Dynamical Diffraction in Periodic Multilayers

Exact reflectivity curves are calculated numerically for various periodic multilayers using the optical matrix method in order to test the dynamical theory of diffraction. The theory is generally valid for values of the bilayer thickness d up to about 100 Å. For somewhat larger values of d, where the theory begins to break down, the initial discrepancy is in the phase of the oscillations in the wings of the peaks. For very large values of d, where the first-order Bragg peak approaches the edge of the mirror reflection, two general types of multilayers can be distinguished. In the first (typified in the present work by Ni/Ti), there is a large (30% or more) reduction in the actual value of the critical wavevector for total reflection while, in the second (typified here by Fe/Ge), there is very little reduction (3% or so). The origin of these two very different types of behavior is explained. It is also shown that, within the dynamical theory of diffraction, the change in the position of the center of the Darwin plateau as d is varied obeys a universal scaling law – universal in the sense that it is the same for all layer materials and all orders of reflection. The width of the Darwin plateau also obeys a universal scaling law. It is verified that the values of the center and width of the Darwin plateau calculated from the optical matrix method for Ni/Ti and Fe/Ge multilayers obey these scaling laws for a wide range of parameters.