## 2 Metamaterial Sensor for eNDE and Theory
EM sensors with MM lenses have been made using two CSRs, the operation frequencies depending both on the constitutive parameters of MM and the polarization of the incident EM field (TE\({}_{\mathrm{z}}\) or TM\({}_{\mathrm{z}}\)). Figure 2 shows the developed sensor with MM [7, 59, 60]. A CSR consists of a number of spiral-wound layers of an insulated conductor on a conical mandrel [31]. The EM sensor is absolute send-receive type; it has the emission part made from one-turn rectangular coil with \(35\times 70\) mm dimensions, using Cu wire with a 1 mm diameter. When an EM TM\({}_{\mathrm{z}}\) polarized waves acts, at normal incidence, the magnetic field being parallel with the y axis such that \(H_{x}=H_{z}=0\) and \(H_{y}\
eq 0\), in very near field, between carbon fibers similar with MSG, evanescent waves can appear. In the focal image point a reception coil with MM lens is placed, having one turn with 1 mm average diameter made of Cu wire with 0.1 mm diameter, to convert localized energy in electromotive force (e.m.f.). An optimized work frequency of 476 MHz assures magnetic effective permeability of 22. At this frequency, the property of CSR to act as an alternative magnetic flux concentrator has been verified. The calculus of the MM lens based on CSR was presented in Ref. [31]; the field in the focal plane of the lens is given by  
\[H\left(x,y,z_{0}+2l\right)=\frac{H_{0}}{\pi^{2}}\,e^{\left(\frac{k_{b}+k_{a}}{2} x+\frac{k_{b}+k_{a}}{2}y\right)}\frac{\sin\left(\frac{k_{b}-k_{a}}{2}\,x \right)}{x}\frac{\sin\left(\frac{k_{b}-k_{a}}{2}\,y\right)}{y}, \tag{3}\]  
where \(z_{0}=R\) (Figure 2a) with \(R<\!\!<\lambda\), and \(H_{0}\) is the amplitude of incident magnetic field. The principle scheme of sensor for the evanescent wave's detection is shown in Figure 2a. In order to enhance the spatial resolution of the sensor, a conductive screen having a circular aperture made from perfect electric conductor (PEC) material with a very small diameter is placed in front of the lens. The circular aperture serves for the diffraction of the evanescent waves that can occur on slits. This ensures paraxial incident beam. The diameter of focal spot provided by MM lens is given by [72]:  
\[D=\frac{4\pi}{k_{b}-k_{a}} \tag{4}\]  
and is equal with the diameter of the small basis of the conical Swiss roll, _i.e._, 3.2 mm. The MM lens with CSR will be displaced along the \(x\)-axis (Figure 2a). When \(k_{a}=0\) and this value is inserted into Equation (4), \(k_{b}\) is obtained and the field in the focal plane is calculated with Equation (3). The detection principle is similar with near-field EM scanning microscopy (NFESM). NFSEM imaging is a sampling technique, _i.e._, the sample (in our case MSG or carbon fibers) is probed point by point by raster scanning with the sensor over the sample surface and recording for energy image pixel a corresponding EM signature. Selecting a region with dimensions \(\left(x_{c}+x_{d}\right)\cdot\left(x_{c}+x_{d}\right)\), (where \(x_{c}\) and \(x_{d}\) are coordinates for conducting/dielectric material), using the Fourier optics methods [72, 73], an object \(O(x,y)\) that can represent the eigenmodes in function of the polarization of an incident EM field, has, while passing through the circular aperture and the lens, an image \(I(x^{\prime},y^{\prime})\), given by [68]. Figure 2: Sensor with MM lens: (**a**) schematic representation; (**b**) photograph.