Abstract

Wavefront reconstruction is carried out using sequentially recorded speckle patterns and an iterative phase retrieval method based on wave propagation. A novel fast-convergent algorithm that maintains the propagation distance in the iteration step equal to the distance between measurement planes is demonstrated. Employing the new algorithm, influences of the number of measurement planes, number of iterations, and uncertainties in the detector’s transverse and axial positions on the rate of phase convergence are analyzed experimentally. A conceptual design for a dynamic wavefront sensor using arrays of beam splitters and detectors for parallel speckle recording is described.

© 2009 Optical Society of America

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2009

2008

2007

2006

2005

2004

D. Monteiro, G. Vdovin, and P. Sarro, “High-speed wavefront sensor compatible with standard CMOS technology,” Sens. Actuators A 109, 220-230 (2004).
[CrossRef]

S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43, 5747-5751 (2004).
[CrossRef]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

2002

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

1998

J. R. Fienup, B. J. Thelen, R. G. Paxman, and D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” Proc. SPIE 3353, 930-940 (1998).
[CrossRef]

D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

1997

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 307-320 (1997).
[CrossRef]

L. McMackin, R. Hugo, R. Pierson, and C. Truman, “High speed optical tomography system for imaging dynamic transparent media,” Opt. Express 1, 302-311(1997).
[CrossRef] [PubMed]

1993

1992

W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, “Phase retrieval through focus variation for ultraresolution in field-emission transmission electron-microscopy,” Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

1988

1984

E. Kirkland, “Improved high resolution image processing of bright field electron micrographs: I. Theory,” Ultramicroscopy 15, 151-172 (1984).
[CrossRef]

1982

1980

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micropraphs,” J. Microsc. Spectrosc. Electron. 5, 661-670 (1980).

1973

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6-L9 (1973).
[CrossRef]

Acton, D.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Acton, S.

B. Dean, D. Aronstein, J. S. Smith, R. Shiri, and S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” Proc. SPIE 6265, 626511 (2006).
[CrossRef]

Almoro, P.

Almoro, P. F.

Anand, A.

Aronstein, D.

B. Dean, D. Aronstein, J. S. Smith, R. Shiri, and S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” Proc. SPIE 6265, 626511 (2006).
[CrossRef]

Bao, P.

Barto, A.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Brady, G. R.

Bunk, O.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Burns, L.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Carrara, D. A.

J. R. Fienup, B. J. Thelen, R. G. Paxman, and D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” Proc. SPIE 3353, 930-940 (1998).
[CrossRef]

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 307-320 (1997).
[CrossRef]

Chidlaw, R.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (l979).

Coene, W.

W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, “Phase retrieval through focus variation for ultraresolution in field-emission transmission electron-microscopy,” Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

Contos, A.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Contreras, J.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

David, C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

de Beek, M. O.

W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, “Phase retrieval through focus variation for ultraresolution in field-emission transmission electron-microscopy,” Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

De Natale, P.

De Nicola, S.

Dean, B.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

B. Dean, D. Aronstein, J. S. Smith, R. Shiri, and S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” Proc. SPIE 6265, 626511 (2006).
[CrossRef]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Ellerbroek, B. L.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 307-320 (1997).
[CrossRef]

Elliott, E.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

Feinberg, L.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Ferraro, P.

Fienup, J. R.

Ford, S. D.

D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (l972).

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 2l, 829-832 (l982).

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32-39 (l979).

Guizar-Sicairos, M.

Gundu, P. N.

Gusev, M. E.

Hansen, K.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Hanson, S.

Hanson, S. G.

Hardy, B.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Hayden, W.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Hugo, R.

Hunt, B. R.

D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Jacquot, P.

P. Jacquot, “Speckle interferometry: a review of the principal methods in use for experimental mechanics applications,” Strain 44, 57-69 (2008).
[CrossRef]

Janssen, G.

W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, “Phase retrieval through focus variation for ultraresolution in field-emission transmission electron-microscopy,” Phys. Rev. Lett. 69, 3743-3746 (1992).
[CrossRef] [PubMed]

Jefimovs, K.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Johnson, I.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Juptner, W.

U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Kirkland, E.

E. Kirkland, “Improved high resolution image processing of bright field electron micrographs: I. Theory,” Ultramicroscopy 15, 151-172 (1984).
[CrossRef]

Knight, J.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Lee, D. J.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 307-320 (1997).
[CrossRef]

Lightsey, P.

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

Marichal-Hernández, J. G.

McMackin, L.

Miccio, L.

Misell, D. L.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200-2225 (l973).
[CrossRef]

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6-L9 (1973).
[CrossRef]

Monteiro, D.

D. Monteiro, G. Vdovin, and P. Sarro, “High-speed wavefront sensor compatible with standard CMOS technology,” Sens. Actuators A 109, 220-230 (2004).
[CrossRef]

Nugent, K. A.

Osten, W.

Paturzo, M.

Paxman, R. G.

J. R. Fienup, B. J. Thelen, R. G. Paxman, and D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” Proc. SPIE 3353, 930-940 (1998).
[CrossRef]

D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 307-320 (1997).
[CrossRef]

R. G. Paxman and J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914-923 (1988).
[CrossRef]

Pedrini, G.

Pfeiffer, F.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Pierson, R.

Roddier, C.

Roddier, F.

Rodenburg, J. M.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

Rodríguez-Ramos, J. M.

Rodríguez-Ramos, L. F.

Roggemann, M. C.

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D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
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D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
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D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

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D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

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D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
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S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43, 5747-5751 (2004).
[CrossRef]

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J. R. Fienup, B. J. Thelen, R. G. Paxman, and D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” Proc. SPIE 3353, 930-940 (1998).
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[CrossRef]

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D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
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D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

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D. Monteiro, G. Vdovin, and P. Sarro, “High-speed wavefront sensor compatible with standard CMOS technology,” Sens. Actuators A 109, 220-230 (2004).
[CrossRef]

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D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

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S. Yang and H. Takajo, “Quantization error reduction in the measurement of Fourier intensity for phase retrieval,” Jpn. J. Appl. Phys. 43, 5747-5751 (2004).
[CrossRef]

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U. Schnars and W. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85-R101 (2002).
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W. Coene, G. Janssen, M. O. de Beek, and D. van Dyck, “Phase retrieval through focus variation for ultraresolution in field-emission transmission electron-microscopy,” Phys. Rev. Lett. 69, 3743-3746 (1992).
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[CrossRef]

A. Contos, D. Acton, A. Barto, L. Burns, J. Contreras, B. Dean, E. Elliott, L. Feinberg, K. Hansen, B. Hardy, W. Hayden, J. Knight, P. Lightsey, C. Starr, and J. Sullivan, “Verification of the James Webb Space Telescope (JWST) wavefront sensing and control system,” Proc. SPIE 7010, 70100S-10(2008).

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[CrossRef]

D. W. Tyler, S. D. Ford, B. R. Hunt, R. G. Paxman, M. C. Roggemann, J. C. Rountree, T. J. Schulz, K. J. Schulze, J. H. Seldin, D. G. Sheppard, B. E. Stribling, W. C. van Kampen, and B. M. Welch, “Comparison of image reconstruction algorithms using adaptive optics instrumentation,” Proc. SPIE 3353, 160-171 (1998).
[CrossRef]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Design of a dynamic wavefront sensor based on speckle phase retrieval. A test wavefront is incident on a phase diffuser, and the generated speckle intensities at multiple planes are measured in parallel to be input into a computer algorithm.

Fig. 2
Fig. 2

(a) Diagram depicting the diffuser plane and the intensity measurement planes separated by Δ z . (b) Diagram of the FF algorithm. (c) Diagram of the FB algorithm.

Fig. 3
Fig. 3

(a)–(d) Reconstructed phases using the FF algorithm, with increasing distance between planes (left to right). (e)–(h) Reconstructed phases using the FB algorithm, with increasing distance between planes (left to right).

Fig. 4
Fig. 4

(a)–(d) Effects of numbers of intensities N and iterations Q on retrieved phases (obtained using the FB algorithm). Down a column, N increases. From left to right, Q increases.

Fig. 5
Fig. 5

(a)–(e) Reconstructed phases for decreasing transverse position errors ( δ x ) (left to right). (f)–(j) Reconstructed phases for decreasing axial position errors ( δ z ) (left to right).

Equations (1)

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U p + 1 ( k Δ x , l Δ y , z ) = I 1 ( I { [ I p ( m Δ x , n Δ y ) ] 1 / 2 exp [ i ϕ p ( m Δ x , n Δ y ) ] } exp [ ( i 2 π z / λ ) ( 1 λ 2 m 2 Δ x 2 λ 2 n 2 Δ y 2 ) 1 / 2 ] ) ,

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