Rapid calculation of the Richards-Wolf vector diffraction integral via a matrix product.

The Richards-Wolf vector diffraction integral is fundamental for modeling tight focusing. However, existing methods for the forward and inverse calculations of tightly focused ultrashort pulsed beams generally involve a trade-off between numerical accuracy and computational efficiency. To overcome this limitation, for the first time, to the best of our knowledge, the matrix product is introduced to reformulate this integral, creating a fast, accurate, and inherently invertible paradigm for forward and inverse simulations of tightly focused fields. Our results demonstrate the high fidelity and direct invertibility of this approach. We believe that this approach can pave the way for promising applications of tight focusing in high-energy laser physics, strong-field optics, and laser-driven particle acceleration.