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

Based on the angular spectrum representation of an optical beam and the method of stationary phase, the analytical vectorial structure of an apertured hollow Gaussian beam in the far-field has been derived without any approximation. The analytical expressions of the energy flux of the TE term, the TM term, and the apertured hollow Gaussian beam are also presented. The energy flux distributions of the TE term, the TM term, and the apertured hollow Gaussian beam are numerically depicted in the far-field reference plane. The influences of the f-parameter, the truncation parameter, and the beam order on the energy flux distributions of the TE term, the TM term, and the apertured hollow Gaussian beam are examined. Also, the effects of the f-parameter, the truncation parameter, and the beam order on the ratios of the total energy fluxes of the TE and the TM terms to that of the whole beam are investigated.