Abstract

A reliable measurement of beam coherence is important for optimal performance of a number of coherence methods being utilized at third-generation synchrotrons and free-electron lasers. Various approaches have been proposed in the past for determining the source size, and hence the degree of coherence; however they often require complex setups with perfect optics and suffer from undefined uncertainties. We present a robust tool for X-ray source characterization with a full quantitative uncertainty analysis for fast on-the-fly coherence measurements. The influence of three multilayer monochromator crystals on the apparent source size is evaluated using the proposed method.

© 2014 Optical Society of America

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2014 (1)

2012 (1)

P. Modregger, F. Scattarella, B. Pinzer, C. David, R. Bellotti, M. Stampanoni, “Imaging the ultrasmall-angle X-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 2–5 (2012).
[CrossRef]

2011 (4)

B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, I. McNulty, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
[CrossRef]

S. Roy, D. Parks, K. A. Seu, R. Su, J. J. Turner, W. Chao, E. H. Anderson, S. Cabrini, S. D. Kevan, “Lensless X-ray imaging in reflection geometry,” Nat. Photonics 5, 243–245 (2011).
[CrossRef]

S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salomé, C. David, “High-efficiency Fresnel zone plates for hard X-rays by 100 keV e-beam lithography and electroplating,” J. Synchrotron Radiat. 18, 442–446 (2011).
[CrossRef] [PubMed]

I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett. 107, 1–5 (2011).
[CrossRef]

2010 (5)

K. S. Morgan, S. C. Irvine, Y. Suzuki, K. Uesugi, A. Takeuchi, D. M. Paganin, K. K. Siu, “Measurement of hard X-ray coherence in the presence of a rotating random-phase-screen diffuser,” Opt. Commun. 283, 216–225 (2010).
[CrossRef]

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

A. Rack, T. Weitkamp, M. Riotte, D. Grigoriev, T. Rack, L. Helfen, T. Baumbach, R. Dietsch, T. Holz, M. Krämer, F. Siewert, M. Meduna, P. Cloetens, E. Ziegler, “Comparative study of multilayers used in monochromators for synchrotron-based coherent hard X-ray imaging,” J. Synchrotron Radiat. 17, 496–510 (2010).
[CrossRef] [PubMed]

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

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[CrossRef] [PubMed]

2009 (1)

R. Kluender, F. Masiello, P. van Vaerenbergh, J. Härtwig, “Measurement of the spatial coherence of synchrotron beams using the Talbot effect,” Phys. Status Solidi A 206, 1842–1845 (2009).
[CrossRef]

2006 (2)

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, R. Abela, “Trends in synchrotron-based tomographic imaging: the SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

T. Weitkamp, C. David, C. Kottler, O. Bunk, F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S (2006).
[CrossRef]

2005 (2)

F. Pfeiffer, O. Bunk, C. Schulze-Briese, A. Diaz, T. Weitkamp, C. David, J. F. van der Veen, I. Vartanyants, I. Robinson, “Shearing interferometer for quantifying the coherence of hard X-ray beams,” Phys. Rev. Lett. 94, 1–4 (2005).
[CrossRef]

T. Weitkamp, B. Nohammer, A. Diaz, C. David, E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
[CrossRef]

2004 (2)

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

W. Leitenberger, H. Wendrock, L. Bischoff, T. Weitkamp, “Pinhole interferometry with coherent hard X-rays,” J. Synchrotron Radiat. 11, 190–197 (2004).
[CrossRef] [PubMed]

2003 (2)

J. Lin, D. Paterson, A. Peele, P. McMahon, C. Chantler, K. Nugent, B. Lai, N. Moldovan, Z. Cai, D. Mancini, I. McNulty, “Measurement of the spatial coherence function of undulator radiation using a phase mask,” Phys. Rev. Lett. 90, 1–4 (2003).
[CrossRef]

T. Panzner, W. Leitenberger, J. Grenzer, Y. Bodenthin, T. Geue, U. Pietsch, H. Möhwald, “Coherence experiments at the energy-dispersive reflectometry beamline at BESSY II,” J. Phys. D: Appl. Phys. 36, A93–A97 (2003).
[CrossRef]

2002 (2)

C. David, B. Nohammer, H. H. Solak, E. Ziegler, “Differential phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287 (2002).
[CrossRef]

S. Yasin, D. Hasko, H. Ahmed, “Comparison of MIBK/IPA and water/IPA as PMMA developers for electron beam nanolithography,” Microelectron. Eng. 61–62, 745–753 (2002).
[CrossRef]

2001 (3)

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

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

D. Paterson, B. Allman, P. McMahon, J. Lin, N. Moldovan, K. Nugent, I. McNulty, C. Chantler, C. Retsch, T. Irving, “Spatial coherence measurement of X-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

2000 (2)

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

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

1999 (2)

I. Robinson, J. Libbert, I. Vartanyants, J. Pitney, D. Smilgies, D. Abernathy, G. Grübel, “Coherent X-ray diffraction imaging of silicon oxide growth,” Phys. Rev. A 60, 9965–9972 (1999).

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation X-rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

1997 (1)

1996 (2)

A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phasecontrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

1995 (1)

S. Dierker, R. Pindak, R. Fleming, I. Robinson, L. Berman, “X-ray photon correlation spectroscopy study of brownian motion of gold colloids in Glycerol,” Phys. Rev. Lett. 75, 449–452 (1995).
[CrossRef] [PubMed]

1981 (1)

1957 (1)

Abbey, B.

B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, I. McNulty, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
[CrossRef]

I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett. 107, 1–5 (2011).
[CrossRef]

Abela, R.

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, R. Abela, “Trends in synchrotron-based tomographic imaging: the SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

Abernathy, D.

I. Robinson, J. Libbert, I. Vartanyants, J. Pitney, D. Smilgies, D. Abernathy, G. Grübel, “Coherent X-ray diffraction imaging of silicon oxide growth,” Phys. Rev. A 60, 9965–9972 (1999).

Acremann, Y.

I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett. 107, 1–5 (2011).
[CrossRef]

Ahmed, H.

S. Yasin, D. Hasko, H. Ahmed, “Comparison of MIBK/IPA and water/IPA as PMMA developers for electron beam nanolithography,” Microelectron. Eng. 61–62, 745–753 (2002).
[CrossRef]

Allman, B.

D. Paterson, B. Allman, P. McMahon, J. Lin, N. Moldovan, K. Nugent, I. McNulty, C. Chantler, C. Retsch, T. Irving, “Spatial coherence measurement of X-ray undulator radiation,” Opt. Commun. 195, 79–84 (2001).
[CrossRef]

Amberger, M.

Anderson, E.

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

Anderson, E. H.

S. Roy, D. Parks, K. A. Seu, R. Su, J. J. Turner, W. Chao, E. H. Anderson, S. Cabrini, S. D. Kevan, “Lensless X-ray imaging in reflection geometry,” Nat. Photonics 5, 243–245 (2011).
[CrossRef]

Attwood, D.

I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett. 107, 1–5 (2011).
[CrossRef]

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

Balaur, E.

B. Abbey, L. W. Whitehead, H. M. Quiney, D. J. Vine, G. A. Cadenazzi, C. A. Henderson, K. A. Nugent, E. Balaur, C. T. Putkunz, A. G. Peele, G. J. Williams, I. McNulty, “Lensless imaging using broadband X-ray sources,” Nat. Photonics 5, 420–424 (2011).
[CrossRef]

Bang, E.

I. Vartanyants, A. Singer, A. Mancuso, O. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. Turner, W. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. Peele, Y. Feng, O. Krupin, S. Moeller, W. Wurth, “Coherence properties of individual femtosecond pulses of an X-ray free-electron laser,” Phys. Rev. Lett. 107, 1–5 (2011).
[CrossRef]

Barnea, Z.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barrett, R.

S. Gorelick, J. Vila-Comamala, V. A. Guzenko, R. Barrett, M. Salomé, C. David, “High-efficiency Fresnel zone plates for hard X-rays by 100 keV e-beam lithography and electroplating,” J. Synchrotron Radiat. 18, 442–446 (2011).
[CrossRef] [PubMed]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation X-rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

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

Bauer, G.

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Figures (8)

Fig. 1
Fig. 1

Experimental setup at the X02DA TOMCAT beamline to measure the coherence properties of the superbending magent source.

Fig. 2
Fig. 2

SEM image of the 2D grating. The grating was fabricated to have a regular gold pattern on a silicon substrate with following parameters: w = 3.65μm (width of gold pillars) with h = 3.39μm (height), dc = 0.53 (duty cycle), α = 4.2° (angle) and 2×2mm2 grating size.

Fig. 3
Fig. 3

Flowchart for calculating the source size. First, intervals for each parameter E, σ, dc and α are nested. In the second subroutine (“Fourier Analysis”) both simulated and experimental Talbot images are loaded which are in return used for calculating the weighted LSE. The program is exited, when the weighted LSE has been calculated for all intervals nmax and all iterations kmax.

Fig. 4
Fig. 4

Demonstration of the fitting algorithm from Fig. 3 for the Si(111) DCM. The experimental Fourier coefficients as well as two iterations from the fitting algorithm are plotted in dependency on the propagation distance z. The “best fit” (solid line) was calculated by loading all parameters obtained from the fitting algorithm and using them as an input for Eqs. (10) and (15).

Fig. 5
Fig. 5

Principal Fourier coefficients in horizontal (a) and vertical (b) direction as functions of the propagation distance z for two multilayer monochromators at approx. 21.4keV X-ray energy. Simulated Fourier coefficients (solid line) were calculated with parameters obtained from the fitting algorithm.

Fig. 6
Fig. 6

Principal Fourier coefficients in horizontal (a) and vertical (b) direction for the [Ru/C]100 monochromator crystal set to three different X-ray energies. Simulated Fourier coefficients (solid line) were calculated with parameters obtained from the fitting algorithm.

Fig. 7
Fig. 7

Principal Fourier coefficients in horizontal (a) and vertical (b) direction for the V/B4C multilayer crystal at 18.0keV X-ray energy. Simulated Fourier coefficients (solid line) were calculated with parameters obtained from the fitting algorithm.

Fig. 8
Fig. 8

Calculated source sizes for the [Ru/C]100 multilayer, where the horizontal and vertical lines denote the source size and energy measurement uncertainties, respectively.

Tables (1)

Tables Icon

Table 1 Horizontal and vertical full-width-at-half-maximum (FWHM) source sizes and fitted energies E for the two experiments conducted at the X02DA TOMCAT beamline. Additionally, the source-to-grating distance R and the maximum detector’s travel range z are listed. For the characterization of the V/B4C multilayer the source-to-grating distance was included as a variable in the fitting algorithm, independently for both the horizontal and the vertical direction. All uncertainties represent maximal error margins.

Equations (27)

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ψ ( r ) = A exp ( i k r ) | r | ,
ψ ( r ) = A 2 i λ G e i k ( R + z ) | R | | z | [ cos ( n , z ) cos ( n , R ) ] d G ,
ψ ( r ) = A 2 i λ G e i k R | R | e i k z | z | [ 1 + cos ( n , z ) ] d G .
ψ ( r ) = A i λ + + e i k R R G ( x 0 , y 0 ) e i k z z d x 0 d y 0 ,
R = ( R 2 + x 0 2 + y 0 2 ) 1 2 ,
ψ ( r ) = A e i k z i λ R + + f ( x 0 , y 0 ) exp { i k 2 z [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } d x 0 d y 0 ,
D : R z R + z = z M ,
ψ ( r ) = C f ( x , y ) exp [ i k 2 z ( x 2 + y 2 ) ] ,
σ proj , i = σ i z R 8 ln 2 ,
I ( r ) = | C f ( x , y ) exp [ i k 2 z ( x 2 + y 2 ) ] | 2 exp [ ( x 2 2 σ proj , H 2 + y 2 2 σ proj , V 2 ) ] .
f ( x , y ) = exp [ i k δ h ( x , y ) ] × exp [ k β h ( x , y ) ] × exp [ i k ( R 2 + x 2 + y 2 ) 1 2 ] ,
h ( x , y ) = { h , for n a x , y < n a + w 0 , else
f H ( x , z ) = 1 n y y I ( x , y , z ) and f V ( y , z ) = 1 n x x I ( x , y , z ) ,
F H ( z ) : = 𝔉 [ f H ( x , z ) ] ( k x ) and F V ( z ) : = 𝔉 [ f V ( y , z ) ] ( k y ) ,
p ( z , E , σ , d c , α ) = i = 1 n z { [ F exp ( z i ) F sim ( z i , E , σ , d c , α ) ] 2 F exp ( z i ) } ,
F exp ( z i ) = F exp ( z i ) ( 1 / n z ) i = 1 n z F exp ( z i ) and F sim ( z i ) = F sim ( z i ) ( 1 / n z ) i = 1 n z F sim ( z i ) ,
N max = 2 n max 4 k max n z ,
p ( z ± Δ z , E , σ , d c , α ) = ! p ( z , E , σ ± Δ σ , d c , α ) .
p ( z ± Δ z , E , σ , d c , α ) = ! p ( z , E ± Δ E , σ , d c , α ) ,
i ( k ± Δ k eff ) h ( δ + i β ) = ! i ( k ± Δ k ) ( h ± Δ h ) [ δ + i β ± ( Δ δ + i Δ β ) ] ,
k ± Δ k eff = [ k ± Δ k ± k Δ h h + Δ k Δ h h ] [ 1 ± Δ δ + i Δ β δ + i β ] ,
Δ δ + i Δ β δ + i β = Δ δ / δ 1 + β 2 / δ 2 + Δ β / β 1 + δ 2 / β 2 + i Δ β / δ 1 + β 2 / δ 2 i Δ δ / β 1 + δ 2 / β 2 .
𝒪 max = 𝒪 ( 10 2 ) 1 + 𝒪 ( 10 3 ) + 𝒪 ( 10 2 ) 1 + 𝒪 ( 10 2 ) + i 𝒪 ( 10 3 ) 1 + 𝒪 ( 10 3 ) i 𝒪 ( 10 1 ) 1 + 𝒪 ( 10 2 ) 𝒪 ( 10 2 ) .
k ± Δ k eff k ± 2 Δ k ± ( k ± 2 Δ k ) 𝒪 ( 10 2 ) ,
| Δ k eff | 3 Δ k Δ E eff = c h ¯ Δ k eff ,
i ( k ± Δ k eff ) 2 z = ! i ( k ± Δ k ) 2 ( z ± Δ z )
| Δ k eff | = | z Δ k k Δ z | z + Δ z .

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