Investigations of different trajectories of X-rays in capillaries

Abstract

Using the methods of geometrical optics, different cases for allowed X-ray trajectories in single capillaries are studied and it is proved that they cannot be described by the same sets of equations. Three main cases should be considered. In two cases, the test X-rays strike both sides of the capillary upon passing, be the second wall convex or concave. In the third case, called the whispering gallery mode, the path is allowed along the single, external wall of the capillary. This case is realistic for some off-axis, close to the wall, trajectories. A trajectory along the capillary with the curved main axis is the particular case, joining the features of the cases mentioned earlier. The separate consideration of different kinds of trajectories has important implications. The first case corresponds to the ideal trajectory in single capillaries, whereas the second and third describe realistic trajectories in the compound polycapillary systems. The transformation of the polycapillary systems into the coaxial polycylindrical systems is considered. In the third case, which constitutes a transition to trajectories of X-rays on the curved mirror surfaces, the removal of the opposite wall of the capillary can be allowed. Surprisingly, very simple angular relationships lead to the differential equations, describing capillary optics in the terms of geometrical parameters of capillary. The equations describing the surfaces on which the X-rays behave in isoreflective way, i.e. they have the same angle of incidence all along the trajectory, are derived. The proposed approach is widened for modeling of surfaces in order to enable the reflections with continuously changing and controlled reflection angle.