Propagation of the Airy beam along the optical axis of a uniaxial medium

We investigate the propagation of an Airy beam along the optical axis of a uniaxial medium, and we find that the propagation property of the Airy beam is determined by the ordinary refractive index of uniaxial crystals and is independent of the ratio of the extraordinary to ordinary refractive index. We also know that the polarization state of linearly polarized Airy beams changes gradually during the propagation. This shows that the propagation properties of the Airy beam in uniaxial crystals along the optical axis is distinctly different from that orthogonal to the optical axis.