The present disclosure describes systems and methods that enable eye-tracking by steering a light signal in a high-density Lissajous pattern over a region of an eye and detecting light reflected from the eye using a non-imaging photodetector configuration. The light signal is scanned by driving each axis of a two-axis MEMS scanner with a periodic signal having a frequency that is based on the resonant frequency of that axis. By choosing periodic signals having frequencies that give rise to precession of the Lissajous pattern at a high rate, a high-density scan pattern is quickly generated, thereby enabling eye tracking with high spatial resolution and low latency.