Abstract

We present two complementary algorithms suitable for using focal-plane measurements to control a wavefront corrector with an extremely high-spatial resolution. The algorithms use linear approximations to iteratively minimize the aberrations seen by the focal-plane camera. The first algorithm, Fast & Furious (FF), uses a weak-aberration assumption and pupil symmetries to achieve fast wavefront reconstruction. The second algorithm, an extension to FF, can deal with an arbitrary pupil shape; it uses a Gerchberg–Saxton (GS)-style error reduction to determine the pupil amplitudes. Simulations and experimental results are shown for a spatial-light modulator controlling the wavefront with a resolution of 170×170pixels. The algorithms increase the Strehl ratio from 0.75 to 0.98–0.99, and the intensity of the scattered light is reduced throughout the whole recorded image of 320×320pixels. The remaining wavefront rms error is estimated to be 0.15rad with FF and 0.10rad with FF-GS.

© 2014 Optical Society of America

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2013 (4)

2012 (3)

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

V. Korkiakoski, C. U. Keller, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Experimental validation of optimization concepts for focal-plane image processing with adaptive optics,” Proc. SPIE 8447, 84475Z (2012).
[CrossRef]

P. Riaud, D. Mawet, and A. Magette, “Nijboer-Zernike phase retrieval for high contrast imaging. Principle, on-sky demonstration with NACO, and perspectives in vector vortex coronagraphy,” Astron. Astrophys. 545, A150 (2012).
[CrossRef]

2010 (4)

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

S. Meimon, T. Fusco, and L. M. Mugnier, “Lift: a focal-plane wavefront sensor for real-time low-order sensing on faint sources,” Opt. Lett. 35, 3036–3038 (2010).
[CrossRef]

O. Guyon, “High sensitivity wavefront sensing with a nonlinear curvature wavefront sensor,” Publ. Astron. Soc. Pac. 122, 49–62 (2010).
[CrossRef]

2007 (2)

2006 (1)

P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488–498 (2006).
[CrossRef]

2004 (2)

J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J., Lett. 604, L117–L120 (2004).
[CrossRef]

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

2003 (2)

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

J. J. Green, D. C. Redding, S. B. Shaklan, and S. A. Basinger, “Extreme wave front sensing accuracy for the Eclipse coronagraphic space telescope,” Proc. SPIE 4860, 266–276 (2003).
[CrossRef]

2001 (1)

1995 (1)

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Abe, L.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Aller-Carpentier, E.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Andrei, R.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

V. Korkiakoski, C. U. Keller, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Experimental validation of optimization concepts for focal-plane image processing with adaptive optics,” Proc. SPIE 8447, 84475Z (2012).
[CrossRef]

Angel, R.

J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J., Lett. 604, L117–L120 (2004).
[CrossRef]

Antichi, J.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Basinger, S. A.

J. J. Green, D. C. Redding, S. B. Shaklan, and S. A. Basinger, “Extreme wave front sensing accuracy for the Eclipse coronagraphic space telescope,” Proc. SPIE 4860, 266–276 (2003).
[CrossRef]

Baudoz, P.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Belikov, R.

Beuzit, J.-L.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Bikkannavar, S.

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

Boccaletti, A.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Bonavita, M.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Bordé, P. J.

P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488–498 (2006).
[CrossRef]

Burruss, R. S.

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

Christou, J. C.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

Codona, J.

Codona, J. L.

J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J., Lett. 604, L117–L120 (2004).
[CrossRef]

Crepp, J. R.

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

den Dekker, A. J.

Doelman, N.

C. S. Smith, R. Marinică, A. J. den Dekker, M. Verhaegen, V. Korkiakoski, C. U. Keller, and N. Doelman, “Iterative linear focal-plane wavefront correction,” J. Opt. Soc. Am. A 30, 2002–2011 (2013).
[CrossRef]

V. Korkiakoski, N. Doelman, J. Codona, M. Kenworthy, G. Otten, and C. U. Keller, “Calibrating a high-resolution wavefront corrector with a static focal-plane camera,” Appl. Opt. 52, 7554–7563 (2013).
[CrossRef]

V. Korkiakoski, C. U. Keller, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Experimental validation of optimization concepts for focal-plane image processing with adaptive optics,” Proc. SPIE 8447, 84475Z (2012).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Dohlen, K.

B. Paul, L. M. Mugnier, J.-F. Sauvage, M. Ferrari, and K. Dohlen, “High-order myopic coronagraphic phase diversity (COFFEE) for wave-front control in high-contrast imaging systems,” Opt. Express 21, 31751–31768 (2013).
[CrossRef]

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Fedrigo, E.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Ferrari, M.

Fienup, J. R.

Fraanje, R.

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

V. Korkiakoski, C. U. Keller, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Experimental validation of optimization concepts for focal-plane image processing with adaptive optics,” Proc. SPIE 8447, 84475Z (2012).
[CrossRef]

Fusco, T.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Give’on, A.

Gonsalves, R. A.

Graham, J. R.

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

Gratton, R. G.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Green, J. J.

J. J. Green, D. C. Redding, S. B. Shaklan, and S. A. Basinger, “Extreme wave front sensing accuracy for the Eclipse coronagraphic space telescope,” Proc. SPIE 4860, 266–276 (2003).
[CrossRef]

Guyon, O.

O. Guyon, “High sensitivity wavefront sensing with a nonlinear curvature wavefront sensor,” Publ. Astron. Soc. Pac. 122, 49–62 (2010).
[CrossRef]

Hanenburg, H.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Hickey, J. P.

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

Hubin, N.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Jager, R.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Kasdin, J.

Kasper, M.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Keller, C. U.

C. S. Smith, R. Marinică, A. J. den Dekker, M. Verhaegen, V. Korkiakoski, C. U. Keller, and N. Doelman, “Iterative linear focal-plane wavefront correction,” J. Opt. Soc. Am. A 30, 2002–2011 (2013).
[CrossRef]

V. Korkiakoski, N. Doelman, J. Codona, M. Kenworthy, G. Otten, and C. U. Keller, “Calibrating a high-resolution wavefront corrector with a static focal-plane camera,” Appl. Opt. 52, 7554–7563 (2013).
[CrossRef]

V. Korkiakoski, C. U. Keller, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Experimental validation of optimization concepts for focal-plane image processing with adaptive optics,” Proc. SPIE 8447, 84475Z (2012).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Kenworthy, M.

Kerber, F.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Korkiakoski, V.

C. S. Smith, R. Marinică, A. J. den Dekker, M. Verhaegen, V. Korkiakoski, C. U. Keller, and N. Doelman, “Iterative linear focal-plane wavefront correction,” J. Opt. Soc. Am. A 30, 2002–2011 (2013).
[CrossRef]

V. Korkiakoski, N. Doelman, J. Codona, M. Kenworthy, G. Otten, and C. U. Keller, “Calibrating a high-resolution wavefront corrector with a static focal-plane camera,” Appl. Opt. 52, 7554–7563 (2013).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

V. Korkiakoski, C. U. Keller, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Experimental validation of optimization concepts for focal-plane image processing with adaptive optics,” Proc. SPIE 8447, 84475Z (2012).
[CrossRef]

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Macintosh, B. A.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

Magette, A.

P. Riaud, D. Mawet, and A. Magette, “Nijboer-Zernike phase retrieval for high contrast imaging. Principle, on-sky demonstration with NACO, and perspectives in vector vortex coronagraphy,” Astron. Astrophys. 545, A150 (2012).
[CrossRef]

Makidon, R. B.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

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

Malbet, F.

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

Marchis, F.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

Marinica, R.

Martinache, F.

F. Martinache, “The asymmetric pupil Fourier wavefront sensor,” Publ. Astron. Soc. Pac. 125, 422–430 (2013).
[CrossRef]

Martinez, P.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Mawet, D.

P. Riaud, D. Mawet, and A. Magette, “Nijboer-Zernike phase retrieval for high contrast imaging. Principle, on-sky demonstration with NACO, and perspectives in vector vortex coronagraphy,” Astron. Astrophys. 545, A150 (2012).
[CrossRef]

Mawet, D. P.

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

Meimon, S.

Mesa, D.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Mugnier, L. M.

Oppenheimer, B. R.

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

Orecchia, M.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Otten, G.

Paul, B.

Perrin, M. D.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

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

Petit, C.

Poyneer, L. A.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

Preis, O.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Rabou, P.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Redding, D. C.

J. J. Green, D. C. Redding, S. B. Shaklan, and S. A. Basinger, “Extreme wave front sensing accuracy for the Eclipse coronagraphic space telescope,” Proc. SPIE 4860, 266–276 (2003).
[CrossRef]

Riaud, P.

P. Riaud, D. Mawet, and A. Magette, “Nijboer-Zernike phase retrieval for high contrast imaging. Principle, on-sky demonstration with NACO, and perspectives in vector vortex coronagraphy,” Astron. Astrophys. 545, A150 (2012).
[CrossRef]

Roberts, J. E.

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

Roberts, L. C.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

Roelfsema, R.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Rousset, G.

Rykoski, K.

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

Salter, G.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Sauvage, J.

Sauvage, J.-F.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schmid, H. M.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Serabyn, E.

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

Shaklan, S.

Shaklan, S. B.

J. J. Green, D. C. Redding, S. B. Shaklan, and S. A. Basinger, “Extreme wave front sensing accuracy for the Eclipse coronagraphic space telescope,” Proc. SPIE 4860, 266–276 (2003).
[CrossRef]

Shao, M.

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

Sivaramakrishnan, A.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

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

Smith, C. S.

Stadler, E.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Tecza, M.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Thatte, N.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Traub, W. A.

P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488–498 (2006).
[CrossRef]

Troy, M.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

van Dam, M. A.

L. C. Roberts, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy, “Is that really your Strehl ratio?” Proc. SPIE 5490, 504–515 (2004).
[CrossRef]

Venema, L.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Verhaegen, M.

C. S. Smith, R. Marinică, A. J. den Dekker, M. Verhaegen, V. Korkiakoski, C. U. Keller, and N. Doelman, “Iterative linear focal-plane wavefront correction,” J. Opt. Soc. Am. A 30, 2002–2011 (2013).
[CrossRef]

V. Korkiakoski, C. U. Keller, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Experimental validation of optimization concepts for focal-plane image processing with adaptive optics,” Proc. SPIE 8447, 84475Z (2012).
[CrossRef]

C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012).
[CrossRef]

Vérinaud, C.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Yaitskova, N.

C. Vérinaud, M. Kasper, J.-L. Beuzit, R. G. Gratton, D. Mesa, E. Aller-Carpentier, E. Fedrigo, L. Abe, P. Baudoz, A. Boccaletti, M. Bonavita, K. Dohlen, N. Hubin, F. Kerber, V. Korkiakoski, J. Antichi, P. Martinez, P. Rabou, R. Roelfsema, H. M. Schmid, N. Thatte, G. Salter, M. Tecza, L. Venema, H. Hanenburg, R. Jager, N. Yaitskova, O. Preis, M. Orecchia, and E. Stadler, “System study of EPICS: the exoplanets imager for the E-ELT,” Proc. SPIE 7736, 77361N (2010).
[CrossRef]

Yu, J. W.

F. Malbet, J. W. Yu, and M. Shao, “High-dynamic-range imaging using a deformable mirror for space coronography,” Publ. Astron. Soc. Pac. 107, 386–398 (1995).
[CrossRef]

Appl. Opt. (2)

Astron. Astrophys. (1)

P. Riaud, D. Mawet, and A. Magette, “Nijboer-Zernike phase retrieval for high contrast imaging. Principle, on-sky demonstration with NACO, and perspectives in vector vortex coronagraphy,” Astron. Astrophys. 545, A150 (2012).
[CrossRef]

Astrophys. J. (2)

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

P. J. Bordé and W. A. Traub, “High-contrast imaging from space: speckle nulling in a low-aberration regime,” Astrophys. J. 638, 488–498 (2006).
[CrossRef]

Astrophys. J., Lett. (1)

J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J., Lett. 604, L117–L120 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (2)

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. SPIE (6)

J. J. Green, D. C. Redding, S. B. Shaklan, and S. A. Basinger, “Extreme wave front sensing accuracy for the Eclipse coronagraphic space telescope,” Proc. SPIE 4860, 266–276 (2003).
[CrossRef]

R. S. Burruss, E. Serabyn, D. P. Mawet, J. E. Roberts, J. P. Hickey, K. Rykoski, S. Bikkannavar, and J. R. Crepp, “Demonstration of on sky contrast improvement using the modified Gerchberg-Saxton algorithm at the Palomar Observatory,” Proc. SPIE 7736, 77365X (2010).
[CrossRef]

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Figures (15)

Fig. 1.
Fig. 1.

Schematic illustration of the FF algorithm.

Fig. 2.
Fig. 2.

Left, a purely sinusoidal WF. Right, resulting image raised to the power of 0.2 to compress the dynamic range.

Fig. 3.
Fig. 3.

Radial cuts of the second-order component ξ, defined in Eq. (10), and an inverted and scaled perfect PSF, a2.

Fig. 4.
Fig. 4.

Radial cuts of the perfect PSF, its improved first-order approximation, and the second-order approximation. The latter is virtually identical to the perfect PSF.

Fig. 5.
Fig. 5.

Schematic illustration of the FF-GS algorithm.

Fig. 6.
Fig. 6.

Schematic view of the used hardware. The lenses are standard 1 inch doublets. The beam diameter is 3.7 mm at the SLM.

Fig. 7.
Fig. 7.

SLM phase and amplitude responses. The dots indicate individual measurements. The lines show fifth-order polynomial fits to the data.

Fig. 8.
Fig. 8.

Modulus of an averaged dOTF array.

Fig. 9.
Fig. 9.

Top row, typical WF shapes (170×170pixels) of the SLM after the convergence of FF and FF-GS. Bottom, radial cuts through the WFs; the shaded area shows the range (minima and maxima) of five independent measurements.

Fig. 10.
Fig. 10.

Top row, pupil amplitudes (170×170pixels) reconstructed with different methods. Left, FF-GS. Middle, dOTF (same as in Fig. 8). Right, GS postprocessing from a converged PSF. Bottom, radial cuts through the pupil amplitudes; five independent measurement runs are shown for FF-GS.

Fig. 11.
Fig. 11.

Tip-tilt-removed residual WF error as a function of time step. Top, simulations (real value). Bottom, measurements (estimation from PSF images).

Fig. 12.
Fig. 12.

Average intensity at different parts of the field. Three cases are shown: the field corresponding to Airy rings 2–4, Airy rings 12–17, and Airy further than 30.

Fig. 13.
Fig. 13.

Residual-WF error as a function of time step. Values calculated from the actual estimates used by the algorithms. Top, simulations. Bottom, measurements.

Fig. 14.
Fig. 14.

Examples of PSF images (320×320pixels) raised to the 0.1 power. A, initial, measured; B, perfect, simulated; C, converged FF-GS, measured; D, converged FF-GS, simulated.

Fig. 15.
Fig. 15.

Averaged radial profiles of PSF images. Upper, simulated, with three study cases shown. Lower, measured, with results from five independent runs shown. The perfect PSF is identical to the one in the upper plot.

Tables (2)

Tables Icon

Table 1. Notations and Symmetries

Equations (37)

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p=|F{Aexp(iϕ)}|2,
p=|F{A+iAϕ0.5Aϕ2}|2.
ϕ=ϕe+ϕo,
p=|F{A+iAϕe+iAϕo0.5Aϕe20.5Aϕo2Aϕeϕo}|2.
a=F{A},
v=F{Aϕe},
y=Im{F{Aϕo}}.
p=|a+ivy0.5v20.5y2iz|2,
p=a2+v2+y22ay+ξ,
ξ=0.25v22+0.25y22+z2av2ay2+0.5v2y2+yv2+yy22vz.
p=a2+y2+v22ay(1max(pn)max(a2))a2,
pn=pmx,ya2(x,y)x,ypm(x,y),
p=pn+(1max(pn)max(a2))a2,
pe=a2+v2+y2,
po=2ay.
y=apo2a2+ϵ,
pe2=a2+(v+vd)2+(y+yd)2,
vs=pepe2vd2yd22yyd2vd.
v=sign(vs)|pea2y2|0.5,
q=NarrNpup,
Aϕ=F1{w(v+iy)},
θk=glθk1gAϕk1,
p1=|e1|2=|F{A+iA(ϕ)}|2,
p2=|e2|2=|F{A+iA(ϕ+ϕd1)}|2,
p3=|e3|2=|F{A+iA(ϕ+ϕd2)}|2,
e1=(ar+α)+i(ai+β),
ar=Re{F{A}},ai=Im{F{A}},α=Im{F{Aϕ}},β=Re{F{Aϕ}}.
[2αd12βd12αd22βd2][αβ]=[c1c2],
αd1=Im{F{Aϕd1}},βd1=Re{F{Aϕd1}},αd2=Im{F{Aϕd2}},βd2=Re{F{Aϕd2}},
c1=p2p1(2arαd1+2aiβd1+αd12+βd12),c2=p3p1(2arαd2+2aiβd2+αd22+βd22).
e1=|p1|0.5exp[iarg((ar+α)+i(ai+β))].
e2=|p2|0.5exp[iarg((ar+α)+i(ai+β))].
E2=F1{e2}.
A=|E2|.
E1=F1{e1}.
ϕ=arg(E1),
A=|E1|.

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