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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 6 — Mar. 12, 2012
  • pp: 6669–6676
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Rotation-free holographic imaging with extended arc reference

Pengfei Lan, Eiji J. Takahashi, and Katsumi Midorikawa  »View Author Affiliations


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

Imaging of the constructs of the samples is of fundamental importance for a wide range of investigations in material, biological and optical sciences. In recent years, lensless imaging techniques have been developed and attracted a growing interest, partially because the potential to overcome the resolution limits introduced by the optical system and partially because of the scarcity of high-efficient lenses and mirrors in the short wavelength x-ray region. Coherent diffractive imaging (CDI) [1

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]

] is one recently developed lensless imaging technique. In this scheme, the diffraction pattern is oversampled and then converted to the object by using an iterative phase-retrieval algorithm [2

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).

,3

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

]. This approach has been successfully utilized to reconstruct the biological and noncrystalline specimens (see the review [4

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]

] and references therein). The great advantage of CDI is that the resolution is, in principle, only limited by the illuminating wavelength. Therefore, very high resolution is possibly realized using the x-ray sources. Nevertheless, we have to face the difficulty of the convergence of phase-retrieval algorithms [5

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

], which start from a random trial and iterate between real and reciprocal spaces. To converge to a unique and satisfactory image, typically, several thousand iterations are required [3

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

], which is very time consuming. On the other hand, a priori knowledge of support constrain, which is the boundary of the sample, in real space is required. Even though 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]

] can speed up the convergence, a tight support is hardly known and then a large number of simulations with different initial trials are still required to confirm the convergence of the reconstruction.

Another lensless imaging scheme is Fourier transform holography, in which the interference of the object and reference lights is recorded and the object can be directly retrieved by an inverse Fourier transform. Conventionally, the reference beam is created using a pinhole. The optical system of this approach is quite simple and has been successfully utilized in many applications [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]

]. The drawback lies in the low visibility of the interference fringe due to the weak reference wave passing through the pinhole. To alleviate this problem, multiple references [8

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

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]

] and specially-designed references [10

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]

] have been used. A recently developed technique, called holography with extended reference by autocorrelation linear differential operation (HERALDO) [11

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

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]

], has significantly alleviated this problem by taking an extended slit or polygonal reference instead of the pinhole. The effectiveness of this approach has been demonstrated both in the visible [13

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]

] and x-ray regions [14

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

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]

]. In HERALDO, because the reference emerges from boundary waves produced by sharp corners of the polygon (or the edge of the slit), the photon flux can become comparable with that of the object wave, therefore high contrast interference can be observed. Moreover, image can be directly retrieved by applying a noniterative differential or integral operation and inverse Fourier transform. Nevertheless, compared with CDI, the resolution of HERALDO is limited by the sharpness of the edge and thickness of the reference. On the other hand, HERALDO depends very sensitively on the straightness of the edge [17

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]

], which therefore requires subtle fabrication of the reference. Moreover, HERALDO requires a priori knowledge of the orientation angle of the corner or slit reference. This information can be estimated from the streaks in the diffraction pattern, unfortunately the accuracy is dependent on the experiment.

In this work, we demonstrate a rotation-free approach of holography by using an extended arc reference, which is called ARC-HERALDO. We show that the object can be easily retrieved with a two-step algorithm without a prior knowledge of the information of the sample and reference. Moreover, this scheme in principle enables us to overcome the resolution limits introduced by the reference and optical system and therefore promises to achieve the diffraction-limited resolution. Also high contrast interference fringe can be recorded.

2. Theoretical model

Fig. 1 (a) Sample under study. The colorbar indicates the intensity in linear scale. (b) Simulated diffraction pattern of the sample. The colorbar indicates the intensity in logarithmic scale. (c) Image reconstructed by the polar derivative operator in step 1. The colorbar is same to (a).

Fig. 2 (a) Image reconstructed with the two-step algorithms. The colorbar is same to Fig. 1(a). (b) The evolution of the error as a function of iteration step. Green dash-dotted line: HIO algorithms and a static support constrain are used. Blue dashed line: HIO and shrinkwrap algorithms are used. Red solid line: our two-step algorithm is used.

3. Experimental results and discussions

To demonstrate our ARC-HERALDO scheme, we have performed an experiment using a linearly polarized 0.5-mW HeNe laser. The central wavelength is 543 nm. The beam size was amplified to 30 mm with a lens pair and an aperture was used to adjust the beam size. The sample was prepared by printing Fig. 1(a) on a transparent paper. The dark area is opaque and the white pattern is transparent. Of course, due to the quality of the printer, the dark area is not perfectly homogeneous, which may induce some experimental noises. To record the diffraction pattern in the far field, we used a lens with a 500 mm focal length. The sample was put at the front focal plane and the CCD camera (Hamamatsu C4742-95) was put at the back focal plane to detect the diffraction intensity. CCD has a dynamic range of 12 bits and an area of 1024 × 1280 pixels with each pixel of 6.7 × 6.7 μm2. The exposure time was adjusted from 132 μs 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.

Fig. 3 (a) Experimental diffraction of the sample number 3 shown in Fig. 1(a). The colorbar in (a) shows the intensity in logarithmic scale. (b) and (c) are the images retrieved from the experimental diffraction in step 1 and 2, respectively. The colorbars are same to Fig. 1(a). (d), (e) and (f) are similar to (a)–(c), but for another sample of LAN.

Figure 3(d) shows the diffraction intensity for another sample, called LAN, which was prepared with the same way. The sample is the pattern shown in Fig. 3(f). Figure 3(e) presents the 4 images obtained by using the polar derivative operator in step 1. Note that the noises in this result are very weak, less than 15% of the image. By starting from this result, the iterative algorithms quickly converge to a unique image shown in Fig. 3(f). We have performed the same experiment with lots of different samples, such as other numbers 1, 2, ... and alphabetic characters A, B, etc. Even though two-step simulations are required, the iterative simulation in the second step always can converge to a unique image in about 200 steps, which does not takes very long time (∼ 1 minute for Figs. 3(c) and 3(f) on a computer with a 3GHz processor). In comparison, several hundred and more than one thousand iteration steps are required to obtain the similar image with CDI method by adopting the shrinkwrap and static supports, respectively. Moreover, for most of the sample in our scheme, the images obtained in step 1 do not contain much stronger noises compared with the final image reconstructed in step 2. This indicates that the system error in step 1 is weaker than the experimental noise. Nevertheless, the quality of the image retrieved from step 2 is much better than that in step 1.

To demonstrate the rotation-independence of our ARC-HERALDO scheme, we have changed the orientation of the reference. The sample is similar to Fig. 1(a), but the reference is rotated to 0 degree. Figure 4(a) shows the diffraction pattern detected by CCD. After the treatment with the polar derivative operator, as shown in Fig. 4(b), 4 images were retrieved with surrounding by some weak noises. However, these noises are less than 10% of the image. Therefore, as shown in Fig. 4(c), the noises are easily removed in the second step. Furthermore, the reference was rotated to 45 degree. Figure 4(d) shows the diffraction pattern. As shown in Fig. 4(e) and 4(f), the object has been successfully retrieved with the two-step algorithms. These results show that the ARC-HERALDO scheme does not depend on the rotation of the reference and sample. It is rotation free, agreeing with our theory. Moreover, we note that the polar derivative operator in step 1 does not require a prior knowledge of the orientation angle and the second step starts from the images retrieved in step 1. Therefore, our two-step algorithm retrieves the object without a prior knowledge of the information of the sample and reference.

Fig. 4 (a) Experimental diffraction of the sample number 3 with the reference rotated to 0 degree. The colorbar in (a) shows the intensity in logarithmic scale. (b) and (c) are the images retrieved from the experimental diffraction in step 1 and 2, respectively. The colorbars are same to Fig. 1(a), 1(d), 1(e), and 1(f) are similar to (a)–(c), but the reference is rotated to 45 degree.

We have also compared our ARC-HERALDO scheme with the conventional holography and HERALDO by using a pinhole and slit instead of the arc reference, respectively. The experiments were performed with the same setup. For the conventional holography and HERALDO schemes, even though the object and its autocorrelation can be reconstructed and separated in space with a noniterative algorithm, the object is much weaker than its autocorrelation. Moreover, the resolution is limited by the size of the pinhole or sharpness and thickness of the slit. In HERALDO, the orientation of the slit reference has to be estimated from the streaks in the diffraction pattern. The accuracy is limited by the signal-to-noise ratio in experiments. A deviation from the true value can introduce some noises and degenerate the resolution of the retrieved images. On the contrary, the two-step algorithm used in ARC-HERALDO scheme enables us to alleviate these disadvantages. On the other hand, we have applied a similar two-step algorithm for the conventional holography and HERALDO schemes. The image was firstly obtained from the diffraction pattern with the inverse Fourier transform and linear differential operator and then was input to the phase-retrieval algorithms. We found that the image quality was improved by the two-step algorithm and the retrieved image becomes smoother. By adopting the two-step algorithm the resolution is similar to that of CDI, which depends on the wavelength of the illuminating light and the highest spatial frequency detected in experiments but does not depend on the reference. Nevertheless, we emphasize that the iteration in the second step can converge faster to a unique result compared with CDI.

Compared with the conventional holography, although high contrast diffraction pattern can be observed in ARC-HERALDO and also HERALDO schemes, one drawback is that the arc or slit reference requires a very subtle fabrication. This is because the ARC-HERALDO scheme depends on the roundness of the arc reference and HERALDO method depends very sensitively on the straightness of slit reference [17

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]

]. Such a feature is very difficult to manufacture, especially for the applications of microsamples. Fortunately the two-step algorithm can relax the requirement of the subtle fabrication. Our test simulation and experiment show that the misshaped reference remarkably influences the image obtained in step 1. Nevertheless, it plays a minor role in the final image retrieved by the iteration in step 2.

4. Conclusions

In summary, we demonstrated a rotation-free holography scheme, called ARC-HERALDO, by using an extended arc references. High contrast diffractive pattern were observed. From the diffraction pattern, the object can be reconstructed with a two-step algorithm without a prior knowledge of the information of the sample and reference. The algorithm also alleviates the convergence and stagnation problem of the iteration. Even though our proof-of-concept experiment is performed with a visible light, it can be straightforwardly extended to the short wavelength region with the synchrotron, free electron laser or high order harmonic x-ray sources. Because the x-ray light and microsample are invisible, our rotation-free scheme allows us to easily align the optical beam and sample. Therefore it is more attractive for the applications of imaging the microsample with x-ray lights.

Acknowledgments

This work was supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) through a grant for Extreme Photonics Research, Japan Society for the Promotion of Science through a Grant-in-Aid for Scientific Research and also was partially supported by the MEXT through a Grants-in-Aid for Scientific Research for Young Scientists (B) No. 23760057. P.F.L. is grateful for the support of the FPR Program of RIKEN.

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 400, 342–344 (1999). [CrossRef]

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).

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]

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]

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. 105, 043901 (2010). [CrossRef]

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]

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]

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

  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,” Nature400, 342–344 (1999). [CrossRef]
  2. 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).
  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. A3, 1897–1907 (1986). [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. B68, 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,” Nature432, 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. Photonics2, 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. Express15, 9954–9962 (2007). [CrossRef] [PubMed]
  12. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express15, 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]
  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.105, 043901 (2010). [CrossRef]
  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. Express3, 102701 (2010). [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. A4, 118–123 (1987). [CrossRef]

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