## Non-iterative solution of the phase retrieval problem using a single diffraction measurement

Optics Express, Vol. 16, Issue 10, pp. 6896-6903 (2008)

http://dx.doi.org/10.1364/OE.16.006896

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### Abstract

Coherent diffractive imaging is a method by which iterative methods are employed to recover image information about a finite object from its coherent diffraction pattern. We employ methods borrowed from density functional theory to show that an image can be recovered in a single non-iterative step for a finite sample subject to phase-curved illumination. The result also yields a new approach to quantitative x-ray phase-contrast imaging.

© 2008 Optical Society of America

*a priori*knowledge of the object, such as its support, to achieve a reconstruction of the complex scattered wave [1

1. J. W. 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]

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

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

5. D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft X-ray diffraction microscopy,” Proc. Nat. Acad. Sci. **102**15343–15346 (2005). [CrossRef] [PubMed]

6. 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]

7. M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature **442**, 63–66 (2006). [CrossRef] [PubMed]

8. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses,” Nature **406**, 752–757 (2000). [CrossRef] [PubMed]

9. H. N. Chapman, A. Barty, M. J. Bogan, S. Boutel, M. Frank, S. P. Hau-Reige, S. Marchesini, B. W. Woods, S. Bajt, W. Henry. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düsterer, 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. 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]

10. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. **66**, 5486–5492 (1995). [CrossRef]

11. M. W. Westneat, O. Betz, R. W. Blob, K. Fezzaa, W. J. Cooper, and W. K. Lee, “Tracheal respiration in insects visualized with synchrotron X-ray imaging,” Science **299**, 558–560 (2003). [CrossRef] [PubMed]

12. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. **42**2015–2025 (1997). [CrossRef] [PubMed]

13. P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M Schlenker, “Holotomography: quantitative phase tomographywith micrometre resolution using hard synchrotron radiation X-rays,” Appl. Phys. Lett. **75**, 2912–2914 (1999). [CrossRef]

14. 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]

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

16. G. J. Williams, H. M. Quiney, B. B. Dhal, 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]

*N*real numbers to recover a wavefield at

*N*points. For example, two intensity measurements are required to constrain properly an algorithm that seeks to recover the complex amplitude; this is the observation that underpins the “oversampling” argument [17

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

18. 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]

19. K. A. Nugent, “X-ray non-interferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A **24**, 536–547 (2007). [CrossRef]

*a priori*information about the sample size and shape (its support), its optical properties or by acquiring two or more sets of intensity data. The central idea of the present paper is the suggestion that a fourth approach, based on ideas borrowed from density functional theory, may be used, in which we use

*a priori*information about the illumination to enable us to assume a functional form for the propagation of the scattered wave.

20. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. **73**, 1434–1441 (1983). [CrossRef]

*z*,

*I*(

**r**,

*z*) and Φ(

**r**,

*z*) respectively, to the longitudinal intensity derivative in that plane:

*λ*is the wavelength and ∇

_{⊥}denotes the transverse gradient operator. Apart from crystallography, all of the methods touched on in the previous paragraph adopt the paraxial approximation, an approximation also made here. In practice, an approximation to the derivative on the right hand side of Eq. (1) is formed from intensity measurements made at two closely spaced planes. Provided the intensity is strictly positive over a simply connected region, the phase distribution may be uniquely recovered to within a physically meaningless additive constant [23

23. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport of intensity equation, and phase uniqueness,” J.Opt. Soc. Am. A. **12**, 1942–1946 (1995). [CrossRef]

22. K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects”, Phys. Rev. Lett. **91**, 203902 (2003). [CrossRef] [PubMed]

*z*

_{1}is communicated to a plane at

*z*

_{2}via the expression

*Z*=

*z*

_{2}-

*z*

_{1}. We assume that the scattered field at

*z*

_{1}has finite support and that the detector plane is at

*z*

_{2}. Our aim is to construct a method by which the intensity data at

*z*

_{2}can be used to construct an expression for the longitudinal intensity derivative in that plane, thereby allowing a non-iterative phase recovery using Eq. (1).

*g*(

**w**) may be regarded both as the autocorrelation of

*ψ*(

**r**

_{1},

*z*

_{1})exp(

*iπr*

^{2}

_{1}/

*λZ*) and the inverse Fourier transform of

*I*(

**r**

_{2},

*z*

_{2}). The region of integration,

*S*(

**w**), is specified completely by the support of

*ψ*(

**r**

_{1},

*z*

_{1}). We may complete the formal correspondence between Eq. (1) and Eq. (2) if we note that

*Z*)

**r**

_{2}·∇

_{⊥}

*I*(

**r**

_{2},

*z*

_{2}), which is readily constructed from

*I*(

**r**

_{2},

*z*

_{2}), isolates the trivial, spherically-expanding component of

*I*(

**r**

_{2},

*z*

_{2}), and the partial derivative of

*g*(

**w**) is taken with respect to

*Z*because the source plane at

*z*

_{1}is regarded as fixed. Since

*ψ*(

**r**

_{1},

*z*

_{1}) and

*S*(

**w**) are independent of

*Z*, then an application of Leibnitz’ Theorem [25, Eq. 3.3.7] yields

*T*[

*S*(

**w**),

*ψ*(

**r**

_{1},

*z*

_{1}),

**w**,

*Z*], whose form is generally too complicated to be written explicitly, but which is defined precisely by Leibnitz’ Theorem through the gradient of the boundary that encloses

*S*(

**w**), and the value of the integrand of

*g*(

**w**) on the interior of that boundary. The first two terms in Eq. (6) are readily obtained by direct manipulation of

*I*(

**r**

_{2},

*z*

_{2}). The third term is sensitive to the gradient of the wavefield and vanishes identically if, for example, the wavefield is detected at a propagation distance

*Z*from a uniformly illuminated aperture of any exterior shape. Similarly, this term vanishes identically in far-field Fresnel diffraction if such an aperture is illuminated by light possessing constant amplitude and a radius of spherical illumination of

*Z*. The fourth term,

*T*[

*S*(

**w**),

*ψ*(

**r**

_{1},

*z*

_{1}),

**w**,

*Z*], depends on the exterior shape of

*S*(

**w**), and on the value of

*ψ*(

**r**1,

*z*

_{1}) along the edges of

*S*(

**w**). This term vanishes identically if

*ψ*(

**r**

_{1},

*z*

_{1}) also vanishes everywhere along the boundaries that define its support, a situation that is readily realized in practice. In general, the third term determines the analytic variation of

*ψ*(

**r**

_{1},

*z*

_{1}) within the interior of the object support, while the fourth term fixes

*ψ*(

**r**

_{1},

*z*

_{1}) on the edges of the support.

*T*[

*S*(

**w**),

*ψ*(

**r**

_{1},

*z*

_{1}),

**w**,

*Z*] in Eq. (6) is negligible, which will be realised if the edges of the object fall smoothly to zero without jump discontinuities. The resulting coherent diffractive imaging problem is then posed as the solution of an integral equation. The left-hand side and the first two terms on the right hand side are known from the experimental data, since

*g*(

**w**) can be recovered from the measured data by a Fourier transformation. The third term on the right hand side, however, contains the wavefunction to be determined. To deal with this issue we take a perturbative approach and write

*ε*is a parameter that we may adjust to change the strength of the perturbation relative to

*ψ*

_{0}(

**r**,

*z*). It is not a parameter that plays any part in the solution algorithm, and is introduced here only to explore the limits of validity of the scheme as a function of the strength of the perturbation. In all cases of practical interest, the scattered wave,

*ψ*(

_{S}**r**,

*z*), is very weak compared to the incident wave,

*ψ*

_{0}(

**r**,

*z*), and so it is safe to assume that

*ε*|

*ψ*(

_{S}**r**,

*z*)|≪|

*ψ*

_{0}(

**r**,

*z*)| for |

*ε*|≤1. We also assume that the field incident on the scattering object has been fully characterized by some independent means, such as those described in [26

26. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focussed X-ray fields,” Nature Physics **2**, 101–104 (2006). [CrossRef]

27. P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev. B , **136**, B864 (1964). [CrossRef]

*∂I*/

*∂z*on

*I*in the plane of the detector, which is sufficient information from which to derive all optical properties of the system through Eq. (1). Since we assume that

*ε*|

*ψ*(

_{S}**r**,

*z*)|≪

*ψ*

_{0}(

**r**,

*z*), we adopt a strategy employed in density functional theory and assume that the functional dependence of ∂

*I*/∂

*z*on I for the illuminating field,

*ψ*

_{0}(

**r**,

*z*), remains valid for the perturbed field,

*ψ*(

**r**,

*z*). This approach is analogous to the use of the free-electron gas model in the specification of an effective zero-order model for the electrostatic exchange potential in density functional theory, which depends only on the electron density.

*ψ*

_{0}(

**r**,

*z*

_{1})=exp(-

*µ*

*r*

^{2}), where the real part of

*µ*is positive. In this case, it is straightforward to obtain the explicit functional relationship for the illuminating beam

*g*

_{0}(

**w**) is the Fourier transform of

*I*

_{0}. This establishes the exact functional relationship between

*∂I*

_{0}/

*∂z*and

*I*

_{0}without making explicit reference to

*ψ*(

**r**) in any plane. The essence of our approach is to use the functional form of Eq. (8) with the experimental measurement of

*g*(

**w**) to devise an estimate of

*∂I*/

*∂z*.

*g*(

**w**) is formed from it by a Fourier transformation.

*ψ*(

_{S}**r**

_{1},

*z*

_{1}) is very close to the input phase, a constant, over the region where the amplitude differs significantly from zero. Some artifacts are visible in Fig. 5(a) which have an intensity of less than 5% of the maximum of

*ψ*(

_{S}**r**

_{1},

*z*

_{1}). These are due both to the use of Eq. (8) to calculate

*∂I*/

*∂z*for the perturbed wave, as well as numerical errors inherent in the method used to solve Eq. (1) [28

28. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. **80**2586–2589 (1998). [CrossRef]

*ε*, has been set at 0.1, corresponding to attenuation of the incident amplitude by 10%. The smaller this parameter, the more closely the approximate solution approaches the exact solution, subject to the approximations inherent in the computational method used to solve Eq. (1).

*T*[

*S*(

**w**),

*ψ*(

**r**

_{1},

*z*

_{1}),

**w**,

*Z*] in Eq. (6), which in practice means that the scattered function contains no piecewise discontinuities in the source plane. Consequently, we have constructed the test sample so that its edges are sufficiently smooth that no new edge contributions are introduced into the functional relationship between

*∂I*

_{0}/

*∂z*and

*I*

_{0}. In this case, the transfer function in the scattering plane remains smooth and of small amplitude relative to the incident beam, and the functional relation defined by Eq. (8) is sufficiently close to the exact form so as to provide an accurate solution of the phase problem. In Fresnel diffractive imaging we are always able to recover the illuminating wavefield from measured intensity data [26

26. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focussed X-ray fields,” Nature Physics **2**, 101–104 (2006). [CrossRef]

*∂I*

_{0}/

*∂z*and

*I*

_{0}that is required in this phase retrieval scheme.

*I*and

*∂I*/

*∂z*in all cases. Indeed, the forms of the functional relations differ markedly from case to case, mainly because of the strong influence of edge effects. The results display a high level of accuracy, and the results may be used to seed refinement by iterative methods. The only restriction that must be observed strictly in practice is that the amplitude of the perturbingwave fall smoothly to zero on its boundary; the wave may be real or complex, subject to this restriction. Iterative methods are necessarily more general, and more stable, because they are not subject to the restrictions inherent in the present algorithm. They are, however, subject to iterative stagnation, and like all non-linear iterative procedures are most useful when the iterative refinement is initiated in the neighbourhood of a solution.

16. G. J. Williams, H. M. Quiney, B. B. Dhal, 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]

26. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focussed X-ray fields,” Nature Physics **2**, 101–104 (2006). [CrossRef]

## References and links

1. | J. W. 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 |

2. | J. R. Fienup, “Phase retrieval algorithms- a comparison,” Appl. Opt. , |

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

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. | D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft X-ray diffraction microscopy,” Proc. Nat. Acad. Sci. |

6. | 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. |

7. | M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature |

8. | R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses,” Nature |

9. | H. N. Chapman, A. Barty, M. J. Bogan, S. Boutel, M. Frank, S. P. Hau-Reige, S. Marchesini, B. W. Woods, S. Bajt, W. Henry. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düsterer, 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. 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 |

10. | A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. |

11. | M. W. Westneat, O. Betz, R. W. Blob, K. Fezzaa, W. J. Cooper, and W. K. Lee, “Tracheal respiration in insects visualized with synchrotron X-ray imaging,” Science |

12. | D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. |

13. | P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M Schlenker, “Holotomography: quantitative phase tomographywith micrometre resolution using hard synchrotron radiation X-rays,” Appl. Phys. Lett. |

14. | K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. |

15. | K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Physics Today |

16. | G. J. Williams, H. M. Quiney, B. B. Dhal, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. |

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

18. | 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 |

19. | K. A. Nugent, “X-ray non-interferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A |

20. | M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. |

21. | D. Paganin, |

22. | K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects”, Phys. Rev. Lett. |

23. | T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport of intensity equation, and phase uniqueness,” J.Opt. Soc. Am. A. |

24. | R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. 1: Underlying theory,” Optik |

25. | M. Abramowitz and I. Stegun (eds), Handbook of Mathematical Functions, Dover Publications, New York (1970). |

26. | H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focussed X-ray fields,” Nature Physics |

27. | P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev. B , |

28. | D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

(110.7440) Imaging systems : X-ray imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: February 11, 2008

Revised Manuscript: March 18, 2008

Manuscript Accepted: April 20, 2008

Published: April 30, 2008

**Citation**

H. M. Quiney, G. J. Williams, and K. A. Nugent, "Non-iterative solution of the phase retrieval problem using a single diffraction measurement," Opt. Express **16**, 6896-6903 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-6896

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### References

- J. W. 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]
- J. R. Fienup, "Phase retrieval algorithms- a comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- V. Elser, "Phase retrieval by iterated projections," J. Opt. Soc. A. 20, 40-55 (2003). [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 (2003). [CrossRef]
- D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, "Biological imaging by soft X-ray diffraction microscopy," Proc. Nat. Acad. Sci. 10215343-15346 (2005). [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]
- M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature 442, 63-66 (2006). [CrossRef] [PubMed]
- R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond X-ray pulses," Nature 406, 752-757 (2000). [CrossRef] [PubMed]
- H. N. Chapman, A. Barty, M. J. Bogan, S. Boutel, M. Frank, S. P. Hau-Reige, S. Marchesini, B. W. Woods, S. Bajt, W. Henry. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft X-ray free-electron laser," Nature Physics 2, 839-843 (2006). [CrossRef]
- A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995). [CrossRef]
- M. W. Westneat, O. Betz, R. W. Blob, K. Fezzaa, W. J. Cooper, and W. K. Lee, "Tracheal respiration in insects visualized with synchrotron X-ray imaging," Science 299, 558-560 (2003). [CrossRef] [PubMed]
- D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, "Diffraction enhanced X-ray imaging," Phys. Med. Biol. 422015-2025 (1997). [CrossRef] [PubMed]
- P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M Schlenker, "Holotomography: quantitative phase tomographywith micrometre resolution using hard synchrotron radiation X-rays," Appl. Phys. Lett. 75, 2912-2914 (1999). [CrossRef]
- 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]
- K. A. Nugent, D. Paganin, and T. E. Gureyev, "A phase odyssey," Physics Today 54, 27-32 (2001). [CrossRef]
- G. J. Williams, H. M. Quiney, B. B. Dhal, 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]
- D. Sayre, "Some implications of a theorem due to Shannon," Acta. Cryst. 5, 843 (1952). [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]
- K. A. Nugent, "X-ray non-interferometric phase imaging: a unified picture," J. Opt. Soc. Am. A 24, 536-547 (2007). [CrossRef]
- M. R. Teague, "Deterministic phase retrieval: a Green�??s function solution," J. Opt. Soc. Am. 73, 1434-1441 (1983). [CrossRef]
- D. Paganin, Coherent X-ray Optics (Oxford University Press, 2005).
- K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, "Unique phase recovery for nonperiodic objects," Phys. Rev. Lett. 91, 203902 (2003). [CrossRef] [PubMed]
- T. E. Gureyev, A. Roberts, and K. A. Nugent, "Partially coherent fields, the transport of intensity equation, and phase uniqueness," J.Opt. Soc. Am. A. 12, 1942-1946 (1995). [CrossRef]
- R. H. T. Bates, "Fourier phase problems are uniquely solvable in more than one dimension. 1: Underlying theory," Optik 61, 247-262 (1982).
- M. Abramowitz and I. Stegun (eds), Handbook of Mathematical Functions, Dover Publications, New York (1970).
- H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, "Diffractive imaging of highly focussed X-ray fields," Nature Physics 2, 101-104 (2006). [CrossRef]
- P. Hohenberg and W. Kohn, "Inhomogeneous electron gas," Phys. Rev. B 136, B864 (1964). [CrossRef]
- D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. 802586-2589 (1998). [CrossRef]

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