Abstract

We propose an iterative method to optimize the phase profile of the initial field so that its intensity profile is observed periodically along the longitudinal (propagation) axis. The new method is inspired from the Gerchberg–Saxton technique, where the Fresnel transform is used, instead of the Fourier transform, for retrieving the phase profile of several light distributions (for example, 15 planes), instead of a Fourier pair of distributions. The additional challenge, with respect to the conventional Gerchberg–Saxton technique, is that the planes where constraints are applied number more than two. It turned out that when the number of periods increased, the spectrum of the obtained initial field converges toward including Montgomery’s rings (self-imaging condition).

© 2010 Optical Society of America

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References

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    [CrossRef]
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2009 (1)

2006 (2)

T. Gureyeva, Y. Nesteretsa, D. Paganina, A. Poganya, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

M. Razzak, S. Guizani, H. Hamam, and Y. Bouslimani, “Optical post-egalization based on self-imaging,” J. Mod. Opt. 53, 1675–1684 (2006).
[CrossRef]

2003 (3)

H. Hamam, “A new measure for optical performance,” Optom. Vision Sci. 80, 175–184 (2003).
[CrossRef]

T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef] [PubMed]

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

2001 (1)

M. Matczak and J. Mamczur, “Degenerate self-imaging Fourier filters,” Optoelectron. Rev. 9, 336–340 (2001).

1999 (1)

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

1998 (2)

1996 (2)

1994 (1)

1990 (1)

1988 (1)

1986 (2)

1984 (2)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

J. Honner and P. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. Gelatt, and M. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1982 (1)

1978 (1)

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44, 208–212 (1978).

1972 (1)

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

1970 (1)

J. Goodman and A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

1967 (1)

1953 (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Bates, R.

R. Bates and M. McDonnell, Image Restoration and Reconstruction (Oxford U. Press, 1986).

Bouslimani, Y.

M. Razzak, S. Guizani, H. Hamam, and Y. Bouslimani, “Optical post-egalization based on self-imaging,” J. Mod. Opt. 53, 1675–1684 (2006).
[CrossRef]

Bryngdahl, O.

Chapman, H.

Chen, D.

Cowley, J.

J. Cowley, Diffraction Physics (North-Holland, 1975).

Fienup, J.

Fienup, J. R.

Gelatt, C.

S. Kirkpatrick, C. Gelatt, and M. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

George, N.

Gerchberg, R.

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

Gianino, P.

Goodman, J.

J. Goodman and A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Greenleaf, J.

T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef] [PubMed]

Guizani, S.

M. Razzak, S. Guizani, H. Hamam, and Y. Bouslimani, “Optical post-egalization based on self-imaging,” J. Mod. Opt. 53, 1675–1684 (2006).
[CrossRef]

Gureyev, T.

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Report 179 (Cooperative Research Centre for Hardwood Fibre and Paper Science, 1999).

Gureyeva, T.

T. Gureyeva, Y. Nesteretsa, D. Paganina, A. Poganya, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Guyot, O.

O. Guyot and H. Hamam, “Logic operations based on the fractional Talbot effect,” Opt. Commun. 127, 96–106 (1996).
[CrossRef]

Hamam, H.

M. Razzak, S. Guizani, H. Hamam, and Y. Bouslimani, “Optical post-egalization based on self-imaging,” J. Mod. Opt. 53, 1675–1684 (2006).
[CrossRef]

H. Hamam, “A new measure for optical performance,” Optom. Vision Sci. 80, 175–184 (2003).
[CrossRef]

O. Guyot and H. Hamam, “Logic operations based on the fractional Talbot effect,” Opt. Commun. 127, 96–106 (1996).
[CrossRef]

H. Hamam, “Intensity based self-imaging,” Java applet (January 2010), www.umoncton.ca/genie/electrique/cours/Hamam/Optics/Fresnel/IntensitySelfImaging/IntensitySelfImaging.htm.

Henderson, C.

Honner, J.

Howells, M.

Jacobsen, C.

Kalinovsky, A.

Kirkpatrick, S.

S. Kirkpatrick, C. Gelatt, and M. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Leger, J.

Lindaas, S.

Mamczur, J.

M. Matczak and J. Mamczur, “Degenerate self-imaging Fourier filters,” Optoelectron. Rev. 9, 336–340 (2001).

Matczak, M.

M. Matczak and J. Mamczur, “Degenerate self-imaging Fourier filters,” Optoelectron. Rev. 9, 336–340 (2001).

McDonnell, M.

R. Bates and M. McDonnell, Image Restoration and Reconstruction (Oxford U. Press, 1986).

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Miao, J.

Montgomery, W. D.

Nakajima, N.

Nesteretsa, Y.

T. Gureyeva, Y. Nesteretsa, D. Paganina, A. Poganya, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Nugent, K.

Paganina, D.

T. Gureyeva, Y. Nesteretsa, D. Paganina, A. Poganya, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol.  27, pp. 1–110.
[CrossRef]

Peele, A.

Pitts, T.

T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef] [PubMed]

Poganya, A.

T. Gureyeva, Y. Nesteretsa, D. Paganina, A. Poganya, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Quiney, H.

Raven, C.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Razzak, M.

M. Razzak, S. Guizani, H. Hamam, and Y. Bouslimani, “Optical post-egalization based on self-imaging,” J. Mod. Opt. 53, 1675–1684 (2006).
[CrossRef]

Roddier, F.

Rolleston, R.

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Saxton, W.

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

Sayre, D.

Silvestri, A.

J. Goodman and A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Smirnov, A. P.

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44, 208–212 (1978).

Snigirev, A.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Snigireva, I.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Teller, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Vecchi, M.

S. Kirkpatrick, C. Gelatt, and M. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Wackerman, C. C.

Wang, Z.

Wilkins, S.

T. Gureyeva, Y. Nesteretsa, D. Paganina, A. Poganya, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Wilkins, S. W.

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Williams, G.

Wyrowski, F.

Appl. Opt. (5)

IBM J. Res. Dev. (1)

J. Goodman and A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

T. Pitts and J. Greenleaf, “Fresnel transform phase retrieval from magnitude,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1035–1045 (2003).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Mod. Opt. (1)

M. Razzak, S. Guizani, H. Hamam, and Y. Bouslimani, “Optical post-egalization based on self-imaging,” J. Mod. Opt. 53, 1675–1684 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Phys. D (1)

T. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Opt. Commun. (4)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

O. Guyot and H. Hamam, “Logic operations based on the fractional Talbot effect,” Opt. Commun. 127, 96–106 (1996).
[CrossRef]

T. Gureyeva, Y. Nesteretsa, D. Paganina, A. Poganya, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Spectrosc. (1)

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44, 208–212 (1978).

Optik (Jena) (1)

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

Optoelectron. Rev. (1)

M. Matczak and J. Mamczur, “Degenerate self-imaging Fourier filters,” Optoelectron. Rev. 9, 336–340 (2001).

Optom. Vision Sci. (1)

H. Hamam, “A new measure for optical performance,” Optom. Vision Sci. 80, 175–184 (2003).
[CrossRef]

Science (1)

S. Kirkpatrick, C. Gelatt, and M. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (6)

T. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Report 179 (Cooperative Research Centre for Hardwood Fibre and Paper Science, 1999).

H. Hamam, “Intensity based self-imaging,” Java applet (January 2010), www.umoncton.ca/genie/electrique/cours/Hamam/Optics/Fresnel/IntensitySelfImaging/IntensitySelfImaging.htm.

J. Cowley, Diffraction Physics (North-Holland, 1975).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol.  27, pp. 1–110.
[CrossRef]

R. Bates and M. McDonnell, Image Restoration and Reconstruction (Oxford U. Press, 1986).

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Figures (9)

Fig. 1
Fig. 1

Diffraction is a modulation of the spectrum of the initial field by the spectrum of the Fresnel kernel (pure phase distribution): (a) real part of the spectrum of the Fresnel kernel, (b) phase profile of the spectrum of the Fresnel kernel, and (c) intensity profile of the spectrum of the initial field (also diffracted field).

Fig. 2
Fig. 2

Any initial spectrum can be sampled to fulfill self-imaging: (a) initial spectrum and sampled version of it and (b) phase profile of the spectrum of the Fresnel kernel. Self-imaging occurs when one takes samples of the initial spectrum where the phase of the spectrum of the Fresnel kernel is 0 (also 2 π ).

Fig. 3
Fig. 3

Generalization of Fig. 2. Any initial spectrum can be sampled to fulfill self-imaging: (a) initial spectrum and sampled version of it and (b) phase profile of the spectrum of the Fresnel kernel. Self-imaging occurs when one take samples of the initial spectrum where the phase of the spectrum of the Fresnel kernel is constant, say α 1 .

Fig. 4
Fig. 4

Any diffracted field may be expressed as a sum of self-imaged components [Fig. 4 includes the two components of Figs. 2, 3]. Each component is weighted by a different phase constant: (a) initial spectrum and sampled version of it and (b) phase profile of the spectrum of the Fresnel kernel.

Fig. 5
Fig. 5

Fourier constraint based iterative algorithm for intensity based self-imaging. The spectrum is forced to be confined in Montgomery’s rings.

Fig. 6
Fig. 6

Fresnel constraint based iterative algorithm for intensity based self-imaging with only one ( K = 1 ) period d z .

Fig. 7
Fig. 7

Fresnel constraint based iterative algorithm for intensity based self-imaging with only two ( K = 2 ) periods d z . Dashed parts present possible options.

Fig. 8
Fig. 8

Diffraction fields in several planes along the z axis. Iterative algorithm applied to obtain alternately one of two images at each period d z until the plane z = 6 d z .

Fig. 9
Fig. 9

Spectrum of the modified initial field for intensity based self-imaging with (a) two alternate images and (b) one image. Frequency 0 is filtered to better visualize the spectrum.

Equations (26)

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

h ( x , z ) = FR z { h ( x , 0 ) } = exp ( i 2 π z / λ ) exp ( i π / 4 ) λ z h ( x ) * f k ( x , z ) ,
f k ( x , z ) = exp ( j π x 2 λ z ) .
F k ( u , z ) = exp ( i π / 4 ) λ z exp ( j π λ z u 2 ) .
H ( u , z ) = H ( u , 0 ) exp ( j π λ z u 2 ) H ( u , 0 ) ,
H ( u , z ) = H ( u , 0 ) exp ( j π λ z u 2 ) = exp ( j α ) H ( u , 0 ) ,
H α 0 ( u , 0 ) = H ( u , 0 ) Δ ( u , α 0 ) ,
Δ ( u , α 0 ) = δ ( u ) + k = 1 + δ ( u 2 k λ z ) + δ ( u 2 k λ z ) .
H α 1 ( u , 0 ) = H ( u , 0 ) Δ ( u , α 1 ) .
H α 1 ( u , z ) = exp ( j α 1 ) H ( u , 0 ) ,
exp ( j α 1 ) = exp ( j π λ z u 1 2 ) .
exp ( j α ) = exp ( j π λ z u 2 ) ,
u 2 = α λ z + k 2 λ z ,
α λ z + k 0 2 λ z 0.
k = k + k 0 .
u 2 + v 2 = R k 2 = ( α + 2 k 0 λ z ) + k 2 λ z .
Δ ( u , α ) = k = 0 + δ ( u + R k ) + δ ( u R k ) .
h ( x , z ) = exp ( j α ) h ( x , 0 ) .
rms = 1 S S ( | h ( x , z ) | | h ( x , 0 ) | ) 2 .
rms = 1 K S k = 1 K S ( | h ( x , k d z ) | | h ( x , 0 ) | ) 2 .
rms = 1 K M k = 1 K M ( | h ( n , m , k d z ) | | h ( n , m , 0 ) | ) 2 .
q α = p 2 π ,
h α ( x , | q z | ) = exp ( j q α ) h α ( x , 0 ) = exp ( j p 2 π ) h α ( x , 0 ) = h α ( x , 0 ) .
H α 01 ( u , 0 ) = H α 0 ( u , 0 ) + H α 1 ( u , 0 ) = H ( u , 0 ) n = 0 n = 1 Δ ( u , α n ) .
H ( u , 0 ) = n = 0 n = N H α n ( u , 0 ) ,
H ( u , z ) = n = 0 n = N exp ( j α n ) H α n ( u , 0 ) .
h ( x , z ) = n = 0 n = N exp ( j α n ) h α n ( x , 0 ) .

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