Two-Dimensional Multi-Ring Dielectric Lens Antenna to Radiate Fan-Shaped Multi-Beams With Optimum Adjacent-Beam Overlapping Crossover by Genetic Algorithm

A two-dimensional (2-D) discrete dielectric lens antenna is designed to radiate fan-shaped multi-beam patterns for gain stability in beam switching. The target is to minimize adjacent-beam overlapping transition regions and provide sufficient and similar gains for all field angles when the antenna is employed in a mobile device. This design starts with a conventional 2-D Luneburg lens antenna, and distorts its dielectric permittivity and the sizes of discrete dielectric rings to defocus the pencil beam patterns into shaped ones with a relatively flat pattern for uniform field distribution. The design is realistically implemented at 38 GHz with both simulation and measurement results shown to validate the concept. Successful validation of feasibility in beam synthesis is achieved. Fabrication discrepancy to result in slight radiation degradation is also discussed.


I. INTRODUCTION
User equipment (UE) or customer-premises equipment (CPE) antennas in mobile or fixed network communications [1] require high gains in radiation to compensate electromagnetic (EM) wave propagation loss in air [2] which is severe at millimeter wave (mmW) frequencies. Conventional implementation of a phased array of antennas to produce electronically steering beams is very cumbersome as it involves numerous costly RF devices. Their realization on dielectric substrates also faces severe challenge of power loss in the lossy substrates when beamforming networks (BFNs) are implemented by microstrip lines. Instead, multi-beam The associate editor coordinating the review of this manuscript and approving it for publication was Chow-Yen-Desmond Sim .
antennas are among the potential solutions to provide high directional beams of easy switch at a much lower cost.
Due to the dynamic changing of position and orientation in UE and CPE usage scenario, multi-beam radiations with a fan-shape pattern [3], [4] (i.e., vertically broad and horizontally narrow to perform horizontal beam switching) is very attractive because a single one-dimensional (1-D) beam scan, instead of applying sophisticated two-dimensional (2-D) ones, is sufficient to provide the required gain. Moreover, a flat-top pattern is most desired along the beam scan dimension to provide seamless connection links with relatively uniform gain performance. This importance has been well-documented in literatures for various antenna designs [5]- [11]. Potential antenna candidates include a 1-D phased array of antennas with proper BFNs and 2-D Luneburg lens antennas [12]- [14]. More advantageous than a phased array of antennas, which suffer from limited beam scan range due to high wide angle scan loss, a 2-D Luneburg lens is rotationally symmetric to provide equal gains for all beams with broad frequency bandwidth in terms of reflection coefficients and radiation characteristics when the feed has broad operational bandwidth.
It is noted that the conventional implementation of multi-or steering beams gives rise to severe drawbacks for producing discrete beams when digital phase shifters and multi-feeds are employed in phased array antennas and Luneburg lens antennas, respectively. For the case of Luneburg lens antennas, the best gain at the crossover point of two adjacent beams is smaller than -6 dB below the beam peak due to limited space for feeds as shown in Fig. 1(a) when they are excited by classic waveguides of fundamental modes. Insufficient gains appear in the transition regions to cause power instability by frequent handovers (i.e., the pingpong effect) [15], where call drops may also happen. It is noted that even though a smaller gain-drop at the crossover point of approximately -3 dB was claimed in [16] by using transmission line excitations for more crowded feeding ports, where numerical simulation results were shown, the interport isolation has been significantly degraded so that it is worse than -15 dB, and the back-lobes are significantly high due to the radiation from the transmission line exciting port. Moreover, experimental validation has not been shown. This increase of excitation ports will make the system implementation more sophisticated.
This paper presents a 2-D lens antenna design for the first time to radiate relatively uniform patterns as illustrated in Fig. 1(b) to reduce the adjacent beam transition region for beam stability at mmW frequencies, also applicable to CPE of front-haul networks. It is noted that front-haul networks are currently viewed as the most important application in the first stage of commercial 5G (the fifth generation) mobile communication system at mmW frequencies. The fan-shaped patterns with relatively flattop patterns provide a very simple connecting mechanism between the base station and CPE antennas by performing a 1-D beam scan. The flat structure of 2-D Luneburg lens can be implemented inside a PCB, which can be integrated with the multi-layered substrate based hardware system for compactness.
The design uses the genetic algorithm (GA) as a tool, and starts with an initial design by a discrete 2-D Luneburg lens [17]- [19] to assure the property of directional beams. Its permittivity and sizes of discrete dielectric rings are then optimized by the GA [20]. In contrast to previous works of three-dimensional (3-D) Luneburg lens optimization, which either minimize the estimation error of permittivity [21] in a least squares error (LSE) sense, or enhance the gain and reduce sidelobe levels (SLLs) [22], [23], we focus on the 2-D main beam synthesis for CPE applications. The major target is to simultaneously reduce the inter-beam transition area [24], and minimize gain variation, i.e., creating a more uniform field. This optimization is more challenging than the uniform permittivity lens optimization on its external surface profile [20], [25] because of the limited parameters available for optimization of retaining symmetric multi-ring structures. After the optimization, the radiation patterns are compared to that of conventional Luneburg lens antennas at 38 GHz. Radiation measurements on an antenna prototype are also presented to validate the design feasibility. Numerical results show that the gain-drop at the crossover point is less than 1 dB in addition to the flat-top main beam pattern for stable coverage, exhibiting much better behavior than that in [16]. The low crossover point gain drop is also validated by measurement results.
The rest of the paper is organized in the following format. Section 2 presents the basic radiation behavior of 1-D multibeams by Luneburg lens, and shows the pattern synthesis for better communication links. Section 3 presents the basic implementation of GA to synthesize the radiation patterns of a 2-D dielectric lens. Parametric studies of GA optimization are also shown. A realistic design is shown in Section 4 for validation of this proposed work of a multi-beam lens antenna design. Finally, a short discussion is presented in Section 5 as a conclusion.

A. ARCHITECTURE OF LUNEBURG LENS AND PROBLEM STATEMENT
A 3-D Luneburg lens is an inhomogeneous dielectric sphere with varying relative permittivity governed by [12]- [14], [17]- [19] ε r (r) where r is the radial distance and a is the sphere radius. In practice, the 2-D Luneburg lens is a dielectric slab sandwiched by a pair of parallel metal plates, where the dielectric slab is formed by a finite number of uniform, concentric dielectric rings of various widths, as shown in Fig. 2 (a), where its potential application as a CPE antenna is also shown in Fig. 2(b). The widths and relative permittivity values of the dielectric slabs are determined by optimization schemes according to different mechanisms as mentioned in [20]- [25]. Based on realistic feed antennas at a separation distance of larger than a half-wavelength, the crossover point of two adjacent beams is more than -6 dB below the beam peaks as shown by the HFSS simulated results in Fig. 1 (a), where a = 40 mm, and 10 rings (including an air ring) are used. Their widths and relative permittivity values of the dielectric slabs are determined by the optimization scheme in [4] and [21]. It is fed by rectangular waveguides (0.7 × 0.35λ 2 in aperture size, where λ is the wavelength of free space) placed tightly to each other.
Numerical experiments show that the larger 6 dB gain drops in the transition regions in Fig. 1(b) are similar for any Luneburg lens regardless of its size of lens except for slight inwardly shifted beam directions due to the smaller angular span occupied by the feeds' physical size when a larger lens is considered. The gain drops are even worse when the two adjacent feeds are further separated to increase the isolation between them. This problem results naturally from the radiation phenomenon of a pencil beam shape, and may cause the pin-pong effect due to insufficient gain. This is most undesired because the possibility of dynamic moving will worsen the stability of connection to the base station antenna.
In this paper, we propose to synthesize a shaped beam to alter the beam overlapping by pencil beams as shown in Fig. 1(a), where the desired one is shown in Fig. 1 In this case, the beam overlapping region is reduced to avoid hand-overs while the beam connection has a high stability due to its uniform field distributions in the main beam regions. This shaped beam pattern synthesis is described below.

B. DESCRIPTION OF RADIATION BEAM PATTERN OPTIMIZATION
The beam shape in Fig. 1

(b) is synthesized in the lens design with a cost function first defined by
where G d m and G m are the desired and computed gains at the M sampled field points. In (2), f m is used to balance the field weightings of main beam to the sidelobes in the transition regions, where f m = 1 is assumed for simplification because we focus on the main beam pattern. The shaped beam is particularly performed to optimize the main beams for better field distribution with minimum overlapping.
The parameters of the lens structure used in the optimization include the overall size, the widths of dielectric rings and their permittivity values. In particular, the relative permittivity is relaxed to be larger than 2 while retaining its sequential decrease from the center to its boundary for air ring matching. Only a single beam is considered for the circular symmetry of the lens, where HFSS is employed to compute the radiation. In GA [20], [26] optimization, each optimization parameter is represented by a 3-bit variable. In order to assure a gradual decrease of relative permittivity, one first selects reference values by performing the algorithm in [4], [21] to minimize the estimation error with respect to (1) as indicated by the black stepped lines in Fig. 3. Afterward, a variation range is selected for the theoretical values between the two boundaries of a selected dielectric ring inside the two red dashed lines in Fig. 3.
To relax the variation of relative permittivity, (1) is modified by where α is the modification factor, which is slightly larger than 1. The relative permittivity of the most outer ring reduces to 1 for the air while that at the central ring may be larger than 2. This range of dielectric constant may allow the lens to retain the basic property of a directional beam as provided by the 2-D Luneburg lens.

III. PARAMETRIC STUDY IN GA 2-D LENS ANTENNA SYNTHESIS
One first examines the effectiveness of optimization for a 2-D Luneburg lens by the GA with a thickness of 2.84 mm which is the thickness of the feeding rectangular waveguide. This lens has an initial radius of 27.5 mm at 38 GHz, and is sandwiched by a pair of circular metal parallel-plates to form a cavity. The target gains are 11dBi for the flat part in Fig. 1 (b), where the fields in this flat region within a beamwidth of 16 degrees are sampled to compute the cost function in (2). As pointed out in Section 1, the desired radiation has fanshaped patterns, i.e. vertically broad and horizontally narrow, to perform a 1-D scan. In particular, a uniform distribution of main beams, as illustrated in Fig. 1 (b), is desired in order to retain gain stability when the multi-beams are discretely switched. It may potentially provide a seamless transition when the beam is switched from one to another.
In the synthesis stage of examining the effectiveness of the GA application, the feeds are rectangular waveguides with a cross-section of 5.68 × 2.84 mm 2 to excite TE 01 modes and investigate the radiation characteristics. A compact embedded probe excitation will be implemented for the later antenna prototype design as well as for the numerical and experimental validation.  dielectric shells. Each chromosome is represented by 3 bits of binary expression. In the case of synthesis, the average CPU time is roughly 16.5 minutes on an i7-7820X processor. In addition, they all start with same values for their geometrical parameters and relative permittivity values as shown in Table 1. This set of initial values is obtained by using the optimized selection of discretized parameters of the Luneburg lens in (1) as suggested in [4], where four rings are considered including three dielectric rings and an air ring for cost consideration.
The four cases examined consider the four combinations of parametric variables by the dielectric constants and the ring thicknesses. In particular, the first case of synthesis considers the dielectric constants of each rings as the optimization variables, where the variation of dielectric constant in (1) is employed to discretize the ranges for optimization of each ring in Fig. 3. In this case, the thicknesses of each ring remains fixed, as in Table 1. Thus the maximum dielectric constant in Fig. 3 is 2. On the other hand, the second case also considers the dielectric constants of each ring as optimization variables under similar conditions except now (3) is employed to specify the ranges of dielectric constants for each ring to optimize their dielectric constants. The other two cases of optimization, which are referred to cases 3 and 4 in Table 2, add the thicknesses of each ring, D n , as the optimization variations to the first two cases.
In the GA implementation of dielectric ring thicknesses as optimization variables, one considers the ratios of thicknesses to a fixed size of lens. It is performed by using 3 digits to discretize the range of variation into 8 steps (1 ∼ 8 th step). Let each ring's thickness be represented by S n in the discretized representation, then the thickness of the n th is given by where a is the selected radius of the lens including the air ring. Once the thicknesses are determined, the ranges of each ring to determine their dielectric constants in Fig. 3 are specified. Similarly, three digits are employed to discretize each of the ranges and represent the variations of dielectric constants in these ranges.

B. EFFECTIVE OF 2-D LUNEBERG LENS OPTIMIZATION
Due to the rotationally symmetric configuration, a single beam is considered for simplification, where the co-polarized (Co-pol) patterns of this initial architecture on the E-and H-planes are shown by the black lines in Fig. 4(a) and (b), respectively, which serve as reference VOLUME 8, 2020 patterns for comparison. The gain is roughly 15.5 dBi with a -3 dB beamwidth of 7.9 degrees along the multibeam dimension. The relative SLL is roughly -17 dB below the main beam peak. As described in Section II, the target beamwidth of the desired flat patterns in Fig 3 is 16 degrees. The roll-off rate on both sides of main beam curve depends on the size of the resulting lens.
The resulting values of variables by GA optimization for the four cases are summarized in Table 2. It is observed that cases 1 and 3 result in a converged relative permittivity of nearly 1.99 for the central dielectric disks. On the other hand, cases 2 and 4 result in an optimized relative permittivity of 2.38 and 2.47, respectively, which are larger than 2 as desired in the original Luneburg lens design in (3). On the other hand, the third rings of cases 1 and 3 are close to free space, where the relative permittivity values are roughly 1.05. In addition, the thicknesses of the first and third shells are almost equal after synthesis.
The resulted radiation patterns on the two principal planes corresponding to the four cases of Table 2 are shown in Fig. 4(a) and (b), respectively by different color lines in comparison to the patterns of the initial design in Table 1. In all cases, the E-plane patterns are relatively broad due to the thin thickness of the 2-D lens. The -3 dB beamwidth is roughly 60 degrees to provide a wide coverage. The gain drops by roughly 3-5 dB caused by the H-plane pattern optimization for the flatness of the central main beam area. The H-plane patterns, which are used to form the multi-beams, have larger variations because of the optimization to achieve flat patterns in Fig. 3.
It is first observed that the case 1 of using (1) with a maximum of 2 at the center of the Luneburg lens only results in a broader pattern without changing the pencil beam shape, which does not fulfill the target pattern in Fig. 3. On the other hand, the remaining three cases may result in relatively flat main beam patterns. In comparison, the main beam shape in the second case, by using (3) to set the variables of optimization, descends very rapidly near the boundary of flat main beam region, where the -1 dB beamwidth is 15.5 degrees while the -3 dB beamwidth is 18.7 degrees. In this case, α in (3) is set to be 1.5 to make a relative permittivity of 2.38 for the central dielectric disk. Also the fourth case of using both (3) and the thicknesses of dielectric shells as the optimization variables results in the largest beamwidth with a smaller slope of descent for its main beam shape near the boundary, which may result in a larger overlapping area between two adjacent beams. All these three cases result in similar SLLs, which are roughly -6 dB below the main beam peak. This side lobe discrepancy can be improved by using H-plane rectangular horns as the feed to create a tapered illumination.

IV. DESIGN IMPLEMENTATION OF VALIDATION A. A REALISTIC DESIGN FOR EXPERIMENTAL VALIDATION
The antenna is designed and prototyped for validation of feasibility at a 38 GHz band, where the achieved geometrical parameters and relative permittivity are summarized in Table 3. The radius of the lens is 50 mm with a thickness of 2 mm, which is larger than these in Table 2 in order to further control the SLLs by a larger aperture. The paired circular metal parallel-plates sandwiching the lens has a radius  of 56 mm, slightly larger than the lens to increase isolation to the nearby RF devices for future applications. In this design, 4 dielectric rings (including an outer air ring) are employed to form the lens. The incident wave illuminating the lens is excited by a probe pin backed by a metal parabolic reflector at 38 GHz as shown by the numerical model and prototype in Fig. 5. The prototype is hand-assembled after every component is fabricated.
For the prototype implementation, the slab of each dielectric ring is equivalently resembled using two stacked layers of Teflon and air [4] whose thickness ratio is selected to make the average relative permittivity equal to the computed ones in Table 1. In this case, five beams are implemented, where the feeding structures are illustrated in Fig. 5. Detailed structures are referred to [4] for brevity.

B. RADIATION CHARACTERISTICS
The simulated and measured radiation patterns are shown in Fig. 6(a) and (b), respectively, where relatively uniform fan-shape patterns are seen in the simulation results as expected. The good agreement between simulation and measurement is achieved with some slight discrepancy in the measurement results due to prototyping discrepancy. The gain variations are relatively uniform on the main beams, where the gain is larger than the targeted value of 11 dBi in the synthesis. The relative cross-polarization level (XPL) is less than -40 dB from the numerical simulations in Fig. 6 (a), which is slightly higher in measurements due to the difficulty of accurate alignment for hand-assembly since the parallelplate antenna structure is well-known to have good polarization characteristics.
The simulated and measured reflection coefficients are shown in Fig. 7 (a) and (b) respectively, where both exhibit very broad frequency bandwidths. The simulation reflection coefficient level is below -15 dB for most frequencies while VOLUME 8, 2020 it is higher in the measurement results due to the fabrication discrepancy and K-type connector implementation. However, the agreement with the simulation results is in general very good, where the -10 dB bandwidth is more than 7.5 GHz in measurement results.

C. CHARACTERISTIC DISCUSSIONS
One first considers the multi-beam distribution in Fig. 8 (a) and (b) for the simulation and measurement results respectively, where good agreement has been achieved. The asymmetric measured patterns of wide-angle beams on both sides are attributed to the discrepancy of hand-assembly and alignment for wide angle patterns in a 2-D far-field measurement system. In this case, the overlapped patterns along the H-plane are relatively uniform, almost 12 dBi in gain in the region covered by these beams. The crossover points are roughly 0.5 dB below the flat top. The roll-off rate of the main beam boundary is much higher than that in Fig. 1 (a), indicating a much smaller transition region. Similar behaviour is also observed on the measurement result even though the first sidelobes are much larger than that of the simulation. Finally, the isolation of beam ports is investigated by considering that between ports 4 and 5, as shown in Fig. 9. It is observed that the isolation is better than -20 dB in both simulation and measurement. The other ports have similar behaviors, which are omitted for simplification.
To further examine the effectiveness of fan-shaped beams, the multi-beam radiations of a Luneburg lens without optimization under the same conditions of size and feeds are shown by the simulation results in Fig. 8 (c). It is seen that   Fig. 8(a) and (c) with and without synthesis for shaped patterns, respectively. the gains are roughly 15.5 dBi, which is 3.5 dB larger than the designed one. However, the crossover point is at 8 dB below the beam peak. Thus, in the transition region near the crossover point, the received EM power by the UE will be 8 dB smaller. On the other hand, in the transmitting mode with power control, the EM power in the beam peak direction is 8 dB larger, and causes severe interferences to the other UEs. The simulated contoured patterns in the cases of Fig. 8 (a) and (c) are shown in Fig. 10 (a) and (b), respectively. If one considers a 10 dBi gain as the threshold of coverage area. It can be seen that Fig. 10 (a) shows a coverage with much better uniform field distribution inside the coverage area, indicating a better fan-shape pattern along beam switching, resulting in a smooth beam transition for CPE connections.
Finally, the frequency stability, by comparing the cases with and without the GA synthesis, is shown in Fig. 11, where simulated gains and reflection coefficients (R.C.) are shown. It is observed that the reflection coefficients remain relatively stable when the lens is synthesized. On the other hand, the synthesized gains exhibit higher fluctuations of roughly 1 dB.

V. CONCLUSION
In this paper, we present a 2-D lens antenna design for fanshaped multi-beam radiations with relatively uniform patterns to minimize the beam overlapping transition regions, where a GA was employed to synthesize the discretized dielectric rings in the widths of dielectric rings and their permittivity. Distinguished radiation characteristics have been shown in both the simulation and measurement results, which provide the advantage of increasing better gain in the transition region of adjacent beams for the stable power control. Currently, only the main beams are employed in the pattern synthesis, and may result in higher sidelobes. Our experimental experience indicates that manufacturing discrepancies may result in significant radiation distortion leading to a loss of uniform shape and a reduction in the conventional pencil beam when the operational frequency is far away from the design central frequency. However, the gain and reflection coefficients have relatively high stability with respect to frequency change. This phenomenon is caused by the limited few available parameters in the GA synthesis.