Degree of polarization of a tightly focused, partially coherent anomalous hollow beam

By using the Richards-Wolf vector diffraction theory, the formulas of the propagation of a linearly polarized, partially coherent anomalous hollow beam (AHB) through a high numerical aperture (NA) thin lens are derived. The tight focusing properties of a partially coherent AHB, including the degree of polarization, the intensity distribution, and the degree of coherence, are studied in detail. It is found that the focusing properties of a partially coherent AHB are closely determined by the initial coherence, initial beam parameters, and NA of the thin lens. However, no matter how high the value of eccentricity, the initial beam will reduce to a Gaussian beam of quasi-circular symmetry at the focal plane. No matter the value of NA, the size of the beam will remain invariant, which is an interesting phenomenon, and the beam has good stability. The results may find application in optical trapping with a partially coherent field.