Abstract

The realization that direct imaging of extrasolar planets could be technologically feasible within the next decade or so has inspired a great deal of recent research into high-contrast imaging. We have contributed several design ideas, all of which can be described as shaped pupil coronagraphs. We offer a complete and unified survey of one-dimensional shaped pupil designs, some of which have been published in our previous papers. We also introduce a promising new design, which we call bar-code masks. With these masks we can achieve the required contrast with a fairly large discovery zone and throughput, but most importantly they are perhaps the easiest to manufacture and might therefore stand up best to refined analyses.

© 2005 Optical Society of America

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  1. R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
    [CrossRef]
  2. D. N. Spergel, “A new pupil for detecting extrasolar planets,” preprint astro-ph/0101142 (2000).
  3. N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J. 582, 1147–1161 (2003).
    [CrossRef]
  4. R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Spiderweb masks for high contrast imaging,” Astrophys. J. 590, 593–603 (2003).
    [CrossRef]
  5. R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Circularly symmetric apodization via starshaped masks,” Astrophys. J. 599, 686–694 (2003).
    [CrossRef]
  6. R. N. Bracewell, Fourier Analysis and Imaging (Kluwer Academic, Dordrecht, The Netherlands, 2003).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. We recognize that the term Airy throughput may be confusing when we refer to square or rectangular pupils. However, it is important to be consistent among all our designs, so we retain the terminology.
  9. P. Jacquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29–186 (1964).
    [CrossRef]
  10. G. Indebetouw, “Optimal apodizing properties of Gaussian pupils,” J. Mod. Opt. 37, 1271–1275 (1990).
    [CrossRef]
  11. S. M. Watson, J. P. Mills, S. L. Gaiser, D. J. Diner, “Direct imaging of nonsolar planets with infrared telescopes using apodized coronagraphs,” Appl. Opt. 30, 3253–3260 (1991).
    [CrossRef] [PubMed]
  12. P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
    [CrossRef]
  13. D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. 55, 1110–1115 (1965).
    [CrossRef]
  14. D. Slepian, H. O. Pollack, H. J. Landau, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–84 (1961).
    [CrossRef]
  15. R. J. Vanderbei, Linear Programming: Foundations and Extensions, 2nd ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001).
    [CrossRef]
  16. R. J. Vanderbei, “LOQO user’s manual—version 3.10,” Optim. Methods Software 12, 485–514 (1999).
    [CrossRef]
  17. M. J. Kuchner, W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570, 900–908 (2002).
    [CrossRef]

2003 (3)

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J. 582, 1147–1161 (2003).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Spiderweb masks for high contrast imaging,” Astrophys. J. 590, 593–603 (2003).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Circularly symmetric apodization via starshaped masks,” Astrophys. J. 599, 686–694 (2003).
[CrossRef]

2002 (1)

M. J. Kuchner, W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570, 900–908 (2002).
[CrossRef]

2001 (1)

P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
[CrossRef]

1999 (1)

R. J. Vanderbei, “LOQO user’s manual—version 3.10,” Optim. Methods Software 12, 485–514 (1999).
[CrossRef]

1991 (1)

1990 (1)

G. Indebetouw, “Optimal apodizing properties of Gaussian pupils,” J. Mod. Opt. 37, 1271–1275 (1990).
[CrossRef]

1965 (1)

1964 (1)

P. Jacquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29–186 (1964).
[CrossRef]

1961 (1)

D. Slepian, H. O. Pollack, H. J. Landau, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–84 (1961).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, Fourier Analysis and Imaging (Kluwer Academic, Dordrecht, The Netherlands, 2003).
[CrossRef]

Brown, R. A.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Burrows, C. J.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Casertano, S.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Clampin, M.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Diner, D. J.

Ebbets, D.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Ford, E. B.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Gaiser, S. L.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Indebetouw, G.

G. Indebetouw, “Optimal apodizing properties of Gaussian pupils,” J. Mod. Opt. 37, 1271–1275 (1990).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29–186 (1964).
[CrossRef]

Jucks, K. W.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Kasdin, N. J.

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Circularly symmetric apodization via starshaped masks,” Astrophys. J. 599, 686–694 (2003).
[CrossRef]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J. 582, 1147–1161 (2003).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Spiderweb masks for high contrast imaging,” Astrophys. J. 590, 593–603 (2003).
[CrossRef]

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Kilston, S.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Kuchner, M. J.

M. J. Kuchner, W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570, 900–908 (2002).
[CrossRef]

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Landau, H. J.

D. Slepian, H. O. Pollack, H. J. Landau, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–84 (1961).
[CrossRef]

Littman, M. G.

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J. 582, 1147–1161 (2003).
[CrossRef]

Mills, J. P.

Nisenson, P.

P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
[CrossRef]

Papaliolios, C.

P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
[CrossRef]

Pollack, H. O.

D. Slepian, H. O. Pollack, H. J. Landau, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–84 (1961).
[CrossRef]

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29–186 (1964).
[CrossRef]

Seager, S.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Slepian, D.

D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. 55, 1110–1115 (1965).
[CrossRef]

D. Slepian, H. O. Pollack, H. J. Landau, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–84 (1961).
[CrossRef]

Sozzetti, A.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Spergel, D. N.

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Circularly symmetric apodization via starshaped masks,” Astrophys. J. 599, 686–694 (2003).
[CrossRef]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J. 582, 1147–1161 (2003).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Spiderweb masks for high contrast imaging,” Astrophys. J. 590, 593–603 (2003).
[CrossRef]

D. N. Spergel, “A new pupil for detecting extrasolar planets,” preprint astro-ph/0101142 (2000).

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Traub, W. A.

M. J. Kuchner, W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570, 900–908 (2002).
[CrossRef]

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Trauger, J. T.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Turner, E. L.

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

Vanderbei, R. J.

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Circularly symmetric apodization via starshaped masks,” Astrophys. J. 599, 686–694 (2003).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Spiderweb masks for high contrast imaging,” Astrophys. J. 590, 593–603 (2003).
[CrossRef]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J. 582, 1147–1161 (2003).
[CrossRef]

R. J. Vanderbei, “LOQO user’s manual—version 3.10,” Optim. Methods Software 12, 485–514 (1999).
[CrossRef]

R. J. Vanderbei, Linear Programming: Foundations and Extensions, 2nd ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001).
[CrossRef]

Watson, S. M.

Appl. Opt. (1)

Astrophys. J. (5)

M. J. Kuchner, W. A. Traub, “A coronagraph with a band-limited mask for finding terrestrial planets,” Astrophys. J. 570, 900–908 (2002).
[CrossRef]

P. Nisenson, C. Papaliolios, “Detection of earth-like planets using apodized telescopes,” Astrophys. J. 548, L201–L205 (2001).
[CrossRef]

N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J. 582, 1147–1161 (2003).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Spiderweb masks for high contrast imaging,” Astrophys. J. 590, 593–603 (2003).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, N. J. Kasdin, “Circularly symmetric apodization via starshaped masks,” Astrophys. J. 599, 686–694 (2003).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollack, H. J. Landau, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–84 (1961).
[CrossRef]

J. Mod. Opt. (1)

G. Indebetouw, “Optimal apodizing properties of Gaussian pupils,” J. Mod. Opt. 37, 1271–1275 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

Optim. Methods Software (1)

R. J. Vanderbei, “LOQO user’s manual—version 3.10,” Optim. Methods Software 12, 485–514 (1999).
[CrossRef]

Prog. Opt. (1)

P. Jacquinot, B. Roizen-Dossier, “Apodization,” Prog. Opt. 3, 29–186 (1964).
[CrossRef]

Other (6)

R. A. Brown, C. J. Burrows, S. Casertano, M. Clampin, D. Ebbets, E. B. Ford, K. W. Jucks, N. J. Kasdin, S. Kilston, M. J. Kuchner, S. Seager, A. Sozzetti, D. N. Spergel, W. A. Traub, J. T. Trauger, E. L. Turner, “The 4-m space telescope for investigating extrasolar Earth-like planets in starlight: TPF is HST2,” in Future EUV and Visible Missions and Instrumentation, J. C. Blades, O. H. W. Siegmund, eds., Proc. SPIE4854, 95–107 (2002).
[CrossRef]

D. N. Spergel, “A new pupil for detecting extrasolar planets,” preprint astro-ph/0101142 (2000).

R. N. Bracewell, Fourier Analysis and Imaging (Kluwer Academic, Dordrecht, The Netherlands, 2003).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

We recognize that the term Airy throughput may be confusing when we refer to square or rectangular pupils. However, it is important to be consistent among all our designs, so we retain the terminology.

R. J. Vanderbei, Linear Programming: Foundations and Extensions, 2nd ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

(a) PSF for the one-dimensional prolate spheroidal apodized square aperture13 of unit area plotted on a logarithmic scale with black areas 10−10 below brightest. (b) On-axis cross section of the PSF showing an inner working distance of 4λ/a. The total throughput is 25%.

Fig. 2
Fig. 2

(a) One-dimensional bar-code mask optimized for a dark region in [0, 40] (all slots are not visible because of resolution limitations). (b) The two-dimensional PSF of the bar-code mask. (c) A cross section of the optimal one-dimensional PSF near the inner working angle. The Airy throughput is 25%.

Fig. 3
Fig. 3

(a) PSF of a bar-code mask designed for high contrast in the range of 3–5 λ/D. (b) The cross section of the PSF. This mask has an Airy throughput of 33.8%.

Fig. 4
Fig. 4

(a) Smooth one-dimensional optimal apodized function. (b) The corresponding cross section of the PSF showing an inner working distance of 4λ/a and a contrast better than 10−10.

Fig. 5
Fig. 5

Optimal apodization with and without a phase shift.

Fig. 6
Fig. 6

PSF for optimal apodization without a phase shift. Here is shown the PSF for the apodization computed without accounting for a phase shift. The upper curve is the PSF one obtains with the phase-shift term of k = 0.1.

Fig. 7
Fig. 7

PSF for optimal apodization with a phase shift. Here is shown the PSF for the apodization computed assuming a phase shift. The inner working angle is 4.5λ/D.

Fig. 8
Fig. 8

(a) Spergel–Kasdin prolate spheroidal mask. (b) The associated two-dimensional PSF and its x-axis slice shown in decibels (10−10 = −100 dB).

Fig. 9
Fig. 9

(a) Sample 10-opening square-aperture pupil by use of the apodization shown in Fig. 1. (b) The PSF for the 10-opening pupil. (c) The PSF for a 100-opening pupil.

Fig. 10
Fig. 10

(a) Multiopening pupil mask designed to open up the high-contrast region. (b) The corresponding PSF. On the x axis, a contrast of 10−10 extends over ρ ≥ 4.

Equations (41)

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

E ( ξ ,     ζ ) = S exp [ - 2 π i ( x ξ + y ζ ) ] A ( x ,     y ) d x d y ,
S = [ x ,     y ] : - 1 / 2 x 1 / 2 ,     - 1 / 2 y 1 / 2 ] .
E ( ξ ,     ζ ) = 2 sin ( π ζ ) π ζ 0 1 / 2 A ( x ) cos ( 2 π ξ x ) d x .
E 2 ( ξ ,     ζ ) / E 2 ( 0 ,     0 ) .
T Airy = 4 0 ξ iwa 0 ζ iwa E 2 ( ξ ,     ζ ) d ζ d ξ ,
E ( 0 ,     0 ) = 2 0 1 / 2 A ( x ) d x ,
minmize     ξ iwa E ( ξ ,     0 ) 2 d ξ , subject to A ( 0 ) = 1.
maximize 0 1 / 2 A ( x ) d x , subject to - 10 - 5 E ( 0 , 0 ) E ( ξ , 0 ) 10 - 5 E ( 0 ,     0 ) , ξ iwa ξ ξ owa , 0 A ( x ) 1 , 0 x 1 / 2.
[ x 2 ,     x 3 ] second opening ,
[ x 4 ,     x 5 ] third opening ,
[ x 6 ,     x 7 ] fourth opening ,
[ x m + 1 x m ] m th opening ,
E ( ξ ,     ζ ) = sin π ζ π ζ [ sin 2 π ξ x 1 π ξ + i = 2 m 1 π ξ ( sin 2 π ξ x i + 1 - sin 2 π ξ x i ) ] .
log ( A ) 0 ,
log ( A ) 0.
maximize 0 1 / 2 A ( x ) d x , subject to - 10 - 5 E ( 0 ,     0 ) E ( ξ ,     0 ) 10 - 5 E ( 0 ,     0 ) , ξ iwa ξ ξ owa , 0 A ( x ) 1 , 0 x 1 / 2 , A ( x ) 0 , 0 x 1 / 2 , A ( x ) A ( x ) A ( x ) 2 , 0 x 1 / 2.
E ( ξ ) = - 1 / 2 1 / 2 exp { - 2 π i x ξ + i k log [ A ( x ) ] } A ( x ) d x ,
E ( ξ ,     ζ ) = - 1 / 2 1 / 2 sin 2 π ζ w ( x ) π ζ exp ( - 2 π i ξ x ) d x .
E ( ξ ,     0 ) = 2 - 1 / 2 1 / 2 w ( x ) exp ( - i 2 π ξ x ) d x .
S = [ ( x ,     y ) : - 1 / 2 x 1 / 2 ,     y ( x ) Y u ] ,
Y u = n = 0 N - 1 [ D 2 n N + w ( x ) 2 ,     D 2 n + 1 N - w ( x ) 2 ] ,
Y 1 = n = 0 N - 1 [ - D 2 n + 1 N + w ( x ) 2 ,     - D 2 n N - w ( x ) 2 ] ,
E ( ξ ,     ζ ) = 2 sin ( π ζ ) π ζ 0 1 / 2 n = 0 N - 1 { sin 2 π ζ [ n + 1 2 N - w ( x ) 2 ] - sin 2 π ζ [ n 2 N + w ( x ) 2 ] } cos ( 2 π ξ x ) sin ( π ζ ) d x ,
E ( ξ ,     ζ ) = 2 sin ( π ζ ) π ζ 0 1 / 2 { cos [ π ζ w ( x ) ] - [ 1 + cos ( π ζ / N ) ] sin ( π ζ / N ) sin [ π ζ w ( x ) ] } cos ( 2 π ξ x ) d x .
w ( x ) = 1 - A ( x ) 2 N ,
E ( ξ ,     ζ ) = 2 sin π ζ π ζ 0 1 / 2 × ( sin { π ζ [ 1 + A ( x ) ] 2 N } - sin { π ζ [ 1 - A ( x ) ] 2 N } sin ( π ζ / N ) ) × cos ( 2 π ξ x ) d x .
E ( ξ ,     ζ ) = 2 sin ( π ζ ) π ζ 0 1 / 2 [ A ( x ) + O ( π ζ N ) 2 ] × cos ( 2 π ξ x ) d x .
Y u = n = 0 N - 1 [ 1 2 n N + b n ( x ) 2 , 1 2 n + 1 N - t n ( x ) 2 ] ,
E ( ξ ,     ζ ) = 2 sin ( π ζ ) π ζ 0 1 / 2 n = 0 N - 1 { sin π ζ [ n + 1 N - t n ( x ) ] - sin π ζ [ n N + b n ( x ) ] } cos ( 2 π ξ x ) sin ( π ζ ) d x .
E ( ξ ,     ζ ) = S exp [ γ ( x ,     y ) A ( x ,     y ) ] exp [ - 2 π i ( x ξ + y ζ ) ] d x d y ,
exp [ α ( x ,     y ) ] = 1 + n = 1 a n cos ( k n · x + φ n ) ,
E ( ξ ,     ζ ) 2 E 0 ( ξ ,     ζ ) 2 + 2 E 0 ( ξ ,     ζ ) Δ E ( ξ ,     ζ ) .
Δ E = S A n cos [ k n · x + φ n ) A ( x ,     y ) exp [ - 2 π i ( x ξ + y ζ ) ] d x d y .
Δ E = E 0 ( ξ - k x ,     ζ - k y ) .
Δ A 2 E 0 ( ξ ,     ζ ) E 0 ( ξ -     k x ,     ζ - k y ) 10 - 10 E 0 ( 0 ,     0 ) 2 .
5 × 10 - 6 E 0 ( 0 ,     0 ) E 0 ( ξ - k x ,     ζ - k y ) .
Δ E ( ξ ,     ζ ) = i = 1 N - 1 / 2 1 / 2 exp ( - 2 π i ζ y ) { [ i f i ( y ) - i f i ) ] sin ( 2 π r i ξ ) + i [ i f i ( y ) + i f i ( y ) ] cos ( 2 π r i ξ ) } d y .
E { E ( ξ ,     ζ ) } 2 } = E 0 ( ξ ,     ζ ) 2 + E { Δ E ( ξ ,     ζ ) 2 } .
E { Δ E ( ξ ,     ζ ) 2 } = σ 2 i = 1 N { [ - 1 / 2 1 / 2 f i ( y ) exp × ( - 2 π i ζ y ) d y ] 2 + [ - 1 / 2 1 / 2 f i ( y ) × exp ( - 2 π i ζ y ) d y ] 2 } 10 - 10 E 0 ( 0 ,     0 ) 2 .
σ 10 - 5 E 0 ( 0 ,     0 ) 2 N .

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