# From Femtoseconds to Hours--Measuring Dynamics over 18 Orders of Magnitude with Coherent X-rays
## 2 XPCS Concepts
### Coherence and Speckle Contrast  
The feasibility of XPCS and XSVS experiments strongly depends on the coherence properties of the X-ray beam and the possibility of resolving speckles. The coherence properties are given by the mutual coherence function of the field \(E\) in spatial and temporal dimensions  
\[\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\Delta t)=\langle E(\mathbf{r}_{1},t)E^{*}( \mathbf{r}_{2},t+\Delta t)\rangle, \tag{13}\]  
where \(\langle\rangle\) denote an ensemble average. Typically, the degree of coherence is used that is given by the normalized mutual coherence as  
\[\psi_{1}(\mathbf{r}_{1},\mathbf{r}_{2},\Delta t)=\frac{\Psi(\mathbf{r}_{1}, \mathbf{r}_{2},\Delta t)}{\sqrt{\Psi(\mathbf{r}_{1},\mathbf{r}_{1},\Delta t) \Psi(\mathbf{r}_{2},\mathbf{r}_{2},\Delta t)}}. \tag{14}\]  
As in X-ray experiments the intensity \(I\) is measured, a second order degree coherence is used as  
\[\psi_{2}(\mathbf{r}_{1},\mathbf{r}_{2},\Delta t)=\frac{\langle I(\mathbf{r}_{1 },t)I(\mathbf{r}_{2},t+\Delta t)\rangle}{\langle I(\mathbf{r}_{1},t)\rangle \langle I(\mathbf{r}_{2},t)\rangle}. \tag{15}\]  
For Gaussians fluctuations, the Siegert relation combines both degrees of coherence to  
\[\psi_{2}(\mathbf{r}_{1},\mathbf{r}_{2},\Delta t)=1+|\psi_{1}(\mathbf{r}_{1}, \mathbf{r}_{2},\Delta t)|^{2}. \tag{16}\]  
This is also the basis for XPCS where the time evolution of correlations is probed. For calculation of the speckle contrast of single speckle patterns, we concentrate now on the spatial coordinates. Following Equation (15), the degree of coherence can be calculated from the speckle pattern with \(\mathbf{r}=\mathbf{r}_{1}=\mathbf{r}_{2}\) as variance of intensity at positions \(\mathbf{r}\) on the detector  
\[\beta(\mathbf{r})\equiv\psi_{2}(\mathbf{r})-1=\frac{\langle I^{2}\rangle}{ \langle I\rangle^{2}}-1=\frac{\langle I^{2}\rangle-\langle I\rangle^{2}}{ \langle I\rangle^{2}}. \tag{17}\]  
The subtraction of 1 yields \(0\leq\psi_{2}\leq 1\). An important quantity in experiments is the speckle size that can be obtained from a spatial autocorrelation of intensity [59]. The speckle size is connected to the coherence volume of the X-rays. Typically, one distinguishes between the transverse coherence, that depends on the source size, and the longitudinal or temporal coherence, that reflect the monochromaticity of the beam. At current storage-ring beamlines the transverse coherence length reaches values of several \(10~{}\upmu\)m to \(100~{}\upmu\)m, while FEL beams are typically fully coherent over their whole size [60]. In contrast, as the longitudinal coherence is bandwidth-dependent, it is typically around \(1~{}\upmu\)m for standard bandwidths in the order of \(10^{-4}\). The coherence volume is connected to the degree of coherence by the number of coherence modes \(M\) via \(\beta=1/M\). If the illuminated sample volume is larger than the coherence volume, the sample can be divided into \(M\) volumes, each of them of the size of the coherence volume. Therefore, a fully coherent illumination consist of one coherent mode and yields thus \(\beta=1\). As in SAXS the path length differences of the X-rays are well below the longitudinal coherence length, the transverse coherence properties can be probed directly by the speckle contrast via Equation (17). In this case the speckle size is given by  
\[s\approx\lambda d_{\text{sd}}/w, \tag{18}\]  
with the sample-detector distance \(d_{\text{sd}}\) and the beam size \(w\). With increasing scattering angle, the longitudinal coherence length affects the speckle contrast resulting in effective reduction. The contrast can be calculated by considering the actual beam shape and coherence lengths, details can be found in [59]. Here we want to discuss several consequences for measuring speckle contrast and thus performing XPCS and XSVS experiments. Besides the bandwidth,coherence length and beam sizes, the sample dimensions and detector pixel sizes define the measured speckle contrast.