# Beam Scanning Capabilities of a 3D-Printed Perforated Dielectric Transmitarray
## 2 Unit-Cell Analysis  
The basic structure of the unit-cell (UC) used for the design of the beam scanning TA was described in detail in [12], where it was proposed as an alternative solution to improve the bandwidth in transmitarray antennas; however, for the sake of clearness, its main features are summarized in the following. As illustrated in Figure 1, the UC consists of three overlapping layers, made of the same dielectric material. The central layer has a square hole, whose size \(d\) is varied to control the phase of the transmission coefficient (\(S_{21}\)). The two external elements are characterized by a truncated pyramid hole that connects the hole in the central layer to the external aperture on the unit-cell external top (bottom) side, whose size \(W\) is the same for all of the unit-cells. The three layer configuration can be described with the equivalent transmission line model shown in Figure 2. The comparison with the lateral view of the unit-cell in Figure 1b highlights that the mid layer behaves as a uniform transmission line with characteristic impedance \(Z_{sqh}\), while the external identical layers are equivalent to tapered transmission lines whose characteristic impedance \(Z_{tap}^{*}(z)\) varies linearly along the \(z\)-axis, and therefore they act as wideband impedance inverters that match \(Z_{sqh}\) to the free-space impedance (\(Z_{0}\)). Both \(Z_{sqh}\) and \(Z_{sqh}\) are related to the effective permittivity of the corresponding layer that in its turn depends on the relative dielectric constant of the material and the size of the hole. The dimension of the tapered section changes linearly with \(z\), and, as a consequence, the characteristic impedance \(Z_{tap}^{*}(z)\) follows the same linear law. As discussed in [10], a good matching can be realized choosing the lengths \(l_{1}\) and \(l_{3}\) of the tapered lines at least equal to \(\lambda_{g}/2\), where \(\lambda_{g}\) is the effective wavelength in the tapered section. The unit-cell was designed to work in Ka-band, at a frequency \(f_{0}=30\) GHz. To make possible its manufacturing with a 3D printer, it was designed using the commercial dielectric VeroWhite Plus (RGD835 provided by Stratasys(r)). This material was electromagnetically characterized using the waveguide method through which it was estimated that the average value of the dielectric constant in the considered frequency range is \(\varepsilon_{r}=2.77\) and the loss tangent is tan\(\delta=0.021\). Considering the requirements introduced by the manufacturing process discussed in [12], and the parametric analysis  
Figure 1: Sketch of a dielectric transmitarray and blow out of the unit-cell structure. (**a**) dielectric TA; (**b**) UC perspective view; (**c**) UC side view; (**d**) UC top view. performed to determine the optimal sizes of the unit-cell, its side is fixed to \(W=0.3\lambda_{0}=3\) mm, while the heights of each layer are \(l_{1}=l_{2}=l_{3}=11\) mm, corresponding to a total cell thickness \(T=l_{1}+l_{2}+l_{3}=3.3\lambda_{0}=33\) mm. Moreover, the resolution of the adopted 3D printer forces the maximum and minimum size of the holes to be equal to 2.65 and 0.5 mm, respectively; this choice does not allow for obtaining a perfect matching between the mid layer equivalent characteristic impedance and \(Z_{0}\), with a consequent decrease of the amplitude of \(S_{21}\), while a range of \(360^{\circ}\) is still achieved for its phase. As also pointed out in [16], the scanning capabilities of a transmitarray depend, in the first instance, on the feature of the unit-cell, whose behavior must be as less as possible subject to the direction of arrival of the unit-cell, i.e., both the phase and the amplitude of \(S_{21}\) have to stay almost the same when the angle of incidence varies.