## 3 Practical Implementation
### Pixel Dispersion Properties  
The transverse magnetic (TM) and transverse electric (TE) modes' spatial and frequency dispersion properties for different patches are shown and compared in this section. In order to facilitate reader understanding, firstly the TM and TE modes' isofrequency dispersion curves for a pixel composed by an elliptical metallic patch printed on a grounded dielectric (i.e., without cross slot) are shown in Figure 3. The dispersion properties for different elliptical patch ratios (\(e_{a}/e_{b}\)) are compared. As can be seen in Figure 3a, the phase constant of the supported TM mode (\(\beta_{TM}\)) increases when the electrical size of the patch is larger. The TM phase constant is always larger when the propagation angle is oriented towards the larger axis of the ellipse (in this case the \(y\)-axis), and the same effect can be observed in the \(x\)-axis when the ratio of the ellipse is increased (Figure 3c). Unlike the TM mode, which is supported in the structure at all frequencies, the TE SW has a cut-off frequency. As can be seen in Figure 3b-d, the cut-off frequency of the TE mode decreases for electrically large patches. For instance, in the patch with larger ellipse ratio (Figure 3d), the TE mode is outside the visible region at 13.5 GHz (shown in bold in the isofrequency curves), while it is inside the visible region for smaller ratios (Figure 3b). The TE phase constant (\(\beta_{TE}\)) increases on the \(x\)-axis for ellipses with larger ratios. When the required MTS must support only the propagation of a TM mode, the geometry and rotations of the selected patches for the MTS implementation must be carefully selected to avoid the propagation of the TE SW working below its cut-off frequency. Nevertheless, in the present work, both  
Figure 3: Frequency and spatial dispersion curves for TM (left column) and TE (right column) SWs supported by a pixel composed by a metallic elliptical patch printed on a grounded dielectric with different ratios: (**a,b**) \(e_{a}/e_{b}=0.6\) and (**c,d**) \(e_{a}/e_{b}=0.85\). The employed dielectric is AD-1000 (\(e_{r}=10.2\), thickness \(h=1.27\) mm) and the unit cell dimension \(u=3.14\) mm. the TM and TE modes must be considered. The main goal is to independently control the propagation properties of both the TM and TE SWs, ensuring, additionally, the synchronized propagation of both modes within a certain impinging angle (\(\phi\)). For this purpose, an asymmetric cross-shaped aperture has been introduced in the center of the previously presented elliptical patch geometry (shown in Figure 2). Frequency and spatial dispersion properties for three geometric parameterizations are shown in Figure 4.  
Figure 4: Frequency and spatial dispersion curves for TM (left column) and TE (right column) SWs supported by three different pixels composed by a metallic elliptical patch with an asymmetric aperture in the center, printed on a grounded dielectric. The ellipse ratio for the patches is \(e_{a}/e_{b}=0.85\) in all cases, and cross apertures are defined by \((w_{a}/u,c_{a}/e_{a},w_{b}/u,c_{b}/e_{b})\) parameters: (**a**,**b**) pixel A (0.15, 0.6, 0.15, 0.9), (**c**,**d**) pixel B (0.25, 0.6, 0.15, 0.9) and (**e**,**f**) pixel C (0.15, 0.85, 0.15, 0.6). The employed dielectric is AD-1000 (\(\varepsilon_{r}=10.2\), thickness \(h=1.27\) mm) and the unit cell dimension \(u=3.14\) mm. Given that the unit cell dimension is very small in terms of wavelength, the inclusion of the asymmetric cross-shaped aperture in the center of the elliptical patch geometry makes it possible to control the phase constant of the different modes (\(\beta_{TM}\), \(\beta_{TE}\)) independently when impinging on the unit cell with different angles. TM mode dispersion properties are affected by the slot of the cross perpendicular to the impinging angle, while the TE SW is affected by the slot aligned with the incidence direction. In Figure 4a, the cross slot along the larger axis of the elliptical patch (\(y\)-axis) has been enlarged, increasing the \(c_{b}\) parameter.