Excess coincidences of reflected and refracted x rays from a synchrotron-radiation beamline

The coincidence rate between x rays reflected and refracted from a silicon crystal after a high-resolution monochromator was measured at the long undulator beam line of SPring-8. The data show excess coincidences relative to the random background, which vanish when the detectors of the reflected and refracted beams probe widely separated parts of the incoming x-ray wave front. The measured coincidence rate with different detector sizes and distances between the probed wave front positions, perpendicular to the scattering plane, agree well with fitting functions derived for the present experimental conditions. Furthermore, it is shown that the coincidence rate measured between probing positions in the scattering plane depends strongly on the angular spread of reflected and refracted x rays. Small angular spread is obtained by detuning from the exact Bragg condition, which results in good quantitative agreement of the coincidence data and theoretical predictions.

Figure 1

Coordinate system of double-slit configuration.

Figure 2

(Color online) Dependence of the excess coincidence rate of Eq. (11) on the distance $d$ between the detectors, displayed for several values of the slit widths.

Figure 3

(Color online) Experimental setup. Hard x-ray beam from the $27\mathrm{m}$ undulator is monochromatized with the DCM and HRM and then divided into two paths by the BS. The x rays are detected with two avalanche photodiodes (APDs). Two apertures, slit o and slit h, are placed in front of the APDs. The scan direction of the apertures is vertical (horizontal) in Setup I (II).

Figure 4

Rocking curves of the refracted (solid line) and reflected (dashed line) beams of the $\mathrm{Si}220$ BS. The diffracted intensities are plotted as a function of deviation angle $\mathrm{\Delta }\theta$ from the exact Bragg condition. Conditions A $(\mathrm{\Delta }\theta =0\mathrm{\mu }\mathrm{rad})$ and B $(\mathrm{\Delta }\theta =-9.2\mathrm{\mu }\mathrm{rad})$ show the two angular alignments used in the measurements.

Figure 5

Dependence of excess coincidence rate $R$ on vertical slit distance for different slit widths: $w_{y1}=w_{y2}=30\mathrm{\mu }\mathrm{m}$ (a), $60\mathrm{\mu }\mathrm{m}$ (b), $120\mathrm{\mu }\mathrm{m}$ (c), and $240\mathrm{\mu }\mathrm{m}$ (d). The solid lines for (a)–(c) are fitted results, while that for (d) is a calculated one.

Figure 6

Example of “conventional” control data obtained with $30\mathrm{\mu }\mathrm{m}$ horizontal width.

Figure 7

Relative excess coincidence rate $R$ measured by scanning the horizontal position $x_1$ of slit o (closed circles). The horizontal widths of the slits were $20\mathrm{\mu }\mathrm{m}$. The deviation angles of the BS are $\mathrm{\Delta }\theta =0$ (a) and $-9.2\mathrm{\mu }\mathrm{rad}$ (b), respectively. The solid line in (a) is a guide for the eyes. The dotted lines are intensity profiles for the o beam. The insets show intensity profiles for the h beam.

Figure 8

(Color online) Effect of departure $\mathrm{\Delta }\theta$ from the Bragg angle on the refracted and reflected x rays. The origin of the $x_1$ axis ($x_2$ axis) is located on the refracted (reflected) ray that satisfies the exact Bragg condition.