Reflectarray Pattern Optimization for Advanced Wireless Communications

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 Gcpis the copolar gain, it should fulfil

where u=sinθcosφand v=sinθsinφare 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 Tminand Tmaxare 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 Tmaxxpis needed in this case, where the superscript indicates that the mask is applied to the crosspolar pattern, fulfilling

However, there are applications in which the figure of merit for cross-polarization 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

where Ωis the coverage zone, XPD is in dB and Gcpand Gxpare in dBi. Usually, the minimum XPD is considered, since it is the value limiting the XPD performance in the coverage zone:

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,minand Gxp,maxare 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 XPD_min and/or XPI. Thus, if TXPDminand TXPIare the minimum requirement templates for XPD_min and XPI, respectively, they should fulfill the following condition:

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 ion the tangential field:

where E→refis the tangential field on the reflectarray surface, calculated as

where xlylare the coordinates of the centre of the reflectarray element l, E→incis the fixed incident field impinging from the feed and

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]. ρxxand ρyyare known as the direct coefficients, while ρxyand ρyxare 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), ℱis 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 Gis the current gain of the reflectarray and G′the trimmed gain, then

This operation is also applied to the crosspolar pattern when performing a direct optimization of the reflectarray layout. If the cross-polarization performance is improved by means of the XPD_min 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 Bin 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:

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

where it was taken into account that the result of the forward projection is the trimmed gain in Eq. (11); wuvis a weighting function; and Λis a subset of the visible region (u2+v2≤1) where the radiation pattern is optimized. The integral in Eq. (13) can be approximated by a sum for the points uvthat belong to Λ:

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

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 geometrical features of the unit cell.

3.1 Phase-only synthesis for the copolar pattern

The first step in the design of a shaped-beam reflectarray antenna is a phase-only 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. 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]:

where ∠ρxlylis the phase of a direct reflection coefficient (ρxxor ρyy, for linear polarizations X and Y, respectively); dlis the distance from the feed to the lth element (see Figure 4); and θ0φ0is the pointing direction of the focused beam. The angle θ0φ0is 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∣=∣ρyy∣=1) and no cross-polarization (ρxy=ρyx=0). Thus, the matrix of reflection coefficients in (11) is simplified to

where ϕlis 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. 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, especially for very large reflectarray antennas, since the phase shift varies with the angle of incidence [5].

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

The third and final stage is optional and consists in improving the cross-polarization 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 requirements are specified by those parameters [17].

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 slower than the POS in the first stage.

4.1 Reflectarray for 5G base station

4.1.1 Antenna specifications

For the first example, the considered reflectarray is circular and comprised of 912 unit cells (34 elements in the main axes). The periodicity is 5.36 mm in both axes, which is half a wavelength at the working frequency, 28 GHz, in order to avoid grating lobes [5]. The feed is placed at (−79.3, 0.0, 200.2) mm with regard to the centre of the reflectarray (see Figure 4), and it is modelled as a cosqθfunction, with q=20.6, generating an illumination taper of −14.6 dB at the reflectarray edges.

The unit cell shown in Figure 4 is used here. The separation between dipoles is set to Sai=Sbi=1mm(i=1,2), while the width of all dipoles is set to 0.3 mm. Variables Tx and Ty are defined as

La4=Tx;Lb1=Lb3=0.63Tx;Lb2=0.93TxLb4=0.95Ty;La1=La3=0.58Ty;La2=Ty.E17

The same substrate is used in both layers of the unit cell, with εr=3.0and tanδ=0.0010, which corresponds to the commercially available Rogers R3003. In addition, the bottom layer has a height of hA=30mil=0.762mm, while the top layer has a height of hB=20mil=0.508mm. Figure 6 presents a unit cell study at central frequency, showing the phase shift produced by the reflectarray element as well as the losses. As it can be seen, the angular stability is good while having low losses better than −0.3 dB. Furthermore, the phase shift provided by the unit cell is more than 720°, which is more than enough for a reflectarray design and subsequent optimization.

Regarding the far field specifications, the chosen pattern for the 5G base station has a 30° sectored beam in azimuth and a squared-cosecant beam in elevation to provide constant power flux in an elevation span of 50°.

4.1.2 Results of the antenna design

The starting point for the POS is a pencil beam pointing at θ=10.4°φ=0. This direction corresponds to a region of the specification masks with high gain. To obtain this radiation pattern, the phase-shift distribution calculated with Eq. (15) is employed, and it is shown in Figure 7a for polarization X, that is, for the direct reflection coefficient ρxx(the phase shift for polarization Y is the same). After the POS, the synthesized phase shift of Figure 7b is obtained, which generates the desired radiation pattern. Then, by using the procedure summarized in Figure 5, the reflectarray layout is found.

The obtained layout was simulated with a MoM-LP [18], and the resulting radiation pattern for polarization X is shown in Figure 8, where the copolar and crosspolar components of the far field are shown in the u−vplane for the whole visible region. In this representation, it can be seen the sectored beam is along the v axis for constant u, while along u the squared-cosecant beam reduces the gain of the antenna from a maximum of 19.6 dBi to roughly 5 dB. This represents a dynamic range of almost 15 dB in which the shaped beam has to smoothly decrease over an angular span of 50°, making it challenging pattern to synthesize. In fact, it is very easy to obtain nulls in this region that penalize performance, even in simulations [21], and they have been avoided with success in the present example. Similar results were obtained for polarization Y.

On the other hand, Figure 9 represents the main cuts in elevation and azimuth for both linear polarizations along with the mask requirements. Here, it can be better appreciated how the specifications are met, with side lobes lower than −2 dB, which represent a SLL better than 20 dB for this shaped pattern. In addition, Figure 10 shows the radiation pattern in 3D perspective, along with a sketch of the reflectarray panel.

Regarding the cross-polarization performance, the initial design presents maximum crosspolar values of −6.1 and −6.8 dBi for polarizations X and Y, respectively, while the maximum copolar gain is 19.6 dBi for both polarizations. This gives a maximum copolar gain/maximum crosspolar gain ratio (CPmax/XPmax from here on) of 25.7 and 26.4 dB for polarizations X and Y, respectively. The following step will be to improve the ratio CPmax/XPmax by minimizing the crosspolar component of the far field while keeping the copolar pattern within specifications and maintaining the maximum copolar gain. To this end, a direct optimization layout will be performed using the generalized intersection approach. Now, the optimizing variables will be variables Tx and Ty as defined in Eq. (17), instead of the phases of the reflection coefficients. In addition, since the starting point already complies with the copolar requirements, all variables will be optimized at the same time. Thus, a total of 1824 variables will be considered. The copolar requirements are the same, while for the crosspolar pattern, a constant template Tmaxxp40 dB below the maximum copolar gain is imposed for both linear polarizations. The goal is to minimize the crosspolar pattern as much as possible.

After the crosspolar optimization, the radiation pattern shown in Figure 11 was obtained for polarization X. When compared with the far field of Figure 8, it can be seen how the crosspolar pattern maximum value has been considerably reduced while keeping the copolar pattern within specifications. In fact, the maximum copolar gain is now 19.7 dBi and 19.6 dBi for polarizations X and Y, respectively. At the same time, the maximum crosspolar values are −15.7 and −15.4 dBi, with CPmax−XPmax values of 35.4 and 35.0 dB for polarizations X and Y, respectively. This represents an improvement of 9.7 and 8.6 dB for both linear polarizations. This information is summarized in Table 1. Finally, Figure 12 shows the layout of the optimized reflectarray for both layers.

	Polarization X			Polarization Y
	CPmax	XPmax	CPmax−XPmax	CPmax	XPmax	CPmax−XPmax
Initial design	19.6	−6.12	25.7	19.6	−6.84	26.4
Optimized design	19.7	−15.75	35.4	19.6	−15.40	35.0

Table 1.

For the reflectarray for 5G base station, summary of the performance of the initial and optimized designs regarding the maximum copolar gain (CP_max), the maximum crosspolar gain (XP_max) and the difference between them (CP_max-XP_max) for both linear polarizations.

CP_max and XP_max are in dBi, while CP_max−XP_max is in dB.

4.2 Reflectarray for direct-to-home satellite application

4.2.1 Antenna specifications

For the second example, an elliptical reflectarray with axes 1128mm×1080mmand comprised of 6640 elements, is considered. The reflectarray cells are arranged in a rectangular grid of 94×90elements for polarization X and 93×89elements for polarization Y, with a periodicity of 12 mm in both axes. The working frequency is 12.5 GHz. The feed is placed at −352.9,0.0,1061.7mmwith regard to the reflectarray centre and is modeled as a cosqθwith q=18, which generates an illumination taper of −17.9dB.

A similar unit cell as in the previous example is used with different dimensions and materials. The separation between dipoles is now set to Sai=Sbi=2.5mm(i=1,2), while the width of all dipoles is set to 0.5 mm and Txand Tyare defined in Eq. (17). Commercial substrates were chosen for both layers, the Arlon AD255C for the top layer, with hA=2.363mmand εr=2.17−j0.0020, and DiClad 880 for the bottom layer, with hB=1.524mmand εr=2.55−j0.0036. Figure 13 presents a unit cell study at central frequency, showing the phase shift produced by the reflectarray element as well as the losses. As it can be seen, the angular stability is good while having low losses better than −0.3dB. The phase shift provided by the cell is slightly larger than 600°.

Figure 14 shows the contour requirements for the Southern Asia mission, similar to that provided by the SES-12 satellite. Zone 1 includes India, Nepal, Bhutan, Bangladesh and Sri Lanka, while zone 2 includes Pakistan and Afghanistan. According to the official specifications [22], the satellite provides an EIRP of 52 dBW for zone 1 and 48 dBW for zone 2. The EIRP can be converted into gain using the following expression:

GdBi=EIRPdBW−PtdBW,E18

where Pt is the power of the transponder. Assuming Pt=150W, it gives a gain specification of 30 dBi for zone 1 and 26 dBi for zone 2. In addition, the design process will take into account typical pointing errors (0.1° in roll and pitch and 0.5° in yaw). The design will be carried out in dual-linear polarization, imposing the same specifications in both polarizations.

4.2.2 Results of the antenna design

For the first step, a POS is carried out to obtain the desired copolar pattern in dual-linear polarization. Figure 15a shows the initial phase shift for the POS obtained with Eq. (15). It generates a focused beam in the direction θ=16.5°φ=0.0°, which corresponds to a high-gain area in India. After the synthesis, the phases shown in Figure 15b were obtained for polarization X. The phases for polarization Y are similar. Once the layout has been obtained, the radiation patterns were computed using a MoM-LP tool. The copolar and crosspolar components for this initial design are shown in Figure 16 for polarization X. In this case, the minimum copolar gain is 31.5 and 28.6 dBi for zones 1 and 2, respectively. Similar results were obtained for polarization Y. Thus, the initial design complies with the requirements in both linear polarizations.

Space missions usually impose very stringent cross-polarization requirements in the form of crosspolar discrimination (XPD) and crosspolar isolation (XPI) for the transmit and receive bands, respectively. Notice that according to the definitions of minimum XPD in Eq. (4) and the XPI in Eq. (5), the XPI is a more stringent parameter than the XPDmin. The first row of Table 2 shows the values of XPDmin and XPI for both coverage zones and polarizations. The initial design presents values of those parameters between 29.5 and 33.0 dB. The goal is thus to improve the cross-polarization performance of this reflectarray by performing a direct optimization of the layout. As in the previous case, Tx and Ty are considered as optimization variables. Thus, a total of 13,097 variables will be considered. In addition, instead of minimizing the crosspolar pattern as in the previous example, now the XPDmin and XPI will be optimized as detailed in [17]. To that end, minimum masks of 37 dB are imposed for both parameters. The goal is to increase as much as possible the XPDmin and XPI while keeping the minimum copolar gain for both coverage zones within specifications.

	Zone 1						Zone 2
	Polarization X			Polarization Y			Polarization X			Polarization Y
	CP	XPDmin	XPI	CP	XPDmin	XPI	CP	XPDmin	XPI	CP	XPDmin	XPI
Initial design	31.52	32.50	32.08	31.36	30.51	29.56	28.63	32.99	31.13	28.91	31.76	29.87
Optimized design	30.07	38.99	38.11	30.04	39.77	37.51	29.15	39.01	37.53	29.17	39.79	38.23

Table 2.

For the reflectarray with southern Asia coverage, comparison of the performance of the initial design after the POS and the optimized layout for improved cross-polarization performance.

CP is the minimum copolar gain in dBi in a coverage area, XPDmin is the minimum crosspolar discrimination in dB, and XPI is the crosspolar isolation in dB.

After the direct layout optimization, the cross-polarization performance of the reflectarray antenna significantly improved. The worst parameter is the XPI for zone 1 and polarization X, which has a value of 37.5 dB. It improved to 8 dB over the value for the initial design. The minimum improvement was 6 dB for the XPI for zone 1 and polarization X and XPDmin for zone 2 and polarization X. At the same time, the copolar minimum gain still complies with the specifications of 30 dBi for zone 1 and 26 dBi for zone 2. A summary of the performance of the initial and optimized layout may be found in Table 2. In addition, Figure 17 shows the copolar and crosspolar pattern for polarization X of the optimized layout. Since the optimization has maximized the cross-polarization performance in the two coverage areas, the maximum crosspolar values are outside both of them. Nevertheless, even its value has decreased, as it can be seen by comparing the crosspolar pattern of Figures 16 and 17.