We present a theoretical study of the emission from a superluminal polarization current whose distribution pattern rotates (with an angular frequency ω) and oscillates (with a frequency Ω) at the same time and that comprises both poloidal and toroidal components. This type of polarization current is found in recent practical machines designed to investigate superluminal emission. We find that the superluminal motion of the distribution pattern of the emitting current generates localized electromagnetic waves that do not decay spherically, i.e., that do not have an intensity diminishing as with the distance from their source. The nonspherical decay of the focused wave packets that are emitted by the polarization currents does not contravene conservation of energy: The constructive interference of the constituent waves of such propagating caustics takes place within different solid angles on spheres of different radii centered on the source. For a polarization current whose longitudinal distribution (over an azimuthal interval of length 2π) consists of m cycles of a sinusoidal wave train, the nonspherically decaying part of the emitted radiation contains the frequencies i.e., it contains only the frequencies involved in the creation and implementation of the source. This is in contrast to recent studies of the spherically decaying emission, which was shown to contain much higher frequencies. The polarization of the emitted radiation is found to be linear for most configurations of the source.
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