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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 9 — Apr. 25, 2011
  • pp: 8073–8078
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Experimental characterization of the coherence properties of hard x-ray sources

Daniele Pelliccia, Andrei Y. Nikulin, Herbert O. Moser, and Keith A. Nugent  »View Author Affiliations


Optics Express, Vol. 19, Issue 9, pp. 8073-8078 (2011)
http://dx.doi.org/10.1364/OE.19.008073


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Abstract

The experimental characterization of the coherence properties of hard X-ray sources is reported and discussed. The source is described by its Mutual Optical Intensity (MOI). The coherent-mode decomposition is applied to the MOI described by a Gaussian-Schell model. The method allows for a direct, quantitative characterization of the degree of coherence of both synchrotron and laboratory sources. The latter represents the first example of characterizing a low coherence hard x-ray source.

© 2011 OSA

1. Introduction

The application of coherence techniques with hard x-ray sources is developing rapidly. Third-generation synchrotron radiation sources and future x-ray Free Electron Lasers (XFELs) display the coherence properties to successfully perform increasingly sophisticated coherent imaging and microscopy experiments. In particular lensless methods have been demonstrated that overcome the resolution limitations of available x-ray optics and so are good candidates for nano-resolution hard x-ray imaging. Such methods invert the diffracted intensity from a sample to form a real-space image of it. A high degree of coherence is necessary for the effective retrieval of images from diffraction data.

The utilization of coherent diffractive imaging (CDI) methods with hard x-rays was successfully demonstrated in 2D [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(6742), 342–344 (1999). [CrossRef]

] and extended to 3D [2

2. J. W. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002). [CrossRef] [PubMed]

,3

3. 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]

]. It was applied to material science problems [2

2. J. W. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002). [CrossRef] [PubMed]

,4

4. A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence, U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege, A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, “Three-dimensional coherent x-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms,” Phys. Rev. Lett. 101(5), 055501 (2008). [CrossRef] [PubMed]

] with important applications to the study of nanocrystals [5

5. M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006). [CrossRef] [PubMed]

] and nanostructures [6

6. I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, T. H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71(24), 245302 (2005). [CrossRef]

,7

7. A. Y. Nikulin, R. A. Dilanian, N. A. Zatsepin, B. M. Gable, B. C. Muddle, A. Y. Souvorov, Y. Nishino, and T. Ishikawa, “3-D X-ray diffraction imaging with nanoscale resolution using incoherent radiation,” Nano Lett. 7(5), 1246–1250 (2007). [CrossRef] [PubMed]

]. Biological applications of CDI were explored [8

8. G. J. Williams, E. Hanssen, A. G. Peele, M. A. Pfeifer, J. Clark, B. Abbey, G. Cadenazzi, M. D. de Jonge, S. Vogt, L. Tilley, and K. A. Nugent, “High-resolution X-ray imaging of Plasmodium falciparum-infected red blood cells,” Cytometry A 73(10), 949–957 (2008). [PubMed]

,9

9. Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent X-ray diffraction,” Phys. Rev. Lett. 102(1), 018101 (2009). [CrossRef] [PubMed]

] and are especially promising for use with XFELs [10

10. 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. Plönjes, M. Kuhlmann, R. Treusch, S. Düsterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-X-ray free-electron laser,” Nat. Phys. 2(12), 839–843 (2006). [CrossRef]

,11

11. 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]

]. The method has been further developed to allow the use of a curved incident wavefront [12

12. 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]

14

14. L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77(8), 081408 (2008). [CrossRef]

] to produce Fresnel, rather than Fraunhofer, diffraction patterns.

The recovery of the real-space image from a single diffraction pattern is accomplished by iterative algorithms [15

15. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 227–246 (1972).

,16

16. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27–29 (1978). [CrossRef] [PubMed]

]. Iterative techniques are typically based on the assumption that the sample is completely coherently illuminated; partial coherence can prevent the convergence of the phase-recovery algorithms [17

17. 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]

]. A recent paper [18

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

] reports the incorporation of spatial coherence into the reconstruction algorithm. The method requires a reliable measurement of the spatial coherence properties of the illuminating field [19

19. C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x-ray source,” Phys. Rev. Lett. 98(22), 224801 (2007). [CrossRef] [PubMed]

].

In recent years several experimental techniques have been developed to characterize the coherence properties of hard x-ray undulator sources. Such techniques are based on the measurement of the first [20

20. P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. 22(14), 1059–1061 (1997). [CrossRef] [PubMed]

23

23. F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. K. Robinson, “Shearing interferometer for quantifying the coherence of hard x-ray beams,” Phys. Rev. Lett. 94(16), 164801 (2005). [CrossRef] [PubMed]

] or second [24

24. M. Yabashi, K. Tamasaku, and T. Ishikawa, “Characterization of the transverse coherence of hard synchrotron radiation by intensity interferometry,” Phys. Rev. Lett. 87(14), 140801 (2001). [CrossRef] [PubMed]

] order coherence function, by estimating the visibility of the interference fringes generated when a know sample or optical element is illuminated by a partially coherent field. The coherence function allows the coherence length at the sample to be estimated.

The method presented here applies the coherent-mode decomposition [25

25. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72(3), 343–351 (1982). [CrossRef]

] to the Mutual Optical Intensity (MOI) described by a Gaussian Schell (GS) model [26

26. A. C. Schell, Multiple Plate Antenna, Ph.D. Thesis (Massachusetts Institute of Technology, 1961).

]. The GS model has been demonstrated to accurately describe synchrotron radiation sources [27

27. R. Coïsson and S. Marchesini, “Gauss-schell sources as models for synchrotron radiation,” J. Synchrotron Radiat. 4(Pt 5), 263–266 (1997). [CrossRef]

]. The combination of the two formalisms introduced here permits the characterization of both laboratory synchrotron radiation hard x-ray sources.

2. Coherent-mode decomposition of a Gaussian-Schell model source

The coherent-mode formulation of Wolf [25

25. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72(3), 343–351 (1982). [CrossRef]

] allows us to write the MOI as a superposition of mutually incoherent modesφn(ξ) in the form:

J(ξ1,ξ2)=n=0ηnφn*(ξ1)φn(ξ2).
(1)

In this work we treat the MOI as an effective radiation source described by the Gaussian-Schell (GS) model [26

26. A. C. Schell, Multiple Plate Antenna, Ph.D. Thesis (Massachusetts Institute of Technology, 1961).

]:
J(ξ1,ξ2)=[I(ξ1)I(ξ2)]1/2g(ξ2ξ1),
(2)
where

I(ξ)=I0exp[ξ22σ2],g(ξ2ξ1)=exp[(ξ2ξ1)22μ2].
(3)

In Eq. (3), σ and μ are a measure of the source size and source coherence respectively and I0 is a positive constant representing the maximum intensity. In this particular case, the analytical expression for the eigenvalues and eigenfunctions in Eq. (1) has been derived [28

28. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72(7), 923–928 (1982). [CrossRef]

]:
φn(ξ)=(2cπ)1/412nn!H(ξ2c)nexp(cξ2),ηn=I0(πa+b+c)1/2(ba+b+c)n,
(4)
where Hn(x)are the Hermite polynomials. Moreover, a=1/(4σ2), b=1/(2μ2) and c=(a2+2ab)1/2. The diffracted intensity from the interaction of a field described by Eq. (1) with a sample, is given by the simple addition of the diffracted intensities from each of the modes, leading to a very simple description of partially coherent diffraction. The effective source GS model incorporates the spatial coherence effects due to the real source and the various optical elements in the beam path, such as windows, primary slits, monochromators, etc. The program of the present experiment is to fit a model described by Eqs. (1) & (4) to diffraction data and thereby recover the relative contributions of the coherent modes and therefore a description of the coherence properties of the incident field.

3. Coherence of laboratory source

The measurements were carried out at the Ultra-Bright x-ray facility located at the School of Physics at Monash University. The characteristic Cu Kα radiation from the source (Rigaku FR-E + Super Bright) was preliminarily collimated by a two-dimensional parabolic multilayer mirror (AXO-Dresden) and subsequently monochromated (and further collimated) by a channel-cut crystal monochromator Ge(111) to select the Cu Kα1 line (x-ray wavelengthλ= 0.154 nm). The schematic of the experimental setup is shown in Fig. 1
Fig. 1 Schematic of the experimental setup (not to scale).
.

Our aim has been to reproduce such diffraction pattern by tuning the effective source parameters of the GS model, Eqs. (2)-(3). The best agreement between the simulation and the data was found to be σ = 30 μm and μ = 1.6 μm. The simulated intensity pattern is shown in Fig. 2(b) (red curve) superimposed to the measurement. The simulation procedure involved the model of a “perfect” grating with transmission function
T(x)={exp[k(iδ+β)t]1ififxSxS,
(5)
where S accounts for the points within the Au lines. In Eq. (5) δ and β are the real and imaginary part of the sample refractive index respectively: n=1δ+iβ. In the case of Au at the selected wavelength we have δ=4.71×105andβ=4.85×106. The contribution of the Si3N4 membrane has been neglected.

Subsequently the transmission function has been smoothed by convolution with a box-car function of width d = 25 nm. The smoothing mimics the relatively smooth edges that would exist in the sample, with the effect of damping the interference fringes beyond some cut-off angle. The used value of d has been chosen to reproduce the measured cut-off angle λ/d0.35deg. The calculated intensity has been subsequently added to the measured background signal (diffraction pattern without the sample), to reproduce the actual envelope of the measured signal. The diffraction pattern has been calculated with Eq. (1), where the summation has been extended to a maximum number of modesnmax=90. The contribution of the modes n>nmax has been neglected, since their relative weight, from Eq. (4), is smaller than approximately 0.01. This can be seen in Fig. 2(c), where the relative weight is plotted as a function of mode order n.

In order to test the validity of this approach, we used the calculated MOI to predict the result of a subsequent, independent, measurement, obtained under identical illumination conditions, using a different sample. A second nano-fabricated pattern, consisting of a square array, with period p = 400 nm and square size of 200 nm, was measured. A SEM of the sample is shown in Fig. 2(d). For our measurement procedure involved the analysis of a one-dimensional pattern, the actual two-dimensional shape of the array cannot be measured and the sample turns out to be equivalent to a one-dimensional grating. Indeed the duty cycle (i.e. the ratio between the width of the gold lines and the period) of the second grating differs from the first one, producing a different set of interference maxima in the diffraction pattern. The comparison between the second measurement and the calculation is shown in Fig. 2(e). The agreement is extremely good at lower angles, while the prediction is less accurate for larger angles. Possible small variations of the actual structure from a perfect mesh are likely to cause deviations in the large angle.

4. Coherence of synchrotron source

The same method was applied to characterize the beam coherence from an undulator source. The experiment was performed on the beamline BL29XU at the Spring-8 synchrotron radiation source (Japan). The measurements were taken at x-ray energy of 9.128 keV (λ = 0.136 nm). As the expected degree of coherence was high, the chosen optics (monochromator and analyzer) were two channel-cut Si(400) crystals, displaying higher resolution. The test sample was a slit of nominal width 2.5 μm, made of two W blades of thickness 2 mm. Therefore the slit blades can be approximated as perfect absorbers for the chosen energy. The experimental result, along with the simulation, is shown in Fig. 3(a)
Fig. 3 (a) Diffraction pattern from a slit of 2.5 μm width (black curve), measured with synchrotron radiation. The red curve is the calculation obtained using a GS model with σ = 3 μm and μ = 3 μm. (b) Relative weight of the first 8 modes. (c) Diffraction pattern from a slit of 1 μm width (black curve), measured with synchrotron radiation. The red curve is the calculation obtained using a GS model with σ = 3 μm and μ = 3 μm. (d) MOI J(ξ1,ξ2) from Eq. (1), calculated using a GS model with σ = 3 μm and μ = 3 μm.
. The simulated pattern has been obtained using a GS model with σ=3 μm and μ=3 μm. The values obtained demonstrate the higher degree of coherence, and a good agreement with the data was achieved by using only 8 modes. Note that only 5 modes have a relative weight larger than 0.01, as plotted in Fig. 3(b). A realistic model of the test samples was constructed by considering the transmission function of an ideal slit and subsequently convoluting it with a box-car function of width 100 nm. This allows imperfections and roughness in the slit blades to be taken into account. The asymmetry in the diffraction pattern is, we believe, caused by the non-perfect alignment of the slit blades with respect to each other and the optical axis. The best agreement was found modeling a misalignment angle of 0.05° between the blades.

5. Conclusions

We have shown the experimental characterization of the coherence properties of hard x-ray sources, described by the Gaussian-Schell model. Good agreement between calculation and experimental results has been found for both laboratory and synchrotron sources. The relatively poor degree of coherence can be modeled although the model seems to be more reliable when only a low number of coherent modes is needed. The possibility of characterizing the properties of sources with any state of coherence has the potential to extend coherent methods based on iterative phase retrieval procedure to sources with non perfect coherence, and eventually laboratory sources.

Acknowledgements

We gratefully acknowledge the support of the Australian Research Council through the Centre of Excellence for Coherent X-Ray Science. We wish to thank Y. Nishino for the experimental collaboration at the beamline BL29XU at Spring-8.

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(6742), 342–344 (1999). [CrossRef]

2.

J. W. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002). [CrossRef] [PubMed]

3.

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]

4.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence, U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege, A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, “Three-dimensional coherent x-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms,” Phys. Rev. Lett. 101(5), 055501 (2008). [CrossRef] [PubMed]

5.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006). [CrossRef] [PubMed]

6.

I. A. Vartanyants, I. K. Robinson, J. D. Onken, M. A. Pfeifer, G. J. Williams, F. Pfeiffer, T. H. Metzger, Z. Zhong, and G. Bauer, “Coherent x-ray diffraction from quantum dots,” Phys. Rev. B 71(24), 245302 (2005). [CrossRef]

7.

A. Y. Nikulin, R. A. Dilanian, N. A. Zatsepin, B. M. Gable, B. C. Muddle, A. Y. Souvorov, Y. Nishino, and T. Ishikawa, “3-D X-ray diffraction imaging with nanoscale resolution using incoherent radiation,” Nano Lett. 7(5), 1246–1250 (2007). [CrossRef] [PubMed]

8.

G. J. Williams, E. Hanssen, A. G. Peele, M. A. Pfeifer, J. Clark, B. Abbey, G. Cadenazzi, M. D. de Jonge, S. Vogt, L. Tilley, and K. A. Nugent, “High-resolution X-ray imaging of Plasmodium falciparum-infected red blood cells,” Cytometry A 73(10), 949–957 (2008). [PubMed]

9.

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent X-ray diffraction,” Phys. Rev. Lett. 102(1), 018101 (2009). [CrossRef] [PubMed]

10.

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. Plönjes, M. Kuhlmann, R. Treusch, S. Düsterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-X-ray free-electron laser,” Nat. Phys. 2(12), 839–843 (2006). [CrossRef]

11.

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]

12.

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]

13.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. D. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4(5), 394–398 (2008). [CrossRef]

14.

L. De Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. Metzger, A. Guagliardi, A. Cedola, I. Burkeeva, and S. Lagomarsino, “In-line holography and coherent diffractive imaging with x-ray waveguides,” Phys. Rev. B 77(8), 081408 (2008). [CrossRef]

15.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 227–246 (1972).

16.

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27–29 (1978). [CrossRef] [PubMed]

17.

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]

18.

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

19.

C. Q. Tran, G. J. Williams, A. Roberts, S. Flewett, A. G. Peele, D. Paterson, M. D. de Jonge, and K. A. Nugent, “Experimental measurement of the four-dimensional coherence function for an undulator x-ray source,” Phys. Rev. Lett. 98(22), 224801 (2007). [CrossRef] [PubMed]

20.

P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. 22(14), 1059–1061 (1997). [CrossRef] [PubMed]

21.

V. Kohn V, I. Snigireva I, and A. Snigirev, “Direct measurement of transverse coherence length of hard X rays from interference fringes,” Phys. Rev. Lett. 85(13), 2745–2748 (2000). [CrossRef] [PubMed]

22.

W. Leitenberger, S. M. Kuznetsov, and A. Snigirev, “Interferometric measurements with hard X-rays using a double slit,” Opt. Commun. 191(1-2), 91–96 (2001). [CrossRef]

23.

F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, and I. K. Robinson, “Shearing interferometer for quantifying the coherence of hard x-ray beams,” Phys. Rev. Lett. 94(16), 164801 (2005). [CrossRef] [PubMed]

24.

M. Yabashi, K. Tamasaku, and T. Ishikawa, “Characterization of the transverse coherence of hard synchrotron radiation by intensity interferometry,” Phys. Rev. Lett. 87(14), 140801 (2001). [CrossRef] [PubMed]

25.

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72(3), 343–351 (1982). [CrossRef]

26.

A. C. Schell, Multiple Plate Antenna, Ph.D. Thesis (Massachusetts Institute of Technology, 1961).

27.

R. Coïsson and S. Marchesini, “Gauss-schell sources as models for synchrotron radiation,” J. Synchrotron Radiat. 4(Pt 5), 263–266 (1997). [CrossRef]

28.

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72(7), 923–928 (1982). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.6600) Coherence and statistical optics : Statistical optics
(340.6720) X-ray optics : Synchrotron radiation

ToC Category:
X-ray Optics

History
Original Manuscript: February 23, 2011
Revised Manuscript: April 7, 2011
Manuscript Accepted: April 9, 2011
Published: April 12, 2011

Citation
Daniele Pelliccia, Andrei Y. Nikulin, Herbert O. Moser, and Keith A. Nugent, "Experimental characterization of the coherence properties of hard x-ray sources," Opt. Express 19, 8073-8078 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8073


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