Quantum calculation of the Vavilov-Cherenkov radiation by twisted electrons

We present a detailed quantum electrodynamical description of Vavilov-Cherenkov radiation emitted by a relativistic twisted electron in the transparent medium. Simple expressions for the spectral and spectral-angular distributions as well as for the polarization properties of the emitted radiation are obtained. Unlike the plane-wave case, the twisted electron produces radiation within the annular angular region, with enhancement towards its boundaries. Additionally, the emitted photons can have linear polarization not only in the scattering plane but also in the orthogonal direction. We find that the Vavilov-Cherenkov radiation emitted by an electron in a superposition of two vortex states exhibits a strong azimuthal asymmetry. Thus, the Vavilov-Cherenkov radiation offers itself as a convenient diagnostic tool of such electrons and complements the traditional microscopic imaging.

Figure 1

Angular distribution of the VC radiation by a Bessel vortex electron with conical angle ${\theta }_p$ for ${\theta }_p>{\theta }_0$ (left) and ${\theta }_p<{\theta }_0$ (right). The dotted circle shows the opening angle of the vortex electron; the solid circles correspond to the VC cones from selected plane-wave configurations inside the vortex electron. Radiation going in every direction inside this region (black dot) receives contributions from two such plane-wave components. The white dot corresponds to the direction along which the polarization is orthogonal to the emission plane, $K=-1$ in Eq. (38).

Figure 2

Geometry of the VC emission by a vortex electron in a three-dimensional side view.

Figure 3

Function $F({\theta }_k,{\theta }_p,{\theta }_0)$ defined in Eq. (29) (top) and the quantity $K$ defined in Eq. (37) (bottom) as functions of the photon polar angle for ${\theta }_0=14.5^{\circ }$.

Figure 4

Degree of linear polarization $P_l^{\mathrm{tw}}$ for a range of angles ${\theta }_k$ and ${\theta }_p$ for the same parameter choice as before.

Figure 5

Spectral-angular distribution as a function of emitted photon spherical angles ${\theta }_k$ and ${\phi }_k$ for ${\theta }_p={\theta }_0/2$ and for the superposition with $\mathrm{\Delta }m=\pm 3, \mathrm{\Delta }\alpha =\pi /2$, and $|c_1|=|c_2|=1/\sqrt{2}$.

Figure 6

Spectral-angular distribution as a function of emitted photon spherical angles ${\theta }_k$ and ${\phi }_k$ for the superposition with $\mathrm{\Delta }m=3, \mathrm{\Delta }\alpha =0$, and $|c_1|=|c_2|=1/\sqrt{2}$. The size of the circle is proportional to ${\theta }_k$.