Fast calculation of tightly focused random electromagnetic beams: controlling the focal field by spatial coherence

Focusing of a vectorial (electromagnetic) optical beam through a high numerical aperture can be investigated by means of the Richards-Wolf diffraction integral. However, such an integral extends from two-dimensional to four-dimensional, greatly increasing the computation time and therefore limiting the applicability, when light with decreased spatial coherence is considered. Here, we advance an effective protocol for the fast calculation of the statistical properties of a tightly focused field produced by a random electromagnetic beam with arbitrary state of spatial coherence and polarization. The novel method relies on a vectorial pseudo-mode representation and a fast algorithm of the wave-vector space Fourier transform. The procedure is demonstrated for several types of radially (fully) polarized but spatially partially coherent Schell-model beams. The simulations show that the computation time for obtaining the focal spectral density distribution with 512 × 512 spatial points for a low coherence beam is less than 100 seconds, while with the conventional quadruple Richards-Wolf integral more than 100 hours is required. The results further indicate that spatial coherence can be viewed as an effective degree of freedom to govern both the transverse and longitudinal components of a tightly focused field with potential applications in reverse shaping of focal fields and optical trapping control.