Electron-Beam Instability in Left-Handed Media

We predict that two electron beams can develop an instability when passing through a slab of left-handed media (LHM). This instability, which is inherent only for LHM, originates from the backward Cherenkov radiation and results in a self-modulation of the beams and radiation of electromagnetic waves. These waves leave the sample via the rear surface of the slab (the beam injection plane) and form two shifted bright circles centered at the beams. A simulated spectrum of radiation has well-separated lines on top of a broad continuous spectrum, which indicates dynamical chaos in the system. The radiation intensity and its spectrum can be controlled either by the beams’ current or by the distance between the two beams.

Figure 1

Waves (solid line) and particles (dashed lines) characteristics in LHM (a) and RHM (b) slabs with thickness $L$. For LHM, particles and waves transport information in the opposite direction, while, for RHM, information can only be transmitted forward. (c),(d) Cherenkov field, emitted by particles (dashed lines), located within Cherenkov cones shown by solid lines. For case (c) the distance between the beams is too large, $D>D_{\max }$, and the radiation emitted by one beam cannot reach the other beam, so there is no interaction between beams. For case (d), $D<D_{\max }$, the radiation emitted by one beam reaches the tail of the other beam, creating a particle-density modulation producing a strong beam coupling.

Figure 2

Spatial distribution of the Cherenkov radiation emitted by a single particle. (a) The field distribution $E_z$ in the ($\rho$, ${\xi }^{\prime }$)-plane, where $\rho =k_{0z}r$ and ${\xi }^{\prime }=k_{0z}(z-vt)$ are the normalized radial and axial distances from the particle; $k_{0z}$ is the axial component of the wave vector ${\overrightarrow{k}}_0$ corresponding to the maximum of the spectrum of radiation. The electric field amplitude is depicted with the factor $\sqrt{\rho }$, which compensates the field decay $E\sim {\rho }^{-1/2}$. (b),(c): axial distribution of the field $E_z$ for $\rho =12.5$ [white line in Fig. 2a] and $\rho =0$ accordingly. Note that $E_z(\xi )$ for $\rho =12.5$ has the form of a wave packet with a distinguishable periodicity (b), whereas $E_z(\xi )$ for $\rho =0$ is fast-decaying and nonperiodic (c).

Figure 3

(a) Amplitude of the beam density modulation (due to bunching of particles) along the beam, divided by the average density. The maximum of the density modulation occurs near the front surface; thus, beams radiate with higher intensity near this surface. (b) Total radiation intensity $I_{\mathrm{tot}}$ versus the separation distance $\Delta$ between two beams. This intensity is nonzero for ${\Delta }_{\min }<\Delta <{\Delta }_{\max }$. (c) Spatial distribution of the radiation intensity $I(\eta ,\zeta )$ at the rear surface of the LHM slab. This distribution has the form of two shifted circles centered at the beams.

Figure 4

(a) Time dependence of the electric field $E(\tau )$ of the emitted wave at the rear surface of the LHM slab. The irregular modulation of the wave amplitude indicates chaotic dynamics of the beams. (b)–(d) Radiation spectra of a two-beam system when the beam current grows. Spectra show well-separated narrow lines on top of a continuous background. The current increases by a factor 1.5 from (b) to (d).