Vectorial nonparaxial flattened gaussian beams and their beam quality in terms of the power in the bucket

The closed-form propagation equations of vectorial nonparaxial flattened Gaussian beams in free space are derived and some special cases are discussed. The power in the bucket (PIB) is extended to the nonparaxial regime, where the intensity is replaced by the z component of the time-averaged Poynting vector, and used to characterize beam quality of nonparaxial beams in the far field. Numerical calculation results show the dependence of the PIB of vectorial nonparaxial FGBs on the waist-width to wavelength ratio w0/λ and beam order N.