In this work, we apply the high-resolution interference microscopy technique to investigate the intensity and phase of light transmitted through a binary phase element. Using a combination of simulation and experimental results, we identify specific features in the intensity and phase maps of the transmitted light that are associated with the positions of the edges of the ridges and grooves in the phase element. Specifically, we identify these features to be minima in the intensity recordings, as well as phase jumps and in some cases phase singularities in the phase maps. We focus on the former two features, as they can reliably be observed in a single z-plane intensity or phase image, respectively, focused at the top of the structure ridges. Using the edge locations extracted from the intensity and phase profiles, we estimate the dimensions of the structure ridges and grooves as well as measure the displacement of the sample on a piezo stage. With both methods, the absolute width of the ridges and grooves is measured with an accuracy of approximately 220 nm, and the sample displacement is detected to approximately 50 nm resolution. (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement