SOFT-X-RAY MULTILAYER COATED ECHELLE GRATINGS - ELECTROMAGNETIC AND PHENOMENOLOGICAL STUDY

High-blaze-angle multilayer coated echelette gratings used in very hig h (i.e., from 10th to 60th) orders and operating with not too small (a pproximate to 5 degrees) glancing angles of incidence turn out to be n ecessary for achieving ultrahigh resolution and collecting important f lux in x-ray fluorescence spectroscopy. Predicting their diffracting b ehavior appears to be a challenge to grating theoreticians, because su ch gratings are associated with the three main difficulties encountere d in grating-efficiency numerical computation, namely, low wavelength- to-groove-spacing ratio, large modulation, and multiple interpenetrati ng profiles. We propose a method to resolve the problem that combines the differential theory of gratings, the R-matrix propagation algorith m, and an asymmetric truncation of the Fourier series of the field. Th e numerical results are checked against many criteria. For a particula r mounting they are compared with those given by a phenomenological fo rmula that was developed for bare echelette gratings and turns out sti ll to apply for multilayer-coated gratings. The theory is developed no t only for periodic stacks but also for stacks made of layers with var ying thicknesses, such as those used in supermirrors. (C) 1996 Optical Society of America