<title>Bragg-Fresnel gratings, zone plates, and periodic stratified media: differential formalism revisited</title>

Abstract

Bragg-Fresnel gratings and zone plates are etched into plane multilayers whose thicknesses may be hundreds of wavelengths. Similar situation is found for modulated multilayers deposited over a X-ray grating. Thus the analysis of such thick periodic stratified media through the classical differential method may present numerical instabilities linked with the growing of exponential functions associated with the evanescent orders during the integration process. There are two ways to avoid this contamination. They consist of cutting the modulated region into thin slices in which the numerical integration is stable and to combine the field diffracted by the thin slices via the Bremmer summation series of the R-matrix propagation algorithm. Both techniques are tried and compared, and the second one turns out to be more stable. It is then possible to analyze gratings with modulation depths ten times the groove spacing and thousand times the wavelength, or deep gratings supporting several hundreds of orders without numerical problems. The use of the R- matrix propagation algorithm opens new possibilities to the existing grating theories and, in particular to the differential method, in analyzing more and more complicated stratified media. As examples of the new possibilities of the Differential Method, Bragg-Fresnel zone plates and deep multilayer coated gratings for soft X-rays are investigated.