Light emission by accelerated electric, toroidal, and anapole dipolar sources

Emission of electromagnetic radiation by accelerated particles with electric, toroidal, and anapole dipole moments is analyzed. It is shown that ellipticity of the emitted light can be used to differentiate between electric and toroidal dipole sources and that anapoles, elementary neutral nonradiating configurations, which consist of electric and toroidal dipoles, can emit light under uniform acceleration. The existence of nonradiating configurations in electrodynamics implies that it is impossible to fully determine the internal makeup of the emitter given only the distribution of the emitted light. Here we demonstrate that there is a loop hole in this “inverse source problem.” Our results imply that there may be a whole range of new phenomena to be discovered by studying the electromagnetic response of matter under acceleration.

Figure 1

Radiation pattern of a pointlike nonradiating configuration (anapole) subjected to acceleration. (a) Electric and toroidal dipoles that together make up the anapole. The electric dipole ($\mathit{p}$) corresponds to two separated opposite charges. The toroidal dipole ($\mathit{T}$) corresponds to a loop of magnetization or, equivalently, to current flowing along the meridians of a torus. (b) Radiation pattern (radius represents power per solid angle) of a dynamic anapole with moment $\mathit{N}$, accelerating with constant acceleration $\mathit{a}$.

Figure 2

World line of the uniformly accelerating particle. The blue curve (hyperbola) shows the path of the uniformly accelerating particle (blue dot). Vectors ${\widehat{\tau }}^{\mu }$ and ${\widehat{a}}^{\mu }$ are orthonormal vectors in the direction of instantaneous four-velocity and four-acceleration. Together they span $tz$ subspace. The inset shows two other orthonormal vector ${\widehat{x}}^{\mu }$ and ${\widehat{y}}^{\mu }$ which span the $xy$ subspace. The four vectors form a complete orthonormal set that spans the space-time. Speed of light is denoted by $c$.

Figure 3

Radiation patterns of accelerating point particles with electric (top row), toroidal (middle row), and anapole dipole moments. The magnitude of acceleration is $a$. In all cases the corresponding dipoles are assumed to be harmonically oscillating along a single axis [e.g., $\mathit{p}=\widehat{\mathit{z}}{p}_{0}cos\left(\omega t\right)$ with $p_0=\mathrm{const}$ and the same for $\mathit{T}$ and $\mathit{N}]$. The key parameter is $a/\omega c$, where $c$ is the speed of light. This parameter corresponds to how many cycles of oscillation (i.e., periods of $2\pi /\omega$) it takes for the accelerating particle to reach relativistic speed (starting from rest). Three cases are considered: The second column shows radiation patterns for the case of low acceleration. For an anapole the acceleration magnitude has to be above zero for nonzero radiation. The third and fourth columns show the radiation patterns for the cases of intermediate and high accelerations. The fourth column also corresponds to the radiation patterns of dc dipoles (i.e., zero-frequency limits $\omega \to 0,a>0$).

Figure 4

Ellipticity of the light emitted by an electric dipole point particle. (a) Definition of the ellipticity $\chi$. The radiation is emitted by a localized particle at the origin. Since polarization of the emitted radiation is transverse, one can decompose it into azimuthal ($\widehat{\mathbf{\varphi }}$) and polar ($\widehat{\mathbf{\theta }}$) components. Elliptical polarization corresponds to the electric field of the emitted radiation tracing an ellipse on the $\varphi \theta$ plane. Ellipticity ($\chi$) is defined as the angle, the tangent of which is the ratio of the semiminor and semimajor radii of the polarization ellipse. The positive angle of ellipticity corresponds to an electric field rotating clockwise [as shown in (a)] around the axis of radiation propagation. (b) Definition of the stereographic projection. The projection allows mapping a scalar field, defined on a surface of a unit sphere, onto a flat plane. Here, for simplicity, we show a circle instead of the sphere and a line instead of the plane. A chosen point on the circle (green circle), which corresponds to angle $\theta$ is projected onto the $x$ axis by plotting a straight line from the south pole of the circle to the chosen point. The point at which the straight line intersects the $x$ axis corresponds to the position of the projected point (orange square). Formally, all points on the surface of a sphere, parametrized by ($\theta ,\varphi$), are projected onto points on a plane with Cartesian coordinates $\left(tan\frac{\theta }{2}cos\varphi ,tan\frac{\theta }{2}sin\varphi \right)$. (c) Color map of the ellipticity of the radiation from the accelerated electric dipole point particle. The electric dipole moment of the particle is $\mathit{p}=p_{0}cos\left(\omega t\right)\widehat{\mathit{z}}$, whereas the normalized acceleration is $a/\omega c={10}^{-7}$ in the $\widehat{\mathit{x}}$ direction. The ellipticity is plotted in a stereographic projection [see (b)] with the polar angle ranging from $\theta =0^{\circ }$ (north pole) to $\theta ={120}^{\circ }$. The blue color on the color map corresponds to negative ellipticity and is clipped for $\chi >-{10}^{-9}\mathrm{rad}$. The red and orange colors correspond to positive ellipticity and are clipped for $\chi <{10}^{-9}\mathrm{rad}$.