Effect of shot noise on X-ray speckle visibility spectroscopy

Ichiro Inoue; Yuya Shinohara; Akira Watanabe; Yoshiyuki Amemiya

doi:10.1364/OE.20.026878

1. Introduction

Coherent X-rays are attracting much attention in various scientific fields [1]. When coherent X-rays impinge upon a disordered system, a grainy scattering pattern called speckle is observed [2]. 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, 4]. 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 [5–7], homopolymer blends [8], block copolymer micelles and vesicles [9, 10], nanoparticles in supercooled liquids [11], alloys [12, 13], and antiferromagnetic materials [14].

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]. 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.

X-ray speckle visibility spectroscopy (XSVS) can overcome these problems. Speckle visibility spectroscopy (SVS) [16, 17] was originally developed in the visible-light region. In SVS or XSVS, the visibility of a speckle pattern is calculated and its dependence on exposure time is analyzed. When the system is static within the exposure time, the speckle patterns are stable. In contrast, when the scatterers move during exposure, the speckle pattern is blurred and the visibility decreases accordingly. Consequently, the dependence of the visibility on the exposure time is closely related to the dynamics of the system. The time resolution of XSVS is determined by the minimum exposure time and is not limited by the frame rate of the detector. XSVS can therefore achieve higher time resolution than XPCS can, even using a 2D detector. Moreover, it is possible to change the irradiation position on the sample for every exposure; this reduces the degree of radiation damage. In this regard, XSVS is expected to be a powerful tool in next-generation synchrotron X-ray facilities, in which much more intense X-rays can be used [18, 19]; this will open up a way of measuring microsecond/nanosecond dynamics, which is hard to achieve with current X-ray/neutron techniques. Compared to conventional SVS, XSVS enables us to study nanoscale structural fluctuation in virtue of a short wavelength of X-rays. Opaque specimens, which often show complex and interesting dynamics, can be investigated with XSVS due to the high penetration power of X-rays.

Previous studies have discussed the relationship between visibility and dynamics [16, 17]. This relationship, however, cannot be directly applied to XSVS analysis; XSVS is different from SVS in that the number of scattered photons is much smaller because of the small available flux of the incident X-ray beam. This will further decrease when XSVS is used for measuring fast dynamics. The effect of shot noise therefore needs to be properly included.

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 $g_{2} (q, t)$ is measured; it is defined as

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

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

f_{r}

are recorded by an area detector with

N

pixels located at scattering vector

q

,

g_{2} (q, t)

is calculated as

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})},

where

Δ t = 1 / f_{r}

and

S_{n, 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.

The function $g_{2} (q, t)$ is related to the intermediate scattering function $f (q, t)$ via the Siegert relation [20]:

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

where

β

is a coherence factor.

2.2 X-ray speckle visibility spectroscopy

In XSVS, the visibility of speckle patterns is calculated; this is defined as

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

where

S (q, T)

is the scattering intensity accumulated for exposure time

T

(

S (q, T) = \int_{0}^{T} I (q, t) d t

). 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) = \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}} .

It has been shown [16] that $v (q, t)$ is related to $f (q, t)$ by

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

Equation (6), however, only holds for the ideal condition, where the number of scattered photons is so large that the effect of shot noise is negligible. Let us consider the situation where the exposure time is much longer than the relaxation time of the system, for example. The speckle images from the system change with time and form various patterns on the detector during exposure to the X-ray beam. If the number of detected photons per pixel is large enough for the shot noise and the dark noise of the detector to be negligible, every

S_{n, k}

is the same, and the visibility calculated by Eq. (4) becomes zero. Conversely, if the number of counted photons per pixel is small,

S_{n, k}

has a nonzero variance because of the shot noise, thus the visibility is larger than zero. Since

f (q, t)

approaches zero as

t

increases, the right-hand side of Eq. (6) becomes zero and therefore Eq. (6) does not hold. In SVS, the effect of the shot noise is negligible because of the high flux from laser sources, and Eq. (6) can simply be used. In XSVS, however, the effect cannot be ignored and a proper treatment of photon shot noise is required.

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, $I (q, t) d t$ is proportional to the counted number of photons during a time $d t$ , $n_{m} (q, t) d t$ . Even if the configuration of the scatterers temporally does not change, $n_{m} (q, t) d t$ is not always the same because of the shot noise. The configuration only determines the expected value of $n_{m} (q, t) d t$ , $n (q, t) d t$ . $n_{m} (q, t) d t$ 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 ${〈〉}_{c}$ and ${〈〉}_{p}$ as averages over configurations and photon statistics for each configuration, respectively. Then, the following relations hold when the relation of $〈〉 = {〈 {〈〉}_{p} 〉}_{c}$ is taken into account:

〈 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}

where

C

is a scale factor between

n_{m} (q, t) d t

and

I (q, t) d t

.

Next, let us consider the case when the electron clouds are shared with different pixels. In this case, $I (q, t) d t$ is proportional to effective counted number of photons during a time $d t$ , , 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, $n_{e} (q, t) d t$ does 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]. 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]. Considering the sharing effect of electron clouds, Eq. (7) and Eq. (8) are rewritten as

〈 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 〉,

where

C

is a scale factor between

n_{e} (q, t) d t

and

I (q, t) d t

.

From Eq. (9) and Eq. (10), the visibility defined by Eq. (4) can be expressed as

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

where

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}},

and

〈 N (q, T) 〉

is the ensemble-averaged number of counted photons during

T

, given by

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

v_{0} (q, T)

corresponds to the visibility for the case when the shot noise is negligible (

〈 N (q, T) 〉 = \infty

), and thus the relation

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

holds. Based on the above discussion, the effect of the shot noise is included as

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

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) = 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}},

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 $1 / T$ ; $2 β / T \int_{0}^{T} (1 - t / T) {| f (q, t) |}^{2} d t$ is also proportional to $1 / T$ at a long exposure time [17], 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 $τ = 1 / (D q^{2})$ , and the visibility is given by

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

Figure 1 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.,

β 〈 N (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.

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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3. Materials and methods

Polystyrene latex nanospheres (Alfa Aesar, USA) in glycerol were used as the sample. The sphere diameter was 100 nm and the spheres were used without further purification. The latex was mixed with glycerol and kept in a vacuum desiccator for 205 h to prepare a colloidal suspension with a volume fraction of 5%. A sample cell of thickness 1 mm covered with Mylar film was filled with the suspension. XPCS and XSVS experiments were performed at BL40XU, SPring-8 (Hyogo, Japan). The experimental setup has been described elsewhere [22]. An X-ray beam of 10.5 keV from a helical undulator [23] was used and a pinhole of diameter 5 μm and a guard pinhole of diameter 25 μm were installed upstream of the sample to make the X-ray beam coherent enough to produce speckle patterns. The temperature of the sample was controlled to be −20 °C using a temperature-controlled stage (THMS600, Linkam Scientific Instruments Ltd., UK) and the speckle patterns were recorded using an interline charge-coupled device detector (CCD) (C4880-80, Hamamatsu Photonics Ltd.) coupled with an X-ray image intensifier [24]. Each pixel of the detector consists of a photodiode and a CCD. Photoelectrons produced by photons incoming to the pixel are accumulated in the photodiode and they are rapidly transferred to the CCD when each exposure is finished. Since the next exposure can be started as soon as the photoelectrons is transferred to CCD, the time gap between the end of a frame and the start of the next one is very small (microseconds) [25]. In our experiment, the exposure time per frame was 36 ms ( $Δ t =$ 36 ms) and the time gap between frames was negligible compared with the exposure time. The distance between the sample and the detector, L, was 3 m and the beam size at the sample, d, was measured by a knife-edge method to be 5 μm. The speckle size s was estimated to be 71 µm using the relation $s = λ L / d$ , where λ is the wavelength of the X-ray. The pixels were binned so that the effective pixel size, 80 µm, corresponds to the speckle size. Two hundred sequential speckle patterns, after subtraction of an averaged dark image, were used as scattering images in the following analysis. In our experimental setup, the scattering intensity from the sample was high enough for the effect of the dark noise of the detector to be negligible in the XSVS analysis.

4. Results and discussion

In XPCS analysis, the normalized intensity correlation function $g_{2} (q, t)$ is calculated using Eq. (2). The calculated $g_{2} (q, t)$ is well fitted by

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

where

τ

is the relaxation time. The relaxation time at each

q

, determined by XPCS [Fig. 3(a)] shows a clear scaling of

τ \propto q^{- 2}

. This indicates that the nanospheres follow Brownian motion.

XSVS analysis is performed as follows. To calculate the visibility of the patterns for exposure time $T = k Δ t$ ( $k = 1, 2, 3 \dots, 55$ ), 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 $T = k Δ t$ , as shown in Fig. 2 , and a series of scattering images is produced. Then, the visibility of the images for the exposure time $k Δ t$ 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]. 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 = 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) 〉} .

Figure 3(b) 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

\pm

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).

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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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 $〈 N (q, Δ t) 〉 / ε$ at each $q$ determined by XSVS analysis. The fact that $〈 N (q, Δ t) 〉 / ε$ 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]. 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, 29], 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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Effect of shot noise on X-ray speckle visibility spectroscopy

Abstract

1. Introduction

2. Methodology

2.1 X-ray photon correlation spectroscopy

2.2 X-ray speckle visibility spectroscopy

3. Materials and methods

4. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (18)

Optics Express