Propagation dynamics of Janus vortex waves

We study the propagation dynamics of Janus vortex wave under the action of a focusing lens based upon the formula of focused circular vortex Airy beams. Two dark foci would be generated under the action of a lens, and thus a perfect light hollow bottle could be formed. By controlling corresponding parameters, we could control the focal position and the relative intensity between the two focal intensities. The off-axis optical vortex (OV) would rotate rapidly in two focal regions, but keep still in the lens focus region. The angular displacement of OV in each focusing process is nearly π/2. (Note that the angular displacement for an off-axis OV in single focusing process of Gaussian beam is nearly π.) Two same OVs would repel to each other, but two opposite OVs would attract each other and annihilate at first focus plane in Janus vortex waves.