Lorentz model for metamaterials: Optical frequency resonance circuits

We illustrate the strong connection between optics and high-frequency electronics by presenting circuit calculations for resonant metamaterials in the optical frequency range. An $RLC$ circuit model is considered. Equations for impedances are derived and used to calculate the resonance positions and quality factors of a variety of samples. These calculations are not quasistatic and include interactions between sites. The calculated resonance values show good agreement with simulations and experiments. Single-site quasistatic formulas are derived from the more general equations.

Figure 1

(a) $RLC$ circuit schematic: resonant antenna model. (b) “$U$”-shaped SRR diagram, single site (top view). The gray region denotes metal.

Figure 2

Source schematic showing the location of the assumed sources in the center of the SRR wires. The assumed charge distribution $\rho$ and current distribution $\mathbf{J}$ are plotted. The current distribution obeys the boundary condition of vanishing at the ends of the source line.

Figure 3

(Color online) Calculated resonance characteristics using the nonquasistatic $RLC$ Lorentz model. (a) Resonance position, $1∕{\lambda }_p={\omega }_p∕2\pi c$. The agreement with simulated values is best for the lower numbered samples where size approximations are most valid. (b) Quality factor $Q$. The agreement with simulation tends to be better for cases where $f_R{R}_{rs}⪢R_{\mathrm{in}}$ (Ref. 15). Simulated (with FMM) and experimental points are from Ref. 6. The lines in the plot are given to guide the eye.

Figure 4

(Color online) Circuit element values. (a) Resistances ($R_{\mathrm{rs}}$, radiative self; $R_{\mathrm{in}}$, internal), network factor $\left(f_R\right)$; (b) Inductance ($L_{\mathrm{rs}}$, $L_{\mathrm{in}}$), network factor $\left(f_L\right)$; and (c) Capacitance $\left(C_{\mathrm{rs}}\right)$, network factor $\left(f_C\right)$. Self refers to a single site. Comments: Internal resistance and inductance are larger than the single-site radiative values. The network factors are most extreme (away from unity) for samples 3 and 9 where the sites are most closely spaced. The fact that $f_R$ is always much larger than $f_L$ and $f_C$ implies that although the network may be ignored for estimates of ${\omega }_p$, it cannot be for estimates of $Q$. ($\mathrm{pH}=\text{picohenry}$, $\mathrm{aF}=\text{attofarad}$.) The lines in the plot are given to guide the eye.