## 2 Materials and Methods
### Numerical Design Procedure  
In order to validate and demonstrate the described approach, we realized a numerical set-up, by employing an electromagnetic full wave solver based on the Method of Moments (Feko Suite, Altair, USA). The overall set-up comprises an active RF loop, the dual-tuned magnetic metasurface and a biological phantom (Figure 2). In detail, the proposed metasurface consists of a passive 5 \(\times\) 5 array, covering a 15 cm \(\times\) 15 cm area, in which each unit-cell is composed of two concentric resonant spiral coils. Specifically, the 5-turn internal spiral is made resonant at 64 MHz, whereas the 4-turn external one is at 17 MHz, taking into particular consideration their mutual coupling. The inner spiral presents an external diameter of 17 mm, with a spiral pitch of 0.9 mm; the outer  
Figure 2: Three-dimensional CAD model of the proposed system, including the radiating elements, the biological phantom, and the transversal plane in which the magnetic field is evaluated. one, instead, has a 29.2 mm external diameter with the same pitch. The passive metasurface is inductively excited by a circular RF loop, presenting a 20 cm diameter and placed 5 mm far from the array. Both the metasurface and the RF loop are fabricated with a 28 AWG single strand lossy copper wire. The corresponding CAD model, including a biological phantom and the transversal plane where the \(B_{1}^{+}\) field is evaluated in the following sections, is depicted in Figure 2. Conversely, all the system geometrical properties are summarized in Table 1.  
As is well known, the mutual coupling term in the inductive regime between two coils is directly proportional to their self-inductances, while being inversely proportional to the relative distance. By considering this phenomenon, we paid special attention to the metasurface unit-cells design and positioning. In particular, a design optimization process was followed to minimize the mutual coupling coefficient between the two spirals constituting the single unit-cell. The different operating frequencies (17 and 64 MHz) are in part helpful in limiting the coupling, but a proper spatial distance between these two elements must also be accomplished. By carefully optimizing the geometry and the periodicity of the unit-cells, an acceptable trade-off can be found. After taking care of the geometry, we then proceeded by applying the analytical algorithm described in the previous section to compensate for the truncation effects. Thus, we loaded each coil with the appropriate capacitor, homogenizing the current flowing in each unit-cell and, consequently, the produced magnetic field. In addition, in order to maximize the power transmitted to the metasurface, we designed two different matching networks for the active RF loop, at 17 MHz and 64 MHz, whose topologies are described in Figure 3. The corresponding lumped elements values are reported in Table 2.  
Finally, we also introduced the presence of a biological phantom to realize a realistic scenario, simulated with a cylinder with an ellipsoidal base presenting a major axis equal to 100 mm and a minor axis equal to 50 mm (Figure 2). The total phantom height is 200 mm, while the corresponding dielectric properties are set to simulate muscle tissues, and they are estimated from [35]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **RF Loop** & **Unit-Cell Inner Spiral** & **Unit-Cell External Spiral** \\ \hline Outer diameter (mm) & 200 & 17 & 29.2 \\ Inner diameter (mm) & 200 & 8 & 22 \\ Number of turns & 1 & 5 & 4 \\ Copper wire & 28 AWG & 28 AWG & 28 AWG \\ Spacing between windings (mm) & N/A & 0.9 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System design parameters. Figure 3: Employed matching networks topologies: (**a**) RF loop and metasurface at 17 MHz; (**b**) RF loop and metasurface at 64 MHz; (**c**) RF loop w/o metasurface at 17 MHz and 64 MHz. In particular, the metasurface can be represented as an equivalent RLC circuit [26].