A beam transforming element for use in an illumination optical apparatus to distribute a beam comprising a first part of the beam and a second part of the beam on a pupil plane of the illumination optical apparatus, wherein the beam transforming element comprises an optical material with optical activity and a diffractive or refractive surface and is constructed and arranged so that: the first part of the beam, having a linear polarization component as a principal component and being incident at a first incidence position on the optical material, and the second part of the beam, having the linear polarization component as a principal component and being incident at a second incidence position on the optical material, pass through different lengths of the optical material in a direction of a crystallographic axis of the optical material providing the optical activity such that the principal polarization components of the first part of the beam and the second part of the beam are transformed to have directions different from each other; the first part of the beam and the second part of the beam are distributed through the diffractive or refractive surface such that the first part of the beam is distributed to a first pair of distribution positions symmetric with respect to a first axis line passing the optical axis, and the second part of the beam is distributed to a second pair of distribution positions symmetric with respect to a second axis line passing the optical axis; the first pair of distribution positions and the second pair of distribution positions are in a predetermined annular region centered around the optical axis; and the lengths of optical material with optical activity for passing the first part of the beam and the second part of the beam are such as to set the polarization directions of the principal polarization components of the first part of the beam and the second part of the beam to be along a circumferential direction around the optical axis. the first part of the beam, having a linear polarization component as a principal component and being incident at a first incidence position on the optical material, and the second part of the beam, having the linear polarization component as a principal component and being incident at a second incidence position on the optical material, pass through different lengths of the optical material in a direction of a crystallographic axis of the optical material providing the optical activity such that the principal polarization components of the first part of the beam and the second part of the beam are transformed to have directions different from each other; the first part of the beam and the second part of the beam are distributed through the diffractive or refractive surface such that the first part of the beam is distributed to a first pair of distribution positions symmetric with respect to a first axis line passing the optical axis, and the second part of the beam is distributed to a second pair of distribution positions symmetric with respect to a second axis line passing the optical axis; the first pair of distribution positions and the second pair of distribution positions are in a predetermined annular region centered around the optical axis; and the lengths of optical material with optical activity for passing the first part of the beam and the second part of the beam are such as to set the polarization directions of the principal polarization components of the first part of the beam and the second part of the beam to be along a circumferential direction around the optical axis.