Abstract

The specification of polishing requirements for the optics in coronagraphs dedicated to exoplanet detection requires careful and accurate optical modeling. Numerical representations of propagated aberrations through the system as well as simulations of the broadband wavefront compensation system using multiple DMs are critical when one devises an error budget for such a class of instruments. In this communication, we introduce an analytical tool that serves this purpose for phase-induced amplitude apodization (PIAA) coronagraphs. We first start by deriving the analytical form of the propagation of a harmonic ripple through a PIAA unit. Using this result, we derive the chromaticity of the field at any plane in the optical train of a telescope equipped with such a coronagraph. Finally, we study the chromatic response of a two-sequential-DM wavefront actuator correcting such a corrugated field and thus quantify the requirements on the manufacturing of PIAA mirrors.

© 2011 Optical Society of America

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  1. A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
    [CrossRef]
  2. S. B. Shaklan and J. J. Green, “Low-order aberration sensitivity of eighth-order coronagraph masks,” Astrophys. J. 628, 474–477(2005).
    [CrossRef]
  3. M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
    [CrossRef]
  4. A. Give’On, N. J. Kasdin, R. Vanderbei, and Y. Avitzour, “On representing and correcting wavefront errors in high-contrast imaging systems,” J. Opt. Soc. Am. A 23, 1063–1073 (2006).
    [CrossRef]
  5. A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” Astrophys. J. 688, 701–708 (2008).
    [CrossRef]
  6. S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006).
    [CrossRef]
  7. L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609–625 (2007).
    [CrossRef]
  8. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404, 379–387 (2003).
    [CrossRef]
  9. D. Mawet, E. Serabyn, K. Liewer, C. Hanot, S. McEldowney, D. Shemo, and N. O’Brien, “optical vectorial vortex coronagraphs using liquid crystal polymers: theory, manufacturing and laboratory demonstration,” Opt. Express 17, 1902–1918(2009).
    [CrossRef] [PubMed]
  10. D. Mawet, L. Pueyo, D. Moody, J. Krist, and E. Serabyn, “The vector vortex coronagraph: sensitivity to central obscuration, low-order aberrations, chromaticism, and polarization,” Proc. SPIE 7739, 773914 (2010).
    [CrossRef]
  11. R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652, 833–844 (2006).
    [CrossRef]
  12. S. B. Shaklan, A. Give’on, R. Belikov, L. Pueyo, and O. Guyon, “Broadband wavefront control in a pupil mapping coronagraph,” Proc. SPIE 6693, 66930R (2007).
    [CrossRef]
  13. W. A. Traub and R. J. Vanderbei, “Two-mirror apodization for high-contrast imaging,” Astrophys. J. 599, 695–701 (2003).
    [CrossRef]
  14. R. J. Vanderbei, “Diffraction analysis of two-dimensional pupil mapping for high-contrast imaging,” Astrophys. J. 636, 528–543(2006).
    [CrossRef]
  15. E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).
  16. L. Pueyo, “Broadband contrast for exo-planet imaging: the impact of propagation effects,” Ph.D. thesis (Princeton University, 2008).
  17. L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
    [CrossRef]
  18. S. B. Shaklan and J. Green, “Reflectivity and optical surface height requirements in a broadband coronagraph. 1.Contrast floor due to controllable spatial frequencies,” Appl. Opt. 45, 5143–5153 (2006).
    [CrossRef] [PubMed]
  19. O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. 622, 744–758 (2005).
    [CrossRef]

2010 (1)

D. Mawet, L. Pueyo, D. Moody, J. Krist, and E. Serabyn, “The vector vortex coronagraph: sensitivity to central obscuration, low-order aberrations, chromaticism, and polarization,” Proc. SPIE 7739, 773914 (2010).
[CrossRef]

2009 (2)

2008 (1)

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” Astrophys. J. 688, 701–708 (2008).
[CrossRef]

2007 (2)

L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609–625 (2007).
[CrossRef]

S. B. Shaklan, A. Give’on, R. Belikov, L. Pueyo, and O. Guyon, “Broadband wavefront control in a pupil mapping coronagraph,” Proc. SPIE 6693, 66930R (2007).
[CrossRef]

2006 (5)

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652, 833–844 (2006).
[CrossRef]

S. B. Shaklan and J. Green, “Reflectivity and optical surface height requirements in a broadband coronagraph. 1.Contrast floor due to controllable spatial frequencies,” Appl. Opt. 45, 5143–5153 (2006).
[CrossRef] [PubMed]

R. J. Vanderbei, “Diffraction analysis of two-dimensional pupil mapping for high-contrast imaging,” Astrophys. J. 636, 528–543(2006).
[CrossRef]

A. Give’On, N. J. Kasdin, R. Vanderbei, and Y. Avitzour, “On representing and correcting wavefront errors in high-contrast imaging systems,” J. Opt. Soc. Am. A 23, 1063–1073 (2006).
[CrossRef]

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006).
[CrossRef]

2005 (3)

A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
[CrossRef]

S. B. Shaklan and J. J. Green, “Low-order aberration sensitivity of eighth-order coronagraph masks,” Astrophys. J. 628, 474–477(2005).
[CrossRef]

O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. 622, 744–758 (2005).
[CrossRef]

2003 (3)

W. A. Traub and R. J. Vanderbei, “Two-mirror apodization for high-contrast imaging,” Astrophys. J. 599, 695–701 (2003).
[CrossRef]

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
[CrossRef]

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404, 379–387 (2003).
[CrossRef]

Avitzour, Y.

Belikov, R.

S. B. Shaklan, A. Give’on, R. Belikov, L. Pueyo, and O. Guyon, “Broadband wavefront control in a pupil mapping coronagraph,” Proc. SPIE 6693, 66930R (2007).
[CrossRef]

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652, 833–844 (2006).
[CrossRef]

Blain, C.

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).

Galicher, R.

O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. 622, 744–758 (2005).
[CrossRef]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).

Give’on, A.

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[CrossRef]

S. B. Shaklan, A. Give’on, R. Belikov, L. Pueyo, and O. Guyon, “Broadband wavefront control in a pupil mapping coronagraph,” Proc. SPIE 6693, 66930R (2007).
[CrossRef]

A. Give’On, N. J. Kasdin, R. Vanderbei, and Y. Avitzour, “On representing and correcting wavefront errors in high-contrast imaging systems,” J. Opt. Soc. Am. A 23, 1063–1073 (2006).
[CrossRef]

Graham, J. R.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
[CrossRef]

Green, J.

Green, J. J.

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006).
[CrossRef]

S. B. Shaklan and J. J. Green, “Low-order aberration sensitivity of eighth-order coronagraph masks,” Astrophys. J. 628, 474–477(2005).
[CrossRef]

Guyon, O.

S. B. Shaklan, A. Give’on, R. Belikov, L. Pueyo, and O. Guyon, “Broadband wavefront control in a pupil mapping coronagraph,” Proc. SPIE 6693, 66930R (2007).
[CrossRef]

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404, 379–387 (2003).
[CrossRef]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).

Hanot, C.

Kasdin, N. J.

L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609–625 (2007).
[CrossRef]

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652, 833–844 (2006).
[CrossRef]

A. Give’On, N. J. Kasdin, R. Vanderbei, and Y. Avitzour, “On representing and correcting wavefront errors in high-contrast imaging systems,” J. Opt. Soc. Am. A 23, 1063–1073 (2006).
[CrossRef]

Krist, J.

D. Mawet, L. Pueyo, D. Moody, J. Krist, and E. Serabyn, “The vector vortex coronagraph: sensitivity to central obscuration, low-order aberrations, chromaticism, and polarization,” Proc. SPIE 7739, 773914 (2010).
[CrossRef]

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[CrossRef]

Liewer, K.

Lloyd, J. P.

A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
[CrossRef]

Makidon, R. B.

A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
[CrossRef]

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
[CrossRef]

Martinache, F.

O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. 622, 744–758 (2005).
[CrossRef]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).

Mawet, D.

D. Mawet, L. Pueyo, D. Moody, J. Krist, and E. Serabyn, “The vector vortex coronagraph: sensitivity to central obscuration, low-order aberrations, chromaticism, and polarization,” Proc. SPIE 7739, 773914 (2010).
[CrossRef]

D. Mawet, E. Serabyn, K. Liewer, C. Hanot, S. McEldowney, D. Shemo, and N. O’Brien, “optical vectorial vortex coronagraphs using liquid crystal polymers: theory, manufacturing and laboratory demonstration,” Opt. Express 17, 1902–1918(2009).
[CrossRef] [PubMed]

McEldowney, S.

Moody, D.

D. Mawet, L. Pueyo, D. Moody, J. Krist, and E. Serabyn, “The vector vortex coronagraph: sensitivity to central obscuration, low-order aberrations, chromaticism, and polarization,” Proc. SPIE 7739, 773914 (2010).
[CrossRef]

O’Brien, N.

Oppenheimer, B. R.

A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
[CrossRef]

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
[CrossRef]

Palacios, D. M.

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006).
[CrossRef]

Perrin, M. D.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
[CrossRef]

Pluzhnik, E. A.

O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. 622, 744–758 (2005).
[CrossRef]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).

Pueyo, L.

D. Mawet, L. Pueyo, D. Moody, J. Krist, and E. Serabyn, “The vector vortex coronagraph: sensitivity to central obscuration, low-order aberrations, chromaticism, and polarization,” Proc. SPIE 7739, 773914 (2010).
[CrossRef]

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[CrossRef]

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” Astrophys. J. 688, 701–708 (2008).
[CrossRef]

L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609–625 (2007).
[CrossRef]

S. B. Shaklan, A. Give’on, R. Belikov, L. Pueyo, and O. Guyon, “Broadband wavefront control in a pupil mapping coronagraph,” Proc. SPIE 6693, 66930R (2007).
[CrossRef]

L. Pueyo, “Broadband contrast for exo-planet imaging: the impact of propagation effects,” Ph.D. thesis (Princeton University, 2008).

R., J.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
[CrossRef]

Ridgway, S. T.

O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. 622, 744–758 (2005).
[CrossRef]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).

Serabyn, E.

D. Mawet, L. Pueyo, D. Moody, J. Krist, and E. Serabyn, “The vector vortex coronagraph: sensitivity to central obscuration, low-order aberrations, chromaticism, and polarization,” Proc. SPIE 7739, 773914 (2010).
[CrossRef]

D. Mawet, E. Serabyn, K. Liewer, C. Hanot, S. McEldowney, D. Shemo, and N. O’Brien, “optical vectorial vortex coronagraphs using liquid crystal polymers: theory, manufacturing and laboratory demonstration,” Opt. Express 17, 1902–1918(2009).
[CrossRef] [PubMed]

Shaklan, S. B.

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[CrossRef]

S. B. Shaklan, A. Give’on, R. Belikov, L. Pueyo, and O. Guyon, “Broadband wavefront control in a pupil mapping coronagraph,” Proc. SPIE 6693, 66930R (2007).
[CrossRef]

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006).
[CrossRef]

S. B. Shaklan and J. Green, “Reflectivity and optical surface height requirements in a broadband coronagraph. 1.Contrast floor due to controllable spatial frequencies,” Appl. Opt. 45, 5143–5153 (2006).
[CrossRef] [PubMed]

S. B. Shaklan and J. J. Green, “Low-order aberration sensitivity of eighth-order coronagraph masks,” Astrophys. J. 628, 474–477(2005).
[CrossRef]

Shao, M.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” Astrophys. J. 688, 701–708 (2008).
[CrossRef]

Shemo, D.

Sivaramakrishnan, A.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” Astrophys. J. 688, 701–708 (2008).
[CrossRef]

A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
[CrossRef]

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
[CrossRef]

Sivaramakrishnan, A. V.

A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
[CrossRef]

Soummer, R.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” Astrophys. J. 688, 701–708 (2008).
[CrossRef]

A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
[CrossRef]

Traub, W. A.

W. A. Traub and R. J. Vanderbei, “Two-mirror apodization for high-contrast imaging,” Astrophys. J. 599, 695–701 (2003).
[CrossRef]

Vanderbei, R.

Vanderbei, R. J.

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652, 833–844 (2006).
[CrossRef]

R. J. Vanderbei, “Diffraction analysis of two-dimensional pupil mapping for high-contrast imaging,” Astrophys. J. 636, 528–543(2006).
[CrossRef]

W. A. Traub and R. J. Vanderbei, “Two-mirror apodization for high-contrast imaging,” Astrophys. J. 599, 695–701 (2003).
[CrossRef]

Wallace, J. K.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” Astrophys. J. 688, 701–708 (2008).
[CrossRef]

Woodruff, R. A.

O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. 622, 744–758 (2005).
[CrossRef]

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).

Appl. Opt. (1)

Astron. Astrophys. (1)

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404, 379–387 (2003).
[CrossRef]

Astrophys. J. (9)

R. Belikov, N. J. Kasdin, and R. J. Vanderbei, “Diffraction-based sensitivity analysis of apodized pupil-mapping systems,” Astrophys. J. 652, 833–844 (2006).
[CrossRef]

A. Sivaramakrishnan, R. Soummer, A. V. Sivaramakrishnan, J. P. Lloyd, B. R. Oppenheimer, and R. B. Makidon, “Low-order aberrations in band-limited Lyot coronagraphs,” Astrophys. J. 634, 1416–1422 (2005).
[CrossRef]

S. B. Shaklan and J. J. Green, “Low-order aberration sensitivity of eighth-order coronagraph masks,” Astrophys. J. 628, 474–477(2005).
[CrossRef]

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R., and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003).
[CrossRef]

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” Astrophys. J. 688, 701–708 (2008).
[CrossRef]

W. A. Traub and R. J. Vanderbei, “Two-mirror apodization for high-contrast imaging,” Astrophys. J. 599, 695–701 (2003).
[CrossRef]

R. J. Vanderbei, “Diffraction analysis of two-dimensional pupil mapping for high-contrast imaging,” Astrophys. J. 636, 528–543(2006).
[CrossRef]

L. Pueyo and N. J. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609–625 (2007).
[CrossRef]

O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. 622, 744–758 (2005).
[CrossRef]

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

Opt. Express (1)

Proc. SPIE (4)

L. Pueyo, S. B. Shaklan, A. Give’on, and J. Krist, “Numerical propagator through PIAA optics,” Proc. SPIE 7440, 74400E (2009).
[CrossRef]

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006).
[CrossRef]

S. B. Shaklan, A. Give’on, R. Belikov, L. Pueyo, and O. Guyon, “Broadband wavefront control in a pupil mapping coronagraph,” Proc. SPIE 6693, 66930R (2007).
[CrossRef]

D. Mawet, L. Pueyo, D. Moody, J. Krist, and E. Serabyn, “The vector vortex coronagraph: sensitivity to central obscuration, low-order aberrations, chromaticism, and polarization,” Proc. SPIE 7739, 773914 (2010).
[CrossRef]

Other (2)

E. A. Pluzhnik, O. Guyon, S. T. Ridgway, F. Martinache, R. A. Woodruff, C. Blain, and R. Galicher, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph III. Hybrid approach: optical design and diffraction analysis,” ArXiv Astrophysics e-prints 0512421 (2005).

L. Pueyo, “Broadband contrast for exo-planet imaging: the impact of propagation effects,” Ph.D. thesis (Princeton University, 2008).

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Figures (11)

Fig. 1
Fig. 1

Setup of the problem and notations. (Top left) Three-dimensional representation of a pupil-to-pupil off-axis PIAA system. (Top right) Side view of the geometrical remapping in a pupil-to-pupil off-axis PIAA system. (Bottom left) Side view of the geometrical remapping in a pupil-to-pupil on-axis PIAA unit; this is the configuration that is studied in this communication. (Bottom right) Side view of all the rays contributing to the diffractive field at a point of coordinates ( x ˜ , y ˜ ) at M2. The ray corresponding to the geometrical remapping, which has coordinates ( x 0 ( x ˜ , y ˜ ) , y 0 ( x ˜ , y ˜ ) ) in the input plane, is highlighted.

Fig. 2
Fig. 2

Phase-induced amplitude errors through a PIAA system. Spatial frequency at M1 N = 6 , with λ = 700 nm , D = 3 cm , and z = 1 m .

Fig. 3
Fig. 3

Phase-induced amplitude errors through a PIAA system. Spatial frequency at M1 N = 10 , with λ = 700 nm , D = 3 cm , and z = 1 m .

Fig. 4
Fig. 4

Amplitude-induced phase errors through a PIAA system. Spatial frequency at M1 N = 6 , with λ = 700 nm , D = 3 cm , and z = 1 m .

Fig. 5
Fig. 5

Amplitude-induced phase errors through a PIAA system. Spatial frequency at M1 N = 10 , with λ = 700 nm , D = 3 cm , and z = 1 m .

Fig. 6
Fig. 6

Cartoon representation of the wavelength data cube at (left panel) M2, and (right panel) at the image plane. The transverse axis is a virtual cut across a wavelength cube. These cubes are obtained by stacking field distributions at M2 and the image plane for several wavelengths across the spectral bandwidth of interest.

Fig. 7
Fig. 7

(Top) Illustration of the fit through the wavelength cube for 1 pixel. (Top left) raw contrast at one pixel in the image plane. (Top right) contrast at 1 pixel in the image plane after a perfect two sequential DM wavefront correction. (Bottom) Residual intensity in the dark zone after subtracting the two dominant terms of the wavelength expansion. Note that the PSF of a ripple propagated through PIAA is much more extended than in the case of a classical coronagraph. The chromaticity of the leakage close to the optical axis has been modified by the propagator that introduced a higher-order wavelength dependence. This drives the best speckle extinction achievable over a broadband. N = 7 , D = 3 cm , z = 1 m , Δ λ / λ = 0.1 .

Fig. 8
Fig. 8

Maximum of the broadband halo in the dark hole created by two sequential DM wavefront controller as a function of Fresnel number and spatial frequency of the wavefront error. The top curve corresponds to the maximum of the noncorrected PSF in the dark hole: note that high spatial frequencies leak in the dark hole due to the spatial extent of the off-axis PIAA PSF. The other three curves show the maximum of the residual halo after correction for, from top to bottom, F = 140 , 1250, 11,250.

Fig. 9
Fig. 9

Maximum of the broadband halo in the dark hole created by two sequential DM wavefront controller as a function of bandwidth and spatial frequency of the wavefront error. The top curve corresponds to the maximum of the noncorrected PSF in the dark hole: note that high spatial frequencies leak in the dark hole due to the spatial extent of the off-axis PIAA PSF. The other three curves show the maximum of the residual halo after correction for, from bottom to top, Δ λ / λ = 0.1 , 0.2, 0.3. The Fresnel number for the PIAA unit is F = 11250 .

Fig. 10
Fig. 10

Maximum of the broadband halo in the dark hole created by two sequential DM wavefront controller as a function of bandwidth and spatial frequency of the wavefront error. The top curve corresponds to the maximum of the noncorrected PSF in the dark hole: note that high spatial frequencies leak in the dark hole due to the spatial extent of the off-axis PIAA PSF. The other three curves show the maximum of the residual halo after correction for, from bottom to top, Δ λ / λ = 0.1 , 0.2, 0.3. The Fresnel number for the PIAA unit is 1250.

Fig. 11
Fig. 11

PSF of two off-axis sources that are separated by 2 λ / D and 4 λ / D from the star.

Equations (46)

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A ( x ˜ , y ˜ ) = A ( r ˜ ) .
r 0 r ˜ = r ˜ r 0 A ( r ˜ ) 2 ,
h r | r 0 = r ˜ ( r 0 ) r 0 Z ,
h ˜ r ˜ | r ˜ = r ˜ r 0 ( r ˜ ) Z ,
E out ( x ˜ , y ˜ ) = 1 i λ Z M 1 E in ( x , y ) e i π λ Z [ ( x x 0 ) 2 + 2 ( x x 0 ) ( y y 0 ) + ( y y 0 ) 2 ] d x d y ,
x 0 ( x ˜ , y ˜ ) = r 0 ( r ˜ ) cos ( θ ˜ ) ,
y 0 ( x ˜ , y ˜ ) = r 0 ( r ˜ ) sin ( θ ˜ ) .
E out ( r ˜ , θ ˜ ) = 1 i λ Z × M 1 E in ( r , θ ) e i π λ Z [ r 0 ( r ˜ ) r ˜ A ( r ˜ ) 2 ( r cos ( θ θ ˜ ) r 0 ( r ˜ ) ) 2 + r ˜ r 0 ( r ˜ ) ( r sin ( θ θ ˜ ) ) 2 ] r d r d θ .
e i 2 π D ( N r cos ( ϕ θ ) ) ,
E in ( r , θ ) = e i 2 π D ( N r cos ( ϕ θ ) ) .
E out , N , ϕ ( r ˜ , θ ˜ ) = A ( r ˜ ) e i 2 π D N r 0 ( r ˜ ) cos ( θ ˜ ϕ ) × e i π λ Z N 2 D 2 [ r ˜ A ( r ˜ ) 2 r 0 ( r ˜ ) cos 2 ( θ ˜ ϕ ) + ( r 0 ( r ˜ ) r ˜ ) 2 sin 2 ( θ ˜ ϕ ) ] ,
E out ( r ˜ , θ ˜ ) = 1 i λ Z 0 R 0 2 π e i 2 π D N ( r cos θ cos ϕ + r sin θ sin ϕ ) × e i π λ Z [ r 0 r ˜ A ( r ˜ ) 2 ( r cos ( θ θ ˜ ) r 0 ) 2 + r ˜ r 0 ( r sin ( θ θ ˜ ) ) 2 ] r d r d θ = 1 i λ Z M 1 e i 2 π D N [ u cos ( θ ˜ ϕ ) + v sin ( θ ˜ ϕ ) ] × e i π λ Z [ r 0 r ˜ A ( r ˜ ) 2 ( u r 0 ) 2 + r ˜ r 0 v 2 ] d u d v ,
E out ( r ˜ , θ ˜ ) = 1 i λ Z e i 2 π D N r 0 cos ( θ ˜ ϕ ) e i π λ z N 2 D 2 [ cos 2 ( θ ˜ ϕ ) r ˜ A ( r ˜ ) 2 r 0 + sin 2 ( θ ˜ ϕ ) r 0 r ˜ ] × M 1 e i π λ Z [ r 0 r ˜ A ( r ˜ ) 2 ( u r 0 λ z N D r ˜ A ( r ˜ ) 2 r 0 cos ( θ ˜ ϕ ) ) 2 + r ˜ r 0 ( v λ z N D r 0 r ˜ sin ( θ ˜ ϕ ) ) 2 ] d u d v .
1 i λ Z M 1 e i π λ Z [ r 0 r ˜ A ( r ˜ ) 2 ( u r 0 λ z N D r ˜ A ( r ˜ ) 2 r 0 cos ( θ ˜ ϕ ) ) 2 + r ˜ r 0 ( v λ z N D r 0 r ˜ sin ( θ ˜ ϕ ) ) 2 ] d u d v = A ( r ˜ ) E λ , r ˜ , θ ˜ e i ρ 2 ρ d ρ d ψ ,
1 i λ Z M 1 e i π λ Z [ r 0 r ˜ A ( r ˜ ) 2 ( u r 0 ) 2 + r ˜ r 0 v 2 ] d u d v A ( r ˜ ) .
h ( x , y ) = m , n λ 0 b m , n e i 2 π D ( m x + n y ) ,
= m , n λ 0 b m , n e i 2 π D m 2 + n 2 ( x cos θ n , m + y sin θ m , n ) ,
e i 2 π λ h ( 1 + i 2 π λ h ) .
E out ( r ˜ , θ ˜ ) = 1 i λ Z M 1 ( 1 + i 2 π λ h ( x , y ) ) × e i π λ Z [ r 0 r ˜ A ( r ˜ ) 2 ( r cos ( θ θ ˜ ) r 0 ) 2 + r ˜ r 0 ( r sin ( θ θ ˜ ) ) 2 ] r d r d θ .
E out ( r ˜ , θ ˜ ) = A ( r ˜ ) ( 1 + 2 π λ 0 λ × m , n b m , n e i 2 π D m 2 + n 2 ( r 0 cos θ ˜ cos θ m , n + r 0 sin θ ˜ sin θ m , n ) × e i π λ Z ( m 2 + n 2 ) D 2 [ ( r ˜ A ( r ˜ ) r 0 ) 2 cos 2 ( θ ˜ θ m , n ) + ( r 0 r ˜ ) 2 sin 2 ( θ ˜ θ m , n ) ] ) .
E in ( x , y ) = r 0 ( 1 + m , n a m , n e i 2 π D ( m x + n y ) ) ,
E out ( r ˜ , θ ˜ ) = A ( r ˜ ) ( 1 + m , n a m , n e i 2 π D m 2 + n 2 ( r 0 cos θ ˜ cos θ m , n + r 0 sin θ ˜ sin θ m , n ) × e i π λ Z ( m 2 + n 2 ) D 2 [ ( r ˜ A ( r ˜ ) r 0 ) 2 cos 2 ( θ ˜ θ m , n ) + ( r 0 r ˜ ) 2 sin 2 ( θ ˜ θ m , n ) ] ) .
E ( x ˜ , y ˜ ) = A ( x ˜ , y ˜ ) ( 1 + m , n k i k f m , n k λ 0 k λ k e i 2 π D ( m x ˜ + n y ˜ ) ) ,
E in ( x , y ) = 1 + m , n k i k f m , n M 1 , k λ 0 k λ k e i 2 π D ( m x + n y ) .
E out ( r ˜ , θ ˜ ) = A ( r ˜ ) [ 1 + m , n k i k f m , n M 1 , k λ 0 k λ k e i 2 π D N m , n ( r 0 cos θ ˜ cos ϕ m , n + r 0 sin θ ˜ sin ϕ m , n ) × e i π λ Z N m , n 2 D 2 [ ( r ˜ A ( r ˜ ) 2 r 0 ) 2 cos 2 ( θ ˜ ϕ m , n ) + ( r 0 r ˜ ) 2 sin 2 ( θ ˜ ϕ m , n ) ] ] .
e i π λ Z N m , n 2 D 2 [ ( r ˜ A ( r ˜ ) 2 r 0 ) 2 cos 2 ( θ ˜ ϕ m , n ) + ( r 0 r ˜ ) 2 sin 2 ( θ ˜ ϕ m , n ) ] = e i λ λ 0 ψ m , n ( r ˜ , θ ˜ ) .
E out ( r ˜ , θ ˜ ) = A ( r ˜ ) [ 1 + m , n k p = 0 i k p f m , n M 1 , k λ 0 k p p ! λ k p × e i 2 π D N m , n ( r 0 cos θ ˜ cos ϕ m , n + r 0 sin θ ˜ sin ϕ m , n ) ψ m , n ( r ˜ , θ ˜ ) p ] E out ( r ˜ , θ ˜ ) = A ( r ˜ ) [ 1 + k p = 0 i k p λ 0 k p λ k p × m , n f m , n M 1 , k p ! ψ m , n ( r ˜ , θ ˜ ) p e i 2 π D N m , n ( r 0 cos θ ˜ cos ϕ m , n + r 0 sin θ ˜ sin ϕ m , n ) ] .
m , n f m , n M 1 , k p ! e i 2 π D N m , n ( r 0 cos θ ˜ cos ϕ m , n + r 0 sin θ ˜ sin ϕ m , n ) ψ m , m ( r ˜ , θ ˜ ) p = m , n f m , n M 2 , ( k p ) e i 2 π D N m , n ( r ˜ cos θ ˜ cos ϕ m , n + r ˜ sin θ ˜ sin ϕ m , n ) .
f m , n M 2 , ( k p ) = m , n f m , n M 1 , k p ! ψ m , n ( r ˜ , θ ˜ ) p e i 2 π D ( m x 0 ( x ˜ , y ˜ ) + n y 0 ( x ˜ , y ˜ ) ) × e i 2 π D ( m x ˜ + n y ˜ ) d x ˜ d y ˜ ,
E out ( x ˜ , y ˜ ) = A ( x ˜ , y ˜ ) ( 1 + m , n k i k f m , n M 2 , k λ 0 k λ k e i 2 π D ( m x ˜ + n y ˜ ) ) ,
N limit C = D λ 0 z ( λ 0 Δ λ ) 1 / 2 .
N limit PIAA = D M λ 0 z ( λ 0 Δ λ ) 1 / 2 .
E in = 1 + i λ 0 λ e i 2 π D ( m x + n y ) .
E λ ( Image DMs ) ( ξ , η ) = ( E λ Re ( ξ , η ) E λ 0 Re ( ξ , η ) ) + i ( E λ Im ( ξ , η ) λ 0 λ E λ 0 Im ( ξ , η ) ) .
x ˜ x 0 = r ˜ r 0 cos 2 θ 0 + r ˜ r 0 sin 2 θ 0 y ˜ y 0 = r ˜ r 0 sin 2 θ 0 + r ˜ r 0 cos 2 θ 0 x ˜ y 0 = ( r ˜ r 0 r ˜ r 0 ) cos θ 0 sin θ 0 ,
( x x 0 ) 2 = r 2 cos 2 θ 2 r r 0 cos θ cos θ 0 + r 0 2 cos 2 θ 0 ( y y 0 ) 2 = r 2 sin 2 θ 2 r r 0 sin θ sin θ 0 + r 0 2 sin 2 θ 0 ( x x 0 ) ( y y 0 ) = r 2 cos θ sin θ r r 0 ( sin θ cos θ 0 + cos θ sin θ 0 ) + r 0 2 cos θ 0 sin θ 0 d x d y = r d r d θ ,
r 2 d r ˜ d r 0 cos 2 ( θ θ 0 ) + r ˜ r 0 sin 2 ( θ θ 0 ) r 0 2 d r ˜ d r 0 r r 0 2 d r ˜ d r 0 cos ( θ θ 0 ) .
E out ( r ˜ ) = 1 i λ Z C ( 0 , 0 ) R e i π λ Z [ r 0 r ˜ A ( r ˜ ) 2 ( r cos θ r 0 ) 2 + r ˜ r 0 ( r sin θ ) 2 ] r d r d θ ,
E out ( r ˜ ) = 1 i λ Z C ( 0 , 0 ) R e i π λ Z [ r 0 r ˜ A ( r ˜ ) 2 ( x r 0 ) 2 + r ˜ r 0 y 2 ] d x d y .
M = γ Camera γ Sky .
M = 1 π R 2 0 2 π 0 R A ( r ˜ ) 2 r 0 r ˜ A ( r ˜ ) cos 2 θ ˜ + r ˜ r 0 sin 2 θ ˜ r ˜ d r ˜ d θ .
E α ( r ˜ , θ ˜ ) = A ( r ˜ ) e i 2 π D γ Sky ( r 0 cos θ ˜ ) = A ( r ˜ ) e i 2 π D γ Sky x 0 ( x ˜ , y ˜ ) .
x 0 x ˜ = r 0 r ˜ A ( r ˜ ) cos 2 θ ˜ + r ˜ r 0 sin 2 θ ˜ ,
x 0 y ˜ = ( r 0 r ˜ A ( r ˜ ) r ˜ r 0 ) cos θ ˜ sin θ ˜ .
N ( r ˜ , θ ˜ ) = γ Sky ( x 0 x ˜ ) 2 + ( x 0 y ˜ ) 2 = γ Sky r 0 r ˜ A ( r ˜ ) cos 2 θ ˜ + r ˜ r 0 sin 2 θ ˜ .
γ Camera = N ( r ˜ , θ ˜ ) = γ Sky 1 π R 2 0 2 π 0 R A ( r ˜ ) 2 r 0 r ˜ A ( r ˜ ) cos 2 θ ˜ + r ˜ r 0 sin 2 θ ˜ r ˜ d r ˜ d θ .

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