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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 9 — May. 6, 2013
  • pp: 11441–11447
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Impact of wave front and coherence optimization in coherent diffractive imaging

X. Ge, W. Boutu, D. Gauthier, F. Wang, A. Borta, B. Barbrel, M. Ducousso, A. I. Gonzalez, B. Carré, D. Guillaumet, M. Perdrix, O. Gobert, J. Gautier, G. Lambert, F. R. N. C. Maia, J. Hajdu, P. Zeitoun, and H. Merdji  »View Author Affiliations


Optics Express, Vol. 21, Issue 9, pp. 11441-11447 (2013)
http://dx.doi.org/10.1364/OE.21.011441


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Abstract

We present single shot nanoscale imaging using a table-top femtosecond soft X-ray laser harmonic source at a wavelength of 32 nm. We show that the phase retrieval process in coherent diffractive imaging critically depends on beam quality. Coherence and image fidelity are measured from single-shot coherent diffraction patterns of isolated nano-patterned slits. Impact of flux, wave front and coherence of the soft X-ray beam on the phase retrieval process and the image quality are discussed. After beam improvements, a final image reconstruction is presented with a spatial resolution of 78 nm (half period) in a single 20 fs laser harmonic shot.

© 2013 OSA

1. Introduction

In many scientific areas, imaging with a high spatial and temporal resolution provides meaningful ways to study and understand physical, chemical or biological processes. Recent advances have been made in combining nanometric resolution to ultrafast time scales. Among the advanced imaging techniques, Coherent X-ray Diffractive Imaging (CDI) is a powerful tool to investigate single particle with potentially atomic resolution in a femtosecond time scale [1

1. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses,” Nature 406(6797), 752–757 (2000). [CrossRef] [PubMed]

,2

2. K. J. Gaffney and H. N. Chapman, “Imaging atomic structure and dynamics with ultrafast x-ray scattering,” Science 316(5830), 1444–1448 (2007). [CrossRef] [PubMed]

].

CDI uses computation algorithms to reconstruct the object image from its far-field diffraction pattern recorded by an X-ray detector. However detectors are only sensitive to field intensities, so CDI uses iterative methods based on oversampling [3

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]

]. This “lens-less” technique is aberration-free so that the theoretical spatial resolution is only limited by the radiation wavelength. In the case of a coherent plane wave illumination, the actual resolution is determined by the maximum scattering angle of the diffraction pattern recorded by the detector. The first demonstration by Miao and associates [4

4. 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(6742), 342–344 (1999). [CrossRef]

] has been followed by many convincing experiments using either X-ray synchrotron radiation or femtosecond free electron laser (FEL) [5

5. 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. 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. 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,” Nat. Phys. 2(12), 839–843 (2006). [CrossRef]

9

9. A. P. Mancuso, T. Gorniak, F. Staier, O. M. Yefanov, R. Barth, C. Christophis, B. Reime, J. Gulden, A. Singer, M. E. Pettit, T. Nisius, T. Wilhein, C. Gutt, G. Grübel, N. Guerassimova, R. Treusch, J. Feldhaus, S. Eisebitt, E. Weckert, M. Grunze, A. Rosenhahn, and I. A. Vartanyants, “Coherent imaging of biological samples with femtosecond pulses at the free-electron laser FLASH,” New J. Phys. 12(3), 035003 (2010). [CrossRef]

]. Thanks to an almost full spatial coherence, high photon flux and ultra-short pulse duration, XFELs have been foreseen as of high potential to resolve down to atomic scale processes occurring on femtosecond time scale [5

5. 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. 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. 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,” Nat. Phys. 2(12), 839–843 (2006). [CrossRef]

,7

7. A. Barty, S. Boutet, M. J. Bogan, S. Hau-Riege, S. Marchesini, K. Sokolowski-Tinten, N. Stojanovic, R. Tobey, H. Ehrke, A. Cavalleri, S. Dusterer, M. Frank, S. Bajt, B. W. Woods, M. M. Seibert, J. Hajdu, R. Treusch, and H. N. Chapman, “Ultrafast single-shot diffraction imaging of nanoscale dynamics,” Nat. Photonics 2(7), 415–419 (2008). [CrossRef]

,10

10. H. N. Chapman, S. P. Hau-Riege, M. J. Bogan, S. Bajt, A. Barty, S. Boutet, S. Marchesini, M. Frank, B. W. Woods, W. H. Benner, R. A. London, U. Rohner, A. Szöke, E. Spiller, T. Möller, C. Bostedt, D. A. Shapiro, M. Kuhlmann, R. Treusch, E. Plönjes, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, and J. Hajdu, “Femtosecond time-delay x-ray holography,” Nature 448(7154), 676–679 (2007). [CrossRef] [PubMed]

]. Laboratory-scale laser driven coherent X-ray sources such as high harmonic generation (HHG) now provide another alternative for ultrafast CDI at nanometer resolutions [11

11. R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hädrich, D. M. Gaudiosi, J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. Song, J. Miao, Y. Liu, and F. Salmassi, “Lensless diffractive imaging using tabletop coherent high-harmonic soft-X-ray beams,” Phys. Rev. Lett. 99(9), 098103 (2007). [CrossRef] [PubMed]

14

14. D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010). [CrossRef] [PubMed]

].

However in CDI the reconstructed image quality and resolution are limited by the signal to noise ratio (SNR) of the diffraction pattern [15

15. X. Huang, H. Miao, J. Steinbrener, J. Nelson, D. Shapiro, A. Stewart, J. Turner, and C. Jacobsen, “Signal-to-noise and radiation exposure considerations in conventional and diffraction x-ray microscopy,” Opt. Express 17(16), 13541–13553 (2009). [CrossRef] [PubMed]

,16

16. G. Williams, M. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63(1), 36–42 (2007). [CrossRef] [PubMed]

] and the beam properties (in particular wave front and coherence) [17

17. J. C. H. Spence, U. Weierstall, and M. Howells, “Coherence and sampling requirements for diffractive imaging,” Ultramicroscopy 101(2-4), 149–152 (2004). [CrossRef] [PubMed]

20

20. I. A. Vartanyants and I. K. Robinson, “Imaging of quantum array structures with coherent and partially coherent diffraction,” J. Synchrotron Radiat. 10(6), 409–415 (2003). [CrossRef] [PubMed]

]. The phase retrieval in the CDI algorithm is an iterative process seeded with a random initial phase. In the present work the convergence is driven by standard constraints: the measured diffraction and a real “support”, usually built from the Fourier transform of the measurement (i.e. the autocorrelation of the object) [3

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]

]. Although those two constraints can be refined, using for instance known properties of the sample, the SNR of the diffraction pattern is always a critical factor for the reconstruction algorithm. However, a high SNR (high flux) cannot always provide high quality diffraction patterns. Since the phase information is encoded in the diffraction pattern by interference from different parts of the object, the beam coherence and the wave front quality are the other key factors to have high quality diffraction patterns. These issues impose constraints on beamline quality, preparation and data collection either on third generation synchrotrons, XFELs or HHG beamlines.

In this letter, we show how the improvement of the beam transport and quality improves the CDI image reconstruction of a nano-scale sample. These studies have been conducted in single shot using an optimized HHG source with a high photon flux and controlling the beam wave front and spatial coherence. In addition, accurate focusing of the soft X-rays onto the sample also impacts on the reconstruction quality.

2. Wave front optimization

The experiments were performed using the table-top infrared femtosecond laser LUCA (Laser Ultra Court Accordable) at the CEA Saclay research center, France. It delivers up to 50 mJ energy pulses at 800 nm with a pulse duration of 50 fs and a repetition rate of 20 Hz. The experimental setup is described elsewhere [13

13. A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys. Rev. Lett. 103(2), 028104 (2009). [CrossRef] [PubMed]

].

The beamline characterization and optimization has been done using a Hartmann type wave front sensor (λEUV HASO, Imagine Optics Corp.) [21

21. P. Mercère, P. Zeitoun, M. Idir, S. Le Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with λEUV/120 accuracy,” Opt. Lett. 28(17), 1534–1536 (2003). [CrossRef] [PubMed]

]. We used the RMS (root mean square) wave front error inferred from the measurements to characterize the quality of the X-ray beam: a lower RMS value indicates a beam close to the diffraction limit (lambda/14 RMS according to the Marechal criterion). The wave front sensor accuracy is λ/50 RMS. The Hartmann sensor was used in two steps; firstly on the direct beam without any focusing optics (Fig. 1
Fig. 1 Experimental setup for the harmonic beam wave front measurement and reconstructed intensity profiles of the beam at the source before (a) and after (b) optimization. Generation parameters in (a): cell length = 8 cm, gas pressure = 8 mbar, beam aperture = 24 mm and laser energy = 15 mJ, in (b), cell length = 8 cm, gas pressure = 8 mbar, beam aperture = 21 mm and laser energy = 15 mJ.
) and then after the focusing optics (Fig. 2
Fig. 2 Experimental setup for the off-axis parabola alignment and profiles of the X-ray focal spot at different stages: (a) Non-optimized X-ray focal spot, presenting strong aberrations. (b) Optimized X-ray focal spot without the pupil. (c) X-ray focal spot after a pupil of 3.8 mm in diameter (reconstructed from the measured full beam wave front and a simulated pupil).
). We recorded both intensity and phase of the soft X-ray wave front and reconstructed the harmonics beam profile using back-propagation functions.

In a second step, the wave front sensor was used to align the off axis parabola (f = 20 cm) and to optimize the 25th harmonic (λ = 32 nm) focal spot using the optimized harmonic beam (Fig. 2). The sensor is set in the far-field after the parabola. The spatial amplitude and phase at focus were reconstructed from the measurements. The RMS value criterion is again used to accurately align the parabola. The initial focal spot has a size of 7.6 μm (diameter at 1/e2) with 50% of the total energy in the DL portion and an RMS value of λ/3 (Fig. 2(a)). After optimization the beam has a diameter of 5 μm (at 1/e2) with 88% of the energy in the DL portion and RMS ~λ/6, i.e. twice the diffraction-limit (Fig. 2(b)). Compared to the 20 μm in diameter focal spot reported by Ravasio and associates [13

13. A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys. Rev. Lett. 103(2), 028104 (2009). [CrossRef] [PubMed]

], the optimized focal spot better matches the objects size (3.2 μm x 2 μm). We consequently increased the total “useful” flux interacting with the object by a factor 25. Finally, we placed a diaphragm 10 cm ahead of the parabola. It acts like a spatial filter and removes the boundary wave of the soft X-ray beam. This leads to a more homogeneous wave front (Fig. 2(c)).

3. Impact of coherence on coherent diffractive imaging

We now explore how the spatial coherence of the HHG beam affects image reconstruction. In CDI, the phase information is encoded in the diffraction pattern through the interference modulations between different parts of the object. Therefore, a high coherence is required to ensure the convergence of the phase retrieval algorithm. Here, we use a Young’s double slits to quantify the beam coherence necessary for the CDI reconstruction. The slits are produced using a nanoscale ion beam focused on a 150 nm thick gold coated silicon nitride membrane. They are separated by 4 μm and are 1.5 μm long by 300 nm wide. We then measured the evolution of the beam coherence with respect to several HHG parameters (gas pressure, cell length and IR beam aperture). As an illustration, we show in Fig. 3(a)
Fig. 3 (a): Black circles: Evolution of the fringe visibility of Young’s double slits with the generating gas pressure. Red triangles: Total photon number diffracted by the Young’s double slit, normalized to the maximum value. Blue squares: Spatial resolution of the reconstruction of the Young’s double slit. All the points are from single shot data. (b): Reconstructed image of the Young’s double slit for gas pressure = 13 mbar and fringe visibility = 0.85 and the corresponding Phase Retrieval Transfer Function. The estimated half period resolution is equal to 138 nm. The pressure range used here is different from the one in Section 2 due to a change of the cell entrance and exit holes.
the beam coherence evolution as a function of the gas pressure. The beam coherence is presented as the fringe visibility (V) of the Young double slits experiment. The fringe visibility varies from 0.45 (non optimized HHG) to 0.84 (optimized HHG), and evolves together with the number of diffracted photons. The total diffracted photon flux is also shown normalized to the maximum measured value for a gas backing pressure of 13 mbar. The pressure has been increased compared to the optimal range reported in the wave front optimization section to compensate for the fact that the cell entrance and output holes diameter are slightly larger. This is due to a laser drilling of these holes so that the pressure has to be adjusted from day to day to maintain a constant optimal pressure inside the cell. Our finding is that the harmonic wave front and the coherence are optimum for the same range of parameters.

Combining all these improvements, we used the soft X-ray harmonic at 32 nm to image a nanometric test sample similar to the one described in ref [13

13. A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys. Rev. Lett. 103(2), 028104 (2009). [CrossRef] [PubMed]

]. The patterned object “λ” is 3.2 μm high and 2 μm wide with sub-100 nm details (see Fig. 4
Fig. 4 Single-shot coherent soft X-ray diffraction pattern and the reconstructed image. (a) Measured diffracted intensity from the sample (log scale), obtained in single-shot (20 fs exposure time). (b) Reconstructed object amplitude with a 56 nm pixel size. On the upper left corner is the SEM image of the sample. The two white bars represent 500 nm. (c) PRTF of the image reconstructed with a half-period resolution of 78 nm, given at 1/e. (d) Lineout of the SEM image (dashed line) and reconstruction (continuous line) along the corresponding dashed line in (b).
). The object placed at the focal plan of the parabola diffracts the X-ray light collected by a back-illuminated CCD camera located 19 mm behind the sample. A single-shot diffraction pattern is shown in Fig. 4(a). The number of photons diffracted by the sample and detected by the CCD is about ~2x107. The high contrast in the diffraction pattern indicates a high coherence length.

By applying an iterative algorithm to the diffraction pattern, we calculated the average from 30 best different reconstructions to get the final reconstruction of the object. Each one was calculated using the relaxed averaged alternating reflectors (RAAR) algorithm [24

24. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005). [CrossRef]

] with 3000 iterations with a starting beta value of 0.9 reduced to 0.6 after 2000 iterations. The initial support of the object was determined using the SHRINKWRAP algorithm combined with the hybrid input-output (HIO) algorithm. It was updated every 20 iterations using a threshold update corresponding to 10% of the maximum pixel intensity [25

25. F. R. N. C. Maia, T. Ekeberg, D. van der Spoel, and J. Hajdu, “Hawk: the image reconstruction package for coherent x-ray diffractive imaging,” J. Appl. Cryst. 43(6), 1535–1539 (2010). [CrossRef]

]. The convergence is very fast, with an error value dropping below 0.1 after 600 iterations and below 0.005 after 1800 iterations. The reconstructed object is shown in Fig. 4(b). The image quality is good with all the edges of the object clearly reconstructed, including small details. We note in the reconstruction image that the upper part of lambda is more intense than the bottom part. This is due to the slight misalignment of the X-ray beam with respect to the object (the X-rays are focused on the upper part of the sample).

The signal far from the center of the pattern corresponds to higher momentum transfer and determines the maximum resolution of the reconstructed image, estimated here to 56 nm (corresponding to a spatial frequency of 8.88 μm−1). The effective image resolution is however lower than this value depending on how the phase retrieval process can be affected (as mentioned in the discussion above). Applying the PRTF criterion leads to a resolution of 78 nm (see Fig. 4(c)). Compared to the 119 nm resolution reported in [13

13. A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys. Rev. Lett. 103(2), 028104 (2009). [CrossRef] [PubMed]

], we have largely improved the resolution down to 2.5 λ, bringing it closer to the theoretical resolution. The resolution here is still limited by the SNR and the coherence. The blur on the two main diffraction directions of the diffraction pattern indicates that the illumination is not fully coherent. In the high spatial frequency region (far from the pattern center), the signal drops quickly. As discussed above this limits the final resolution of the reconstructed images. However some methods have been proposed to improve the reconstruction for a partially coherent source, such as using a “Multi-modal propagation” algorithm [19

19. L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009). [CrossRef] [PubMed]

].

4. Conclusion

In summary, we have presented the improvement of a soft X-ray laser harmonic beamline and how the beam properties can affect the image reconstruction process in coherent diffractive imaging. Control over the high harmonic generation source parameters and the focusing of the beam has been achieved. A wave front error of λ/9 has been obtained corresponding to 1.5 times the diffraction limit according to the Marechal criterion. A regular focal spot of 5 μm is also obtained using control given by a wave front sensor feedback. We have then investigated how CDI phase retrieval convergence depends on the spatial coherence of the HHG source. We demonstrated that a substantial degree of coherence is required to ensure an image reconstruction. No image has been obtained for a fringe visibility equal to or lower than 0.5. Finally, with the highest flux, highest coherence and the best focal spot we have reconstructed a nanoscale object with a spatial resolution of 78 nm in a single 20 fs duration shot. These results confirm the high potential of the table-top soft X-ray laser harmonic source for dynamic studies at a femtosecond temporal scale with a sub-80 nm spatial resolution. Indeed, high control of all HHG parameters and pump-probe operation of our table-top source offer a good alternative to FEL source for ultrafast imaging of nanoscale objects.

Acknowledgments

We acknowledge support from the Saclay SLIC laser team. We acknowledge financial support from the European Union through the EU-LASERLAB and the EU–FP7 X-Motion and ATTOFEL programs, from the French ministry of research through the 2009 ANR grants ”I-NanoX” and ”Femto-X-Mag”, from the ”Triangle de la Physique” through the COX grant and the C’NANO research program through the X-NANO grants. Further support came from the Swedish Research Council, The European Research Council, Knut och Alice Wallenbergs Stiftelse, and the DFG Cluster of Excellence at the Munich Centre for Advanced Photonics.

References and links

1.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses,” Nature 406(6797), 752–757 (2000). [CrossRef] [PubMed]

2.

K. J. Gaffney and H. N. Chapman, “Imaging atomic structure and dynamics with ultrafast x-ray scattering,” Science 316(5830), 1444–1448 (2007). [CrossRef] [PubMed]

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.

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(6742), 342–344 (1999). [CrossRef]

5.

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. 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. 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,” Nat. Phys. 2(12), 839–843 (2006). [CrossRef]

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H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23(5), 1179–1200 (2006). [CrossRef] [PubMed]

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10.

H. N. Chapman, S. P. Hau-Riege, M. J. Bogan, S. Bajt, A. Barty, S. Boutet, S. Marchesini, M. Frank, B. W. Woods, W. H. Benner, R. A. London, U. Rohner, A. Szöke, E. Spiller, T. Möller, C. Bostedt, D. A. Shapiro, M. Kuhlmann, R. Treusch, E. Plönjes, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, and J. Hajdu, “Femtosecond time-delay x-ray holography,” Nature 448(7154), 676–679 (2007). [CrossRef] [PubMed]

11.

R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hädrich, D. M. Gaudiosi, J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. Song, J. Miao, Y. Liu, and F. Salmassi, “Lensless diffractive imaging using tabletop coherent high-harmonic soft-X-ray beams,” Phys. Rev. Lett. 99(9), 098103 (2007). [CrossRef] [PubMed]

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R. L. Sandberg, C. Song, P. W. Wachulak, D. A. Raymondson, A. Paul, B. Amirbekian, E. Lee, A. E. Sakdinawat, C. La-O-Vorakiat, M. C. Marconi, C. S. Menoni, M. M. Murnane, J. J. Rocca, H. C. Kapteyn, and J. Miao, “High numerical aperture tabletop soft x-ray diffraction microscopy with 70-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 105(1), 24–27 (2008). [CrossRef] [PubMed]

13.

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys. Rev. Lett. 103(2), 028104 (2009). [CrossRef] [PubMed]

14.

D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett. 105(9), 093901 (2010). [CrossRef] [PubMed]

15.

X. Huang, H. Miao, J. Steinbrener, J. Nelson, D. Shapiro, A. Stewart, J. Turner, and C. Jacobsen, “Signal-to-noise and radiation exposure considerations in conventional and diffraction x-ray microscopy,” Opt. Express 17(16), 13541–13553 (2009). [CrossRef] [PubMed]

16.

G. Williams, M. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63(1), 36–42 (2007). [CrossRef] [PubMed]

17.

J. C. H. Spence, U. Weierstall, and M. Howells, “Coherence and sampling requirements for diffractive imaging,” Ultramicroscopy 101(2-4), 149–152 (2004). [CrossRef] [PubMed]

18.

G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Coherent diffractive imaging and partial coherence,” Phys. Rev. B 75(10), 104102 (2007). [CrossRef]

19.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009). [CrossRef] [PubMed]

20.

I. A. Vartanyants and I. K. Robinson, “Imaging of quantum array structures with coherent and partially coherent diffraction,” J. Synchrotron Radiat. 10(6), 409–415 (2003). [CrossRef] [PubMed]

21.

P. Mercère, P. Zeitoun, M. Idir, S. Le Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with λEUV/120 accuracy,” Opt. Lett. 28(17), 1534–1536 (2003). [CrossRef] [PubMed]

22.

J. Gautier, P. Zeitoun, C. Hauri, A.-S. Morlens, G. Rey, C. Valentin, E. Papalarazou, J.-P. Goddet, S. Sebban, F. Burgy, P. Mercère, M. Idir, G. Dovillaire, X. Levecq, S. Bucourt, M. Fajardo, H. Merdji, and J.-P. Caumes, “Optimization of the wave front of high order harmonics,” Eur. Phys. J. D 48(3), 459–463 (2008). [CrossRef]

23.

J. Lohbreier, S. Eyring, R. Spitzenpfeil, C. Kern, M. Weger, and C. Spielmann, “Maximizing the brilliance of high-order harmonics in a gas jet,” New J. Phys. 11(2), 023016 (2009). [CrossRef]

24.

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005). [CrossRef]

25.

F. R. N. C. Maia, T. Ekeberg, D. van der Spoel, and J. Hajdu, “Hawk: the image reconstruction package for coherent x-ray diffractive imaging,” J. Appl. Cryst. 43(6), 1535–1539 (2010). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(140.7090) Lasers and laser optics : Ultrafast lasers
(140.7240) Lasers and laser optics : UV, EUV, and X-ray lasers
(340.7440) X-ray optics : X-ray imaging
(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

ToC Category:
X-ray Optics

History
Original Manuscript: August 10, 2012
Revised Manuscript: October 26, 2012
Manuscript Accepted: November 16, 2012
Published: May 3, 2013

Citation
X. Ge, W. Boutu, D. Gauthier, F. Wang, A. Borta, B. Barbrel, M. Ducousso, A. I. Gonzalez, B. Carré, D. Guillaumet, M. Perdrix, O. Gobert, J. Gautier, G. Lambert, F. R. N. C. Maia, J. Hajdu, P. Zeitoun, and H. Merdji, "Impact of wave front and coherence optimization in coherent diffractive imaging," Opt. Express 21, 11441-11447 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-11441


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References

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  12. R. L. Sandberg, C. Song, P. W. Wachulak, D. A. Raymondson, A. Paul, B. Amirbekian, E. Lee, A. E. Sakdinawat, C. La-O-Vorakiat, M. C. Marconi, C. S. Menoni, M. M. Murnane, J. J. Rocca, H. C. Kapteyn, and J. Miao, “High numerical aperture tabletop soft x-ray diffraction microscopy with 70-nm resolution,” Proc. Natl. Acad. Sci. U.S.A.105(1), 24–27 (2008). [CrossRef] [PubMed]
  13. A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys. Rev. Lett.103(2), 028104 (2009). [CrossRef] [PubMed]
  14. D. Gauthier, M. Guizar-Sicairos, X. Ge, W. Boutu, B. Carré, J. R. Fienup, and H. Merdji, “Single-shot femtosecond x-ray holography using extended references,” Phys. Rev. Lett.105(9), 093901 (2010). [CrossRef] [PubMed]
  15. X. Huang, H. Miao, J. Steinbrener, J. Nelson, D. Shapiro, A. Stewart, J. Turner, and C. Jacobsen, “Signal-to-noise and radiation exposure considerations in conventional and diffraction x-ray microscopy,” Opt. Express17(16), 13541–13553 (2009). [CrossRef] [PubMed]
  16. G. Williams, M. Pfeifer, I. A. Vartanyants, and I. K. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A63(1), 36–42 (2007). [CrossRef] [PubMed]
  17. J. C. H. Spence, U. Weierstall, and M. Howells, “Coherence and sampling requirements for diffractive imaging,” Ultramicroscopy101(2-4), 149–152 (2004). [CrossRef] [PubMed]
  18. G. J. Williams, H. M. Quiney, A. G. Peele, and K. A. Nugent, “Coherent diffractive imaging and partial coherence,” Phys. Rev. B75(10), 104102 (2007). [CrossRef]
  19. L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett.103(24), 243902 (2009). [CrossRef] [PubMed]
  20. I. A. Vartanyants and I. K. Robinson, “Imaging of quantum array structures with coherent and partially coherent diffraction,” J. Synchrotron Radiat.10(6), 409–415 (2003). [CrossRef] [PubMed]
  21. P. Mercère, P. Zeitoun, M. Idir, S. Le Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with λEUV/120 accuracy,” Opt. Lett.28(17), 1534–1536 (2003). [CrossRef] [PubMed]
  22. J. Gautier, P. Zeitoun, C. Hauri, A.-S. Morlens, G. Rey, C. Valentin, E. Papalarazou, J.-P. Goddet, S. Sebban, F. Burgy, P. Mercère, M. Idir, G. Dovillaire, X. Levecq, S. Bucourt, M. Fajardo, H. Merdji, and J.-P. Caumes, “Optimization of the wave front of high order harmonics,” Eur. Phys. J. D48(3), 459–463 (2008). [CrossRef]
  23. J. Lohbreier, S. Eyring, R. Spitzenpfeil, C. Kern, M. Weger, and C. Spielmann, “Maximizing the brilliance of high-order harmonics in a gas jet,” New J. Phys.11(2), 023016 (2009). [CrossRef]
  24. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl.21(1), 37–50 (2005). [CrossRef]
  25. F. R. N. C. Maia, T. Ekeberg, D. van der Spoel, and J. Hajdu, “Hawk: the image reconstruction package for coherent x-ray diffractive imaging,” J. Appl. Cryst.43(6), 1535–1539 (2010). [CrossRef]

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