Two-Dimensional Beam Steering Using a Stacked Modulated Geodesic Luneburg Lens Array Antenna for 5G and Beyond

Antennas for future communication systems are required to be highly directive and steerable to compensate for the high path loss in the millimeter-wave band. In this work, we propose a linear array of modulated geodesic Luneburg lens (the so-called water drop lens) antennas operating at 56–62 GHz. The lens array antenna features 2-D beam scanning with low structural complexity. The lenses are fully metallic and implemented in parallel plate waveguides (PPWs), meaning that they are highly efficient. Each lens is fed with 13 rectangular waveguides surrounded by glide-symmetric holes to suppress leakage. The lenses provide <inline-formula> <tex-math notation="LaTeX">$\mathrm{110}^{\circ} $ </tex-math></inline-formula> beam coverage in the H-plane with scan losses below 1 dB. In order to scan in the E-plane, we use a feeding network based on a 1:4 power divider and three phase shifters. In this configuration, the array can scan <inline-formula> <tex-math notation="LaTeX">$\mathrm{60}^{\circ} $ </tex-math></inline-formula> in the E-plane, albeit with higher scanning losses than in the H-plane. The lens array is manufactured and a good agreement between simulated and experimental results is obtained.


I. INTRODUCTION
N EW generations of communication systems intend to cover the requirements that higher data rates and lower latency than many emerging applications demand. In order to do that, a wider frequency spectrum allocation is needed [1]. Specifically for 5G, wide bands around 30 and 60 GHz were proposed to achieve high data rates [2]. The band at 60 GHz is mainly considered for short-range indoor scenarios due to its intrinsic high path losses. Therefore, antennas used in these applications need to compensate for these losses with high gain. To provide adequate angular coverage, the antennas must also be able to steer their directive beam.
A widely used antenna solution for directive beam scanning applications is planar phased arrays [3]. However, at millimeter-wave (mm-wave) frequencies, their feeding networks become complex and lossy [4]. Several methods have been proposed in the literature to simplify the feeding network of planar arrays. The most common one is to use beamforming networks, such as Butler matrices [5], [6], Blass matrices [7], [8], and Nolen matrices [9], [10], among others. However, for 2-D beam steering, these devices are often narrowband and complex. Alternatively, lenses can be added to a sparse array to simplify the feeding network while suppressing the grating lobes [11]. However, the scanning range is limited due to large scan losses. Another alternative is to use radomes that extend the scan angle [12], [13], which compromises the directivity in the broadside direction.
Fully metallic planar lens antennas have also been proposed to provide high-gain and wide scanning at mm-wave frequencies [14]. In [15], a bifocal lens antenna implemented in a parallel plate waveguide (PPW) is proposed. The presented antenna can produce 11 fixed beams in a range of ±30 • . In [16], a double bifocal lens design is proposed, which increases the scanning range of this type of antenna (±50 • ). An even wider scanning range can be reached with a Luneburg lens antenna [17], [18]. This is due to the fact that the Luneburg lens is rotationally symmetric, which implies that the scan losses can be small. This lens antenna can be implemented with discretized layers of dielectrics [19] or through use of metasurfaces [20], [21]. The metasurfaces can be fully metallic to avoid the high material losses from dielectrics at high frequencies [22], [23]. However, due to the subwavelength size required for the unit cells, metasurface Luneburg lenses can be difficult to manufacture at higher frequencies.
Fully metallic Luneburg lenses can also be implemented using geodesic profiles. These profiles are smooth, which alleviates the manufacturing constraints. Geodesic lenses are typically implemented using curved PPWs filled with a homogeneous material (such as air) that mimics the ray path of its This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ refractive index counterpart [24]. This concept was introduced by Rinehart [25] for the specific case of the Luneburg lens. Moreover, in [26], the analytical solution of the generalized geodesic Luneburg lens profile was presented. One drawback of this solution however is that the height of the lens is about one third of the lens diameter. As a result, conventional geodesic Luneburg lenses are bulky. In [27], it was proposed that this height can be reduced by folding the lens profile and in [28], an implementation of a folded geodesic lens antenna was presented in which the sharp folding points in the lens profile are approximated with arcs to reduce reflections at the cost of moderate phase aberrations. This implementation reduced the height by 40% when compared to the conventional geodesic Luneburg lens. Recently, an efficient method to design modulated geodesic lens antennas has been presented in [18]. This modulated lens is referred to as the water drop lens and has contributed to a regain of interest in geodesic lenses for a number of applications at mm-wave frequencies [29]. The profile is defined by continuously differentiable functions, meaning that the folding is accounted for in the design, thus reducing the phase aberrations compared to [28]. Moreover, in [30], the integration of a polarizer at the flare of a geodesic lens antenna was proposed for dual-polarization applications. While geodesic lens antennas provide relatively simple manufacturing and high efficiency, these PPW lens antennas can only scan in one plane, which is not sufficient in some applications.
Here, we present a vertically stacked array of four water drop lens antennas. Stacking geodesic lenses was previously proposed in [30], but no demonstration of the scanning capabilities has been reported until now. The lens element is a scaled version of the lens in [18]. The lens antenna array is able to scan in two planes through beam switching in the horizontal plane and beam scanning in the vertical plane. Each lens antenna can produce 13 independent beams in an angular range of ±55 • in the horizontal plane with negligible scan losses thanks to the rotational symmetry of the lenses. Moreover, the linear array of four lenses allows to scan ±30 • in the vertical plane. This antenna benefits of the high efficiency and simplicity in the feeding network associated with geodesic lens antennas while enabling 2-D beam steering.
This article is organized as follows. Section II explains the design process of a single water drop lens antenna. In Section III, the lens array antenna is presented, including the design of a feeding network that includes a power divider and phase shifters. Section IV describes the experimental results of the lens array antenna, and Section V summarizes the main conclusions.

II. WATER DROP LENS ANTENNA DESIGN
The designed array element is composed of a water drop lens integrated with 13 waveguide feeds and a flare. The waveguide feeds match the impedance of a standard WR19 waveguide to the PPW that forms the lens. On the other side, the flare impedance matches the lens to free space and its height is constrained by the separation between array elements.

A. Lens Design
Luneburg proposed a rotationally-symmetric graded-index lens that has one focal point at the contour of the lens and one at infinity [17]. This lens transforms a cylindrical wave at any point on its surface into a plane wave at the opposite side. The refractive index of this lens is n(ρ) = 2 − ρ 2 , where ρ is the normalized radial position. The lens has a refractive index of √ 2 at the center that gradually changes to 1 at the contour, meaning that it is matched to the free space impedance.
This graded refractive index can be mimicked by a geodesic profile with constant refractive index. The geodesic profile for a Luneburg lens is [25], [26] where z the normalized height. This profile is plotted in Fig. 1 in blue together with its 3-D surface. The height of this geodesic curve is around 0.63 times the radius of the lens, which can be too bulky for some applications. The lens response can be mimicked more compactly with a modulated surface profile (water drop) as the one illustrated in pink in Fig. 1 [18]. This modulated profile has a height ∼0.24 times more compact than the original geodesic profile.
In this work, we use the modulated profile from [18], with a radius of 4λ 0 at 57 GHz (21 mm). A PPW is formed with the pink profile in Fig. 1 corresponding to the profile of the mean surface between the two metallic surfaces, which are separated here by a distance of 0.8 mm. This separation is less than 0.2λ 0 at the highest frequency (62 GHz), ensuring that only the fundamental PPW mode can propagate over the frequency band of operation.

B. Ports and Flare Design
Rectangular waveguides are used to feed the lens. Each rectangular waveguide is stepped, as illustrated in Fig. 2, in order to match the impedance of the waveguide to the PPW. The stepped transition is designed in two sections to match the dimensions of a standard WR19 waveguide for test purposes. In the section closest to the PPW (see Section I), the height is modified to match the impedances of the waveguide and the PPW, while in the section close to the port (see Section II), the width of the waveguide that is modified. The width of the waveguide is not modified close to the lens due to the limited space. The two matching sections are connected by a uniform waveguide whose length does not have a relevant impact on its operation. The steps that increase the width are rounded in the corners with a radius of 0.3 mm to account for manufacturing using milling. The optimized dimensions of the transition are reported in Table I and the reflection coefficient is presented with the purple dashed line in Fig. 3. The stepped waveguide has a reflection coefficient below −15 dB in the band of operation (56-62 GHz).
To mitigate the reflections at the output of the lens, a flare is used to match the impedance of the PPW to the free space impedance. The lens is intended to be used in a vertical array, and therefore the height of the flare must be sufficiently small to avoid grating lobes when scanning. In our case, the targeted vertical scanning range is ±30 • and so the height at the output of the flare is chosen to be 3.5 mm (0.7λ at 60 GHz). The flare has a spline curved shape and a length of 5.24 mm. This flare is illustrated in the inset of Fig. 3. The reflection coefficient of the flare is presented with blue solid line in Fig. 3 and is below −20 dB in the band of interest. The z component of the electric field, E z , is illustrated in Fig. 5 for port 1 and 7 (numbering indicated in Fig. 4) at 56 and 62 GHz. A plane wave is produced at 55 • from port 1 and 0 • from port 7 in the horizontal plane. Fig. 6 shows the simulated radiation patterns at 60 GHz. The maximum directivity and sidelobe level (SLL) are 19.04 and −12 dB, respectively. A scanning range of ±55 • is achieved with a  Table I.

III. LENS ARRAY ANTENNA
The water drop lens antenna presented in Section II can only scan in one plane. In this section, we propose stacking these single elements in order to produce an array antenna that can scan in both azimuth (thanks to the lens) and elevation (thanks to the array), corresponding to the angles φ and θ , respectively, with the notations in Fig. 4. In this section, we propose an array composed of four lenses. This array is fed with a power divider and phase shifters that control the   pointing angle in elevation. We note that, as a consequence of manufacturing tolerances, air gaps can occur between the stacked layers. In order to make our design robust to manufacturing errors, we introduce an electromagnetic band gap (EBG) structure to the design to mitigate against leakage, as previously implemented in [31].

A. Lens Array Design
The designed lens array antenna with four identical stacked lenses is illustrated in Fig. 7. It is interesting to note that the stack of lenses has a height of 16.7 mm, which is smaller than a single conventional geodesic Luneburg lens with the same diameter. Fig. 8 presents the radiation patterns in the E-plane at 60 GHz for a single lens element and the stacked array. The amplitudes and phases of the array are uniformly distributed and the patterns correspond to the ports that produce a beam at 0 • in azimuth. The array has 6 dB higher directivity and the half-power beamwidth decreases from 53.1 • to 18.3 • , compared to the single lens element. The SLL in the E-plane of the array is −14 dB, as expected from the uniform amplitude distribution.

B. Array Feeding Network
A feeding network, composed by a 1:4 power divider and phase shifters, is designed to demonstrate the scanning capabilities of the lens array antenna. The main design parameters of the power divider and phase shifters are illustrated in Fig. 9.   The simulated S-parameters of the power divider are shown in Fig. 10. The reflection coefficient at the input port [port 1 in Fig. 9(a)] is below −20 dB in the band of interest, under the condition that ports 2 to 5 are matched. To reduce SLL in the E-plane of the lens array antenna, we design the power divider to produce a tapered amplitude distribution at its output ports. In our design, the transmission coefficients for ports 3 and 4 are equal to −5 dB at the design frequency whereas for ports 2 and 5, their value is set to −7 dB. The path length between the input port and all output ports is equalized resulting in a phase difference between output ports  lower than 0.15 • , which is in the order of the numerical accuracy of the model. This means that the power divider can be used to produce beams at 0 • in elevation, without the external phase shifters.
To scan the beam in elevation, phase shifting waveguides are connected to each of the output ports of the power divider. For demonstration purposes, fixed phase shifters are designed. In Fig. 10, the top waveguide is invariant in the x -direction and is used as a reference. In the remaining waveguides, different phase shifts are achieved by changing their width and the length of thin dielectric slabs placed at y = 0, as proposed in [32]. The parameters are L P S = 40 mm, d = 35 mm, w 1 = 4.8 mm, w 2 = 5.6 mm and w 3 = 6 mm. The dielectric slabs are made of Rogers RT5880 with thickness 0.508 mm. To scan 15 • , slabs of length L 1 = 3.5 mm, L 2 = 2.5 mm and L 3 = 5.6 mm are used. To scan 30 • , slabs of length L 1 = 12 mm, L 2 = 16 mm and L 3 = 26.5 mm are used. Fig. 11 shows the phase difference between each phase shifting waveguide and the reference waveguide. A progressive phase  of 65 • and 126 • is required for scanning to 15 • and 30 • , respectively. Fig. 12 shows the simulated E-plane radiation patterns for the lens array when connecting the feeding network to the waveguide feeds that produce a beam at 0 • in azimuth. The beam at 0 • in elevation is obtained without the phase shifters. The other directions are obtained connecting the lens array antenna with the power divider and its corresponding set of phase shifters. The maximum directivity is 23.9 dB and the scan losses at 30 • are 2.2 dB, following the cosine-like scan losses in linear arrays.

C. EBG Implementation
An EBG structure can be placed surrounding each waveguide feed to avoid undesired leakage due to manufacturing tolerances. For example, metallic pins have already been proposed in a technology known as gap waveguides [33], [34]. Avoiding this leakage is especially critical in mm-wave bands [35]. Here, instead of metallic pins we have chosen to use glide-symmetric holes as the EBG structure [36], [37], [38]. These holes have demonstrated to be a robust solution for waveguides at high frequencies [39]. The proposed unit cell is shown in Fig. 13(a). The holey structure used in this design has a period p = 2.5 mm, a radius r = 1 mm and a height h = 1.5 mm.
To illustrate the effect on the antenna performance due to undesired leakage, we have simulated a single water drop lens with an intentional uniform gap of 50 μm between the two plates. This uniform gap is obviously a worst case from a mechanical point of view as gaps are expected to be smaller and more localized; however the sensitivity of the structure to very small gaps shows the importance of the EBG structure. Fig. 14 reports the simulated radiation patterns of port 1 and 7 at 60 GHz for three different cases: 1) ideal case (no gap); 2) uniform gap of 50 μm between plates; and 3) intentional gap of 50 μm including a glide-symmetric holey EBG structure. Fig. 13(b) shows the placement of the holes on the top and bottom plates that form a single lens antenna. To minimize the crossover between beams in azimuth, the waveguides must be placed as close as possible to each other on the contour of the lens. Therefore, at the vicinity of the lens, there is not enough space for the EBG structure. For these simulations, we assume that there is no gap in this region. This is a good assumption in a real implementation, since good contact between the two plates near the lens can be ensured if an intentional gap in the area protected by the glide-symmetric holes is imposed [31].
The results in Fig. 14 show that the uniform gap of 50 μm has a clear impact on the radiation patterns. For example, in port 1, this gap produces an increase of the SLL to −5.3 dB, and for port 7 to −3.8 dB, which is unacceptable for the targeted applications. Instead, the radiation patterns with the EBG structure have a similar performance than the no-gap case, validating the proposed mechanical design.

IV. EXPERIMENTAL RESULTS
A prototype is manufactured using high-precision milling to verify the concept of the water drop lens array antenna. The prototype is shown in Fig. 15 with four array elements. More elements could be added at the top or bottom of the stack to increase the aperture size in elevation if required thanks to the generic layer design. The lens array antenna is composed of five identical plates milled on both top and bottom faces with an intentional gap of 20 μm in the areas protected by the glide-symmetric holes [see Fig. 13(b)]. The manufactured feeding network is also shown in Fig. 15. Both power divider and phase shifters are produced in two parts, and the parts are joined in the E-plane. Therefore, the leakage resulting from manufacturing tolerances will be minimized thanks to symmetries.
The simulated and measured reflection coefficient of seven ports of a single lens array element are presented in Fig. 16. The results of the remaining six ports are omitted due to symmetry. Fig. 16(a) shows that the simulated reflection coefficients are below −17 dB in the band 56-62 GHz. Fig. 16(b) shows that the measured reflection coefficients for one lens antenna are below −10 dB in the band of interest. These discrepancies are attributed to the complexity of the antenna and manufacturing tolerances at these high frequencies. Similar results were achieved for the other array elements. The reflection coefficients of the lens antenna are measured connecting each port to a commercial coax-towaveguide transition. The results are time-gated to remove the effect of the transition. Fig. 17 shows the simulated and measured phase shifts required for scanning to 15 • and 30 • in elevation. The phase shifts are relative to the reference waveguide. The simulations and measurements agree, apart from small peaks in the measurements. These peaks are due to the difficulty in placing the dielectric slabs straight.
The lens array antenna prototype is tested in the anechoic chamber of KTH. The measurement setup is depicted in Fig. 18. The radiation pattern is measured in a half sphere. Fig. 19 shows the normalized H-plane radiation patterns when the array is fed with uniform phase distribution at 56 and 62 GHz. This means that no phase shifter is used. The SLL is below −10 dB in all ports at both frequencies. Fig. 20 shows the simulated (solid lines) and measured (dashed lines) 3 dB contour plots. The measurements are done for seven ports and the phases of the lens elements are configured to point to 0 • , 15 • and 30 • in the E-plane. The remaining contour plots are symmetric and were not acquired as a complete characterization is time consuming. The beams have an average beamwidth of 6 • in the H-plane and 20 • in the E-plane. The simulations and measurements agree at both 56 and 62 GHz, apart from some discrepancies in the pointing angle. The measured beams intended to be pointing to 30 • in elevation instead point at 25 • . The discrepancies are attributed to manufacturing tolerances in the prototype that can cause undesired phase shifts between the elements. The simulated (solid lines) and measured (dashed lines) realized gains are shown in Fig. 21 for the considered beam pointing directions in the band of 56-62 GHz. When the array points at 0 • in elevation [see Fig. 21(a)], the simulated peak gain varies from 22.7 to 24.1 dBi. When the array points at 15 • [see Fig. 21(b)], the simulated peak gain varies from 22.2 to 23.7 dBi. When the array points at 30 • [see Fig. 21(c)], the simulated peak gain varies from 20.2 to 21.3 dBi. This means that the maximum scan losses are within 2.5-2.8 dB. The measured gain is 1-2.5 dB lower than in the corresponding simulation for all beam pointing directions. These discrepancies in gain are attributed to the surface roughness in the prototype, which was not considered in the initial simulations and proved to be higher than anticipated. A simulation of port 1 with 3 μm of roughness provides a better agreement with the measurements. These results demonstrate the ability of the lens array to produce a high gain beam which can be steered in two planes.

V. CONCLUSION
In this work, we demonstrated the possibility of steering a beam both in azimuth and elevation using an array of stacked water drop lens antennas. The array consists of four lens antennas with an interelement separation of 0.7λ at 60 GHz. Each lens has 13 waveguide feeds and a flare is used to provide impedance matching to free space. Glide-symmetric holes have been added around the waveguide feeds to avoid leakage due to manufacturing tolerances. The lens antennas produce beams covering a ±55 • range with 9.2 • separation in the H-plane and have negligible scan losses. A feeding network is used to steer the beam ±30 • in the E-plane. The feeding network is composed of a 1:4 power divider and phase shifters. To prove the concept, the phase shifters are designed to steer 15 • and 30 • in elevation.
The proposed array antenna is manufactured using milling and tested. The antenna operates in the band 56-62 GHz with reflection coefficients lower than −10 dB and measured peak gain of 22.5 dB. The lens antennas can steer ±55 • in the H-plane, with 13 discrete beams. The array antenna can scan ±30 • in elevation. The maximum scan losses are 2.5-2.8 dB across the band. The simulations and measurements agree well.
The proposed solution features low complexity since the beamforming in one plane is done by the lens. A switching matrix may be added for applications requiring single beam. Moreover, the scan losses in one plane are also reduced thanks to the rotational symmetry of the lens. He has authored or coauthored over 60 peerreviewed journal articles and conference papers. His research interests include transformation optics, lens antennas, metamaterials possessing higher symmetries, and leaky-wave antennas.
Mr Since November 2020, he also holds an Honorary Appointment as a Professional Fellow with the University of Technology Sydney (UTS), Ultimo, NSW, Australia. He is currently an Antenna Engineer at the Antenna and Sub-Millimetre Waves Section, European Space Agency (ESA), Noordwijk, The Netherlands. He has authored or coauthored more than 270 papers in peer-reviewed journals and conferences and holds over 50 patents issued or pending. His research interests include multiple beam antennas for space missions, beam-former theory and design, ground terminal antennas and novel manufacturing techniques.
Dr. Fonseca has been a Board Member of the European School of Antennas and Propagation (ESoA) since January 2019 and is also serving as a Coordinator of the ESA/ESoA course on Antennas for Space Applications. He is the elected EurAAP Regional Delegate representing Benelux for the He has coauthored more than 110 articles in international journals and 180 papers at international conferences. He has made scientific contributions to higher symmetries, transformation optics, lens antennas, metasurfaces, leaky wave antennas, and high-impedance surfaces.
Dr. Quevedo-Teruel has been a member of the EurAAP Board of Directors since January 2021. He was the EurAAP Delegate for Sweden, Norway, and Iceland, from 2018 to 2020. Since January 2022, he has been the Vice-Chair for EurAAP. He has also been the Chair for the IEEE APS Educational Initiatives Programme since 2020. He was a Distinguished Lecturer of the IEEE Antennas and Propagation Society for the period of 2019-2021. He has been an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 2018 to 2022 and a Track Editor since 2022. He has also been the Founder and the Editor-in-Chief of the EurAAP journal Reviews of Electromagnetics since 2020.