## Measuring temporal speckle correlations at ultrafast x-ray sources

Optics Express, Vol. 17, Issue 1, pp. 55-61 (2009)

http://dx.doi.org/10.1364/OE.17.000055

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### 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

## 1. Introduction

*g*

_{2}(τ) = <

*I*(

*t*)

*I*(

*t*+ τ) > / <

*I*(

*t*) >

^{2}. Assuming Gaussian fluctuations the intensity autocorrelation function can be rewritten with the help of the Siegert relation [1] as

*f*(τ) is the intermediate scattering function reflecting the dynamics of the sample.

^{-2}to 10

^{6}Hz [2

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

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

*direct*information about the intermediate scattering function of the sample.

## 2. Methodology

### 2.1. Sequential technique

*N*pixels. Suppose that we take a series of K images with an exposure time

*t*at an interval Δ

_{e}*t*which are read out and stored on a hard drive. The normalized intensity autocorrelation function sampled in the classical sequential way can then be calculated for each pixel

*n*as

*k*is the lag time,

*I*(

_{n}*t*) the intensity of pixel

_{i}*n*at time

*t*and

_{i}*I*̄ the temporal averaged intensity of pixel

*n*. Averaging over pixels leads to the intensity autocorrelation function

*g*

_{2}(

*k*) = 1/

*N*∑

^{N}

_{n = 1}

*g*

_{2}(

*n,k*). Effects of partial coherence are accounted for by introducing a contrast factor

*β*

^{2}in the Siegert relation, yielding

*k*= Δ

_{min}*t*+

*t*. Therefore the sequential scheme relies on area detectors with fast enough read out times and it is restricted to time scales on the order of 10

_{e}^{-6}seconds [4

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

### 2.2. Split-pulse technique

*S*(

*k*)

*S*

^{2}(

*k*)〉 can be expressed as

*f*(

*k*) is the intermediate scattering function of the system. So we find for the normalized speckle contrast

*β*) and shot noise (

*α*) into account Eq. (11) yields

*r*denoting the ratio between the two intensity pattern. Maximum contrast can be achieved with a ratio

*r*= 1.

### 2.3. Photon statistic of the split-pulse technique

*S*in terms of their counting statistics. The probability distribution function

*P*(

*i*) for observing

*i*counts in a speckle pattern is

*i*̄ is the mean count rate and

*M*is the number of modes [1]. The number of modes is a measure of the contrast of a speckle pattern and therefore reflects the underlying dynamics of the sample. That is for a delay time smaller than the sample correlation time the number of modes M contributing to the speckle pattern is smaller than for delay times larger than the sample correlation time. Thus the probability distribution function

*P*(

*i*) will change as a function of delay time

*k*.

## 3. Experiment and analysis

*ξ*=

_{l}*λ*

^{2}/Δ

*λ*≈ 1

*μ*m. A pinhole with a diameter of 20

*μm*is placed upstream of the sample in order to obtain a collimated and transversely coherent beam. The beam is tilted vertically by a mirror onto the sample surface resulting in an incident angle

*α*= 0.12°. The sample investigated was a colloidal suspension of silica particles immersed in a high viscosity polymeric liquid PPG-4000 [6

_{i}6. S. Streit-Nierobisch, C. Gutt, M. Paulus, and M. Tolan, “Cooling rate dependence of the glass transition at free surfaces,” Phys. Rev. B **77**, 041410 (R) (2008). [CrossRef]

*μm*

^{2}. The exposure time was 1s and typically series of 500 -1000 images were recorded and stored on hard disk. In addition a series of dark images with the same exposure time was recorded and its average subtracted from the speckle patterns. The sample was placed in a housing consisting of a two chamber design, where the outer cell was evacuated for thermal isolation and the inner one contained a stainless steel trough of 120 mm diameter and 0.2 mm height. The sample was filled into the trough ensuring a homogenous and flat surface. The air above the sample was replaced by helium gas. Sample cooling was achieved by evaporating liquid nitrogen in a heat exchanger underneath the sample chamber with a constant flow rate.

*β*

^{2}= 0.21. At large lag times the intensity correlation function lies below the unbiased value of 1.0 because Eq. (1) represents a

*biased*estimator of the normalized intensity autocorrelation function [8

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]

*c*

_{2}(

*k*) obtained with the help of Eq. (4). This correlation function decays from a maximum value of 0.48 to a baseline of 0.35. As in the case of the sequential autocorrelation function a long term intensity drift is apparent and also values of

*c*

_{2}below the apparent baseline at 0.35. The optical contrast of the correlation function

*c*

_{2}(

*k*) is

*β*

^{2}= 0.25 which is in agreement with the contrast obtained from the sequential method. The bottom of Fig. 2 displays both correlation functions normalized to each other. The good agreement between both correlation functions for the short time scale dynamics is apparent. The long-term intensity fluctuations at lag times larger than 30 s are however slightly different in both correlation approaches which is attributed to the differences in the averaging procedure.

*P*(

*i*) of the summed speckle patterns is shifting to the right as a function of lag time. This implies that the number of modes

*M*of the summed speckle patterns is increasing with increasing lag time, i.e., the contrast is decreasing. Thus one may as well analyze the statistics of the photon distribution function to retrieve the lag-time dependent degree of coherence

*M*(

*k*)

^{-1}. As an illustration we plot in Fig. 4 the number of photons contained in

*S*(

*t*,

_{i}*k*) as a function of delay time between the two summed intensities. An inspection of Eq. (14) reveals that

*P*(

*i*) is not directly proportional to the number of modes

*M*and therefore does not give direct access to the intermediate scattering function. However, the similarity between the curves is striking and it becomes clear that the basic feature of intensity correlation functions, namely the correlation time τ

_{0}, is already accessible by counting the number of zero photons in a speckle pattern as a function of pulse delay time.

## 4. Conclusion

## References and links

1. | J. Goodman, |

2. | G. Grübel and F. Zontone, “Correlation spectroscopy with coherent X-rays,” J. Alloys Compd. |

3. | M. Altarelli, Editor, |

4. | G. Grübel, G. B. Stephenson, C. Gutt, H. Sinn, and T. Tschentscher, “XPCS at the European X-ray free electron laser facility,” Nucl. Instrum. Methods. B |

5. | P. K. Dixon and D. J. Durian, “Speckle Visibility Spectroscopy and Variable Granular Fluidization,” Phys. Rev. Lett. |

6. | S. Streit-Nierobisch, C. Gutt, M. Paulus, and M. Tolan, “Cooling rate dependence of the glass transition at free surfaces,” Phys. Rev. B |

7. | T. Seydel, A. Madsen, M. Sprung, M. Tolan, and G. Grübel, “Setup for in situ surface investigations of the liquid/glass transition with coherent x rays,” Rev. Sci. Instrum. |

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. |

**OCIS Codes**

(140.2600) Lasers and laser optics : Free-electron lasers (FELs)

(300.6480) Spectroscopy : Spectroscopy, speckle

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: September 5, 2008

Revised Manuscript: November 14, 2008

Manuscript Accepted: November 17, 2008

Published: December 22, 2008

**Citation**

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**, 55-61 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-1-55

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### References

- J. Goodman, Statistical Optics, (John Wiley & Sons, New York, 1985).
- G. Grübel and F. Zontone, "Correlation spectroscopy with coherent X-rays," J. Alloys Compd. 362, 3-11 (2004). [CrossRef]
- M. Altarelli, Editor, XFEL: Technical Design Report 2006, DESY 2006-097.
- 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]
- P. K. Dixon and D. J. Durian, "Speckle Visibility Spectroscopy and Variable Granular Fluidization," Phys. Rev. Lett. 90, 184302 (2003). [CrossRef] [PubMed]
- 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]
- 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]

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