Applicability of nonresonant artificial diamagnetics

Artificial diamagnetics are prominent for achieving extraordinarily strong diamagnetism in a wide frequency range. However, as far as the magnetic fields outside the artificial medium are concerned, bulk conductors show a very similar pattern. The question arises whether the complicated internal structure of artificial diamagnetics can, to this end, be replaced by a simpler object. We show that for an electrically small body, internal structuring is likely to make the effective diamagnetic response weaker than that of a simple conducting object.

Figure 1

(a) Equivalent permeability ${\mu }_{\mathrm{r}}^{\mathrm{eq}}$ for a copper ($\sigma =5.6\times {10}^7\mathrm{S}/\mathrm{m}$) sphere for several values of radius $a$. (b) Effective permeability of a metamaterial made of close copper loops with different radii $r_0$ and the same relative geometry and lattice. The radii of the rings are such that there are 10 unit cells per radius of the corresponding sphere in panel (a).

Figure 2

Magnetic levitation figure of merit for the two cases with the same parameters as in Fig. 1: (a) for a copper spherical shell of thickness $3\delta$ and (b) for a metamaterial made of closed copper loops; the loops are either completely metallic loops (solid lines) or, where possible (when $r_{\mathrm{w}}>3\delta$), hollow toroidal shells with wall thickness of $3\delta$ (dashed lines).

Figure 3

The magnetic levitation figure of merit of a metamaterial made of either complete metal loops (solid lines) or hollow toroidal shells with wall thickness of $3\delta$, provided that $r_{\mathrm{w}}>3\delta$ (dashed lines). The three depicted curves are described by the following sets of parameters: $\{r_0=4.545\mu \mathrm{m}$, $w=0.1$, $b_1=0.2$, $b_2=2.2\}$, $\{r_0=4.854\mu \mathrm{m}$, $w=0.03$, $b_1=0.06$, $b_2=2.06\}$, $\{r_0=4.95\mu \mathrm{m}$, $w=0.01$, $b_1=0.02$, $b_2=2.02\}$.