The Non-Vortex Inverse Propagation of Energy in a Tightly Focused High-Order Cylindrical Vector Beam

Abstract

In this paper, the tight focusing of high-order cylindrical vector beams (beams with polarization singularity) was investigated. Using the Richards-Wolf formalism, there were obtained expressions for all projections of the electric and magnetic light fields with m-order polarization singularity in the focus of the aplanatic system. Also expressions for the longitudinal component of the Poynting vector were obtained. It was shown that these beams produce in the focal areas with the direction of the Poynting vector opposite to the direction of propagation of the beam. Moreover, the negative values could be comparable in absolute value with positive values; however, this strong inverse energy flow is obtained only while laser light is focused by a lens with high numerical aperture. Of particular interest is the case when the beam order is two (m = 2). In this case, the region where the Poynting vector longitudinal projection is negative is located on the optical axis. If the order of the beam is more than two (m > 2), than the reverse flow occurs near the optical axis and has a shape of a “tube.” Moreover, the width of the negative values region (the diameter of the “tube”) increases with increasing order of the beam, however, the absolute value of energy backflow decreases. Earlier, we reported on the observation of a spiral negative energy flow from the center of the focal plane of a focused vortex beam. In this paper, the negative propagation of a laminar near-axis energy flow is reported.

Keywords

• Poynting vector
• energy backflow
• cylindrical vector beam
• tight focusing
• Richards-Wolf formulae
• FDTD-method