Propagation of nonparaxial vectorial anomalous hollow beams

The theoretical model of anomalous hollow beams proposed by Cai in the paraxial approximation is extended to the nonparaxial regime. The closed-form propagation expression for nonparaxial vectorial anomalous hollow beams in free space is derived and used to study their propagation properties in free space. It is found that anomalous hollow beams can not preserve their shape upon propagation, but the paraxial approximation condition for anomalous hollow beams additionally depends on the propagation distance, which is different from Gaussian beams. The far-field beam quality of nonparaxial vectorial anomalous hollow beams can be described by the power in the bucket(PIB), which decreases with increasing the f parameter (the ratio between wavelength and beam maist width).