Analysis of the behavior of highly focused, radially coherent Dirichlet sources

We introduce a novel class of electromagnetic light sources that exhibit radial coherence, a degree of coherence characterized by a Dirichlet kernel, and with radial or azimuthal polarization. We provide a mathematical description of the spectral irradiance, 3D degree of polarization, and electromagnetic degree of coherence for such sources when they are focused using a high numerical aperture microscope lens. Since the incident beam can be described as a modal expansion, we also consider how the number of terms affects the behavior of the field in the focal area. The formal analysis is complemented with numerical simulations. The study reveals interesting properties in the focal area, despite the focused beam is no longer radially coherent.