## Novel Fourier-domain constraint for fast phase retrieval in coherent diffraction imaging |

Optics Express, Vol. 19, Issue 20, pp. 19330-19339 (2011)

http://dx.doi.org/10.1364/OE.19.019330

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### Abstract

Coherent diffraction imaging (CDI) for visualizing objects at atomic resolution has been realized as a promising tool for imaging single molecules. Drawbacks of CDI are associated with the difficulty of the numerical phase retrieval from experimental diffraction patterns; a fact which stimulated search for better numerical methods and alternative experimental techniques. Common phase retrieval methods are based on iterative procedures which propagate the complex-valued wave between object and detector plane. Constraints in both, the object and the detector plane are applied. While the constraint in the detector plane employed in most phase retrieval methods requires the amplitude of the complex wave to be equal to the squared root of the measured intensity, we propose a novel Fourier-domain constraint, based on an analogy to holography. Our method allows achieving a low-resolution reconstruction already in the first step followed by a high-resolution reconstruction after further steps. In comparison to conventional schemes this Fourier-domain constraint results in a fast and reliable convergence of the iterative reconstruction process.

© 2011 OSA

## 1. Introduction

1. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. **5**(6), 843–843 (1952). [CrossRef]

3. 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**(6), 1662–1669 (1998). [CrossRef]

4. B. W. J. McNeil and N. R. Thompson, “X-ray free-electron lasers,” Nature Photon. **4**(12), 814–821 (2010). [CrossRef]

5. H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nature Photon. **4**(12), 833–839 (2010). [CrossRef]

6. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**(15), 2758–2769 (1982). [CrossRef] [PubMed]

8. G. Oszlányi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. A **60**(2), 134–141 (2004). [CrossRef] [PubMed]

9. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier-transform using a support constraint,” J. Opt. Soc. Am. A **4**(1), 118–123 (1987). [CrossRef]

10. 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. Natl. Acad. Sci. U.S.A. **102**(43), 15343–15346 (2005). [CrossRef] [PubMed]

12. W. J. Huang, J. M. Zuo, B. Jiang, K. W. Kwon, and M. Shim, “Sub-angstrom-resolution diffractive imaging of single nanocrystals,” Nature Phys. **5**(2), 129–133 (2009). [CrossRef]

13. 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**(2), 025506 (2006). [CrossRef] [PubMed]

15. S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szoke, C. W. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nature Photon. **2**(9), 560–563 (2008). [CrossRef]

16. R. A. Dilanian, G. J. Williams, L. W. Whitehead, D. J. Vine, A. G. Peele, E. Balaur, I. McNulty, H. M. Quiney, and K. A. Nugent, “Coherent diffractive imaging: a new statistically regularized amplitude constraint,” New J. Phys. **12**(9), 093042 (2010). [CrossRef]

17. F. C. Zhang and J. M. Rodenburg, “Phase retrieval based on wave-front relay and modulation,” Phys. Rev. B **82**(12), 121104 (2010). [CrossRef]

*I*

^{γ}, and noticed that if γ>0.5 the iterative routine converges much faster. Below we present a novel constraint in the Fourier-domain which exhibits two major advantages: it allows dropping the requirement of a well-defined support around the object and it provides fast, reliable and unambiguous reconstructions.

## 2. Theoretical background

*I*

_{0}=

*U*

_{0}

*U*

_{0}*, where

*U*

_{0}is the complex-valued wave in the detector plane. The iterative reconstruction process based on the Gerchberg-Saxton algorithm [2] is schematically illustrated in Fig. 1 . As in common phase retrieval algorithms, the iterative loop in our method starts with the complex-valued field in the detector plane

*U*

_{1}: its amplitude is given by the square root of the measured intensity, and its phase is chosen to be randomly distributed. In the iterative process

*U*

_{i}', i = 1,2,... is the updated complex-valued wavefront in the detector plane after each iteration. In common phase retrieval methods, the phase of the iterated field is adapted for the next iterative loop, while the amplitude is replaced by the square root of the measured intensity:

*H*is given by

*H*~

*R**

*O*+

*RO**. Here,

*R*denotes the reference wave and

*O*the object wave. In the reconstruction process, the hologram is multiplied with the reference wave resulting in the straightforward reconstructed object wave O~

*RH*/|

*R*|

^{2}. This approach can be applied for reconstructing a diffraction pattern if we formally treat the measured intensity of the diffraction pattern

*U*

_{0}

*U*

_{0}* as a hologram

*H*and

*U*

_{i}' as the reference wave

*R*, see Eq. (2). This novel constraint unlike the one given by Eq. (1), does not only account for the phase of

*U*

_{i}', but also for its amplitude. The latter must gradually approach the amplitude of

*U*

_{0}. In order to avoid division by zero, a constant ε is added in the denominator. For measured intensities of up to 10

^{7}counts per pixel, ε is selected in the range of 0.1-1000. In Eq. (2),

*L*

_{i}denotes the Fourier-domain filter, a complex-distributed function altered at each iteration. Any additional constraint in the object plane can still be selected independently of the Fourier-domain constraint.

## 3. Simulation results

3. 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**(6), 1662–1669 (1998). [CrossRef]

_{1}is set to some value 0<μ

_{1}<<1, and the solution approaches

_{Ui(1)→μ1U0}. This modified

*L*-filteracts as a band-pass filter for the measured intensity distribution, as illustrated in Fig. 3 . For the 1st step μ is selected such that the filter acts as a low-pass filter, see Fig. 3(e-f). For the first 90 iterations μ was set to μ

_{1}= 0.001 and ε

_{1}= 1. The result of this iterative run is a reconstruction of a low-resolution image of the object, presented in Fig. 2(c).The error function, calculated aswhere |

*U*

_{0}| is the square root of the measured intensity and |

*U*

_{i}´| is the updated amplitude of the complex wave at the detector plane after each iteration, shows fluctuations around some constant value, as evident from the plot in Fig. 2(e). The obtained in the first step complex-valued distribution

_{2}is set to μ

_{2}= 1, and thus

_{Ui(2)→U0}. The result of the second iterative run (step two), presented in Fig. 2(d), where μ

_{2}= 1 and ε

_{2}= 1 were used, required just less than 30 iterations to refine the low-resolution reconstruction. The error function quickly decreases, as shown in Fig. 2(e) and reaches 0.25 after a total of 120 iterations. The iterative process was terminated once the error reached a stable minimum. Alternatively, the low-resolution result obtained from the first iterative run can be used as an initial distribution for any other phase retrieval method.

### 3.1 Simulated diffraction pattern of a ribosome

18. M. Valle, A. Zavialov, W. Li, S. M. Stagg, J. Sengupta, R. C. Nielsen, P. Nissen, S. C. Harvey, M. Ehrenberg, and J. Frank, “Incorporation of aminoacyl-tRNA into the ribosome as seen by cryo-electron microscopy,” Nat. Struct. Mol. Biol. **10**(11), 899–906 (2003). [CrossRef] [PubMed]

^{12}photons with a wavelength of λ = 1.4 Å that was focused onto a 100 nm region. A low-resolution reconstruction was obtained after 105 iterations in the 1st step, using μ

_{1}= 0.0001 and ε

_{1}= 10. The final high-resolution reconstruction was obtained in the 2nd step after 101 iterations with μ

_{2}= 1 and ε

_{2}= 1000. A total of 206 iterations were required and the final error amounts to 0.25. The reconstructed electron density, see Fig. 4(d), is in good agreement with the original object, Fig. 4(a), taking into account shift and flip invariance.

### 3.2 Diffraction pattern of a particle field with missing information in the central region

### 3.3 Diffraction pattern of a phase-shifting object

7. 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**(14), 140101 (2003). [CrossRef]

_{1}= 0.3 and ε

_{1}= 1000 were chosen for the 1st step, and the constraint in the object plane is that the phase of the transmission function is set to 0. This allows obtaining a low-resolution reconstruction of the object after only 48 iterations. In the 2nd step, μ

_{2}= 1 and ε

_{2}= 10000 are selected and the phase in the object domain is free from any constraint at those pixels where the amplitude exceeds 30% of the maximum of the amplitude, and it is set to 0 at the other pixels. In the 2nd step, 43 iterations are required to retrieve both, the amplitude and the phase distribution of the object, as illustrated in Fig. 6 (c)-(d). The evolution of the error is displayed in Fig. 6(g), and reaches a final value of 0.40. The very same diffraction pattern was also reconstructed employing the shrinkwrap method (Fig. 6(e)-(f)), using the hybrid input-output iterative scheme with the feedback parameter β = 0.9. The supporting masks were created at 5% threshold of the maximum of the amplitude in the object plane and convoluted with a Gaussian (σ = 3). The supporting mask was updated every 20 iterations. A total of 750 iterations were necessary to retrieve the complex-valued object with a final error of 0.47. The evolution of the error is shown in Fig. 6(h).

## 4. Experimental results

^{2}is sampled with 500x500 pixels. The recorded diffraction pattern was centered by the symmetry of its peaks and the noise level of the camera (about 50 counts per pixel) was subtracted. Finally, taking the modulus of the resulting diffraction pattern allows the elimination of small negative values due to noise subtraction. The reconstruction was accomplished in three steps with the results displayed in Fig. 7 . In all three steps the constraint in the object plane was that the phase of the retrieved transmission function was set to zero; no other constraints such as masks or “supports” were applied. In step one, 200 iterations were done using ε

_{1}= 10 and μ

_{1}= 0.0001. In step two, 360 iterations were done using ε

_{2}= 5000 and μ

_{2}= 1. Finally, in step three, 40 iterations were performed using ε

_{3}= 500 and μ

_{3}= 1 to decrease the value of ε. Still, this third and last step resulted in only a minor improvement of the reconstruction, and could as well be skipped. By applying a larger number of iterations, the reconstructed image remained unchanged except for slight oscillations in the noise distribution. Thus, the stable reconstruction was already achieved after a total of 600 iterations. For comparison, we also tried to employ the “shrink-wrap” algorithm to the experimental diffraction pattern; however it failed to achieve a stable reconstruction resembling the object. The reason could be that the “shrink-wrap” algorithm relies on a continuously updated support mask, which could not be well defined for our experimental sample as there were some fine scratches around the object as verified by optical microscopy. Instead, we employed a reconstruction method using a support in the form of a fixed tight mask. In this case, about 4000 iterations were needed in a successfully convergent run to achieve a presentable reconstruction, as shown in Fig. 7(f).

## 5. Conclusion

5. H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nature Photon. **4**(12), 833–839 (2010). [CrossRef]

19. O. Kamimura, T. Dobashi, K. Kawahara, T. Abe, and K. Gohara, “10-kV diffractive imaging using newly developed electron diffraction microscope,” Ultramicroscopy **110**(2), 130–133 (2010). [CrossRef] [PubMed]

20. E. Steinwand, J.-N. Longchamp, and H.-W. Fink, “Coherent low-energy electron diffraction on individual nanometer sized objects,” Ultramicroscopy **111**(4), 282–284 (2011). [CrossRef] [PubMed]

## Acknowledgements

## References and links

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

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

3. | 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 |

4. | B. W. J. McNeil and N. R. Thompson, “X-ray free-electron lasers,” Nature Photon. |

5. | H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nature Photon. |

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

7. | 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 |

8. | G. Oszlányi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. A |

9. | J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier-transform using a support constraint,” J. Opt. Soc. Am. A |

10. | 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. Natl. Acad. Sci. U.S.A. |

11. | P. Thibault and I. C. Rankenburg, “Optical diffraction microscopy in a teaching laboratory,” Am. J. Phys. |

12. | W. J. Huang, J. M. Zuo, B. Jiang, K. W. Kwon, and M. Shim, “Sub-angstrom-resolution diffractive imaging of single nanocrystals,” Nature Phys. |

13. | 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. |

14. | D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A |

15. | S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szoke, C. W. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nature Photon. |

16. | R. A. Dilanian, G. J. Williams, L. W. Whitehead, D. J. Vine, A. G. Peele, E. Balaur, I. McNulty, H. M. Quiney, and K. A. Nugent, “Coherent diffractive imaging: a new statistically regularized amplitude constraint,” New J. Phys. |

17. | F. C. Zhang and J. M. Rodenburg, “Phase retrieval based on wave-front relay and modulation,” Phys. Rev. B |

18. | M. Valle, A. Zavialov, W. Li, S. M. Stagg, J. Sengupta, R. C. Nielsen, P. Nissen, S. C. Harvey, M. Ehrenberg, and J. Frank, “Incorporation of aminoacyl-tRNA into the ribosome as seen by cryo-electron microscopy,” Nat. Struct. Mol. Biol. |

19. | O. Kamimura, T. Dobashi, K. Kawahara, T. Abe, and K. Gohara, “10-kV diffractive imaging using newly developed electron diffraction microscope,” Ultramicroscopy |

20. | E. Steinwand, J.-N. Longchamp, and H.-W. Fink, “Coherent low-energy electron diffraction on individual nanometer sized objects,” Ultramicroscopy |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(110.7440) Imaging systems : X-ray imaging

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: July 19, 2011

Revised Manuscript: August 25, 2011

Manuscript Accepted: August 27, 2011

Published: September 20, 2011

**Citation**

Tatiana Latychevskaia, Jean-Nicolas Longchamp, and Hans-Werner Fink, "Novel Fourier-domain constraint for fast phase retrieval in coherent diffraction imaging," Opt. Express **19**, 19330-19339 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19330

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### References

- D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr.5(6), 843–843 (1952). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttg.)35, 237–246 (1972).
- J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15(6), 1662–1669 (1998). [CrossRef]
- B. W. J. McNeil and N. R. Thompson, “X-ray free-electron lasers,” Nature Photon.4(12), 814–821 (2010). [CrossRef]
- H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nature Photon.4(12), 833–839 (2010). [CrossRef]
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982). [CrossRef] [PubMed]
- 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. B68(14), 140101 (2003). [CrossRef]
- G. Oszlányi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. A60(2), 134–141 (2004). [CrossRef] [PubMed]
- J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier-transform using a support constraint,” J. Opt. Soc. Am. A4(1), 118–123 (1987). [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. Natl. Acad. Sci. U.S.A.102(43), 15343–15346 (2005). [CrossRef] [PubMed]
- P. Thibault and I. C. Rankenburg, “Optical diffraction microscopy in a teaching laboratory,” Am. J. Phys.75(9), 827–832 (2007). [CrossRef]
- W. J. Huang, J. M. Zuo, B. Jiang, K. W. Kwon, and M. Shim, “Sub-angstrom-resolution diffractive imaging of single nanocrystals,” Nature Phys.5(2), 129–133 (2009). [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(2), 025506 (2006). [CrossRef] [PubMed]
- D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic Fresnel coherent diffractive imaging,” Phys. Rev. A80(6), 063823 (2009). [CrossRef]
- S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szoke, C. W. Cui, D. A. Shapiro, M. R. Howells, J. C. H. Spence, J. W. Shaevitz, J. Y. Lee, J. Hajdu, and M. M. Seibert, “Massively parallel X-ray holography,” Nature Photon.2(9), 560–563 (2008). [CrossRef]
- R. A. Dilanian, G. J. Williams, L. W. Whitehead, D. J. Vine, A. G. Peele, E. Balaur, I. McNulty, H. M. Quiney, and K. A. Nugent, “Coherent diffractive imaging: a new statistically regularized amplitude constraint,” New J. Phys.12(9), 093042 (2010). [CrossRef]
- F. C. Zhang and J. M. Rodenburg, “Phase retrieval based on wave-front relay and modulation,” Phys. Rev. B82(12), 121104 (2010). [CrossRef]
- M. Valle, A. Zavialov, W. Li, S. M. Stagg, J. Sengupta, R. C. Nielsen, P. Nissen, S. C. Harvey, M. Ehrenberg, and J. Frank, “Incorporation of aminoacyl-tRNA into the ribosome as seen by cryo-electron microscopy,” Nat. Struct. Mol. Biol.10(11), 899–906 (2003). [CrossRef] [PubMed]
- O. Kamimura, T. Dobashi, K. Kawahara, T. Abe, and K. Gohara, “10-kV diffractive imaging using newly developed electron diffraction microscope,” Ultramicroscopy110(2), 130–133 (2010). [CrossRef] [PubMed]
- E. Steinwand, J.-N. Longchamp, and H.-W. Fink, “Coherent low-energy electron diffraction on individual nanometer sized objects,” Ultramicroscopy111(4), 282–284 (2011). [CrossRef] [PubMed]

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