Propagation of Airy beams in the quadratic-index medium based on matrix optics

We investigate the propagation of the Airy beam in the quadratic-index medium based on matrix optics. It shows there is a periodic beam profile change induced by the inhomogeneity of medium, which displays the periodic change of the field distribution from the Airy profile to the Gaussian profile. We also know the duration time of the Gaussian profile of the beam is determined by the decay factor of the Airy beam. Yet, the accelerating trajectory of the Airy beam is independent of decay factor of the Airy beam. Moreover, we show that the beam size of the Gaussian profile is different for the Airy beam with the different decay factor.