Abstract

We extend the Gerchberg–Saxton algorithm to phase retrieval in a nonlinear system. Using a tunable photorefractive crystal, we experimentally demonstrate the noninterferometric technique by reconstructing an unknown phase object from optical intensity measurements taken at different nonlinear strengths.

© 2013 Optical Society of America

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  1. A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
    [CrossRef]
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    [CrossRef]
  3. 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).
  4. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
    [CrossRef]
  5. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
    [CrossRef]
  6. F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
    [CrossRef]
  7. H. R. Ingleby and D. R. McGaughey, “Parallel multiframe blind deconvolution using wavelength diversity,” Proc. SPIE 5562, 58–64 (2004).
    [CrossRef]
  8. B. R. Hunt, T. L. Overman, and P. Gough, “Image reconstruction from pairs of Fourier-transform magnitude,” Opt. Lett. 23, 1123–1125 (1998).
    [CrossRef]
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    [CrossRef]
  10. H. A. Hauptman, “The phase problem of X-ray crystallography,” Rep. Prog. Phys. 54, 1427–1454 (1991).
    [CrossRef]
  11. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
    [CrossRef]
  12. M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
    [CrossRef]
  13. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
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    [CrossRef]
  17. Y.-R. E. Tan, D. M. Paganin, R. P. Yu, and M. J. Morgan, “Wave-function reconstruction of complex fields obeying nonlinear parabolic equations,” Phys. Rev. E 68, 066602 (2003).
    [CrossRef]
  18. M. Puida and F. Ivanauskas, “Light beam phase retrieval in nonlinear media: a computer simulation,” Liet. Matem. Rink 45, 504 (2005).
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  23. C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3, 211–215 (2009).
    [CrossRef]
  24. A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83, 031802R (2011).
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    [CrossRef]
  27. E. Peli, “Contrast in complex images,” J. Opt. Soc. Am. A 7, 2032–2040 (1990).
    [CrossRef]
  28. G. Liu and P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
    [CrossRef]
  29. C. Barsi, and J. W. Fleischer, “Digital reconstruction of optically induced potentials,” Opt. Express 17, 23338–23343 (2009).
    [CrossRef]
  30. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
    [CrossRef]
  31. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
    [CrossRef]
  32. W. Wan, S. Jia, and J. W. Fleischer, “Dispersive, superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
    [CrossRef]
  33. C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
    [CrossRef]
  34. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [CrossRef]
  35. S. Jia, W. Wan, and J. W. Fleischer, “Forward four-wave mixing with defocusing nonlinearity,” Opt. Lett. 32, 1668–1670 (2007).
    [CrossRef]
  36. L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
    [CrossRef]

2012 (1)

M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
[CrossRef]

2011 (2)

E. Serabyn, J. K. Wallace, and D. Mawet, “Speckle-phase measurement in a tandem-vortex coronagraph,” Appl. Opt. 50, 5453–5456 (2011).
[CrossRef]

A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83, 031802R (2011).
[CrossRef]

2010 (1)

2009 (3)

C. Barsi, and J. W. Fleischer, “Digital reconstruction of optically induced potentials,” Opt. Express 17, 23338–23343 (2009).
[CrossRef]

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3, 211–215 (2009).
[CrossRef]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

2008 (1)

2007 (4)

A. V. Martin and L. J. Allen, “Measuring the phase of a Bose-Einstein condensate,” Phys. Rev. A 76, 053606 (2007).
[CrossRef]

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive, superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[CrossRef]

S. Jia, W. Wan, and J. W. Fleischer, “Forward four-wave mixing with defocusing nonlinearity,” Opt. Lett. 32, 1668–1670 (2007).
[CrossRef]

2005 (1)

M. Puida and F. Ivanauskas, “Light beam phase retrieval in nonlinear media: a computer simulation,” Liet. Matem. Rink 45, 504 (2005).

2004 (3)

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

H. R. Ingleby and D. R. McGaughey, “Parallel multiframe blind deconvolution using wavelength diversity,” Proc. SPIE 5562, 58–64 (2004).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

2003 (1)

Y.-R. E. Tan, D. M. Paganin, R. P. Yu, and M. J. Morgan, “Wave-function reconstruction of complex fields obeying nonlinear parabolic equations,” Phys. Rev. E 68, 066602 (2003).
[CrossRef]

2002 (2)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

D. Paganin and K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
[CrossRef]

2000 (1)

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

1998 (3)

1992 (1)

1991 (1)

H. A. Hauptman, “The phase problem of X-ray crystallography,” Rep. Prog. Phys. 54, 1427–1454 (1991).
[CrossRef]

1990 (1)

1987 (2)

G. Liu and P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
[CrossRef]

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef]

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

1982 (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef]

1981 (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1978 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

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).

Agrawal, G. P.

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef]

Allen, L. J.

A. V. Martin and L. J. Allen, “Measuring the phase of a Bose-Einstein condensate,” Phys. Rev. A 76, 053606 (2007).
[CrossRef]

Bahk, S.-W.

Barbastathis, G.

Bardroff, P. J.

P. J. Bardroff, U. Leonhardt, and W. P. Schleich, “Adaptive phase retrieval of nonlinear waves,” Opt. Commun. 147, 121–125 (1998).
[CrossRef]

Barsi, C.

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3, 211–215 (2009).
[CrossRef]

C. Barsi, and J. W. Fleischer, “Digital reconstruction of optically induced potentials,” Opt. Express 17, 23338–23343 (2009).
[CrossRef]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Chen, C.-C.

M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
[CrossRef]

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Conti, C.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Dahmen, U.

M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
[CrossRef]

Di Porto, P.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Ercius, P.

M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
[CrossRef]

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

Fienup, J. R.

Fischer, B.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

Fleischer, J. W.

C. Barsi, and J. W. Fleischer, “Digital reconstruction of optically induced potentials,” Opt. Express 17, 23338–23343 (2009).
[CrossRef]

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3, 211–215 (2009).
[CrossRef]

S. Jia, W. Wan, and J. W. Fleischer, “Forward four-wave mixing with defocusing nonlinearity,” Opt. Lett. 32, 1668–1670 (2007).
[CrossRef]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive, superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[CrossRef]

Fratalocchi, A.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

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).

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[CrossRef]

Gough, P.

Goy, A.

A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83, 031802R (2011).
[CrossRef]

Grier, D. G.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Hauptman, H. A.

H. A. Hauptman, “The phase problem of X-ray crystallography,” Rep. Prog. Phys. 54, 1427–1454 (1991).
[CrossRef]

Heebner, J.

Hunt, B. R.

Ingleby, H. R.

H. R. Ingleby and D. R. McGaughey, “Parallel multiframe blind deconvolution using wavelength diversity,” Proc. SPIE 5562, 58–64 (2004).
[CrossRef]

Ivanauskas, F.

M. Puida and F. Ivanauskas, “Light beam phase retrieval in nonlinear media: a computer simulation,” Liet. Matem. Rink 45, 504 (2005).

Ivanov, V. Yu.

Jia, S.

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive, superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[CrossRef]

S. Jia, W. Wan, and J. W. Fleischer, “Forward four-wave mixing with defocusing nonlinearity,” Opt. Lett. 32, 1668–1670 (2007).
[CrossRef]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

Leonhardt, U.

P. J. Bardroff, U. Leonhardt, and W. P. Schleich, “Adaptive phase retrieval of nonlinear waves,” Opt. Commun. 147, 121–125 (1998).
[CrossRef]

Lim, J. S.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Liu, G.

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

Martin, A. V.

A. V. Martin and L. J. Allen, “Measuring the phase of a Bose-Einstein condensate,” Phys. Rev. A 76, 053606 (2007).
[CrossRef]

Mawet, D.

McGaughey, D. R.

H. R. Ingleby and D. R. McGaughey, “Parallel multiframe blind deconvolution using wavelength diversity,” Proc. SPIE 5562, 58–64 (2004).
[CrossRef]

Mecklenburg, M.

M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
[CrossRef]

Miao, J.

M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Morgan, M. J.

Y.-R. E. Tan, D. M. Paganin, R. P. Yu, and M. J. Morgan, “Wave-function reconstruction of complex fields obeying nonlinear parabolic equations,” Phys. Rev. E 68, 066602 (2003).
[CrossRef]

Nugent, K. A.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Osten, W.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Overman, T. L.

Paganin, D.

Paganin, D. M.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Y.-R. E. Tan, D. M. Paganin, R. P. Yu, and M. J. Morgan, “Wave-function reconstruction of complex fields obeying nonlinear parabolic equations,” Phys. Rev. E 68, 066602 (2003).
[CrossRef]

Peccianti, M.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

Pedrini, G.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Peli, E.

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Psaltis, D.

A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83, 031802R (2011).
[CrossRef]

Puida, M.

M. Puida and F. Ivanauskas, “Light beam phase retrieval in nonlinear media: a computer simulation,” Liet. Matem. Rink 45, 504 (2005).

Regan, B. C.

M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
[CrossRef]

Rodenburg, J. M.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

Ruocco, G.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

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).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Schleich, W. P.

P. J. Bardroff, U. Leonhardt, and W. P. Schleich, “Adaptive phase retrieval of nonlinear waves,” Opt. Commun. 147, 121–125 (1998).
[CrossRef]

Scott, M. C.

M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483, 444–447 (2012).
[CrossRef]

Scott, P. D.

Serabyn, E.

Sivokon, V. P.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1978).
[CrossRef]

Tan, Y.-R. E.

Y.-R. E. Tan, D. M. Paganin, R. P. Yu, and M. J. Morgan, “Wave-function reconstruction of complex fields obeying nonlinear parabolic equations,” Phys. Rev. E 68, 066602 (2003).
[CrossRef]

Tian, L.

Trillo, S.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

Vinetskii, V. L.

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Appl. Opt. (3)

Ferroelectrics (1)

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W. Wan, S. Jia, and J. W. Fleischer, “Dispersive, superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
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http://ervinlaszlo.com/forum/2010/07/12/designing-a-multiperson-planetary-consciousness/ .

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Figures (4)

Fig. 1.
Fig. 1.

Pseudocode for nonlinear generalization of the GS algorithm.

Fig. 2.
Fig. 2.

Numerical reconstruction of a phase object. (a) Amplitude of simulated linear output. (b) and (c) Amplitude of simulated nonlinear output of (b) Δn/n0=1.38·105 and (c) 3.16·105. (d) Neural network model. (e) and (f) Reconstructed input phase using (e) the original GS algorithm, which requires the amplitudes of the input and linear output, and (f) the generalized GS algorithm, which uses the two nonlinear outputs (b) and (c).

Fig. 3.
Fig. 3.

Schematic of experimental setup. An extraordinarily polarized plane wave (532 nm) is modulated by a phase object and imaged onto a photorefractive SBN:60 crystal. Output intensity patterns are measured as a function of nonlinearity (applied voltage). The input phase is then reconstructed numerically.

Fig. 4.
Fig. 4.

Reconstruction of a phase object from experimental measurements. (a) Amplitude of measured linear output. (b) and (c) Amplitude of measured nonlinear output of (b) 310V/cm and (c) 420V/cm. (d) Input phase, measured by phase-shifting holography. (e) and (f) Reconstructed input phase using (e) the original GS algorithm, which requires the amplitudes of the input and linear output, and (f) the generalized GS algorithm, which uses the two nonlinear outputs (b) and (c). (g) Line profiles of the cross sections shown in (d)–(f).

Equations (3)

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ψz=[i12k02+iγ|ψ|2]ψ=[D+N(|ψ|2)]ψ,
ER=rSg(r)||f(r)rS|f(r)|,
iuzωu=ivux+12k0(2u2x+iμuxμ2u)+γ|u|2u=0,

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