# Development of a Metalens for Radio Wave Ground- Penetrating Radars
## 2 Materials and Methods  
Let us consider the case of a plane wave falling onto a single hollow cylinder with an annular cross-section located in the plane of the electric field component. A hollow cylinder with an annular cross-section is made of a composite with an electrical conductivity of 3.4 \(\times\) 10 S/m, with an outer diameter of 8 mm and an inner diameter of 4 mm. The composite material was obtained by mixing ABS plastic with copper and aluminum powders in a ratio of 60:20:20 %, respectively. Figure 1 shows the distribution of the modulus of the electric component of the field with a wavelength of \(\lambda\) = 119 mm, when a plane electromagnetic wave is incident perpendicular to the ring axis. In this case, the electric component intensity vector is perpendicular to the ring axis, and the magnetic intensity vector is parallel to the conductor axis. The maximum values of the field are distributed approximately at an angle of 45\({}^{\circ}\) from the direction of propagation of the plane incident wave. In this case, the minimum value of the electric field is observed in the center of the hollow cylinder with an annular cross-section. The analysis of the electromagnetic field near this element allows us to choose the location of the elements when building a matrix structure. A complete analysis of the secondary field near the conductor for constructing the structure was carried out earlier in [6]. The results of this study make it possible to evaluate the interaction of the ring elements with each other and achieve the greatest interaction between elements when modeling structures with a negative refractive index. When creating a composite environment, hollow cylinders with an annular cross-section must be placed line by line, making up some layers, which in turn are displaced in the transverse direction, as shown in Figure 2a. Four elementary layers were used, so that the thickness of the full layer is h = 40 mm, and its width is b = 70 mm. In order to determine whether a given structure has the properties of a metamaterial, it is necessary to determine the phase shift along the line passing through the center of the structure. It is known that a material with a negative value of dielectric and magnetic permeability has a negative phase incursion. Therefore, it is necessary to analyze the change in the phase characteristic of the field along this line passing through the center of the model. Figure 2b shows the dependence of the unwrapped phase, recalculated into an equivalent distance, along this straight line. The dashed lines indicate the lattice boundaries. One can see a negative phase shift inside the layer, which indicates the occurrence of the phase front movement in the negative direction. To calculate the real part of the equivalent refractive index, we use the formula:  
\[n=\frac{\Delta L}{h} \tag{1}\]  
where \(\Delta L=\Delta\varphi/k\) is the additional incursion of the wave path in the layer, expressed in terms of the phase incursion \(\Delta\varphi\), and \(k=2\pi/\lambda\) is the wave number. The resulting equivalent refractive index for this model is \(Re(n)=-0.036\). This means that, as a result, we have obtained a metamaterial model. The negative phase incursion at the center of the  
Figure 1: Interaction of an electromagnetic wave with a single hollow cylinder with an annular cross-section. Figure 2: Matrix structure of hollow cylinders with an annular cross-section (**a**). Phase advance of the total field in the matrix of hollow cylinders with an annular cross-section (**b**). synthesized structure inevitably generates a curvature of the wave front of the resulting field at the output. To analyze this phenomenon, consider the spatial distribution of the real part of the electric field (Figure 3). It is clearly seen that a curvature of the phase front occurs behind the layer. This means that there is some dip in the phase shift inside the layer.