Abstract

Recently we have proposed a noniterative and analytic phase retrieval method using the filter of an aperture array to reconstruct a complex-valued object from a diffraction intensity pattern [Phys. Rev. Lett. 98, 223901 (2007) ], but this method suffers from the restriction of the far-field condition of two distances between the object and the filter and between the filter and the detector for the intensity measurement. An improved method, which extends the adaptable condition of those distances to the region of Fresnel diffraction, is proposed here. In addition, the procedure for reducing the influence of noises on the phase retrieval is presented, in which the phase information contained in multiple groups of sampling data of a single intensity distribution is utilized. The usefulness of this method is shown in computer-simulated examples of the object reconstructions, including an object with phase vortices.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
    [CrossRef]
  29. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65-75 (2001).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2007 (3)

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

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]

N. Nakajima, “Noniterative phase retrieval from a single diffraction intensity pattern by use of an aperture array,” Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

2006 (2)

2005 (2)

N. Nakajima, “Phase retrieval from diffraction intensities by use of a scanning slit aperture,” Appl. Opt. 44, 6228-6234 (2005).
[CrossRef] [PubMed]

L. D. Turner, K. F. E. M. Domen, and R. E. Scholten, “Diffraction-contrast imaging of cold atoms,” Phys. Rev. A 72, 031403(R) (2005).
[CrossRef]

2004 (5)

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, “Femtosecond and subfemtosecond x-ray pulses from a self-amplified spontaneous-emission-based free-electron laser,” Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

N. Nakajima, “Lensless imaging from diffraction intensity measurements by use of a noniterative phase-retrieval method,” Appl. Opt. 43, 1710-1718 (2004).
[CrossRef] [PubMed]

W. McBride, N. L. O'Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

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] [PubMed]

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

2003 (2)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

2002 (1)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

2001 (3)

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27-32 (2001).
[CrossRef]

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65-75 (2001).
[CrossRef]

2000 (1)

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

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 speciments,” Nature 400, 342-344 (1999).
[CrossRef]

1996 (2)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

N. Nakajima and B. E. A. Saleh, “Reconstruction of a vibrating object from its time-averaged image intensities by the use of exponential filtering,” Appl. Opt. 35, 4581-4588 (1996).
[CrossRef] [PubMed]

1995 (3)

P. J. Bones, C. R. Parker, B. L. Satherley, and R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 9, 1842-1857 (1995).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[CrossRef]

A. Singirev, I. Snigireva, V. Kohn, S. Kuznestsov, and I. Schelokov, “On the possibility of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

1990 (1)

1988 (1)

1987 (2)

1983 (1)

1982 (1)

1980 (1)

1976 (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, “The phase problem,” Proc. R. Soc. London, Ser. A 350, 191-212 (1976).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[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 (Stuttgart) 35, 237-246 (1972).

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (7)

Nature (2)

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

S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432, 885-888 (2004).
[CrossRef] [PubMed]

Opt. Commun. (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65-75 (2001).
[CrossRef]

Optik (Stuttgart) (1)

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

Phys. Rev. A (1)

L. D. Turner, K. F. E. M. Domen, and R. E. Scholten, “Diffraction-contrast imaging of cold atoms,” Phys. Rev. A 72, 031403(R) (2005).
[CrossRef]

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstrall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Phys. Rev. Lett. (9)

P. Emma, K. Bane, M. Cornacchia, Z. Huang, H. Schlarb, G. Stupakov, and D. Walz, “Femtosecond and subfemtosecond x-ray pulses from a self-amplified spontaneous-emission-based free-electron laser,” Phys. Rev. Lett. 92, 074801 (2004).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, S. Nayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy x rays,” Phys. Rev. Lett. 86, 5827-5830 (2001).
[CrossRef] [PubMed]

N. Nakajima, “Noniterative phase retrieval from a single diffraction intensity pattern by use of an aperture array,” Phys. Rev. Lett. 98, 223901 (2007).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

W. McBride, N. L. O'Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

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] [PubMed]

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]

Phys. Today (1)

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27-32 (2001).
[CrossRef]

Proc. R. Soc. London, Ser. A (2)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, “The phase problem,” Proc. R. Soc. London, Ser. A 350, 191-212 (1976).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Singirev, I. Snigireva, V. Kohn, S. Kuznestsov, and I. Schelokov, “On the possibility of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486-5492 (1995).
[CrossRef]

Science (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1419-1421 (2003).
[CrossRef] [PubMed]

Ultramicroscopy (1)

S. Bajt, A. Barty, K. A. Nugent, M. MaCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67-73 (2000).
[CrossRef] [PubMed]

Other (1)

N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P.W.Hawkes, ed. (Academic, 1995), Vol. 93, pp. 109-171.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the measurement system for object reconstruction by the phase-retrieval method. The object function is reconstructed from a single intensity distribution of a diffracted wave through an array of square apertures under the Fresnel approximation.

Fig. 2
Fig. 2

Reconstruction of a complex-valued object function from a single intensity distribution by the phase-retrieval method: (a) modulus and (b) phase of the original object (where the values of the phase are in the range of 2.58 to 2.02 rad ), and (c) modulus in the detector plane of Fig. 1. (d) and (e) [or (f) and (g)] are the modulus and phase, respectively, of a reconstructed object from the noiseless (or noisy) modulus in (c). The only picture of (c) is represented on a gray scale saturated with 20% of the maximum value of the modulus for display.

Fig. 3
Fig. 3

Reconstruction of an object function with vortices from a single intensity distribution by the phase-retrieval method: (a) modulus and (b) phase of the original object having three first-order vortices of Laguerre–Gaussian mode, and (c) modulus in the detector plane of Fig. 1. (d) and (e) [or (f) and (g)] are the modulus and phase, respectively, of a reconstructed object from the noiseless (or noisy) modulus in (c). The only picture of (c) is represented on the same gray scale as in Fig. 2c.

Equations (29)

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

K ( ξ , η ) = n = N 2 N 2 1 m = M 2 M 2 1 F ( x , y ) exp [ i π λ z ( x 2 + y 2 ) ] R ( x x n , y y m ) exp { i π λ l [ ( x ξ ) 2 + ( y η ) 2 ] } d x d y ,
F ( x , y ) = σ f ( u , v ) exp [ i π λ z ( u 2 + v 2 ) ] exp [ i 2 π λ z ( x u + y v ) ] d u d v ,
K ( ξ n , η m ) = n = N 2 N 2 1 m = M 2 M 2 1 q F ( x , y ) R ( x x n , y y m ) exp { i π λ l ( 1 + l z ) [ ( x x n ) 2 + ( y y m ) 2 ] } d x d y ,
K ( ξ n , η m ) 2 F ( x , y ) R ( x x n , y y m ) d x d y 2 ,
R ( x , y ) = R ( x , y ) exp [ i π λ l ( 1 + l z ) ( x 2 + y 2 ) ] .
K ( ξ n ± τ , η m ) 2 F ( x , y ) R ( x x n , y y m ) exp ( i 2 π λ l x τ ) d x d y 2 ,
K ( ξ n , η m ± τ ) 2 F ( x , y ) R ( x x n , y y m ) exp ( i 2 π λ l y τ ) d x d y 2 .
N F = ( w 2 ) 2 λ l ( 1 + l z ) .
R ( x , y ) exp [ i 2 π λ z ( u x + v y ) ] d x d y w 2 exp [ π 2 6 ( a w λ z ) 2 ( u 2 + v 2 ) ] exp [ i b ( u 2 + v 2 ) ( λ z ) 2 ] ,
K ( ξ n , η m ) 2 F ( x , y ) G ( x x n , y y m ) d x d y 2 ,
K ( ξ n ± τ , η m ) 2 p F ( x , y ) G [ x ( x n ± s i c ) , y y m ] d x d y 2 ,
K ( ξ n , η m ± τ ) 2 p F ( x , y ) G [ x x n , y ( y m ± s i c ) ] d x d y 2 .
F ( x , y ) G ( x x n , y y m ) d x d y = M ( x n , y m ) exp [ i ϕ ( x n , y m ) ] .
F ( x , y ) G [ x ( x n i c ) , y y m ] d x d y = M ( x n i c , y m ) exp { i [ ϕ R ( x n i c , y m ) + i ϕ I ( x n i c , y m ) ] } ,
K [ ( x n s ) ( 1 + l z ) + τ , y m ( 1 + l z ) ] 2 p M ( x n i c , y m ) 2 exp [ 2 ϕ I ( x n i c , y m ) ] ,
K [ ( x n + s ) ( 1 + l z ) τ , y m ( 1 + l z ) ] 2 p M ( x n + i c , y m ) 2 exp [ 2 ϕ I ( x n + i c , y m ) ] .
D ( x n , y m ; τ ) ϕ I ( x n i c , y m ) + ϕ I ( x n + i c , y m ) ,
D ( x n , y m ; τ ) = ln [ K [ ( x n s ) ( 1 + l z ) + τ , y m ( 1 + l z ) ] K [ ( x n + s ) ( 1 + l z ) τ , y m ( 1 + l z ) ] ] .
j = 1 J D ( x n , y m ; τ j ) = ln [ j = 1 J K [ ( x n s j ) ( 1 + l z ) + τ j , y m ( 1 + l z ) ] j = 1 J K [ ( x n + s j ) ( 1 + l z ) τ j , y m ( 1 + l z ) ] ] ,
ϕ ( x n , y m ) I 1 [ Γ ( α , y m ) j = 1 J 2 i sinh ( 2 π c j α ) ] + Γ ( 0 , y m ) 2 j = 1 J c j x n .
ER = [ u , v σ f ( u , v ) f r ( u , v ) 2 u , v σ f ( u , v ) 2 ] 1 2 ,
j = 1 J D ( x n , y m ; τ j ) j = 1 J [ ϕ I ( x n i c j , y m ) + ϕ I ( x n + i c j , y m ) ] .
j = 1 J D ( x n , y m ; τ j ) 1 2 i j = 1 J { [ ϕ ( x n i c j , y m ) ] * ϕ ( x n i c j , y m ) + ϕ ( x n + i c j , y m ) [ ϕ ( x n + i c j , y m ) ] * } ,
Γ ( α , y m ) 1 2 i j = 1 J [ ϕ ( x n + i c j , y m ) ϕ ( x n i c j , y m ) + ϕ ( x n + i c j , y m ) ϕ ( x n i c j , y m ) ] exp ( i 2 π α x n ) d x n ,
ϕ ( x n ± i c j , y m ) = Φ ( α , y m ) exp [ i 2 π ( x n ± i c j ) α ] d α ,
I [ ϕ ( x n ± i c j , y m ) ] = exp ( 2 π c j α ) Φ ( α , y m ) .
Γ ( α , y m ) 1 i j = 1 J [ exp ( 2 π c j α ) exp ( 2 π c j α ) ] Φ ( α , y m ) = j = 1 J 2 i sinh ( 2 π c j α ) I [ ϕ ( x n , y m ) ] ,
ϕ ( x n , y m ) I 1 [ Γ ( α , y m ) j = 1 J 2 i sinh ( 2 π c j α ) ] ( α 0 ) ,
ϕ ( x n , y m ) I 1 [ Γ ( α , y m ) j = 1 J 2 i sinh ( 2 π c j α ) ] + Γ ( 0 , y m ) 2 j = 1 J c j x n .

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