Abstract

We have studied the spatial coherence properties of a nano-focused x-ray beam by grating (Talbot) interferometry in projection geometry. The beam is focused by a fixed curvature mirror system optimized for high flux density under conditions of partial coherence. The spatial coherence of the divergent exit wave emitted from the mirror focus is measured by Talbot interferometry The results are compared to numerical calculations of coherence propagation. In view of imaging applications, the magnified in-line image of a test pattern formed under conditions of partial coherence is analyzed quantitatively. Finally, additional coherence filtering by use of x-ray waveguides is demonstrated. By insertion of x-ray waveguides, the beam diameter can be reduced from typical values of 200 nm to values below 15 nm. In proportion to the reduction in the focal spot size, the numerical aperture (NA) of the projection imaging system is increased, as well as the coherence length, as quantified by grating interferometry.

© 2011 OSA

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2010 (8)

K. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 8732 (2010).
[CrossRef]

I. A. Vartanyants and A. Singer, “Coherence properties of hard x-ray synchrotron sources and x-ray free-electron lasers,” N. J. Phys. 12, 035004 (2010).
[CrossRef]

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

S. P. Krger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18, 13492–13501 (2010).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” N. J. Phys. 12, 035008 (2010).
[CrossRef]

2008 (5)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinearoptimization approach,” Opt. Express 16, 7264–7278 (2008).
[CrossRef] [PubMed]

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

T. Salditt, S. P. Krger, C. Fuhse, and C. Bhtz, “High-transmission planar x-ray waveguides,” Phys. Rev. Lett. 100, 184801–4 (2008).
[CrossRef] [PubMed]

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

2007 (4)

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

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

2006 (3)

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006).
[CrossRef]

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

C. Fuhse and T. Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. 45, 4603–4608 (2006).
[CrossRef] [PubMed]

2005 (2)

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

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, 164801 (2005).
[CrossRef] [PubMed]

2004 (1)

J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synch. Radiat. 11, 476–482 (2004).
[CrossRef]

2003 (2)

L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).
[CrossRef] [PubMed]

1999 (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]

1995 (1)

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

1978 (1)

1848 (1)

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, “Wave-field formation in a hollow x-ray waveguide,” Phys. Rev. Lett. 97, 184801 (2006).

Abbey, B.

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

Bartels, M.

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

S. P. Krger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18, 13492–13501 (2010).
[CrossRef]

Bauer, G.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

Bhtz, C.

T. Salditt, S. P. Krger, C. Fuhse, and C. Bhtz, “High-transmission planar x-ray waveguides,” Phys. Rev. Lett. 100, 184801–4 (2008).
[CrossRef] [PubMed]

Boye, P.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Bukreeva, I.

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, “Wave-field formation in a hollow x-ray waveguide,” Phys. Rev. Lett. 97, 184801 (2006).

Bunk, O.

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

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, 164801 (2005).
[CrossRef] [PubMed]

Burghammer, M.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Burkeeva, I.

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

Cai, Z.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006).
[CrossRef]

Caro, L. D.

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

Cedola, A.

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, “Wave-field formation in a hollow x-ray waveguide,” Phys. Rev. Lett. 97, 184801 (2006).

Charalambous, P.

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]

Clark, J. N.

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

Cloetens, P.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synch. Radiat. 11, 476–482 (2004).
[CrossRef]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

Dabagov, S. B.

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, “Wave-field formation in a hollow x-ray waveguide,” Phys. Rev. Lett. 97, 184801 (2006).

David, C.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

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, 164801 (2005).
[CrossRef] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synch. Radiat. 11, 476–482 (2004).
[CrossRef]

Davis, T.

de Jonge, M.

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

Diaz, A.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

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, 164801 (2005).
[CrossRef] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

Dierolf, M.

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

Feldkamp, J. M.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Fienup, J. R.

Fonzo, S. D.

L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

Fuhse, C.

T. Salditt, S. P. Krger, C. Fuhse, and C. Bhtz, “High-transmission planar x-ray waveguides,” Phys. Rev. Lett. 100, 184801–4 (2008).
[CrossRef] [PubMed]

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

C. Fuhse and T. Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. 45, 4603–4608 (2006).
[CrossRef] [PubMed]

Giannini, C.

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

Giewekemeyer, K.

S. P. Krger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18, 13492–13501 (2010).
[CrossRef]

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” N. J. Phys. 12, 035008 (2010).
[CrossRef]

Guagliardi, A.

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

Guigay, J.-P.

J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synch. Radiat. 11, 476–482 (2004).
[CrossRef]

Guizar-Sicairos, M.

Gulden, J.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Gureyev, T.

Hartmann, B.

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

Hoppe, R.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

Ishikawa, T.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Jefimovs, K.

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

Johnson, I.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

Kalbfleisch, S.

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” N. J. Phys. 12, 035008 (2010).
[CrossRef]

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

S. P. Krger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18, 13492–13501 (2010).
[CrossRef]

Keplinger, M.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

Kewish, C. M.

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

Kirz, J.

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]

Kopylov, Y. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

Krger, S. P.

S. P. Krger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18, 13492–13501 (2010).
[CrossRef]

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” N. J. Phys. 12, 035008 (2010).
[CrossRef]

T. Salditt, S. P. Krger, C. Fuhse, and C. Bhtz, “High-transmission planar x-ray waveguides,” Phys. Rev. Lett. 100, 184801–4 (2008).
[CrossRef] [PubMed]

Lagomarsino, S.

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, “Wave-field formation in a hollow x-ray waveguide,” Phys. Rev. Lett. 97, 184801 (2006).

Leupold, O.

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

Mancuso, A. P.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Marchesini, S.

S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef] [PubMed]

Matsuyama, S.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Mayo, S.

McNulty, I.

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

Menzel, A.

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Metzger, T. H.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

Miao, J.

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]

Miller, P.

Mimura, H.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Mocuta, C.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

Mokso, R.

J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synch. Radiat. 11, 476–482 (2004).
[CrossRef]

Mori, Y.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Neubauer, H.

S. P. Krger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18, 13492–13501 (2010).
[CrossRef]

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” N. J. Phys. 12, 035008 (2010).
[CrossRef]

Nishino, Y.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Nugent, K.

K. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 8732 (2010).
[CrossRef]

Nugent, K. A.

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

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

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006).
[CrossRef]

Ollinger, C.

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

Osterhoff, M.

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

Paganin, D.

Paganin, D. M.

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

Paterson, D.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006).
[CrossRef]

Patommel, J.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Peele, A. G.

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

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

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006).
[CrossRef]

Pelliccia, D.

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, “Wave-field formation in a hollow x-ray waveguide,” Phys. Rev. Lett. 97, 184801 (2006).

Pfeifer, M. A.

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

Pfeiffer, F.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

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, 164801 (2005).
[CrossRef] [PubMed]

Pogany, A.

Popov, A.

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, “Wave-field formation in a hollow x-ray waveguide,” Phys. Rev. Lett. 97, 184801 (2006).

Popov, A. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

Quiney, H. M.

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

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006).
[CrossRef]

Robinson, I. K.

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, 164801 (2005).
[CrossRef] [PubMed]

Rodenburg, J. M.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

Salditt, T.

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” N. J. Phys. 12, 035008 (2010).
[CrossRef]

S. P. Krger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by twocrossed x-ray waveguides,” Opt. Express 18, 13492–13501 (2010).
[CrossRef]

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

T. Salditt, S. P. Krger, C. Fuhse, and C. Bhtz, “High-transmission planar x-ray waveguides,” Phys. Rev. Lett. 100, 184801–4 (2008).
[CrossRef] [PubMed]

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

C. Fuhse and T. Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. 45, 4603–4608 (2006).
[CrossRef] [PubMed]

Samberg, D.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Sano, Y.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Sayre, D.

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]

Schlenker, M.

J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synch. Radiat. 11, 476–482 (2004).
[CrossRef]

Schrder, S.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Schroer, C. G.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Schropp, A.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Schulze-Briese, C.

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, 164801 (2005).
[CrossRef] [PubMed]

Siewert, F.

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

Singer, A.

I. A. Vartanyants and A. Singer, “Coherence properties of hard x-ray synchrotron sources and x-ray free-electron lasers,” N. J. Phys. 12, 035004 (2010).
[CrossRef]

Sprung, M.

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

Stampanoni, M.

Stangl, J.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

Stephan, S.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Stevenson, A.

Tamasaku, K.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Thibault, P.

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

van der Veen, J. F.

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, 164801 (2005).
[CrossRef] [PubMed]

Vartanyants, I.

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, 164801 (2005).
[CrossRef] [PubMed]

Vartanyants, I. A.

I. A. Vartanyants and A. Singer, “Coherence properties of hard x-ray synchrotron sources and x-ray free-electron lasers,” N. J. Phys. 12, 035004 (2010).
[CrossRef]

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Vila-Comamala, J.

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

Vinogradov, A. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

Weckert, E.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Weitkamp, T.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

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, 164801 (2005).
[CrossRef] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

Wilke, R. N.

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

Wilkins, S.

Williams, G. J.

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

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

Wolf, E.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Oxford University Press, 2007).

Yabashi, M.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Yamamura, K.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Yamauchi, K.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Yark, W.

L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

Yumoto, H.

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

Zabler, S.

J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synch. Radiat. 11, 476–482 (2004).
[CrossRef]

Ziegler, E.

Adv. Phys. (1)

K. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 8732 (2010).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

H. Mimura, H. Yumoto, S. Matsuyama, Y. Sano, K. Yamamura, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “Efficient focusing of hard x-rays to 25 nm by a total reflection mirror,” Appl. Phys. Lett. 90, 051903 (2007).
[CrossRef]

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schrder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[CrossRef]

J. Synch. Radiat. (2)

J.-P. Guigay, S. Zabler, P. Cloetens, C. David, R. Mokso, and M. Schlenker, “The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays,” J. Synch. Radiat. 11, 476–482 (2004).
[CrossRef]

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synch. Radiat. 17, 299–307 (2010).
[CrossRef]

N. J. Phys. (2)

I. A. Vartanyants and A. Singer, “Coherence properties of hard x-ray synchrotron sources and x-ray free-electron lasers,” N. J. Phys. 12, 035004 (2010).
[CrossRef]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” N. J. Phys. 12, 035008 (2010).
[CrossRef]

Nat. Phys. (2)

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nat. Phys. 2, 101–104 (2006).
[CrossRef]

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

Nature (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]

Opt. Commun. (2)

L. D. Caro, C. Giannini, S. D. Fonzo, W. Yark, A. Cedola, and S. Lagomarsino, “Spatial coherence of x-ray planar waveguide exiting radiation,” Opt. Commun. 217, 31–45 (2003).
[CrossRef]

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. B (2)

L. D. Caro, C. Giannini, D. Pelliccia, C. Mocuta, T. H. 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, 081408 (2008).
[CrossRef]

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

Phys. Rev. Lett. (5)

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, 164801 (2005).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801– (2007).
[CrossRef] [PubMed]

T. Salditt, S. P. Krger, C. Fuhse, and C. Bhtz, “High-transmission planar x-ray waveguides,” Phys. Rev. Lett. 100, 184801–4 (2008).
[CrossRef] [PubMed]

I. Bukreeva, A. Popov, D. Pelliccia, A. Cedola, S. B. Dabagov, and S. Lagomarsino, “Wave-field formation in a hollow x-ray waveguide,” Phys. Rev. Lett. 97, 184801 (2006).

C. Fuhse, C. Ollinger, and T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

Rev. Sci. Instrum. (1)

S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef] [PubMed]

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

SRI 2009, AIP Conf. Proc. (1)

S. Kalbfleisch, M. Osterhoff, K. Giewekemeyer, H. Neubauer, S. P. Krger, B. Hartmann, M. Bartels, M. Sprung, O. Leupold, F. Siewert, and T. Salditt, “The holography endstation of beamline P10 at PETRA III,” SRI 2009, AIP Conf. Proc. 1234, 433–436 (2010).
[CrossRef]

Ultramicroscopy (1)

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[CrossRef] [PubMed]

Other (3)

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

T. Weitkamp, “Imaging and tomography with high resolution using coherent hard synchrotron radiation,” Ph.D. thesis, Universitt Hamburg (2002).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Oxford University Press, 2007).

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Figures (8)

Fig. 1
Fig. 1

Experimental setup: A parallel hard x-ray wave front is focused by the KB mirror system. A Talbot grating is illuminated at a (defocus) distance z 1 behind the KB focal plane, and the magnified projection Talbot image is recorded by an area detector positioned in the far-field at a distance z 1 + z 2. The schematic shows the example of a grating period a = 1 μm with 500 nm lines and spaces (l&s) recorded by a pixel detector with 172 μm (Pilatus, Dectris), while smaller grating periods corresponding to 200 nm l&s, and 50 nm l&s required a CCD (LCX, RoperScientific) with smaller pixel size of 20μm.

Fig. 2
Fig. 2

Beam width (FWHM) along the (a) horizontal (x) and (b) vertical (y) direction, as a function of z along the optical axis, as measured by scanning the waveguide through the focus, along with the Fresnel-Kirchhoff simulations (solid black line) and a fit to a Gaussian beam profile (solid red line), convolved with a residual width (see text). (c) Schematic of the waveguide scan. (d) Focus intensity profile along the vertical direction in the focal plane after iterative alignment, along with a Lorentzian fit (solid red line). Side minima and maxima are observed around the central focus. Two-dimensional (e) simulated and (f) experimental intensity distribution in the focal region (vertical plane), after logarithmic color encoding (see colorbar).

Fig. 3
Fig. 3

Simulation results. (a) Degree of coherence |j(x,z)| in the horizontal (xz) plane, evaluated for point (1) on the optical axis (0,z), and point (2) at (x,z). (b) The degree of coherence |j| as a function of x for selected constant values of z, along with a fit (red solid line) to a Gaussian. (c) The full width at half maximum ξ FWHM of the Gaussian fits as a function of z shows linear scaling (solid red line), defining the (full) opening angle of the coherent beam cone ξ FWHM /z ≃ 0.733 mrad, which is smaller than the 2 mrad (FWHM) angle of the full beam in the horizontal direction. In the vertical plane, simulations predict full coherence, as a result of the significantly smaller vertical undulator source size.

Fig. 4
Fig. 4

(a) Intensity distribution (linear gray scale) of the 200nm lines and spaces (l&s) at z 1 = 8.4 mm recorded with the CCD detector, after empty beam correction (scale bar: 0.8 mm in the detector coordinates, corresponding to 1.3μm in the sample plane. (b) same as in (a) but for z 1 = 8.5 mm, where high contrast is observed. (c) Intensity profile obtained by vertical averaging of (b) and subtraction of the mean. (d) Modulus of the Fourier transform of the profile (spectral density, PSD) as a function of inverse pixel, showing the strong m = 1 order.

Fig. 5
Fig. 5

(a) The magnitude of the first (m = 1) order Fourier component |f m =1| of the 200 nm l&s grating (solid symbols), as a function of the defocus distance z 1, computed by Fourier analysis of the recorded Fresnel patterns, along with a least-square fit to Eq. (5) (solid line). The intensity oscillates periodically as the grating position is scanned, with a period of the expected Talbot distance Z. (b) The same for the 500 nm l&s grating, with (solid symbols) and without (open symbols) a refined image analysis procedure (see text). As expected, the amplitude of the |f m =1| oscillation is higher for the 500 nm l&s grating than for the 200 nm l&s grating.

Fig. 6
Fig. 6

(a) Degree of coherence |j(z 1 , d)| in the xz-plane as a function of normalized lateral distance X = (x/z), with the two experimental j values corresponding to the two gratings (open squares), along with a one-parameter fit to a Gaussian of unit amplitude exp(–X 2/(2Σ2)), yielding a normalized coherence length Σ = 0.30± 0.02, or correspondingly a FWHM value o ξ F W H M = 2 2 ln ( 2 ) Σ z = 0.707 10 - 3 z . Along with the experimental values, the degree of coherence | j| from the simulations is shown (thin black line), as evaluated for the plane z 1 = 10 mm, scaled to the X = x/z coordinate. The j values corresponding to the vertical direction (filled triangles) are also included for comparison. A corresponding Gaussian fit yields Σ = 0.33 ± 0.02 (not shown). (b) The function |j(z 1 , x)| = exp(–(x/z 1)2/(2Σ2)) as determined for the horizontal plane, shown as a contour plot, with selected contour levels, along with the two oblique lines (white dotted lines) in the (xz)-plane probed by the two defocus scans, and the edge of the beam (solid white lines) indicating the divergence of the KB-beam, here for the horizontal (xz) plane. (c) The lateral coherence length ξ FWHM (z 1) corresponding to the experimental values (open circles) (with errors from the Σ fit), and the numerical simulations (solid red squares), as well as the coherence length corresponding to a fully incoherent source of size s, observed at distance z 1. The fitted slope corresponds to s = 194.6 nm (FWHM), see text.

Fig. 7
Fig. 7

(a) Near-field propagation image (linear gray scale) of a double slit pattern with predominant phase contrast after correction of dark current and empty beam (flat field). The DS is illuminated by the partially coherent and divergent KB beam. Scale bar denotes 0.8 mm in the detector plane, corresponding to 3.025μm in the sample plane. (b) Horizontal intensity trace showing the characteristic Fresnel oscillations (dark circles) with least-square fit (red line), see text.

Fig. 8
Fig. 8

(a) Schematic of the waveguide Talbot setup. (b) Numerical simulation of beam propagation in the waveguide, assuming a plane incidence wave. Transmission, beam confinement and the coupling of the field into and out of the waveguide can be studied. Due to damping of the higher order modes, the field in the exit plane is highly confined, with a width (FWHM) smaller than the guiding layer D, resulting in a broad diffracted emission cone used to illuminate the grating. (c) Illustrative example of the Talbot image as measured with the detector at the same position as above, with the 50 nm l&s grating at a defocus value z 1 = 0.5880 mm behind the exit plane of the waveguide. The vertical scale bar represents 1.8 mm in the detector plane, corresponding to 200 nm in the sample plane. (d) Talbot scan yielding the magnitude of the first (m = 1) order Fourier component of the 50 nm l&s grating (circles), as determined from Fourier analysis of the recorded Fresnel patterns, along with a least-square fit to Eq. (5) (solid line), as a function of the defocus distance (here with respect to KB focus).

Equations (6)

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FWHM  = 2ln(2) z 0 λ π ( 1 + z 2 / z 0 2 ) + ζ 2 .
γ 1 , 2 ( τ ) = u 1 ( t ) u 2 * ( t + τ ) I 1 I 2 ,
J 1 , 2 ( x 1 , x 2 ) = I 1 ( x 1 ) I 2 ( x 2 ) exp ( - ( x 1 - x 2 ) 2 / ( 2 ξ 2 ) ) ,
f m = 1 = j ( λ z e f f / a ) 1 a 0 a d x exp [ - i 2 π x / a ] g ( x ) g * ( x + λ z e f f / a ) ,
| f m = 1 | = c | j ( λ z e f f / a ) sin ( π z e f f / Z T ) | ,
I / I 0 ( x D ) = | T + 1 2 ( e i ϕ - T ) ( Erf [ π 4 λ z ( w - 2 ( l + x D ) ) ] + Erf [ π 4 λ z ( w + 2 ( l + x D ) ) ] + Erf [ π 4 λ z ( w + 2 ( l - x D ) ) ] + Erf [ π 4 λ z ( w - 2 ( l - x D ) ) ] ) | 2 .

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