Refractive lenses are quite **insensitive to surface roughness and contamination** compared to x-ray optics based on mirrors. The damping of the intensity due to surface roughness is given by exp(-Q^{2}σ^{2}), where ħQ is the momentum transfer (see Figure 1) and σ is the standard deviation quantifying the surface roughness; see Lengeler et al. (1999): *Imaging by parabolic refractive lenses in the hard x-ray range*, Section 3.2.2 and Appendix B. We have

Q = 2k sin ϑ ≈ 2k ϑ and Q = N^{1/2} k δ

for the mirror and for the stack of N beryllium lenses, respectively; see again Appendix B. Assuming a wavelength λ = 1 Å and ϑ = 0.6 ° we obtain for the mirror Q ≈ 1.3 10^{-1} Å^{-1}, while for N = 100 the lens stack gives Q ≈ 1.4 10^{-4} Å^{-1}.

Hence, the compound refractive lens is about **1000 times less sensitive to surface roughness** than a mirror. A surface roughness of 100 nm yields in the above case a damping factor of about 0.98 which is tolerable.