  The quantum theory of rotation angles (S. M. Barnett and D. T. Pegg, Phys.
Rev. A, 41, 3427-3425 (1990)) is generalised to non-integer values of the
orbital angular momentum. This requires the introduction of an additional
parameter, the orientation of a phase discontinuity associated with fractional
values of the orbital angular momentum. We apply our formalism to the
propagation of light modes with fractional orbital angular momentum in the
paraxial and non-paraxial regime.
