Focal shift in tightly focused Laguerre-Gaussian beams

We study the nonparaxial propagation behavior of Laguerre-Gaussian (LG) beams under the tight focusing condition by using the Rayleigh-Sommerfeld integrals. We obtain an analytical formula and show the focal shift with respect to the geometric focus of a high numerical-aperture objective. An analytical expression for the focal shift of a tightly focused LG beam increases with the focal length while decreases with the beam waist. This approach can be extended to deal with the tight focusing field and the focal shift of LG vector fields with space-variant polarization distributions or other focusing behaviors such as the 4π focusing and the Fresnel zone plates.