Representation theory for vector electromagnetic beams

TL;DR

This paper develops a representation theory for finite electromagnetic beams using a matrix and vector factorization, revealing how the beam's polarization and displacement depend on a key angular parameter.

Contribution

It introduces a novel representation framework for electromagnetic beams that incorporates a fixed unit vector and analyzes their polarization and displacement properties.

Findings

Beam displacement depends on the angle between the fixed vector and propagation axis.

Elliptically polarized beams are transversely displaced from the center.

Theoretical expressions for displacement are derived in paraxial approximation.

Abstract

A representation theory of finite electromagnetic beams in free space is formulated by factorizing the field vector of the plane-wave component into a $3 \times 2$ mapping matrix and a 2-component Jones-like vector. The mapping matrix has one degree of freedom that can be described by the azimuthal angle of a fixed unit vector with respect to the wave vector. This degree of freedom allows us to find out such a beam solution in which every plane-wave component is specified by the same fixed unit vector $\mathbf{I}$ and has the same normalized Jones-like vector. The angle $\theta_I$ between the fixed unit vector and the propagation axis acts as a parameter that describes the vectorial property of the beam. The impact of $\theta_I$ is investigated on a beam of angular-spectrum field scalar that is independent of the azimuthal angle. The field vector in position space is calculated in the first-order approximation under the paraxial condition. A transverse effect is found that a beam of elliptically-polarized angular spectrum is displaced from the center in the direction that is perpendicular to the plane formed by the fixed unit vector and the propagation axis. The expression of the transverse displacement is obtained. Its paraxial approximation is also given.