Inverse energy flux of focused radially polarized optical beams

We report the formation of a negative energy flux in the case of tight focusing of an arbitrary-order radially polarized annular beam. We consider not only the longitudinal component of the Poynting vector (energy flux density) and the region of its negative values, but also the local inverse energy flux as an integral characteristic over the region of negative values. Theoretically, the Richards-Wolf formulas in the Debye approximation are used to maximize the negative value of the energy flux density on the optical axis in the focal region for the second-order radial polarization ($m=2$). In this case, the local inverse energy flux, an integral characteristic in the bulk region of negative values, increases with increasing radial polarization order, i.e., for $m>2$. Jones matrices are employed to show that the result obtained will be valid for azimuthal polarization, as well as for a circularly polarized vortex beam. The results of numerical simulation are in good agreement with theoretical calculations. In addition, taking into account the approximate nature of the Richards-Wolf formulas, we also additionally model the tight focusing of a circular beam based on the solution of Maxwell's equation using the finite element method. It is shown that the local inverse energy flux in the case of the second-order radial polarization will be 1.5 times less than that in the case of the third-order radial polarization. In turn, the use of the fourth-order radial polarization makes it possible to increase the local inverse energy flux by a factor of 2.

Figure 1

$S_z$ calculated in the Debye approximation under focusing a radially polarized circular field at $m<0$: (a) $m=-1$, (b) $m=-2$, (c) $m=-3$, (d) $m=-4$; panel (e) shows the normalized cross sections along the $X$ axis. In panels (a) through (e), $x$ and $y$ range from $-2.5\lambda$ to $+2.5\lambda$.

Figure 2

$S_z$ calculated in the Debye approximation under focusing a radially polarized circular field at $m\ge 2$: (a) $m=2$, (b) $m=3$, (c) $m=4$, (d) $m=8$; panel (e) shows the normalized cross sections along the $X$ axis. In panels (a) through (e), $x$ and $y$ range from $-2.5\lambda$ to $+2.5\lambda$.

Figure 3

Square of the electric field amplitude calculated by the finite element method under the focusing of a radially polarized annular field at $m\ge 2$: (a) $m=2$, (b) $m=3$, (c) $m=4$. $x$ and $y$ range from $-5\lambda$ to $+5\lambda$.

Figure 4

$S_z$ calculated by the finite element method under the focusing of a radially polarized circular field at $m\ge 2$: (a) $m=2$, (b) $m=3$, (c) $m=4$; panel (d) shows the normalized cross sections along the $X$ axis. In panels (a) through (c), $x$ and $y$ range from $-5\lambda$ to $+5\lambda$.