# A Split Ring Resonator-Based Metamaterial for Microwave Impedance Matching with Biological Tissue
## 2 Recall of Theory  
In this section, the analytical model and physics of the split ring resonators is recalled since it has been employed in order to roughly obtain preliminary geometric parameters, before the actual design, which is numerically refined via CST Microwave Studio(r) as described in Section 3. The metamaterial based on SRR exhibits an effective magnetic permeability described by the Lorentz model approximation [31; 32]:  
\[\mu_{\text{eff}}(\omega)=1-\frac{F\omega^{2}}{\omega^{2}-\omega_{0m}{}^{2}+\ j \gamma\omega} \tag{1}\]where \(\omega=2\pi f\) is the angular frequency; \(\omega_{0m}\) is the resonant frequency; \(F\) is the fractional of the unit cell occupied by interior ring; \(\gamma\) is the damping factor due to metal losses. Starting from (1), a frequency range in which \(\text{Re}(\mu_{\text{refl}})\) is negative can be identified:  
\[\omega_{0m}\ <\ \omega\ <\ \frac{\omega_{0m}}{\sqrt{1-F}}\ =\ \omega_{pm} \tag{2}\]  
where \(\omega_{pm}\) is the plasma magnetic frequency. Both \(\omega_{pm}\) and \(\omega_{0m}\) are finely tunable by optimizing the geometry of the unit cell. When an electromagnetic wave propagates orthogonally to the SRR plane, the SRR behavior can be approximatively described by an equivalent LC circuit, where the inductance and the capacitance are related to the currents induced in the metal rings and to the capacitive phenomena between the split terminations (capacitive gap), respectively [32]. If the dimension of unit cell is very small with respect to wavelength \(\lambda\), the metamaterial layer exhibits an effective capacitance and an effective inductance at the macroscopic/average level, according to the effective medium theory, which leads to the calculation of an effective permeability [31]. Considering a single broadside coupled SRR with square form rings in the \(xy\) plane, the effective magnetic permeability is described by (3) [14]:  
\[\mu_{\text{refl}}=1-\frac{j\ \omega\ L_{\text{eff}}\ S}{\Delta_{x}\Delta_{y} \Big{(}R_{\text{eff}}\ -\ \frac{j}{\omega\ \text{C}_{\text{eff}}}\ +j\ \omega\ \ L_{\text{eff}}\Big{)}} \tag{3}\]  
\[S=l_{x}l_{y} \tag{4}\]  
where \(\Delta_{x}\) and \(\Delta_{y}\) are the dimensions of the SRR unit cell in \(xy\) plane; \(l_{x}\) and \(l_{y}\) are the lengths of the metallic ring in the \(x\) and \(y\) directions, respectively; \(R_{\text{eff}}\), \(C_{\text{eff}}\) and \(L_{\text{eff}}\) are the effective resistance, effective capacitance and effective inductance of the metamaterial, respectively. The effective capacitance is approximatively obtained from the formula for the capacitance per unit length of a strip line [33]. \[C_{\text{eff}}=\frac{\varepsilon_{r}\ \varepsilon_{0}\ l}{4}\ \frac{K(k)}{K^{ \prime}(k)} \tag{5}\]  
\[k=\tan\ h\frac{\pi\ w}{2\ t} \tag{6}\]  
\[l\ =\ 2\Big{(}l_{x}+l_{y}-s\Big{)} \tag{7}\]  
where \(K(k)\) and \(K^{\prime}(k)\) are elliptic integrals [34], \(w\), \(s\) are the width and the split gap of the SRR; \(t\) is the distance between the two SRR in the \(z\) direction. The effective inductance is [33]:  
\[L_{\text{eff}}=\frac{\mu_{0}\ S}{t} \tag{8}\]  
The effective resistance \(R_{\text{eff}}\) includes the radiation resistance and the loss resistance [33]. In other words, the control of the effective magnetic permeability, by varying both the inductive and capacitive properties of metamaterial, via a proper SRR optimization, is the basic physics principle of the electromagnetic field focusing. A planar metamaterial lens based on SRR, formed by an array of the unit cell in the rings plane, exhibits an anisotropic negative permeability; if the wave propagates in the z direction, the permeability tensor is [14]:  
\[\mu=\mu_{0}\left( \begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&\mu_{\text{refl}}\end{array}