Abstract

An occulter is a large diffracting screen which may be flown in conjunction with a telescope to image extrasolar planets. The edge is shaped to minimize the diffracted light in a region beyond the occulter, and a telescope may be placed in this dark shadow to view an extrasolar system with the starlight removed. Errors in position, orientation, and shape of the occulter will diffract additional light into this region, and a challenge of modeling an occulter system is to accurately and quickly model these effects. We present a fast method for the calculation of electric fields following an occulter, based on the concept of the boundary diffraction wave: the 2D structure of the occulter is reduced to a 1D edge integral which directly incorporates the occulter shape, and which can be easily adjusted to include changes in occulter position and shape, as well as the effects of sources—such as exoplanets—which arrive off-axis to the occulter. The structure of a typical implementation of the algorithm is included.

© 2012 OSA

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  1. N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
    [CrossRef]
  2. S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
    [CrossRef]
  3. T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
    [CrossRef]
  4. R. J. Vanderbei, E. J. Cady, and N. J. Kasdin, “Optimal occulter design for finding extrasolar planets,” Astrophys. J. 665, 794–798 (2007).
    [CrossRef]
  5. E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE 7010, 70101X (2008).
    [CrossRef]
  6. C. J. Copi and G. D. Starkman, “The Big Occulting Steerable Satellite [BOSS],” Astrophys. J. 532, 581–592 (2000).
    [CrossRef]
  7. W. Cash, “Detection of earth-like planets around nearby stars using a petal-shaped occulter,” Nature 442, 51–53 (2006).
    [CrossRef] [PubMed]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  9. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  10. R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15(24), 15935–15951, (2007).
    [CrossRef] [PubMed]
  11. E. Cady, “Design, tolerancing, and experimental verification of occulters for finding extrasolar planets,” PhD thesis, Princeton University, 2010.
  12. A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. of Phys. 67, 87–92 (1999).
    [CrossRef]
  13. 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]
  14. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am 52, 615–622 (1962).
    [CrossRef]
  15. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am 52, 626–636 (1962).
    [CrossRef]
  16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing (Cambridge University Press, 2007).

2011 (1)

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

2010 (2)

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[CrossRef]

2009 (1)

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)

E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE 7010, 70101X (2008).
[CrossRef]

2007 (2)

R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15(24), 15935–15951, (2007).
[CrossRef] [PubMed]

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 (1)

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

2000 (1)

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

1999 (1)

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. of Phys. 67, 87–92 (1999).
[CrossRef]

1962 (2)

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am 52, 615–622 (1962).
[CrossRef]

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am 52, 626–636 (1962).
[CrossRef]

Bach, V.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Cady, E.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[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]

E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE 7010, 70101X (2008).
[CrossRef]

E. Cady, “Design, tolerancing, and experimental verification of occulters for finding extrasolar planets,” PhD thesis, Princeton University, 2010.

Cady, E. J.

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[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]

Cash, W.

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

Copi, C. J.

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

Dailey, D.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[CrossRef]

Dubra, A.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. of Phys. 67, 87–92 (1999).
[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]

Dumont, P. J.

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

Ferrari, J. A.

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. of Phys. 67, 87–92 (1999).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing (Cambridge University Press, 2007).

Glassman, T.

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Johnson, A.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[CrossRef]

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.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE 7010, 70101X (2008).
[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]

Lawson, P. R.

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

Lisman, D.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Lo, A.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[CrossRef]

Lo, A. S.

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

Lynch, D.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Macintosh, B.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Marchen, L.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Martin, S.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Mikula, J.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Miyamoto, K.

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am 52, 615–622 (1962).
[CrossRef]

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am 52, 626–636 (1962).
[CrossRef]

Noecker, M. C.

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

Oakes, E.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing (Cambridge University Press, 2007).

Pueyo, L.

E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE 7010, 70101X (2008).
[CrossRef]

R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15(24), 15935–15951, (2007).
[CrossRef] [PubMed]

Rudd, R. E.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Shaklan, S.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[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]

Shaklan, S. B.

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

Shelton, H.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[CrossRef]

Sivaramakrishnan, A.

Soummer, R.

E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE 7010, 70101X (2008).
[CrossRef]

R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15(24), 15935–15951, (2007).
[CrossRef] [PubMed]

Spergel, D. N.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Starkman, G. D.

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

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing (Cambridge University Press, 2007).

Thomson, M.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Vanderbei, R.

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[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]

Vanderbei, R. J.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15(24), 15935–15951, (2007).
[CrossRef] [PubMed]

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

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing (Cambridge University Press, 2007).

Vogrin, J.

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[CrossRef]

Walkemeyer, P.

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

Wolf, E.

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am 52, 626–636 (1962).
[CrossRef]

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am 52, 615–622 (1962).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Am. J. of Phys. (1)

A. Dubra and J. A. Ferrari, “Diffracted field by an arbitrary aperture,” Am. J. of Phys. 67, 87–92 (1999).
[CrossRef]

Astrophys. J. (2)

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

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

J. Opt. Soc. Am (2)

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part I,” J. Opt. Soc. Am 52, 615–622 (1962).
[CrossRef]

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave part II,” J. Opt. Soc. Am 52, 626–636 (1962).
[CrossRef]

Nature (1)

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

Opt. Express (1)

Proc. SPIE (5)

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]

E. Cady, L. Pueyo, R. Soummer, and N. J. Kasdin, “Performance of hybrid occulters using apodized pupil Lyot coronagraphy,” Proc. SPIE 7010, 70101X (2008).
[CrossRef]

N. J. Kasdin, D. N. Spergel, R. J. Vanderbei, D. Lisman, S. Shaklan, M. Thomson, P. Walkemeyer, V. Bach, E. Oakes, E. Cady, S. Martin, L. Marchen, B. Macintosh, R. E. Rudd, J. Mikula, and D. Lynch, “Advancing technology for starlight suppression via an external occulter,” Proc. SPIE 8151, 81510J (2011).
[CrossRef]

S. B. Shaklan, M. C. Noecker, T. Glassman, A. S. Lo, P. J. Dumont, N. J. Kasdin, E. J. Cady, R. Vanderbei, and P. R. Lawson, “Error budgeting and tolerancing of starshades for exoplanet detection,” Proc. SPIE 7731, 77312G (2010).
[CrossRef]

T. Glassman, A. Johnson, A. Lo, D. Dailey, H. Shelton, and J. Vogrin, “Error analysis on the NWO starshade,” Proc. SPIE 7731, 773150 (2010).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

E. Cady, “Design, tolerancing, and experimental verification of occulters for finding extrasolar planets,” PhD thesis, Princeton University, 2010.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing (Cambridge University Press, 2007).

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Figures (5)

Fig. 1
Fig. 1

An example of a typical occulter with 20 petals, with the associated Cartesian and polar coordinate systems.

Fig. 2
Fig. 2

A diagram of the points, coordinate systems, and vectors used in this paper. The left grid is in the plane of the occulter, and the right grid in the plane of the telescope aperture.

Fig. 3
Fig. 3

A diagram showing the projected extent of the aperture given by geometric optics. A plane wave incident on the aperture in plane A with vector direction p produces a shifted aperture in plane B. Two angles which parameterize the direction of the plane wave, ψ1 and ψ2, are shown as well.

Fig. 4
Fig. 4

A comparison of the time to run the two propagation algorithms, as a function of number of points along the edge of the entrance aperture.

Fig. 5
Fig. 5

Intensity at the plane of the telescope aperture following the occulter, plotted on a base-10 logarithmic scale. The telescope aperture is shown by a dashed circle.

Tables (1)

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Algorithm 1: Pseudocode representation of BDW calculation

Equations (49)

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Ω = { ( r , θ ) : 0 r R , θ Θ ( r ) }
where Θ ( r ) = n = 0 N p 1 [ 2 π n N p π N p A ( r ) , 2 π n N p + π N p A ( r ) ] ,
z 3 π 4 λ [ ( x ξ ) 2 + ( y η ) 2 ] max 2 .
U ( ξ , η ) = A exp ( 2 π i z λ ) ( 1 1 i λ z Ω exp { π i λ z [ ( r cos θ ξ ) 2 + ( r sin θ η ) 2 ] } r d r d θ ) = A exp ( 2 π i z λ ) ( 1 1 i λ z ) Ω exp { π i λ z [ ( x ξ ) 2 + ( y η ) 2 ] } d x d y ) ,
U ( P ) = 1 4 π S ( U ( Q ) Q exp ( i k s ) s exp ( i k s ) s Q U ( Q ) ) n d S 1 4 π S V ( Q , P ) n d S .
W ( Q , p ) = 1 4 π A exp ( i k p r ) exp ( i k s ) s s ^ × p 1 + s ^ p ,
U ( P ) = S × W ( Q , P ) n d S .
s r r ,
s = s ,
s ^ = s s .
U 0 ( P ) = A exp ( i k p r ) ,
U ap ( P ) = Ω × W ( Q , P ) n d Ω ,
U ap ( P ) = A exp ( i k p r ) + U ( B ) ( P ) for points inside the extent of the aperture = U ( B ) ( P ) otherwise ,
U ( B ) ( P ) = Ω W ( Q , P ) d .
U 0 ( P ) = U ( P ) + U ap ( P ) ,
U ( P ) = Ω W ( Q , P ) d for points in the geometric shadow of the occulter , = A e i k p r Ω W ( Q , P ) d otherwise .
U ( G ) ( P ) = 0 for points inside the extent of the aperture = U 0 ( P ) otherwise .
s r r = ( x ξ , y η , z ) ,
s = s = [ z 2 + ( x ξ ) 2 + ( y η ) 2 ] 1 / 2 ,
s ^ = s s .
p = ( 0 , 0 , 1 ) ,
p r = 0 ,
s ^ × p = 1 s ( y η , ( x ξ ) , 0 ) ,
1 + s ^ p = 1 z s ,
exp ( i k s ) = exp ( i k z ) exp [ i k ( s z ) ] ,
W ( Q , P ) = 1 4 π A exp ( i k z ) exp [ i k ( s z ) ] s z s ^ × p .
p = ( sin ψ 1 cos ψ 2 , sin ψ 1 sin ψ 2 , cos ψ 1 ) ,
p r = x sin ψ 1 cos ψ 2 + y sin ψ 1 sin ψ 2 .
f = z sin ψ 1 + ( x ξ ) cos ψ 2 cos ψ 1 + ( y η ) sin ψ 2 cos ψ 1 ,
g = ( x ξ ) sin ψ 2 + ( y η ) cos ψ 2 ,
h = z cos ψ 1 ( x ξ ) cos ψ 2 sin ψ 1 ( y η ) sin ψ 2 sin ψ 1 s p .
s = [ f 2 + g 2 + h 2 ] 1 / 2 ,
1 + s ^ p = 1 h s ,
s ^ × p = 1 s ( f sin ψ 2 + g cos ψ 2 cos ψ 1 , f cos ψ 2 + g sin ψ 2 cos ψ 1 , g sin ψ 1 )
exp ( i k h ) = exp [ i k z cos ψ 1 i k ( x ξ ) cos ψ 2 sin ψ 1 i k ( y η ) sin ψ 2 sin ψ 1 ] = exp ( i k p r ) exp ( i k z cos ψ 1 ) exp [ i k ( ξ cos ψ 2 + η sin ψ 2 ) sin ψ 1 ] ,
W ( Q , P ) = 1 4 π A exp ( i k z cos ψ 1 ) exp [ i k ( ξ cos ψ 2 + η sin ψ 2 ) sin ψ 1 ] exp ( i k ( s h ) ) s h s ^ × p ,
1 s ( s h ) = 1 f 2 + g 2 ( 1 + 1 [ 1 + f 2 + g 2 h 2 ] 1 / 2 ) 2 f 2 + g 2 ,
s h f 2 + g 2 2 h ,
W ( Q , P ) 1 2 π A exp ( i k z cos ψ 1 ) exp [ i k ( ξ cos ψ 2 + η sin ψ 2 ) sin ψ 1 ] exp ( i k f 2 + g 2 2 h ) f 2 + g 2 v ( f , g ) ,
v ( f , g ) = ( f sin ψ 2 + g cos ψ 2 cos ψ 1 , f cos ψ 2 + g sin ψ 2 cos ψ 1 , g sin ψ 1 ) = s × p .
U ( G ) ( ξ 0 , η 0 , z ) = A exp ( i k z cos ψ 1 ) exp [ i k ( ξ 0 cos ψ 2 + η 0 sin ψ 2 ) sin ψ 1 ] , for points outside the shadow = 0 otherwise .
( x j ( m ) , y j ( m ) , 0 ) = ( [ x j + x j + 1 ] / 2 , [ y j + y j + 1 ] / 2 , 0 ) ,
( x j ( ) , y j ( ) , 0 ) j = ( x j + 1 x j , y j + 1 y j , 0 ) .
f j = z sin ψ 1 + ( x j ( m ) ξ 0 ) cos ψ 2 cos ψ 1 + ( y j ( m ) η 0 ) sin ψ 2 cos ψ 1 ,
g j = ( x j ( m ) ξ 0 ) sin ψ 2 + ( y j ( m ) η 0 ) cos ψ 2 ,
h j = z cos ψ 1 ( x j ( m ) ξ 0 ) cos ψ 2 sin ψ 1 ( y j ( m ) η 0 ) sin ψ 2 sin ψ 1 ,
T 1 j = f j 2 + g j 2 ,
T 2 j = v ( f j , g j ) d = x j ( ) ( z sin ψ 1 sin ψ 2 + ( y j ( m ) η 0 ) cos ψ 1 ) y j ( ) ( z sin ψ 1 cos ψ 2 + ( x j ( m ) ξ 0 ) cos ψ 1 ) ,
U ( B ) ( ξ 0 , η 0 , z ) = 1 2 π A exp ( i k z cos ψ 1 ) exp [ i k ( ξ 0 cos ψ 2 + η 0 sin ψ 2 ) sin ψ 1 ] × j = 1 N exp ( i k T 1 j 2 h j ) T 1 j T 2 j .

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