This article presents a novel, lightweight, and full-polarized Luneburg lens (LL) antenna featuring a diamond lattice structure for the first time. While retaining the essential attributes of a full-scale LL, such as pencil-beam generation capability, high gain, wide bandwidth, and omnidirectional scanning, this LL leverages the isotropic nature of its diamond crystal structure to seamlessly adapt to various polarizations and a wide range of incident wave angles. Utilizing advanced three-dimensional-printing (3DP) technology, an intricately designed structure was fabricated. An elaborate wideband feeding system was also developed to serve as the primary feed for the lens. The proposed system supports typical polarization states and consists of a dual-polarized horn antenna integrated with a sextuple-polarized feeding network. The results indicate that the operation bandwidth of the proposed LL antenna covers the entire X-band (8-12 GHz) under any polarization scenario. Furthermore, the antenna exhibits consistent and stable gains, thereby simplifying the design complexity of signal processing systems. Compared with conventional LLs constructed with a cubic lattice structure, this innovative LL antenna boasts a reduced weight and diminished sidelobe levels. These advancements position the proposed design as a promising candidate for satellite communications, integrated sensing and communication, microwave imaging, and diverse future endeavors.
Multibeam antennas are indispensable devices across diverse domains, such as wireless communication, radar systems, and microwave imaging, and consistently surpass performance expectations [[1], [2], [3], [4]]. Such antennas are required to exhibit exceptional characteristics, including high gain, excellent isolation, negligible sidelobe, broad operational bandwidth, and versatile multi-polarization capabilities, to streamline signal processing systems and reduce costs. As an alternative to their active counterparts, passive multibeam antennas leveraging lenses for beamforming have sparked significant research interest, owing to their cost-effective manufacturing processes and lower energy consumption.
Typically, lenses are classified into homogeneous dielectric lenses and gradient-index (GRIN) lenses [5]. The Luneburg lens (LL), a GRIN lens based on geometrical optics, effectively converts incoming spherical waves into outgoing plane waves and vice versa, while maintaining a constant aperture across a wide range of incident angles. This lens offers a viable approach for achieving precise wide-angle beam control with low cross-polarization discriminations [6]. Nevertheless, the realization of an ideal LL is currently impeded by the absence of suitable materials, due to the requirement that the refractive index n must follow a radial distribution [7]:
$n(R)=\sqrt{2-{R}^{2}}$
where R is the normalized radius. Therefore, traditional LLs are crafted using a series of concentric shells that approximates the theoretical permittivity curve in a stepped manner. This manufacturing process poses significant challenges stemming from the stringent demands for shape accuracy and interlayer permittivity precision [[8], [9], [10]]. Although various two-dimensional (2D) designs—such as printed circuit board (PCB)-stacked LL antennas [[11], [12], [13]], columnar LL antennas [[14], [15], [16]], and parallel-plate waveguide/dielectric slab configurations [[17], [18], [19]]—reduce the overall profile and processing difficulty, they encounter limitations in achieving narrow beamwidths and wide scanning angles simultaneously across both the plane parallel to the direction of the electric field (E-plane) and the plane parallel to the direction of the magnetic field (H-plane). Therefore, wide-angle beam scanning in both the elevation and azimuth planes remains elusive [20]. Nevertheless, in complex scenarios demanding robust and high-capacity data transmission across various directions, such as in satellite communications, full-scale LLs have unmatched advantages [21]. Based on the effective medium theory (EMT), three-dimensional-printing (3DP) technology has facilitated the cost-effective implementation of GRIN materials by filling periodic unit cells with precalculated volumes of materials. This advancement has enabled the production of LLs with expansive operational bandwidths and simplified processing methods.
In communication systems, isotropic antennas—which exhibit radiation characteristics independent of excitation polarization and direction—can achieve wide-angle steering while simplifying signal processing systems [22]. A variety of unit structures have been employed in three-dimensional (3D)-printed full-scale LL designs, including dielectric holes [5,23,24], cubic lattices [25,26], dielectric slabs [27], rod configurations [[28], [29], [30]], and curved elements [31]. The asymmetry present in dielectric holes [23,24], dielectric slabs [27], and rod configurations [28,29] across different directions results in an effective permittivity that varies with the angle of incidence or polarization states, making these LLs unsuitable for the simultaneous provision of a stable linearly polarized and circularly polarized performance [28]. Furthermore, the bandwidth and radiation characteristics of the feed directly influence the performance of LL antennas. To the best of our knowledge, no report has validated the concurrent performance of linearly polarized and circularly polarized radiation. Therefore, the design of a full-polarized LL antenna holds significant importance.
Compact lightweight antennas are the primary goal of modern wireless communication and sensing systems. Table 1 summarizes the spatial occupancy rate of common crystal structures [32]. While a 3D-printed LL with a cubic lattice structure exhibits satisfactory isotropic performance [26], its simple cubic packing structure results in a spatial occupancy rate of 52%, imparting substantial weight to the LL. Conversely, in the unit cell of a diamond crystal structure, each atom is surrounded by four atoms, forming a 3D spatial structure through tetrahedral covalent bonding. This arrangement not only offers excellent isotropic performance but also boasts a space packing efficiency of just 34%, conferring substantial weight-reduction benefits.
This article introduces for the first time a novel diamond-lattice-structure-inspired lightweight LL that incorporates full polarizations—that is, horizontal linear polarization (H-LP), vertical linear polarization (V-LP), ±45° linear polarization (LP), left-hand circular polarization (LHCP), right-hand circular polarization (RHCP), and so forth. The proposed design retains numerous advantages of traditional full-scale LLs, including the ability to generate a pencil beam, high gain, wide bandwidth coverage, and omnidirectional scanning capabilities. By leveraging the isotropy of its diamond crystal structure, the proposed LL can accommodate variations in arbitrary polarization states and incident wave angles. A wideband feeding system with six general polarizations is developed as the primary feed of the LL, consisting of a dual-polarized horn antenna and a sextuple-polarized feeding network. The experimental results demonstrate that, regardless of the polarization scenario, the operational bandwidth of the proposed antenna covers the entire X-band while maintaining consistent and smooth gains. This simplifies the design complexity of signal processing systems. Furthermore, the proposed LL showcases lower sidelobe levels, reduced weight, and simplified fabrication compared with conventional LLs constructed with a cubic lattice structure. As shown in Fig. 1, thanks to its exceptional characteristics of lightness, high gain, flexible beam steering, and polarization-switching capabilities, the proposed design holds significant potential for a wide array of advanced applications, including satellite communications, radar sensing, wireless power transfer, and microwave imaging. The integrated single-lens system with multiple feed sources achieves functional isolation through spatial separation and polarization orthogonality to prevent mutual interference. These distinctive features facilitate superior signal transmission efficiency, accurate target detection, effective energy delivery, and high-resolution imaging, thereby establishing the system as a highly versatile and robust solution for diverse applications in contemporary electromagnetic systems. Moreover, this work introduces a crystallography-inspired design strategy that facilitates the comprehensive and integrated optimization of electromagnetic components. By concurrently considering structural, material, and wave-related properties, this approach drives forward the progress of wireless communication technology and strengthens the connection between the realms of materials science and electromagnetic engineering.
The remainder of this article proceeds as follows: Section 2 presents a detailed description of the proposed LL, encompassing its operating mechanism and design methodology. Section 3 focuses on the design and analysis of the sextuple-polarized feeding system. The simulated and measured results of the proposed LL antenna are detailed in Section 4. Section 5 provides a discussion, while a concise conclusion is presented in Section 6.
2. Realization of the LL
2.1. Realization of varying permittivity
Given the varying relative permittivity demands for a spherical LL ranging from 2 to 1, the material of choice was the photopolymer VeroClear, which boasts a permittivity of 2.9 and a loss tangent of 0.01 [15]. The model was simulated using the periodic boundaries in a full-wave electromagnetic simulator (High Frequency Structure Simulator, Ansys, USA). Due to constraints in computer memory, a prismatic configuration was favored over a curved surface structure to reduce the complexities of mesh generation. Fig. 2 illustrates the unit cell of the diamond crystal structure, which comprises four hexagonal prisms characterized by angles of arccos(-1/3) and a cube with variable side lengths. Limited by the minimum accuracy of 3DP technology, the basal side length of the hexagonal prism lm was specified at 0.75 mm to meet the minimal effective permittivity for the LL design. Subsequently, the effective permittivity of the unit cell was extracted using the S-parameter retrieval method [33], and the effective refractive index ne of the unit cell was modeled and computed by means of Eq. (2):
where |k| is the wavenumber of the electromagnetic wave; a is the side length of the unit cell, which is 7.5 mm (less than half of the free-space wavelength) to mitigate any potential phase ambiguity [34]. Therefore, the equivalent permittivity εe = ne2 of the unit structure can be determined. The effective permittivity of the unit cell with different polymer cube sizes wc, depicted in Fig. 2, harmonizes perfectly with the requirements of the lens design.
To demonstrate the isotropy of the proposed crystal lattice structure, the unit cell is rotated around the x-, y-, and z-axis to explore the variation of the effective permittivity at different incident angles. Without loss of generality, Fig. 3 illustrates that the effective permittivity remains relatively constant within the range of 8-16 GHz for the cell configurations at rotation angles (α = 0°, γ = 0°), (β = 30°, γ = 0°), and (α = 0°, β = 60°), where α, β, and γ refer to the rotation angles of the cell around the x-, y-, and z-axis, respectively. A comparison between our diamond lattice structure and various classical structures—including post [28], cubic lattice [26], and body-centered cubic (BCC) lattice [35]—along with results is illustrated in Table 2 and Fig. 4, revealing that our lens exhibits more stable equivalent permittivity across different incidence angles and a lower variance in permittivity than its counterparts.
2.2. Discretization of the lens
Compared with common LLs achieved through unit cell translation duplication, the proposed diamond-lattice-structure-inspired LL necessitates replication across the entire crystal cell, as shown in Fig. 5(a). This is because the crystal cell serves as the smallest repeating unit that encapsulates all symmetries of the crystal structure. Given the symmetry properties of the proposed LL, it is sufficient to consider merely one eighth of a sphere in order to attain the complete spherical configuration.
Moreover, the enhanced level of discretization of the LL sphere, characterized by an increasing number of stepwise gradient square units, results in a closer approximation to the ideal LL. However, as the quantity of discretized small units grows, there is a corresponding increase in the size of the LL sphere due to the fixed dimensions of each unit. As a result, the performance improvements gained from this increased discretization begin to diminish in returns. To strike a balance between the performance and size of the lens, as depicted in Fig. 5(b), the cell is translated fourfold along the x-, y-, and z-axis to configure one eighth of the overall structure, which is subsequently divided into ten layers, as shown in Fig. 5(c). By calculating the distance from the unit cell to the lens center, the side length of the cube—corresponding to the equivalent permittivity εe—is determined, as outlined in Table 3. To ensure the electromagnetic waves gather from all incident directions, the one-eighth structure is triplicated through mirroring, resulting in a spherical lens with a diameter of 147.8 mm, as depicted in Fig. 5(d).
3. Design of the feeding system
To ascertain the electromagnetic wave-gathering capability of the LL across multiple polarizations within the X-band at a lower cost, without loss of generality, Fig. 6(a) presents a schematic diagram of the wideband sextuple-polarized feeding system. This setup consists of a wideband dual-polarized horn antenna and a wideband sextuple-polarized feeding network. The feeding network encompasses various components, including a Wilkinson power divider (WPD), a 90° wideband phase shifter, and a 180° wideband phase shifter. Through strategic channel manipulation within the network, six distinct polarizations are attained for the horn antenna, as shown in Fig. 6(b). A comprehensive analysis of the design principles and detailed measured results is provided below.
3.1. Horn antenna
A dual-polarized quadruple-ridged horn antenna is designed to serve as the feeding source of the proposed LL, providing wideband operation and multiple polarized incident waves. As depicted in Fig. 6(c), the horn antenna can be segmented into three parts: a quadruple-ridged waveguide, the horn with linear-graded quadruple-ridges, and a coaxial-to-waveguide structure. The overall length of the horn is set to 73.7 mm, with an aperture size width of 46 mm. Acknowledging the constraints imposed by the computer numerical control (CNC) milling machine on intricate structures [36], the designed horn antenna was fabricated using aluminum alloy 3DP technology and covers an operating frequency range of 7.8-13.8 GHz (|S11| < −10 dB).
3.2. Feeding network
Compared with traditional T-junction power dividers, the WPD has a distinct advantage in terms of impedance matching and high isolation at the output ports, which can reduce the crosstalk between output signals [37]. Based on the theory that the bandwidth of a power divider can be enhanced by employing multi-section transmission lines [38], a two-section WPD was detailed on a Rogers 4350 substrate, with a relative permittivity εr of 2.2, a loss tangent tanδ of 0.0009, and a thickness ht of 0.76 mm.
A basic phase shifter is composed of two sections of transmission lines with different lengths—a configuration that only guarantees the necessary phase difference within a limited frequency range [39]. As shown in Fig. 6(d), the circuit configuration of the proposed 90° phase shifter was established by loading shorted stubs to expand the operational bandwidths, which can be analyzed using the ABCD matrix approach, as follows.
The electrical lengths θx of the shorted stubs S1 and S2, transmission lines LA and LB, and reference line at operating frequency f, can be defined as follows:
${\theta }_{x}={\theta }_{x0}\frac{f}{{f}_{0}}$
where θx0 is the corresponding electrical length at center frequency f0.
The ABCD matrices of the transmission lines (Mt) and shorted lines (Ms) can be written as follows [40]:
where j is the imaginary part; Zt,Yt, and θt are the characteristic impedance, characteristic admittance, and electrical length of the transmission lines and reference line, respectively; Ys and θs are the characteristic admittance and electrical length of the shorted stubs, respectively. Therefore, the ABCD matrices of the main line and reference line shown in Fig. 6(d) can be deduced as follows:
where Mm and Mr are the ABCD matrices of the main line and reference line, respectively; Ma and Mb are the ABCD matrices of the transmission lines LA and LB, respectively; Ms1 and Ms2 are the ABCD matrices of the shorted stubs S1 and S2, respectively. To streamline the design and reduce complexity, the impedance Zb is set to characteristic impedance Z0 = 50 Ω. Am, Bm, Cm, and Dm are the voltage transfer ratio, transfer impedance, transfer admittance, and current rransfer ratio of the main line, respectively; Ar, Br, Cr, and Dr are the voltage transfer ratio, transfer impedance, transfer admittance, and current rransfer ratio of the reference line, respectively.
The insertion phase shift of the main line (Φm) and the reference line (Φr) can be derived as follows [39]:
The differential phase shift is attributed to the transmission phase disparity between the main line and the reference line. To keep the phase shift value Δϕ = 90° in the passband, the slopes of the phase shift need to be equal to zero; that is,
Based on the provided design analysis, an iterative method was implemented in MATLAB (The MathWorks Inc., USA). The substrate chosen for this device was consistent with a WPD, as shown in Fig. 6(e). Design flexibility and size reduction can be achieved using curved reference lines. By connecting two 90° phase shifters (Fig. 6(f)) in series and appropriately adjusting the size of the lines, a 180° phase shifter was also developed, as depicted in Fig. 6(g). Further optimizations with a WPD were performed, and a sextuple-polarized feeding network was designed, as shown in Fig. 6(h); it was then fabricated through standard PCB processing. Employing a vector network analyzer (Keysight N5225B, Keysight Technologies, USA), the amplitude and phase responses of the fabricated sextuple-polarized feeding network were measured and are delineated in Fig. 7, exhibiting a notable concordance with the simulated results. It can be observed that the measured reflection coefficient is below −12 dB within the required frequency range of 8-12 GHz. The phase difference is equal to a specific phase shift with a ripple of less than ±6°, while the amplitude difference is less than 1 dB. These results provide the basis for the demonstration of a full-polarized LL antenna.
4. Simulated and experimental results
The dual-polarized horn antenna, introduced in Section 3, serves as the primary feeding source for the LL devised in Section 2. Fig. 8 illustrates the normalized radiation patterns and electric field distributions, both with and without the lens. The transformation of the spherical waves emitted by the horn feed into plane waves after passing through the lens, as observed in both the E-plane and H-plane, validates the effectiveness of the proposed design methodology inspired by a diamond-lattice structure for the LL.
The LL antenna and its corresponding support were fabricated using 3DP technology. In order to meet the manufacturing requirements, the diameter of the lens was adjusted to 138 mm—a modification that was verified to have minimal impact on performance. Fig. 9(a) presents the simulated and measured S-parameters for the two ports of the lens antenna. The results indicate a significant bandwidth of up to 46.3% (7.8-13.8 GHz), with reflection coefficients consistently below −10 dB. Furthermore, the isolation between the two ports exceeds 29 dB.
The feeding network introduced in Section 3 was utilized to produce distinct excitations for the dual-polarized horn, enabling the achievement of feeding sources with different polarizations. The radiation performance of the feeding source in different radiation states was tested in an anechoic far-field chamber, as shown in Fig. 9(b). The simulated and measured normalized radiation patterns of the entire LL antenna for six polarizations at three frequencies (i.e., 8, 10, and 12 GHz) are plotted in Fig. 10. The simulated and measured gains for the six polarizations of the proposed LL antenna, along with axial ratios (ARs) for circular polarization (CP), are shown in Fig. 11(a). It should be noted that the insertion loss induced by the coaxial cable adapter and power divider network was meticulously calibrated. For the LP, the measured gains fluctuate between 16.7 and 20.3 dB across the operating band, ranging from 7.8 to 12.7 GHz. In the case of the CP, the slight deficiencies in the fabrication process of the sextuple polarization feeding network and the horn antenna, as well as measurement errors such as misalignment, and so forth, lead to the degradation of the AR within the 8-12 GHz spectrum.
Upon a comprehensive review, it was found that various factors—such as fabrication errors, limitations in the dynamic range of the testing equipment, excitation imbalances, and environmental influences—had collectively contributed to a slight degradation in the obtained results. Nevertheless, the measured results convincingly demonstrate the insensitivity of the gain to polarization variations, thereby confirming the isotropy of the proposed LL antenna design. Fig. 11(b) presents the simulated and measured normalized scanning patterns of the proposed lens antenna at 10 GHz for various polarizations (i.e., H-LP, V-LP, LHCP, and RHCP) in the xoz-plane. By rotating the horn antenna to arbitrary angles, the innovative LL antenna can seamlessly achieve a scanning range of 360° in the azimuth plane and ±90° in the elevation plane.
5. Discussion
In order to showcase the merits of the proposed lightweight LL antenna, we have devised a comparative study with a conventional LL antenna constructed using cubic lattices with an identical radius, as shown in Figs. 12(a) and (b), which was demonstrated to exhibit near isotropy in Ref. [26]. The design and placement of the feeding horn antenna are consistent with our previous settings. Fig. 12(c) compares the normalized radiation patterns in different polarizations (i.e., H-LP and V-LP) between the two LLs. The normalized radiation patterns of both the traditional LL and our design at 10 GHz exhibit remarkable similarities in terms of cross-polarization discriminations. However, the simulated sidelobe level of our design is more than −14 dB in both the E-plane and H-plane, whereas that of the traditional conventional LL only reaches −10.2 dB. This phenomenon stems from the fact that the spatially varying permittivity distributions in different lens structures create distinct electromagnetic wave propagation paths through differential refraction and reflection, leading to characteristic phase/amplitude modulations that produce different far-field interference patterns and radiation characteristics. The commercial software SolidWorks was employed to analyze the weight, and the results revealed that the proposed lens weighs only 55.8% of the mass of the conventional LL—clearly demonstrating the outstanding weight-saving performance achieved through the utilization of a diamond lattice structure.
Table 4 [5,11,12,14,16,18,19,22,[25], [26], [27], [28],30,35,[41], [42], [16], [43]] showcases a performance comparison between the proposed antenna design and several previously reported representative LL antennas. When compared with spherical LLs [5,22,[25], [26], [27], [28],30,35], cylindrical configurations [14,16], parallel-plate waveguide/dielectric slab configurations [18,19,42,43], PCB-based implementations [11,12], and especially specialized configurations such as the semi-compressed elliptical LL in Ref. [41] significantly undermine the 2D multibeam capability, even though they offer reduced profiles. Hence, full-scale lenses have irreplaceable advantages. In the design of spherical lenses, a diverse range of unit structures have been adopted. However, some of these structures are based on asymmetric cell designs, such as slabs [27], rings [28], and posts [26,30], which prevents them from achieving full-angle scanning in both the elevation and azimuth planes. As a result, some of these structures have been repurposed for polarization conversion applications [27,30]. An innovative radial conical hole-type lens introduced in Ref. [5] is capable of collecting electromagnetic waves across various incidence angles and polarizations. Nevertheless, it operates within a notably limited bandwidth. Similarly, the lenses presented in Ref. [22] have a relatively large physical size, extending up to 5.12λ0 (where λ0 is defined as the wavelength in free space at central frequency). The lens design in Ref. [35], which is based on a BCC lattice configuration, requires the insertion of conductive cubes into foam support structures, leading to significant manufacturing complexity. In comparison, our design not only retains the advantages of conventional LL antennas, such as a wide bandwidth, high gain, and the capability for multibeam radiation, but also achieves stable full-polarized radiation while featuring a remarkably lightweight, environmentally friendly and fabrication-simplified architecture. In addition, this work presents an innovative design methodology that harnesses crystallographic principles to design and optimize electromagnetic components. Design goals can be achieved through a multidimensional strategy that seamlessly integrates considerations of structural design, material properties, and wave manipulation. This approach provides invaluable insights for propelling the development of wireless communication technologies and serves as a catalyst for fostering the synergistic connection between the fields of materials science and electromagnetic engineering.
6. Conclusions
This article demonstrates a lightweight, fabrication-simplified, and full-polarized LL antenna that boasts a wide bandwidth, high gain, and lower sidelobe levels. The diamond lattice structure introduced in the LL design increases the robustness of the antenna against incident angles while simultaneously reducing its overall weight. By leveraging 3DP technology, a sophisticated structure was brought to fruition. The integration of a meticulously designed dual-polarized horn antenna, in conjunction with a sextuple-polarized broadband feeding network as the source, validates the ability of the LL to exhibit consistent isotropic radiation potential characteristics across the entire X-band. These distinctive features not only position the proposed LL antenna as a promising technology for integrated communication, sensing, and transmission applications but also provide valuable insights for the ongoing convergence of materials science and electromagnetic engineering.
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