# Beam Scanning Capabilities of a 3D-Printed Perforated Dielectric Transmitarray
## 2 Unit-Cell Analysis
In fact, this feature guarantees that, changing the angle of incidence, no higher modes resonate, as occurs for other unit-cell (see for instance [13]), degrading the antenna radiation performances and also that, when the direction of arrival changes, due to a variation of the position of the feed, the behavior of the unit-cell is not completely different from the presumed one. To study what happens in the unit-cell considered here, it has been analyzed using the Floquet excitations in CST MW Studio, used to compute the transmission coefficient as a function of \(d\), for several angles of incidence \(\theta_{i}\) (see inset in Figure 3a for its definition). The curves representing the variation of the amplitude and the phase of \(S_{21}\) are shown in Figure 3a,b, respectively. They have been calculated at \(f_{0}\) and for \(\theta_{i}=10^{\circ}\), \(20^{\circ}\), \(30^{\circ}\), \(40^{\circ}\). Since the analyses for TE (\(\phi_{i}=~{}0^{\circ}\)) and TM (\(\phi_{i}=~{}90^{\circ}\)) modes provided the same results, only a single curve is plotted for each incidence angle. As it can be observed in Figure 3a, the amplitude of \(S_{21}\) is slightly affected by direction of arrival of the incident wave, with a reduction not greater than 0.5 dB when \(\theta_{i}=40^{\circ}\). For what concerns the phase, from Figure 3b, it emerges that both its linearity and full range of variation are not affected by \(\theta_{i}\): in fact, the different curves are almost a translated copy of the other, with small changes in the slope. This means that the dielectric unit-cell behavior is not particularly sensitive to the angle of incidence, and this makes it a potential good candidate for the realization of a beam scanning TA, more suitable than other unit-cells, as some of those using metallic printed elements, where the oblique incident wave excites superior modes, at their turns responsible for a strong reduction in the amplitude of the transmission coefficient and discontinuities in the phase variation. Figure 2: Equivalent transmission line model of the perforated dielectric unit-cell. The value of the incidence angles \(\theta_{f}(m,n)\) in correspondence with the TA elements in the case considered here is shown in Figure 5a. It has been computed considering a focal length \(F=150\) mm (\(F/D=1\)). The angle is maximum at the extremes of the aperture, achieving a peak value of about \(34^{\circ}\). The phase error introduced when the normal incidence is assumed for all the unit cells instead of the real direction of arrival of the incidence field is given by  
\[\psi_{error}(m,n)=\angle S_{21}(m,n)-\angle S_{21}(\theta_{f}(m,n)), \tag{4}\]  
where \(\angle S_{21}(m,n)\) is the transmission phase computed with a normal incidence (\(\theta_{f}=0^{\circ}\)), while \(\angle S_{21}(\theta_{f}(m,n))\) takes into account the different behavior of the UC with varying \(\theta_{f}\) and it has been evaluated from the curves obtained in Figure 3b. In Figure 5b, the resulting phase error distribution \(\psi_{error}(m,n)\) for the \(50\times 50\) TA is plotted: as it can be observed, at the center, the phase error is lower than \(5^{\circ}\) and it can be considered negligible, while beyond the phase jump, the error increases arriving up to about \(30^{\circ}\). It is higher when the position of the feed changes, as occurs during the mechanical beam scanning considered here. For instance, if we consider a scanning angle of \(-30^{\circ}\), the distribution of the incident angles becomes that in Figure 5c, in which the maximum angle achieved on the right corners is \(51^{\circ}\). In Figure 5d, it can be observed that the phase error related to this case has increased, reaching a maximum of about \(45^{\circ}\). Moreover, another important aspect has to be taken into account when the feed is rotating.