The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays

The Talbot effect is the self-imaging, at distances D multiple of D(R), of the intensity downstream of a periodic object. Earlier work with hard synchrotron radiation X-rays showed the variation with D of the fundamental Fourier component of intensity to be a good measurement of beam coherence. Any higher-order Fourier coefficients I (D, m > 1) would be periodic with a reduced period D(Rm) = D(R)/m for an ideally coherent incident beam (partial Talbot effect). The degree of coherence gamma(x) is sampled through the ratio of I (D, m) at D = 0 and multiples of D(Rm). This requires the Fourier coefficient for D = 0, which is not accessible for a phase object (no contrast at D = 0). However, the ratio of the slopes of I (D, m) at D = 0 and D = pD(Rm) also provides this information. Furthermore, a characterization of gamma(x) is possible, provided an assumption is made on its shape, using only the ratio of the Fourier coefficient I (D, m) of two images a distance pD(Rm) apart. Experiments with one- and two-dimensional phase gratings and a mixed (amplitude and phase) two-dimensional grating confirm that the partial Talbot effect approach is viable. It requires a reduced range of distances, and yields important results directly, obviating the need for computer fits. In particular, 8% of the beam intensity was found to have very low coherence in the vertical direction, probably due to monochromator imperfection.