Laser beam tight focusing: Optimal algorithm for focal-region field computation within a plane perpendicular to the objective axis

We suggest an objective-axis-expansion single-integral algorithm based on the Richards-Wolf vectorial diffraction theory to calculate tight focusing of a collimated continuous-wave coherent laser beam by a high-numerical-aperture microscope objective into a planar layered microcavity. The mathematical basis of the algorithm is formed by (i) expansion of a focused beam into a linear superposition of circularly polarized vortex vector beams and (ii) expansion of the focal amplitudes of the circularly polarized vortex vector beams into power series of the focal distance coordinate $\ensuremath{\rho}$. The algorithm's computational speed is from four to seven orders of magnitude greater than that of the direct double-integral Richards-Wolf algorithm and up to two orders of magnitude greater than that of the Bluestein-method-based algorithm [Light: Sci. Appl. 9, 119 (2020)], the fastest existing method, for typical focal-plane field calculations. We show that such computational speeds are the fastest possible for focal field calculations in planes perpendicular to the microscope objective axis. These speeds are able to allow for the real-time prediction of focal fields. The accuracy of the suggested algorithm is completely controllable and unlimited. The reported computational speed enhancement is valid for three-dimensional focal field calculations as well because a three-dimensional region can be scanned slice by slice in the longitudinal direction.