2.1 Pattern requirements in the optimization procedure

Before describing in detail the optimization algorithm, we will establish the different pattern requirements that can be imposed in the optimization procedure and how they are implemented in the generalized IA. For the case of radiation pattern optimization, the requirements may be imposed in the copolar and crosspolar components. When performing a POS, only copolar requirements are considered due to the simplifications in the analysis of the unit cell [9]. However, a direct optimization of the layout may consider both copolar and crosspolar requirements. In the generalized IA, the copolar requirements are given by means of two mask templates, which impose the minimum (Tmin) and maximum (Tmax) values that the far field must achieve. Thus, if Gcp is the copolar gain, it should fulfil

$$T\_{\min}(\mathfrak{u}, \boldsymbol{v}) \le \mathcal{G}\_{\text{cp}}(\mathfrak{u}, \boldsymbol{v}) \le T\_{\max}(\mathfrak{u}, \boldsymbol{v}),\tag{1}$$

where u ¼ sin θ cos φ and v ¼ sin θ sin φ are the angular coordinates where the far field is computed. Figure 1 shows an example of typical copolar requirement templates for a squared-cosecant pattern and a sectored-beam pattern in a plane, where Tmin and Tmax are the minimum and maximum specifications between which the copolar pattern must lie. Alternatively, these requirements can be provided in terms of minimum gain and maximum ripple.

On the other hand, there are several methodologies to implement crosspolar requirements. A typical approach is to minimize the crosspolar far field component by means of templates [16], similarly to the procedure followed with the copolar pattern. However, the crosspolar pattern does not need a lower bound in the optimization. Thus, only the maximum mask Txp max is needed in this case, where the superscript indicates that the mask is applied to the crosspolar pattern, fulfilling

$$G\_{\rm xp}(u,v) \le T\_{\rm max}^{\rm xp}(u,v). \tag{2}$$

Figure 1. Typical requirement templates for (a) squared-cosecant pattern and (b) sectored-beam pattern.

squared-cosecant pattern [3]. Traditionally, shaped parabolic reflectors or phased arrays have been employed for these applications [2, 3]. However, shaped parabolic reflectors are bulky and expensive to manufacture, while phased arrays require complex feeding networks which introduce high losses. Nonetheless, with the popularization of the microstrip technology, reflectarray antennas have become a potential

antenna that combines the simplicity of reflectors and the versatility of arrays, using waveguides as the reflecting element. This resulted in a bulky and expensive structure. However, reflectarrays were not widely studied until the development of low-profile printed antennas in the 1980s, when the printed planar reflectarray was developed [5]. It consists of an array of radiating elements that are spatially fed by a primary feed, which is usually a horn antenna. Its working principle is based on altering the properties of the electromagnetic field impinging from the feed. By adjusting the dimensions of the reflectarray elements, a phase shift is introduced in

the impinging field [6], allowing to obtain the desired radiation pattern.

direction may be achieved with analytical equations [5], the synthesis of

The concept of reflectarray antenna was first introduced in 1963 [4] as a type of

Although designing reflectarrays for high-gain pencil beam patterns at a certain

crosspolar direct optimization of reflectarray antennas was presented in [10], using a method of moments based on local periodicity (MoM-LP) for the analysis of the unit cell. However, the algorithm was slow and only handled 1 polarization and small reflectarrays (225 elements). Other approaches for the minimization of the crosspolar component of the far field include a proper arrangement of the elements [11] and the minimization of the undesired tangential field adjusting the dimensions of the element [12] or through rotation [13]. These techniques are faster, but they work at the

In this chapter, we present a general framework for the efficient and accurate pattern optimization of reflectarray antennas for advanced wireless communications, including copolar and crosspolar specifications. It is based on the use of the generalized intersection approach (IA) algorithm [14] for the optimization and a MoM-LP [15] for the accurate characterization of the reflectarray unit cell. The design procedure is divided in several stages. First, a phase-only synthesis (POS) is carried out, to efficiently obtain the desired copolar pattern. Then, by using a zerofinding routine and the MoM-LP, the layout of the reflectarray is obtained adjusting the dimensions of each unit cell. Finally, an optional stage to improve the crosspolarization performance may be carried out. It employs the MoM-LP directly in the optimization loop to accurately characterize the crosspolar pattern. Both the POS and direct layout optimization are carried out with the generalized IA, demonstrating the versatility of the algorithm. Two relevant examples are provided to demonstrate the capabilities of the proposed framework. First, a shaped-beam reflectarray for future 5G base stations at millimeter waveband is proposed. It radiates a sectored-beam pattern in azimuth and a squared-cosecant pattern in elevation. The second example is a very large contoured-beam, spaceborne

reflectarray for direct-to-home (DTH) broadcasting based on a real space mission. The rest of the chapter is divided as follows. Section 2 introduces the optimization framework based on the generalized IA algorithm. Section 3 describes the

noncanonical beams is a challenging task and requires the use of an optimization algorithm, especially in cases with tight requirements, such as space applications [2]. Since reflectarrays are usually comprised of hundreds or even thousands of elements, the employed algorithm must be computationally efficient. Until recently, the dominant approach was the phase-only synthesis (POS) [5], which employs a simplified analysis of the unit cell. This results in an extremely efficient synthesis [7–9] but has no control over the cross-polarization performance. The first approach to the

substitute to parabolic reflector dishes and phased arrays.

Advances in Array Optimization

element level and thus provide suboptimal results.

54

However, there are applications in which the figure of merit for crosspolarization performance is not the crosspolar pattern. In particular, some space missions [2] give the requirements for the crosspolar discrimination (XPD) and/or crosspolar isolation (XPI).

The XPD is defined for a certain coverage zone as the difference (in logarithmic scale) point by point of the copolar gain and the crosspolar gain. Mathematically it is expressed as

$$\text{XPD}(\boldsymbol{\mu}, \boldsymbol{\nu}) = \mathbf{G}\_{\text{cp}}(\boldsymbol{\mu}, \boldsymbol{\nu}) - \mathbf{G}\_{\text{xp}}(\boldsymbol{\mu}, \boldsymbol{\nu}), \quad (\boldsymbol{\mu}, \boldsymbol{\nu}) \in \Omega,\tag{3}$$

where Ω is the coverage zone, XPD is in dB and Gcp and Gxp are in dBi. Usually, the minimum XPD is considered, since it is the value limiting the XPD performance in the coverage zone:

$$\text{XPD}\_{\text{min}} = \min\{\text{XPD}(u, v)\}, \quad (u, v) \in \Omega. \tag{4}$$

Similarly, the XPI is defined for a certain coverage zone as the difference (in logarithmic scale) of the minimum copolar gain and the maximum crosspolar gain:

$$\text{XPI} = G\_{\text{cp\\_min}}(u, v) - G\_{\text{xp\\_max}}(u, v), \quad (u, v) \in \Omega,\tag{5}$$

where XPI is in dB and Gcp, min and Gxp, max are in dBi. Notice that, unlike the XPD, the XPI is defined as a single value for a given coverage area. Also, the XPI is a stricter parameter than the XPD. Figure 2 shows graphically how the XPD and XPI are defined.

The optimization procedure should maximize the XPDmin and/or XPI. Thus, if TXPDmin and TXPI are the minimum requirement templates for XPDmin and XPI, respectively, they should fulfill the following condition:

$$T\_{\text{XPD}\_{\text{min}}} \leq \text{XPD}\_{\text{min}}.\tag{6}$$

$$T\_{\text{XPI}} \leq \text{XPI}.\tag{7}$$

is the matrix of reflection coefficients which define the electromagnetic behaviour of the unit cell. These coefficients are complex numbers and are computed by a full-wave analysis tool assuming local periodicity [5]. ρxx and ρyy are known as the direct coefficients, while ρxy and ρyx are known as the cross-coefficients. In addition,

In (8), ℱ is the forward projector. As shown in Figure 3, it is divided into two steps. First, starting from the tangential field, which depends on the optimizing variables, either the phases of the direct coefficients in a phase-only synthesis or the reflectarray element geometry in the case of a direct optimization, it computes the current far field radiated by the reflectarray. In its second step, it trims the far field according to the specification masks. For the power pattern synthesis, the specifications may be given in gain. Thus, if G is the current gain of the reflectarray and G<sup>0</sup>

> Tmaxð Þ u, v , Tmaxð Þ u, v < G u, v ð Þ Tminð Þ u, v , G u, v ð Þ<Tminð Þ u, v

(11)

G u, v ð Þ, otherwise:

This operation is also applied to the crosspolar pattern when performing a direct

optimization of the reflectarray layout. If the cross-polarization performance is

the copolar pattern mainly depends on the direct coefficients phase, and the

Graphical definition of the parameters for co- and cross-polarization performance: The crosspolar discrimination (XPD), which is defined point by point as the difference between the copolar gain and the crosspolar gain, and the crosspolar isolation (XPI), which is defined for the coverage zone as the difference between the minimum copolar gain and the maximum crosspolar gain. The copolar and crosspolar patterns are

Reflectarray Pattern Optimization for Advanced Wireless Communications

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

crosspolar pattern depends on all coefficients.

in dBi, while the XPD and the XPI are in dB.

G0

ð Þ¼ u, v

8 >><

>>:

the trimmed gain, then

57

Figure 2.

### 2.2 Generalized intersection approach

The framework for the optimization of the radiation pattern of reflectarray antennas is based on the generalized intersection approach (IA) [4]. A flowchart of the algorithm is shown in Figure 3. It is an iterative algorithm which performs two operations at each iteration i on the tangential field:

$$
\overrightarrow{E}\_{\text{ref},i+1} = \mathcal{B}\left[\overrightarrow{\mathcal{F}}\left(\overrightarrow{E}\_{\text{ref},i}\right)\right],\tag{8}
$$

where E ! ref is the tangential field on the reflectarray surface, calculated as

$$
\overrightarrow{E}\_{\text{ref}}\left(\mathbf{x}\_{l},\boldsymbol{y}\_{l}\right) = \mathbf{R}^{l}\ \overrightarrow{E}\_{\text{inc}}\left(\boldsymbol{\varkappa}\_{l},\boldsymbol{y}\_{l}\right),\tag{9}
$$

where xl, yl � � are the coordinates of the centre of the reflectarray element l, E ! inc is the fixed incident field impinging from the feed and

$$\mathbf{R}^l = \begin{pmatrix} \rho^l\_{\mathbf{x}\mathbf{x}} & \rho^l\_{\mathbf{x}\mathbf{y}}\\ \rho^l\_{\mathbf{y}\mathbf{x}} & \rho^l\_{\mathbf{y}\mathbf{x}} \end{pmatrix} \tag{10}$$

Reflectarray Pattern Optimization for Advanced Wireless Communications DOI: http://dx.doi.org/10.5772/intechopen.88909

Figure 2.

However, there are applications in which the figure of merit for crosspolarization performance is not the crosspolar pattern. In particular, some space missions [2] give the requirements for the crosspolar discrimination (XPD) and/or

The XPD is defined for a certain coverage zone as the difference (in logarithmic scale) point by point of the copolar gain and the crosspolar gain. Mathematically it is

where Ω is the coverage zone, XPD is in dB and Gcp and Gxp are in dBi. Usually, the minimum XPD is considered, since it is the value limiting the XPD performance

Similarly, the XPI is defined for a certain coverage zone as the difference (in logarithmic scale) of the minimum copolar gain and the maximum crosspolar gain:

where XPI is in dB and Gcp, min and Gxp, max are in dBi. Notice that, unlike the XPD, the XPI is defined as a single value for a given coverage area. Also, the XPI is a stricter parameter than the XPD. Figure 2 shows graphically how the XPD and XPI

The optimization procedure should maximize the XPDmin and/or XPI. Thus, if TXPDmin and TXPI are the minimum requirement templates for XPDmin and XPI,

The framework for the optimization of the radiation pattern of reflectarray antennas is based on the generalized intersection approach (IA) [4]. A flowchart of the algorithm is shown in Figure 3. It is an iterative algorithm which performs two

ref,iþ<sup>1</sup> ¼ B ℱ E

� � <sup>¼</sup> <sup>R</sup><sup>l</sup> <sup>E</sup>

<sup>R</sup><sup>l</sup> <sup>¼</sup> <sup>ρ</sup><sup>l</sup>

! ref,i h i � �

ref is the tangential field on the reflectarray surface, calculated as

!

� � are the coordinates of the centre of the reflectarray element l, E

xx ρ<sup>l</sup> xy

!

ρl yx ρ<sup>l</sup> yx

inc xl, yl

respectively, they should fulfill the following condition:

operations at each iteration i on the tangential field:

E !

E !

is the fixed incident field impinging from the feed and

ref xl, yl

2.2 Generalized intersection approach

XPDð Þ¼ u, v Gcpð Þ� u, v Gxpð Þ u, v , u, v ð Þ∈ Ω, (3)

XPDmin ¼ min XPD f g ð Þ u, v , u, v ð Þ∈ Ω: (4)

XPI ¼ Gcp, min ð Þ� u, v Gxp, max ð Þ u, v , u, v ð Þ∈ Ω, (5)

TXPDmin ≤XPDmin, (6)

TXPI ≤XPI: (7)

, (8)

� �, (9)

! inc

(10)

crosspolar isolation (XPI).

Advances in Array Optimization

in the coverage zone:

expressed as

are defined.

where E !

where xl, yl

56

Graphical definition of the parameters for co- and cross-polarization performance: The crosspolar discrimination (XPD), which is defined point by point as the difference between the copolar gain and the crosspolar gain, and the crosspolar isolation (XPI), which is defined for the coverage zone as the difference between the minimum copolar gain and the maximum crosspolar gain. The copolar and crosspolar patterns are in dBi, while the XPD and the XPI are in dB.

is the matrix of reflection coefficients which define the electromagnetic behaviour of the unit cell. These coefficients are complex numbers and are computed by a full-wave analysis tool assuming local periodicity [5]. ρxx and ρyy are known as the direct coefficients, while ρxy and ρyx are known as the cross-coefficients. In addition, the copolar pattern mainly depends on the direct coefficients phase, and the crosspolar pattern depends on all coefficients.

In (8), ℱ is the forward projector. As shown in Figure 3, it is divided into two steps. First, starting from the tangential field, which depends on the optimizing variables, either the phases of the direct coefficients in a phase-only synthesis or the reflectarray element geometry in the case of a direct optimization, it computes the current far field radiated by the reflectarray. In its second step, it trims the far field according to the specification masks. For the power pattern synthesis, the specifications may be given in gain. Thus, if G is the current gain of the reflectarray and G<sup>0</sup> the trimmed gain, then

$$G'(u,v) = \begin{cases} T\_{\max}(u,v), & T\_{\max}(u,v) < G(u,v) \\\\ T\_{\min}(u,v), & G(u,v) < T\_{\min}(u,v) \\\\ G(u,v), & \text{otherwise} \end{cases} \tag{11}$$

This operation is also applied to the crosspolar pattern when performing a direct optimization of the reflectarray layout. If the cross-polarization performance is

Finally, the generalized IA can be applied to perform a phase-only synthesis (POS), where the optimizing variables are the phase shift introduced by each reflectarray element corresponding to the phases of the direct coefficients in Eq. (10), or a direct layout optimization, where the optimizing variables are the

Reflectarray Pattern Optimization for Advanced Wireless Communications

This section briefly describes the design methodology employing the optimization framework presented in the previous section. It is applied to a reflectarray in single-offset configuration, as shown in Figure 4. The procedure is divided into three stages: first, a phase-only synthesis to obtain the desired radiation pattern; then, a design procedure to adjust the element dimensions yielding a reflectarray layout; and the last and optional stage is the optimization of the cross-polarization

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.

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].

geometrical features of the unit cell.

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

3. Design and optimization methodology

performance of the reflectarray antenna.

Figure 4.

59

3.1 Phase-only synthesis for the copolar pattern

Figure 3.

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

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

$$\overrightarrow{E}\_{\text{ref},i+1} = \mathcal{B}\left[\overrightarrow{\mathcal{F}}\left(\overrightarrow{E}\_{\text{ref},i}\right)\right] = \min \text{dist}\left[\mathbf{G}\_{i\nu}\,\overrightarrow{\mathcal{F}}\left(\overrightarrow{E}\_{\text{ref},i}\right)\right].\tag{12}$$

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:

$$d\_i = \text{dist}^2\left(\mathcal{F}\left(\overrightarrow{E}\_{\text{ref},i}\right), \ G\_i(u,v)\right) = \iint\_{\Lambda} w(u,v) \left(G\_i'(u,v) - G\_{ii}(uv)\right)^2 du \, dv,\tag{13}$$

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:

$$d\_i = \sum\_{u\_\nu v \in \Lambda} \left[ \sqrt{\nu(u, v) \Delta u \Delta \nu} \left( G\_i'(u, v) - G\_{i\nu}(u v) \right) \right]^2. \tag{14}$$

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