Weakly diverging to tightly focused Gaussian beams: a single set of analytic expressions: continuation-symmetric...

The always diverging-converging laser beams, more rigorously referred to as Gaussian beams, are part of many physical and electro-optical systems. Obviously, a single set of analytic expressions describing these beams in a large span of divergence-convergence angles at the focal plane, and at any distance away from the focal plane, will prove very handy. We have recently published three such analytic sets, one set for linearly polarized beams and two sets for radially polarized beams. However, our published analytic set for linearly polarized beams describes nonsymmetric electric-magnetic field components. Specifically, the strong transverse magnetic field component does not become elliptic at very large divergence angles as it should be, and the other transverse magnetic component, indeed very weak, is missing altogether. Here we present an analytic set of expressions symmetrically describing linearly polarized Gaussian beams. The symmetry applies to the x-electric y-magnetic components and vice versa and to the two electric-magnetic z-components. An important property of the presented set of expressions is power conservation. That is, the electromagnetic power crossing a plane transverse to the propagation direction in a unit time is conserved. Power conservation assures beam description accuracy at any axial distance. The presented analytic expressions, although not strictly satisfying Maxwell's equations, describe Gaussian beams with very reasonable accuracy from low divergence angles up to divergence angles as large as 0.8 rad in a medium with refractive index of 1.5, i.e., up to a NA of 1.1. These expressions should then readily assist in the design of practically all laser-related systems and in the research of diverse physics and electro-optic fields.