Rotation formulas in the jones calculus extended to axially symmetrically polarized beam

The Jones calculus is widely used for analyzing the polarization of a fully polarized beam. By the Jones calculus, usually, a beam having spatially homogeneous polarization is analyzed. Recently, axially symmetrically polarized beams have attracted great interest, whose polarization is not homogeneous in its cross section. The Jones calculus can be extended to such beams by adding angularly variant terms. We show the necessity of the correct rotation formulas in the Jones calculus extended to axially symmetrically polarized beam, and deduce them. We also show the effectiveness of this calculus by calculating the generation of two types of axially symmetrically polarized beams with an angularly variant optical element.