Nonparaxial propagation of a vectorial apertured off-axis anomalous hollow beam

Based on the vectorial Rayleigh-Sommerfeld integral formula and the complex Gaussian expansion of the hard-edge aperture function, an analytical propagation expression for a nonparaxial vectorial off-axis anomalous hollow beam (AHB) passing through a rectangular aperture is derived. The corresponding closed-forms for unapertured case, the far field expression and the scalar paraxial result are also given as special cases of the general formulae. By using the derived formulae, some numerical examples are given to illustrate the nonparaxial propagation properties of a vectorial off-axis AHB through a rectangular aperture. It is shown that the f parameter, the off-axis displacement and the truncation parameter all play an important role in determining nonparaxial propagation behavior.