Calculation of intensities in grazing-emission x-ray fluorescence

The sensitivity of x-ray fluorescence spectroscopy to surface and subsurface layers in the nanometer regime can be greatly enhanced by measuring the fluorescence radiation that is emitted at grazing angles. In this paper, we present a formalism for the calculation of x-ray fluorescence intensities that is also valid under grazing emission conditions. By applying asymptotics to plane-wave expansions, an approximate solution to Maxwell's equations for a radiating point source in a layered system is derived, without the use of the optical reciprocity theorem. In the computation of the fluorescence intensity, secondary and higher-order fluorescence effects are taken into account. The total fluorescence of a particular layer is obtained by integrating the contributions of point sources at different depths. The derived expressions compare well with the measured angular dependence of the fluorescence intensity in a number of typical examples.