Far-field properties and beam quality of vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation

The far-field properties and beam quality of vectorial nonparaxial Hermite–Laguerre–Gaussian (HLG) beams are studied in detail, where, instead of the second-order-moments-based M2 factor, the extended power in the bucket (PIB) and βparameter are used to characterize the beam quality in the far field and the intensity in the formulae is replaced by the z   component of the time-averaged Poynting vector 〈Sz〉〈Sz〉. It is found that the 〈Sz〉〈Sz〉 PIB and βparameter of vectorial nonparaxial HLG beams depend on the mode indices n, m, αparameter and waist-width-to-wavelength ratio w0/λ and the PIB and βparameter are additionally dependent on the bucket's size taken.