Fast-speed algorithm to compute tight focusing of laser beams: The effectiveness of circularly polarized vortex beam series as a mathematical basis

We suggest a time-effective algorithm to calculate tight focusing of a collimated continuous-wave laser beam with an arbitrary cross-section light vector distribution by a high-aperture microscope objective into a planar microcavity. This algorithm is based on the mathematical fact that any beam can be decomposed into a superposition – either finite or infinite – of circularly polarized vortex vector beams, which allows one to factorize focal field into two parts, one of which depends only on distance coordinates $\rho$ and $z$ and the other one only on an azimuth $\phi$ in cylindrical coordinates. We compare the suggested algorithm with that based on the direct use of the double-integral Richards-Wolf method and demonstrate that the former is at least 5 times faster for single-point computations and at least two orders faster for typical focal-region computations.