## 2 Metamaterials and Frequency Dependent Effective Index of Refraction  
The metamaterial treated here is constructed from a two-dimensional photonic crystal made of parallel coated rods, see Figure 1. There can be one or more parallel coated rods inside the crystal period. All materials used here are nonmagnetic and have relative magnetic permeability equal to 1. We will obtain an effective magnetic response using resonant sub-wavelength geometries. The time harmonic field is TE-polarized and the magnetic field inside the crystal is parallel to the rods \(\mathbf{H}=H_{z}(\mathbf{x})\exp{(-i\omega t)}\hat{z}\) where \(\mathbf{x}=(x,y)\) in the \(xy\)-plane transverse to the rods. The period for the crystal is \(d\) and the dielectric coefficient \(\epsilon_{d}(\mathbf{x})\) takes the values  
\[e_{d}(\mathbf{x})= \begin{cases}e_{R}&\text{in the rod,}\\ e_{p}&\text{in the coating,}\\ e_{h}&\text{in the host material.}\end{cases}  \tag{1}\]  
The coating is a cylindrical shell of plasmonic material with dielectric constant \(\epsilon_{p}(\omega^{2}/c^{2})=1-\frac{\omega_{p}^{2}/c^{2}}{\omega^{2}/c^{2}}\). Here \(\omega_{p}\) is the plasma frequency associated with the coating material and \(c\) is the speed of light in vacuum. The dielectric constant of the rod is chosen according to \(\epsilon_{R}=\frac{\epsilon_{r}}{d^{2}}\) where \(\epsilon_{r}\) has units of area. The idea is to choose the dielectric permittivity of the rod to be large relative to the period size so that the corresponding Mie resonances are excited in the sub-wavelength limit. The dielectric constant of the host material is given by \(\epsilon_{h}=1\). The time harmonic magnetic field \(H_{z}\) for the \(d\)-periodic crystal is a Bloch wave  
\[H_{z}=h(\mathbf{x})\exp{(ik_{m}\hat{\kappa}\cdot\mathbf{x})}, \tag{2}\]  
Figure 1: Photonic crystal made from coated rods. where \(h\) is \(d\)-periodic in \(\mathbf{x}\), the wave number in the metamaterial is given by \(k_{m}=2\pi/\lambda\), where \(\lambda\) is the wavelength in metamaterial. The direction of propagation in the \(xy\)-plane is described by the unit vector \(\hat{\kappa}=(\hat{\kappa}_{x},\hat{\kappa}_{y})\). The magnetic field satisfies the Helmholtz equation  
\[\
abla\cdot\epsilon_{d}^{-1}\
abla H_{z}=-\frac{\omega^{2}}{c^{2}}H_{z}, \tag{3}\]  
with propagation bands described by dispersion relations relating \(\omega/c\) to \(k_{m}\hat{\kappa}\). We examine the sub-wavelength case when \(d<2\pi c/\omega=\lambda_{f}\), where \(\lambda_{f}\) is the wavelength in free space. The power series approach [34; 36; 37] is used to find the explicit Bloch wave solution of [PERSON]'s equation as a convergent multiscale power series in \(\eta=2\pi d/\lambda_{f}\). The \(\eta=0\) term in the expansions for dispersion relation and field deliver the\"homogenized sub-wavelength\" dispersion relation for plane waves inside an effectively magnetic double negative medium, see [34; 36; 37]. In the effective medium, the dispersion relation relating frequency \(\omega\) to wave number \(k_{m}\) is  
\[k_{m}=\frac{\omega}{c}n_{eff}\left(\frac{\omega^{2}}{c^{2}}\right), \tag{4}\]  
where \(n_{eff}\) is the effective index of refraction, and this is derived rigorously in [34]. The homogenized wave is a plane \(B\) field wave:  
\[B_{z}^{Hom} = B_{0}e^{i(k_{m}\hat{\kappa}\cdot\mathbf{x}-\omega t)},\]  
where \(B_{0}\) is a constant in space and time. The homogenized \(B\) field is related to the homogenized \(H\) field by \(B_{z}^{Hom}=\mu_{eff}(\frac{\omega^{2}}{c^{2}})H_{z}^{Hom}\) where \(\mu_{eff}\) is the effective magnetic permeability [36].