Abstract

We present a new method for numerical propagation through Lyot-style coronagraphs using finite occulting masks. Standard methods for coronagraphic simulations involve Fast Fourier Transforms (FFT) of very large arrays, and computing power is an issue for the design and tolerancing of coronagraphs on segmented Extremely Large Telescopes (ELT) in order to handle both the speed and memory requirements. Our method combines a semi-analytical approach with non-FFT based Fourier transform algorithms. It enables both fast and memory-efficient computations without introducing any additional approximations. Typical speed improvements based on computation costs are of about ten to fifty for propagations from pupil to Lyot plane, with thirty to sixty times less memory needed. Our method makes it possible to perform numerical coronagraphic studies even in the case of ELTs using a contemporary commercial laptop computer, or any standard commercial workstation computer.

© 2007 Optical Society of America

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  1. B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Veran, "The Gemini Planet Imager," Proc. SPIE 6272, (2006).
  2. J. L. Beuzit, D. Mouillet, C. Moutou, K. Dohlen, T. Fusco, P. Puget, S. Udry, R. Gratton, H. M. Schmid, M. Feldt, M. Kasper, and The Vlt-Pf Consortium, "A "Planet Finder" instrument for the VLT," in Tenth Anniversary of 51 Peg-b: Status of and prospects for hot Jupiter studies, L. Arnold, F. Bouchy, and C. Moutou, eds., pp. 353-355 (2006).
  3. M. Tamura, K. Hodapp, H. Takami, L. Abe, H. Suto, O. Guyon, S. Jacobson, R. Kandori, J.-I. Morino, N. Murakami, V. Stahlberger, R. Suzuki, A. Tavrov, H. Yamada, J. Nishikawa, N. Ukita, J. Hashimoto, H. Izumiura, M. Hayashi, T. Nakajima, and T. Nishimura, "Concept and science of HiCIAO: high contrast instrument for the Subaru next generation adaptive optics," Proc. SPIE 6269, 62690V (2006)., vol. 6269 presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference (2006).
  4. B. Macintosh, M. Troy, R. Doyon, J. Graham, K. Baker, B. Bauman, C. Marois, D. Palmer, D. Phillion, L. Poyneer, I. Crossfield, P. Dumont, B. M. Levine, M. Shao, G. Serabyn, C. Shelton, G. Vasisht, J. K. Wallace, J.-F. Lavigne, P. Valee, N. Rowlands, K. Tam, and D. Hackett, "Extreme adaptive optics for the Thirty Meter Telescope," Proc. SPIE, 6272, 62720N (2006)., vol. 6272 presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference (2006).
  5. L. Close, "Extrasolar Planet Imaging with the Giant Magellan Telescope," in Proceedings of the conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007. University of California, Berkeley, CA, USA. Edited by Paul Kalas., P. Kalas, ed. (2007).
  6. C. Verinaud, M. Kasper, J.-L. Beuzit, N. Yaitskova, V. Korkiakoski, K. Dohlen, P. Baudoz, T. Fusco, L. Mugnier, and N. Thatte, "EPICS Performance Evaluation through Analytical and Numerical Modeling," in Proceedings of the conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007. University of California, Berkeley, CA, USA. Edited by Paul Kalas., P. Kalas, ed. (2007).
  7. O. Guyon, J. R. P. Angel, C. Bowers, J. Burge, A. Burrows, J. Codona, T. Greene, M. Iye, J. Kasting, H. Martin, D. W. McCarthy, Jr., V. Meadows, M. Meyer, E. A. Pluzhnik, N. Sleep, T. Spears, M. Tamura, D. Tenerelli, R. Vanderbei, B. Woodgate, R. A. Woodruff, and N. J. Woolf, "Telescope to observe planetary systems (TOPS): a high throughput 1.2-m visible telescope with a small inner working angle," Proc. SPIE 6265, 62651R (2006)., vol. 6265 presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference (2006).
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    [CrossRef] [PubMed]
  9. A. Sivaramakrishnan, C. D. Koresko, R. B. Makidon, T. Berkefeld, and M. J. Kuchner, "Ground-based Coronagraphy with High-order Adaptive Optics," Astrophys. J. 552, 397-408 (2001). astro-ph/0103012.
    [CrossRef]
  10. C. Aime and R. Soummer, "The Usefulness and Limits of Coronagraphy in the Presence of Pinned Speckles," Astrophys. J. 612, L85-L88 (2004).
    [CrossRef]
  11. C. Cavarroc, A. Boccaletti, P. Baudoz, T. Fusco, and D. Rouan, "Fundamental limitations on Earth-like planet detection with extremely large telescopes," A&A 447, 397-403 (2006).
  12. 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. 167, 81-99 (2006). arXiv:astro-ph/0608506.
    [CrossRef]
  13. R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, "Speckle noise and dynamic range in coronagraphic images," ApJ 669, 642-656 (2007). arXiv:0706.1739v1.
  14. B. Lyot, " ´ Etude de la couronne solaire en dehors des ´eclipses. Avec 16 figures dans le texte." Zeitschrift fur Astrophysics 5, 73-+ (1932).
  15. C. Aime, R. Soummer, and A. Ferrari, "Total coronagraphic extinction of rectangular apertures using linear prolate apodizations," A&A 389, 334-344 (2002).
  16. R. Soummer, C. Aime, and P. E. Falloon, "Stellar coronagraphy with prolate apodized circular apertures," A&A397, 1161-1172 (2003).
  17. R. Soummer, "Apodized Pupil Lyot Coronagraphs for Arbitrary Telescope Apertures," Astrophys. J. 618, L161- L164 (2005).
    [CrossRef]
  18. R. Soummer, L. Pueyo, A. Ferrari, C. Aime, A. Sivaramakrishnan, and N. Yaitskova, "Apodized Pupil Lyot Coronagraphs for Arbitrary Telescope Apertures. II. Application to Extremely Large Telescopes," submitted to ApJ (2007).
  19. F. Roddier and C. Roddier, "Stellar Coronagraph with Phase Mask," PASP 109, 815-820 (1997).
    [CrossRef]
  20. R. Soummer, K. Dohlen, and C. Aime, "Achromatic dual-zone phase mask stellar coronagraph," A&A403, 369-381 (2003).
  21. D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, "The Four-Quadrant Phase-Mask Coronagraph. I. Principle," PASP 112, 1479-1486 (2000).
    [CrossRef]
  22. L. Abe, F. Vakili, and A. Boccaletti, "The achromatic phase knife coronagraph," A&A 374, 1161-1168 (2001).
  23. M. J. Kuchner and W. A. Traub, "A Coronagraph with a Band-limited Mask for Finding Terrestrial Planets," ApJ 570, 900-908 (2002).
  24. G. Foo, D. M. Palacios, and G. A. Swartzlander, Jr., "Optical vortex coronagraph," Opt. Lett. 30, 3308-3310 (2005).
    [CrossRef]
  25. J. Goodman, Introduction to Fourier Optics (Mac Graw Hill, 1996).
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  27. A. Ferrari, R. Soummer, and C. Aime, "An introduction to stellar coronagraphy," Comptes Rendus Physique 8, 277-287 (2007). arXiv:astro-ph/0703655.
    [CrossRef]
  28. D. H. Bailey and P. N. Swarztrauber, "The Fractional Fourier Transform and Applications," SIAM Review 33 (3), 389-404 (1991).
    [CrossRef]
  29. L. R. Rabiner, R. W. Schafer, and C. M. Rader, "The Chirp z-Transform Algorithm and its Application," Bell Sys. Tech. J. 48 (5), 1249-1291 (1969).
  30. J. O. Smith, Mathematics of the Discrete Fourier Transform (DFT) (W3K Publishing, http://www.w3k.org/books/>, 2007).
  31. C. Aime, "Principle of an Achromatic Prolate Apodized Lyot Coronagraph," PASP 117, 1012-+ (2005).
    [CrossRef]
  32. M. Frigo and S. G. Johnson, "The Design and Implementation of FFTW3," Proceedings of the IEEE 93(2), 216-231 (2005). Special issue on "Program Generation, Optimization, and Platform Adaptation".
    [CrossRef]
  33. J. P. Lloyd and A. Sivaramakrishnan, "Tip-Tilt Error in Lyot Coronagraphs," Astrophys. J. 621, 1153-1158 (2005). arXiv:astro-ph/0503661.
    [CrossRef]
  34. A. Sivaramakrishnan and J. P. Lloyd, "Spiders in Lyot Coronagraphs," Astrophys. J. 633, 528-533 (2005). arXiv:astro-ph/0506564.
    [CrossRef]
  35. L. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," IEEE Transactions on Audio and Electroacoustics 18, 451-455 (1970).
    [CrossRef]
  36. A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, "Amplitude and phase correction for high-contrast imaging using Fourier decomposition," Proc. SPIE, 5905, 368-378 (2005).
  37. A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, "On representing and correcting wavefront errors in high-contrast imaging systems," J. Opt. Soc. Am. A 23 (2006).
  38. N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar Planet Finding via Optimal Apodized-Pupil and Shaped-Pupil Coronagraphs," Astrophys. J. 582, 1147-1161 (2003). URL http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v582n2/56580/56580.web.pdf>.
    [CrossRef]
  39. URL http://www.primatelabs.ca/geekbench/>.
  40. L. A. Poyneer and B. Macintosh, "Spatially filtered wave-front sensor for high-order adaptive optics," J. Opt. Soc. Am. A 21, 810-819 (2004).
    [CrossRef]
  41. R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
    [CrossRef]

2007 (2)

J. T. Trauger and W. A. Traub, "A laboratory demonstration of the capability to image an Earth-like extrasolar planet," Nature 446, 771-773 (2007).
[CrossRef] [PubMed]

A. Ferrari, R. Soummer, and C. Aime, "An introduction to stellar coronagraphy," Comptes Rendus Physique 8, 277-287 (2007). arXiv:astro-ph/0703655.
[CrossRef]

2006 (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. 167, 81-99 (2006). arXiv:astro-ph/0608506.
[CrossRef]

2005 (5)

R. Soummer, "Apodized Pupil Lyot Coronagraphs for Arbitrary Telescope Apertures," Astrophys. J. 618, L161- L164 (2005).
[CrossRef]

G. Foo, D. M. Palacios, and G. A. Swartzlander, Jr., "Optical vortex coronagraph," Opt. Lett. 30, 3308-3310 (2005).
[CrossRef]

M. Frigo and S. G. Johnson, "The Design and Implementation of FFTW3," Proceedings of the IEEE 93(2), 216-231 (2005). Special issue on "Program Generation, Optimization, and Platform Adaptation".
[CrossRef]

J. P. Lloyd and A. Sivaramakrishnan, "Tip-Tilt Error in Lyot Coronagraphs," Astrophys. J. 621, 1153-1158 (2005). arXiv:astro-ph/0503661.
[CrossRef]

A. Sivaramakrishnan and J. P. Lloyd, "Spiders in Lyot Coronagraphs," Astrophys. J. 633, 528-533 (2005). arXiv:astro-ph/0506564.
[CrossRef]

2004 (1)

C. Aime and R. Soummer, "The Usefulness and Limits of Coronagraphy in the Presence of Pinned Speckles," Astrophys. J. 612, L85-L88 (2004).
[CrossRef]

2002 (1)

M. J. Kuchner and W. A. Traub, "A Coronagraph with a Band-limited Mask for Finding Terrestrial Planets," ApJ 570, 900-908 (2002).

2001 (1)

A. Sivaramakrishnan, C. D. Koresko, R. B. Makidon, T. Berkefeld, and M. J. Kuchner, "Ground-based Coronagraphy with High-order Adaptive Optics," Astrophys. J. 552, 397-408 (2001). astro-ph/0103012.
[CrossRef]

2000 (1)

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, "The Four-Quadrant Phase-Mask Coronagraph. I. Principle," PASP 112, 1479-1486 (2000).
[CrossRef]

1997 (1)

F. Roddier and C. Roddier, "Stellar Coronagraph with Phase Mask," PASP 109, 815-820 (1997).
[CrossRef]

1996 (1)

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

1991 (1)

D. H. Bailey and P. N. Swarztrauber, "The Fractional Fourier Transform and Applications," SIAM Review 33 (3), 389-404 (1991).
[CrossRef]

1970 (1)

L. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," IEEE Transactions on Audio and Electroacoustics 18, 451-455 (1970).
[CrossRef]

1969 (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, "The Chirp z-Transform Algorithm and its Application," Bell Sys. Tech. J. 48 (5), 1249-1291 (1969).

ApJ (1)

M. J. Kuchner and W. A. Traub, "A Coronagraph with a Band-limited Mask for Finding Terrestrial Planets," ApJ 570, 900-908 (2002).

Astrophys. J. (6)

J. P. Lloyd and A. Sivaramakrishnan, "Tip-Tilt Error in Lyot Coronagraphs," Astrophys. J. 621, 1153-1158 (2005). arXiv:astro-ph/0503661.
[CrossRef]

A. Sivaramakrishnan and J. P. Lloyd, "Spiders in Lyot Coronagraphs," Astrophys. J. 633, 528-533 (2005). arXiv:astro-ph/0506564.
[CrossRef]

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. 167, 81-99 (2006). arXiv:astro-ph/0608506.
[CrossRef]

A. Sivaramakrishnan, C. D. Koresko, R. B. Makidon, T. Berkefeld, and M. J. Kuchner, "Ground-based Coronagraphy with High-order Adaptive Optics," Astrophys. J. 552, 397-408 (2001). astro-ph/0103012.
[CrossRef]

C. Aime and R. Soummer, "The Usefulness and Limits of Coronagraphy in the Presence of Pinned Speckles," Astrophys. J. 612, L85-L88 (2004).
[CrossRef]

R. Soummer, "Apodized Pupil Lyot Coronagraphs for Arbitrary Telescope Apertures," Astrophys. J. 618, L161- L164 (2005).
[CrossRef]

Bell Sys. Tech. J. (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, "The Chirp z-Transform Algorithm and its Application," Bell Sys. Tech. J. 48 (5), 1249-1291 (1969).

Comptes Rendus Physique (1)

A. Ferrari, R. Soummer, and C. Aime, "An introduction to stellar coronagraphy," Comptes Rendus Physique 8, 277-287 (2007). arXiv:astro-ph/0703655.
[CrossRef]

IEEE Transactions on Audio and Electroacoustics (1)

L. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," IEEE Transactions on Audio and Electroacoustics 18, 451-455 (1970).
[CrossRef]

Journal of Modern Optics (1)

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

Nature (1)

J. T. Trauger and W. A. Traub, "A laboratory demonstration of the capability to image an Earth-like extrasolar planet," Nature 446, 771-773 (2007).
[CrossRef] [PubMed]

Opt. Lett. (1)

PASP (2)

F. Roddier and C. Roddier, "Stellar Coronagraph with Phase Mask," PASP 109, 815-820 (1997).
[CrossRef]

D. Rouan, P. Riaud, A. Boccaletti, Y. Clenet, and A. Labeyrie, "The Four-Quadrant Phase-Mask Coronagraph. I. Principle," PASP 112, 1479-1486 (2000).
[CrossRef]

Proceedings of the IEEE (1)

M. Frigo and S. G. Johnson, "The Design and Implementation of FFTW3," Proceedings of the IEEE 93(2), 216-231 (2005). Special issue on "Program Generation, Optimization, and Platform Adaptation".
[CrossRef]

SIAM Review (1)

D. H. Bailey and P. N. Swarztrauber, "The Fractional Fourier Transform and Applications," SIAM Review 33 (3), 389-404 (1991).
[CrossRef]

Other (24)

J. O. Smith, Mathematics of the Discrete Fourier Transform (DFT) (W3K Publishing, http://www.w3k.org/books/>, 2007).

C. Aime, "Principle of an Achromatic Prolate Apodized Lyot Coronagraph," PASP 117, 1012-+ (2005).
[CrossRef]

J. Goodman, Introduction to Fourier Optics (Mac Graw Hill, 1996).

R. N. Bracewell, The Fourier Transform & Its Applications, McGraw-Hill series in electrical and computer engineering., 3rd ed. (McGraw-Hill Science/Engineering/Math, Boston, 1999).

A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, "Amplitude and phase correction for high-contrast imaging using Fourier decomposition," Proc. SPIE, 5905, 368-378 (2005).

A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, "On representing and correcting wavefront errors in high-contrast imaging systems," J. Opt. Soc. Am. A 23 (2006).

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, "Extrasolar Planet Finding via Optimal Apodized-Pupil and Shaped-Pupil Coronagraphs," Astrophys. J. 582, 1147-1161 (2003). URL http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v582n2/56580/56580.web.pdf>.
[CrossRef]

URL http://www.primatelabs.ca/geekbench/>.

L. A. Poyneer and B. Macintosh, "Spatially filtered wave-front sensor for high-order adaptive optics," J. Opt. Soc. Am. A 21, 810-819 (2004).
[CrossRef]

L. Abe, F. Vakili, and A. Boccaletti, "The achromatic phase knife coronagraph," A&A 374, 1161-1168 (2001).

R. Soummer, K. Dohlen, and C. Aime, "Achromatic dual-zone phase mask stellar coronagraph," A&A403, 369-381 (2003).

R. Soummer, L. Pueyo, A. Ferrari, C. Aime, A. Sivaramakrishnan, and N. Yaitskova, "Apodized Pupil Lyot Coronagraphs for Arbitrary Telescope Apertures. II. Application to Extremely Large Telescopes," submitted to ApJ (2007).

C. Cavarroc, A. Boccaletti, P. Baudoz, T. Fusco, and D. Rouan, "Fundamental limitations on Earth-like planet detection with extremely large telescopes," A&A 447, 397-403 (2006).

R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, "Speckle noise and dynamic range in coronagraphic images," ApJ 669, 642-656 (2007). arXiv:0706.1739v1.

B. Lyot, " ´ Etude de la couronne solaire en dehors des ´eclipses. Avec 16 figures dans le texte." Zeitschrift fur Astrophysics 5, 73-+ (1932).

C. Aime, R. Soummer, and A. Ferrari, "Total coronagraphic extinction of rectangular apertures using linear prolate apodizations," A&A 389, 334-344 (2002).

R. Soummer, C. Aime, and P. E. Falloon, "Stellar coronagraphy with prolate apodized circular apertures," A&A397, 1161-1172 (2003).

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Veran, "The Gemini Planet Imager," Proc. SPIE 6272, (2006).

J. L. Beuzit, D. Mouillet, C. Moutou, K. Dohlen, T. Fusco, P. Puget, S. Udry, R. Gratton, H. M. Schmid, M. Feldt, M. Kasper, and The Vlt-Pf Consortium, "A "Planet Finder" instrument for the VLT," in Tenth Anniversary of 51 Peg-b: Status of and prospects for hot Jupiter studies, L. Arnold, F. Bouchy, and C. Moutou, eds., pp. 353-355 (2006).

M. Tamura, K. Hodapp, H. Takami, L. Abe, H. Suto, O. Guyon, S. Jacobson, R. Kandori, J.-I. Morino, N. Murakami, V. Stahlberger, R. Suzuki, A. Tavrov, H. Yamada, J. Nishikawa, N. Ukita, J. Hashimoto, H. Izumiura, M. Hayashi, T. Nakajima, and T. Nishimura, "Concept and science of HiCIAO: high contrast instrument for the Subaru next generation adaptive optics," Proc. SPIE 6269, 62690V (2006)., vol. 6269 presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference (2006).

B. Macintosh, M. Troy, R. Doyon, J. Graham, K. Baker, B. Bauman, C. Marois, D. Palmer, D. Phillion, L. Poyneer, I. Crossfield, P. Dumont, B. M. Levine, M. Shao, G. Serabyn, C. Shelton, G. Vasisht, J. K. Wallace, J.-F. Lavigne, P. Valee, N. Rowlands, K. Tam, and D. Hackett, "Extreme adaptive optics for the Thirty Meter Telescope," Proc. SPIE, 6272, 62720N (2006)., vol. 6272 presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference (2006).

L. Close, "Extrasolar Planet Imaging with the Giant Magellan Telescope," in Proceedings of the conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007. University of California, Berkeley, CA, USA. Edited by Paul Kalas., P. Kalas, ed. (2007).

C. Verinaud, M. Kasper, J.-L. Beuzit, N. Yaitskova, V. Korkiakoski, K. Dohlen, P. Baudoz, T. Fusco, L. Mugnier, and N. Thatte, "EPICS Performance Evaluation through Analytical and Numerical Modeling," in Proceedings of the conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007. University of California, Berkeley, CA, USA. Edited by Paul Kalas., P. Kalas, ed. (2007).

O. Guyon, J. R. P. Angel, C. Bowers, J. Burge, A. Burrows, J. Codona, T. Greene, M. Iye, J. Kasting, H. Martin, D. W. McCarthy, Jr., V. Meadows, M. Meyer, E. A. Pluzhnik, N. Sleep, T. Spears, M. Tamura, D. Tenerelli, R. Vanderbei, B. Woodgate, R. A. Woodruff, and N. J. Woolf, "Telescope to observe planetary systems (TOPS): a high throughput 1.2-m visible telescope with a small inner working angle," Proc. SPIE 6265, 62651R (2006)., vol. 6265 presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference (2006).

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Figures (9)

Fig. 1.
Fig. 1.

Illustration of the four coronagraphic planes: the pupil corresponds to Plane A (possibly apodized). A focal masks (hard-edged, or phase mask) is placed in the focal plane B, and a Lyot stop (possibly undersized) in plane C.

Fig. 2.
Fig. 2.

Sampling relationship between pupil (left column) and image plane (center), using FFTs. The padded pupil plane array is γ times larger than the actual pupil (here γ=3), and its Fourier transform features three pixels per unit of angular resolution, as shown in the zoomed image of the core (right). In this case a 4~5 resolution element mask would not be sufficiently sampled. A general consensus for coronagraphic calculations is to use γ=6 or γ=8.

Fig. 3.
Fig. 3.

Computing steps in each plane A,B,C using FFTs with a six-fold zero padding (4500×4500 FFTs), and the semi-analytical method (SAM), for a possible ELT geometry and an APLC. With SAM, only the points of the FT that are inside of the occulting mask are computed as opposed to the FFT method, where the points outside the mask are needed. In this example we obtain a speed improvement of about 15 using 36 times less memory.

Fig. 4.
Fig. 4.

Gain between SAM and FFT for a propagation from pupil to Lyot plane, as a function of the number of pixels across the pupil. The gain is based on the evaluation of the computation costs for both methods. We use common zero-padding coefficients (γ=4,6,8) and a mask size m=5λ/D. This corresponds to the regime of simulations of Lyot coronagraphs for 8-meter class telescopes.

Fig. 5.
Fig. 5.

Same as Fig.4, but in the case of an ELT where a very large number of pixel is used in the pupil.

Fig. 6.
Fig. 6.

Same as Fig.4, but for a Roddier or Dual Zone coronagraph with a mask size of m=λ/D and an oversizing γ=20. This corresponds to a maximum FFT size of 10000×10000 for the 500 pixel pupil.

Fig. 7.
Fig. 7.

Non-coronagraphic, direct imaging with FFT and MFT where the PSF is calculated to a field of view of 50 resolution element (25 in radius). Both MFT and FFT have similar costs with a slight advantage to MFTs when higher sampling is used.

Fig. 8.
Fig. 8.

Non-coronagraphic, direct imaging with FFT and MFT where the PSF is calculated to a field of view of 20 resolution element (10 in radius). For a limited field of view, the MFT can be significantly faster than the FFT. This can be interesting for coronagraphic and extreme adaptive optics simulations.

Fig. 9.
Fig. 9.

Normalized speed expressed in Mflop/s using the FFTW3 package [32] on a 2Ghz dual-processor 2003 Apple G5 with 4Gb RAM. The normalized speed is defined as the number of operations divided by the execution time: 10N 2 log2(N)/T where T is the time for one FFT (single-threaded, double precision complex arrays). The red dots correspond to power of twos arrays. This hardware delivers approximately 1Gflop/s over the range of array sizes, which translates for example into 2.3s for 4096×4096 transforms, and 10s for 8192×8192 transforms.

Tables (2)

Tables Icon

Table 1. Performance comparison of FFT-based and semi-analytical methods (SAM). N A is the number of pixels across the pupil diameter, and N B is the size of the array in the focal plane. The FFT timings correspond to 2 FFTs calculated with FFTW3, and the SAM timings correspond to an actual propagation from pupil to Lyot plane, with Mathematica.

Tables Icon

Table 2. Performance of semi-analytical simulations for double precision complex arrays, with their possible applications. These calculations cannot be performed using the FFT method on commercial workstations, as they would correspond respectively to 200000×200000, 25600×25600, and 48000×48000 FFTs.

Equations (20)

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Ψ A ( r ) = P ( r ) Φ ( r )
Ψ B ( r ) = Ψ ̂ A ( r ) ( 1 ε M ( r ) )
Ψ C ( r ) = ( Ψ A ( r ) ε Ψ A ( r ) * M ̂ ( r ) ) L ( r )
Ψ D ( r ) = ( Ψ ̂ A ( r ) ε Ψ ̂ A ( r ) M ( r ) ) * L ̂ ( r )
F ̂ ( u k ) = γ D 2 γ D 2 f ( x ) e i 2 π x u k dx
δ x n = 0 N A 1 f ( x n ) e i 2 π x n u k
δ x δ u = 1 N ,
F ̂ ( u k ) = γ D N ( 1 ) N 2 k n = 0 N 1 ( 1 ) n f ( x n ) e i 2 π kn N ,
Ψ C ( r ) = ( Ψ A ( r ) ε [ [ Ψ A ( r ) ] M ( r ) ] ) L ( r ) ,
( F ̂ ( u 0 ) F ̂ ( u k ) F ̂ ( u N B 1 ) ) = ( e 2 i π x 0 u 0 e 2 i π x k u 0 e 2 i π x N A 1 u 0 e 2 i π x 0 u k e 2 i π x k u k e 2 i π x N A 1 u k e 2 i π x 0 u N B 1 e 2 i π x k u N B 1 e 2 i π x N A 1 u N B 1 ) ( f ( x 0 ) f ( x k ) f ( x N A 1 ) ) ,
F ̂ ( U ) = e 2 i π U X T · f ( X ) ,
F ̂ ( U , H ) = m N A N B e 2 i π U X T · f ( X , Y ) · e 2 i π Y V T ,
F = E 1 · f · E 2 ,
n ( MFT ) = 8 ( N A 2 N B + N A N B 2 ) 2 N A N B 2 N B 2 .
n ( FFT ) n ( MFT ) = 5 log 2 ( N ) 8 N .
F ̂ ( u k ) = D N e i ( π γ ) ( k N B 2 ) n = 0 N A e i π n N B ( γ N A ) f ( x n ) e i 2 π kn ( γ N A )
F ̂ ( u k ) = D N e i ( π γ ) ( k N B 2 ) n = 0 N A e i π n N B ( γ N A ) f ( x n ) e i π k 2 ( γ N A ) e i π ( n k ) 2 ( γ N A )
n ( FFT ) = 20 ( γ N A ) 2 log 2 ( γ N A ) + 6 ( γ m ) 2 ,
n ( SAM ) = 16 ( N A 2 γ m + N A ( γ m ) 2 ) + 4 ( γ m ) 2 4 N A N B ,
n ( FFT ) n ( SAM ) = 5 4 γ N A log 2 ( γ N A ) m N A + γ m 2 ,

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