Asymmetric Cherenkov emission in a topological plasmonic waveguide

Here, we investigate the Cherenkov emission by an array of moving electric charges in the vicinity of a topologically nontrivial gyrotropic material. It is shown that the nonreciprocal material response may result in a robustly asymmetric Cherenkov emission, such that the spectrum of the emitted radiation and the stopping power depend strongly on the sign of the particle velocity. It is demonstrated that the main emission channels are determined by the unidirectional edge states supported by the topological material. We consider as examples both magnetized plasmas and Weyl semimetals. The latter may exhibit a spontaneous nonreciprocal response without a biasing magnetic field.

Figure 1

(a) A linear array of charged particles moves with a constant velocity $v$ at a distance $d$ from either a (i) magnetized plasma (biased along the $z$ direction) or a Weyl semimetal. (b) Band diagram of the natural plane-wave modes for a bulk magnetized plasma. (c) Band diagram of the natural plane-wave modes for a bulk Weyl semimetal.

Figure 2

(a) Dispersion diagrams of the edge (surface) plasmons propagating at an interface between air and a magnetized plasma with ${\omega }_0=0.5{\omega }_p$ (blue thick lines). The plasmon resonance occurs at $\omega ={\omega }_{+}$, for $k_x>0$, and $\omega ={\omega }_{-}$ for $k_x<0$. The thin black lines represent the curves $\omega =k_{x}v$ for the velocities $v_1=\pm 0.1c$ and $v_2=\pm 0.4c$. Their intersection with the dispersion diagram determines the main emission channels in the Cherenkov problem. (b) Amplitude of the reflection coefficient (in a logarithmic scale) for $v=\pm 0.7c$. The vertical gridlines correspond approximately to $\omega /\phantom{\omega {\omega }_p\approx 0.78}{\omega }_p\approx 0.78{\omega }_{+}$ (curve $v=0.7c$) and $\omega /\phantom{\omega {\omega }_p}{\omega }_p\approx 0.87{\omega }_{-}$ (curve $v=-0.7c$). The collision frequency was taken equal to $\mathrm{\Gamma }=0.01{\omega }_p$.

Figure 3

Real part of the magnetic field emission spectrum $H_z\left(x,y,\omega \right)$ (in arbitrary units) for an array of moving charges in the vicinity of a magnetized plasma with ${\omega }_0=0.5{\omega }_p, {\omega }_{p}d/c=1$, and $\mathrm{\Gamma }=0^{+}$. (a) $v=0.4c$ and (i) $\omega =0.94{\omega }_{+}$, (ii) $0.7{\omega }_{-}$, (iii) $0.97{\omega }_{-}$. (b) Similar to (a) but for $v=-0.4c$.

Figure 4

Time snapshot ($t=0$) of the magnetic field $H_z$ (in arbitrary units) for an array of moving charges in the vicinity of a magnetized plasma with cyclotron frequency ${\omega }_0=0.5{\omega }_p$ and for ${\omega }_{p}d/c=1$. (ai) $v=0.7c, \mathrm{\Gamma }=0.001{\omega }_p$ and (aii) $v=-0.7c, \mathrm{\Gamma }=0.001{\omega }_p$. (b) Similar to (a) but with $\mathrm{\Gamma }=0.1{\omega }_p$.

Figure 5

(a) Normalized power spectral density $G$ as a function of the normalized frequency for ${\omega }_{p}d/c=1$. The vertical gridlines indicate the points that satisfy the selection rule (11), relatively near the plasmon resonances $\omega \approx {\omega }_{\pm }$. (b) Normalized stopping power $P_{ext}$ as a function of the charge velocity $v$, for $d=d_0, d=2d_0$, and $d=4d_0$ with ${\omega }_p{d}_0/c=1$.

Figure 6

(a) Dispersion diagrams of the edge (surface) plasmons propagating at an interface between air and a Weyl semimetal (purple thick lines). The plasmon resonance occurs at ${\omega }_{+}\approx 0.59{\omega }_p$, for $k_x>0$, and ${\omega }_{-}\approx 0.84{\omega }_p$, for $k_x<0$. The thin black lines represent the curves $\omega =k_{x}v$ for the velocities $v_1=\pm 0.1c$ and $v_2=\pm 0.4c$. (b) Amplitude of the reflection coefficient (in a logarithmic scale) for $v=\pm 0.3c$. The vertical gridlines correspond approximately to $\omega \approx 0.99{\omega }_{+}$ (curve $v=0.3c$) and $\omega \approx 0.98{\omega }_{-}$ (curve $v=-0.3c$).

Figure 7

Time snapshot ($t=0$) of the magnetic field $H_z$ (in arbitrary units) for an array of moving charges in the vicinity of a Weyl semimetal with ${\varepsilon }_{\infty }=1, {\omega }_b=0.5{\omega }_p$, and ${\omega }_{p}d/c=1$. (ai) $v=0.7c$ and (aii) $v=-0.7c$.

Figure 8

(a) Normalized power spectral density $G$ as a function of the normalized frequency for ${\omega }_{p}d/c=1$.The vertical gridlines indicate the points that satisfy the selection rule (11). (b) Normalized stopping power $P_{\mathrm{ext}}$ as a function of the charge velocity $v$, for $d=d_0, d=2d_0$, and $d=4d_0$ with ${\omega }_p{d}_0/c=1$. In the simulations, we used ${\varepsilon }_{\infty }=1$ and ${\omega }_b=0.5{\omega }_p$.

Figure 9

Time snapshot ($t=0$) of the magnetic field $H_z\left(x,y,t\right)$ (in arbitrary units) radiated by a linear array of moving charges for ${\omega }_b=0.5{\omega }_p$ and ${\omega }_{p}d/c=1$. (a) The charges velocity is $v=0.7c$ and the high-frequency permittivity is (i) ${\varepsilon }_{\infty }=3$, (ii) 5, and (iii) 13. (b) Similar to (a) but with $v=-0.7c$.

Figure 10

(a) Comparison between the exact stopping power (solid line) and the stopping power obtained with the quasistatic approximation [Eq. (16)] (dashed line) for a “pencil” beam. (b) Stopping power obtained with the quasistatic approximation [Eq. (17)] for a “pointlike” beam. In both plots we use ${\omega }_{p}d/c=0.1$ and ${\omega }_0/{\omega }_p=0.4$.