# Chipless Radio Frequency Identification (RFID) Sensor for Angular Rotation Monitoring
## 2 Cross Polar Features and Chipless RFID Sensor Design  
The considered chipless RFID sensor is a truncated periodic surface printed on the top of a grounded dielectric slab. The surface is obtained by repeating the _unit cell_ on a lattice of periodicity (\(T_{x}\) and \(T_{y}\)), along the \(x\) and \(y\) direction, respectively. The number of the employed unit cells determines the amplitude level of the backscattered signal. Therefore, the larger the tag, the easier the detection procedure, with a consequent reduction of the Bit Error Rate (BER) [40]. The frequency response is tailored by properly defining the shape of the element in the unit cell, the lattice, and the substrate material. The dipole has proved to be the most suitable shape for the defined task [22; 37], therefore it will be adopted as the element (Figure 1). The frequency response of the investigated structure will be calculated using a Periodic Method of Moments (PMoM) [41], which considers an infinite periodic surface. The employed PMoM requires the discretization of the unit cell into a \(M\times M\) pixel matrix, in which each pixel has dimension \(T_{x}/M\) and \(T_{y}/M\). The adopted discretization for the design, is the one obtained by using \(M=64\). Let us consider a plane wave that is normally impinging and whose electric field is \(E_{0}\) (Figure 1). It is sufficient to know the 2 \(\times\) 2 reflection matrix \(\Gamma\) in order to prove the cross polar response of the structure as a function of the phi angle (\(\phi\)). To explain in more detail, the two components of the reflected field \(E_{R}\) along the principal axes can be expressed as:  
\[ \begin{array}{l}E_{R,x}(\phi)=E_{0,x}(\phi)\,\Gamma_{xx}+E_{0,y}(\phi)\, \Gamma_{xy}\\ E_{R,y}(\phi)=E_{0,y}(\phi)\,\Gamma_{yy}+E_{0,x}(\phi)\,\Gamma_{yx}\end{array} , \tag{1}\]  
whereas the co-polar (_copol_) and cross polar (_Xpol_) component with respect to the incident field \(E_{0}\) are:  
\[ \begin{array}{l}E_{R,copol}(\phi)=\underline{E}_{0}(\phi)\cdot\underline{E} _{R}(\phi)\\ E_{R,Xpol}(\phi)=\underline{z}\cdot(\underline{E}_{0}(\phi)\times\underline{E} _{R}(\phi))\end{array} . \tag{2}\]  
It is important to underline that in the proposed scenario, the elements of the reflection matrix \(\Gamma\) are only a function of frequency. Therefore, the behavior of the co-polar and cross polar responses of a structure can be quickly evaluated starting from a simulation or a measurement that allows the evaluation of \(\Gamma_{xx}\), \(\Gamma_{yy}\) and \(\Gamma_{xy}\) (which is equal to \(\Gamma_{yx}\)). Let us focus the attention on the angular response of the single-dipole unit cell presented in (Figure 1) in correspondence to the frequency \(f_{1}=5.25\) GHz, for which the cross-polar reflected field is maximized. For the sake of simplicity, a 90-degree span is considered in this first part of the analysis, but a 180 degree span will be considered for the final performance assessment. Looking at the cross polar response of the single-dipole unit cell (Figure 2a) as a function of \(\phi\), it is apparent that there are some cases in which the magnitude of the scattered field is quite low (e.g., around 40 degrees), and above all, that in some cases two different angles exhibit the same cross polar level. For example, \(\phi_{1}\) and \(\phi_{2}\) both assume a value of XP\({}_{1}\), thus causing an ambiguity in the correct estimate of rotation angle \(\phi\). Figure 1: Example of a unit cell of the periodic surface employed for the design of the chipless RFID sensor: (**a**) A unit cell with a dipole element and (**b**) the correspondent finite chipless RFID sensor tag with \(4\times 4\) unit cells. To reduce the ambiguity and increase the level of the received signal, a second dipole is added in the unit cell (inset Figure 2b). It is important to notice that the length of this dipole and its orientation is different than the initial one.