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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 12 — Dec. 19, 2012
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Effect of shot noise on X-ray speckle visibility spectroscopy

Ichiro Inoue, Yuya Shinohara, Akira Watanabe, and Yoshiyuki Amemiya  »View Author Affiliations


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


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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.

© 2012 OSA

1. Introduction

Coherent X-rays are attracting much attention in various scientific fields [1

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

]. When coherent X-rays impinge upon a disordered system, a grainy scattering pattern called speckle is observed [2

2. 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]

]. Such a speckle pattern reflects the exact configuration of the scatterers inside the system, whereas scattering patterns obtained by conventional scattering methods reflect the spatially averaged configuration of the scatterers. When the system evolves with time, the corresponding speckle pattern also changes. Temporal changes in the speckle patterns therefore provide information on the system dynamics. This technique is called X-ray photon correlation spectroscopy (XPCS) [3

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

, 4

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

]. In XPCS, time correlation of the X-ray intensities in speckle patterns is analyzed. XPCS has been applied to various disordered systems, such as colloidal suspensions [5

5. 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]

7

7. 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]

], homopolymer blends [8

8. 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]

], block copolymer micelles and vesicles [9

9. 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]

, 10

10. 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]

], nanoparticles in supercooled liquids [11

11. 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]

], alloys [12

12. 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]

, 13

13. 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]

], and antiferromagnetic materials [14

14. 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]

].

However, the application of XPCS is limited in two ways. First, its time resolution is limited by the frame rate of the detector. A two-dimensional (2D) detector is commonly used in XPCS, so the time resolution of XPCS experiments is at present limited to the order of milliseconds. Although the frame rates of zero- and one-dimensional detectors surpass those of 2D detectors, the use of a 2D detector is preferred because averaging over pixels significantly reduces the experimental time and is much more suitable for measuring transient phenomena [15

15. 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]

]. Secondly, the long exposure to X-rays often gives rise to radiation damage of the sample, which affects its dynamics. In XPCS, fixed positions on the sample are irradiated and the sample is easily damaged.

To the best of our knowledge, this is the first report of XSVS. In this paper, we quantitatively describe the effect of photon shot noise on the relationship between visibility and dynamics. Then, the importance of the effect is shown for the measurement of Brownian nanospheres.

2. Methodology

2.1 X-ray photon correlation spectroscopy

In XPCS, a normalized intensity correlation function g2(q,t) is measured; it is defined as
g2(q,t)=I(q,t')I(q,t+t')I(q,t')2,
(1)
where I(q,t) is the scattering intensity at scattering vector q at time t, and the brackets represent an ensemble-average. The ensemble-average is substituted by averaging the intensity values over pixels and times as follows: assuming that K sequential speckle patterns from a disordered sample with a frame rate fr are recorded by an area detector with N pixels located at scattering vector q, g2(q,t) is calculated as
g2(q,t=k'Δt)=1N(Kk')n=1Nk=1Kk'Sn,kSn,k+k'(1N(Kk')n=1Nk=1Kk'Sn,k)(1Nk'n=1Nk=Kk'+1KSn,k),
(2)
where Δt=1/fr and Sn,k is the value of the scattering intensity at the n-th pixel of the k-th image after subtraction of the average dark noise of the detector.

2.2 X-ray speckle visibility spectroscopy

In XSVS, the visibility of speckle patterns is calculated; this is defined as
v(q,T)=S2(q,t)S(q,t)2S(q,t)2,
(4)
where S(q,T) is the scattering intensity accumulated for exposure time T (S(q,T)=0TI(q,t)dt). In the same way as in XPCS, the ensemble average of Eq. (4) is substituted by the averages over pixels and time: assuming that the exposure time of the sequential images is T, the visibility of the speckle pattern v(q,T) is calculated as

v(q,T)=1NK1n=1Nk=1K(Sn,k1NKn=1Nk=1KSn,k)2(1NKn=1Nk=1KSn,k)2.
(5)

Now we show how to include the effect of shot noise in Eq. (6). First, let us assume that (i) the dark noise of the detector is much smaller than the scattering intensity and (ii) the size of an electron cloud on the detector produced by an incoming photon is small enough for the charge not to be shared between different pixels. Then, I(q,t)dt is proportional to the counted number of photons during a time dt, nm(q,t)dt. Even if the configuration of the scatterers temporally does not change, nm(q,t)dt is not always the same because of the shot noise. The configuration only determines the expected value of nm(q,t)dt, n(q,t)dt. nm(q,t)dt follows the Poisson distribution when the configuration of the scatterers temporally does not change. Ensemble average in Eq. (5) corresponds to average over all configurations of the scatterers and all cases of the photon statistics for each configuration. Let us define cand p as averages over configurations and photon statistics for each configuration, respectively. Then, the following relations hold when the relation of =pcis taken into account:
S(q,T)=C0Tnm(q,t)dtpc==C0Tn(q,t)dtpc=C0Tn(q,t)dt,
(7)
S2(q,T)=C2(0Tnm(q,t)dt)2pc=C2(0Tn(q,t)dt)2+0Tn(q,t)dtpc=C2(0Tn(q,t)dt)2+C20Tn(q,t)dt,
(8)
where C is a scale factor between nm(q,t)dt and I(q,t)dt.

Next, let us consider the case when the electron clouds are shared with different pixels. In this case, I(q,t)dt is proportional to effective counted number of photons during a time dt, , which is defined based on the ratio of the accumulated charge in a pixel to the total charge on the detector produced by a single X-ray photon. When the configuration of the scatterers temporally does not change, ne(q,t)dtdoes not follow the Poisson distribution. For example, in the case of a spatially uniform exposure, the deviation of is smaller than by a factor of ε (0 ≤ ε ≤ 1), while ensemble-average of equals to [21

21. 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]

]. The factor ε is the constant determined by the spread of an electron cloud on the detector and equals to unity when the cloud is small enough for the charge not to be shared with different pixels [21

21. 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]

]. Considering the sharing effect of electron clouds, Eq. (7) and Eq. (8) are rewritten as
S(q,T)=C0Tne(q,t)dt=C0Tn(q,t)dt,
(9)
S2(q,T)=C2(0Tne(q,t)dt)2=C2(0Tn(q,t)dt)2+C2ε0Tn(q,t)dt,
(10)
where C is a scale factor between ne(q,t)dt and I(q,t)dt.

From Eq. (9) and Eq. (10), the visibility defined by Eq. (4) can be expressed as
v(q,T)=v0(q,T)+εN(q,T),
(11)
where
v0(q,t)=(0Tn(q,t)dt)20Tn(q,t)dt20Tn(q,t)dt2,
(12)
and N(q,T) is the ensemble-averaged number of counted photons during T, given by
N(q,T)=0Tne(q,t)dt=0Tn(q,t)dt.
(13)
v0(q,T) corresponds to the visibility for the case when the shot noise is negligible (N(q,T)=), and thus the relation v0(q,T)=2β/T0T(1t/T)|f(q,t)|2dt holds. Based on the above discussion, the effect of the shot noise is included as
v(q,T)=2β/T0T(1t/T)|f(q,t)|2dt+εN(q,T).
(14)
Taking into account the dark noise of the detector, the relationship between the visibility of the speckle pattern after subtracting the average dark noise and f(q,t) is given by
v(q,T)=2β/T0T(1t/T)|f(q,t)|2dt+εN(q,T)+σ2(T)0TI(q,t)dt2,
(15)
where σ(T) is the standard deviation of the dark images with an exposure time T.

When the intensity of the incident X-ray beams is constant, 1/N(q,T) is proportional to 1/T; 2β/T0T(1t/T)|f(q,t)|2dt is also proportional to 1/T at a long exposure time [17

17. 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]

], so v(q,T) is proportional to 1/T at large values of T, regardless of the forms of f(q,t), if the standard deviation of the dark noise is negligible. For a Brownian motion system with a diffusion coefficient D, f(q,t)=exp(t/τ), with a relaxation time τ=1/(Dq2), and the visibility is given by
v(q,T)=βexp(2T/τ)1+2T/τ2(T/τ)2+εN(q,T).
(16)
Figure 1
Fig. 1 Dependence of visibility on exposure time based on Eq. (16). Four cases of a Brownian motion system are calculated: βN(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.
shows the dependence of the visibility on the exposure time when ε = 1, calculated using Eq. (16), for a Brownian system: four cases, i.e., βN(q,T)=0.1, 1, 10, and ∞, are compared. When T is comparable to or shorter than τ, the dependence of v(q,T) on T is greatly affected by the shot noise. The effect of the shot noise deserves careful attention, to obtain information on the dynamics of the system in XSVS experiments.

3. Materials and methods

4. Results and discussion

In XPCS analysis, the normalized intensity correlation function g2(q,t) is calculated using Eq. (2). The calculated g2(q,t) is well fitted by
g2(q,t)=1+βexp(2t/τ),
(17)
where τis the relaxation time. The relaxation time at each q, determined by XPCS [Fig. 3(a)] shows a clear scaling of τq2. This indicates that the nanospheres follow Brownian motion.

XSVS analysis is performed as follows. To calculate the visibility of the patterns for exposure time T=kΔt (k=1,2,3,55), the scattering images are divided into units of k successive images. The k images in each unit are accumulated to produce a single scattering image of T=kΔt, as shown in Fig. 2
Fig. 2 Typical scattering images for T = (a) 36 ms (Δt), (b) 180 ms (5Δt), and (c) 900 ms (25Δt). The lower-right part is covered by a beamstop.
, and a series of scattering images is produced. Then, the visibility of the images for the exposure time kΔt is calculated according to Eq. (5). More than 3600 pixels are used for the calculation of each v(q,T). Recently, it was pointed out that the number of pixels and the relation between the speckle size and the pixel size lead to differences between the variances of the expected values of the pixel-averaged and the ensemble-averaged intensities, whereas the expected value of the intensity averaged over pixels is the same as that of the ensemble-averaged intensity [26

26. 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]

]. The difference in the intensity variance is, however, negligible in our XSVS experiments because the relative difference scales as the inverse of the total pixels used for the calculation. The calculated visibility is therefore considered to be a good approximation to the real visibility defined by Eq. (4). The calculated visibility at each q is fitted by a model function that includes the effect of the shot noise for a Brownian motion system:
v(q,T=kΔt)=βexp(2kΔt/τ)1+2kΔt/τ2(kΔt/τ)2+εkN(q,Δt).
(18)
Figure 3(b)
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).
is a typical fitting result for q = 0.0335 nm−1. The fitting result for the case when the effect of shot noise is not considered ( in Eq. (18) is infinity) is also shown for comparison. The fitted curve well describes the calculated visibility and this result is favorable compared with the data analysis, which does not include the effect of the shot noise. Relaxation time determined by the Eq. (18) is 198.7 s ± 19.5 s and well agrees that by XPCS (206.6 s ± 21.7 s). On the contrary, determined by XSVS without consideration of the effect of the shot noise is 38.2 s ± 1.5 s and this disagreement with the result of XPCS means that the effect of the shot noise deserves careful attention in XSVS analysis. The results at other q values are also fitted by Eq. (18).

The relaxation times determined by XPCS and XSVS are shown in Fig. 3(b). The good agreements of the relaxation times determined by both methods indicate the validity of Eq. (14). Figure 3(c) shows the circularly averaged intensity of the speckle patterns and N(q,Δt)/ε at each q determined by XSVS analysis. The fact that N(q,Δt)/ε is proportional to the circularly averaged intensity also supports the validity of Eq. (14). In order to obtain the value of ε, it is required to measure the spread of the electron cloud on the detector produced by a single counted photon. However, this is out of the scope of the present work.

As described above, the time resolution of XSVS is not limited by the frame rate of detector. This makes it possible to study microsecond/nanosecond dynamics which is hard to be studied using conventional XPCS, although the present study has demonstrated XSVS in millisecond dynamics using the technique summing successive images. Recently, a split-pulse technique using an X-ray free-electron laser was reported and is also expected to improve the time resolution of XPCS [27

27. 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]

]. The timescale over which the split-pulse technique can perform measurements is determined by the minimum and maximum delay times of the split pulses. Since the path difference of the pulses determines the delay time [28

28. 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]

, 29

29. 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]

], the split-pulse technique covers dynamics on a timescale shorter than sub-microseconds. XSVS is expected to measure nano- and micro-second dynamics by controlling the exposure time. Complementary use of XSVS and the split-pulse technique therefore gives us further opportunities for understanding hierarchical dynamics in various disordered systems. Moreover, XSVS can reduce the degree of radiation damage by changing the irradiation position on the sample for every exposure. The effect of the damage will become severe in next-generation synchrotron X-ray facilities and XSVS will be a powerful method in the future.

4. Conclusion

In this paper, we quantitatively describe the effect of shot noise on XSVS. It is shown that the shot noise significantly affects the visibility of speckle patterns. The effect of shot noise on visibility is properly included in the relationship between visibility and the corresponding intermediate scattering function. XSVS is successfully used to measure the relaxation times of polystyrene nanospheres in glycerol. The effect of shot noise is important in XSVS since the number of X-ray photons available is limited. The present study shows that the effect is properly accounted for in analyses, even when the number of photons is small. By combining XSVS and future X-ray sources, it will be possible to fill the gap between XPCS and quasi-elastic scattering techniques without radiation damage to samples.

Acknowledgments

The authors thank Drs. N. Yagi and N. Ohta (JASRI) and Mr. H. Kishimoto (Sumitomo Rubber Industry Ltd.) for their support in performing the experiments at SPring-8. The authors also thank the anonymous reviewers for their valuable comments. The XSVS experiments were performed under the approval of JASRI (Proposal Number: 2011A1112, 2011B1131). This study has been partly supported by JSPS KAKENHI (Grant Number 24710092).

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2.

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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]

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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]

26.

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]

27.

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]

28.

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]

29.

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]

OCIS Codes
(030.5290) Coherence and statistical optics : Photon statistics
(300.6480) Spectroscopy : Spectroscopy, speckle

ToC Category:
Spectroscopy

History
Original Manuscript: October 10, 2012
Revised Manuscript: November 6, 2012
Manuscript Accepted: November 7, 2012
Published: November 14, 2012

Virtual Issues
Vol. 7, Iss. 12 Virtual Journal for Biomedical Optics

Citation
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)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-24-26878


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  29. 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]

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