We show that if a nonzero $E$-field intensity is found in the focus of a circularly polarized optical vortex with phase singularity or a higher-order azimuthally polarized beam with polarization singularity, an energy backflow will occur at the center of the focal plane. Using the Richards-Wolf formulas, which describe the electromagnetic field near the tight focus of an aplanatic system containing an ideal spherical lens, we prove that the negative on-axis propagation of light occurs when focusing (i) a left-hand circularly polarized optical vortex with the topological charge $m=2$ or (ii) a conventional second-order azimuthally polarized radially symmetric laser beam. Notably, in the former case, the negative energy flow rotates on a spiral about the optical axis, while in the latter case, the Poynting vector is perpendicular to the focal plane at any point. The theoretical conclusions are confirmed by numerical simulation.

Based on the Richards-Wolf formalism, we obtain for the first time a set of explicit analytical expressions that completely describe a light field with a double higher-order singularity (phase and polarization), as well as distributions of its intensity and energy flux near the focus. A light field with the double singularity is an optical vortex with a topological charge m and with nth-order cylindrical polarization (azimuthal or radial). From the theory developed, rather general predictions follow. 1) For any singularity orders m and n, the intensity distribution near the focus has a symmetry of order 2(n – 1), while the longitudinal component of the Poynting vector has always an axially symmetric distribution. 2) If n = m + 2, there is a reverse energy flux on the optical axis near the focus, which is comparable in magnitude with the forward flux. 3) If m ≠0, forward and reverse energy fluxes rotate along a spiral around the optical axis, whereas at m = 0 the energy flux is irrotational. 4) For any values of m and n, there is a toroidal energy flux in the focal area near the dark rings in the distribution of the longitudinal component of the Poynting vector.

Using Richards–Wolf formulas, we show analytically that when strongly focusing a circularly polarized axisymmetric non-vortex field (e.g., a Gaussian beam or zero-order Bessel–Gaussian beam), a spiral energy flow is produced around the sub-wavelength focal spot. This phenomenon can be explained by the conversion of spin angular momentum associated with the circular polarization state into the near-focus orbital angular momentum, with the on-axis angular momentum being strictly zero. It is also shown that a linearly polarized optical vortex with topological charge 2 produces an on-axis near-focus energy backflow whose amplitude (defined by the negative longitudinal component of the Poynting vector) is comparable with that of the incident beam.

By decomposing a linearly polarized light field in terms of plane waves, the elliptic intensity distribution across the focal spot is shown to be determined by the $E$-vector’s longitudinal component. Considering that the Poynting vector’s projection onto the optical axis (power flux) is independent of the $E$-vector’s longitudinal component, the power flux cross section has a circular form. Using a near-field scanning optical microscope (NSOM) with a small-aperture metal tip, we show that a glass zone plate (ZP) having a focal length of one wavelength focuses a linearly polarized Gaussian beam into a weak ellipse with the Cartesian axis diameters ${\mathrm{FWHM}}_x=(0.44\pm 0.02)\lambda$ and ${\mathrm{FWHM}}_y=(0.52\pm 0.02)\lambda$ and the (depth of focus) $\mathrm{DOF}=(0.75\pm 0.02)\lambda$, where $\lambda$ is the incident wavelength. The comparison of the experimental and simulation results suggests that NSOM with a hollow pyramidal aluminum-coated tip (with 70° apex and 100 nm diameter aperture) measures the transverse intensity, rather than the power flux or the total intensity. The conclusion that the small-aperture metal tip measures the transverse intensity can be inferred from the Bethe–Bouwkamp theory.

Based on the Richards-Wolf theory, it is strictly shown that in the sharp focus of a linearly polarized laser beam, the flux of a spin vector has only transverse components (the effect of photonic wheels or a photonic helicopter). For a linearly polarized optical vortex, the orbit-spin conversion leads to the appearance of both longitudinal and transverse components of the spin angular momentum (SAM) vector in the focus. We show that in the strong focus of a circularly polarized Gaussian beam, the longitudinal component of the SAM is maximal on the optical axis, with the longitudinal component of the orbital angular momentum (OAM) being maximal on a ring. In this way, the effect of SAM and OAM on the motion of a trapped microparticle can be evaluated separately. Spin-orbit conversion is experimentally demonstrated for a circularly polarized Gaussian beam when transverse energy flux (orbital angular momentum) arises in the focus, which is transmitted to a microparticle and causes it to rotate. Switching the handedness of circular polarization (from left to right) switches the direction of microparticle rotation. It is also shown here that an azimuthally polarized vortex beam with an arbitrary integer topological charge generates in the focus a SAM vector with only an axial component (pure magnetization), whereas there is no transverse spin flux.