Probing the Transverse Coherence of an Undulator X-Ray Beam Using Brownian Particles

We present a novel method to map the two-dimensional transverse coherence of an x-ray beam using the dynamical near-field speckles formed by scattering from colloidal particles. Owing to the statistical nature of the method, the coherence properties of synchrotron radiation from an undulator source is obtained with high accuracy. The two-dimensional complex coherence function is determined at the sample position and the imaging optical scheme further allowed us to evaluate the coherence factor at the undulator output despite the aberrations introduced by the focusing optics.

Figure 1

The direct image of the beam depicting two brighter stripes which are distorted images of the undulator source. The mean square width of the intensity distribution is $\sim 100\mu \mathrm{m}$ which is comparable with the calculated value by ray tracing ($\sim 90\mu \mathrm{m}$). The beam profile appears roughly as a Gaussian with usual SAXS detector resolution ($\sim 60\mu \mathrm{m}$). Insets (a) and (b) display magnified regions of the beam without and with colloidal particles, respectively.

Figure 2

Simulation of the interference patterns between a spherical wave ($\lambda =1\AA$) and a single coherent patch of beam with limited transverse coherence. The size of the coherence patch at the origin ($z=0$) is approximately $8\mu \mathrm{m}$ in diameter.

Figure 3

Azimuthally averaged power spectra of the speckle intensity distribution generated by the colloidal suspension recorded at $z=68\mathrm{mm}$, sensor power transfer function, and cumulative shot and readout noise.

Figure 4

Modulus of complex coherence factor (visibility) as a function of transverse displacements $x$ and $y$ represented by curves (a) and (b), respectively. These two curves are obtained from a stack of power spectra taken at various distances ($5\mathrm{mm}<z<250\mathrm{mm}$) denoted by different symbols and represent the maxima of the oscillating Talbot function. Inset shows the full two-dimensional distribution.