## 2 Circular Array Synthesis Approach  
Steering a transmitted beam in a desired direction is a critical function in telecommunication systems, enabling targeted coverage of specific areas within the surrounding environment. Traditionally, phased antenna arrays have been used to achieve this [34]. However, these systems depend on complex phase-shifting networks, which are often energy-intensive and can lead to signal losses [35, 36, 37]. A novel approach to overcoming these limitations involves the use of metamaterial covers and coatings, which offer an alternative design strategy by introducing abrupt phase variations in the transmitted wave [30]. In particular, we focus on the specific class of HMS, a type of engineered surface that uses arrays of subwavelength elements to achieve precise control of electromagnetic waves, including manipulation of phase, amplitude, and polarization, based on Huygens' principle of wavefront shaping. HMSs are composed of unit cells that ensure full transmission and allow for precise control over the phase of the transmitted field by aligning electric and magnetic dipole moments [38; 39; 40]. The flexibility provided by the Huygens cells makes it possible to fine-tune the antenna's performance in dynamic environments, enhancing its effectiveness in targeted communication and noise reduction. A straightforward way to synthesize the HMS cells relies on the use of stacked impedance sheets separated by dielectric layers, characterized by a surface impedance value \(Z_{s,ni}\), where \(i=1\), \(2\), \(3\ldots\) represents the number of stacked layers [19]. This configuration ensures high transmission values and phase coverage that strongly depends on the thickness of the dielectric layer [41]. In our specific case, we have chosen three metasurface layers and two thin dielectric substrates. The substrate has a thickness \(t=\lambda_{0}/60\) and a relative permittivity \(\varepsilon_{r}=10\), while \(Z_{s,ni}\) is assumed to be purely reactive (i.e., \(Z_{s,ni}=jX_{s,ni}\) with \(j\) being the imaginary unit) in the range [\(-990\), \(+100\)] \(\Omega/\)sq. In this work, the HMS coating structure is designed in Figure 1a. In [25], we have shown that for the synthesis of the metasurface, the cylindrical HMS can be segmented into N cells, with each cell functioning as an individual antenna element in an equivalent circular array, as depicted in Figure 1b. We propose to design the cylindrical metasurfaces by leveraging the synthesis techniques of circular antenna arrays. Specifically, the cylindrical metasurface is treated as operating in the H-plane, making its behavior analogous to that of a circular array. This methodology allows for the simplification of the metasurface synthesis process while enabling precise control over the radiation characteristics. In this way, the HMS can be described through the synthesis of a circular array consisting of N elements with a radius \(a\) on the xy-plane. This correspondence allows for the effective representation and manipulation of the HMS's electromagnetic behavior by treating it as an array, facilitating precise control over its phase and amplitude characteristics. In this modeling scenario, the total electric field expression of the circular array is defined as the field of a single radiating element multiplied by the array factor AF that reads as (Equation (1)):  
Figure 1: (**a**) Design of the HMS coating of the omnidirectional antenna. In the inset are the details of the unit cell implementing the HMS, which consists of stacked impedance sheets separated by dielectric layers. (**b**) Equivalent circular array model used for the metasurface synthesis procedure. \[AF(\theta,\phi)=\sum_{n=1}^{N}I_{n}e^{i[ka\ sin\theta cos(\phi-\phi_{n})+a_{n}]}, \tag{1}\]  
In particular, \(I_{n}\) represents the excitations amplitude, while \(\alpha_{n}\) describes the excitation phases of the array elements, and it is expressed as \(\alpha_{n}=-ka\ sin\theta_{0}\ cos(\phi_{0}-\phi_{n})\) when the array points toward (\(\theta_{0}\), \(\phi_{0}\)) direction.