Direct calculation of tightly focused field in an arbitrary plane

The Debye–Wolfdiffraction integral is a powerful tool for computing the focused field of an illumination by a high numerical aperture (NA) objective lens. With the help of the fast Fourier Transform (FT) algorithm, this integral can be evaluated at a high processing speed. In general, the FT pair is the illumination field at the entrance pupil and the focused field in a transverse plane in the focal region, i.e., the xy-plane. To analyze the field distribution of special beams in a titled plane, conventional calculations require a stack of slices of fields in the xy-planes plane by plane. Here, we propose a modified FT-based diffraction integral algorithm that exhibits a direct FT connection between the illumination field and the field in an arbitrary plane in the focal volume. As a result, significant time saving is achieved with the proposed algorithm. The algorithm is verified by numerous simulations and should be helpful to quickly design holograms to shape complex focus used in holographic optical tweezers.