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

We suggest a time-effective single-integral algorithm to calculate tight focusing of a collimated continuous-wave coherent 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 the 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 from several times to two orders faster for single-point computations and from two to five orders faster for typical focal-region computations. Some existing algorithms based on the discrete Fourier transform can exceed the single-integral algorithm in computational speed up to two orders of magnitude but their accuracy is significantly less and error level is poorly controllable, whereas the single-integral algorithm is able to reproduce field values obtained by the direct Richards-Wolf integration with unlimited accuracy and completely controllable error level.