## 3 Controlling Frequency Dependent Effective Properties and Positive and Negative Dispersion
It is clear that we can control the frequency dependence of \(\mu_{eff}\) and \(\epsilon_{eff}\) by controlling the poles and zeroes of these functions which depend on the Dirichlet resonances (7) and generalized electrostatic resonances (10) respectively. The graphs of \(\mu_{eff}\) and \(\epsilon_{eff}^{-1}\) as functions of \(\omega^{2}/c^{2}\) are displayed in Figures 3 and 4. Here the intervals \  
It follows that dispersion relation is also controlled by the poles and zeros of \(\mu_{eff}\) and \(e_{eff}^{-1}\) as explicitly determined through the Dirichlet spectra and generalized electrostatic resonances. The Formulas (4) and (6) show that the frequency intervals associated with pass bands and stop bands are governed by the poles and zeros of the effective magnetic permittivity and dielectric permittivity tensors. Pass bands of fixed center frequency and band width occur when the effective magnetic permittivity and dielectric permeability have the same sign. This includes frequency intervals where effective tensors are either simultaneously positive or negative. When both are positive one has positive dispersion and negative dispersion when both are negative. Bands of positive and negative dispersion are separated by stop bands, see Figure 5.  
In the following sections we will choose the the high contrast core phase to be a disk centered in the period cell surrounded by an annular coating of Drude dielectric. The ratios of the disk radius and the outer diameter of the coating to the period length are fixed. We then calculate the Dirichlet and electrostatic eigenvalues to calculate a frequency dependent effective index of refraction. For the concentric Drude dielectric coating-high dielectric core geometries we can control the spectra by changing the ratio of core to coating material used, see Figure 6. Here we denote the core radius by \(a\) and the outer radius of the coating by \(b\) where these numbers are in dimensionless units relative to period length. In this context we can change the outer radius of the coating and radius of the core relative to the period size. The band width and center frequency of the negative index band and its dependence on \(a\) and \(b\) listed in Table 1. This demonstrates that we can control the frequency bands where the effective index of refraction is negative (or positive) by changing the inner and outer radii of the coating material. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\mathbf{a=0.5b}\) & \(\mathbf{a=0.55b}\) & \(\mathbf{a=0.6b}\) & \(\mathbf{a=0.65b}\) \\ \hline \multirow{2}{*}{\(b=0.3\)} & \multirow{2}{*}{0} & \multirow{2}{*}{0} & \multirow{2}{*}{0} & \multirow{2}{*}{0} & 0.0332 \\  & & & & (0.8919) \\ \hline \multirow{2}{*}{\(b=0.35\)} & \multirow{2}{*}{0} & \multirow{2}{*}{0.02733} & \multirow{2}{*}{0.03824} & \multirow{2}{*}{0.04425} \\  & & (0.9003) & & (0.8315) & (0.7716) \\ \hline \multirow{2}{*}{\(b=0.4\)} & \multirow{2}{*}{0.03541} & \multirow{2}{*}{0.04204} & \multirow{2}{*}{0.04893} & \multirow{2}{*}{0.05579} \\  & (0.8707) & & & (0.7345) & (0.6830) \\ \hline \multirow{2}{*}{\(b=0.45\)} & \multirow{2}{*}{0.04366} & \multirow{2}{*}{0.05143} & \multirow{2}{*}{0.05944} & \multirow{2}{*}{0.06801} \\  & (0.7795) & & & (0.6605) & (0.6161) \\ \hline \end{tabular}
\end{table} 
Table 1: The changes of the negative index band when inner radius and outer radius vary. In each cell, the upper and lower numbers denote the band width and center frequency, respectively. Figure 5: The leading order dispersion relation over two selected intervals \(I_{n^{\prime}}\) and \(I_{n^{\prime}}\). These calculations highlight the frequency dependent refraction of signals composed of waves with frequencies inside different frequency bands by a free space - metamaterial system. In the following sections we demonstrate how to calculate the explicit refraction of signals passing between a free space - metamaterial interface in the time domain.