Analytical description of propagation of vectorial apertured off-axis Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh–Sommerfeld diffraction integrals, an analytical description of propagation of vectorial nonparaxial off-axis Gaussian beams propagating through a rectangular aperture is presented. The on-axis, far-field and paraxial cases are dealt with as special cases of our general result and the corresponding mathematical formulations are given. It is shown that for the apertured off-axis cases, besides the f parameter, which determines the beam nonparaxiality, the truncation and off-axis parameters have to be introduced, which additionally affect the beam nonparaxial evolution behavior.