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 o
f 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.
A unified description of the free-space cylindrical vector beams is presented, which is an integral transformation solution to the vector Helmholtz equation and the transversality condition. The amplitude 2-form of the angular spectrum involved in th
is solution can be arbitrarily chosen. When one of the two elements is zero, we arrive at either transverse-electric or transverse-magnetic beam mode. In the paraxial condition, this solution not only includes the known $J_1$ Bessel-Gaussian vector beam and the axisymmetric Laguerre-Gaussian vector beam that were obtained by solving the paraxial wave equations, but also predicts two new kinds of vector beam, called the modified-Bessel-Gaussian vector beam.
