Effects of partial coherence on correlation functions measured by x-ray photon correlation spectroscopy

We present a rigorous description of the effects of partial coherence and detector resolution on intensity autocorrelation functions as they can be measured by x-ray photon correlation spectroscopy (XPCS). Based on the Huygens-Fresnel propagation law and on the first Born approximation, we derive a general expression for the normalized intensity autocorrelation function. We calculate how the mutual coherence function of the x-ray beam propagates from an aperture to the sample and how it propagates after the scattering process to the detector area and consequently influences the intensity autocorrelation function. We illustrate our calculation with examples of XPCS intensity autocorrelation functions of liquid surfaces calculated for grazing incidence geometry.

Figure 1

(Color online) Schematic view of the scattering geometry. The beam emerges from the aperture A. The line $L_a$ is parallel to the mean beam direction and connects the center of the aperture and the center of the sample. $L_b$ is parallel to the mean outgoing beam direction and connects the center of the sample with the center of the detector aperture B. An x-ray beam emerges from A at position $\mathbf{a}$, propagates along the line $R_a$, falls onto the sample at position $\mathbf{r}$, and is then scattered along the line $R_b$ into the detector at position $\mathbf{b}$.

Figure 2

(Color online) Calculated propagating intensity autocorrelation functions at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$. The lower curve represents $g_2\left(q_{\parallel },\tau \right)$ without any effects of partial coherence, i.e., at infinitely good resolution. The upper curve (shifted $+1$ for clarity) represents $g_2\left(q_{\parallel },\tau \right)$ as calculated from Eq. (25) with a relative resolution $\Delta q/q=0.1$. The two functions are calculated for a pointlike detector.

Figure 3

(Color online) Calculated overdamped intensity autocorrelation functions at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$. The lower curve represents $g_2\left(q_{\parallel },\tau \right)$ without any effects of partial coherence, i.e., at infinitely good resolution. The upper curve (shifted $+1$ for clarity) represents $g_2\left(q_{\parallel },\tau \right)$ as calculated from Eq. (25) with a relative resolution $\Delta q/q=0.1$. The two functions are calculated for a pointlike point detector.

Figure 4

(Color online) Calculated propagating intensity autocorrelation functions at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$ for different values of the wave vector spread (from bottom to top, $\Delta q/q=0.197$, 0.2, 0.203, 0.207, and 0.21). The correlation functions are shifted for reasons of clarity. All functions are calculated for a pointlike detector.

Figure 5

(Color online) Calculated overdamped intensity autocorrelation functions at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$ for different values of the wave vector spread (from left to right, $\Delta q/q=0.197$, 0.2, 0.203, 0.207, and 0.21). All functions are calculated for $\beta =0$, i.e., for a pointlike detector.

Figure 6

(Color online) Calculated propagating intensity autocorrelation functions at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$ for different values of the detector opening (from top to bottom, $\beta =0.3$, 30, 60, 90, and $120\mu \text{m}$). All functions are calculated for fixed value of the wave vector spread $\Delta q/q=0.06$.

Figure 7

(Color online) Calculated overdamped intensity autocorrelation functions at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$ for different values of the detector opening (from top to bottom, $\beta =0.3$, 30, 60, 90, and $120\mu \text{m}$). All functions are calculated for fixed value of the wave vector spread $\Delta q/q=0.06$. Note that the correlation function is now plotted on a logarithmic scale with the time axis being linear.

Figure 8

(Color online) Contrast $\left[g_2\left(q_{\parallel },0\right)-1\right]$ as a function of detector opening of two surface correlation functions with different $q$ dependence. The upper curve represents the contrast for $C_{zz}=\text{cos}\left(\omega \tau \right)$ while the lower curve is calculated with $C_{zz}=q_{\parallel }^{-2}\text{cos}\left(\omega \tau \right)$. Calculated at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$ with a fixed wave vector spread $\Delta q/q=0.06$.

Figure 9

(Color online) Calculated propagating intensity autocorrelation functions at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$ and fixed wave vector spread $\Delta q/q=0.206$ for different values of the detector opening $\beta$.

Figure 10

(Color online) Calculated overdamped intensity autocorrelation functions at $q_{\parallel }=5\times {10}^{-6}{\text{Å}}^{-1}$ and fixed wave vector spread $\Delta q/q=0.206$ for different values of the detector opening.