Modified vectorial angular spectrum formula for propagation of non-paraxial beams

We presented modified forms of full-vector angular spectrum formula without singular points for longitudinal spectral components. In paraxial limitation, the modified angular spectrum formula has the same form as usual angular spectrum formula. Since there are not any singular points at the angular spectrum formula, we can apply it to analyze the propagation of non-paraxial beams with subwavelength-size waist and give results that strictly satisfy Maxwell’s equations because the calculating process includes contributions from all field components and angular spectral components.