Dual-focusing behavior of a one-dimensional quadratically chirped Pearcey-Gaussian beam

In this paper, we provide analytical solutions describing the dynamic behavior of the Pearcey-Gaussian beams propagating in free space. Based on the analytical solutions, explicit expressions governing the focusing distances of the Pearcey-Gaussian beams are found and verified by numerical simulations. For the linearly chirped Pearcey-Gaussian beam, it exhibits a uni-focusing behavior during propagation. Particularly, the focusing distance is independent on the linear chirp parameter and remains z_f = 2 unchanged. Of particular interest is that the quadratically chirped Pearcey-Gaussian beam focuses twice when the quadratic chirp parameter β < 0. The first and the second focusing distances are determined by z_f1 = 2/(1 - 4β) and z_f2 = -1/(2β), respectively. Furthermore, we numerically investigate the peak powers at the different focusing positions and find that as β increases, the peak powers at z_f1 and z_f2 linearly decrease. It is expected that the characteristics can be used for manipulating the focusing distances and the peak powers to generate an optical beam with high peak power by adjusting the chirp parameter β.