Abstract

We develop a deterministic algorithm for coherent diffractive imaging (CDI) that employs a modified Fourier transform of a Fraunhofer diffraction pattern to quantitatively reconstruct the complex scalar wavefield at the exit surface of a sample of interest. The sample is placed in a uniformly-illuminated rectangular hole with dimensions at least two times larger than the sample. For this particular scenario, and in the far-field diffraction case, our non-iterative reconstruction algorithm is rapid, exact and gives a unique analytical solution to the inverse problem. The efficacy and stability of the algorithm, which may achieve resolutions in the nanoscale range, is demonstrated using simulated X-ray data.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  21. S. G. Podorov, G. Hölzer, E. Förster, and N. N. Faleev, "Fourier analysis of X-ray rocking curves from superlattices," Phys. Stat. Sol. B 213, 317-324 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  26. M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University Press, Cambridge, 1999).
  27. Q. Shen, I. Bazarov, and P. Thibault, "Diffractive imaging of nonperiodic materials with future coherent sources,". J. Synchr. Rad. 11, 432-438 (2004).
    [CrossRef]
  28. D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
    [CrossRef] [PubMed]
  29. J. T. Winthrop and C. R. Worthington, "X-ray microscopy by successive Fourier transformation," Phys. Lett. 15, 124-126 (1965).
    [CrossRef]
  30. S. Mallick and M. L. Roblin, "Fourier transform holography using a quasimonochromatic incoherent source," Applied Optics 10, 596-598 (1971).
    [CrossRef] [PubMed]
  31. W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, "Fourier-transform holographic microscope," Appl. Opt. 31, 4973-4978 (1992).
    [CrossRef] [PubMed]
  32. G. W. Stroke, "Lensless Fourier-transform method for optical holography," Appl. Phys. Lett. 6(10), 201-203 (1965).
    [CrossRef]
  33. G. W. Stroke and D. G. Falconer, "Attainment of high resolution in wavefront-reconstruction imaging," Phys. Lett. 13, 306-309 (1964).
    [CrossRef]
  34. G. W. Stroke, An introduction to coherent optics and holography, (Academic Press, New York, London, 1966).
  35. J. W. Goodman, Introduction to Fourier optics, 3rd edition (Roberts & Company, Englewood, Colorado, 2005).

2006 (4)

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, "Neutron phase imaging and tomography," Phys. Rev. Lett. 96, 215505 (2006).
[CrossRef] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature Physics 2, 258-261 (2006).
[CrossRef]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, "Fresnel coherent diffractive imaging," Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düesterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N.C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nature Physics 2, 839-843 (2006).
[CrossRef]

2004 (1)

Q. Shen, I. Bazarov, and P. Thibault, "Diffractive imaging of nonperiodic materials with future coherent sources,". J. Synchr. Rad. 11, 432-438 (2004).
[CrossRef]

2003 (2)

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

V. Elser, "Phase retrieval by iterated projections," J. Opt. Soc. Am. A 20, 40-55 (2003).
[CrossRef]

2002 (2)

J. Miao, T. Ohsuna, O. Tereasaki, K. O. Hodgson, and M. A. O'Keefe, "Atomic resolution three-dimensional electron diffraction microscopy," Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

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

2001 (1)

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

1999 (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 specimens," Nature 400, 342-344 (1999).
[CrossRef]

S. G. Podorov, G. Hölzer, E. Förster, and N. N. Faleev, "Fourier analysis of X-ray rocking curves from superlattices," Phys. Stat. Sol. B 213, 317-324 (1999).
[CrossRef]

1998 (2)

S. G. Podorov, G. Hölzer, E. Förster, and N. N. Faleev, "Semidynamical solution of the inverse problem of X-ray Bragg diffraction on multilayered crystals," Phys. Stat. Sol. A 169, 9-16 (1998).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, "Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects," J. Opt. Soc. Am. A 15, 1662-1669 (1998).
[CrossRef]

1992 (1)

1988 (1)

1982 (2)

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
[CrossRef] [PubMed]

R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory," Optik 61(3), 247-262 (1982).

1972 (1)

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 (1972).

1971 (1)

S. Mallick and M. L. Roblin, "Fourier transform holography using a quasimonochromatic incoherent source," Applied Optics 10, 596-598 (1971).
[CrossRef] [PubMed]

1965 (2)

G. W. Stroke, "Lensless Fourier-transform method for optical holography," Appl. Phys. Lett. 6(10), 201-203 (1965).
[CrossRef]

J. T. Winthrop and C. R. Worthington, "X-ray microscopy by successive Fourier transformation," Phys. Lett. 15, 124-126 (1965).
[CrossRef]

1964 (1)

G. W. Stroke and D. G. Falconer, "Attainment of high resolution in wavefront-reconstruction imaging," Phys. Lett. 13, 306-309 (1964).
[CrossRef]

1952 (1)

D. Sayre, "Some implications of a theorem due to Shannon,"Acta Cryst. 5, 843-843 (1952).
[CrossRef]

1948 (1)

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

1931 (1)

P. A. M. Dirac, "Quantised singularities in the electromagnetic field," Proc. R. Soc. A 133, 60-72 (1931).
[CrossRef]

Acta Cryst. (1)

D. Sayre, "Some implications of a theorem due to Shannon,"Acta Cryst. 5, 843-843 (1952).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

G. W. Stroke, "Lensless Fourier-transform method for optical holography," Appl. Phys. Lett. 6(10), 201-203 (1965).
[CrossRef]

Applied Optics (1)

S. Mallick and M. L. Roblin, "Fourier transform holography using a quasimonochromatic incoherent source," Applied Optics 10, 596-598 (1971).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

J. Synchr. Rad. (1)

Q. Shen, I. Bazarov, and P. Thibault, "Diffractive imaging of nonperiodic materials with future coherent sources,". J. Synchr. Rad. 11, 432-438 (2004).
[CrossRef]

Nature (2)

D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948).
[CrossRef] [PubMed]

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]

Nature Physics (2)

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düesterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N.C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nature Physics 2, 839-843 (2006).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature Physics 2, 258-261 (2006).
[CrossRef]

Opt. Lett. (1)

Optik (2)

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 (1972).

R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory," Optik 61(3), 247-262 (1982).

Phys. Lett. (2)

J. T. Winthrop and C. R. Worthington, "X-ray microscopy by successive Fourier transformation," Phys. Lett. 15, 124-126 (1965).
[CrossRef]

G. W. Stroke and D. G. Falconer, "Attainment of high resolution in wavefront-reconstruction imaging," Phys. Lett. 13, 306-309 (1964).
[CrossRef]

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, 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. (5)

J. Miao, T. Ohsuna, O. Tereasaki, K. O. Hodgson, and M. A. O'Keefe, "Atomic resolution three-dimensional electron diffraction microscopy," Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

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

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, "Neutron phase imaging and tomography," Phys. Rev. Lett. 96, 215505 (2006).
[CrossRef] [PubMed]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, "Fresnel coherent diffractive imaging," Phys. Rev. Lett. 97, 025506 (2006).
[CrossRef] [PubMed]

Phys. Stat. Sol. A (1)

S. G. Podorov, G. Hölzer, E. Förster, and N. N. Faleev, "Semidynamical solution of the inverse problem of X-ray Bragg diffraction on multilayered crystals," Phys. Stat. Sol. A 169, 9-16 (1998).
[CrossRef]

Phys. Stat. Sol. B (1)

S. G. Podorov, G. Hölzer, E. Förster, and N. N. Faleev, "Fourier analysis of X-ray rocking curves from superlattices," Phys. Stat. Sol. B 213, 317-324 (1999).
[CrossRef]

Proc. R. Soc. A (1)

P. A. M. Dirac, "Quantised singularities in the electromagnetic field," Proc. R. Soc. A 133, 60-72 (1931).
[CrossRef]

Other (9)

M. V. Berry, "Singularities in waves and rays," in R. Balian et al. (eds), Les Houches Lecture Series, session XXXV, Physics of defects (North Holland, Amsterdam, 1981) pp. 453-543

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, New York, 2006).
[CrossRef]

G. A. Korn and T. M. Korn, Mathematical handbook for scientists and engineers, 2nd ed (McGraw-Hill Book Company, 1968).

J. W. Goodman, Statistical optics (John Wiley & Sons, Inc., 2000).

G. T. Herman, Image reconstruction from projections (Academic Press, 1980).

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University Press, Cambridge, 1999).

G. W. Stroke, An introduction to coherent optics and holography, (Academic Press, New York, London, 1966).

J. W. Goodman, Introduction to Fourier optics, 3rd edition (Roberts & Company, Englewood, Colorado, 2005).

W. Pauli, "Die allgemeinen Prinzipien der Wellenmechanik" in Handbuch der Physik, ed. H. Geiger and K. Scheel (Springer, Berlin, 1933) 24(1), 83-272.

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Figures (3)

Fig. 1.
Fig. 1.

Setup for non-iterative Coherent Diffractive Imaging.

Fig. 2.
Fig. 2.

Images corresponding to 45 dB SNR for the brightest pixel (Nmax =109): 2a (noisy Fraunhofer pattern), 2b - reconstructed function a(x, y) with d=0.255 and r=0.066, 2c - reconstructed function φ(x, y) with d=0.189 and r=0.050.

Fig. 3.
Fig. 3.

Images corresponding to 50 dB SNR for the brightest pixel (Nmax =1010): 3a (noisy Fraunhofer pattern), 3b - reconstructed function a(x, y) with d=0.063 and r=0.016, 3c - reconstructed function φ(x, y) with d=0.044 and r=0.012.

Equations (25)

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

ψ ( x , y ) = ( ψ o ( x , y ) + ψ s ( x , y ) ) I 0 ,
I D ( q x = kx d z , q y = ky d z ) = k 2 z 2 I 0 ψ ˜ ( q x , q y ) 2 = k 2 z 2 I 0 { ψ ˜ o ( q x , q y ) 2
1 π q x q y Re { ψ ˜ o ( q x , q y ) × [ exp ( iq x ( x 0 + Δ x ) ) exp ( iq x ( x 0 Δ x ) ) ] × [ exp ( iq y ( y 0 + Δ y ) ) exp ( iq y ( y 0 Δ y ) ) ] }
+ ( 2 sin ( q x Δ x ) sin ( q y Δ y ) ) 2 ( π q x q y ) 2 } ,
U ( x , y ) = z 2 2 π k 2 I 0 q x q y I ̂ D ( q x , q y ) exp ( iq x x + iq y y ) dq x dq y .
2 π U ( x , y ) = xy 2 dx dy ψ o ( x , y ) ψ o * ( x x , y y )
+ ψ o ( x + x 0 + Δ x , y + y 0 + Δ y ) + ψ o ( x + x 0 Δ x , y + y 0 Δ y )
ψ o ( x + x 0 + Δ x , y + y 0 Δ y ) ψ o ( x + x 0 Δ x , y + y 0 + Δ y )
+ ψ o * ( x + x 0 + Δ x , y + y 0 + Δ y ) + ψ o * ( x + x 0 Δ x , y + y 0 Δ y )
ψ o * ( x + x 0 + Δ x , y + y 0 Δ y ) ψ o * ( x + x 0 Δ x , y + y 0 + Δ y )
+ sgn ( x ) [ θ ( x + 2 Δ x ) θ ( x 2 Δ x ) ] sgn ( y ) [ θ ( y + 2 Δ y ) θ ( y 2 Δ y ) ]
ψ ( x , y , z ) = ik 2 π z exp ( ikz ) I 0 { dx dy ψ o ( x , y , 0 ) exp ( i k 2 z ( x x ) 2 ) exp ( i k 2 z ( y y ) 2 )
+ x 0 Δ x x 0 + Δ x dx y 0 Δ y y 0 + Δ y dy exp ( i k 2 z ( x x ) 2 ) exp ( i k 2 z ( y y ) 2 ) }
U ( x , y ) = z 2 2 π k 2 I 0 q x q y I ̂ D ( q x , q y ) exp ( iq x x + iq y y ) dq x dq y .
= A + B 1 , 2 , 3 , 4 + C 1 , 2 , 3 , 4 + D
A = x y 1 2 π dq x dq y ψ ˜ o ( q x , q y ) 2 exp ( iq x x + iq y y )
= x y 1 2 π dx dy ψ o ( x , y ) ψ o * ( x x , y y )
D = x y 1 2 π 4 π 2 dq x dq y ( sin ( q x Δ x ) sin ( q y Δ y ) q x q y ) 2 exp ( iq x x + iq y y ) = q x = 2 π f x ; q y = 2 π f y
= x y 4 π 2 2 π 4 π 2 ( Δ x Δ y ) 2 df x df y ( sin ( π f x 2 Δ x ) sin ( π f y 2 Δ y ) 4 π 2 f x f y ) 2 exp ( i 2 π f x x + i 2 πf y y )
= x y 4 π 2 2 π 4 π 2 ( Δ x Δ y ) 2 1 2 Δ x 1 2 Δ y Λ ( x 2 Δ x ) Λ ( x 2 Δ y )
= 1 2 π sgn ( x ) [ θ ( x + 2 Δ x ) θ ( x 2 Δ x ) ] sgn ( y ) [ θ ( y + 2 Δ y ) θ ( y 2 Δ y ) ] .
B 1 = 1 4 π 2 dq x dq y ψ ˜ o ( q x , q y ) exp ( iq x ( x + x 0 + Δ x ) + iq y ( y + y 0 + Δ y ) )
= 1 2 π ψ o ( x + x 0 + Δ x , y + y 0 + Δ y )
C 1 = 1 4 π 2 dq x dq y ψ ˜ o * ( q x , q y ) exp ( iq x ( x x 0 Δ x ) + iq y ( y y 0 Δ y ) ) .
= 1 2 π ψ o * ( x + x 0 + Δ x , y + y 0 + Δ y )

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