# Optimization Environment Definition for Beam Steering Reflectarray Antenna Design
## 1 Introduction  
In recent years, Evolutionary Algorithms (EAs) have been successfully applied to many engineering problems thanks to their capability to find optimal solutions in nonlinear and multimodal problems [1]. Another important aspect of EAs is their flexibility: similar frameworks can be applied to different engineering problems [2]. Most of the problems faced with EAs require complex models to simulate the system and to define performance parameters. The model complexity, required for providing an accurate output of the optimization process, is usually time expensive; consequently, it requires an effective optimization process management to achieve the optimal result in reasonable time [3]. The design optimization of Reflectarray Antennas (RAs) is a clear example of this problem: in fact, these systems are composed by hundreds or even thousands of elements that can be optimized, leading to a very large non-linear multimodal optimization problem [4]. RAs are a group of powerful and efficient high-gain antennas; they are highly adopted in many different conditions thanks to their numerous advantages, such as low profile, low cost, good radiation performance, and ease of manufacturing [5]. Compared to traditional phased arrays, RA have a less complex feeding system and, thus, lead to a reduction in the losses introduced by the feeding networks [6]. Among all the antenna configurations, reflector antennas have been exploited for the high demand of radar and satellite communication, where a point-to-point connection was needed and, consequently, a high gain was required. In order to reduce the space required by the reflector, Reflectarray Antennas have been introduced, especially in aerospace applications: in fact, they have low weight, low profile, and the possibility to be easily folded [7]. Reflectarray Antennas consist in a planar array made up of different re-radiating elements illuminated by a feed source (typically a horn antenna) placed in central or offsetposition. In these antennas, changing one or more geometrical parameter of each patch is used to control the phase of the re-radiated field changes and to obtain the desired radiation pattern [8]. The design process complexity for RA depends on the type of radiation pattern that is required: for the simple case of a pencil beam RA, the design can be carried out analytically [9]. In the case of more complex requirements in terms of radiation pattern, optimization methods are fundamental. In this context, Evolutionary Algorithms (EAs) are very valuable tools. Genetic Algorithms (GAs) have been widely adopted in the field of reflectarray optimization. In [10] the authors propose the application of GA to create the reflection layer of an RA: with respect to other works, in this one the reflection element shape is not fixed and can be tuned with the GA. A similar approach has been implemented in [11] for a dual band RA. In [12], a reconfigurable RA composed by a two layer pixel patches is proposed. The second layer height can be mechanically adjustable to change the beam direction. Both the pixel patches are optimized by means of a binary GA.  
Other EAs have been applied to this problem: for example, Particle Swarm Optimization has been implemented in [13] to design a dual band RA and also Differential Evolution has been applied for the design of RAs [14]. In order to improve the optimization performance, variations of known algorithms are proposed: in [15] an dynamic clustering process is introduced in PSO to improve the trade-off between exploration and exploitation. The application of a single algorithm is not always sufficient due to the problem complexity: for this reason, hybrid approaches have been proposed in literature. The first tested hybridization consists in the combination of GA and PSO [16], while further improvements can be achieved combining EAs with deterministic methods, like the Taguchi one [17; 18]. Finally, local search techniques can be applied for the final refinement of the solutions found by the EA.