The effect of longer-range waviness on X-ray reflectivity measurements

A model for calculating the X-ray reflectivity (XRR) of surfaces to extract both roughness and waviness features is presented. Expressions of reflectivity intensity are derived as a function of root-mean-square (RMS) roughness σ, RMS waviness σ_L, and the cut-off frequency between the features ω₀. Experiments were conducted at the Advanced Light Source at Lawrence Berkeley National Laboratory, beamline 8.3.2, on BK7 glass manufactured with a multi-step polishing process to validate the model, and were compared with atomic force microscopy (AFM), Fizeau interferometry and surface profilometry measurements. The parameter results and their deviations for XRR measurements were σ = 2.9 ± 0.2 nm and σ_L = 14.6 ± 0.5 nm with a wavelength cut-off of 1/(18 ± 2) µm^-1, while the results from the AFM, Fizeau and profilometry measurements were σ_AFM = 3.4 ± 0.4 nm, σ_L,Fizeau = 21.6 nm, σ_prof = 4.0 ± 0.1 nm, and σ_L,prof = 21.4 ± 0.1 nm with cut-offs for the profilometry and Fizeau measurements limited to frequencies of (1/16) µm^-1 to (1/4) mm^-1.