An apparatus (AS) measures positions of marks (202) on a lithographic substrate (W). An illumination arrangement (940, 962, 964) provides off-axis radiation from at least first and second regions. The first and second source regions are diametrically opposite one another with respect to an optical axis (O) and are limited in angular extent. The regions may be small spots selected according to a direction of periodicity of a mark being measured, or larger segments. Radiation at a selected pair of source regions can be generated by supplying radiation at a single source feed position to a self-referencing interferometer. A modified half wave plate is positioned downstream of the interferometer, which can be used in the position measuring apparatus. The modified half wave plate has its fast axis in one part arranged at 45° to the fast axis in another part diametrically opposite.