Analytical and stochastic modeling of surface topography in time-dependent sub-aperture processing

Time-dependent sub-aperture surface processing is widely used in industry for finishing of optical surfaces. It can often be viewed as the convolution process between a tool influence function (TIF, also known as footprint) and the equivalent dwell time derived from the velocity at discrete locations across the tool path. While the direct convolution problem has been extensively studied through numerical computation, this approach leads to limited and inaccurate prediction of the processed surface. Therefore, this paper proposes a simple and universal analytical model that uncovers the intrinsic relationship between process parameters and processed surface, and which has potential for a wide range of manufacturing processes. Furthermore, sensitivity of the process to TIF fluctuations is considered, which leads to highly accurate stochastic predictions of the processed surface waviness variation by Monte Carlo simulation. Experimental results in fluid jet polishing confirm correctness of the proposed analytical solution and effectiveness of the waviness prediction model. The presented solution provides a better understanding and insight into time-dependent sub-aperture surface processing and opens the door to efficiently tackling the related direct and inverse problem.