Abstract

We present a new method to extract the intermediate scattering function from series of coherent diffraction patterns taken with 2D detectors. Our approach is based on analyzing speckle patterns in terms of photon statistics. We show that the information obtained is equivalent to the conventional technique of calculating the intensity autocorrelation function. Our approach represents a route for correlation spectroscopy on ultrafast timescales at X-ray free-electron laser sources.

© 2009 Optical Society of America

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References

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  1. J. Goodman, Statistical Optics, (John Wiley & Sons, New York, 1985).
  2. G. Grübel and F. Zontone, "Correlation spectroscopy with coherent X-rays," J. Alloys Compd. 362, 3-11 (2004).
    [CrossRef]
  3. M. Altarelli, Editor, XFEL: Technical Design Report 2006, DESY 2006-097.
  4. G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, "XPCS at the European X-ray free electron laser facility," Nucl. Instrum. Methods. B 262, 357-367 (2007).
    [CrossRef]
  5. P. K. Dixon and D. J. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003).
    [CrossRef] [PubMed]
  6. S. Streit-Nierobisch, C. Gutt, M. Paulus, M. Tolan, "Cooling rate dependence of the glass transition at free surfaces," Phys. Rev. B 77, 041410 (R) (2008).
    [CrossRef]
  7. T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Gr¨ubel, "Setup for in situ surface investigations of the liquid/glass transition with coherent x rays," Rev. Sci. Instrum. 74, 4033-4040 (2003).
    [CrossRef]
  8. D. Lumma, L.B. Lurio, S. G. J. Mochrie, and M. Sutton, "Area detector based photon correlation in the regime of short data batches: Data reduction for dynamic x-ray scattering," Rev. Sci. Instrum. 71, 3274-3289 (2000).
    [CrossRef]

2008 (1)

S. Streit-Nierobisch, C. Gutt, M. Paulus, M. Tolan, "Cooling rate dependence of the glass transition at free surfaces," Phys. Rev. B 77, 041410 (R) (2008).
[CrossRef]

2007 (1)

G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, "XPCS at the European X-ray free electron laser facility," Nucl. Instrum. Methods. B 262, 357-367 (2007).
[CrossRef]

2004 (1)

G. Grübel and F. Zontone, "Correlation spectroscopy with coherent X-rays," J. Alloys Compd. 362, 3-11 (2004).
[CrossRef]

2003 (2)

P. K. Dixon and D. J. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Gr¨ubel, "Setup for in situ surface investigations of the liquid/glass transition with coherent x rays," Rev. Sci. Instrum. 74, 4033-4040 (2003).
[CrossRef]

2000 (1)

D. Lumma, L.B. Lurio, S. G. J. Mochrie, and M. Sutton, "Area detector based photon correlation in the regime of short data batches: Data reduction for dynamic x-ray scattering," Rev. Sci. Instrum. 71, 3274-3289 (2000).
[CrossRef]

Dixon, P. K.

P. K. Dixon and D. J. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

Durian, D. J.

P. K. Dixon and D. J. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

Gr¨ubel, G.

T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Gr¨ubel, "Setup for in situ surface investigations of the liquid/glass transition with coherent x rays," Rev. Sci. Instrum. 74, 4033-4040 (2003).
[CrossRef]

Grübel, G.

G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, "XPCS at the European X-ray free electron laser facility," Nucl. Instrum. Methods. B 262, 357-367 (2007).
[CrossRef]

G. Grübel and F. Zontone, "Correlation spectroscopy with coherent X-rays," J. Alloys Compd. 362, 3-11 (2004).
[CrossRef]

Gutt, C.

S. Streit-Nierobisch, C. Gutt, M. Paulus, M. Tolan, "Cooling rate dependence of the glass transition at free surfaces," Phys. Rev. B 77, 041410 (R) (2008).
[CrossRef]

G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, "XPCS at the European X-ray free electron laser facility," Nucl. Instrum. Methods. B 262, 357-367 (2007).
[CrossRef]

Lumma, D.

D. Lumma, L.B. Lurio, S. G. J. Mochrie, and M. Sutton, "Area detector based photon correlation in the regime of short data batches: Data reduction for dynamic x-ray scattering," Rev. Sci. Instrum. 71, 3274-3289 (2000).
[CrossRef]

Lurio, L.B.

D. Lumma, L.B. Lurio, S. G. J. Mochrie, and M. Sutton, "Area detector based photon correlation in the regime of short data batches: Data reduction for dynamic x-ray scattering," Rev. Sci. Instrum. 71, 3274-3289 (2000).
[CrossRef]

Madsen, A.

T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Gr¨ubel, "Setup for in situ surface investigations of the liquid/glass transition with coherent x rays," Rev. Sci. Instrum. 74, 4033-4040 (2003).
[CrossRef]

Mochrie, S. G. J.

D. Lumma, L.B. Lurio, S. G. J. Mochrie, and M. Sutton, "Area detector based photon correlation in the regime of short data batches: Data reduction for dynamic x-ray scattering," Rev. Sci. Instrum. 71, 3274-3289 (2000).
[CrossRef]

Paulus, M.

S. Streit-Nierobisch, C. Gutt, M. Paulus, M. Tolan, "Cooling rate dependence of the glass transition at free surfaces," Phys. Rev. B 77, 041410 (R) (2008).
[CrossRef]

Seydel, T.

T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Gr¨ubel, "Setup for in situ surface investigations of the liquid/glass transition with coherent x rays," Rev. Sci. Instrum. 74, 4033-4040 (2003).
[CrossRef]

Sinn, H.

G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, "XPCS at the European X-ray free electron laser facility," Nucl. Instrum. Methods. B 262, 357-367 (2007).
[CrossRef]

Sprung, M.

T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Gr¨ubel, "Setup for in situ surface investigations of the liquid/glass transition with coherent x rays," Rev. Sci. Instrum. 74, 4033-4040 (2003).
[CrossRef]

Stephenson, G. B.

G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, "XPCS at the European X-ray free electron laser facility," Nucl. Instrum. Methods. B 262, 357-367 (2007).
[CrossRef]

Streit-Nierobisch, S.

S. Streit-Nierobisch, C. Gutt, M. Paulus, M. Tolan, "Cooling rate dependence of the glass transition at free surfaces," Phys. Rev. B 77, 041410 (R) (2008).
[CrossRef]

Sutton, M.

D. Lumma, L.B. Lurio, S. G. J. Mochrie, and M. Sutton, "Area detector based photon correlation in the regime of short data batches: Data reduction for dynamic x-ray scattering," Rev. Sci. Instrum. 71, 3274-3289 (2000).
[CrossRef]

Tolan, M.

S. Streit-Nierobisch, C. Gutt, M. Paulus, M. Tolan, "Cooling rate dependence of the glass transition at free surfaces," Phys. Rev. B 77, 041410 (R) (2008).
[CrossRef]

T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Gr¨ubel, "Setup for in situ surface investigations of the liquid/glass transition with coherent x rays," Rev. Sci. Instrum. 74, 4033-4040 (2003).
[CrossRef]

Tschentscher, T.

G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, "XPCS at the European X-ray free electron laser facility," Nucl. Instrum. Methods. B 262, 357-367 (2007).
[CrossRef]

Zontone, F.

G. Grübel and F. Zontone, "Correlation spectroscopy with coherent X-rays," J. Alloys Compd. 362, 3-11 (2004).
[CrossRef]

J. Alloys Compd. (1)

G. Grübel and F. Zontone, "Correlation spectroscopy with coherent X-rays," J. Alloys Compd. 362, 3-11 (2004).
[CrossRef]

Nucl. Instrum. Methods. B (1)

G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, T. Tschentscher, "XPCS at the European X-ray free electron laser facility," Nucl. Instrum. Methods. B 262, 357-367 (2007).
[CrossRef]

Phys. Rev. B (1)

S. Streit-Nierobisch, C. Gutt, M. Paulus, M. Tolan, "Cooling rate dependence of the glass transition at free surfaces," Phys. Rev. B 77, 041410 (R) (2008).
[CrossRef]

Phys. Rev. Lett. (1)

P. K. Dixon and D. J. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (2)

T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Gr¨ubel, "Setup for in situ surface investigations of the liquid/glass transition with coherent x rays," Rev. Sci. Instrum. 74, 4033-4040 (2003).
[CrossRef]

D. Lumma, L.B. Lurio, S. G. J. Mochrie, and M. Sutton, "Area detector based photon correlation in the regime of short data batches: Data reduction for dynamic x-ray scattering," Rev. Sci. Instrum. 71, 3274-3289 (2000).
[CrossRef]

Other (2)

J. Goodman, Statistical Optics, (John Wiley & Sons, New York, 1985).

M. Altarelli, Editor, XFEL: Technical Design Report 2006, DESY 2006-097.

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Figures (4)

Fig. 1.
Fig. 1.

Schematics of the split-pulse technique. A delay line unit consisting of mirrors and beam-splitters produces two equal intensity pulses travelling along the same path but delayed in time. Each pulse produces a speckle pattern and the sum is recorded on an area detector and analyzed in terms of speckle contrast.

Fig. 2.
Fig. 2.

Top: Normalized intensity autocorrelation function g 2(τ) - 1 deduced in the sequential mode. Middle: Speckle contrast correlation function c 2(τ) as calculated from the split-pulse technique. Bottom: Both correlation functions normalized to each other.

Fig. 3.
Fig. 3.

Intensity distribution function P(i) for the sum of two speckle patterns with different delay times τ.

Fig. 4.
Fig. 4.

Number of photons observed in the summed speckle patterns as a function of delay time. Curves are shifted and renormalized for reasons of clarity.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

g2(τ)=1+f(τ)2 ,
g2nk=1Kk i=1KkIn (ti)In (ti+k) / Īn2
g2(k)=1+β2f(k)2 .
c2(k)=S2(k)S(k)2S(k)2 .
S2(k)=[I(t)+I(t+k)][I(t)+I(t+k)]=2I2+2I(t)I(t+k) ,
S(k)2=I(t)+I(t+k)2=4I2 .
c2(k)=2I2+2I(t)I(t+k)4I24I2 ,
c2(k)=2σ2(I)+2I(t)I(t+k)2I24I2 .
12σ2(I)I2=12 .
I(t)I(t+k)2I2=12 (1+f(k)2) ,
c2(k)=12 (1+f(k)2) .
c2(k)=β22 (1+f(k)2) + α .
c2(k)=β2(r2+1+2rf(k)2r2+1+2r) ,
P(i)=Γ(i+M)Γ(M)Γ(i+1)(1+Mi)i(1+īM)M

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