# Methodology for Improving Scanning Performance Loading an Array Element with a 3D All-Metal WAIM
## 3 WAIM Design Methodology
### General Design Methodology  
Figure 6 depicts the global EC of one array element extracted from an infinite periodic phased array together with its associated WAIM, as represented in Figure 7b. Note that the phased array scans the radiating beam, producing the oblique incidence on the WAIM. Henceforth, the corresponding incident direction is named the scanning direction. It is defined by \(\theta\) and \(\phi\), where \(\phi\) is either equal to 0 (\(H\)-_plane_ in previous section) or \(\pi/2\), (\(E\)-_plane_). Here, the periodicity of the WAIM unit-cell is the half of an individual array element, which means each array element is covered by four WAIM unit-cells (Figure 7a). The array element is represented by a 2-port network expressing the power exchanges between its feeding port and the fundamental radiated Floquet mode. It is assumed that all higher-order modes are evanescent at the frequency of design (array periodicity is less than 0.5\(\lambda_{0}\), being \(\lambda_{0}\) the free-space wavelength at 18 GHz) and they do not contribute to the interactions with the WAIM structure. This 2-port network is followed by a transmission line that represents the separation distance between the array and the WAIM (\(d_{WAIM}\)). As will be shown later, this distance is a possible degree of freedom in the design process. The other characteristics of the line, \(Z_{0}\) and \(\beta_{0}\), are defined by the scanning direction and can be derived analytically. The WAIM with its EC, as defined in Section 2.2, is connected with the 2-port network through this transmission line. The design methodology can be divided into four main steps. As a preliminary step, a data-base is computed to obtain the circuit elements of the unloaded WAIM unit-cell, using full-wave simulations. It consists of deriving \(Z_{1}(\theta)\) and \(Z_{cap}(\theta)\) for a discretized set of geometries (only varying \(w_{y}\)) and for a discretized set of scanning directions. A full-wave simulation is needed for each geometry and each direction, but these simulations are relatively quick. In the second step, the S-matrix of the array element is also computed using full-wave simulations for the same set of directions. Once again, a new full-wave simulation is needed for each direction. Figure 6: Global EC: antenna _S-Parameters_ plus space between antenna and _WAIM_ plus _WAIM EC_. Figure 7: (**a**) Phased array of stepped horns with four unit-cells with half-periodicity (one unit-cell represented in red) per one array element (represented in green) (illustrative) and (**b**) global structure: phased array plus WAIM structure with infinite periodic conditions. In the third step, the optimization of the WAIM EC is carried out. To do so, the global EC in Figure 6 is analyzed using a circuit simulator (Keysight ADS) and the possible degrees of freedom (\(l_{1},\ l_{2},d_{WAIM},\ Z_{x}\) and \(Z_{1}\)) are optimized in order to minimize the input active reflection coefficient. To be noted, the optimization is run for all scanning directions simultaneously, so that, a unique WAIM design is obtained in the end. Moreover, for the sake of simplicity, we assume \(Z_{x}\) does not depend on \(\theta\) as explained before. Finally, it is important to point out that the choice of \(Z_{1}\) enforces the cross-section geometry of the cell. As a consequence, \(Z_{cap}\) and \(\beta\) must be defined correspondently. This is of course automatically managed through the WAIM data-base that opportunely links all interdependent parameters in the EC. On the contrary, \(l_{1},\ l_{2}\) and \(d_{WAIM}\) are completely independent degrees of freedom which can be optimized quite freely. It should be emphasized that this optimization step does not require any supplementary full-wave simulation because it is performed with the EC only. This is a definitive advantage of this methodology regarding the huge number of WAIM configurations that may be involved along the successive iterations of the process. The fourth and last step is a post-processing for \(Z_{x}\).