3.1 Phase-only synthesis for the copolar pattern

The first step in the design of a shaped-beam reflectarray antenna is a phaseonly synthesis (POS). The aim of the POS is to obtain a phase-shift distribution that generates the desired shaped radiation pattern, which in general cannot be obtained through analytical means since that approach presents some limitations [9]. Since we are interested in dual-linear polarized reflectarrays, two phase-shift distributions are necessary, one for each linear polarization. In addition, the generalized IA is a local search algorithm. Thus, a good starting point is of utmost importance.

#### Figure 4.

Diagram of the planar reflectarray antenna optics and the considered unit cell based on two sets of parallel and coplanar dipoles. © 2018 IEEE. Reprinted, with permission, from [17].

improved by means of the XPDmin or XPI optimization, a similar expression to Eq. (11) is used but only taking into account the minimum masks, as in Eqs. (6) and (7). The second operation of the generalized IA, denoted by B in Eq. (8), is the backward projector. It minimizes the distance between the trimmed gain and the current gain radiated by the antenna (see Figure 3), obtaining a reflected tangential

The latter operation is performed by a general minimizing algorithm [14]. In addition, as a distance definition, we employ the Euclidean norm for squareintegrable functions [8], which is implemented by the weighted Euclidean metric:

w u, v ð Þ G<sup>0</sup>

i ð Þ� u, v Gi, uv ð Þ � � h i<sup>2</sup>

This sum can be minimized by the Levenberg-Marquardt algorithm (LMA) [9].

where it was taken into account that the result of the forward projection is the trimmed gain in Eq. (11); w u, v ð Þ is a weighting function; and Λ is a subset of the visible region (u<sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>≤</sup>1) where the radiation pattern is optimized. The integral in Eq. (13) can be approximated by a sum for the points ð Þ u, v that belong to Λ:

¼ min dist Gi, ℱ E

i

! ref,i h i � �

ð Þ� u, v Gi, uv ð Þ � �<sup>2</sup>

: (12)

du dv, (13)

: (14)

field that generates a radiation pattern that is closer to fulfil specifications:

Flowchart of the generalized intersection approach algorithm as applied to the far field in gain.

¼ ðð Λ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w u, v ð ÞΔuΔ<sup>v</sup> <sup>p</sup> <sup>G</sup><sup>0</sup>

! ref,i h i � �

E !

Advances in Array Optimization

! ref,i � �

di <sup>¼</sup> dist<sup>2</sup> <sup>ℱ</sup> <sup>E</sup>

58

Figure 3.

ref,iþ<sup>1</sup> ¼ B ℱ E

, Gið Þ u, v

� �

di <sup>¼</sup> <sup>X</sup> u, <sup>v</sup>∈<sup>Λ</sup> It has been demonstrated that a properly focused pattern is sufficient for the POS [7]. In that case, the initial phase distribution for the POS may be obtained analytically [5]:

$$
\Delta\rho(\mathbf{x}\_l, y\_l) = k\_0 \left( d\_l - \left( \mathbf{x}\_l \cos \varphi\_0 + y\_l \sin \varphi\_0 \right) \sin \theta\_0 \right), \tag{15}
$$

It requires the use of a full-wave technique based on local periodicity (FW-LP) to analyse the unit cell. Here, we employ the MoM-LP described in [18] to analyse the unit cell shown in Figure 4. In this step, a common procedure in the literature is to use a design curve obtained at normal incidence to seek the size of the reflectarray element that matches the required phase shift. However, it is recommended to consider the real angle of incidence to increase accuracy,

Reflectarray Pattern Optimization for Advanced Wireless Communications

DOI: http://dx.doi.org/10.5772/intechopen.88909

especially for very large reflectarray antennas, since the phase shift varies with the

This procedure is divided into three steps. Firstly, a phase-shift table is gener-

ated, increasing the size of the element (for instance, the patch size or dipole length) in little intervals. For the case at hand and using the unit cell based on two sets of parallel dipoles of Figure 4, two variables,Tx and Ty, are defined that allow to control the phase shift for linear polarizations X and Y, respectively. Thus, the phase-shift table is generated modifying at the same time Tx and Ty. Then, we select two sizes of the element that provide a phase shift a little above and below the exact value. This is done independently for the two linear polarizations. Next, a linear equation is used to approximate the value of the element size that provides the required phase shift. Finally, by using a zero-finding routine (for instance, the Newton-Raphson method as indicated in Figure 5), the exact value for both polarizations is sought at the same time, taking into account the coupling between polarizations. This is done for every reflectarray element, obtaining a layout which generates the desired radiation pattern obtained in the POS of the first stage.

3.3 Improvement of the cross-polarization performance through direct

The third and final stage is optional and consists in improving the crosspolarization performance of the synthesized reflectarray by directly optimizing its layout using a FW-LP tool. This is especially important for applications with tight cross-polarization requirements, such as space missions [2], since the layout obtained in the previous stage most likely will only comply with copolar specifications. As a starting point, the layout obtained in the previous stage is employed. Also, the copolar specification masks are maintained to keep the copolar pattern within specifications while the cross-polarization performance is improved.

There are a number of approaches that can be followed in this stage depending on the application. A common approach in the literature is to directly minimize the crosspolar component of the far field [19, 20]. This is done by applying Eq. (2) for the crosspolar pattern masks in the forward projector of the generalized IA. Another approach is to impose Eqs. (6) and (7) in order to maximize the XPDmin or the XPI. This is especially convenient for space applications in which cross-polarization

Nevertheless, this stage requires the use of a FW-LP tool to obtain the full matrix of reflection coefficients in Eq. (10) in order to correctly characterize the crosspolar radiation pattern. Thus, the improvement in cross-polarization performance will be

Here, we present two examples of application of the optimization framework presented in the previous sections. First, a medium-sized reflectarray is designed to work in a base station for future 5G application in the millimeter band at 28 GHz. The second example is a very large contoured-beam reflectarray for

requirements are specified by those parameters [17].

slower than the POS in the first stage.

4. Examples of application

61

angle of incidence [5].

optimization

where ∠ρ xl, yl � � is the phase of a direct reflection coefficient (ρxx or ρyy, for linear polarizations X and Y, respectively); dl is the distance from the feed to the lth element (see Figure 4); and θ0, φ<sup>0</sup> ð Þ is the pointing direction of the focused beam. The angle θ0, φ<sup>0</sup> ð Þ is usually selected in a direction where the desired shaped beam has maximum gain.

Then, the generalized IA is employed to synthesize the desired pattern. For the POS, the elements are modelled as ideal phase shifters, in which there are no losses (∣ρxx∣ ¼ ∣ρyy∣ ¼ 1) and no cross-polarization (ρxy ¼ ρyx ¼ 0). Thus, the matrix of reflection coefficients in (11) is simplified to

$$\mathbf{R}^l = \begin{pmatrix} \exp\left(j\,\phi^l\_{\text{xx}}\right) & \mathbf{0} \\ \mathbf{0} & \exp\left(j\,\phi^l\_{\text{yy}}\right) \end{pmatrix},\tag{16}$$

where ϕ<sup>l</sup> is the phase of the corresponding reflection coefficient. Thus, the optimizing variables are the phases of the direct coefficients. In addition, the POS is carried out in several steps, gradually increasing the number of optimizing variables as suggested in [14] to further improve the convergence of the algorithm. Once the desired phase-shift distributions are obtained, the following step is to obtain the reflectarray layout.

#### 3.2 Obtaining a reflectarray layout from a phase-shift distribution

The procedure to obtain a reflectarray layout from the two phase-shift distributions obtained after the POS is summarized in the flowchart of Figure 5.

#### Figure 5.

Flowchart of the procedure to obtain a reflectarray layout from the synthesized phase-shift distribution for two linear polarizations.
