Effective spacetime description of light propagation in linear magnetoelectric media

TL;DR

The paper investigates light propagation in linear magnetoelectric media through an effective spacetime framework, deriving a metric-like tensor g^{ab} = (1/sqrt(n)) [ eta^{ab} - (n^2 - 1 - alpha^2) v^a v^b + v^a alpha^b + v^b alpha^a ] that encodes optical response. In the geometric optics limit, light obeys the eikonal equation g^{mu nu} k_mu k_nu = 0 and travels along geodesics k^mu nabla_mu k^alpha = 0, enabling a kinematic analogy with curved spacetime. The authors present slab configurations yielding horizon-like one-way propagation regions and even a time-machine–like spacetime related to a spinning cosmic string, including explicit eikonal solutions such as S = -omega t + n_0 omega x + (sqrt(pi) omega ell / 2) alpha_0 erf(x/ell). They discuss the regime of validity, namely the non-dispersive approximation, and suggest that metamaterials with enhanced magnetoelectric effects could realize richer analogs of gravitational phenomena.

Abstract

Formal analogies between gravitational and optical phenomena have been explored for over a century, providing valuable insights into kinematic aspects of general relativity. Here, this analogy is employed to study light propagation in linear magnetoelectric media from an effective spacetime perspective. Starting from Maxwell's equations in covariant form, it is shown that an effective metric can always be identified for linear, non-dispersive magnetoelectric materials. The effective metric is then used to construct analog models in the limit of geometric optics. Among the optical effects analyzed, it is shown that under reasonable assumptions on the magnitude of the magnetoelectric response, a one-way propagation region can be established, which behaves analogously to an event horizon.

Figures (2)

• Figure 1: Light ray propagation in an analog model displaying event horizons. Top: The dashed red curve corresponds to a single light ray obtained from the negative branch, $f^{(-)}$, in Eq. \ref{['eqdifBH']}. All such rays have the same property: they propagate leftwards and cross the region $x=0$ with a small phase velocity. The rays determined by the positive branch $f^{(+)}$ possess a more interesting behavior. Rays determined by this branch cannot cross the surface $x=0$. Bottom: Geodesics close to $x=0$. Two distinct surfaces at $x\approx\pm 10^{-8} \ell$, determined by $\dot{x}=0$, separate the space into three regions with distinct properties. Light rays in $x<-10^{-8}\ell$ propagate rightwards and accumulate at $x=-10^{-8}\ell$, whereas rays in $10^{-8}\ell<x$ also propagate rightwards but scape to $x\rightarrow\infty$. In the region $|x|<10^{-8}\ell$ light rays propagate only leftwards, and are dragged towards the surface $x=-10^{-8}\ell$. The depicted geodesic profile resembles the principal null geodesics of a Kerr black hole Carter1968Gralla. The surfaces $x=\pm10^{-8}\ell$ play the role of analog event horizons, in the sense that they can only be crossed by light in one direction.
• Figure 2: Examples of level curves (i.e., light rays) of the eikonal. Here the parameters, $\alpha_0=-1.5, n_0=1$, are such that near $x=0$ the magnitude of the magnetoelectric effects dominate light ray propagation. The green dot-dashed curves represent geodesics determined by the branch $t^{(+)}$, whereas the red dashed curve corresponds to a curve determined by $t^{(-)}$. The phase velocity is given by the inverse of the slopes of the curves, and thus, the light ray modeled by the dashed curve crosses $x=0$ with a smaller phase velocity. For the dot-dashed curve, because the slopes vanish at $x\approx\pm0.64\ell$ (the vertical dotted lines), near these points the phase velocity is superluminal.