Effect of shot noise on X-ray speckle visibility spectroscopy

Ichiro Inoue; Yuya Shinohara; Akira Watanabe; Yoshiyuki Amemiya

doi:10.1364/OE.20.026878

Effect of shot noise on X-ray speckle visibility spectroscopy

Ichiro Inoue, Yuya Shinohara, Akira Watanabe, and Yoshiyuki Amemiya

Optics Express
Vol. 20,
Issue 24,
pp. 26878-26887
(2012)
•https://doi.org/10.1364/OE.20.026878

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Ichiro Inoue, Yuya Shinohara, Akira Watanabe, and Yoshiyuki Amemiya, "Effect of shot noise on X-ray speckle visibility spectroscopy," Opt. Express 20, 26878-26887 (2012)

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Related Topics
Table of Contents Category
- Spectroscopy
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Photon statistics (030.5290)
- Spectroscopy, speckle (300.6480)

About this Article
History
- Original Manuscript: October 10, 2012
- Revised Manuscript: November 6, 2012
- Manuscript Accepted: November 7, 2012
- Published: November 14, 2012
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 7, Iss. 12

Accessible

Open Access

Abstract

X-ray speckle visibility spectroscopy (XSVS) is a method for studying dynamics in disordered systems. This method was originally developed in the visible-light region, in which an intense laser can be used. When applied in the X-ray region, where the number of photons is much smaller than in the visible-light region, it suffers from photon statistics. In this paper, we quantitatively discuss the effect of photon shot noise on XSVS analyses. The effect is experimentally confirmed using sequential speckle patterns from Brownian polystyrene nanospheres in glycerol.

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References

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Year
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Publication

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W. Roseker, H. Franz, H. Schulte-Schrepping, A. Ehnes, O. Leupold, F. Zontone, S. Lee, A. Robert, and G. Grübel, “Development of a hard X-ray delay line for X-ray photon correlation spectroscopy and jitter-free pump-probe experiments at X-ray free-electron laser sources,” J. Synchrotron Radiat. 18(3), 481–491 (2011).
[Crossref] [PubMed]

2010 (3)

Y. Shinohara, R. Imai, H. Kishimoto, N. Yagi, and Y. Amemiya, “Indirectly illuminated X-ray area detector for X-ray photon correlation spectroscopy,” J. Synchrotron Radiat. 17(6), 737–742 (2010).
[Crossref] [PubMed]

S. E. Skipetrov, J. Peuser, R. Cerbino, P. Zakharov, B. Weber, and F. Scheffold, “Noise in laser speckle correlation and imaging techniques,” Opt. Express 18(14), 14519–14534 (2010).
[Crossref] [PubMed]

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[Crossref]

2009 (3)

M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, and G. Vogl, “Atomic diffusion studied with coherent X-rays,” Nat. Mater. 8(9), 717–720 (2009).
[Crossref] [PubMed]

C. Gutt, L.-M. Stadler, A. Duri, T. Autenrieth, O. Leupold, Y. Chushkin, and G. Grübel, “Measuring temporal speckle correlations at ultrafast x-ray sources,” Opt. Express 17(1), 55–61 (2009).
[Crossref] [PubMed]

W. Roseker, H. Franz, H. Schulte-Schrepping, A. Ehnes, O. Leupold, F. Zontone, A. Robert, and G. Grübel, “Performance of a picosecond x-ray delay line unit at 8.39 keV,” Opt. Lett. 34(12), 1768–1770 (2009).
[Crossref] [PubMed]

2008 (2)

C. Caronna, Y. Chushkin, A. Madsen, and A. Cupane, “Dynamics of nanoparticles in a supercooled liquid,” Phys. Rev. Lett. 100(5), 055702 (2008).
[Crossref] [PubMed]

M. Sutton, “A review of X-ray intensity fluctuation spectroscopy,” C. R. Phys. 9(5-6), 657–667 (2008).
[Crossref]

2007 (2)

O. G. Shpyrko, E. D. Isaacs, J. M. Logan, Y. Feng, G. Aeppli, R. Jaramillo, H. C. Kim, T. F. Rosenbaum, P. Zschack, M. Sprung, S. Narayanan, and A. R. Sandy, “Direct measurement of antiferromagnetic domain fluctuations,” Nature 447(7140), 68–71 (2007).
[Crossref] [PubMed]

Y. Shinohara, H. Kishimoto, T. Maejima, H. Nishikawa, N. Yagi, and Y. Amemiya, “X-ray photon correlation spectroscopy of filler in rubber,” Jpn. J. Appl. Phys. 46(4), L300–L302 (2007).
[Crossref]

2005 (2)

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093110 (2005).
[Crossref]

P. Falus, M. A. Borthwick, and S. G. J. Mochrie, “Fluctuation dynamics of block copolymer vesicles,” Phys. Rev. Lett. 94(1), 016105 (2005).
[Crossref] [PubMed]

2004 (1)

G. Grübel and F. Zontone, “Correlation spectroscopy with coherent X-rays,” J. Alloy. Comp. 362(1-2), 3–11 (2004).
[Crossref]

2003 (2)

M. Sutton, K. Laaziri, F. Livet, and F. Bley, “Using coherence to measure two-time correlation functions,” Opt. Express 11(19), 2268–2277 (2003).
[Crossref] [PubMed]

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90(18), 184302 (2003).
[Crossref] [PubMed]

2002 (1)

S. M. Gruner, M. W. Tate, and E. F. Eikenberry, “Charge-coupled device area x-ray detectors,” Rev. Sci. Instrum. 73(8), 2815–2842 (2002).
[Crossref]

2001 (2)

T. Hara, T. Tanaka, T. Seike, T. Bizen, X. Maréchal, T. Kohda, K. Inoue, T. Oka, T. Suzuki, N. Yagi, and H. Kitamura, “In-vacuum X-ray helical undulator for high flux beamline at SPring-8,” Nucl. Instrum. Methods Phys. Res. Sect. A 467–468, 165–168 (2001).
[Crossref]

D. Lumma, M. A. Borthwick, P. Falus, L. B. Lurio, and S. G. J. Mochrie, “Equilibrium dynamics in the nondiffusive regime of an entangled polymer blend,” Phys. Rev. Lett. 86(10), 2042–2045 (2001).
[Crossref] [PubMed]

2000 (1)

L. B. Lurio, D. Lumma, A. R. Sandy, M. A. Borthwick, P. Falus, S. G. J. Mochrie, J. F. Pelletier, M. Sutton, L. Regan, A. Malik, and G. B. Stephenson, “Absence of scaling for the intermediate scattering function of a hard-sphere suspension: Static and dynamic x-ray scattering from concentrated polystyrene latex spheres,” Phys. Rev. Lett. 84(4), 785–788 (2000).
[Crossref] [PubMed]

1997 (1)

S. G. J. Mochrie, A. M. Mayes, A. R. Sandy, M. Sutton, S. Brauer, G. B. Stephenson, D. L. Abernathy, and G. Grübel, “Dynamics of block copolymer micelles revealed by x-ray intensity fluctuation spectroscopy,” Phys. Rev. Lett. 78(7), 1275–1278 (1997).
[Crossref]

1996 (1)

T. Thurn-Albrecht, W. Steffen, A. Patkowski, G. Meier, E. W. Fischer, G. Grübel, and D. L. Abernathy, “Photon correlation spectroscopy of colloidal palladium using a coherent x-ray beam,” Phys. Rev. Lett. 77(27), 5437–5440 (1996).
[Crossref] [PubMed]

1995 (3)

S. B. Dierker, R. Pindak, R. M. Fleming, I. K. Robinson, and L. Berman, “X-Ray photon correlation spectroscopy study of Brownian motion of gold colloids in glycerol,” Phys. Rev. Lett. 75(3), 449–452 (1995).
[Crossref] [PubMed]

S. Brauer, G. B. Stephenson, M. Sutton, R. Brüning, E. Dufresne, S. G. J. Mochrie, G. Grübel, J. Als-Nielsen, and D. L. Abernathy, “X-ray intensity fluctuation spectroscopy observations of critical dynamics in Fe3Al,” Phys. Rev. Lett. 74(11), 2010–2013 (1995).
[Crossref] [PubMed]

E. Dufresne, R. Brüning, M. Sutton, B. Rodricks, and G. B. Stephenson, “A statistical technique for characterizing X-ray position sensitive detectors,” Nucl. Instrum. Methods Phys. Res. Sect. A 364(2), 380–393 (1995).
[Crossref]

1991 (1)

M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, and G. B. Stephenson, “Observation of speckle by diffraction with coherent X-rays,” Nature 352(6336), 608–610 (1991).
[Crossref]

Abernathy, D. L.

Aeppli, G.

Als-Nielsen, J.

Amemiya, Y.

Autenrieth, T.

Bandyopadhyay, R.

Berman, L.

Berman, L. E.

Bizen, T.

Bley, F.

M. Sutton, K. Laaziri, F. Livet, and F. Bley, “Using coherence to measure two-time correlation functions,” Opt. Express 11(19), 2268–2277 (2003).
[Crossref] [PubMed]

Borthwick, M. A.

P. Falus, M. A. Borthwick, and S. G. J. Mochrie, “Fluctuation dynamics of block copolymer vesicles,” Phys. Rev. Lett. 94(1), 016105 (2005).
[Crossref] [PubMed]

Brauer, S.

Brüning, R.

Caronna, C.

C. Caronna, Y. Chushkin, A. Madsen, and A. Cupane, “Dynamics of nanoparticles in a supercooled liquid,” Phys. Rev. Lett. 100(5), 055702 (2008).
[Crossref] [PubMed]

Cerbino, R.

Chushkin, Y.

C. Caronna, Y. Chushkin, A. Madsen, and A. Cupane, “Dynamics of nanoparticles in a supercooled liquid,” Phys. Rev. Lett. 100(5), 055702 (2008).
[Crossref] [PubMed]

Cupane, A.

C. Caronna, Y. Chushkin, A. Madsen, and A. Cupane, “Dynamics of nanoparticles in a supercooled liquid,” Phys. Rev. Lett. 100(5), 055702 (2008).
[Crossref] [PubMed]

Dierker, S. B.

Dixon, P. K.

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90(18), 184302 (2003).
[Crossref] [PubMed]

Dufresne, E.

Duri, A.

Durian, D. J.

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90(18), 184302 (2003).
[Crossref] [PubMed]

Ehnes, A.

Eikenberry, E. F.

S. M. Gruner, M. W. Tate, and E. F. Eikenberry, “Charge-coupled device area x-ray detectors,” Rev. Sci. Instrum. 73(8), 2815–2842 (2002).
[Crossref]

Falus, P.

P. Falus, M. A. Borthwick, and S. G. J. Mochrie, “Fluctuation dynamics of block copolymer vesicles,” Phys. Rev. Lett. 94(1), 016105 (2005).
[Crossref] [PubMed]

Feng, Y.

Fischer, E. W.

Fleming, R. M.

Franz, H.

Gittings, A. S.

Greytak, T.

Grübel, G.

G. Grübel and F. Zontone, “Correlation spectroscopy with coherent X-rays,” J. Alloy. Comp. 362(1-2), 3–11 (2004).
[Crossref]

Gruner, S. M.

S. M. Gruner, M. W. Tate, and E. F. Eikenberry, “Charge-coupled device area x-ray detectors,” Rev. Sci. Instrum. 73(8), 2815–2842 (2002).
[Crossref]

Gutt, C.

Hara, T.

Held, G. A.

Imai, R.

Inoue, K.

Isaacs, E. D.

Jaramillo, R.

Kim, H. C.

Kishimoto, H.

Kitamura, H.

Kohda, T.

Laaziri, K.

M. Sutton, K. Laaziri, F. Livet, and F. Bley, “Using coherence to measure two-time correlation functions,” Opt. Express 11(19), 2268–2277 (2003).
[Crossref] [PubMed]

Lee, S.

Leitner, M.

M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, and G. Vogl, “Atomic diffusion studied with coherent X-rays,” Nat. Mater. 8(9), 717–720 (2009).
[Crossref] [PubMed]

Leupold, O.

Livet, F.

M. Sutton, K. Laaziri, F. Livet, and F. Bley, “Using coherence to measure two-time correlation functions,” Opt. Express 11(19), 2268–2277 (2003).
[Crossref] [PubMed]

Logan, J. M.

Lumma, D.

Lurio, L. B.

Madsen, A.

C. Caronna, Y. Chushkin, A. Madsen, and A. Cupane, “Dynamics of nanoparticles in a supercooled liquid,” Phys. Rev. Lett. 100(5), 055702 (2008).
[Crossref] [PubMed]

Maejima, T.

Malik, A.

Maréchal, X.

Mayes, A. M.

Meier, G.

Mochrie, S. G. J.

P. Falus, M. A. Borthwick, and S. G. J. Mochrie, “Fluctuation dynamics of block copolymer vesicles,” Phys. Rev. Lett. 94(1), 016105 (2005).
[Crossref] [PubMed]

Nagler, S. E.

Narayanan, S.

Nishikawa, H.

Nugent, K. A.

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[Crossref]

Oka, T.

Patkowski, A.

Pelletier, J. F.

Peuser, J.

Pfau, B.

M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, and G. Vogl, “Atomic diffusion studied with coherent X-rays,” Nat. Mater. 8(9), 717–720 (2009).
[Crossref] [PubMed]

Pindak, R.

Regan, L.

Robert, A.

Robinson, I. K.

Rodricks, B.

Roseker, W.

Rosenbaum, T. F.

Sandy, A. R.

Scheffold, F.

Schulte-Schrepping, H.

Seike, T.

Sepiol, B.

M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, and G. Vogl, “Atomic diffusion studied with coherent X-rays,” Nat. Mater. 8(9), 717–720 (2009).
[Crossref] [PubMed]

Shinohara, Y.

Shpyrko, O. G.

Skipetrov, S. E.

Sprung, M.

Stadler, L.-M.

M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, and G. Vogl, “Atomic diffusion studied with coherent X-rays,” Nat. Mater. 8(9), 717–720 (2009).
[Crossref] [PubMed]

Steffen, W.

Stephenson, G. B.

Suh, S. S.

Sutton, M.

M. Sutton, “A review of X-ray intensity fluctuation spectroscopy,” C. R. Phys. 9(5-6), 657–667 (2008).
[Crossref]

M. Sutton, K. Laaziri, F. Livet, and F. Bley, “Using coherence to measure two-time correlation functions,” Opt. Express 11(19), 2268–2277 (2003).
[Crossref] [PubMed]

Suzuki, T.

Tanaka, T.

Tate, M. W.

S. M. Gruner, M. W. Tate, and E. F. Eikenberry, “Charge-coupled device area x-ray detectors,” Rev. Sci. Instrum. 73(8), 2815–2842 (2002).
[Crossref]

Thurn-Albrecht, T.

Vogl, G.

M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, and G. Vogl, “Atomic diffusion studied with coherent X-rays,” Nat. Mater. 8(9), 717–720 (2009).
[Crossref] [PubMed]

Weber, B.

Yagi, N.

Zakharov, P.

Zontone, F.

G. Grübel and F. Zontone, “Correlation spectroscopy with coherent X-rays,” J. Alloy. Comp. 362(1-2), 3–11 (2004).
[Crossref]

Zschack, P.

Adv. Phys. (1)

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[Crossref]

C. R. Phys. (1)

M. Sutton, “A review of X-ray intensity fluctuation spectroscopy,” C. R. Phys. 9(5-6), 657–667 (2008).
[Crossref]

J. Alloy. Comp. (1)

G. Grübel and F. Zontone, “Correlation spectroscopy with coherent X-rays,” J. Alloy. Comp. 362(1-2), 3–11 (2004).
[Crossref]

J. Synchrotron Radiat. (2)

Jpn. J. Appl. Phys. (1)

Nat. Mater. (1)

M. Leitner, B. Sepiol, L.-M. Stadler, B. Pfau, and G. Vogl, “Atomic diffusion studied with coherent X-rays,” Nat. Mater. 8(9), 717–720 (2009).
[Crossref] [PubMed]

Nature (2)

Nucl. Instrum. Methods Phys. Res. Sect. A (2)

Opt. Express (3)

M. Sutton, K. Laaziri, F. Livet, and F. Bley, “Using coherence to measure two-time correlation functions,” Opt. Express 11(19), 2268–2277 (2003).
[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Rev. Lett. (9)

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90(18), 184302 (2003).
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Fig. 1

Dependence of visibility on exposure time based on Eq. (16). Four cases of a Brownian motion system are calculated: $β 〈 N (q, T) 〉 =$ 0.1, 1, 10, and ∞ when $ε$ = 1. The vertical and horizontal axes represent visibility divided by β and exposure time normalized by τ, respectively. For clarity, the plots are moderately shifted.

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Fig. 2

Typical scattering images for $T$ = (a) 36 ms ( $Δ t$ ), (b) 180 ms ( $5 Δ t$ ), and (c) 900 ms ( $25 Δ t$ ). The lower-right part is covered by a beamstop.

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Fig. 3

(a) Relaxation times determined by XSVS (red circles) and XPCS (blue diamonds). (b) Dependence of the visibility of speckle patterns at $q$ = 0.0335 nm⁻¹ on exposure time (red circles) and its fitted result using Eq. (18) (solid line) and that for the case where the effect of shot noise is neglected (dotted line). (c) Average number of counted photons per frame per pixel normalized by $ε$ determined by fitting Eq. (18) to experimental data (red circles) and circularly averaged intensity of speckle patterns (black line).

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Equations (18)

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g_{2} (q, t) = \frac{〈 I (q, t') I (q, t + t') 〉}{{〈 I (q, t') 〉}^{2}},

g_{2} (q, t = k' Δ t) = \frac{\frac{1}{N (K - k')} \sum_{n = 1}^{N} \sum_{k = 1}^{K - k'} S_{n, k} S_{n, k + k'}}{(\frac{1}{N (K - k')} \sum_{n = 1}^{N} \sum_{k = 1}^{K - k'} S_{n, k}) (\frac{1}{N k'} \sum_{n = 1}^{N} \sum_{k = K - k' + 1}^{K} S_{n, k})},

g_{2} (q, t) = 1 + β {| f (q, t) |}^{2},

v (q, T) = \frac{〈 S^{2} (q, t) 〉 - {〈 S (q, t) 〉}^{2}}{{〈 S (q, t) 〉}^{2}},

v (q, T) = \frac{\frac{1}{N K - 1} \sum_{n = 1}^{N} {\sum_{k = 1}^{K} (S_{n, k} - \frac{1}{N K} \sum_{n = 1}^{N} \sum_{k = 1}^{K} S_{n, k})}^{2}}{{(\frac{1}{N K} \sum_{n = 1}^{N} \sum_{k = 1}^{K} S_{n, k})}^{2}} .

v (q, t) = 2 β / T \int_{0}^{T} (1 - t / T) {| f (q, t) |}^{2} d t .

〈 S (q, T) 〉 = C {〈 {〈 \int_{0}^{T} n_{m} (q, t) d t 〉}_{p} 〉}_{c} = = C {〈 {〈 \int_{0}^{T} n (q, t) d t 〉}_{p} 〉}_{c} = C 〈 \int_{0}^{T} n (q, t) d t 〉,

\begin{matrix} 〈 S^{2} (q, T) 〉 = C^{2} {〈 {〈 {(\int_{0}^{T} n_{m} (q, t) d t)}^{2} 〉}_{p} 〉}_{c} = C^{2} {〈 {〈 {(\int_{0}^{T} n (q, t) d t)}^{2} + \int_{0}^{T} n (q, t) d t 〉}_{p} 〉}_{c} \\ = C^{2} 〈 {(\int_{0}^{T} n (q, t) d t)}^{2} 〉 + C^{2} 〈 \int_{0}^{T} n (q, t) d t 〉, \end{matrix}

〈 S (q, T) 〉 = C 〈 \int_{0}^{T} n_{e} (q, t) d t 〉 = C 〈 \int_{0}^{T} n (q, t) d t 〉,

〈 S^{2} (q, T) 〉 = C^{2} 〈 {(\int_{0}^{T} n_{e} (q, t) d t)}^{2} 〉 = C^{2} 〈 {(\int_{0}^{T} n (q, t) d t)}^{2} 〉 + C^{2} ε 〈 \int_{0}^{T} n (q, t) d t 〉,

v (q, T) = v_{0} (q, T) + \frac{ε}{〈 N (q, T) 〉},

v_{0} (q, t) = \frac{〈 {(\int_{0}^{T} n (q, t) d t)}^{2} 〉 - {〈 \int_{0}^{T} n (q, t) d t 〉}^{2}}{{〈 \int_{0}^{T} n (q, t) d t 〉}^{2}},

〈 N (q, T) 〉 = 〈 \int_{0}^{T} n_{e} (q, t) d t 〉 = 〈 \int_{0}^{T} n (q, t) d t 〉 .

v (q, T) = 2 β / T \int_{0}^{T} (1 - t / T) {| f (q, t) |}^{2} d t + \frac{ε}{〈 N (q, T) 〉} .

v (q, T) = 2 β / T \int_{0}^{T} (1 - t / T) {| f (q, t) |}^{2} d t + \frac{ε}{〈 N (q, T) 〉} + \frac{〈 σ^{2} (T) 〉}{{〈 \int_{0}^{T} I (q, t) d t 〉}^{2}},

v (q, T) = β \frac{\exp (- 2 T / τ) - 1 + 2 T / τ}{2 {(T / τ)}^{2}} + \frac{ε}{〈 N (q, T) 〉} .

g_{2} (q, t) = 1 + β \exp (- 2 t / τ),

v (q, T = k Δ t) = β \frac{\exp (- 2 k Δ t / τ) - 1 + 2 k Δ t / τ}{2 {(k Δ t / τ)}^{2}} + \frac{ε}{k 〈 N (q, Δ t) 〉} .