1. Introduction

In modern wireless communication systems, the ever-increasing demand for higher data rates, improved coverage, and reduced latency has sparked intense research efforts to develop innovative RF technologies. A clear solution for these demands is to use higher frequency bands such as mmWave, however, the loss in performance at these higher frequencies hinders their use. To offset the performance loss introduced from tapping into mmWave frequencies, lens-embedded antenna systems have been proposed to offer a path forward through the development of high gain, beam-steering lenses which may also achieve multiple-input multiple-output (MIMO) capabilities [1,2]. Traditional RF beam steering techniques have often relied on mechanical mechanisms that suffer from speed, size, and durability constraints. Electronically steered phased array structures offer an excellent solution, however, the quantity and complexity of active elements and RF components needed to achieve such a design with subwavelength spacings hinders cost-effectiveness and practicality [2]. Additionally, the scan loss induced by such systems produces a practical limit of around ±60 degrees for planar configurations. As a result, there is an urgent need to explore novel approaches that transcend these constraints. Passive beam steering using graded dielectric lenses has emerged as a promising candidate, presenting a fascinating blend of optical principles and microwave engineering to achieve beam steering over a wide frequency band and steering angles.

Like its optical counterpart, the RF gradient dielectric lens exploits the principle of a spatially varying permittivity distribution within the lens material to control the direction and phase of propagating waves. This graded profile allows the lens to focus or steer electromagnetic waves by bending their paths smoothly, akin to how a traditional optical lens manipulates light rays in the visible spectrum. By suitably designing the permittivity profile, the lens can focus or deflect RF signals with high efficiency and precision over a wide range of angles and frequencies [3–5]. While these devices have demonstrated significant theoretical value, their intricate analysis and challenging fabrication have hindered their realization in practice. It was not until the proliferation of modern computational electromagnetic methods, iterative design algorithms, and the ability to manifest those designs with advanced manufacturing that these devices could be practically realized.

Recent literature has highlighted the success of these advancements within the domain of RF beam steering lenses [5–11], such as full 3D Luneburg lenses for antenna beam steering. A Luneburg lens is a spherically symmetric gradient dielectric lens. It operates by focusing plane waves into a focal point on a corresponding point along its surface [12]. Conversely, if a small low-gain feed antenna is positioned on the spherical surface, the lens produces a high-gain beam in the same direction. Because the lens is spherically symmetric, altering the feed locations along its surface results in beam steering at any angle without loss of antenna gain. However, the traditional Luneburg lens's spherical geometry presents challenges in devising a conformal feed configuration. To address this issue, several investigators have leveraged advancements in transformational optics to design modified Luneburg lenses with flat surfaces that allow for the integration of planar feed arrays. In [11], the authors used a transformational optics approach with the addition of an anti-reflective surface layer to achieve wideband impedance matching over a range of feed locations. These lens designs were subsequently fabricated using various additive manufacturing approaches and experimentally characterized. While excellent broadband RF performance was demonstrated, scan angles were typically limited to approximately -60° to 60°. Thus, achieving wide-angle beam steering over a wide spectral bandwidth while allowing for a planar integration of antenna feeds has remained an ongoing challenge.

For this purpose, it is worth investigating other lens designs and materials. For example, the Gutman lens provides the same function as a Luneburg lens in a smaller volume [13] and can be truncated along a plane to allow for planar feeds [14]. However, the Gutman lens also suffers from a limited scanning range. The Eaton lens, on the other hand, is capable of rotating beams at right angles but requires high permittivity materials and cannot support small scan angles [5,15]. In [16], Lu et. al. investigated the use of a partial Maxwell fisheye (PMFE) lens. They demonstrated how a PMFE lens has features that combine some of the best attributes of the Gutman and Eaton lenses (see Fig. 1). Specifically, this design incorporates the focusing ability of the Gutman lens at its core and the beam-rotating capability of the Eaton lens towards the outer radius. Like the partial Gutman lens, the spherical geometry of the PMFE can be sliced along the focal plane of maximum boresight gain, allowing for the placement of feeds along a convenient, flat plane. Like the traditional Maxwell fisheye lens, however, the PMFE requires a higher maximum permittivity than the Luneburg lens. Several designs achieved this using both 2D metamaterial and dielectric wedge structures. For example, in [16], the authors demonstrated the ultrawide beam steering ability of a 2D PMFE lens using a fully metallic bed of nails design. This approach was extended in [17] to realize an E-plane-focused lens antenna by creating a 2D dielectric wedge structure fed with a series of rectangular waveguides along a straight line for multiple beams.

 

Fig. 1. PMFE Lens Structure and Profile. As described in [12], (a) the PMFE lens design and (b) dielectric constant comparison between the partial Maxwell fisheye, Gutman, and Eaton lenses with a model of the PFME permittivity profile to demonstrate how the design combines attractive features of both lens types.

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Fig. 2. Iterative design methodology used in this work to design, modify, optimize, and fabricate MPMFE lens.

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To the best of our knowledge, we present the first full 3D, optimized modified partial Maxwell fisheye (MPMFE) that achieves broadband, wide-angle passive beam steering. The implications of this research suggest a near hemispherical range of beam steering capability in elevation across 360° in azimuth with a single lens and antenna feed system. Our design methodology utilizes the PMFE lens as an initial approximation within an iterative optimization algorithm. Modifying various lens parameters and introducing an anti-reflective (AR) surface along the plane of truncation allows us to achieve wide-angle beam steering, minimize scan-loss, and minimize side lobe levels while ensuring manufacturability. As a part of the design process, material, system, and computational design constraints are all taken under consideration and are a part of our iterative method, as shown in Fig. 2. To fabricate the MPMFE lens, a novel additively manufactured polar lattice structure is introduced, which reproduces an effective permittivity distribution that closely resembles the desired MPMFE lens characteristics.

The experimental validation of the fabricated MPMFE lens demonstrates its broadband beam steering capabilities across a range of steering angles from -80° to 80°. The following sections of this paper provide a detailed overview of (1) the optimized design methodology and simulated results, (2) the fabrication process of the lens, including material characterization using a novel space-filling curve pattern and effective medium theory, (3) experimental validation that showcases the agreement between our predicted and measured results, and (4) a concise discussion and conclusion of our work.

2. Computational modeling and design

The design of the MPMFE lens draws inspiration from the work conducted by Lu et al. [16], who described the novel characteristics of the PMFE lens. Towards the outer radius, the lens takes on the permittivity profile of the Eaton lens with its sharp beam rotating feature. Towards the core, the lens takes on the permittivity profile of the Gutman lens, enabling it to produce high-gain beams with gradual beam steering properties. By truncating the spherical geometry of the lens at some optimal location, high-gain steered beams can be achieved by simply feeding the lens at their corresponding locations along a convenient, flat surface (see Fig. 1). With this modified permittivity profile, the PMFE effectively mimics both lens variants with a shift of the feed along a radial line. Doing so exploits the benefits of each lens while minimizing their drawbacks.

Our objective was to optimize the performance of the PMFE lens by modifying several critical parameters. The primary consideration was to ensure that the lens remains manufacturable. To that end, we adjusted both the maximum permittivity, ε_max, at the lens's core and the minimum permittivity, ε_min, along the outer spherical surface to meet material and fabrication constraints. Another design goal was to generate a lens capable of operating efficiently over a wide band of frequencies using a standard feed antenna. The original PMFE lens, shown in Fig. 1, has spatially varying dielectric properties along the planar cut line where the feeds are mounted. This contrast in permittivity creates a variable insertion loss depending on the location of the feed antennas. To address this issue, our design incorporates a half-wavelength thick anti-reflective (AR) layer along the surface of the cut plane (shown in Fig. 3). The AR layer provides a wide-band impedance matching property from air to the planar surface of the MPMFE lens. Lastly, we discovered a strong co-dependence between the location of the cutline, z_o, and the maximum and minimum permittivity when evaluating these lenses’ steering performance and gain. To a lesser degree, the electrical size must also be considered in this dependency. Thus, our design strategy was to select a minimum and maximum permittivity based on the available materials and fabrication constraints. The electrical size of the lens was then calculated based on the desired maximum gain. Finally, a simple iterative optimization algorithm was used to determine the optimal cut line distance given an integrated half-wavelength thick AR layer. While a variety of objective functions could be used to evaluate different lens profiles, we chose a weighted function that evaluates performance based on boresight gain (G_BS), gain roll-off at 45°, 60°, and 80° steering angles (G_RO), and boresight relative side-lobe levels (RSLL_BS) as described in Eq. (1).

Given these design considerations, the rotationally symmetric permittivity profile of the MPMFE lens and the cylindrical anti-reflective region, respectively are given mathematically by Eq. (2) and Eq. (3);

 

Fig. 3. MPMFE Structure and Profile. Illustration of the MPMFE lens showing graded permittivity profile and parameters used to optimize performance.

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To assess the MPMFE lens's performance, we utilized COMSOL's commercial finite element solver to create models of lenses with radii of 2.5λ_ο, 3.75λ_ο, and 5λ_o. The lens profile was defined using Eq. (2) and (3), and a parametric sweep was conducted to investigate the optimal design parameters for each size lens. To comply with the constraints imposed by the available materials and fabrication methods, the permittivity was limited to the range 1.5 < ε_r < 5.0. It was found that in an ideal scenario where the minimum effective permittivity can be matched to air (ε_r= 1), the performance increase was negligible in comparison to the tuning of the other parameters. Additionally, it was determined through parametric sweeps that the optimal cutline distance, z_o, varied within the range of 0.15R to 0.35R depending on the electrical size of the lens. We assumed an open-ended rectangular WR-28 waveguide port operating at 30 GHz to feed the lenses. To approximate an infinite free-space region, perfectly matched layer (PML) absorbing boundary conditions were place λ_o/2 from the lens surface. Once the near fields were calculated, the Stratton-Chu equations were used to determine the far-field radiation patterns.

Given these constraints, the optimal parameters for each lens size are outlined in Table 1. For instance, when considering a lens of size 5λ_o, the most favorable performance was achieved with a cutline of 0.275R and a maximum relative permittivity of 4.7. The simulated far-field gain of this optimized lens is shown in Fig. 4 for several feed locations. In Fig. 4, we also provide a comparison between our MPMFE lens and the original PMFE lens design, referred to as the baseline, as described in [16]. The results revealed that our MPMFE lens exhibited improved characteristics, with slightly greater steering of approximately 5°, reduced gain roll-off for the intermediary steering angles, and lower side lobe levels. Furthermore, the addition of the antireflection (AR) region contributed to an average reduction of 3 dB in insertion loss at all feed locations. Remarkably, this lens design showcased a realized gain of 15 dBi at a steering angle of 85° and nearly 13 dBi at 90°, indicating a wide beam-steering capability with a moderate degree of gain roll-off.

 

Fig. 4. Simulated MPMFE Realized Far-Field, Optimal and Baseline. Example MPMFE lens design with R = 5λo (solid) compared to the baseline [16] performance (dashed). The MPMFE design results in lower relative side lobe levels, reduced gain roll-off, and strong steering performance.

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Table 1. Optimal Lens Parameters for, R = 2.5λ_ο, 3.75λ_ο, 5λ_ο

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To highlight the wide-angle beam steering capabilities of the MPMFE lens we compared its performance to two common graded index lenses. Namely, the traditional Luneburg lens and a modified Luneburg lens using a quasi-conformal transformational optics approach [10,11]. For all three lenses full wave simulations were performed using COMSOL with an open-ended waveguide feed placed at the center, middle, and edge of each of the lenses (see Fig. 5(d)). As shown in Fig. 5, the maximum achievable scan angle for the Luneburg and modified Luneburg lenses are 45° and 60° respectively compared to >80° for the MPMFE. In Fig. 6, we present the predicted scan loss as a function of scan angle for these three lenses along with an equivalently sized electronically scanned 2D phased array. The phased array was assumed to be a simple 18 × 18 dipole array with half-wavelength spacing between elements to avoid grating lobes at high scan angles. It should be noted that for a phased array the scan loss at wide angles is highly dependent on the specific antenna element pattern and is known to approximately follow a cosine dependence for most antenna elements. As demonstrated in Fig. 5 and Fig. 6, the MPMFE lens further extends the useable scanning range of typical dielectric beam-steering lenses. Given that the manufacturing constraints related to feature size and material permittivity were easily met, we experimentally verified our design approach using this example.

 

Fig. 5. Realized far-field gain comparison of 3D dielectric lenses. With a center feed (a), intermediary feed (b), and edge feed (c). Compared to the two Luneburg lens variants, the MPMFE lens exhibits wider steering capabilities with comparable side lobe levels and scan loss, (d) illustration of the Luneburg and modified Luneburg lenses, and feed locations used for comparison.

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Fig. 6. One-dimensional scan loss of recent dielectric lenses in literature, an equivalent electronically steered phased array, and the proposed MPMFE design.

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3. Experimental validation

3.1 Lens fabrication

While gradient index (GRIN) lenses, such as the Luneburg lens, have been known for over 50 years, the ability to fabricate them has been aided by recent advancements in additive manufacturing (AM). Before AM, fabricating a structure with spatially graded dielectric properties was an expensive and a more challenging manufacturing problem. Some modern non-AM approaches devise clever methods, such as fabrication through the casting and assembling of dielectric cylinders as performed in [18], to develop a flat Luneburg lens using a transformative optics approach. Another approach in [19] discusses a 3D metamaterial design composed of radially diverging, cast dielectric rods. While subtractive manufacturing methods have been successfully used to create gradient dielectrics through the perforation of high dielectric substrates for achieving effective gradient properties such as in [20,21], geometries with 3D varying permittivity require many of these substrate layers to be processed and stacked vertically, increasing fabrication time and complexity [5]. These alternative manufacturing methods each present their own advantages and disadvantages and should be taken into consideration when designing gradient index structures. Material characteristics, scalability, cost, and performance are among these factors that should be considered. The use of FFF processes to achieve the same goal is considered as a means of reducing production cost and time [22]. Using AM, the graded electromagnetic properties are achieved “effectively” by printing a subwavelength lattice structure where the effective dielectric properties are determined by the printed material's local geometry, permittivity, and volume fraction relative to the background material (usually air). Of the various AM technologies used to realize GRIN lenses, fused filament fabrication (FFF) is one of the most popular [5]. FFF is an additive manufacturing technique that involves the extrusion of a melted thermoplastic filament through a nozzle mounted on a moving mechanical system. Recent literature has emphasized the advantages of FFF for fabricating GRIN devices [23–26]; however, one drawback is the limited dielectric properties of commonly available thermoplastic filaments. Most standard thermoplastics such as PLA, ABS, and polycarbonate, have permittivity values less than 3.0 and thus are unsuitable for our MPMFE lens designs, thus necessitating alternative options, such as loaded polymers. To that end, our study used the commercially available Avient PREPERM, an ABS-based loaded filament incorporating custom additives with a relative permittivity of 5.4 [27].

In this work, we present a novel custom infill pattern developed to print the spatially varying lattice structure via FFF. Sharing the same principle as the space-filling structure described in [10], this polar-space filling curve (PSFC) takes advantage of the lens's spherical symmetry and radial varying gradient structure to realize the desired effective permittivity distribution, shown in Fig. 7(a). The unit cells of the PSFC, shown in Fig. 7(b), resembles a square-wave pattern adapted in polar coordinates and is mathematically described by the following expression.

In Eq. (4), R, A, and k represent the curve's radial offset, ring thickness, and angular periodicity, respectively. An advantage of the PSFC is that it exploits the lens's rotational symmetry by ensuring identical unit cells in a ring-like fashion and a permittivity gradient in the same radial direction as the MPMFE lens design. This pattern has the added benefits of improved layer-to-layer structural support due to its periodic and adaptive cell sizes promoting interlayer adhesion. Additionally, we found it to be a convenient toolpath requiring no extra movement or retraction for faster print times. By altering A and k, different volume fractions can be achieved by shrinking or increasing the periodicity of the pattern. Moreover, if A is constrained such that its arc length is equal to the radial length of a single cell, we can adequately ensure isotropy along the 2D plane of the curve.

 

Fig. 7. PSFC Unit Cell Render and Design. (a) Illustration depicting rotationally symmetric effective graded dielectric using polar space filling curves (PSFC), (b) PSFC unit cells.

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Through geometrical approximations, the ratio of local to host material, or the volume fraction, is given by

After fabrication, the discs were mounted and subjected to transmission measurements using a free-space-focused beam system. By analyzing the S₂₁ scattering parameter, we determined the effective dielectric permittivity through the use of an iterative inversion method. This inversion method assumes a relative permeability of 1 with an initial guess for the dielectric permittivity. By using the relations between the scattering parameter and the reflection and transmission coefficients in [28], this method iteratively solves for the correct inversion using Newton’s method [29,30].

The resulting calibration curve for the PSFC is shown in Fig. 8. To our knowledge, the PREPERM material is relatively non-dispersive from X-band, according to the manufacturer, to K_a band where we characterized it. Characterization beyond these frequencies was beyond the scope of this paper. It was found that a Maxwell-Garnett mixture formula [31] proved sufficient to model the effective behavior of the test samples and was subsequently used to relate the volume fraction to the relative permittivity at each radial location. Specifically, the expression given by Eq. (6), is the fitted Maxwell-Garnett formula for the PSFC where V_f is the volume fraction of the printed dielectric material.

 

Fig. 8. Material Characterization of Premix Dk5.4. Resulting permittivity curve following characterization with fitted Maxwell-Garnett curve.

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Fig. 9. Render and Manufacture of MPMFE Lens. (a) Rendered model of the sliced MPMFE lens and (b) photograph of the lens during printing.

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After a suitable calibration curve was determined, the 3D MPFME lens was subsequently sliced layer-by-layer using a custom slicer developed in MATLAB that integrated the lens's gradient profile with the PSFC geometry. Finally, the lens was manufactured using a 300 micron diameter nozzle at an extrusion temperature of 230°C and a bed temperature of 90°C over the course of three days. A model of the lens along with a photograph of its fabrication in process is displayed in Fig. 9.

3.2 Experimental characterization

To assess the lens's performance, we developed a measurement setup to scan the lens at various elevation angles for each feed location (Fig. 10(c)). To this end, we fabricated a custom mount capable of securely holding the lens (Fig. 10(a)). The mount was designed to be attached to a motorized rotary table, allowing sweeping motion from -90° to 90°. For calibration, we utilized a 24 dBi standard gain pyramidal horn as the reference. The lens was fed on the custom mount by fixing a WR-28 waveguide adapter at discrete positions along a line passing through the center of the lens (Fig. 10(b)). A K_a-band horn was positioned opposite the lens in the far field, serving as the receive antenna. This arrangement allowed us to sweep the lens across the entire K_a-band frequency range. At each discrete feed location, which ranged from 0 to 0.9 times the radius (R) away from the lens's center, we measured the antenna gain from 26 GHz to 40 GHz. While the antenna feeds positions were mechanically set for the purpose of experimentally characterizing the lens, it is important to note that in practice, a variety of approaches may be used. Among these potential approaches is the placement of a stationary array of low-gain antennas along the planar surface of the lens and activating them to steer in the desired direction, such as the direction-finding system described in [32]. Multiple antennas may even be excited at the same time to steer multiple beams. It is also important to note that this is distinct from phased array antenna systems which require more sophisticated and expensive RF components. An inexpensive, lightweight mechanical scanning system may still be used to scan the planar surface on the back of the lens as compared to traditional mechanical beam steering mechanisms which require gimbals to rotate the entire antenna system.

 

Fig. 10. Photographs of Mounted Lens and Measurement Setup Illustration. Experimental characterization system showing (a), front surface of the fabricated lens and custom mount (b), back side of the lens with the adjustable open-ended waveguide feed and (c) graphic illustrating the overall setup.

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As shown in Fig. 11 and 12, the measured results exhibit a favorable agreement with the simulated performance of the lens, demonstrating a wide range of beam-steering capability. At the highest steering angle, about 80°, the achieved maximum gain is 13 dBi, with a roll-off of less than 10.0 dB from its maximum value at boresight. However, upon comparing the simulated and experimental results, noticeable deviations from the expected values are observed as the steering angle increases, becoming more pronounced with steeper steering angles. These deviations can be mostly be attributed to the anisotropy of the layered filaments in the lens structure. While the adaptive pattern of the space-filling curve effectively controls the 2D isotropy along the curve, the actual cross-section structure of the 3D printed filament reveals slight anisotropy in the through direction of the unit cells. LaRocca [33] has previously discussed the impacts of this uniaxial anisotropy and noted that as the steering angles become steeper, this anisotropy becomes more significant and, consequently, affects the beam-steering ability of the lens. Notwithstanding the observed shift due to anisotropy, it is important to highlight that the lens achieved wide-angle steering with commendable gain. The measured insertion loss at various feed locations is shown in Fig. 13. As shown in the figure, the AR layer successfully impedance-matched the feed (|S11| < -10 dB) over the entire K_a-band at each feed location.

 

Fig. 11. K_a-band measurements. Measured realized gain over the K_a-band for discrete feed locations (a) 0R, (b) 0.2R, (c) 0.4R, (d) 0.6R, (e) 0.8R, and (f) 0.9R as shown in (g).

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Fig. 12. Simulated and Experimental Gain Comparison. Simulated and realized far-field measurements at 30 GHz.

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Fig. 13. Feed Dependent S11 Experimental Measurements. Measured S11 along feed distances from the center of the lens.

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4. Discussion and conclusion

In this work, we have successfully designed and fabricated a novel beam-steering gradient index lens based on a modified PMFE lens concept. With a remarkable scanning range of -80° to 80°, the lens demonstrates an impressive boresight gain of 23 dBi, accompanied by a gain roll-off of 10 dB. Our approach modified the original PMFE design described in [16] to include a wider range of permittivity values and an integrated AR layer for wideband impedance matching of a planar array of feed antennas. We successfully implemented the full 3D lens design by combining the FFF additive manufacturing approach with commercially available high dielectric filaments and novel graded infill patterns based on a polar space-filling curve. The fabricated lens was experimentally characterized within the K_a-band (26.5 GHz – 40 GHz) and successfully demonstrated a wide range of beam steering. While the lens was designed and experimentally fabricated for the K_a-band frequency range, this class of lens may be theoretically scaled to any desired frequency range using our design methodology. However, there are practical limitations for the fabrication of lenses at higher and lower frequencies. To produce a lens at lower frequencies of the same electrical length (10λ), the size of the lens must be scaled in all directions. At 3 GHz, this would require a lens diameter of roughly 40 inches. This may prove to be impractical with the use of FFF additive manufacturing. Conversely, at higher frequencies, the lens diameter shrinks. The limitation at higher frequencies comes from the ability to define subwavelength unit cells for realizing a permittivity range adequate to produce the lens. As the dimensions of the unit cells begin to approach the minimum feature size the FFF system can produce, this requirement also becomes impractical with this method. We are currently working on methods to improve the lens performance, including using multi-materials and investigating new lattice geometries to reduce the effect of anisotropy at high scan angles. While improvements to our design are warranted, the lens discussed in this work exhibits satisfactory performance and holds promise for advancing the development of low-cost, wide scan angle, and manufacturable passive beam-steering antenna systems.