Phase-space analysis and experimental results for secondary focusing at X-ray beamlines.

Micro-focusing optical devices at synchrotron beamlines usually have a limited acceptance, but more flux can be intercepted if such optics are used to focus secondary sources created by the primary optics. Flux throughput can be maximized by placing the secondary focusing optics close to or exactly at the secondary source position. However, standard methods of beamline optics analysis, such as the lens equation or matching the mirror surface to an ellipse, work poorly when the source-to-optics distance is very short. In this paper the general characteristics of the focusing of beams with Gaussian profiles by a ;thin lens' are analysed under the paraxial approximation in phase space, concluding that the focusing of a beam with a short source-to-optics distance is distinct from imaging the source; slope errors are successfully included in all the formulas so that they can be used to calculate beamline focusing with good accuracy. A method is also introduced to use the thin-lens result to analyse the micro-focusing produced by an elliptically bent trapezoid-shaped Kirkpatrick-Baez mirror. The results of this analysis are in good agreement with ray-tracing simulations and are confirmed by the experimental results of the secondary focusing at the 18-ID Bio-CAT beamline (at the APS). The result of secondary focusing carried out at 18-ID using a single-bounce capillary can also be explained using this phase-space analysis. A discussion of the secondary focusing results is presented at the end of this paper.