Obtaining of smooth high-precision surfaces by the mechanical lapping method

We describe in detail a method for obtaining high-precision smooth spherical substrates using mechanical lapping and metrology used for these purposes. We consider a modified version of a two-probe interferometer with a diffraction reference wave, which ensures leveling of the intensities of the arms of the interferometer and rearrangement of the working aperture without instrument resetting. We report on the experimental results obtained during lapping of a concave spherical fused silica substrate using this technique with numerical aperture NA = 0.30, which has been prepared by traditional deep grinding–polishing. The initial characteristics of the substrate are the root-mean-square shape error of 36 nm ($\sim\lambda$/20) and effective roughness $\sigma_{\operatorname{eff}}$ = 1.1 nm in the spatial frequency range of 0.025–65 $\mu$m$^{-1}$. After substrate lapping, the surface parameters were improved to a root-mean-square error of 3.3 nm ($\sim\lambda$/200) and $\sigma_{\operatorname{eff}}$ = 0.26 nm. We have analyzed the effect of the grain size in suspension on the roughness and shape of the substrate.