A simple closed-form expression for the X-ray reflectivity from multilayers with cumulative roughness

Abstract

We present a simple closed-form expression for the reflectivity from a multilayer which includes the effects of absorption, refraction, surface and substrate reflections but neglects dynamical effects. This expression reproduces the exact dynamical calculation except for the regions near the critical angle and for intense Bragg reflections. The expression is generalized to include cumulative interface roughness which follows a $\text{t}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ power-law growth.