We present a method for fast calculation of the electromagnetic field near the focus of an objective with a high numerical aperture (NA). Instead of direct integration, the vectorial Debye diffraction integral is evaluated with the fast Fourier transform for calculating the electromagnetic field in the entire focal region. We generalize this concept with the chirp z transform for obtaining a flexible sampling grid and an additional gain in computation speed. Under the conditions for the validity of the Debye integral representation, our method yields the amplitude, phase and polarization of the focus field for an arbitrary paraxial input field in the aperture of the objective. Our fast calculation method is particularly useful for engineering the point-spread function or for fast image deconvolution. We present several case studies by calculating the focus fields of high NA oil immersion objectives for various amplitude, polarization and phase distributions of the input field. In addition, the calculation of an extended polychromatic focus field generated by a Bessel beam is presented. This extended focus field is of particular interest for Fourier domain optical coherence tomography because it preserves a lateral resolution of a few micrometers over an axial distance in the millimeter range
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