Conformal Parallel Plate Waveguide Polarizer Integrated in a Geodesic Lens Antenna

Here, we propose a low-profile polarizing technique integrated in a parallel plate waveguide (PPW) configuration, compatible with fully metallic geodesic lens antennas. The geodesic shape of the antenna is chosen to resemble the operation of a Luneburg lens. The lens is fed with 11 waveguide ports with 10° separation producing 11 switchable beams in an angular range of <inline-formula> <tex-math notation="LaTeX">$\pm \mathrm{50}^{\circ }$ </tex-math></inline-formula>. Two metallic polarizing screens are loaded into the aperture of the antenna to rotate the electric field from a vertical linear polarization, which is the polarization of the transverse electromagnetic (TEM) mode supported in the lens, to a <inline-formula> <tex-math notation="LaTeX">$+ \mathrm{45}^{\circ }$ </tex-math></inline-formula> linear polarization. Since the polarizing unit cells are integrated into the aperture of the antenna, the final design is compact. In addition, the size of the polarizing unit cells is about <inline-formula> <tex-math notation="LaTeX">$0.55\lambda $ </tex-math></inline-formula> at the central frequency of operation making the antenna suitable to produce an array formed of stacked lenses. A prototype of the antenna in the <inline-formula> <tex-math notation="LaTeX">$\text{K}_{\text {a}}$ </tex-math></inline-formula>-band was manufactured and tested, verifying the performance obtained in simulations.

electronically scanned arrays becomes complex and lossy [10]. Losses in the feeding network can be avoided by having a fully mechanically scanned array; this, however, results in a slower scanning speed and a bulky design [8].
Lens antennas have received attention due to their wideband behavior and their ability to produce high-gain switchable beams over a wide angular range without requiring a complex feeding network [11]. Lens antennas were broadly studied in the 1940s-1960s [12], [13] but deemed unpractical at low frequencies due to their large size and high manufacturing cost. Nevertheless, when the frequency increases, the physical size of the lenses is reasonably small while being large in terms of wavelength meaning they can provide high-gain beams [14]. The Luneburg lens [15] is an interesting case for antenna designs since it can be used to produce multiple beams in a large angular range and a wide frequency bandwidth. The planar version of the Luneburg lens is a compact device with a wide scanning range in one plane [16]. The required graded refractive index of the Luneburg lens can be achieved using dielectrics [17], metasurfaces [18], [19], [20], [21] or geodesic surfaces [22], [23], [24], [25], [26].
Geodesic surfaces can mimic the refractive index of rotational symmetric lenses using a profiled surface [22], [27], [28]. By making use of the direction orthogonal to the beamforming plane, a parallel plate region can be deformed into a geodesic surface [29]. The resulting surface is a device that can be easily manufactured and eliminates the need for inhomogeneous materials to achieve the required refractive index of the lens. However, this is done at a cost of an increased height of the lens, hence sacrificing compactness. Solutions to this problem have been addressed in [23], [24], and [30].
Investigating ways to implement dual-polarized lens antennas is of special interest for both the satellite and terrestrial communication systems, where it is needed to transmit or receive dual-polarized signals to or from the antennas [31], [32]. Polarization manipulation is indeed a challenge in fully metallic parallel plate waveguide (PPW) beamformers due to the supported fundamental modes. This is less challenging in 3-D lenses since the polarization can be readily implemented in the feed or before the lens [33], [34]. The typical solution in PPW beamformers is to have an array of polarizing unit cells at some distance away from the beamformer. One such designs is proposed in [35] where the polarizer array transformed the linear polarization of the PPW beamformer to circular This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ polarization. In addition, the polarizer array acted as a reflector which could be mechanically steered. However, for some antenna applications it is not suitable to use big reflectors for polarization transformation, and hence compact solutions are needed [36].
Using septum polarizers integrated within a square waveguide is one method to achieve circular polarization for a PPW beamformer in a compact way [37]. In septum polarizers, a square waveguide is fed by two rectangular waveguides, each of which supports one propagating mode. The square waveguide, on the other hand, supports two orthogonal propagating modes. The polarization of the incident signal is converted by adjusting the phase between these two orthogonal propagating modes. In [38], septum polarizers integrated within a square waveguide were used to generate circular polarization in a PPW multiple beam quasi-optical beamformer operating in K a -band. This was done by discretizing the aperture of the beamformer which results in a reduced scanning range. In [39], a modification of the standard septum is used where the vertical walls added to create square waveguides are removed, and the stepped septum is replaced with a saw-like design based on periodic teeth. This results in a continuous aperture avoiding the problem of the limited scanning range in [38]. However, this concept comes at a cost of limited bandwidth because the TEM mode is maintained in the teeth area, while the orthogonal mode is a TE mode. The two modes have very different propagation characteristics, meaning the required phase shift between both the modes will only be obtained at the design central frequency.
In this article, we propose a low-profile solution to obtain a linear-to-linear polarization transformation in a fully metallic geodesic Luneburg lens operating at the K a -band. The concept is based on a PPW polarizer with wideband matching to free space and polarization conversion. The implementation of the presented solution relies on thin polarizing sheets that may be bent to adapt to the curvature of the aperture (e.g., combination with a Luneburg or geodesic lens), hence supporting wide scanning. The polarizing screen provides linear polarization rotation (±45 • ) to enable polarization diversity from a stack of PPW beamformers. Even though this concept is being demonstrated using a geodesic Luneburg lens, this method can be applied to any kind of PPW beamformer such as a shaped parallel plate delay lens [40] or a pillbox antenna [11], [41].
This article is organized as follows. In Section II, the geodesic surface of the lens antenna is introduced, and a flare and feeding design are presented. This section also reports the simulation results of the whole lens antenna. Section III is dedicated to the polarizing unit cell design and a discussion on how to integrate the polarizer together with the antenna. Section IV includes the experimental results of the integrated antenna, and finally, the main conclusions are drawn in Section V.

A. Geodesic Lens Design
The Luneburg lens is a rotationally symmetric graded index lens [15]. The refractive index of the lens varies from (2) 1/2 in the center to 1 at the border of the lens. When the Luneburg lens is excited with a cylindrical source on its boundary, it produces a wave with a planar phase front in the diametrically opposite direction of the lens. Due to the rotational symmetry, when this lens is integrated in an antenna, the scan losses are low and scanning is readily achieved by feeding the lens at different locations along its boundary, eliminating the need for a complex feeding network.
In [29], a surface equivalence of the Luneburg lens is proposed and implemented with a PPW which is deformed to mimic the gradient refractive index of the Luneburg lens. The resulting PPW lens is referred to as a geodesic Luneburg lens. The main advantage of the geodesic lens is that it can be realized using a homogeneous material, such as vacuum or air.
One disadvantage of the geodesic lens is the required height of the equivalent surface which is a third of its diameter. This issue is addressed in [30] where it was proposed to fold the geodesic surface to achieve a more compact device while preserving the optical characteristics of the lens. Such structure was not studied again in the literature until the recent implementation presented in [23], where a height reduction of 2.5 from the conventional geodesic Luneburg lens is reported.
In [24], a rigorous design procedure for implementing a compact geodesic lens is described. The resulting lens is referred to as a water drop lens. We have followed such a design procedure in the current investigation. The shape of the designed water drop lens used in this work is shown in Fig. 1. The dashed red lines show the surface profile of the water drop lens. The distance between parallel plates is 2 mm, which is about 0.2λ at the central frequency. This distance was chosen so only the TEM mode can propagate within the frequency range of interest. In Fig. 1, the solid blue lines show modifications applied to the initial profile, in terms of cuts, to improve the matching at the folding points. These cuts are C 1 = 0.5 mm and C 2 = 0.35 mm. Since they are small, the beamforming capability of the lens is not affected. This lens is designed to operate from 25 to 31 GHz. The diameter of the lens is chosen as 107 mm, which is approximately 10λ at the center frequency. The inset of Fig. 1 shows how the lens  transforms a cylindrical wave excited at its border to a planar wavefront in the opposite direction.

B. Flare and Feeding Design
To feed the water drop lens, we use a standard WR28 waveguide. Since the distance between parallel plates in the water drop lens does not match the height of a WR28, a stepped waveguide transition is needed, as illustrated in Fig. 2. To reduce the computational time of the simulations, only a portion of the lens is included in the optimization of the stepped waveguide, as shown in Fig. 3(a).
On the other side of the lens, a flare design is required to allow efficient radiation to free space. The flare has an exponential tapering from the height of the parallel plate of the lens, 2 mm, to a final aperture height of 10 mm which is approximately one wavelength at the central frequency. The reflection coefficients for the designed feeding and flare are depicted in Fig. 3(c). The flare has a reflection coefficient below −20 dB in the frequency band of interest, and the feeding reflection coefficient is below −18 dB.   Fig. 5(a), demonstrating that all the reflection coefficients are below −16 dB and portto-port coupling is below −19 dB in the frequency band of interest. The simulated 2-D radiation patterns for ports 1, 6, and 11 at 25, 28, and 31 GHz, respectively, are shown in Fig. 5(b). The antenna produces directive beams, and the performance is stable over the scanning range and the frequency band. The sidelobe levels are as low as −20 dB at the highest frequency, 31 GHz, and never higher than −14 dB at the lowest frequency, 25 GHz. Fig. 5(c) shows the maximum directivity of all 11 feeding ports at 25, 28, and 31 GHz. A low directivity variation is observed between pointing directions with a maximum of 0.2 dB.

A. Polarization Diversity in PPW Beamformers
Polarization transformation in fully metallic PPW beamformers introduces challenges, mainly due to the fundamental modes supported in such devices. The height of the PPW used to form the required geodesic surface must be chosen so that only the TEM mode is allowed. Introducing the second mode, the TE 01 mode, would not result in the desired performance of the lens due to the dispersive relationship between the two modes. This means that any polarization transformation must be done after the beamforming stage of the antenna. Techniques to achieve dual-polarized PPW antennas are of special interest for both the satellite and terrestrial communication systems as polarization diversity doubles the link capacity for a given frequency bandwidth allocation. Here, we want to investigate a way to transform the nominal vertical polarization of the TEM mode supported in the geodesic Luneburg lens to a +45 • linear polarization, so polarization diversity may be achieved stacking two lenses with polarization transformation of +45 • and −45 • . In this article, we propose a compact polarizer that does not limit the bandwidth or the beam scanning capabilities of the antenna.

B. Complementary Split-Ring Resonators
In [42], it is shown that twist-symmetric complementary split-ring resonators (CSRRs), perforated from a sheet of aluminum, can be used to construct a flat fully metallic lens. This type of perforated sheets with CSRRs can also be used to generate polarization transformation with a wideband performance as demonstrated in [43]. In the aforementioned studies, the CSRRs were used in a 2-D array configuration. In this work, we load the CSRRs in a PPW with a 1-D design for polarization transformation in a geodesic Luneburg lens.
The CSRR unit cell is shown in Fig. 6(a). The unit cell is characterized with a rectangular section with dimensions P z and P x , a center bar of width g, and a slot of width w made by two concentric circles with radii S and S-w. The center bar is rotated by an angle δ n , so the electric field traveling through the CSRR rotates as well. This rotation of the electric field is smoothly achieved in steps as illustrated in Fig. 6(b), where the center bar of every consequent unit cell is rotated by an incremental value. This polarization conversion in steps improves the matching between screens when rotating the center bar. This means that the fewer the screens, the larger the reflections. A study to determine the minimum required number of polarizing screens to achieve a good matching to free space must be carried out.

C. Design Procedure for the Polarizer
In [42] and [43], the CSRRs were studied at a unit cell level, as commonly done when designing a 2-D transmit array. That design environment does not fit the purpose of this work since the polarizer will not be used in a planar configuration but rather in a PPW environment. The CSRRs will be repeated in the x-direction but there is only one CSRR unit cell in the z-direction. To mimic this environment, the polarizer is analyzed directly in its PPW environment as shown in Fig. 7, where a PPW is loaded with n = 4 polarizer screens. For each screen, the unit cell is repeated 16 times in the x-direction. In this way, when the PPW is excited at the waveguide port with a TE 10 mode, the excited wave will behave like a quasi-TEM wave. Furthermore, with 16 unit cells, the width of this simulation model is roughly the same as the lens diameter. The PPW height is tapered from the height of the PPW of the lens to the height of the CSRR unit cells. As can be seen in the cross-sectional view of the loaded PPW in Fig. 7(b), the polarizer screens are placed in grooves for assembly purposes.
Two, three, and four polarizer screens are considered for the study, as shown in Fig. 8(a). In all the cases, the outer radius and the slot width are kept constant as S = 2.65 mm and w = 1 mm, respectively. The lateral dimensions are P x = P z = 5.7 mm, roughly 0.55λ at the central frequency. These lateral dimensions were chosen to be small enough so the final design is suitable for a vertical stacked antenna array. It must  be noted that with this lateral dimension, the cutoff of the TE 01 PPW mode is around 26 GHz, which means that this mode (horizontal polarization) is operating below cutoff. Therefore, in order not to suppress this mode in the PPW region between the screens, they must be closely spaced. The thickness of the perforated metal sheets, t, is chosen as 0.3 mm since at this thickness aluminum is still quite pliable and the sheets can, thereby, be bent into a circular shape without damage. For n = 2, 3, and 4, the rotation of the last polarizing screen is set to 45 • . The rest of the parameters are then tuned to get the best matching over the frequency range. The reflection coefficients for the different numbers of screens are shown in Fig. 8(b). The best result is achieved with the highest number of screens, n = 4. However, decreasing the number of screens down to two still yields acceptable results, with a reflection coefficient which is below −15 dB over the frequency band of interest. Therefore, we continue with the design process using only two polarizer screens.
To estimate the performance of the polarizer screens, we study the far-field using Ludwig's third definition for co-polarization (co-pol) and cross-polarization (x-pol), which is suitable for a linear directional radiation pattern [44]. The inset of Fig. 9 defines the co-pol and x-pol. The parameters of the unit cell and screens were optimized, and their values are given in Table I. Fig. 9 shows a parametric sweep of the rotation angle of δ 2 at 45 • , 47 • , 49 • , and 50 • while keeping δ 1 = 29 • . The bar needs to be rotated more than 45 • since the metallic plates in the z-direction suppress the TE 01 mode, especially at the lowest frequency where this mode is in cutoff. By rotating δ 2 beyond 45 • , it is possible to get a lower level of x-pol. Therefore, the chosen δ 2 is 50 • .
The numerical results clearly indicate that the performance of the polarizer improves as the frequency increases. This is because the modal response becomes less dispersive when we move further away from the cutoff of the TE 01 PPW mode. The overall x-pol level over the design frequency band could therefore be improved by increasing the unit cell height so that the TE 01 PPW mode operates further away from cutoff. However, this would result in increased spacing between the antenna elements if the design is implemented in a vertically stacked antenna array, limiting beam steering in elevation.

D. Polarizer Under Oblique Incidence
The polarizer is intended to be installed conformally in the aperture of the PPW beamformer. As a result, the performance  of the polarizer is not dependent on the scanning angle and is maintained when scanning to any direction. On the other hand, this implies that the polarizer partially operates under oblique incidence for all the scan angles, which may lead to reduced performance [45], [46]. To evaluate the performance under oblique incidence, we simulate the truncated planar polarizer that is rotated relative to the waveguide feed. The model used for the study is illustrated in Fig. 10(a). Specifically, we evaluate how rotation affects the polarization isolation under incident angles, β, from 0 • to 40 • in 10 • steps. The polarization isolation is defined as the difference between the peak gains of the co-pol and x-pol in the −3 dB beamwidth of the main beam. The results of the study are shown in Fig. 10(b). We observe that the polarizer maintains the performance under incident angles up to β = 20 • . At β = 30 • , the performance at higher frequencies deteriorates as we observe that the polarization isolation is above −15 dB. At β = 40 • , the polarizer is no longer performing as intended, with the polarization isolation above −15 dB over the whole frequency  Fig. 11(a). range. With this study, we conclude that the angular stability of the polarizer is approximately ±30 • . Fig. 11 illustrates the electric field distribution in the lens designed in Section II when fed with port 6 at 28 GHz. We sample the power density on the dashed line as shown in Fig. 11(a). The dashed line corresponds to the position where the polarizer screens will be integrated in the final antenna design. We plot the result in Fig. 11(b). It is interesting to note that approximately 85% of the total power is confined within the ±30 • range, corresponding to the shaded area in Fig. 11(b). This means that the majority of the power is incident at the polarizer with angles lower than 30 • . This analysis provides further confidence in the operation of the conformal polarizer design prior to integration with the lens.

E. Comparison Between Antenna With a Flare and Polarizer Screens
In this section, we compare the simulation results of the antenna with a flare and the antenna with the polarizer in the aperture. As noted above, the vertical aperture size in the lens with the polarizer is smaller than that in the lens without the polarizer. To provide a fair comparison of the two antennas, we therefore compare the normalized radiation patterns. Fig. 12 shows a cross-sectional view of the two lenses, and Fig. 13 shows the simulation model and integration of the polarizer in the lens antenna. As illustrated in Fig. 13, the polarizer screens are bent to the shape of the lens and then placed between the top and bottom plates of the antenna. Fig. 14(a) presents normalized radiation patterns for port 6 in the beamforming plane (H-plane) and the plane orthogonal to the beamforming plane (E-plane) at 28 GHz. As can be observed, the beamwidth in the H-plane is roughly the same for both the antennas. However, since the antenna with the radiating flare has a larger aperture in the E-plane than the antenna with polarizer screens, it has a narrower radiation pattern in this plane. In Fig. 14(b), it can be seen that the radiation efficiency for both the antennas is above 95%. In the same figure, we plot the 3 dB beamwidth in the H-plane of the two antennas when exciting the center port. The beamwidth of the two lenses is similar up to 30 GHz where we observe an increase in beamwidth for the antenna with polarizer screens. This is most likely due to phase aberrations caused by the polarizer screens.

IV. EXPERIMENTAL RESULTS OF THE FULLY METALLIC INTEGRATED ANTENNA AND POLARIZER
In this section, we present the simulations and experimental verification of the integrated lens antenna. The antenna is simulated using the time-domain solver of CST Microwave Studio. The polarizing screens are curved into a circular shape with a radius that matches the placement grooves allocated in the flare of the antenna. The manufactured prototype is shown in Fig. 15. Fig. 15(a) shows the top and bottom plates of the geodesic lens antenna before assembly with the polarizer screens placed in the bottom plate. The two plates that form the geodesic surface were manufactured using CNC milling, and the polarizing screens were manufactured by water-jet cutting aluminum sheets. Fig. 15(b) illustrates the assembled antenna during measurements in the anechoic chamber at KTH Royal Institute of Technology.
The measured and simulated reflection coefficients are represented in Fig. 16(a). The measured reflection coefficients are below −10 dB in the frequency band of interest, but are a bit higher than the simulated results, below −15 dB. The measured and simulated port-to-port coupling between selected ports are depicted in Fig. 16(b). Similar levels of portto-port coupling are obtained from the measurement and simulation with most of the values below −20 dB. The measured and simulated normalized radiation patterns of the co-pol, in the scanning plane at 25, 28, and 31 GHz are depicted  in Fig. 17. There is a good agreement between the simulated and measured results, especially at 25 and 28 GHz. In the measurements, the beamwidth at 31 GHz slightly increases. The sidelobe levels in both the simulations and measurements are below −15 dB for all the ports at 25 and 28 GHz. Slightly higher side lobes are observed at 31 GHz but always below −13 dB.
The measured and simulated peak gains of ports 5, 6, 8, and 11 are shown in Fig. 18(a). Although omitted here, the other ports had a similar performance. The gain is stable between ports although some differences are noticeable between the measured and simulated results. This discrepancy is due to the surface roughness of the materials. A simulation with a root mean square (Rq) surface roughness of 5 μm is illustrated in Fig. 18(a), demonstrating a good agreement with the measurements. Port 6 experienced more losses than port 1, especially at 31 GHz. The main difference between these two ports is in the waveguide length, which for port 6 is substantially longer. The longer the waveguide, the higher the undesired leakage and material losses due to surface roughness. It is important to note that the waveguides are only for testing purposes. In a real implementation, these waveguides are not necessary since the lens will be directly connected to the PCB. Fig. 18(b) presents the polarization isolation of the integrated antenna. The isolation obtained with the truncated planar polarizer (Fig. 9) is included as reference. The polarization isolation of the antenna is defined in the same manner as in Section III. Good agreement is observed between the x-pol of the truncated planar polarizer and the simulated and measured x-pol levels in the integrated lens antenna. The polarization isolation is below −18 dB over the whole frequency range and mostly below −20 dB in both the simulations and measurements.

V. CONCLUSION
A fully metallic and low-profile method to achieve polarization transformation in a geodesic Luneburg lens antenna has been proposed and validated experimentally. The polarization transformation is achieved by placing two metallic screens of CSRRs in the aperture of the antenna. The polarizing screens rotate the electric field from a vertical linear polarization to a linear +45 • polarization while providing a good impedance matching to free space. Since the screens are placed conformally in the aperture of the antenna, the antenna performance is unaffected when the beam is steered in the angular range of ±50 • .
The proposed concept can be extended to design PPW antennas that generate circular polarization by replacing the linear-to-linear polarizer with a linear-to-circular polarizer similar to the designs in [32], [47], and [48]. To the authors' best knowledge, no polarizer design, applicable for a PPW beamformer, has been reported that transforms the polarization while providing impedance matching to free space. In addition, the reported designs on low-profile polarization transformation in PPW beamformer suffer from either limited bandwidth or limited scanning range [38], [39] which is not the case for the proposed design.
The integrated antenna operates at a center frequency of 28 GHz with over 20% bandwidth having measured reflection coefficients below −10 dB, a port-to-port coupling below −18 dB, and a measured peak gain of 17.6 dBi. Good agreement is achieved between the measured and simulated radiation patterns, apart from slightly lower gain observed in measurements. The lower gain is attributed to the surface roughness of the metal, which was not considered in the simulations. The antenna supports a beam scanning of ±50 • , which is demonstrated with 11 discrete beams with 10 • separation.
By vertically stacking the proposed integrated antenna, polarization diversity can be exploited by mirroring the polarizing screens in the antennas. One of the antennas would then radiate a +45 • polarized wave, while the other one a −45 • polarized wave. Furthermore, the presented design has an aperture height of 0.55λ, and as a result, 2-D beam steering (switching in azimuth and scanning in elevation) is enabled by stacking antennas with the same polarization. For the 2-D beam steering antenna stack, the polarization diversity can be exploited using two independent systems, one for each polarization. Although the polarization technique proposed in this article was only applied to a geodesic Luneburg lens, it is general and it can be applied to both flat PPW lenses and other versions of conformal beamformers. He has authored or coauthored over 60 peerreviewed journals and conference papers. His research has been focused on transformation optics, lens antennas, metamaterials possessing higher symmetries, and leaky-wave antennas Mr. Zetterstrom is a member of the newly formed EurAAP Working Group for Early Careers in Antennas and Propagation (ECAP). He has been awarded the first price in the Student Design Competition at APS/URSI in 2016, the Best Student Paper Award at URSI Spain in 2020, and the Best Antenna Technology Paper Award at EuCAP 2022 for his works.