## Rotation-free holographic imaging with extended arc reference |

Optics Express, Vol. 20, Issue 6, pp. 6669-6676 (2012)

http://dx.doi.org/10.1364/OE.20.006669

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### Abstract

We proposed and experimentally demonstrated a rotation-free approach of holographic imaging by using an extended arc reference. From the diffraction intensity, the objects were retrieved using a two-step algorithm without a prior knowledge of the information of the sample and reference. This scheme alleviates the convergence problem of coherent diffractive imaging and also promises to achieve a high resolution.

© 2012 OSA

## 1. Introduction

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 specimens,” Nature **400**, 342–344 (1999). [CrossRef]

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

4. J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending X-ray Crystallography to allow the imaing of noncrystalline materials, cells and single protein Complexes,” Annu. Rev. Phys. Chem. **59**, 387–410 (2008). [CrossRef]

5. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A **3**, 1897–1907 (1986). [CrossRef]

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

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

7. S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature **432**, 885–888 (2004). [CrossRef] [PubMed]

8. W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. **89**, 163112 (2006). [CrossRef]

9. R. L. Sandberg, D. A. Raymondson, C. La-o-vorakiat, A. Paul, K. S. Raines, J. Miao, M. M. Murnane, C. H. Kapteyn, and W. F. Schlotter, “Tabletop soft-x-ray Fourier transform holography with 50 nm resolution,” Opt. Lett. **34**, 1618–1620 (2009). [CrossRef] [PubMed]

10. S. Marchesini, S. Boutet, A. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. 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,” Nat. Photonics **2**, 560–563 (2008). [CrossRef]

11. S. G. Podorov, K. M. Pavlov, and D. M. Paganin, “A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaing,” Opt. Express **15**, 9954–9962 (2007). [CrossRef] [PubMed]

12. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express **15**, 17592–17612 (2007). [CrossRef] [PubMed]

13. M. Guizar-Sicairos and J. R. Fienup, “Direct image reconstruction from a Fourier intensity pattern using HERALDO,” Opt. Lett. **33**, 2668–2670 (2008). [CrossRef] [PubMed]

14. D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu1, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot Femtosecond X-ray holography using extended references,” Phys. Rev. Lett. **105**, 093901 (2010). [CrossRef] [PubMed]

16. M. Guizar-Sicairos, D. L. Zhu, J. R. Fienup, B. Wu, A. Scherz, and J. Stöhr, “Holographic x-ray image reconstruction through the application of differential and integral operators,” Opt. Lett. **35**, 928–930 (2010). [CrossRef] [PubMed]

17. Y. Nishino, Y. Tanaka, M. Okada, M. Okaya, Y. Uozaki, K. Nozaki, M. Yabashi, M. Nagasono, K. Tono, H. Kimura, H. Ohashi, S. Matsui, T. Ishikawa, and E. Matsubara, “Femtosecond snapshot holography with extended reference using extreme ultraviolet free-electron laser,” Appl. Phys. Express **3**, 102701 (2010). [CrossRef]

## 2. Theoretical model

12. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express **15**, 17592–17612 (2007). [CrossRef] [PubMed]

*U*

*in the far field is the Fourier transform of the wave in the object plane, which can be expressed as*

_{f}*U*

*= (*

_{f}*o*+

*r*), where

*o*and

*r*denote the object and reference waves, respectively. The inverse Fourier transform of the diffraction intensity is given by

^{−1}(|

*U*

*|*

_{f}^{2}) =

*o*⊗

*o*+

*r*⊗

*r*+

*r*⊗

*o*+

*o*⊗

*r*, where the cross correlation

*o*⊗

*r*is defined by

*∫ o*(

*x*′)

*r*

^{*}(

*x*′ −

*x*)

*dx*′. To reconstruct the object, a proper operation

*L*can be applied to the extended reference such that

*L*(

*r*) =

*δ*. For the arc reference shown in Fig. 1(a), we introduce an operator of the derivative with respect to the polar angle,

*L*to the diffraction pattern. The retrieved images are shown in Fig. 1(c). We can know that

*L*(

*r*) =

*δ*(

*x*−

*x*

_{0},

*y*−

*y*

_{0}) +

*δ*(

*x*−

*x*

_{1},

*y*−

*y*

_{1}), where (

*x*

_{0},

*y*

_{0}) and (

*x*

_{1},

*y*

_{1}) denote the coordinates of two ends of the arc reference. Moreover, according to the definition of cross correlation and Fourier transform theory, we have

*x*(

*r*⊗

*o*) =

*x ∫ r*(

*x*′)

*o*

^{*}(

*x*′ −

*x*)

*dx*′ =

*∫ x*′

*r*(

*x*′)

*o*

^{*}(

*x*′ −

*x*)

*dx*′ −

*∫ r*(

*x*′)(

*x*′ −

*x*)

*o*

^{*}(

*x*′ −

*x*)

*dx*′ = (

*xr*) ⊗

*o*−

*r*⊗ (

*xo*) and

*o*⊗

*o*,

*r*⊗

*r*) and object (i.e.

*o*

^{*},

*o*), respectively. As shown in Fig. 1(c), these terms can be separated in real space. Therefore, 4 images of the object, which correspond to

*o*

^{*}(

*x*

_{0}−

*x*,

*y*

_{0}−

*y*),

*o*

^{*}(

*x*

_{1}−

*x*,

*y*

_{1}−

*y*),

*o*(

*x*−

*x*

_{0},

*y*−

*y*

_{0}),

*o*(

*x*−

*x*

_{1},

*y*−

*y*

_{1}) of Eq. (3), can be observed. Note that to clear distinguish these 4 images, the distance between the positions (

*x*

_{0},

*y*

_{0}) and (

*x*

_{1},

*y*

_{1}), i.e., two ends of the arc reference, should be larger than the size of the sample. Otherwise, the images will overlap with each other. Moreover, to separate these 4 images from the autocorrelation, the distance between the sample and reference must be larger than twice of the size of the sample. These separation requirements are similar to those of the HERALDO method [12

12. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express **15**, 17592–17612 (2007). [CrossRef] [PubMed]

**15**, 17592–17612 (2007). [CrossRef] [PubMed]

*L*does not satisfy Eq. (7) in [12

**15**, 17592–17612 (2007). [CrossRef] [PubMed]

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

*E*

*= ∑*

_{r}_{∉}

*|*

_{S}*ρ*

*|*

_{n}^{2}/∑

*|*

_{S}*ρ*

*|*

_{n}^{2}, where

*S*denotes the real-space support and

*ρ*

*denotes the image retrieved in n-th iteration. The red solid line in Fig. 2(b) shows the error as a function of iteration step of our two-step algorithms. We can see that the result rapidly converges and the error decreases to less than 10*

_{n}^{−3}in 50 steps. Note that in CDI scheme, a large static support is usually adopted. For comparison, we have performed the simulation by adopting a static support of the autocorrelation calculated by the inverse Fourier transform of the diffraction intensity. However, it requires approximately 1300 iteration steps to converge to the similar result [see the green dot-dashed line in Fig. 2(b)]. This simulation can be speeded up if updating the support with the shrinkwrap algorithm [6

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

18. 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**, 118–123 (1987). [CrossRef]

## 3. Experimental results and discussions

*μ*m

^{2}. The exposure time was adjusted from 132

*μ*

*s*to 2

*ms*to find the high contrast diffraction pattern. However, all the figures shown in this work are single frame measurement by the CCD. Average with several different exposure times was not applied.

*μ*s. To reconstruct the object, we applied the polar derivative operator in step 1. As shown in Fig. 3(b), 4 images are retrieved. These images are surrounded by some noises, which include both the system noises and experimental noises. The amplitude of the noise is about 20% of the image. Moreover, unlike the previous holography schemes [7

7. S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature **432**, 885–888 (2004). [CrossRef] [PubMed]

**15**, 17592–17612 (2007). [CrossRef] [PubMed]

*x,y*has significantly suppressed the autocorrelation and noise in the center part. To remove the system noise, in the second step, we started the phase-retrieval algorithm with the image obtained in step 1. As shown in Fig. 3(c), the object has been successfully reconstructed.

17. Y. Nishino, Y. Tanaka, M. Okada, M. Okaya, Y. Uozaki, K. Nozaki, M. Yabashi, M. Nagasono, K. Tono, H. Kimura, H. Ohashi, S. Matsui, T. Ishikawa, and E. Matsubara, “Femtosecond snapshot holography with extended reference using extreme ultraviolet free-electron laser,” Appl. Phys. Express **3**, 102701 (2010). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

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 specimens,” Nature |

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

3. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” App. Opt. |

4. | J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending X-ray Crystallography to allow the imaing of noncrystalline materials, cells and single protein Complexes,” Annu. Rev. Phys. Chem. |

5. | J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A |

6. | S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. H. Spence, “X-ray image reconstrunction from a diffraction pattern alone,” Phys. Rev. B |

7. | S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature |

8. | W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett. |

9. | R. L. Sandberg, D. A. Raymondson, C. La-o-vorakiat, A. Paul, K. S. Raines, J. Miao, M. M. Murnane, C. H. Kapteyn, and W. F. Schlotter, “Tabletop soft-x-ray Fourier transform holography with 50 nm resolution,” Opt. Lett. |

10. | S. Marchesini, S. Boutet, A. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. 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,” Nat. Photonics |

11. | S. G. Podorov, K. M. Pavlov, and D. M. Paganin, “A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaing,” Opt. Express |

12. | M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express |

13. | M. Guizar-Sicairos and J. R. Fienup, “Direct image reconstruction from a Fourier intensity pattern using HERALDO,” Opt. Lett. |

14. | D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu1, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot Femtosecond X-ray holography using extended references,” Phys. Rev. Lett. |

15. | D. L. Zhu, M. Guizar-Sicairos, B. Wu, A. Scherz, Y. Acremann, T. Tyliszczak, P. Fischer, N. Friedenberger, K. Ollefs, M. Farle, J. R. Fienup, and J. Stöhr, “High-resolution X-ray lensless imaging by differential holographic encoding,” Phys. Rev. Let. |

16. | M. Guizar-Sicairos, D. L. Zhu, J. R. Fienup, B. Wu, A. Scherz, and J. Stöhr, “Holographic x-ray image reconstruction through the application of differential and integral operators,” Opt. Lett. |

17. | Y. Nishino, Y. Tanaka, M. Okada, M. Okaya, Y. Uozaki, K. Nozaki, M. Yabashi, M. Nagasono, K. Tono, H. Kimura, H. Ohashi, S. Matsui, T. Ishikawa, and E. Matsubara, “Femtosecond snapshot holography with extended reference using extreme ultraviolet free-electron laser,” Appl. Phys. Express |

18. | 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 |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(090.1995) Holography : Digital holography

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Holography

**History**

Original Manuscript: December 19, 2011

Revised Manuscript: January 20, 2012

Manuscript Accepted: January 25, 2012

Published: March 7, 2012

**Citation**

Pengfei Lan, Eiji J. Takahashi, and Katsumi Midorikawa, "Rotation-free holographic imaging with extended arc reference," Opt. Express **20**, 6669-6676 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6669

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### References

- 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,” Nature400, 342–344 (1999). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 237–246 (1972).
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” App. Opt.21, 2758–2769 (1982). [CrossRef]
- J. Miao, T. Ishikawa, Q. Shen, and T. Earnest, “Extending X-ray Crystallography to allow the imaing of noncrystalline materials, cells and single protein Complexes,” Annu. Rev. Phys. Chem.59, 387–410 (2008). [CrossRef]
- J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A3, 1897–1907 (1986). [CrossRef]
- S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. H. Spence, “X-ray image reconstrunction from a diffraction pattern alone,” Phys. Rev. B68, 140101 (2003). [CrossRef]
- S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen1, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature432, 885–888 (2004). [CrossRef] [PubMed]
- W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt, Ch. Günther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference Fourier transform holography with soft x rays,” Appl. Phys. Lett.89, 163112 (2006). [CrossRef]
- R. L. Sandberg, D. A. Raymondson, C. La-o-vorakiat, A. Paul, K. S. Raines, J. Miao, M. M. Murnane, C. H. Kapteyn, and W. F. Schlotter, “Tabletop soft-x-ray Fourier transform holography with 50 nm resolution,” Opt. Lett.34, 1618–1620 (2009). [CrossRef] [PubMed]
- S. Marchesini, S. Boutet, A. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N. Chapman, M. Frank, S. P. Hau-Riege, A. Szöke, C. 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,” Nat. Photonics2, 560–563 (2008). [CrossRef]
- S. G. Podorov, K. M. Pavlov, and D. M. Paganin, “A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaing,” Opt. Express15, 9954–9962 (2007). [CrossRef] [PubMed]
- M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express15, 17592–17612 (2007). [CrossRef] [PubMed]
- M. Guizar-Sicairos and J. R. Fienup, “Direct image reconstruction from a Fourier intensity pattern using HERALDO,” Opt. Lett.33, 2668–2670 (2008). [CrossRef] [PubMed]
- D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu1, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot Femtosecond X-ray holography using extended references,” Phys. Rev. Lett.105, 093901 (2010). [CrossRef] [PubMed]
- D. L. Zhu, M. Guizar-Sicairos, B. Wu, A. Scherz, Y. Acremann, T. Tyliszczak, P. Fischer, N. Friedenberger, K. Ollefs, M. Farle, J. R. Fienup, and J. Stöhr, “High-resolution X-ray lensless imaging by differential holographic encoding,” Phys. Rev. Let.105, 043901 (2010). [CrossRef]
- M. Guizar-Sicairos, D. L. Zhu, J. R. Fienup, B. Wu, A. Scherz, and J. Stöhr, “Holographic x-ray image reconstruction through the application of differential and integral operators,” Opt. Lett.35, 928–930 (2010). [CrossRef] [PubMed]
- Y. Nishino, Y. Tanaka, M. Okada, M. Okaya, Y. Uozaki, K. Nozaki, M. Yabashi, M. Nagasono, K. Tono, H. Kimura, H. Ohashi, S. Matsui, T. Ishikawa, and E. Matsubara, “Femtosecond snapshot holography with extended reference using extreme ultraviolet free-electron laser,” Appl. Phys. Express3, 102701 (2010). [CrossRef]
- 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, 118–123 (1987). [CrossRef]

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