A head-mounted device may have catadioptric lenses that each include a partial mirror, a quarter wave plate, and a polarizer. An optical system in the head-mounted device may have an infrared light-emitting device and an infrared light-sensing device. The optical system may illuminate a user's eyes in eye boxes and may gather measurements from the illuminated eye boxes for eye tracking and other functions. The optical system may operate through the catadioptric lenses. To enhance optical system performance, the polarizers may be wire grid polarizers that are formed from conductive lines that exhibit enhanced infrared transmission and/or the quarter wave plates may be formed from cholesteric liquid crystal layers that serve as quarter wave plates at visible wavelengths and that do not serve as quarter wave plates at infrared wavelengths.