V-Band Fully Metallic Geodesic Luneburg Lens Antenna

Antennas in emerging millimeter-wave (mm-wave) applications are often required to have low losses and produce a steerable directive beam. These properties are achievable with fully metallic geodesic Luneburg lens antennas. In this communication, we report the first experimental verification of a geodesic Luneburg lens antenna in the V-band. The designed lens antenna is fed with 13 waveguides providing beam switching capability in a 110° range. The lens is implemented in the parallel plate waveguide (PPW) technology. The antenna is manufactured in two pieces, and a tolerance analysis indicates that gaps between the pieces can cause a severe performance degradation. Based on this tolerance analysis, two measures are taken to alleviate the manufacturing tolerances for the prototype. First, electromagnetic band gap (EGB) structures are placed around the feeding waveguides. Second, the electrical contact between the two pieces is improved in critical regions. Two prototypes are manufactured, one without and one with the extra measures implemented. The measured radiation patterns of the prototype without these measures have high side lobes and low realized gain compared with the simulation. The measurements of the robust version of the prototype agree well with the simulations and demonstrate the applicability of geodesic Luneburg lens antennas for applications in the V-band.


I. INTRODUCTION
Many emerging applications are intended to operate at millimeterwave (mm-wave) frequencies, including next-generation cellular network communications [1], also integrating non-terrestrial networks [2], and radars [3], to extend the capabilities of the existing systems. Presently, the K-to W-bands are being considered due to the wide available spectrum [4]. To overcome the high propagation losses at these frequencies, the antennas are required to produce directive beams, and wide angle beam scanning is often requested. Luneburg lens antennas have been proposed to meet these requirements [5].
A Luneburg lens is a rotationally symmetric gradient index lens that when fed from the edge of the lens produces a directive beam in the diametrically opposite direction [6]. The directive beam can be steered without scan losses by moving the feeding point on the surface of the lens. Due to the gradient refractive index, the realization of Luneburg lenses can be challenging. 3-D Luneburg lenses have been manufactured in layers [7] or using quasiperiodic structures [8]. However, most reported 3-D Luneburg lenses require dielectric materials that introduce losses often prohibitive at mm-wave frequencies. A fully metallic 3-D Luneburg lens, based on wired media, was proposed in [9]; however, its matching to free-space is challenging, making this configuration difficult to be used as an antenna.
Geodesic Luneburg [18], [21] and half-Luneburg [22] lens antennas have been reported at the K a -band. Beam-steering up to ±60 • in one plane with less than 1 dB scan losses have been demonstrated [18]. To improve the crossover gain between beams when beam-steering, a geodesic near-field focused lens was proposed in [27]. The improvement in the beam crossover gain comes at a cost of reduced peak gain. In [28], a PPW is deformed to design a lens antenna based on nonrotational symmetric lenses. The feeding is mechanically moved to scan the beam continuously ±50 • in one plane; however, since the lens is not rotationally symmetric, scan losses of almost 3 dB are observed.
In the present work, we demonstrate the applicability of geodesic Luneburg lens antennas in the V-band. Specifically, we design a geodesic Luneburg lens antenna operating over the band 56-62 GHz. The presented design is based on the antenna developed in [29]. This is the first time that a geodesic lens antenna is experimentally demonstrated at this high frequency, which imposes challenges in the manufacturing and assembling of the lens. We discuss the importance of using robust and cost-effective methods to improve the isolation between the feeding waveguides. The insights from this study are expected to also benefit future developments at even higher frequencies. The communication is outlined as follows. In Section II, the components of a geodesic Luneburg lens antenna are presented. In Section III, a tolerance analysis is provided demonstrating the need for a design that is robust to manufacturing and assembling tolerances. Corrective measures are implemented resulting in a design more tolerant to assembly gaps. In Section IV, the simulation and measurement results for the lenses without and with corrective measures are presented. A comparison of the measured response of the two antennas further validates the need for a robust implementation. Finally, the conclusions of the work are presented in Section V.

II. GEODESIC LUNEBURG LENS ANTENNAS
The presented geodesic Luneburg lens antenna consists of a lens, feeding, and flare. In the following, we outline the design of these components.

A. Geodesic Luneburg Lens
The rotationally symmetric gradient refractive index of the Luneburg lens is given by n(ρ) = 2 − ρ 2 /R 2 , where ρ is the radial position in the lens, and R is the radius of the lens [6]. Parallel rays impinging on this refractive index distribution are focused in a point at the surface of the lens. Since the optical path is equal for all the rays, they arrive in-phase at the focal point. In other words, the Luneburg lens has no phase abberations.
In the 1940s, Rinehart demonstrated that the optical paths in a 2-D Luneburg lens can be mimicked with a curved surface [10] filled with a homogeneous medium (typically air). The height profile derived by Rinehart (assuming z = 0 at ρ = R) is illustrated with the red dashed line in Fig. 1. A geodesic lens is designed by rotating the Rinehart-Luneburg profile around the z-axis (ρ = 0). The obtained surface is used as a mean surface for the two PPW conductors (illustrated with blue solid lines in Fig. 1). To avoid higher order PPW modes in the geodesic lens, the distance between the two conductors must be smaller than λ/2, where λ is the free-space wavelength at the highest operating frequency. In practice, this value is generally set around λ/4 to provide some margin. In this work, the distance between the two conductors is 0.8 mm. A circular transition is added around ρ = R to mitigate the reflections between the lens and a surrounding planar PPW. Its design is optimized to minimize phase aberrations. The time-domain solver of CST Microwave Studio [30] is used to demonstrate the focusing property of a geodesic lens in the inset of Fig. 1. The profile in Fig. 1 is used to design a geodesic Luneburg lens with a radius of R = 20 mm (i.e., 4λ at 60 GHz).

B. Feeding and Flare Design
Rectangular waveguides are used to feed the proposed lens. The waveguides are stepped to match their impedance to the impedance of the PPW lens. The stepped transitions are the same as in [29]. Due to the small spacing between the conductors of the PPW, a flare is used to mitigate the reflections at the output of the lens, providing a smooth transition to free-space. In this work, we use an exponential flare that is placed around the contour of the lens. The flare design is the same as in [29].

III. MANUFACTURING AND ASSEMBLING CONSIDERATIONS
In this section, the integration of geodesic lens antennas in the V-band is presented. While such lenses have demonstrated excellent performance in the K a -band using milling techniques [18], [21], a tolerance analysis indicates that manufacturing errors can significantly reduce the performance in the V-band. While part of the degradation is due to the higher operating frequency, another important aspect is the use of waveguide flanges here requiring longer feeding waveguides in place of the more compact coaxialto-waveguide transitions used in previous works. The lens antenna design is thus adapted to be more robust to manufacturing errors. A tolerance analysis of the robust lens antenna demonstrates that the performance impairment from manufacturing and assembling errors is notably reduced.

A. Tolerance Analysis in V-Band
The integrated lens antenna is illustrated in Fig. 2. About 13 waveguide feeds are placed along the contour of the lens, of which seven are numbered in Fig. 2, starting with the most scanned one as port 1 and with port 7 as the central one. The remaining ports are symmetric to ports 1-6. Each feed produces a directive beam at the diametrically opposite side of the lens. The feeds are mutually displaced by 9.2 • , and thus, the antenna covers a 110 • range in one plane. Depending on the complete antenna system implementation including electronics, these feeding ports may be used to implement beam-steering through port switching or simultaneous multiple beam operation. The waveguide feeds are extended out to the edges of a rectangular base where WR19 standard waveguides can be connected for test purposes. The feeding waveguides are designed to accommodate the flanges of the coaxial-to-waveguide transitions used in the measurements. These are significantly longer than those necessary at K a -band [18], [21], increasing the susceptibility of the design to manufacturing errors.
The antenna is manufactured in two parts that are joined together with screws. In Fig. 2, the top part is transparent and partly hidden for illustration purposes. Screw holes for joining the two pieces are indicated in Fig. 2. It is worth noting that due to the small distance between the feeding waveguides close to the lens, more screws cannot be added in that region. Perfect contact between the two pieces is difficult to ensure everywhere as a result of manufacturing and assembling tolerances despite the use of high-precision milling. These errors are likely to produce randomly distributed gaps between the two metallic pieces. The mechanical design of the lens in two blocks due to the PPW configuration forces to cut the feeding waveguides through their H-plane, thus cutting through strong currents. This obviously results in localized leakage in the areas of poor contact. The effects of mechanical errors on antenna parameters are difficult to evaluate exhaustively. We provide here an assessment of the sensitivity of the design by comparing the simulated realized gain for port 7 of the lens antenna assuming perfect contact and a constant 30 μm gap between the two pieces. Although not representative of a practical case, this constant gap is easier to implement in the full-wave model. The obtained patterns are presented in Fig. 3. We observe that the presence of the gap significantly reduces the gain and distorts the radiation pattern of the lens antenna. The significant performance degradation despite the very small gap value considered shows how critical tolerances may be at the V-band.
It is worth noting that the rectangular base needed for testing purposes can increase the unintentional gap. A transmitter or receiver based on the geodesic lens concept may not need to connect to coaxial-to-waveguide transitions, which means that the size of the rectangular base may be reduced in a practical device. However, integration of additional functionalities (e.g., filtering) and the electronics require some space, and, as a result, similar performance degradation as observed can be expected in a complete device. In Section III-B, we discuss methods to mitigate the effect of the unintentional gap. These methods are then used to design a robust version of the lens antenna.

B. Robust V-Band Geodesic Luneburg Lens Antenna
Periodic structures can exhibit an electromagnetic band gap (EBG) that is useful for suppressing leakage in imperfect waveguide components [31]. Gap waveguides are designed to be manufactured in two pieces that are joined together with screws [31], [32]. EBG structures are placed along the two lateral sides of the waveguide to suppress the leakage that can occur if the two pieces are not in perfect electrical contact. Gap waveguides are attractive at mm-wave frequencies since small imperfections in the manufacturing can result in significant leakage as observed in Section III-A.
Glide-symmetric holey periodic structures have demonstrated similar EBG response [33], [34], [35]. These EBGs have been used to design gap waveguides at mm-waves [36]. Here, we use a 1-D glidesymmetric holey EBG structure to suppress the leakage in the lens antenna. The EBG structure and the arrangement of the holes along the lateral sides of the waveguides are illustrated in Fig. 4.
In some regions, the spacing between the waveguides is small, which must be considered in the design of the EBG structure.  Fig. 4(a) illustrates the simulation setup used to design the EBG structure. The coupling between ports 1 and 2 is simulated for different parameters of the glide-symmetric structure and gaps of 30 and 60 μm. The power transfer with glide-symmetric holes along the waveguide walls is compared with the shielded waveguide and waveguides with 30 and 60 μm gaps and no holes. One large and one small set of holes are designed. The simulated results are presented in Fig. 4 It is observed that the holes significantly reduce the leakage from the waveguide when compared with the same configuration with no holes. The results reported also demonstrate that the leakage reduction is preserved for larger gaps, indicating the response of the EBG is tolerant to this parameter. Furthermore, we observe that the larger holes suppress the leakage more than the smaller holes. Fig. 5 illustrates the integration of the holes in the lens antenna. Due to their larger leakage suppression, large holes are placed between the majority of the waveguides. Small holes are placed where large holes cannot be used. Furthermore, in the region nearest the lens, the waveguides are too closely spaced to place any holes. To improve the electrical contact in this region, an intentional 30 μm air gap is implemented in the highlighted region in Fig. 5. As the screws are tightened, they will apply higher pressure in the areas without 30 μm air gap, resulting in improved contact.
We repeat the above tolerance analysis for the robust lens antenna. Again, two scenarios are compared. First, the lens antenna is simulated with only the intentional gap (highlighted in Fig. 5). We note that the two pieces are in contact in the region without the intentional   gap. Second, the antenna is simulated with an extra 30 μm gap (i.e., in addition to the intentional gap). In this case, there is no contact between the pieces. The obtained patterns are presented in Fig. 6. We observe that the effect of the unwanted gap is significantly reduced in the robust antenna, compared with the original design. This comes at the expense of a slightly degraded sidelobe level, but this does preserve the design from more severe performance degradation.

IV. SIMULATION AND MEASUREMENT RESULTS
To further demonstrate the importance of the measures taken in the robust lens antenna, we manufacture, test, and compare the original and robust lens antennas. The manufactured prototype of the original and robust lens antennas is presented in Figs. 7 and 8(a). Fig. 8(b) presents a close-up view of the glide-symmetric EBG structure, and Fig. 8(c) shows the assembled robust lens antenna. The prototypes are manufactured with computer numerical control (CNC) milling.
The reflection coefficients of the lens antennas are evaluated using CST and measured using an Anritsu MS4647B VNA. The results for ports 1-7 are presented in Figs. 9 and 10 for the original and robust lens antennas, respectively. We target a reflection coefficient lower than −10 dB in the operational band (indicated with the   gray lines). Note that the 30 μm air gap (highlighted in Fig. 5) is included in the simulations for the robust lens antenna. The measured results are time-gated to remove the effect of the connecting coaxial-to-waveguide transition (i.e., the reflections in the transition are compensated for). The reflection coefficients in the operational band are below −10 dB in simulations and measurements. A better agreement between simulations and measurements is observed with the robust design.
The simulated and measured normalized radiation patterns in the scanning plane for ports 1-7 of the original lens at 56, 58, 60, and 62 GHz are presented in Fig. 11. High sidelobe levels (SLLs) are observed in measurements. Furthermore, it is observed that the SLL is the highest at the lowest operational frequency. In Fig. 12, the simulated and measured peak realized gains for ports 1-7 are  presented. The measured gain varies significantly between the ports and is roughly 1.5-4 dB lower than the simulated gain across the operational band. The discrepancies between the simulated and measured results are attributed to the manufacturing and assembling errors discussed in Section III. Due to the gap between the two pieces, leakage between the waveguides occurs, which results in increased SLL and reduced gain. We also observe that the proximity of port 1 to the edge of the rectangular base causes some of the leaked energy from port 1 to radiate at positive φ angles. The leakage (and performance impairment) is the highest at the lowest operational frequency since the TE 10 waveguide mode is more confined at higher frequencies.
The simulated and measured normalized radiation patterns in the scanning plane for ports 1-7 of the robust lens antenna at 56, 58, 60, and 62 GHz are presented in Fig. 13. The measured SLL is below −10 dB for all the ports at all the frequencies and around −15 dB for most ports and frequencies reported. In Fig. 14, the simulated and measured peak realized gains for ports 1-7 are presented. The measured gain is 1-1.5 dB lower than the corresponding simulated gain. These results are significantly improved when compared with the results of the original antenna, demonstrating the importance of the extra measures taken in the robust lens antenna. These results    Table I shows comparison of the presented geodesic Luneburg lens antenna with reported planar Luneburg lens antennas operating above 50 GHz. It is observed that the geodesic Luneburg lens antenna provides the highest radiation efficiency, which makes it attractive for antenna applications in the V-band and above.

V. CONCLUSION
In this work, we design and experimentally demonstrate a geodesic Luneburg lens antenna in the V-band. The geodesic lens, implemented in the PPW technology, is fed by 13 waveguides and terminated with a flare to mitigate the reflections at the PPW to free-space interface. The antenna can steer its directive beam in a ±55 • angular range in the azimuth plane. The antenna is intended to be manufactured in two parts, and a tolerance analysis indicates that a gap between the two parts can severely impair the performance of the antenna. The gap results in leakage between the feeding waveguides. Therefore, two cost-effective measures are taken to increase the robustness of the antenna assembly. First, EBG structures are placed between the waveguides. Second, the electrical contact is improved in critical regions.
Two prototypes are manufactured, one without the extra measures and one with. The measured radiation patterns of the prototype without the extra measures have high SLLs and low realized gain. The measurements of the robust geodesic lens antenna are in better agreement with the simulations. The measures taken to improve the performance of the lens antenna provide valuable lessons for future developments. The designed antenna is attractive for mm-wave satellite and mobile communication, as well as radar applications.