Angular-momentum-selective nanofocusing with Weyl semimetals

TL;DR

This work tackles nanoscale control of orbital angular momentum (OAM) in light by exploiting a magnetic Weyl semimetal (WS) conical tip. It combines a cylindrical WS waveguide model with an adiabatic conical geometry, leveraging the axion term in the WS electrodynamics to yield a sign-dependent SPP dispersion for the OAM number $m$. The main result is that, within a finite frequency window set by the plasma frequency $\omega_p$ and Weyl node separation (through the parameter $\beta$), all modes with a given sign of $m$ propagate toward the cone apex while modes with the opposite sign radiate at finite radii, enabling selective OAM nanofocusing. This mechanism promises enhanced near-field control and potential applications in NSOM, TERS, and quantum information where nanoscale twisted light is advantageous. All mathematical notation is provided in $...$ to facilitate rigorous interpretation and reproduction.

Abstract

We investigate the theory of surface plasmon polaritons on a magnetic Weyl semimetal conical tip. We show that the axion term in the effective electrodynamics modifies the surface plasmon polariton dispersion relation and allows all modes with a given sign of the orbital angular momentum to be focused at the end of the tip. This is in contrast with normal metals, in which only one mode can reach the end. We discuss how this orbital angular momentum nanofocusing expands the potential of technologies that use this degree of freedom.

Figures (6)

• Figure 1: Conical waveguide scheme: a magnetic Weyl semimetal core surrounded by a dielectric medium. The vector $\mathbf{b}$ oriented along the $z$ direction describes the Weyl node separation in momentum space. The cone has an opening angle $\theta_c$. For further details, see main text.
• Figure 2: Size dependence of the SPP dispersion relation in a cylindrical WS waveguide. The parameters were chosen as $\beta=16$, $\epsilon_W=25$, $\epsilon_d=1$ and $\rho=4$ (left), $\rho=2.4$ (right). Here we show the bands $m=0$ (solid red), $m=1$ (dashed green), $m=-1$ (solid green), $m=2$ (dashed blue) and $m=-2$ (solid blue). The dotted purple line is the frequency $\omega_-$, see Eq. \ref{['eq:omega-']}. The dotted black line denotes a given frequency, here $\omega=0.65\omega_p$. For the larger section, the waveguide supports the $m=2$ mode at that frequency, while for the smaller one, this mode is not supported.
• Figure 3: Phase and group velocities \ref{['eq:vpvg']} along the cone (red and blue curves, respectively). Main panel: modes with $m=+2$ (dashed) and $m=-2$ (solid). The blue dashed curve reaches the horizontal dashed line. Inset: same for $m=\pm 1$. Dashed curves asymptotically approach the horizontal dashed line, but numerical accuracy prevents resolution of their final segment near this asymptote. The parameters used here are: $\omega/\omega_p=0.5$, $\epsilon_d=10$, $\epsilon_W=10$, and $\beta=10$. The $m=-2$ mode reaches the cone tip with vanishing phase and group velocities, therefore focusing the electromagnetic field energy, analogous to the $m=0$ mode of a conventional metal. In contrast, the $m=2$ mode is radiated into the dielectric at the radius $R_m$ (see Sec. \ref{['sec:nanofocusing']}), indicated by the vertical dashed line. The horizontal dashed line marks $c/\sqrt{\epsilon_d}$, the velocity at which the $m=+2$ mode is radiated. We note that $\delta_m$ in Eq. \ref{['eq:delta']} remains very small throughout the shown range of parameters.
• Figure 4: Refractive index of the $m=0,-1,-2,-3$ modes as a function of the position along the axis of the cone. The dots represent the numerical solutions of \ref{['eq:SecularEquation']}, while the solid lines are the best fits to the expression \ref{['eq:nsmallR']} with the single parameter $g_m$. In this plot, $\omega/\omega_p=0.4$, $\epsilon_d=10$, $\epsilon_W=10$ and $\beta=10$.
• Figure 5: Logarithm of the LFIE Eq. \ref{['eq:LFIE']} in the region $0.5R(z)<r<1.5R(z)$ around the surface of the cone, for material parameters $\beta=16$, $\epsilon_W=25$ and frequency $\omega=0.75\omega_p$. Lengths are in units of $c/\omega_p$. Left: $m=-1$, right $m=-2$.