An illumination optical apparatus for illuminating a pattern on a mask with illumination light through a pupil plane of the illumination optical apparatus, the illumination optical apparatus comprising: an optical integrator arranged in an optical path of the illumination light; the apparatus being characterized by: a birefringent member including a birefringent material characterized by a first refractive index for linearly polarized light polarized in a direction parallel to a fast axis of the birefringent material and a second refractive index for linearly polarized light polarized in a direction parallel to a slow axis of the birefringent material, the birefringent member being arranged in the optical path on an incidence side of the optical integrator and configured to change a polarization state of the illumination light so that first and second rays of the illumination light are polarized in different directions on the pupil plane, the first ray being a ray which passes through a first region on the pupil plane away from an optical axis of the illumination optical apparatus, the second ray being a ray which passes through a second region different from the first region on the pupil plane away from the optical axis, wherein the birefringent member is configured such that an optical path length of the first ray in the birefringent material is different from an optical path length of the second ray in the birefringent material, and wherein the birefringent member is configured to change the polarization state of the illumination light so that each of the first and second rays is polarized in a substantially circumferential direction about the optical axis on the pupil plane. an optical integrator arranged in an optical path of the illumination light; the apparatus being characterized by: a birefringent member including a birefringent material characterized by a first refractive index for linearly polarized light polarized in a direction parallel to a fast axis of the birefringent material and a second refractive index for linearly polarized light polarized in a direction parallel to a slow axis of the birefringent material, the birefringent member being arranged in the optical path on an incidence side of the optical integrator and configured to change a polarization state of the illumination light so that first and second rays of the illumination light are polarized in different directions on the pupil plane, the first ray being a ray which passes through a first region on the pupil plane away from an optical axis of the illumination optical apparatus, the second ray being a ray which passes through a second region different from the first region on the pupil plane away from the optical axis, wherein the birefringent member is configured such that an optical path length of the first ray in the birefringent material is different from an optical path length of the second ray in the birefringent material, and wherein the birefringent member is configured to change the polarization state of the illumination light so that each of the first and second rays is polarized in a substantially circumferential direction about the optical axis on the pupil plane. a birefringent member including a birefringent material characterized by a first refractive index for linearly polarized light polarized in a direction parallel to a fast axis of the birefringent material and a second refractive index for linearly polarized light polarized in a direction parallel to a slow axis of the birefringent material, the birefringent member being arranged in the optical path on an incidence side of the optical integrator and configured to change a polarization state of the illumination light so that first and second rays of the illumination light are polarized in different directions on the pupil plane, the first ray being a ray which passes through a first region on the pupil plane away from an optical axis of the illumination optical apparatus, the second ray being a ray which passes through a second region different from the first region on the pupil plane away from the optical axis, wherein the birefringent member is configured such that an optical path length of the first ray in the birefringent material is different from an optical path length of the second ray in the birefringent material, and wherein the birefringent member is configured to change the polarization state of the illumination light so that each of the first and second rays is polarized in a substantially circumferential direction about the optical axis on the pupil plane.