Mathematics of vectorial Gaussian beams

Uri Levy, Yaron Silberberg, and Nir Davidson

Since the development of laser light sources in the early 1960s, laser beams are everywhere. Laser beams are central in many industrial applications and are essential in ample scientific research fields. Prime scientific examples are optical trapping of ultracold atoms, optical levitation of particles, and laser-based detection of gravitational waves. Mathematically, laser beams are well described by Gaussian beam expressions. Rather well covered in the literature to date are basic expressions for scalar Gaussian beams. In the past, however, higher accuracy mathematics of scalar Gaussian beams and certainly high-accuracy mathematics of vectorial Gaussian beams were far less studied. The objective of the present review then is to summarize and advance the mathematics of vectorial Gaussian beams. When a weakly diverging Gaussian beam, approximated as a linearly polarized two-component plane wave, say (𝐸𝑥,𝐵𝑦), is tightly focused by a high-numerical-aperture lens, the wave is “depolarized.” Namely, the prelens (practically) missing electric field 𝐸𝑦,𝐸𝑧 components suddenly appear. This is similar for the prelens missing 𝐵𝑥,𝐵𝑧 components. In fact, for any divergence angle (𝜃𝑑<1), the ratio of maximum electric field amplitudes of a Gaussian beam 𝐸𝑥:𝐸𝑧:𝐸𝑦 is roughly 1:𝜃2𝑑:𝜃4𝑑. 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 nearly exactly 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 here-presented nearly exact six electric–magnetic Gaussian-beam-like field components are symmetric, meaning that the cross-sectional amplitude distribution of 𝐸𝑥(𝑥,𝑦) at any distance (𝑧) is similar to the 𝐵𝑦(𝑥,𝑦) distribution, 𝐸𝑦(𝑥,𝑦) is similar to 𝐵𝑥(𝑥,𝑦), and a 90° rotated 𝐸𝑧(𝑥,𝑦) is similar to 𝐵𝑧(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 at half-maximum intensity of the dominant vector field component does not exceed 60°. The highest accuracy field distributions to date 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 the approximation quality of different vector sets, and 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 a Hermite–Gaussian beam, and the other for computing the six practically Maxwell’s-equations-satisfying components of a focused laser beam, supplement this review.

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