Abstract

Segmented mirror telescope designs address issues of mechanical rigidity, but introduce the problem of aligning, or cophasing, the separate segments to conform to the optimum mirror shape. While several solutions have been widely adopted, a few difficulties persist — the introduction of non-common path errors and an artificial division of the problem into coarse and fine regimes involving separate dedicated hardware solutions. Here we propose a novel method that addresses many of these issues. Fizeau Interferometric Cophasing of Segmented Mirrors (FICSM) uses non-redundant sparse aperture interferometry to phase mirror segments to interferometric precision using unexceptional science hardware. To show the potential of this technique we numerically simulate conditions on NASA’s James Webb Space Telescope (JWST), showing that the FICSM method has the potential to phase the primary mirror from an initial state with segment-to-segment pistons as large as 150 microns and tilts as large as 0.5 arcseconds, to produce a final state with 0.75 nm rms segment-to-segment pistons and 3.7 mas rms segment tilts. The image undergoes monotonic improvement during this process. This results in a rms wavefront error of 3.65 nm, well below the 100 nm requirement of JWST’s coarse phasing algorithm.

© 2012 OSA

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2012

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE8442 (2012), 84423D.

2009

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009).
[CrossRef]

2007

R. J. Allen, “Station-keeping requirements for constellations of free-flying collectors used for astronomical imaging in space,” Pub. Ast. Soc. Pac.119, 914–922 (2007).
[CrossRef]

2006

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

2005

2001

J. Rodríguez-Ramos and J. Fuensalida, “Phasing of segmented mirrors: a new algorithm for piston detection,” MNRAS328, 167–173 (2001).
[CrossRef]

2000

P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac.112, 555–565 (2000).
[CrossRef]

G. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt.39, 4706–4714 (2000).
[CrossRef]

1999

1998

Abraham, R.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Acton, D.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Albert, L.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Allen, R. J.

R. J. Allen, “Station-keeping requirements for constellations of free-flying collectors used for astronomical imaging in space,” Pub. Ast. Soc. Pac.119, 914–922 (2007).
[CrossRef]

Aronstein, D.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Barto, A.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Beaulieu, M.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

Beichman, C.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

Bowers, C.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Bracewell, R.

R. Bracewell, The Fourier Transform & its Applications, 3rd ed. (McGraw-Hill Science/Engineering/Math, 2000).

Burge, J.

Chanan, G.

Chayer, P.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Cheetham, A.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

A. Cheetham, Cophasing JWST’s segmented mirror using sparse aperture interferometry University of Sydney Hons. Thesis, (2011).

Cohen, R.

J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009).
[CrossRef]

Comeau, T.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Contos, A.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Danchi, W.

P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac.112, 555–565 (2000).
[CrossRef]

Dean, B.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Dekens, F.

Doyon, R.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

Elliot, E.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Elliott, E.

M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE8442 (2012), 84423D.

Feinberg, L.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Ferrarese, L.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge Univ Press, 1986).

Ford, S. K. E.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Fuensalida, J.

J. Rodríguez-Ramos and J. Fuensalida, “Phasing of segmented mirrors: a new algorithm for piston detection,” MNRAS328, 167–173 (2001).
[CrossRef]

Fullerton, A. W.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Greenbaum, A. Z.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Grimaldi, S.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Haniff, C.

P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac.112, 555–565 (2000).
[CrossRef]

Hayden, W.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Hutchings, J.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Ireland, M.

J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009).
[CrossRef]

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

Ireland, M. J.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Jayawardhana, R.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Johnstone, D.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Kirkman, D.

Knight, J.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Koekemoer, A.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Lafrenière, D.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Lallo, M.

M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE8442 (2012), 84423D.

League, B.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Lightsey, P.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Lloyd, J.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

Lloyd, J. P.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Makidon, R.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

Martel, A. R.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Martinache, F.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

Mast, T.

Maszkiewicz, M.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

McKernan, B.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Meyer, M. R.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Michaels, S.

Monnier, J.

J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009).
[CrossRef]

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac.112, 555–565 (2000).
[CrossRef]

Nelson, J.

Ohara, C.

Perrin, M.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE8442 (2012), 84423D.

J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009).
[CrossRef]

Pipher, J. L.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Pravdo, S.

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

Press, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge Univ Press, 1986).

Rodríguez-Ramos, J.

J. Rodríguez-Ramos and J. Fuensalida, “Phasing of segmented mirrors: a new algorithm for piston detection,” MNRAS328, 167–173 (2001).
[CrossRef]

Rowlands, N.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Sabatke, D.

Sabatke, E.

Sawicki, M.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Shaklan, S.

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

Sirko, E.

Sivaramakrishnan, A.

M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE8442 (2012), 84423D.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

Smith, J.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Soummer, R.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE8442 (2012), 84423D.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

Starr, C. W.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Tannirkulam, A.

J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009).
[CrossRef]

Terry, J.

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

Teuben, P.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge Univ Press, 1986).

Touahri, D.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Troy, M.

Tuthill, P.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009).
[CrossRef]

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac.112, 555–565 (2000).
[CrossRef]

Tuthill, P. G.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge Univ Press, 1986).

Volk, K.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Willott, C.

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

Wishnow, E.

P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac.112, 555–565 (2000).
[CrossRef]

ApJ

J. Monnier, P. Tuthill, M. Ireland, R. Cohen, A. Tannirkulam, and M. Perrin, “Mid-infrared size survey of young stellar objects: Description of Keck segment-tilting experiment and basic results,” ApJ700, 491 (2009).
[CrossRef]

ApJ Lett.

J. Lloyd, F. Martinache, M. Ireland, J. Monnier, S. Pravdo, S. Shaklan, and P. Tuthill, “Direct detection of the brown dwarf GJ 802B with adaptive optics masking interferometry,” ApJ Lett.650, L131 (2006).
[CrossRef]

Appl. Opt.

MNRAS

J. Rodríguez-Ramos and J. Fuensalida, “Phasing of segmented mirrors: a new algorithm for piston detection,” MNRAS328, 167–173 (2001).
[CrossRef]

Proc. SPIE

D. Acton, J. Knight, A. Contos, S. Grimaldi, J. Terry, P. Lightsey, A. Barto, B. League, B. Dean, J. Smith, C. Bowers, D. Aronstein, L. Feinberg, W. Hayden, T. Comeau, R. Soummer, E. Elliot, M. Perrin, and C. W. Starr, “Wavefront sensing and controls for the James Webb Space Telescope,” Proc. SPIE8442 (2012), 84422H.

M. Perrin, R. Soummer, E. Elliott, M. Lallo, and A. Sivaramakrishnan, “Simulating point-spread functions for the James Webb Space Telescope with WebbPSF,” Proc. SPIE8442 (2012), 84423D.

A. Sivaramakrishnan, P. Tuthill, M. Ireland, J. Lloyd, F. Martinache, R. Soummer, R. Makidon, R. Doyon, M. Beaulieu, and C. Beichman, “Planetary system and star formation science with non-redundant masking on JWST,” Proc. SPIE7440 (2009), 74400Y.
[CrossRef]

R. Doyon, J. Hutchings, M. Beaulieu, L. Albert, D. Lafrenière, C. Willott, D. Touahri, N. Rowlands, M. Maszkiewicz, A. W. Fullerton, K. Volk, A. R. Martel, P. Chayer, A. Sivaramakrishnan, R. Abraham, L. Ferrarese, R. Jayawardhana, D. Johnstone, M. R. Meyer, J. L. Pipher, and M. Sawicki, “The JWST fine guidance sensor (FGS) and near-infrared imager and slitless spectrograph (NIRISS),” Proc. SPIE8442 (2012), 84422R.

A. Sivaramakrishnan, D. Lafrenière, S. K. E. Ford, B. McKernan, A. Cheetham, A. Z. Greenbaum, P. G. Tuthill, J. P. Lloyd, M. J. Ireland, R. Doyon, M. Beaulieu, A. R. Martel, A. Koekemoer, F. Martinache, and P. Teuben, “Non-redundant aperture masking interferometry (AMI) and segment phasing with JWST-NIRISS,” Proc. SPIE8442 (2012), 84422S.

Pub. Ast. Soc. Pac.

R. J. Allen, “Station-keeping requirements for constellations of free-flying collectors used for astronomical imaging in space,” Pub. Ast. Soc. Pac.119, 914–922 (2007).
[CrossRef]

P. Tuthill, J. Monnier, W. Danchi, E. Wishnow, and C. Haniff, “Michelson interferometry with the Keck I telescope,” Pub. Ast. Soc. Pac.112, 555–565 (2000).
[CrossRef]

Other

JWST Mission Operations Concept Document, Revision C (2008).

A. Cheetham, Cophasing JWST’s segmented mirror using sparse aperture interferometry University of Sydney Hons. Thesis, (2011).

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes (Cambridge Univ Press, 1986).

R. Bracewell, The Fourier Transform & its Applications, 3rd ed. (McGraw-Hill Science/Engineering/Math, 2000).

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Figures (3)

Fig. 1
Fig. 1

Simulated images generated from a 3 hole mask (inset: phases added in the pupil plane) and their corresponding Fourier phase spectra, illustrating the effects summarized in table 1. Note that the fringes (and splodges) in narrowband images in the absence of aberrations have zero phase, and so the background has been darkened to show the splodge outlines.

Fig. 2
Fig. 2

Left Panel: An example of a simulated interferogram as imaged with the NIRCam instrument using a non-redundant mask. Right Panel: the aperture mask with seven circular holes overlaid on the hexagonal segments of JWST’s primary mirror. The envelope of the interferogram is sometimes referred to as the primary beam in synthesis imaging. The shape of the primary beam depends on the hole shape.

Fig. 3
Fig. 3

Output residuals from 100 complete cophasing simulations, each with noise as described above. The left panel plots histograms of the initial pistons and the residuals after each step of the piston measurement process. After phasing, one simulation produced visible outliers around 5 nm, while the remainder are well below 1 nm. The right panel shows histograms of the initial mirror tilts, and the residuals after the tilt measurement process. All tilts are reduced to well below a single pixel (65 mas).

Tables (2)

Tables Icon

Table 1 Effects generated by our two classes of segment alignment errors (tilt and piston), on different fringe observables for Narrowband (monochromatic) and broadband interferograms.

Tables Icon

Table 2 Specifications used for simulations

Equations (14)

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

I ( ξ , η ) = λ 1 λ 2 f ( λ ) | E ( ξ , η ) | 2 d λ ,
E i = e 4 π i λ p i [ C i ( x x i , y y i ) e 2 π i λ ( m i x + n i y ) ] .
E i ( ξ , η ) = e 2 π i λ ( 2 p i x i ξ y i η ) A i ( ξ m i , η n i )
A i ( ξ m i , η n i ) = [ C ( x x i , y y i ) ] = r i J 1 ( 2 π r i ( ξ m i ) 2 + ( η n i ) 2 λ ) ( ξ m i ) 2 + ( η n i ) 2
I ( ξ , η ) = λ 1 λ 2 f ( λ ) i = 1 N j = 1 N E i E j * d λ = λ 1 λ 2 f ( λ ) i = 1 N j = 1 N I i , j d λ ,
I i , j = e 4 π i λ ( p i p j ) e 2 π i λ ( ξ ( x i x j ) + η ( y i y j ) ) A i ( ξ m i , η n i ) A j * ( ξ m i , η n i ) .
L c = λ c 2 Δ λ .
χ I 2 = ξ , η | I model ( ξ , η ) I measured ( ξ , η ) | 2
χ V 2 = u , v | V model ( u , v ) V measured ( u , v ) | 2
δ i , j = p j p i
M i , j = 1 2 ( t i + t j )
A p = δ
p = v diag ( 1 / w j ) U T δ
δ i , j = p j p i = p a p i + p j p a = δ i , a + δ a , j

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