Focusing of higher-order radially polarized Laguerre–Gaussian beam

We consider the formation of a small focal spot, which has been reported in [J. Opt. Soc. Am. A 24, 1793 (2007)], by separating a higher-order radially polarized Laguerre-Gaussian beam into two parts: a Bessel-like multi-ring part and an annular part. The latter forms a dominant small spot of longitudinal electric field component near the focus, while the former mainly contributes to a weak annular pattern of radial component.