The factor of nonparaxial Hermite–Gaussian beams and related problems

On the basis of the second-order moment of the power density and in the use of the series expansion, the expressions for the beam width, far-field divergence angle and M² factor of nonparaxial Hermite–Gaussian (H–G) beams are derived and expressed in a sum of the series of the Gamma function. The theoretical results are illustrated with numerical examples. The M² factor of nonparaxial H–G beams depends not only on the beam order m, but also on the waist-width to wavelength ratio w₀/λ. The far-field divergence angles of nonparaxial H–G beams with even and odd orders approach their upper limits θ_max=63.435 and 73.898, respectively, which results in M²<1 as w₀/λ→0. For the special case of m=0 our results reduce to those of nonparaxial Gaussian beams. Some problems related to the characterization of the nonparaxial beam quality are also discussed.