Non-paraxial investigation in the far field properties of controllable dark-hollow beams diffracted by a circular aperture.

Based on the angular spectrum representation and the method of stationary phase, an analytically asymptotic expression of the far field of a controllable dark-hollow beam (CDHB) diffracted by a circular aperture is derived. The normalized intensity distributions of an apertured CDHB in the far field are numerically demonstrated. The influence of the f-parameter on the non-paraxiality of the far field of an apertured CDHB is the largest, and the effect of the truncation parameter delta on the non-paraxiality of the far field is the smallest. The difference between the non-paraxial and the paraxial results is independent of the beam order N. The far field pattern of apertured CDHBs sometimes is only a central bright spot. Under certain conditions, a weakly brightly outer ring around the central bright spot is detected in the far field pattern.