The Mathematics of Vectorial Gaussian Beams

On September 14, 2015 at 09:50:45 UTC, the Livingston, Louisiana, USA LIGO detector and 7 milliseconds later the Hanford, Washington, USA detector, sensed extremely weak gravitational waves. Sensing the exerted distance-change of 10-18 meter (proton diameter/1000) between two 40-kilogram masses 4000 meters apart was based on the interference of two coherent Gaussian beams. When a weakly diverging Gaussian beam, approximated as a linearly polarized two-component planewave, say (E_x,B_y), is tightly focused by a high numerical-aperture lens, the wave is “depolarized”. Namely – the pre-lens (practically) missing electric field E_y,E_z components suddenly appear. Similarly for the pre-lens missing B_x,B_z components. In-fact, for any divergence angle (θ_d<1), the ratio of maximum electric-field amplitudes of a Gaussian beam E_x:E_z:E_y is roughly 1:θ_d^2:θ_d^4. It follows that if a research case involves a tightly focused laser beam, then the case analysis calls for the mathematics of vectorial Gaussian beams. Gaussian-beam-like distributions of the six electric-magnetic vector field components that satisfy Maxwell’s equations are presented. We show that the near-field distributions with and without evanescent waves are markedly different from each other. The presented exact six electric-magnetic Gaussian-beam-like field components are symmetric. Meaning – the cross-sectional amplitude distribution of E_x (x,y) at any distance (z) is similar to the B_y (x,y) distribution, E_y (x,y) is similar to B_x (x,y) and a ninety degrees rotated E_z (x,y) is similar to B_z (x,y). Components’ symmetry was achieved by executing the steps of an outlined symmetrization procedure. Regardless of how tightly a Gaussian beam is focused, its divergence angle is limited. We show that the full cone angle to full-width-half-max intensity of the dominant vector field component does not exceed 60^0. Highest-accuracy to date field distributions of the less familiar higher-order Hermite-Gaussian vector components are also presented. Hermite-Gaussian E"-" B vectors only approximately satisfy Maxwell’s equations. We have defined a Maxwell’s-residual power measure to quantify approximation quality of different vector sets, each set approximately (or exactly) satisfies Maxwell’s equations. Several vectorial “applications”, i.e. research fields that involve vector laser beams are briefly discussed. The mathematics of vectorial Gaussian beams is particularly applicable to the analysis of the physical systems associated with such applications. Two user-friendly “Mathematica” programs, one for computing six high accuracy vector components of an Hermite-Gaussian beam, and the other for computing the six Maxwell’s-equations-satisfying components of a focused laser beam, supplement this review.