Reconstructing surface profiles from curvature measurements

Abstract

Recently, the measurement of curvature has been suggested as a promising new technique for the highly accurate determination of large-area surface profiles on the nanometer scale. It was shown that the curvature can be measured with highest accuracy and high lateral resolution. However, the reconstruction of surface profiles from curvature data involves the numerical solution of an ordinary differential equation for which initial or boundary values must be specified. This paper investigates the accuracy with which surface profiles can be reconstructed from curvature data. The stability of the reconstructions is examined with respect to the presence of measurement noise and the accuracy of the initial values. The assessment of the reconstruction accuracy is based on an analytical solution (up to numerical integration) derived for the case when the measurement results are given in Cartesian coordinates, and on numerical results in the polar case. The results presented for the latter case allow, in particular, conclusions to be drawn regarding the minimum accuracy of data and initial values required for reconstructing aspheres from curvature measurements with nanometer accuracy.