Dual Abruptly Focus of Modulated Circular Airy Beams

In this paper, we propose a new kind of circular Airy beam whose spectrum is imposed an initial quadratic phase modulation (QPM). Unlike common abruptly autofocusing beams, we find that such modulated circular Airy beams (MCAB) at image plane, i.e., initial plane, may be morphed into conventional Airy patterns or Bessel-like patterns in spatial regime, depending on the magnitude of QPM. We further investigate the propagation characteristics of the MCAB both analytically and numerically. It is found that the focal intensity, as well as the size of focal spot, do not change with the QPM, but the latter greatly affects the focal trajectory and the focal position of MCAB. In particular, we show that, for the case of Airy pattern, MCAB will exhibit a dual abruptly focusing behaviors having the same focal intensity as well as size of focal spot, where there is one focus behind the image plane, and the another one is in front of it. On the other hand, for the case of Bessel-like pattern, the dual abruptly foci are simultaneously observed behind the image plane for positive QPM but in front of it for negative QPM, thus leading to the formation of an elegant optical bottle with paraboloidal shape. Such novel dual focusing behaviors are expected to be applied in the field of optical trapping and particle acceleration.