Analytical vectorial structure of an apertured hollow Gaussian beam in the far field

Based on the vector angular-spectrum and the complex Gaussian expansion of the aperture function, the structure of an apertured hollow Gaussian beam in the far field is given in a integral form. In virtue of the method of stationary phase approximation, the analytical vectorial structures are derived. Starting from the analytical expressions, the propagation properties of apertured hollow Gaussian beams with different order in the far field are illustrated graphically. In addition, the influence of the truncation parameter on far field distribution is studied detailedly. This research can shed light on the further understanding of the vectorial property of an apertured hollow Gaussian beam.