The analytical expressions of one-dimensional cosh-Airy and cos-Airy beams in the parabolic potential are derived in the general and the phase transition points. The expression in the phase transition point shows a symmetric Gaussian intensity profile and is independent of any Airy features, which is completely different from that in the general point. The intensity, the center of gravity, and the effective beam size of the cosh-Airy and cos-Airy beams in the parabolic potential are periodic and have the same period. The effects of the transverse displacement, the cosh factor, and the cosine factor on these periodic behaviors are also investigated. The direction of self-acceleration reverses every half-period. The phase transition point is also the inversion point of the intensity distribution, which indicates that the intensity distributions before and after the phase transition point are mirror symmetrical. The periodic behaviors of the normalized intensity, the center of gravity, and the effective beam size of the cosh-Airy and cos-Airy beams in the parabolic potential are attractive and well displayed. The results obtained here may have potential applications in particle manipulation, signal processing, and so on.
Periodic inversion and phase transition of finite energy Airy beams in a medium with parabolic potential