Fig. 1. Momentum transfer for mirror and CRL

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(-Q2σ2), 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. Denoting the x-ray wave number by k and the refractive decrement of the lens material by δ we have

Q = 2k sin ϑ ≈ 2k ϑ   and   Q = N1/2 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.