A Method for Optical Measurements of Height Distributions in Transparent Media

As light passes the interface between two media with different densities, the light will be refracted which leads to an optical beam displacement. In this paper we present a simple optical and non-invasive measurement technique and a reconstructing algorithm needed to determine the height distribution of a transparent medium. The local height of one medium is reconstructed using Snell's law after measuring the beam displacement. Thereto, the optical displacement on a two-dimensional background plane is obtained by using background oriented schlieren and spatial cross-correlation with reference imaging. Moreover, to rebuilt the three dimensional height distribution from the image plane, a camera calibration is required. As long as the interface between the two media is plan-parallel to the background plane, the presented method provides a unique solution with only one camera. In this paper we show that this simple reconstruction algorithm can also be used for reconstructing weakly curved interfaces as well. We also present results of measurement of water droplets on a planar surface, using this algorithm. Furthermore, we discuss the errors, which result from the non plan-parallelity of the interface and show that for strongly curved interfaces the method must be advanced by using spatial displacement gradients or multiple cameras, which allows to reconstruct curvy interfaces more accurately.