Optical element positioning apparatus, projection optical system and exposure apparatus

An apparatus that includes an optical element and positions the optical element, the apparatus comprising:

• a holder configured to hold the optical element;
• a back plate configured to support the optical element via the holder;
• a mechanism configured to move the optical element in a six-degree-of-freedom;
• a base plate configured to support the back plate via the mechanism;
• three first displacement sensors disposed on the base plate and configured to measure respective displacement amounts of three mutually different points on the optical element in a first direction;
• a second displacement sensor disposed on the base plate and configured to measure a displacement amount of one point on the optical element in a second direction different from the first direction;
• two third displacement sensors disposed on the base plate and configured to measure respective displacement amounts of two mutually different points on the optical element in a third direction different from the first and second directions;
• a transformation processor configured to transform the six displacement amounts measured by the first, second and third displacement sensors into displacement amounts of the optical element in the six-degree-of-freedom;
• a calibration processor configured to calibrate the displacement amounts transformed by the transformation processor;
• and
• a controller configured to output command values to the mechanism based on differences between the displacement amounts calibrated by the calibration processor and target displacement amounts of the optical element.