A projection exposure apparatus (10) for microlithography comprises a measuring system (50) for measuring an optical element of the projection exposure apparatus. The measuring system (50) comprises: an irradiation device (54), which is configured to radiate measuring radiation (62) in different directions (64) onto the optical element (20), such that the measuring radiation (62) covers a respective optical path length (68) within the optical element (20) for the different directions (64) of incidence, a detection device (56), which is configured to measure, for the respective direction (64) of incidence, the corresponding optical path length covered by the measuring radiation (62) in the optical element (20), and an evaluation device, which is configured to determine a spatially resolved distribution of the refractive index in the optical element (20) by computed-tomographic back projection of the measured path lengths taking account of the respective direction of incidence.