INSENSITIVITY OF LENSES TO SURFACE ROUGHNESS AND CONTAMINATION

Momentum transfer for mirror and compound refractive lens
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 the article "Imaging by parabolic refractive lenses in the hard x-ray range", Appendix B. 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.