# Development of a Metalens for Radio Wave Ground- Penetrating Radars
## 2 Materials and Methods
It is important that the layer itself has the shape of a rectangular parallelepiped, and the negative phase incursion arises not due to any geometric curvature of the shape of the focusing system, but due to the negative refractive index inside the layer. Now, let us pay attention to the distribution of the power flow behind the constructed structure. Figure 4 shows the dependence of the power flow along the longitudinal axis of the structure. The dependence is shown by the distance expressed in wavelengths. The second boundary of the constructed metamaterial layer is taken as the origin. The ordinate shows the power flux normalized to the maximum power of the transmitted wave. It can be seen that, at a distance F = 27 mm, i.e., about 0.8 of the wavelength \(\lambda\), a maximum power flux is observed. The average longitudinal distribution of the power flux in the plane of the electric field is shown in Figure 5. It can be seen that the longitudinal localization of the focused field at the half-power level is about \(\Delta H\) = 125 mm, i.e., 3.7 of the wavelength \(\lambda\). Figure 4: Power distribution behind the layer. Figure 5: Focusing the power flow in the plane of the electric field. Figure 3: Distribution of the real part of the electric component of the field. Figure 6 shows the power flow distribution in the transverse plane. It can be seen that the dimension region has an approximately elliptical cross-section with a maximum located in the center. The width of this zone according to the level of the half-power in the transverse direction is \(\Delta R\) = 27.0 mm. This corresponds to 0.8 of the wavelength \(\lambda\). The diffraction limit is characterized by the wavelength \(\lambda\), so that the resulting flat metamaterial focusing lens gives better focusing. The diffraction limit exceeded: \(\Delta R\) < \(\lambda\). The transverse size of the matrix structure of the rings is 70 mm, which is close to the size of the first Fresnel zone \(D\) = 69 mm (2.0 \(\lambda\)) for the focal length F = 27 mm. The size of the first Fresnel zone is given by the equation:  
\[\sqrt{\left(\frac{D}{2}\right)^{2}+F^{2}}-F=\frac{\lambda}{2} \tag{2}\]  
It is important to note that conventional dielectric lenses must have a transverse dimension on which not one, but many Fresnel zones fit. Since the ring structure of closed conductors is not sharply resonant, then, as the calculations show, the focusing properties of the synthesized matrix are preserved in a fairly wide frequency range of 1.5-3 GHz or wavelengths \(\lambda\) = 100-199 mm. Figure 7 shows the longitudinal distribution of the focused power flux with a change in wavelength. A change in the wavelength leads to a decrease in the power of the focused radiation and to some longitudinal shift of the focus area. The noted effect is explained by the change in the receiving size of the Fresnel zone \(D\) from the value of the given transverse size of the synthesized structure of the metamaterial \(b\) = 70 mm. This effect leads to a change in the transverse width of the focus area (Figure 8). Figure 6: Transverse distribution of power flow in the focus area. Figure 7: Longitudinal distribution of focused power flow with wavelength change.