ) 2 ( R c 2 + I c 2 R s 2 I s 2 ) + sin ( 2 ) 2 ( 2 R s R c + 2 I s I c ) .
_
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 ] 1 2 = 1 2 ( I c 4 + 2 I c 2 I s 2 + I s 4
+ 2 I c 2 R c 2 2 I s 2 R c 2 + R c 4 + 8 I c I s R c R s
2 I c 2 R s 2 + 2 I s 2 R s 2 + 2 R c 2 R s 2 + R s 4 ) 1 2 .
8 I c I s R c R s 2 I s 2 R c 2 2 I c 2 R s 2 2 I s 2 R c 2 + 2 I c 2 R s 2 ,
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 ] 1 2 1 2 ( I c 4 + 2 I c 2 I s 2 + I s 4
+ 2 I c 2 R c 2 + 2 I s 2 R c 2 + R c 4 + 2 I c 2 R s 2
+ 2 I s 2 R s 2 + 2 R c 2 R s 2 + R s 4 ) 1 2
= 1 2 ( R c 2 + I c 2 + R s 2 + I s 2 )
E j = 1 2 ( R c 2 + I c 2 + R s 2 + I s 2 ) .
max ϕ E j = 1 = 1 2 ( R c 2 + I c 2 + R s 2 + I s 2 ) + 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 ] 1 2 .
Minimize : f ̄ ( β ) p ,
Ω b = { ( r , θ ) : 0 r R , θ Θ b ( r ) }
where Θ b ( r ) = n = 0 N 1 { m = 1 M [ 2 πn N π N A 2 m 1 ( r ) , 2 πn N π N A 2 m ( r ) ]
m = 1 M [ 2 πn N + π N A 2 m ( r ) , 2 πn N + π N A 2 m 1 ( r ) ] } .
E b ( ρ , ϕ ) = E 0 exp ( 2 πiz λ ) ( 1 1 iλz
× 0 R Θ b ( r ) exp [ πi λz ( r 2 + ρ 2 ) ] exp [ 2 πirρ λz cos ( θ ϕ ) ] r d r d θ )
= E 0 exp ( 2 πiz λ ) ( 1 2 iλz
× 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J 0 ( 2 πrρ λz ) [ A 1 ( r ) A 2 ( r ) + ] r d r )
E 0 exp ( 2 πiz λ ) j = 1 i j N 4 π iλz cos ( jNϕ )
× 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J jN ( 2 πrρ λz ) sin A 1 ( r ) sin A 2 ( r ) + r d r )
A ( r ) = A 1 ( r ) A 2 ( r ) + A 3 ( r ) ,
r 1 r 2 exp [ πi λz ( r 2 + ρ 2 ) ] J N ( 2 πrρ λz ) sin [ πS ( r ) ] sin [ πS ( r ) πA ( r ) ] sin [ πA ( r ) ] π r d r
Δ * ( r ) = cos ( π Δ ( r ) + πA ( r ) 2 ) ,
sin π ( A ( r ) + Δ ( r ) ) sin π Δ ( r ) = sin πA ( r ) sec πA ( r ) 2 cos ( π Δ ( r ) + πA ( r ) 2 ) ,
( 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J N ( 2 πrρ λz ) sin πA ( r ) sec πA ( r ) 2 Δ * ( r ) π r d r ) .
Δ ( r ) = arccos Δ * ( r ) π A ( r ) 2 .
E β ( ρ , ϕ ) = E 0 exp ( 2 πiz λ ) ( 1 2 π iλz
0 R exp [ πi λz ( r 2 + ρ 2 ) ] J 0 ( 2 πrρ λz ) [ A 1 ( r ) A 2 ( r ) + ] r d r )
E 0 exp ( 2 πiz λ ) j = 1 i jN 4 π iλz ( 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J jN ( 2 πrρ λz )
× sin A 1 ( r ) sin A 2 ( r ) + j π cos [ jNϕ ( r ) ] r d r ) .
Ω f = { ( r , θ ) : 0 r R , θ Θ f ( r ) }
where Θ f ( r ) = n = 0 N 1 [ 2 πn N + π N f 1 ( r ) , 2 πn N + π N f 2 ( r ) ]
[ 2 πn N + π N f 3 ( r ) , 2 πn N + π N f 4 ( r ) ]
[ 2 πn N + π N f 2 m 1 ( r ) , 2 πn N + π N f 2 m ( r ) ]
1 f 1 ( r ) f 2 ( r ) f 2 m ( r ) 1 ,
E β ( ρ , ϕ ) = E 0 exp ( 2 πiz λ ) ( 1 2 π iλz
× 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J 0 ( 2 πrρ λz ) 1 2 = 1 m [ f 2 ( r ) f 2 1 ( r ) ] r d r )
E 0 exp ( 2 πiz λ ) j = 1 4 π i jN iλz ( 0 R exp [ πi λz ( r 2 + ρ 2 ) ] J jN ( 2 πrρ λz ) F ( r , ϕ ) r d r ) ,
F ( r , ϕ ) = 1 2 = 1 m [ sin ( f 2 ( r ) ) sin ( f 2 1 ( r ) ) cos ( jNϕ )
cos ( f 2 ( r ) ) cos ( f 2 1 ( r ) ) sin ( jNϕ ) ] .

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