Geodesic Half-Maxwell Fish-Eye-Lens Antenna

We propose and implement a geodesic half-Maxwell fish-eye (MFE)-lens antenna. The lens was optimized using an in-house physical optics (PO) code adapted for generalized geodesic lenses. The final antenna design was validated with commercial electromagnetic simulation software. The antenna combines a modulated geodesic half-MFE lens and a transition to a linear flare, which is needed to preserve the linear polarization in the aperture. The antenna prototype, designed to operate in the $\text{K}_{\mathrm {a}}$ -band, was manufactured with computer numerical control (CNC) milling and measured in an anechoic chamber. The design provides continuous beam scanning because of a mechanically actuated feed. Promising beam scanning properties are demonstrated in an angular range of ±45° with a scan loss below 3 dB, as well as good frequency stability from 26 to 32 GHz. Since the antenna is fully metallic, its radiation efficiency is high (approximately 90%).

Abstract-We propose and implement a geodesic half-Maxwell fish-eye (MFE)-lens antenna. The lens was optimized using an in-house physical optics (PO) code adapted for generalized geodesic lenses. The final antenna design was validated with commercial electromagnetic simulation software. The antenna combines a modulated geodesic half-MFE lens and a transition to a linear flare, which is needed to preserve the linear polarization in the aperture. The antenna prototype, designed to operate in the K a -band, was manufactured with computer numerical control (CNC) milling and measured in an anechoic chamber. The design provides continuous beam scanning because of a mechanically actuated feed. Promising beam scanning properties are demonstrated in an angular range of ±45 • with a scan loss below 3 dB, as well as good frequency stability from 26 to 32 GHz. Since the antenna is fully metallic, its radiation efficiency is high (approximately 90%).

I. INTRODUCTION
B EAM-scanning antennas are key technologies for terrestrial communications and space applications. The most popular solution for these antennas is phased arrays, given their flexibility and low cost in low-frequency bands. However, at higher frequencies, typically millimeter waves and above, the required feed networks and active components increase their cost and losses considerably. Therefore, quasioptical systems, such as reflectors and lenses, are gaining popularity as an alternative to conventional arrays [1], [2]. Unless the entire antenna system is rotated, reflectors typically have limited scanning capabilities [3], [4], restricting their application scenarios, while lenses are generally more flexible. The lenses commonly used in these applications are the Manuscript  Luneburg lens [5] and the Rotman lens [6]. Luneburg lenses provide a wider angular scanning range due to their rotational symmetry, but require graded-index materials [7], [8], [9]. The behavior of these materials can be mimicked with periodic structures, which can be dielectric and/or metallic, such as bed-of-nails [10], [11], [12] or holey structures [13], [14], [15]. However, these subwavelength structures become a challenge at very high frequencies, leading to costly implementations. An alternative to graded-index materials is the use of geodesic lenses that are realized with a curved parallel plate waveguide (PPW) structure. The idea is to map the effective refractive index to equivalent physical paths [5], [16]. Taking advantage of their fully metallic nature, geodesic lens antennas have high efficiency and are good candidates for high-frequency applications [17], [18]. They also have potential for space applications, as they provide high-power handling and can work in extreme environments [19]. Examples of geodesic lens antennas that mimic the performance of a Luneburg lens are reported in [20], [21], and [22]. These are particularly attractive, because they provide directive beams in different directions with low scan losses. However, an important drawback of these solutions is their large footprint, which may be impractical in applications with limited space. To reduce the size of these lenses, a mirroring plane has been used, resulting in half-Luneburg lens antennas [23], [24], [25], which are more compact but have a reduced scanning range and higher scan losses.
An alternative solution to the half-Luneburg lens is the half-Maxwell fish-eye (MFE) lens. The MFE lens derives from a graded-index medium proposed by Maxwell [26]. This lens can focus the rays excited at one point of its contour to the diametrically opposite point of the contour. If an MFE lens is cut into two half-MFE lenses, all rays that reach the middle cut have the same phase and are collimated. This property was used in [27], [28], and [29] to create feeding networks and in [7], [30], and [31] to produce directive antennas. However, since the rotational symmetry of the lens is broken, the lens only has one true focal point, leading to significant scan losses, as well as reflections/refractions at its aperture due to the discontinuous refractive index.
In this article, we propose a geodesic half-MFE lens antenna as a good compromise solution for the above drawbacks. Unlike the graded-index counterpart, the proposed solution does not have refractions/reflections at the aperture caused by nonhomogeneous materials. To reduce the antenna height and optimize its performance, a modulated lens profile is implemented and optimized using an in-house physical optics (PO) tool [32], [33]. A mechanically scanning feed with optimized performance is also implemented to provide the antenna with a continuous beam-scanning functionality.
This article is organized as follows. Section II-A introduces the half-MFE lens, and Section II-B describes the proposed mechanically scanning half-MFE lens antenna. In Section III-A, we design an example of a geodesic half-MFE lens antenna operating in the K a -band. The mechanically scanning feed is detailed in Section III-B. Section IV shows the experimental results. The conclusions are drawn in Section V.

II. OPERATING PRINCIPLES A. Graded-Index Half-MFE Lens
An MFE lens focuses the rays excited from one point on its contour to a single point placed on the diametrically opposite side of the lens. As shown in Fig. 1, the refractive index profile of a planar MFE lens with a normalized radius is defined in the cylindrical coordinate system (r, φ, z) as follows: The ray trajectories of such a lens are shown in Fig. 1. Each ray describes a portion of a circle whose center is located in the median plane between the point source and the point image [26]. As a consequence, all rays are perpendicular to the median plane when crossing it, and a half-MFE lens produces collimated outgoing rays, as shown in Fig. 2(a). In this representation, all rays have the same electrical length, such that their ends represent a wave front, clearly illustrating the outgoing plane wave in Fig. 2(a). This means that a half-MFE lens can be used to design highly directive antennas with half the size of an equivalent Luneburg lens. However, this focusing property only works for a single point. As the feed scans, aberrations will appear, since the lens is no longer rotationally symmetric. The graded refractive index also introduces refraction that changes along the aperture, further contributing to the degradation of the wave front as the feed scans. This is illustrated in Fig. 2(b) and (c) for feeding points at 15 • and 30 • . The angle value 30 • is precisely the critical angle for the ray that passes through the center of the lens, where the refractive index is equal to 2. Beyond that point, some rays will begin to be totally reflected, as illustrated in Fig. 2(d) for a feeding point at 45 • , degrading significantly the performance. For example, the graded-index half-MFE lens design reported in [34] has the scan losses of about 6 dB over a ±40 • angular range. Due to refraction in the aperture, this range is covered with a feed displacement of ±25 • . Geodesic lenses can provide a convenient solution to these problems when the propagation medium in the PPW is the same as the surrounding medium.

B. Geodesic Half-MFE Lens Antenna
The proposed antenna is shown in Fig. 3 and consists of a fully metallic geodesic lens and its mechanically scanning feed. The geodesic lens is used to map the equivalent refractive index to a curved surface, so that the wave fronts are modulated by manipulating the physical paths of the rays in a homogeneous medium [35]. The lens itself can be divided into three distinct parts: 1) a modulated half-MFE lens; 2) a transition to a flat parallel plate; and 3) a flare. The detailed hollow structure of the lens is illustrated in Fig. 3(a) (air region). The modulated lens and transition are used to transform the cylindrical wave fronts into plane waves, resulting in highly directional beams over a wide scanning range. In the present case, the transition is essential, as it amends the curved aperture of the lens resulting from the shaped profile and keeps the field polarization vertical (more details are given in Section III-A). A mechanically scanning feed with a leak-preventing choke structure is designed to produce continuous azimuthal beam scanning. A detailed view of this feeding structure can be seen in Fig. 3(b). The structure is folded, so that when connected to a coaxial-towaveguide transition, a fixed feeding position can be ensured during mechanical scanning. The feeding transition, similar to the one used in [20], is set to fit the smaller waveguide dimensions considered here. To reduce the reflections caused by the folds, chamfers are implemented. A gap of 0.2 mm is designed between the feed and the antenna to ensure a smooth rotation that accounts for manufacturing and assembly errors. To prevent leakage at operating frequencies, a doublechoke structure is implemented (more details are given in Section III-B).

III. ANTENNA DESIGN IN K a -BAND A. Modulated Geodesic Half-MFE Lens Antenna
From the refractive index distribution of a rotationally symmetric MFE lens given in (1), the vertical profile of its equivalent geodesic lens shape is derived as follows [35]: with the definition of z and ρ illustrated in Fig. 4. The original geodesic MFE lens is, in fact, a sphere with a relatively high vertical profile equal to its radius, as shown by the dotted line in Fig. 4(a). To have a compact design, the profile can be folded, as suggested by Kunz [16]. The thicker green line shown in Fig. 4(a) gives an example of a profile folded twice to reduce the height to 1/3 of the radius. In practical applications, sharp bending points will result in spurious reflections.
To provide greater flexibility in shape modulation with smooth folds, arcs can be used as proposed in [20] for the design of a modulated geodesic Rinehart-Luneburg lens. Differently, here, we use five circles, C 1 , . . . , C 5 , to modify the shape of the lens, as shown in Fig. 4(b). The adjacent circles are connected by arcs with a normalized radius of 1, corresponding to portions of the original profile. The radius of the antenna is set to 10λ at 30 GHz in this design. By changing the position and radius of the circles, the profile can be easily modified. However, these arcs introduce phase aberrations, and an optimization of the profile is required to retrieve the focusing properties of the original geodesic MFE. An important point to consider here is that the half-MFE lens has an aperture that follows its profile. A PPW has the polarization of its fundamental transverse electromagnetic (TEM) mode orthogonal to the mean surface. Thus, the electric field in the aperture has a varying orientation, leading to poor polarization purity. This is the case in both the original and modulated half-MFE lenses. This problem does not occur in geodesic half-Luneburg lenses, as the cutting plane is a reflecting wall, and the aperture remains in the beamforming plane [24]. To solve this issue in the case of the proposed geodesic half-MFE lens, a transition is introduced between the lens aperture and the flare to generate a linear radiation aperture, as illustrated in Fig. 3(a) with the transition highlighted in darker blue. As this transition will also contribute to the modulation of the wave front, its shape should be conveniently chosen. In this case, the transition is generated by connecting the curved lens profile to the straight flare profile using the "Loft" function in CST Microwave Studio. The length of the transition is chosen to be 2λ at 30 GHz, with its end located at z = 0. To minimize the deterioration of the radiation performance caused by the transition, the lens should be further modified, so that it has a flatter central part, where most of the energy is confined. This aspect drives the design of the modulated profile here rather than the height reduction. The resulting geodesic design has a total of 15 variables, 3 per arc, namely, the in-plane coordinates of the center and the radius, leading to a rather cumbersome model and a slow optimization process using commercially available full-wave software. For this reason, we resorted to an in-house PO tool specifically developed for the design of generalized geodesic lens antennas [32], [33]. The ray-tracing (RT) part of the code is based on an open source library for the computation of the ray trajectories on a geodesic surface [36]. This approach is compatible with generalized geodesic lenses, not necessarily rotationally symmetric, as is the case for the proposed combined lens and transition. The surface is approximated with a 3-D triangular mesh, as shown in Fig. 5(a). The rays help to evaluate the electric field in the aperture. These are reported in the case of a feed at 0 • and 30 • in Fig. 5(b) and (c). The representation in Fig. 5(c) is particularly illustrative of the benefits of geodesic implementation compared with the equivalent graded index case in Fig. 2(c). Because of the transition design, scalar diffraction theory can be used to evaluate far-field patterns, as described in [34]. The optimized variables are reported in Table I. The final design is validated using CST. While the RT model analyzes only the mean geodesic surface, the full wave requires introducing the actual PPW cavity design. This is implemented by thickening the mean surface, so that the PPW section has a constant spacing of 2 mm between the two metallic plates. This distance is selected well below half-wavelength at the design frequency, ensuring that the PPW operates in its fundamental TEM mode band. Furthermore, a straight flare is added after the transition to ensure good matching to free space. The flare, visible in Fig. 3(b), has a length of 25 mm and a height of 15 mm, similar to the one used in [21]. A representation of the normalized E-field amplitude distribution at 30 GHz is shown in Fig. 6 for feed positions ranging from 0 • to 45 • in steps of 15 • . It can be clearly observed that the aperture is well illuminated in the considered angular range.
The radiation patterns in the azimuth plane when the antenna is fed from 0 • and 30 • are given in Fig. 7. The results provided by the RT method are compared with the full-wave CST simulation and measurement data. The RT method predicts very well the shape of the main beam and even the sidelobe levels (SLLs) below −10 dB, validating the focusing properties of the proposed antenna even at large scan angles.

B. Mechanically Scanning Feed Assembly
Directive antennas for communication applications should preferably have beam-steering capabilities. Thus, we next explore the potential of using a mechanically scanning feed assembly to provide continuous beam scanning over a wide angular range. A way to implement a mechanical scan feed combined with a PPW antenna is reported in [37]. It takes advantage of the flat PPW section comprising the focal arc to achieve feed scanning by relative rotation of the two PPW plates. This approach is not applicable here due to the volumetric design of the lens. Therefore, a separate feed system is used instead. To account for tolerances for assembly and manufacturing, a gap is intentionally implemented between the feed and the lens, and leakage is minimized due to a doublechoke structure.
An isometric exploded view of the manufactured mechanically scanning feed and lens antenna is shown in Fig. 8. The mechanical design of the lens comprises two parts that correspond to the top and bottom plates of the PPW. The feed structure is also manufactured in two parts, conveniently cutting through the E-plane of the feeding waveguide to minimize unwanted leakage throughout the electrically long waveguide run. As mentioned above, the feeding waveguide is folded twice to fit the measurement setup. This design places the feeding point at a central location, so that a fixed feeding position can be ensured during the mechanical scan. This arrangement secures a stable calibration of the measurement setup, which is particularly crucial when evaluating gain over the scanning range, and, thus, scan losses. Reflections from the folding points are reduced with chamfers. The results of the full-wave simulation of the manufactured feeding structure alone are shown in Fig. 9(a), demonstrating a reflection  coefficient lower than −20 dB over a wide band ranging from 24 up to 34 GHz. To make the feed rotate smoothly, a small gap of 0.2 mm is designed between the feed and the half-MFE lens antenna. In the K a -band, the degradation of performance that results from this gap cannot be neglected. To solve this problem, a double-choke structure is added around the waveguide port. Both chokes have a width of 1 mm. The inner radius of each choke is 5 and 7 mm, with a depth of 0.9 and 3 mm, respectively. A full-wave simulation has been carried out to analyze the response of the choke, the simulation setup being as shown in Fig. 9(b). A PPW with open boundary conditions is used to emulate a geodesic lens and is fed by a choked waveguide at a distance of G gap . The simulation results of the amplitude pattern of the E-field located in the PPW are compared in Fig. 9(b). The results show that the choke structure is essential to maintain almost the same E-field distribution in the PPW with a designed gap, compared with an ideal reference design without a gap. Furthermore, the double-choke structure expands the operation bandwidth of the  design compared with a single-choke design, ensuring stable performance of the antenna within the operating band.

IV. EXPERIMENTAL RESULTS
The prototype of the half-MFE lens antenna was manufactured in aluminum using computer numerical control (CNC) milling, as illustrated in Fig. 10. The prototype was measured in the anechoic chamber of the Division of Electromagnetic Engineering at KTH in a far-field setup. The distance between the probe and the antenna under test was 4.094 m. A standard gain horn, calibrated in the frequency range 22-34 GHz, is used as a probe. In the measurements, a coaxialto-waveguide transition that works up to 34 GHz is used as input port. The raw measured reflection coefficients of the antenna are given in Fig. 11(a) when all reflections are considered. The reflection coefficients are largely below −15 dB in the band of interest and all below −10 dB from 24 to 32 GHz. The VNA was calibrated using a coaxial line kit; thus, the effect of the commercial coaxial-to-waveguide transition is included in the response. For a better comparison with the full-wave model that does not include the transition, time gating is applied to filter out the associated reflection. The results are reported in Fig. 11(b), which show better agreement with the simulated results in Fig. 11(c). The simulated and measured results show stable gain performance with feed scanning, as illustrated in Fig. 12(a). The dashed black line shows the cosine variation of the gain with reference to the measured boresight realized gain due to the projected aperture in the beamforming plane. For the simulated results at 30 GHz, the realized gain varies from 24.59 to 22.27 dBi in the angular range ±45 • , corresponding to a scan loss of 2.32 dB. The gain measured at 30 GHz ranges from 24.26 to 21.65 dBi, with a scan loss of 2.61 dB. This is significantly lower than the scan loss of about 6 dB reported over a similar angular range for the graded-index half-MFE lens in [34], which confirms the benefits of using geodesic lenses. At 50 • , the realized gain drops to 20.52 and 21.09 dBi in simulation and measurement, respectively, which still remains in good agreement. Regarding the SLL, the simulation results show a variation from −20.5 to −11.2 dB when scanning from 0 • to 45 • at 30 GHz, in line with the SLL performance reported for graded-index half-MFE lenses [34]. The corresponding measured SLL ranges from −17.86 to −9.4 dB. Small deviations are attributed to manufacturing and assembly errors. This sidelobe degradation is quite limited and is consistent with previously reported work on compact geodesic lenses [24]. Measured co-and cross-polarized realized gain patterns are compared in Fig. 12(b). Patterns when the antenna is fed from 0 • , 20 • , and 40 • at 30 GHz are exemplified. The cross-polarized realized gains are largely below −10 dB, showing good polarization purity because of the transition design. Fig. 13 shows the contour maps of the measured realized gain at 30 GHz when the antenna is fed from different angles. The realized-gain patterns from both simulations and measurements are compared in the frequency range of 26-32 GHz for selected beams when the antenna is fed from 0 • to 50 • with a step of 10 • , in Fig. 14. Similar to Fig. 12, the black dashed lines represent the cosine variation of the gain due to the projected aperture with a peak reference corresponding to the boresight gain at 30 GHz. Some variations in the sidelobes are observed in the measurement of the most  deviated beam (50 • ), but the overall frequency behavior of the proposed lens has good stability within the extended angular range of ±50 • . The produced beams have a fan shape and have a stable pointing angle over frequency.
The realized gain of the half-MFE lens antenna for the central beam in the operating band is reported in Fig. 15. These results are compared with the gain of a rectangular reference aperture with a height of 15 mm and a width of 204 mm, which are equal to the size of the lens radiation aperture, assuming a uniform electrical field distribution. The width of the aperture is slightly larger than the diameter of the lens, as it also includes the transition to the feed on the outer rim. Ranges of aperture efficiency are shaded in grays. When assuming a perfect electric conductor (PEC), the simulated antenna has an aperture efficiency largely above 70% across the band. The drop in the measured realized gain is mainly attributed to losses coming from the following: 1) mismatch; 2) ohmic loss in aluminum (Al); and 3) its surface roughness (SR).  The full-wave simulation with lossy aluminum assuming a conductivity of 3.56 × 10 7 S/m and a root-mean-square SR of 0.3 µm (provided by the manufacturer) agrees well with the measured results. The remaining discrepancies are in the order of measurement uncertainties. We show the simulated efficiencies with the curves and the proportion of each contribution in the total loss using colored areas in Fig. 16. Both the total and the radiation efficiencies are around 90%. The main contribution is the ohmic loss in Al, which can be reduced by coating the surface with a better conductor, such as gold. The mismatch contributes a small proportion of the total losses, but is overall higher than the SR loss at these frequencies. Although the SR loss generally increases with frequency, it remains low in our band of interest and can be further reduced by polishing the antenna if needed.
A comparison with the state-of-the-art half-lens designs is summarized in Table II. The proposed antenna exhibits the lowest scan loss within 2.3 dB over a wider scan range up to ±45 • . The aperture efficiency of 70% is slightly lower in comparison with the geodesic half-Luneburg lens [24], which is attributed to the insufficient illumination on the edges of the aperture [seen in Fig. 6(a)] caused by the irregular shape of the transition. However, this may be acceptable for some applications considering the significant improvement in scanning range. The dielectric MFE lens in [31] also reports an extended scanning range up to ±44 • , but the implementation leads to significantly lower aperture efficiency. Its volumetric design has the advantage to enable beam scanning in both azimuth and elevation directions. This may be implemented with the proposed design using a stacked lens configuration as described in [38].

V. CONCLUSION
This work has proposed a mechanically scan-fed geodesic half-MFE lens antenna operating at the K a -band. The antenna is theoretically analyzed and optimized with an in-house PO method and validated with full-wave simulations and measurements. Our measurement results demonstrate that the proposed half-MFE lens antenna can produce a continuous beam scan over ±45 • within 3 dB scan loss with a radiation efficiency of approximately 90% in the operating band from 26 to 32 GHz. At 30 GHz, the measured realized gain varies from 24.3 to 21.65 dBi over an angular range of ±45 • , with corresponding SLLs from −17.9 to −9.39 dB. Thus, the proposed design shows satisfactory frequency stability, sufficient for many practical applications, and greater than that provided by an equivalent graded-index half-MFE [34] or by a half-Luneburg lens [24]. It also has a smaller size compared with a Luneburg lens [20], [21]. These advantages make the geodesic half-MFE lens antenna a good candidate for highfrequency communication/satellite applications.
The in-house PO tool was successfully used for the optimization of the combined lens and transition surface; this transition being essential in the design of geodesic half-MFE lens antennas to achieve good polarization purity. This PO-based approach can also be used for any type of geodesic lens design with the aim of minimizing its height or reducing its footprint without degrading its focusing property.
Furthermore, this article explored the possibility of using a continuous mechanical scan in the K a -band. With a design gap of 0.2 mm between the antenna and the feed, which is intended to guarantee smooth rotation, a double choke was implemented in the waveguide to prevent degradation of feed performance. Although the presented design operated in the K a -band for 5G/6G or space communication systems, it can be further promoted at other frequencies and applications.