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### Representation theory for vector electromagnetic beams

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.Comment: 16 pages. The final version appears in the Phys. Rev.

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