Ion beam figuring is suitable for the final correction of the surface figure error of aspherical substrates in an extreme ultraviolet lithography apparatus. In ion beam processing, however, the surfaces are considered to become rougher. This paper reports the investigation of the surface roughness of Zerodur (R) machined with an Ar+ ion beam having an energy of 3-10 keV. For an Ar+ ion beam with energies in the range of 3-10 keV, the mid-spatial frequency roughness of the surfaces machined to a depth of less than 50 nm was comparable to the surface roughness of an unprocessed one. The high-spatial frequency roughness of the unprocessed surface was 0.26 nm rms; whereas the high-spatial frequency roughness of the surface machined with energies of 3, 5, 7, and 10 keV was 0.46, 0.54, 0.63, and 0.65 nm rms, respectively. The HSFR of the machined surface increases with an increase in the energy of the ion beam. (c) 2007 American Vacuum Society.