Magnetic Energy Harvesting and Concentration at a Distance by Transformation Optics

Based on transformation optics, we introduce a magnetic shell with which one can harvest magnetic energy and distribute it as desired in space with unprecedented efficiency at an arbitrary scale. It allows a very large concentration of magnetic energy in a free space region, which can be used for increasing the sensitivity of magnetic sensors, and the transfer of magnetic energy from a source to a given distant point separated by empty space, with possible applications in wireless transmission of energy.

Figure 1

Magnetic field lines and their density (in color; in units of applied magnetic field) for (a) the ideal cylindrical magnetic concentrator shell with homogenous anisotropic permeabilities ${\mu }_{\rho }\to \infty$ and ${\mu }_{\theta }\to 0$ and radii ratio $R_2/R_1=4$, and three discretized versions: (b) 36 wedges of alternating homogenous and isotropic ideal superconducting ($\mu =0.0001$) and ideal soft ferromagnetic ($\mu =10000$) wedges; (c) same with 72 wedges; and (d) same with 72 rectangular prisms, for which a very good behavior is obtained even though materials do not fill the whole shell volume.

Figure 2

Common strategies for field concentration use the gap between two superconductors. In (a) and (b) we show the magnetic energy density (normalized to that of applied field) for two ideal superconducting blocks ($\mu =0.0001$) with a thickness $w$ separated by a gap $b$ and decreasing heights $h=$ (a) $4b$, and (b) $b/10$, in a vertical applied magnetic field. In (c), our optimum homogeneous anisotropic concentrating shell (${\mu }_{\rho }\to \infty$ and ${\mu }_{\theta }\to 0$) with $R_2/R_1=4$ shows a large uniform energy density in its hole (for the comparison, notice that $2R_1=b$ and $w=R_2-R_1$). In (d), a homogeneous isotropic shell made of ideal soft ferromagnetic material with the same dimensions as (c) attracts more field lines from outside but cannot bring them to the hole, so the inner energy density is zero.

Figure 3

Left panels: the magnetic energy density of a cylindrical dipole (e.g., a small magnet) (a) is spatially redistributed towards larger radial distances by the use of our shell with $R_2/R_1=8$ (b); in (c) a second concentrator harvests the enhanced magnetic energy in its volume and transfers it to its hole, where a large value of magnetic energy is achieved. Right panels: in (d) magnetic energy density of two identical cylindrical dipoles separated a given gap; when separating and enclosing them with two of our shells with $R_2/R_1=4$ [(e)] the magnetic energy density in the middle free space is similar to that in (d); when the inner radii of the shells are reduced to $R_2/R_1=10$ [(f)] the magnetic energy is concentrated in the free space between the enclosed dipoles, enhancing the magnetic coupling.

Figure 4

Four magnetic dipoles show a weak interaction when they are separated by free space (a) and the field in the region between them is very small (light yellow color), even when they are closer (b). However, when all four dipoles are surrounded by our magnetic shells [$R_2/R_1=10$] (c) the field in their exterior becomes greatly enhanced yielding a larger energy density in the intermediate region between the dipoles.