Study of the focusing features of spatial amplitude and phase modulated radially polarized vortex beams in a 4pi focusing system

The focusing properties of phase and amplitude modulated radially polarized vortex beams in a 4pi focusing system are theoretically investigated near the focal plane by using Richards-Wolf vectorial diffraction method. The amplitude modulation of vortex beams can be adjusted by changing the start integration value. The phase modulation of vortex beams can be realized by adding liquid crystal variable retarder with the phase delay angle δ. The simulated results show that multiple spherical spots can be obtained near the focus of the 4pi focusing system with the decrease of amplitude. The phase delay angle δ of the input beams can generate extruding effect for the electrical field distribution near the focus of the 4pi focusing system. Some special intensity distributions can be obtained by changing topological charge m and phase delay angle δ. Optical chain can be generated in the case of m=1. Dark channel can be obtained in the case of m=2. These special focusing beams can also transform with phase modulation. With the increase of phase δ, the multiple spherical spots at m=0 change slowly into an optical chain, and finally become a dark channel. In contrast, the optical chain at m=1 changes slowly into multiple spherical spots; and the dark channel at m=2 changes into the superposition of optical spherical spots and the optical chain. These special focusing beams have potential applications in optical trapping and micro-manipulation.