The effect of astigmatism induced by refraction on the orbital angular momentum of light

We use the Fourier transform and Snell's law to demonstrate how refraction at a flat interface induces astigmatism and transforms the spatial distribution of a stigmatic beam. Refraction makes the beam parameters for the transverse dimensions perpendicular and parallel to the plane of incidence grow differently and gives rise astigmatism. The decompositions of the orbital angular momentum of the beam before and after refraction are different. A single-value state of orbital angular momentum of the incident photon in a Laguerre-Gaussian mode is transformed into a superposition state.