## 3 Controlling Frequency Dependent Effective Properties and Positive and Negative Dispersion  
The frequency dependent effective magnetic permeability \(\mu_{eff}\) and effective dielectric permittivity \(\epsilon_{eff}\) are determined explicitly in terms of local resonances. We first rescale to a unit period cell to define effective properties. The unit cell is \(Y=\{0<x<1;\,0<y<1\}\). The subset of \(Y\) containing the high contrast dielectric is \(R\), the coating containing the Drude dielectric is \(P\) and the host material occupies \(H\). These three regions make up the unit cell \(Y\), see Figure 2.  
The Dirichlet eigenfunctions \(\phi,\mu\)  
\[-\Delta\phi(\mathbf{x})=\mu\phi(\mathbf{x}),\;\mathbf{x}\;\text{in}\;R \tag{7}\] \[\phi(\mathbf{x})=0,\;\mathbf{x}\;\text{on the boundary of}\;R\]  
with nonzero mean and associated eigenvalues of the Laplacian defined on \(R\). The set of eigen functions and eigenvalues are denoted by \(\phi_{n}\) and \(\mu_{n}\) respectively for \(n=1,2,\cdots\). The effective magnetic permeability is given by  
\[\mu_{eff}\left(\frac{\omega^{2}}{c^{2}}\right)=\theta_{H}+\theta_{P}+\sum_{n= 1}^{\infty}\frac{\mu_{n}<\phi_{n}>_{R}^{2}}{\mu_{n}-\frac{\omega^{2}}{c^{2}}}, \tag{8}\]  
where \(\langle\cdot\rangle\) denotes the average over \(R\), \(\theta_{H}\) and \(\theta_{P}\) are the areas occupied by regions \(H\) and \(P\), respectively. The generalized electrostatic resonance of the coating is the eigenfunction \(\psi\) eigenvalue \(\gamma\) pair solving  
\[\left\{ \begin{array}{ll}\Delta\psi_{=}0&\text{in}\;H,\\ \Delta\psi=0&\text{in}\;P,\end{array} \right. \tag{9}\]  
with the boundary and transmission conditions  
\[\left\{ \begin{array}{ll}\psi|^{-}=\psi|^{+}&\text{on the boundary of}\;P,\\ \partial_{\bar{n}}\psi=0&\text{on the boundary of}\;R,\\ \gamma[\partial_{\bar{n}}\psi]_{=}^{+}=-\frac{1}{2}(\partial_{\bar{n}}\psi|^{ -}+\partial_{\bar{n}}\psi|^{+})&\text{on the boundary of}\;P,\\ \psi\;\text{is}\;Y\text{-periodic}.\end{array} \right. \tag{10}\]  
Here \(\partial_{\bar{n}}\) is the outward directed normal derivative. The generalized electrostatic eigenvalues \(\{\gamma_{h}\}_{h=0}^{\infty}\) lie in the open interval \((-\frac{1}{2},\frac{1}{2})\) with zero being the only accumulation point, see [34]. The eigenfunctions \(\{\psi_{\gamma_{h}}\}_{h=0}^{\infty}\) form a complete orthonormal set of functions in the space of mean zero square integrable periodic functions on the set \(Y\setminus R\). Similar types of electrostatic resonances are well known in the context of effective coefficients of DC two phase media [38; 39]. We form the integrals  
\[\alpha_{\gamma_{h}}^{1}=\int_{H}\
abla\psi_{\gamma_{h}}\,d\mathbf{x}\;\;\text {and}\;\;\alpha_{\gamma_{h}}^{2}=\int_{P}\
abla\psi_{\gamma_{h}}\,d\mathbf{x}. \tag{11}\]  
Figure 2: Unit cell containing a single rod cross section. \(R\) represents the rod cross section, \(P\) the plasmonic coating and \(H\) the host. and the frequency dependent effective dielectric constant is  
\[\epsilon_{eff}^{-1}\left(\frac{\omega^{2}}{c^{2}}\right) = \theta_{H}+\frac{\xi_{0}}{\frac{\omega^{2}}{c^{2}}-\frac{\epsilon _{r}\omega_{F}^{2}}{c^{2}}}\theta_{P}\] \[-\frac{1}{2}\sum_{-\frac{1}{2}<\gamma_{h}<\frac{1}{2}}\left(\frac{ \left(\frac{\omega^{2}}{c^{2}}-\frac{\epsilon_{r}\omega_{F}^{2}}{c^{2}}\right) |\alpha_{\gamma_{h}}^{(1)}|^{2}+2\frac{\epsilon_{r}\omega_{F}^{2}}{c^{2}}\alpha _{\gamma_{h}}^{(1)}\cdot\alpha_{\gamma_{h}}^{(2)}+\frac{\left(\frac{\epsilon \omega_{F}^{2}}{c^{2}}\right)^{2}}{\frac{\omega^{2}}{c^{2}}-\frac{\epsilon \omega_{F}^{2}}{c^{2}}}|\alpha_{\gamma_{h}}^{(2)}|^{2}}{\frac{\omega^{2}}{c^{2} }-(\gamma_{h}+\frac{1}{2})\frac{\epsilon_{r}\omega_{F}^{2}}{c^{2}}}\right).\]  
The index of refraction for the metamaterial is given by (6), and the dispersion relation (4). Here the the link between dispersion relation and the subwavelength geometry is explicitly given in terms of the local resonances defining the effective properties through the Formulas (8) and (12). These formulas are derived in [36; 37].