## A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging

Optics Express, Vol. 15, Issue 16, pp. 9954-9962 (2007)

http://dx.doi.org/10.1364/OE.15.009954

Acrobat PDF (3168 KB)

### 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

## 1. Introduction

1. J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrival from experimental far-field speckle data,” Opt. Lett. **13**, 619–621 (1988). [CrossRef] [PubMed]

9. 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]

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

1. J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrival from experimental far-field speckle data,” Opt. Lett. **13**, 619–621 (1988). [CrossRef] [PubMed]

3. 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]

13. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Cryst. **5**, 843–843 (1952). [CrossRef]

2. 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]

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]

9. 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]

14. 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]

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

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

**21**, 2758–2769 (1982). [CrossRef] [PubMed]

15. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A **20**, 40–55 (2003). [CrossRef]

2. 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]

1. J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrival from experimental far-field speckle data,” Opt. Lett. **13**, 619–621 (1988). [CrossRef] [PubMed]

9. 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]

14. 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]

16. P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. R. Soc. A **133**, 60–72 (1931). [CrossRef]

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

## 2. Method

*I*. The sample is assumed to lie within a square support of length and width no greater than 2Δ

_{0}_{0}, this support lying entirely within a rectangular-shaped hole with centre at (

*x*) and sizes of 2Δ

_{0}, y_{0}_{x}≥4Δ

_{0}and 2Δ

*≥4Δ*

_{y}_{0}in the horizontal and vertical directions, respectively. We emphasize that the constraining rectangular hole must be at least twice the size of the object, in both lateral directions, to avoid overlapping of the reconstructed object functions [see Eq. (4)].

*ψ*, defined within the hole, is the difference between the coherent complex scalar field at the exit surface of the object and unity, and

_{0}*ψ*is a ‘support function’ that is by definition equal to unity within the entire area of the rectangular hole, and equal to zero outside the rectangular hole. Further, (

_{S}*x, y*) denotes Cartesian coordinates in the object plane. Note that harmonic time dependence, of the coherent scalar field, is suppressed throughout.

*I*will be (see Appendix):

_{D}*U(x, y)*, which will be seen to provide a closed-form solution to the inverse problem of reconstructing the amplitude and phase of

*ψ*from

_{0}*I*, via the definition:

_{D}*ÎD(qx,qy)*is the intensity with noise. The procedure of introducing noise in the simulated intensity is described below. For real experimental data this will correspond to the registered intensity.

*U(x, y)*in the following form:

*x*) is the signum function, and

*θ(x)*is the step function [19]. Equations (3) and (4) are the main result of the article. The first term in Eq. (4) is the derivative of the object auto-correlation function, while the last term in Eq. (4) is the auto-correlation of the rectangular hole. Both of these functions are placed in the centre of the reconstructed image [see Fig.1]. The remaining terms represent eight laterally displaced reconstructions of the desired object function, half of which are complex conjugated, and at least six of which are separated from the object auto-correlation. The lateral displacements of the phase–amplitude reconstructions are determined by the position, (

*x*), and the dimensions, (2Δ

_{0}, y_{0}*,2Δ*

_{x}*), of the rectangular hole.*

_{y}*ψ*shown in Eq. (4) are unique and unambiguous, apart from three “trivial” ambiguities irresolvable when only the modulus of the wavefunction can be registered [20]: a transverse spatial shift

_{0}(x, y)*ψ*, a constant additional phase

_{0}(x+x_{1}, y+y_{1})*ψ*and a complex conjugation

_{0}(x, y)e^{iθ}*ψ**. Note also that Podorov

_{0}(-x,-y)*et al*. have already used the technique of differentiation of the Fourier transform [see Eq. (3)], in a different context, to analyse the strain distribution in thin surface layers [21

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]

22. 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]

## 3. Numerical results

*a(x, y)*and phase

*φ(x, y)*, so that

*ψ*=

_{0}(x,y)*0(x,y)exp(iφ(x,y))-1*. We use the “Lena” image as the phase function, with 0≤

*φ(x, y)*≤3 radians, and a photo of Einstein as the amplitude function, with 0≤

*a(x, y)*≤1 [see Figs 2, 3].

*I*(i.e. the simulated Fraunhofer diffraction pattern) using the following procedure. First, we calculate the noise-free intensity

_{D}(q_{x}=kx_{d}/z,qy=ky_{d}/z)*I*[see Eq. (2)] for each detector pixel position, which is determined by the appropriate coordinates (

_{D}(qx=kx_{d}/z,qx=ky_{d}/z)*x*) in the detector plane. Second, we normalise this 2D map by multiplying by a constant

_{d}, y_{d}*N*, where

_{max}/max[I_{D}]*N*is the number of photons collected in the pixel registering the highest intensity max[

_{max}*I*]. This produces a 2D map,

_{D}*N(x*, of the registered number of photons without the effects of noise. We then replace each of these numbers,

_{d},y_{d})*N(x*, by a random deviate,

_{d},y_{d})*N̂(x*, produced by a Poisson generator, for which 〈

_{d},y_{d})*N̂(x*〉. Finally, we renormalise the new 2D map,

_{d},y_{d})*N〈(x*, to the new “noisy” intensity map Î

_{d},y_{d})*, by multiplying by the factor max[*

_{D}(q_{x}=kx_{d}/z,qy=ky_{d}/z)*I*]/

_{D}*N*. Note that the signal-to-noise ratio (

_{max}*SNR*) of the resulting image will vary with position in the detector plane, with the maximum

*SNR*corresponding to the point in the diffraction pattern that has the maximum intensity

*N*. Typically, this point will be in the vicinity of the centre of the pattern.

_{max}*N*in our simulations, namely 10

_{max}^{9}and 10

^{10}photons. This corresponds to

*SNR*=10log

_{10}(

*Signal Noise*), for the brightest pixel, of 45 dB and 50 dB, respectively. The “noisy” simulated Fraunhofer diffraction patterns along with reconstructed images of the amplitude,

*a(x, y)*, and phase distribution,

*φ(x, y)*, for both levels of noise, are shown in Figs. 2(a), 2(b), 2(c) and Figs. 3(a), 3(b), 3(c), respectively. To estimate errors in the reconstructed images we employ two different metrics [24], namely a normalised root-mean-square (

*RMS*) error criterion, defined as:

*A*and

^{ideal}_{jk}*A*are ideal and reconstructed two-dimensional images, respectively, and 〈

^{rec}_{jk}*A*〉 is the mean of the original image, with all sums being taken over the integer pixel coordinates (

^{ideal}*j,k*). The reconstruction results [Figs. 2(b), 2(c) and 3(b), 3(c)] show that a high-quality reconstruction requires a sufficiently intense registered signal, which, in the context of coherent X-ray optics, will typically require a filtered synchrotron source. Note that the reconstruction quality of the amplitude function, on,

*a(x, y)*, and the phase function,

*φ(x, y)*, is comparable to one another regardless of the level of noise. For instance, the

*d*and

*r*criteria for the

*a(x, y)*images are 0.255 and 0.066 (for

*N*=10

_{max}^{9}); compare to 0.063 and 0.016 (for

*N*=10

_{max}^{10}). The reconstructed phase images,

*φ(x, y)*, have

*d*and

*r*criteria of 0.189 and 0.050 (for

*N*=10

_{max}^{9}); compare to 0.044 and 0.012 (for

*N*), respectively. This is only marginally worse than for the amplitude images,

_{max}=10^{10}*a(x, y)*, for the lower

*SNR*, and better for the higher

*SNR*.

## 4. Discussion

25. 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]

*λ/NA*. Here,

*NA*is the numerical aperture, i.e. half of the detector acceptance angle. Our simulations performed using the Fast Fourier Transform (FFT) algorithm, as well as the analytical form of the solution itself, show that there are several factors which can significantly deteriorate the quality of the reconstructed images. They are as follows: (i) the signal-to-noise ratio determined (apart from the object characteristics) by the intensity of the incident beam [see, e.g., [27

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]

^{12}–2

^{14}dynamic range of CCD detectors necessitates a large number of subsequent images to be collected and averaged to get an acceptable

*SNR*as well as using a beam stop to avoid saturation of the central part of the Fraunhofer pattern; (iii) edge smoothness and rectangularity of hole in the screen, which is important to obtain the second and third terms in Eq. (2) in the current form; (iv) the distance between the object and the detector, the detector size and number of pixels in the detector [see e.g., reference [25

25. 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]

*q*= 0 or

_{x}*q*=0 can be easily excluded from the reconstruction process, because these terms are multiplied by 0. This means that a beam stop can cover the central vertical (

_{y}*q*=0) and horizontal (

_{x}*q*=0) lines of the detector pixels without affecting the reconstruction process. Alternatively, the analysis may proceed by “stitching together” two images: (i) the Fraunhofer pattern obtained with a beam stop, and (ii) a short-exposure image of the central portion of the Fraunhofer pattern.

_{y}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]

*,Δ*

_{x}*→∞, we see that the introduction of a confining rectangular hole allows us to remove the twin-image problem [28*

_{y}28. D. Gabor, “A new microscopic principle,” Nature **161**, 777–778 (1948). [CrossRef] [PubMed]

## 5. Conclusions

## Appendix

*I*, [see Eq. (2)], we calculate the propagated wavefield using Fresnel diffraction theory as [26]:

_{D}*z*is the propagation distance. After taking the squared modulus of Eq. (A1) we arrive at Eq. (2) for the far-field intensity, after making the usual approximation that (

*x*′)

^{2}≪

*λz*2 and (

*y*′)

^{2}≪

*λz*.

*B*

_{1,2,3,4}and

*C*

_{1,2,3,4}are straight-forward results of the Fourier shift theorem [see e.g., p. 8 in [35]]. For instance,

## Acknowledgments

## References and links

1. | J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, “Phase retrival from experimental far-field speckle data,” Opt. Lett. |

2. | 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 |

3. | 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 |

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 |

5. | J. Miao, T. Ohsuna, O. Tereasaki, K. O. Hodgson, and M. A. O’Keefe, “Atomic resolution threedimensional electron diffraction microscopy,” Phys. Rev. Lett. |

6. | 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. |

7. | 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. |

8. | F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. |

9. | F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Physics |

10. | W. PauliH. Geiger and K. Scheel, “Die allgemeinen Prinzipien der Wellenmechanik” in |

11. | R. W. Gerchberg and W.O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik , |

12. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

13. | D. Sayre, “Some implications of a theorem due to Shannon,” Acta Cryst. |

14. | 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 |

15. | V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A |

16. | P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. R. Soc. A |

17. | M. V. BerryR. Balian “Singularities in waves and rays,” et al. (eds), Les Houches Lecture Series, session XXXV, |

18. | D. M. Paganin, |

19. | G. A. Korn and T. M. Korn, |

20. | R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik |

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) |

22. | 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) |

23. | J. W. Goodman, |

24. | G. T. Herman, |

25. | 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. |

26. | M. Born and E. Wolf, |

27. | Q. Shen, I. Bazarov, and P. Thibault, “Diffractive imaging of nonperiodic materials with future coherent sources,”. J. Synchr. Rad. |

28. | D. Gabor, “A new microscopic principle,” Nature |

29. | J. T. Winthrop and C. R. Worthington, “X-ray microscopy by successive Fourier transformation,” Phys. Lett. |

30. | S. Mallick and M. L. Roblin, “Fourier transform holography using a quasimonochromatic incoherent source,” Applied Optics |

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. |

32. | G. W. Stroke, “Lensless Fourier-transform method for optical holography,” Appl. Phys. Lett. |

33. | G. W. Stroke and D. G. Falconer, “Attainment of high resolution in wavefront-reconstruction imaging,” Phys. Lett. |

34. | G. W. Stroke, |

35. | J. W. Goodman, |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(100.5070) Image processing : Phase retrieval

(340.0340) X-ray optics : X-ray optics

(340.7470) X-ray optics : X-ray mirrors

**ToC Category:**

Image Processing

**History**

Original Manuscript: May 4, 2007

Revised Manuscript: July 3, 2007

Manuscript Accepted: July 23, 2007

Published: July 24, 2007

**Citation**

S. G. Podorov, K. M. Pavlov, and D. M. Paganin, "A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging," Opt. Express **15**, 9954-9962 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-9954

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### References

- J. N. Cederquist, J. R. Fienup, J. C. Marron, and R. G. Paxman, "Phase retrival from experimental far-field speckle data," Opt. Lett. 13, 619-621 (1988). [CrossRef] [PubMed]
- 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]
- 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. 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]
- 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]
- 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]
- W. Pauli, "Die allgemeinen Prinzipien der Wellenmechanik" in Handbuch der Physik, ed. H. Geiger and K. Scheel (Springer, Berlin, 1933) 24(1), 83-272.
- 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).
- J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- D. Sayre, "Some implications of a theorem due to Shannon,"Acta Cryst. 5, 843-843 (1952). [CrossRef]
- 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]
- V. Elser, "Phase retrieval by iterated projections," J. Opt. Soc. Am. A 20, 40-55 (2003). [CrossRef]
- P. A. M. Dirac, "Quantised singularities in the electromagnetic field," Proc. R. Soc. A 133, 60-72 (1931). [CrossRef]
- 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).
- R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory," Optik 61(3), 247-262 (1982).
- 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]
- 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. W. Goodman, Statistical optics (John Wiley & Sons, Inc., 2000).
- G. T. Herman, Image reconstruction from projections (Academic Press, 1980).
- 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]
- M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University Press, Cambridge, 1999).
- Q. Shen, I. Bazarov, and P. Thibault, "Diffractive imaging of nonperiodic materials with future coherent sources,". J. Synchr. Rad. 11, 432-438 (2004). [CrossRef]
- D. Gabor, "A new microscopic principle," Nature 161, 777-778 (1948). [CrossRef] [PubMed]
- J. T. Winthrop and C. R. Worthington, "X-ray microscopy by successive Fourier transformation," Phys. Lett. 15, 124-126 (1965). [CrossRef]
- S. Mallick and M. L. Roblin, "Fourier transform holography using a quasimonochromatic incoherent source," Applied Optics 10, 596-598 (1971). [CrossRef] [PubMed]
- 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]
- G. W. Stroke, "Lensless Fourier-transform method for optical holography," Appl. Phys. Lett. 6(10), 201-203 (1965). [CrossRef]
- G. W. Stroke and D. G. Falconer, "Attainment of high resolution in wavefront-reconstruction imaging," Phys. Lett. 13, 306-309 (1964). [CrossRef]
- 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).

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