The ULE (R) substrates used in projection optics of extreme ultraviolet lithography (EUVL) tools are mechanically prefinished with shape accuracy of several nanometer rms (specification: under 0.15 nm rms) with high-spatial frequency roughness (HSFR) (spatial wavelength: under 1 mu m) of 0.06 nm rms. The ion beam figuring is used for the final shape error correction of the substrates at low spatial wavelength of greater than 1 mm using high energy (5-10 keV) Ar+ ion beam with 1 mm beam diameter. However, the surface roughness values of the ULE (R) substrates when machined by 5 and 10 keV Ar+ ion beams result in 0.15 and 0.17 nm rms, respectively; also it is to be noted that these values are greater than the required 0.15 nm rms specification of HFSR. Therefore, the authors developed a method where low energy (0.3 and 0.5 keV) ion beams are used for smoothening the surface of ULE (R) substrates, previously treated with high energy (5 and 10 keV) ion beams for figuring. By this two-stage operation of ion beam figuring and followed by smoothening, ULE (R) substrates with HSFR of 0.1 nm rms were obtained. Therefore, the authors conclude that the method presented here is one of the most effective methods for the figure error correction of ULE (R) substrate and will be useful in EUVL projection optics.