Generation of hollow beam with radially polarized vortex beam and complex amplitude filter

The generation of hollow beams with a long focal depth from a radially polarized Bessel-Gaussian beam with a second-order vortex phase and an amplitude filter is theoretically investigated by Richards-Wolf's integral. The null intensity on the optical axis is achieved by introducing the second-order vortex. The long focal depth is a result of the amplitude filtering based on the cosine function and Euler transformation. Numerical results indicate that the focal depth of a hollow beam is improved from 0.96λ to 2.28λ with a slight increase of the transverse size for the simplest amplitude filter design. The intensity distribution twist phenomenon of the x- and y-polarized components around the optical axis due to the introducing of the vortex phase is also discussed. It is believed that the proposed scheme can be used to achieve particle acceleration and optical trapping.