Mechanism of formation of an inverse energy flow in a sharp focus

Abstract

A reverse energy flow is theoretically demonstrated to occur in the interference pattern generated by four plane waves with linear polarization. In some regions of the interference pattern, the right-handed triplets of plane-wave vectors k $({k}_{z}g0)$, E, and H (comprising a wave vector and $\mathbf{E}$- and H-field vectors) sum up to form an electromagnetic field described by a right-handed triplet of vectors k $({k}_{z}l0)$, E, and H. It is in these regions that the negative propagation of light occurs. On the optical axis the orbital energy flow, proportional to the light intensity, is shown to be positive, while the spin flow is negative and exceeds the orbital flow in magnitude. That is why the on-axis summary energy flow is negative. The magnitude of the reverse flow on the optical axis is two times lower than that of the intensity. A similar mechanism may apply to the case of sharply focusing a laser beam with second-order polarization or phase singularity. Using two identical micro-objectives with a numerical aperture of 0.95, it has been demonstrated experimentally that the intensity on the optical axis near the focus of an optical vortex with a topological charge of 2 is zero for right circular polarization and nonzero for left circular polarization. This confirms that in the latter case there is the reverse flow of light energy on the optical axis, since in the center of the measured energy flow distribution there is a very weak local maximum (the Arago spot) aroused due to diffraction of the forward flow by a circle with a diameter of 300 nm (the diameter of the tube with the reverse flow). Comparing the numerical and experimental intensity distributions, it is possible to determine the diameter of a ``tube'' with the reverse flow. For a numerical aperture of 0.95 and a wavelength of 532 nm, the diameter of the tube of the reverse flow along the optical axis is 300 nm. It is also shown experimentally that when an optical beam with second-order cylindrical polarization is focused with a numerical aperture of 0.95, there is a circularly symmetric energy flow in the focus with a very weak flow in the center (the Arago spot), whose distribution is determined by diffraction of the forward flux by an $\ensuremath{\sim}300$-nm-diameter circular area, where the energy flow is reverse. This also confirms that in the latter case, there is a reverse energy flow on the optical axis.