The reflection efficiencies for the various orders of diffraction produced by a grating are calculated under the following assumptions. The grating is for the most part flat, but with a fraction ε of the periodicity interval <i>a</i> described by a profile function <i>z</i>= ƒ(<i>x</i>). The reflection and transmission efficiencies are calculated as a series expansion in ε. Numerical calculations have been performed for the case of ultrasoft x rays incident at near grazing angles, using a complex index of refraction η = 1-δ<sub>0</sub>λ<sup>2</sup>+<i>i</i>β<sub>0</sub>λ<sup>3.25</sup> or tabulated values. No effects of shadowing by the grating profile have been considered, and the numerical work has been done for a symmetrical triangular trench incised in the flat surface.
JACK C. MILLER, "Reflection Efficiencies of a Periodic Absorbing Surface," J. Opt. Soc. Am. 54, 353-356 (1964)