Reflectarray Feeds Augmented With Size-Reducing GRIN Lenses for Improved Power Handling and Aperture Efficiency

The feed for reflectarrays and transmitarrays is crucial for determining the overall antenna aperture efficiency, gain, and power handling. In the past, several methods, such as altering the horn shape to achieve higher modes, have been employed to modify the feed radiation pattern to improve these key parameters. However, another approach is to use gradient index (GRIN) lenses, which offer unprecedented control over the radiation pattern and can be used to shrink the overall antenna. In this paper, we demonstrate highly effective approaches for optimizing GRIN lenses to achieve different design objectives including high gain radiation with a compact feed as well as improved aperture efficiency and power handling. Additive manufacturing is employed to fabricate the lenses, and measured results agree well with simulations.


I. INTRODUCTION
R EFLECTARRAY and transmitarray antennas have many advantages over traditional high-gain designs, such as parabolic dishes and phased-array systems [1]. Compared to parabolic dishes, reflectarray (RA) and transmitarray (TA) antennas offer faster reconfigurable beam steering and multibeam operation. Meanwhile, compared to phased-array systems, RA and TA antennas are often simpler to construct. For these reasons, RA and TA antenna designs that achieve high aperture efficiency and handle high power levels are of significant interest for radar and long-distance communications applications.
For any aperture antenna, maximizing the aperture efficiency is highly desirable as it directly affects the tradeoff between overall footprint and realized gain; maximizing aperture efficiency can lead to considerable SWaP-C (size, weight, power, and cost) reductions for a targeted realized gain. While the overall aperture efficiency depends on many variables, the main limiting factors are illumination and spillover efficiencies [2]. Other factors include blockage by the feed and support structures, material losses, and phase discretization effects.
The illumination efficiency is a measurement of the field amplitude uniformity across the antenna aperture, while the spillover efficiency describes the ratio of the total radiated power to that which is incident on the reflector [3]. These two factors are determined based on the shape of the reflector and the feed source. Roughly fifty years ago, considerable effort was made to develop feeds that maximized aperture efficiency [4], [5], [6]. Some of the approaches explored since then include multimode [7], [8], corrugated [9], [10], and dielectric-loaded horns [11], [12], [13], as well as homogeneous dielectric lenses [14] and array feeds. Recently, metamaterials [15] and dielectric materials [16], [17], [18], [19], [20] have been used to improve the far-field gain of horn antennas. However, these previous design studies have not considered the aperture efficiency improvement of RA antennas.
In this paper, we propose using a Gradient Index (GRIN) lens to augment a conventional horn antenna to reduce the overall size of the reflector feed while improving the aperture efficiency of a representative planar RA antenna. GRIN lenses possess spatially varying (i.e., inhomogeneous) permittivities which give them greater control over the propagation of electromagnetic waves than traditional homogeneous lenses. GRIN lenses can be fabricated using additive [21] or subtractive manufacturing [22], [23] techniques. Moreover, they can be constructed from metamaterials [15] when anisotropic behaviors and/or inhomogeneous permeabilities are required. The predominant approach for GRIN lens design is transformation optics [24], [25], which has been successfully employed to create lenses that enhance gain [26], [27], [28], [29], reduce sidelobes [18], [30], and produce multibeam emission [31], [32], [33]. Other approaches for GRIN lens design include ray-tracing [34] and the field transformation technique [27]. In this work, we combine multiple optimization approaches with full-wave simulation to design bespoke GRIN lenses for offset-fed planar square RAs. While the remainder of this paper deals only with RA antennas, the same principles also apply to TA antennas. This paper describes two pathways for GRIN lens integration into RA systems. The first approach, examined in Section II, presents a tradeoff study of different GRIN lens designs with varying sizes and realized gain levels. A design was fabricated to demonstrate how the feed size for an RA antenna system could be reduced with a lens mounted on top of a short horn. This SWaP-reduced feed source could be useful for minimizing blockage, shrinking the reflectarray's overall dimensions, and for feeding smaller reflectors or reflectarrays where conventional horns are often quite large. The design approach discussed in Section III concerns GRIN lenses which are optimized to maximize the aperture efficiency and power handling capability of the RA. These lenses are useful for reducing system size for a targeted gain value, which can lead to fabrication cost reductions as well as a more evenly distributed incident field across the RA surface. This can increase the safety margin for operation (an important factor for high-power microwave (HPM) applications) or allow the RA to handle more power with a simultaneously reduced size.
In these studies, the GRIN lens is placed directly on and mounted to the mouth of the horn. This allows the lens to serve as a radome for the horn feeds and enables them to be pressurized. While this behavior was not pursued in this study, a simple modification of the lenses would be to introduce a dielectric shell to eliminate any air gaps and ensure pressurization of the augmented lens/horn feed system.

II. SIZE-REDUCING GRIN LENSES
In this first section, we will employ custom numerical optimization techniques to design GRIN lens augmented horn feeds to achieve similar gains as larger horns, albeit with a smaller cross-sectional area and height.

A. OPTIMIZING LENSES WITH DIFFERENT SIZES
For large RAs, the horn size is of little consequence since it is typically small compared to the reflector. However, for smaller RAs (on the order of 10 wavelengths), utilizing a high-gain feed is still desired to achieve uniform illumination and reduce spillover. Yet, high gain horns can be quite large and cause significant blockage. To mitigate these detrimental effects, we explored reducing the cross-sectional area of the GRIN and minimizing the feed's overall height.
Previously, an adjoint sensitivity method was developed to design GRIN lenses with a similar aperture size as a 20 dB standard gain horn, albeit with reduced total height [16]. Using this approach, various-sized GRIN lenses were optimized for maximum broadside gain. However, unlike in the previous study, which only sought to reduce the overall height, here we desired to shrink the aperture dimensions as well. Figure 1(a) shows the set of lenses optimized for maximum gain but with a significantly reduced size. Each lens design represented by a circle in Figure 1(a) has a different cross-sectional width, length, and thickness. The feed horn for the lens was fixed at 5 in (127 mm) tall and had the same aperture dimensions as the GRIN lens. The permittivity was allowed to vary between 1 and 10 for each discrete block that made up the lens. The blocks were 2 x 2 x 0.5 in 3 (50.8 x 50.8 x 12.7 mm 3 ) with the intention that the designs could be realized using Preperm 3D printer materials [35]. Unsurprisingly, larger feeds generally do better than smaller ones. However, all solutions are much smaller than the standard gain horn (SGH). The inset shows the SGH for comparison and the best GRIN lens that achieves similar gain as the SGH. This GRIN lens design is 6 in (152.4 mm) thick and has an aperture size of 14 x 10 in 2 (355.6 x 254 mm 2 ). The SGH has an aperture size of 19.7 x 13.8 in 2 (500.38 x 350.52 mm 2 ) and a height of 29.4 in (746.76 mm). The relative permittivity of each block is between 2 and 5.1, with most dielectric blocks having a relative permittivity of less than 3. Figure 1(b) shows the simulated (COMSOL Multiphysics v5.6) realized gain for the best GRIN lens. The peak gain is around 16.9 dBi, which is comparable to that of the SGH. This design also possesses a peak side lobe level of 0.7 dB, lower than that of the SGH (see Figure 5(a)).
This design demonstrates that we can significantly reduce the overall size of the feed horn and therefore reduce the associated blockage for a RA system. In addition, we reduced the side lobe levels, which should result in lower spillover. The effective aperture area is 240 in 2 (0.155 m 2 ), which is much greater than the physical cross-sectional area of 140 in 2 (0.090 m 2 ). Figures 2(a) and (b) show the electric field magnitude 5 cm above the SGH aperture and the GRIN lens, respectively. The SGH has most of the field concentrated in the center of the horn as would be expected for the TE 10 mode of operation. By comparison, the GRIN lens more evenly distributes the electromagnetic wave across the smaller aperture. Figures 2(c) and (d) show the phase of the electric field in both the xz-and yz-planes for the SGH and the GRIN lens, respectively. For the SGH, the phase contours are flat in both planes directly above the horn aperture and include some mild wavefront curvature towards the edges. In contrast, the GRIN lens design has planar wavefronts directly above the GRIN lens and in the regions beyond the edges of the lens. The radiation from the sides of the lens helps to spread out the electric field and create a larger effective area where the wavefront is planar. In some ways, the proposed GRIN lens is similar to both a conventional lens and a dielectric rod antenna (e.g., [36]) with radiation coming from both the front and sides of the lens, yet it is much narrower than a conventional lens while also being much shorter than a dielectric rod antenna. In the next section, we realize a compact feed with a lens and horn optimized for a chosen fabrication method.

B. FABRICATED LENS DESIGN
Several modifications were implemented into the GRIN setup to make the lens fabrication easier and more robust, after which the lens was reoptimized in HFSS. The first modification was to choose a commercially available horn with dimensions of 10.75 x 9.05 x 8.7 in 3 (273.05 x 229.87 x 220.98 mm 3 ) with a 0.9 in (22.86 mm) flange around the aperture, which allowed the lens to be directly attached to the horn with bolts (see Figure 3  two-fold symmetric structure. Holes of varying sizes were drilled into the Delrin 150 White material ( r = 3.1, tan δ = 0.03) along one axis to achieve the different permittivity values by varying the volume filling fraction of the combined dielectric/free-space media. These materials and production strategies made the lens easier and more affordable to fabricate with the final design being robust enough to support pressurization if required (a simple solid dielectric covering could be added to the design to fully enclose it). The final fabricated lens is displayed in Figure 4(a), which is seen to have similar size dimensions to the lens optimized in Section II-A.
The newly fabricated lens was reoptimized for maximum gain, lower side lobe levels and a symmetric beamwidth in the E-and H-planes. A symmetric beam pattern is important for attaining uniform illumination in the two orthogonal cuts and for maximizing aperture efficiency. Figure 4(b) shows that the realized gain of the fabricated lens agrees well with simulations from 1.2 to 1.7 GHz (the lens was designed for an operational frequency of 1.5 GHz), which confirms the efficacy of the chosen fabrication strategy. As further confirmation, Figure 4(c) shows the E-and H-plane realized gain patterns at 1.22, 1.50, and 1.63 GHz. The simulated and measured patterns show good agreement, which further validates our approach.
Lastly, we compare the fabricated lens design to the corresponding SGH. Figure 5 shows the realized gain pattern in the E-and H-plane for (a) the SGH and (b) the fabricated lens design. The SGH has a peak gain of 16.8 dBi, while the fabricated lens has a peak gain of 14.7 dBi. We note that the peak gain levels are not as crucial for RA antennas since the overall system gain is mainly determined by the RA's size. Further, a higher gain horn needs to be placed farther from the RA, which increases the overall system size. In addition, our fabricated design has lower side lobes that are at least 21 dB below the main beam. On the other hand, the SGH has side lobes that are 14 dB below the main beam. Both designs have symmetric patterns in the E-and H-planes. This suggests that the fabricated design should have lower spillover than the SGH at a considerably smaller size. The area of the aperture and height of the fabricated lens is 50% and 48% less than that of the SGH, respectively. Thus, we were able to accurately fabricate a lens which could serve as a more compact feed for a RA antenna.

III. APERTURE EFFICIENCY IMPROVING GRIN FEEDS
A second avenue of this research is to explore GRIN lens designs that yield greater aperture efficiency and support higher power handling capability. Traditional feeds such as horns have a gradual taper where the feed location has to be optimized to balance spillover and illumination efficiency. The ideal radiation pattern would have no spillover and illuminate the aperture uniformly. However, standard feeds do not have these properties. Rather, the beam intensity usually starts high at the center and then tapers towards the edge with significant spillover. Furthermore, to reduce blockage the feed often needs to be offset from the center of the reflector. This results in an asymmetric pattern, which makes finding the best feed source for a particular application even more challenging.
GRIN lenses provide a considerable ability to control the radiation coming from the source and can be used to improve the total aperture efficiency. However, what perhaps is more interesting is their ability to evenly distribute the power over the RA surface. In order to maximize power delivered to a location, either the gain or feed power level can be increased. Improving the gain can be achieved easily by simply increasing the size of the RA. However, this is undesirable, especially at low frequencies, since the RA can quickly become too large for many practical applications. Therefore, increasing the power levels is the better alternative. Yet this can easily lead power levels incident on the reflector that many RA antennas can no longer survive [37], [38], [39]. Even for more conservative power levels, power handling improvement could be valuable for electrically reconfigurable designs (e.g., using varactordiodes) to prevent component failure. Therefore, improving aperture efficiency can improve gain and, hence, the amount of power delivered to a target. More evenly distributing the power across the reflectarray improves the power handling capability of the system.

A. COMPUTATIONAL METHODS
The aperture efficiency, η ap , is calculated by multiplying the illumination and spillover efficiencies, which are obtained by integrating the fields over the surface where the RA would be positioned (see [3]). The spillover efficiency η sp is where S is the reflectarray surface to integrate over, P z is the Poynting vector normal to the surface, and P in is the total input power. Since the feed is pointed in the negative direction, we take the absolute value of the integration of the Poynting vector to remove the negative sign. The illumination efficiency η il can be computed using: where |E| is the magnitude of the electric field, and A is the area of the RA. We note that this approach is slightly different than approaches based on far-field radiation patterns but is easier to compute for rectangular RAs.
Besides aperture efficiency, we also computed two other metrics in our analysis of the GRIN lens fed antenna systems. The first metric, K, is the peak electric field relative to the no-lens case where E NoGRIN and E GRIN are the electric fields across the hypothetical reflectarray surface without and with the GRIN lens, respectively, for which we find the max value across the surface, S. This metric is very important when considering the power handling capability of the reflectarray since the location with the largest electric field will correspond to the most likely location for electrical breakdown. Therefore, minimizing the relative peak electric field is one of the main design objectives. The second metric of interest is similar to illumination efficiency but better corresponds to characterizing a uniform field distribution. Here the proposed metric is where we compute what we call the uniformity efficiency, η u , from the mean electric field over the peak field across the surface. As will be shown later, this metric corresponds to a more uniform distribution of the electric field and therefore can be optimized to find designs that are better for handling high power levels. Illumination efficiency has the fault that it averages out peak values so, as will be shown later, two GRIN lens designs may have very similar illumination efficiencies but drastically different uniformity efficiencies. Lastly, we compute what we call the relative effective radiated power (RERP), which is the effective radiated power relative to the no-GRIN lens case.
This expression indicates that a higher aperture efficiency with the GRIN lens will improve the gain of the reflectarray and result in higher power delivered to the target direction. Similarly, if the GRIN lens reduces the peak electric field magnitude across the RA surface, one can increase the feed power to increase the effective radiated power delivered to the target direction. Alternatively, a higher K value provides the RA with a larger safety factor before dielectric breakdown would occur.

B. REFLECTARRAY SETUP AND NO GRIN PERFORMANCE
There are many ways to set up the feed and RA system. For this proof-of-concept study, a 1 m (10 wavelength) per side square aperture is illuminated by a commercial off the shelf (COTS) pyramidal horn. The horn dimensions a, b, and l are 188 mm, 137.2 mm, and 211.3 mm, respectively. The horn flange was 13.5 mm. A horizontal ground plane of the same width as the RA was placed under the horn feed and extends from the RA surface to the end of the horn (see Figure 6).
Initially, the horn's location and direction were varied in a FEKO simulation framework to find the optimal position/orientation to achieve maximum aperture efficiency. Then, a GRIN lens was placed in front of the horn and the permittivity profile was optimized to achieve maximum effective radiated power compared to the no-lens case. Figure 7 shows the aperture efficiency for the no-lens case when (a) the horn is pointed at the center of the reflectarray and (b) when the horn is pointed −4 degrees below the center. Figure 7(a) unsurprisingly shows that the centerfed configuration is the optimal design choice, providing the largest total aperture efficiency. However, this horn positioning would result in increased blockage. On the other hand, with an offset feed, similar aperture efficiency to that of the center-fed system is achieved when the horn is positioned at −0.450 m below the center of the RA and 0.9 m away from the RA. This performance is enabled by tilting the feed horn slightly down from the center of the RA. After finding the feed location corresponding to the greatest aperture efficiency of the no-lens system, the GRIN lens was optimized to maximize aperture efficiency and to better distribute the power across the surface according to the metrics defined in Eqs. (1), (2), and (4).

C. OPTIMIZATION PROCEDURE
As pointed out in the introduction, several methods have been used to design GRIN lenses including transformation optics, field transformation methods, and ray tracing approaches. However, none of these methods have been used to tailor the fields on a RA surface. It is difficult to even know a priori what the optimal field pattern for the RA would even need to be since an ideal uniform field with zero spillover cannot be achieved in practice. Therefore, we used an optimization approach that would maximize the aperture efficiency and RERP instead of trying to target a specific near-field pattern on the RA. This technique is based on the GRIN lens generation method [40] but extended to support permittivity variations in all three orthogonal directions. The variables from the optimizer control the permittivity of a GRIN lens with a fixed size and position, while the objective function maximizes the RERP with the constraint that the aperture efficiency is greater than 65%. This constraint was necessary to prevent designs where there was high spillover, which would also result in a high K value (i.e., since with high spillover there will also be less power incident on the RA).
To optimize the GRIN lens, we employed the CMA-ES algorithm to update the permittivity of 12 x 11 x 5 dielectric cubes that were each 2 cm on each side for a total lens size of 24 x 22 x 10 cm 3 . The GRIN profile was symmetric across the yz-plane. To efficiently optimize the permittivity of the lens, 18 control points were evenly distributed in a uniform 3 x 3 x 2 grid. The permittivity of each block was then found by performing a cubic interpolation between the control points. Since there were 660 different dielectric blocks, controlling the permittivity of each block would require too many variables for global optimization. Thus, the interpolation scheme was necessary to reduce the number of variables down to 18. The angular tilt of the feed horn was also optimized. With this setup, we were able to create the design in CADFEKO using macros to build each of the dielectric blocks. The model was solved using the Multi-Level Fast Multipole Method (MLFMM) employing the surface equivalence principle setting in FEKO. The full-wave simulation of a GRIN lens, horn, and ground plane can be quite computationally expensive and would take 3-5 minutes per simulation on an Intel Xeon Gold 6258R with 56 cores and 1.5 TB of RAM. The optimizer ran 4765 different designs over the course of two weeks until the relative improvement from each iteration was negligible. Using this procedure, the optimizer found exemplary GRIN lens designs for this computationally intensive problem.

D. OPTIMIZATION RESULTS
Using the optimization procedure described, we were able to successfully find GRIN lenses that improved the aperture efficiency and power handling capability of the RA. Figures 8(a) and (b) show the set of best solutions that have varying amounts of illumination and spillover with the color bar indicating the aperture efficiency and RERP, respectively. Design 1, designated by the 5-pointed star, is the no-lens case, while all circles represent GRIN lens designs that have different amounts of improvement in illumination, spillover, or aperture efficiency. Design 2 corresponds to the design that was selected to be fabricated. It has the same aperture efficiency as the no-lens case but an improved power handling capability. This is a result of the more uniform distribution of the electric field across the surface of the RA, which can be seen by comparing the fields in subplots (1) and (2). We note that Design 2 has the same aperture efficiency but higher illumination and lower spillover efficiency than the no-lens case. Design 3, on the other hand, improves the aperture efficiency by maintaining both a better spillover and illumination efficiency than the no-lens case. However, this design does not distribute the electric field more uniformly across the RA and does not have better power handling capability than the no-lens design. Nevertheless, the RERP is slightly improved because of the increase in aperture efficiency. Lastly, Design 4 shows the design with maximum RERP. Here the aperture efficiency is very similar to the no-lens case, but like Design 2, the illumination efficiency is better while the spillover efficiency is lower. Even so, because of the more uniform distribution of the electric field across the surface, the design can handle higher power levels, which results in greater power delivered to the target direction. The computed efficiencies and metrics for the four different cases are provided in Table 1.
It seems logical to think that the illumination efficiency would provide a good objective for making the electric field more uniform. However, because this metric averages the field across the entire surface, the reported illumination efficiency can be high while the fields over the aperture  still possess hotspots, thereby degrading the overall system performance. Instead, an alternative objective measuring field uniformity was devised and included in the optimization objectives. Figure 9(a) shows the illumination and uniformity efficiencies for designs simulated during the optimization. Clearly, there is a positive correlation between these two metrics. However, to demonstrate the differences between the two efficiencies, we picked two designs, Designs 5 and 6, with similar illumination and aperture efficiency, but differing uniformity efficiency. The field plots in Figure 9 show that Design 6 exhibits a lower peak electric field than that of Design 5. This peak field reduction makes Design 6 more suitable for handling high power and for corresponding applications.

E. FABRICATION AND CHARACTERIZATION
To demonstrate that the GRIN lens works according to the simulations, we fabricated the selected design and measured the corresponding radiation pattern of the synthesized feed in an anechoic chamber. A design was selected based on similar aperture efficiency to that of the no-lens case and maximum field handling capability as well as ease of fabrication. The permittivity of each block was used to determine (based on Figure 10) the ratio of 3D printer material to air needed. Using an automated script, the structure was built in Ansys High Frequency Simulation Software (HFSS) and the result was exported as an STL file. We then used a Stratasys Object260 Connex3 printer which uses a VeroWhitePlus material with a permittivity of approximately 3.15 at 3 GHz and a loss tangent of 0.02. We performed a simulation where the permittivity was changed to 3 and found only minor deviations in the radiation pattern. The GRIN lens was printed with support material that then had to be cleaned away. To maintain structural integrity, the minimum feature size for the 3D printer or holes was restricted to be greater than 4 mm. The unit cell was designed to be thicker on the bottom to give better support for attaching the lens to the horn. However, the lens' weight and size slightly warped it as it was printed, which resulted in a slight curvature of the lens. Nevertheless, this did not change the performance significantly.
The fabricated lens was measured in an anechoic chamber. Figures 11(a) and (b) show the FEKO and HFSS models, with the HFSS model using the 3D printed lattice. Figure 11(c) shows photographs of the fabricated lens with the horn antenna attached. The simulation as well as the measured results of the two models are provided in Figures 11(d)-(g). Figure 11(d) shows that the S 11 is below −10 dB from 2.5 to 3.5 GHz for both the two simulation models and the fabricated design. Figure 11(e) shows that the broadside realized gain agrees well between the simulations and the measured data. Figures 11(f) and (g) show the radiation pattern of the lens and horn feed system at 3 GHz in the E-and H-plane, respectively. We see that there is very good agreement between the simulations and measurement indicating that the lens will perform as expected.

IV. CONCLUSION
There is a current need for RA antenna systems capable of handling high-power radiation in a condensed size. This requires feeds that have high gain but are more compact, reducing the overall size of the system and minimizing blockage caused by the feed. Applications involving the realworld deployment of high-power systems may have space constraints, which means the ERP cannot be improved simply by increasing the aperture size and thereby the system gain. Alternatively, this necessitates that the feed more evenly spread out the energy to achieve a higher aperture efficiency and to prevent "hotspots" from being formed that could damage the RA unit cells. While previous approaches for designing feeds have focused on different kinds of metallic radiators, this work demonstrated methods for optimizing GRIN lenses that can better tailor the fields across the reflector. We designed and fabricated GRIN lenses with horn antennas that have high gain, but in a much more compact size and smaller aperture than a conventional standard gain horn. In addition, this work showed that GRIN lenses can be optimized to achieve higher aperture efficiency while more uniformly distributing the energy across the reflector surface to minimize electric field hotspots. GRIN lenses offer greater control over the radiation from a feed and can thereby significantly improve the performance of reflectors resulting in smaller and more powerful RA and TA systems. sonal communication systems (including on-body networks), wearable and e-textile antennas, RFID tag antennas, conformal antennas, reconfigurable antennas, frequency selective surfaces, electromagnetic wave interactions with complex media, metamaterials, electromagnetic bandgap materials, zero and negative index materials, transformation optics, nanoscale electromagnetics (including nanoantennas), fractal and knot electrodynamics, and nature-inspired optimization techniques (genetic algorithms, clonal selection algorithms, particle swarm, wind driven optimization, and various other evolutionary programming schemes).
Prof. Werner received the Pennsylvania State University Applied Research Laboratory Outstanding Publication Award in 1994. He was presented with the 1993 Applied Computational Electromagnetics Society Best Paper Award and was also the recipient of a 1993 International Union of Radio Science Young Scientist Award. He was a coauthor (with one of his graduate students) of a paper published in the IEEE TRANSACTIONS ON