Disclosed are an optical system for conditioning a beam of radiation, and an illumination system and metrology apparatus comprising such an optical system. The optical system comprises one or more optical mixing elements in an optical system. The optical system defines at least a first optical mixing stage, at least a second optical mixing stage, and at least one transformation stage, configured such that radiation entering the second optical mixing stage includes a transformed version of radiation exiting the first optical mixing stage. The first and second optical mixing stages can be provided using separate optical mixing elements, or by multiple passes through the same optical mixing element. The transformation stage can be a Fourier transformation stage. Both spatial distribution and angular distribution of illumination can be homogenized as desired.