Abstract

Historically, phase retrieval algorithms have relied on linear propagation between two different amplitude (intensity) measurements. While generally successful, these algorithms have many issues, including susceptibility to noise, local minima, and indeterminate initial and final conditions. Here, we show that nonlinear propagation overcomes these issues, as intensity-induced changes to the index of refraction create additional constraints on the phase. More specifically, phase-matching conditions (conservation of wave energy and momentum) induce an object-dependent resonance between the measured amplitudes and the unknown phase. The result is a non-classical convergence profile in the reconstruction algorithm that contains a zero crossing, where the observable minimum in amplitude error and the unobservable minimum in phase error align at the same iteration number. We demonstrate this convergence experimentally in a photorefractive crystal, showing that there is a clear rule for stopping iterations. We find that the optimum phase retrieval occurs for a nonlinear strength that gives minimal correlation between the linear and nonlinear output amplitudes, i.e. a condition that maximizes the information diversity between linear and nonlinear propagation. The corresponding algorithm greatly improves the conventional Gerchberg-Saxton result and holds much potential for enhancing other methods of diffractive imaging.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (1)

2014 (1)

J. Yang, V. Makhija, V. Kumarappan, and M. Centurion, “Reconstruction of three-dimensional molecular structure from diffraction of laser-aligned molecules,” Struct. Dyn. 1(4), 044101 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (1)

L. Waller, D. V. Dylov, and J. W. Fleischer, “Nonlinear restoration of diffused images via seeded instability,” IEEE J. Quantum Electron. 18(2), 916–925 (2012).
[Crossref]

2011 (2)

2010 (1)

D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010).
[Crossref]

2009 (2)

C. Barsi and J. W. Fleischer, “Digital reconstruction of optically-induced potentials,” Opt. Express 17(25), 23338–23343 (2009).
[Crossref] [PubMed]

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

2008 (2)

V. L. Lo and R. P. Millane, “Reconstruction of compact binary images from limited Fourier amplitude data,” J. Opt. Soc. Am. A 25(10), 2600–2607 (2008).
[Crossref] [PubMed]

J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. 150, 87–184 (2008).
[Crossref]

2007 (1)

2006 (1)

2005 (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

2004 (2)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[Crossref] [PubMed]

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

2003 (1)

2000 (2)

R. Hillenbrand and F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85(14), 3029–3032 (2000).
[Crossref] [PubMed]

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(1-6), 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(6742), 342–344 (1999).
[Crossref]

1998 (2)

1997 (1)

1996 (1)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: Snake instability and creation of optical vortices,” Phys. Rev. Lett. 76(13), 2262–2265 (1996).
[Crossref] [PubMed]

1993 (2)

J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32(10), 1747–1767 (1993).
[Crossref] [PubMed]

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29(2), 571–579 (1993).
[Crossref]

1987 (1)

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59(8), 880–883 (1987).
[Crossref] [PubMed]

1982 (2)

1981 (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981).
[Crossref]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22(1), 949–960 (1979).
[Crossref]

1972 (1)

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

1896 (1)

K. Pearson, “Mathematical contributions to the theory of evolution. III. Regression, heredity, and panmixia,” Philos. Trans. Royal Soc. London Ser. A 187(0), 253–318 (1896).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59(8), 880–883 (1987).
[Crossref] [PubMed]

Barsi, C.

Barty, A.

Beetz, T.

Centurion, M.

J. Yang, V. Makhija, V. Kumarappan, and M. Centurion, “Reconstruction of three-dimensional molecular structure from diffraction of laser-aligned molecules,” Struct. Dyn. 1(4), 044101 (2014).
[Crossref] [PubMed]

Chapman, H. N.

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(6742), 342–344 (1999).
[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(1-6), 9–13 (2000).
[Crossref]

Crimmins, T. R.

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(1-6), 9–13 (2000).
[Crossref]

Cui, C.

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(1-6), 9–13 (2000).
[Crossref]

Dylov, D. V.

L. Waller, D. V. Dylov, and J. W. Fleischer, “Nonlinear restoration of diffused images via seeded instability,” IEEE J. Quantum Electron. 18(2), 916–925 (2012).
[Crossref]

D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010).
[Crossref]

Elser, V.

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(2), 023903 (2004).
[Crossref] [PubMed]

Fienup, J. R.

Fleischer, J. W.

C. Barsi and J. W. Fleischer, “Model of anisotropic nonlinearity in self-defocusing photorefractive media,” Opt. Express 23(19), 24426–24432 (2015).
[Crossref] [PubMed]

C.-H. Lu, C. Barsi, M. O. Williams, J. N. Kutz, and J. W. Fleischer, “Phase retrieval using nonlinear diversity,” Appl. Opt. 52(10), D92–D96 (2013).
[Crossref] [PubMed]

C. Barsi and J. W. Fleischer, “Nonlinear Abbe theory,” Nat. Photonics 7(8), 639–643 (2013).
[Crossref]

L. Waller, D. V. Dylov, and J. W. Fleischer, “Nonlinear restoration of diffused images via seeded instability,” IEEE J. Quantum Electron. 18(2), 916–925 (2012).
[Crossref]

D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010).
[Crossref]

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

C. Barsi and J. W. Fleischer, “Digital reconstruction of optically-induced potentials,” Opt. Express 17(25), 23338–23343 (2009).
[Crossref] [PubMed]

S. Jia, W. Wan, and J. W. Fleischer, “Forward four-wave mixing with defocusing nonlinearity,” Opt. Lett. 32(12), 1668–1670 (2007).
[Crossref] [PubMed]

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 (Stuttg.) 35, 237–246 (1972).

Gough, P.

Hau-Riege, S. P.

He, H.

Herrera-Fernandez, J. M.

Hillenbrand, R.

R. Hillenbrand and F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85(14), 3029–3032 (2000).
[Crossref] [PubMed]

Holsztynski, W.

Howells, M. R.

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]

Jacobsen, C.

Jia, S.

Kane, D. J.

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29(2), 571–579 (1993).
[Crossref]

Keilmann, F.

R. Hillenbrand and F. Keilmann, “Complex optical constants on a subwavelength scale,” Phys. Rev. Lett. 85(14), 3029–3032 (2000).
[Crossref] [PubMed]

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(6742), 342–344 (1999).
[Crossref]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22(1), 949–960 (1979).
[Crossref]

Kumarappan, V.

J. Yang, V. Makhija, V. Kumarappan, and M. Centurion, “Reconstruction of three-dimensional molecular structure from diffraction of laser-aligned molecules,” Struct. Dyn. 1(4), 044101 (2014).
[Crossref] [PubMed]

Kutz, J. N.

Lim, J. S.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981).
[Crossref]

Lo, V. L.

Lu, C.-H.

Luke, D. R.

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

Makhija, V.

J. Yang, V. Makhija, V. Kumarappan, and M. Centurion, “Reconstruction of three-dimensional molecular structure from diffraction of laser-aligned molecules,” Struct. Dyn. 1(4), 044101 (2014).
[Crossref] [PubMed]

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: Snake instability and creation of optical vortices,” Phys. Rev. Lett. 76(13), 2262–2265 (1996).
[Crossref] [PubMed]

Marchesini, S.

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22(1), 949–960 (1979).
[Crossref]

Marron, J. C.

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]

Miao, 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(6742), 342–344 (1999).
[Crossref]

Millane, R. P.

Noy, A.

Odoulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22(1), 949–960 (1979).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69(5), 529–541 (1981).
[Crossref]

Overman, T. L.

Pearson, K.

K. Pearson, “Mathematical contributions to the theory of evolution. III. Regression, heredity, and panmixia,” Philos. Trans. Royal Soc. London Ser. A 187(0), 253–318 (1896).
[Crossref]

Rodenburg, J. M.

J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. 150, 87–184 (2008).
[Crossref]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[Crossref] [PubMed]

Rosen, R.

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: Snake instability and creation of optical vortices,” Phys. Rev. Lett. 76(13), 2262–2265 (1996).
[Crossref] [PubMed]

Sanchez-Brea, L. M.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 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(6742), 342–344 (1999).
[Crossref]

Schulz, T. J.

Seldin, J. H.

Serabyn, E.

Shapiro, D.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22(1), 949–960 (1979).
[Crossref]

Spence, J. C. H.

Trebino, R.

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29(2), 571–579 (1993).
[Crossref]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22(1), 949–960 (1979).
[Crossref]

Wallace, J. K.

Waller, L.

L. Waller, D. V. Dylov, and J. W. Fleischer, “Nonlinear restoration of diffused images via seeded instability,” IEEE J. Quantum Electron. 18(2), 916–925 (2012).
[Crossref]

Walmsley, I. A.

Wan, W.

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

S. Jia, W. Wan, and J. W. Fleischer, “Forward four-wave mixing with defocusing nonlinearity,” Opt. Lett. 32(12), 1668–1670 (2007).
[Crossref] [PubMed]

Weierstall, U.

Wild, W. J.

Williams, M. O.

Wong, V.

Yang, J.

J. Yang, V. Makhija, V. Kumarappan, and M. Centurion, “Reconstruction of three-dimensional molecular structure from diffraction of laser-aligned molecules,” Struct. Dyn. 1(4), 044101 (2014).
[Crossref] [PubMed]

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: Snake instability and creation of optical vortices,” Phys. Rev. Lett. 76(13), 2262–2265 (1996).
[Crossref] [PubMed]

Adv. Imaging Electron Phys. (1)

J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. 150, 87–184 (2008).
[Crossref]

Appl. Opt. (5)

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22(1), 949–960 (1979).
[Crossref]

IEEE J. Quantum Electron. (2)

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29(2), 571–579 (1993).
[Crossref]

L. Waller, D. V. Dylov, and J. W. Fleischer, “Nonlinear restoration of diffused images via seeded instability,” IEEE J. Quantum Electron. 18(2), 916–925 (2012).
[Crossref]

Inverse Probl. (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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

Nat. Photonics (3)

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010).
[Crossref]

C. Barsi and J. W. Fleischer, “Nonlinear Abbe theory,” Nat. Photonics 7(8), 639–643 (2013).
[Crossref]

Nature (1)

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Figures (8)

Fig. 1
Fig. 1 Pseudocode for phase retrieval with nonlinear propagation.
Fig. 2
Fig. 2 Numerical demonstration of nonlinear phase retrieval. (a) Input amplitude image (golf ball). (b) Input phase image (Princeton Tiger). (c) Amplitude of simulated linear output (d) Amplitude of simulated nonlinear output with ∆n/n0 = −1.1 * 10−5. (e,f) Reconstructed phase image with the GS algorithm for (e) linear and (f) nonlinear propagation.
Fig. 3
Fig. 3 Convergence characteristics of phase retrieval. (a-d) Convergence of the phase retrieval algorithm for (a,c) linear and (b,d) nonlinear propagation. (a,b) are in a linear scale and the red dashed lines represent zeros, while (c,d) show the absolute value of the convergence in a logarithmic scale. (e,f) Reconstructed phase error of the GS algorithm for (e) linear and (f) nonlinear propagation. (g) Phase reconstruction error vs. noise for linear (blue) and nonlinear (red) propagation.
Fig. 4
Fig. 4 Convergence (a,b) and reconstructed phase error (c,d) of the nonlinear algorithm with self-focusing nonlinearity of (a,c) ∆n/n0 = 2.5*10−6 and (b,d) 3.5*10−6.
Fig. 5
Fig. 5 Experimental setup for nonlinear phase retrieval.
Fig. 6
Fig. 6 Reconstruction of a Princeton Tiger phase image from experimental measurements. (a) Amplitude of measured linear output (b) Amplitude of measured nonlinear output of −350V/cm (c,d) Reconstructed phase image for (c) linear and (d) nonlinear propagation.
Fig. 7
Fig. 7 Experimental convergence characteristics. (a-d) show the experimental convergence for (a,c) linear and (b,d) nonlinear propagation. (a,b) are in a linear scale and the red dashed lines represent zeros, while (c,d) show the absolute value of the convergence in a logarithmic scale. (e,f) Corresponding reconstructed phase error using (e) linear and (f) nonlinear propagation.
Fig. 8
Fig. 8 Numerical simulations of phase retrieval as a function of nonlinear coupling strength. (a) Convergence and (b) Reconstructed phase error of the nonlinear algorithm for r =0.7 (blue), 0 (red), and −0.2 (green) (c) Relation between reconstructed phase error and correlation r.

Equations (5)

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ψ z =[ i 1 2k 2 +iΔn( | ψ | 2 ) ]ψ[D+N( | ψ | 2 )]ψ
ψ( z+dz )= e dz 2 .D e dz.N e dz 2 .D ψ( z )
ψ( z i )= e dz 2 .D e dz.N e dz 2 .D  ψ( z f = z i +dz )
ER= rS || g( r ) ||f(r)|| rS |f(r)|
r= ij ( A ij A ¯ )( B ij B ¯ ) ij ( A ij A ¯ ) 2 ij ( B ij B ¯ ) 2 .

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