Weakly diverging to tightly focused Gaussian beams: a single set of analytic expressions

Analytic expressions describing all vector components of Gaussian beams, linearly polarized as well as radially polarized, are presented. These simple expressions, to high powers in divergence angle, were derived from a single-component vector potential. The vector potential itself, as in the 1979 work of Davis [Phys. Rev. A19, 1177 (1979)PLRAAN1050-294710.1103/PhysRevA.22.1159], was approximated by the first two terms of an infinite series solution of the Helmholtz equation. The expressions presented here were formulated to emphasize the dependence of the amplitude of the various field components on the beam's divergence angle. We show that the amplitude of the axial component of a linearly polarized Gaussian beam scales as the divergence angle squared, whereas the amplitude of the cross-polarized component of a linearly polarized Gaussian beam scales as the divergence angle to the fourth power. Weakly diverging Gaussian beams as well as strongly focused Gaussian beams can be described by exactly the same set of mathematical expressions, up to normalization constant. For a strongly focused linearly polarized Gaussian beam, the ellipticity of the dominant electric field component, typically calculated by the Debye-Wolf integral, is reproduced. For yet higher accuracy, terms with higher powers in divergence angle are presented, but the inclusion of these terms is limited to low divergence angles and short axial distances.