Figure 1

(Color online) $\mathrm{Re}\mu$ vs ${\log }_{10}(a∕\lambda )$ obtained by simulations for a periodic assembly of (a) open or (b) closed square metallic rings. The corresponding $\mathrm{Im}\mu$ vs ${\log }_{10}(a∕\lambda )$ are shown in panels (c) and (d), respectively. The simulations have been done using the commercial software packet COMSOL MULTIPHYSICS, which is essentially a finite-element method frequency domain solver. The complex current distribution was obtained directly from the simulated field distributions and, averaged over the unit cell, gave the magnetization (this averaging produces unphysical results when the wavelength $\lambda$ becomes smaller than the unit cell size as evidenced by $\mathrm{Im}\mu <0$). Using Eq. (4) together with the incident field $H_0$, we found the effective permeability shown. Unit cell size is $a\times a\times a_{\perp }=10\times 10\times 2\mu {\mathrm{m}}^3$. Size of the ring is $\ell \times \ell =8\times 8\mu {\mathrm{m}}^2$; its cross section is $d\times w$, where the depth $d=400\mathrm{nm}$ and the width $w$ takes the values $1\mu \mathrm{m}$ (red, solid curve), $2\mu \mathrm{m}$ (green, dashed curve), and $3\mu \mathrm{m}$ (blue, dotted curve). The metal is the Drude-model gold with experimental values for the plasma frequency, ${\omega }_p=2\pi \times 2184\mathrm{THz}$, and collision frequency, ${\omega }_{\tau }=2\pi \times 6.5\mathrm{THz}$. (Ref. 9).Reuse & Permissions

Figure 2

(Color online) $\mathrm{Re}\mu$ vs ${\log }_{10}(a∕\lambda )$ according to Eq. (1) for (a) open rings for the values of the width $w=1\mu \mathrm{m}$ (red, solid curve), $2\mu \mathrm{m}$ (green, dashed curve), and $3\mu \mathrm{m}$ (blue, dotted curve) and for (b) closed rings of any width. The corresponding $\mathrm{Im}\mu$ vs ${\log }_{10}(a∕\lambda )$ is shown in (c) and (d), respectively. All parameters are as in Fig. 1.Reuse & Permissions

Figure 3

(Color online) Current distribution in the split-ring-resonator ring (the color or gray level is the azimuthal current density, the arrows indicate current direction at the edges) for the three regimes shown in Fig. 1 when the external magnetic field is normal to the plane of the rings and pointing out of it: (a) In the diamagnetic regime below resonance, the current is confined near the outer and the inner edges flowing in the opposite directions enclosing the metal filled area of the ring trace; (b) at resonance, the current is mainly confined near the inner surface; (c) above resonance, the current flows clockwise and essentially at the outer edges of the metallic ring.Reuse & Permissions

Figure 4

Equivalent circuit for an open ring. Loop 1 (2) represents the current path roughly one skin depth wide along the outer (inner) edge of the ring. The two loops have a mutual inductance $L_{12}=L_{21}$.Reuse & Permissions

Figure 5

(Color online) (a) $\mathrm{Re}\mu$ vs ${\log }_{10}(a∕\lambda )$ for the open SRR rings with $w=1\mu \mathrm{m}$ (red, solid curve) together with the contribution of inner (blue, dotted curve) and the outer (green, dashed curve) edges; (b) shows the corresponding $\mathrm{Im}\mu$ vs ${\log }_{10}(a∕\lambda )$ (red, solid curve) together with the contribution of inner (blue, dotted curve) and the outer (green, dashed curve) edges. (c) and (d) show $\mathrm{Re}\mu$ vs ${\log }_{10}(a∕\lambda )$ and $\mathrm{Im}\mu$ vs ${\log }_{10}(a∕\lambda )$, respectively, for the three cases discussed in Fig. 1a with ring widths of $1\mu \mathrm{m}$ (red, solid curve), $2\mu \mathrm{m}$ (green, dashed curve), and $3\mu \mathrm{m}$ (blue, dotted curve). These results were obtained according to Eqs. (3, 4).Reuse & Permissions