Reversed Cherenkov-Transition Radiation by a Charge Crossing a Left-Handed Medium Boundary

We analyze the radiation from a charged particle crossing the boundary between an ordinary medium and a “left-handed” metamaterial. We obtain exact and approximate expressions for the field components and develop algorithms for their computation. The spatial radiation in this system can be separated into three distinct components, corresponding to ordinary transition radiation having a relatively large magnitude, Cherenkov radiation, and reversed Cherenkov-transition radiation (RCTR). The last one is explained by reflection and refraction of reversed Cherenkov radiation at the interface. Conditions for generating of RCTR are obtained. We note properties of this radiation that have potential applications in the detection of charged particles and accelerator beams and for the characterization of metamaterial macroscopic parameters ($\epsilon$, $\mu$).

Figure 1

Geometry of the problem.

Figure 2

Dependence of the electric field Fourier harmonic $\mathrm{Re}{E}_{\rho \omega }(\mathrm{V}{\mathrm{m}}^{-1}\mathrm{s})$ on the angle $\theta$ for a charged particle crossing the vacuum-RHM (top) and the vacuum-LHM (bottom) boundary for velocities below the Cherenkov threshold $\beta <{\beta }_{\mathrm{CR}}=0.7$; $\epsilon =1.7$, $\mu =1.2$ for RHM, $\epsilon =-1.7$, $\mu =-1.2$ for LHM; $q=-1\mathrm{nC}$, $\nu =10\mathrm{GHz}$, and $R=15\mathrm{cm}$. The values of $\beta$ are indicated near the curves.

Figure 3

Dependence of the electric field Fourier harmonic $\mathrm{Re}{E}_{\rho \omega }(\mathrm{V}{\mathrm{m}}^{-1}\mathrm{s})$ on the angle $\theta$ for vacuum-RHM transition and for $\beta >{\beta }_{\mathrm{CR}}=0.7$. Other parameters are the same as in Fig. 2. The values of $\beta$ are indicated near the curves.

Figure 4

Modification of the spatial distribution of the electric field Fourier harmonic $\mathrm{Re}{E}_{\rho \omega }(\mathrm{V}{\mathrm{m}}^{-1}\mathrm{s})$ with an increase in $\beta$ for a vacuum-LHM interface. Parameters are the same as in Fig. 2; ${\beta }_{\mathrm{CR}}=0.7$, ${\beta }_{\mathrm{TIR}}=0.98$.

Figure 5

Top: wave vectors and power flux densities of CR and RCTR (reflected and transmitted waves). Bottom: the lines are parallel to the power flux density $\overrightarrow{S}$ of CR, the reflected wave, and the transmitted wave; one can see areas of interference.