Introduction
The area of passive retroreflective devices for mm-wave (30-300 GHz) and THz wave (0.3-10 THz) communication, objects tracking and indoor localization and identification has recently received increased attention [1]–[14]. Retroreflectors are devices that reflect an incoming electromagnetic (EM) wave into the direction of its arrival and serve for radar cross-section (RCS) enhancement, thus increasing detectability of the specified objects, e. g. drones, satellites, or airplanes. Retroreflectors based on different technological realizations in available literature are presented. For example, Van Atta arrays are formed by planar patches [1] or substrate integrated waveguides [2]. Moreover, frequency-coded corner reflectors are achieved by employing frequency selective surfaces [3], [4] or dielectric resonator arrays [5]–[7]. A large number of retroreflective structures are based on lenses by incorporating a reflective layer, such as frequency-coded fused silica spherical lenses [8], [9], or lenses backed by a photonic crystal (PhC)-based structures [10], such as a polyethylene lens with a Bragg grating [11], and planar [12] or spherical [13] gradient-index Luneburg lens backed by planar PhCs. Finally, the Luneburg lens can also be employed to achieve an omnidirectional retroreflector when it is surrounded by slant polarizers [14]. Most of the aforementioned references use frequency coding, enabling their employment in different areas. For instance, in chipless indoor self-localization systems [3], the ability to distinguish between retroreflectors inside the building allows for a precise position calculation, which is achieved by incorporating different and distinguishable frequency-coded signatures to the reflected wave of each retroreflector [3]–[13]. The advances in those researches bring unconventional retroreflectors based on metasurfaces [15]–[18], or transformation optics (TO) principles [19]–[23]. A summary of different retroreflectors and their operational angular range is presented in Table 1. However, the metasurfaces and TO-based retroreflectors have been proposed and realized only up to centimeter frequency bands (0.3-30 GHz) or in optics, outside the mm-wave region.
The use of retroreflectors at mm-wave frequencies allows for wider absolute bandwidths, i.e., a better ranging accuracy and smaller devices, as well as higher antenna gains [6]. More details can be found in our previous work [13], where a combination of a spherical 3D Luneburg lens with 2D PhC coding particles is introduced at 80 GHz. Further, in [24], we addressed the current limitations of a lens-based system due to the limited angular response and overall reduced compactness of the coding particles, i.e., having to adapt the employed planar coding particles (2D PhC, resonating element arrays) to the spherical surface of a lens introduces undesired effects. For example, in [4] the curvature of the employed cross-dipole frequency surface decreases frequency selectivity. In addition, in 2D PhC-based coding [12], the curvature of the lens introduces structural distortions that results in a larger required angular separation between coding particles. To answer these challenges, we proposed a theoretical concept of a 3D flattened quasi-conformal transformation optics (QCTO) Luneburg lens backed by a 3D PhC coding structure [24], which showed a large potential for developing integrated wide-angle passive frequency-coded retroreflective devices since the flat bottom of the QCTO lens allows for easier integration of different coding particles. However, the required high permittivity of the lens (
This paper presents the design and manufacturing of a ceramic wide-angle 3D QCTO Luneburg lens working in the Ka-band, as the first step towards frequency-coded monolithic ceramic-based tags for indoor localization applications. The design incorporates an impedance matching layer (IML) to the air as well. The high permittivity of alumina allows wide coverage angles while ensuring very stable operation in harsh environments (high-temperature, ionizing, chemically polluted) [3]. To the knowledge of the authors, this is the first time that an alumina-based QCTO Luneburg lens antenna with wide 3D operating angular range is presented, and its performance as a retroreflector is evaluated. The paper is organized as follows: Section II presents a study on QCTO Luneburg lenses; their maximum required permittivity and angular range. Section III deals with the manufacturing restrictions, their effects on the lens performance and fabrication process realized by ceramic 3D printing, and the measurement results. In Section IV, the achieved results and future improvements are discussed.
Quasi-Conformal Transformation Optics Enabled Luneburg Lens
The transformation optics theory has gained popularity in the last two decades for providing a systematic approach to the design of invisible cloaks [27]–[31], conformal antenna arrays [32], [33], directivity enhancers [34], [35], and waveguide bends and couplers [36]–[38]. TO has also been employed to modify the geometry of various dielectric lenses [39]–[53].
The transformation optics establishes a relation between the fields and the material between two spaces: the virtual space where the wave propagation properties are known, and the physical space. A spatial deformation is applied to the virtual space to attain a desired behavior for the electromagnetic waves. Using the transformation optics recipe, one can calculate a transformation medium for the physical space that mimics the desired space transformation. The transformation medium is quite complex if the underlying transformations are general. However, employing conformal and quasi-conformal transformations simplifies the derived material and leads to an all-dielectric, isotropic solution [42], [43], [51].
The QCTO has been used to modify the geometry of the Luneburg lens [54], by flattening a portion of the lens contour [40], [44], [50]–[53]. This creates a planar surface suitable for placing a 3D PhC medium with an array of embedded high-Q resonators for frequency coding as proposed in [24]. However, by applying QCTO to the Luneburg lens, the required maximum relative permittivity values increases drastically compared to the conventional Luneburg lens [40], [44], [50]–[53]. This drawback results in a significant impedance mismatch at the boundaries between the modified lens and the surrounding medium (usually free space). Furthermore, this mismatch is amplified if a larger portion of the lens contour is flattened. The solution for this defect was proposed by Biswas et al. in [52] where an anti-reflective (AR) layer was added to the flattened area of the QCTO lens resulting in an improved antenna impedance matching and an enhancement of the antenna gain in an angular range of ± 55° with a maximum required relative permittivity of 2.9. The work [52] also reported that by increasing the AR layer thickness, one could improve the impedance matching at the expense of decreasing the maximum antenna gain for very broad scan angles. Therefore, it is necessary to pay more attention to the AR layer design to attain broad scan angles over ± 70°, as shown in [24], [40], [53]. This scenario is further investigated in section C.
The design process of the flattened QCTO Luneburg lens can be divided into 5 steps:
Quasi-conformal mapping of the original Luneburg lens permittivity profile in the given virtual space (Eq. 1) onto the permittivity profile in the physical space (Eqs. 2–3).
Limiting the resulting permittivity profile to the maximum and minimum values achievable by a ceramic 3D printing process using the alumina ceramics for the selected building unit cell (cross).
Creating the discretized 3D model (solid cubes in Fig. 3b) by using customized MATLAB code for the full-wave simulations in CST Studio Suite.
Assigning the targeted 3D permittivity distribution of the QCTO lens to the closest available permittivity values for the selected unit cell (Fig. 11).
Replacing the solid cubes by the crosses with corresponding dimensions for creation of the suitable 3D printing fabrication model (Fig. 13).
The geometry of the virtual space containing the Luneburg lens with a diameter of 60 mm, centered at the origin. The relative permittivity profile of Eq. 1 is presented. The white lines are the vertical
(a) Rectangular physical space and its permittivity profile. The white lines are the images of the
(a) Discretized QCTO Luneburg lens in MATLAB, and (b) its geometry in CST Studio Suite. The unit cell size is 0.7 mm
Relation between the maximum required relative permittivity of the transformed Luneburg lens and the half beam steering angle. The dotted curves correspond to an internal half-angle
Simulated H-plane radiation pattern of the QCTO Luneburg lens excited by a WR-28 waveguide. Position at 0 mm corresponds to the lens center and position at 15 mm corresponds to the lens edge.
Simulated reflection coefficient of the QCTO Luneburg lens excited by a WR-28 waveguide.
2D discretized impedance matching profiles: (a) Klopfenstein, (b) exponential, (c) Gaussian.
Simulated radiation patterns of the QCTO Luneburg lens with an exponential, a Klopfenstein, and a Gaussian IML at a frequency of 40 GHz.
Simulated reflection coefficients of the QCTO Luneburg lens with an exponential, a Klopfenstein, and a Gaussian IML.
Unit cell types used in the ceramic 3D printing and their minimum inner structure’s dimensions: squared hole block (not in scale), cross, cube with rods (from left to right), (a) top view, (b) 3D view. The unit cell size is 0.7 mm
Effective relative permittivity of the selected unit cells and the corresponding eigenfrequencies for which the permittivity value is valid. The green area shows the 3D printable region.
Absolute difference between the ideal and achievable effective relative permittivity of the QCTO Luneburg lens (fabricated version) in 3D version composed of the cross unit cells.
Fabrication model of the QCTO Luneburg lens composed of cross unit cells is shown in red color and the exponential IML is shown in blue color. The structure is cut in the middle.
A. Quasi-Conformal Mapping of the Luneburg Lens
We assume that the Cartesian coordinates that define the virtual and physical spaces are (\begin{align*} \varepsilon _{\textrm {r}} =\begin{cases} \displaystyle 2-\frac {u^{2}+v^{2}}{R^{2}}&\textrm {for} \left |{ {u^{2}+v^{2}} }\right |\le R^{2} \\ \displaystyle 1&\textrm {elsewhere}, \\ \displaystyle \end{cases}\tag{1}\end{align*}
The physical space is a rectangle with side lengths equal to AB and AG. The curved bottom boundary of the Luneburg lens CDE is flattened by the transformation. The quasi-conformal transformation between the virtual and physical spaces is calculated by solving the Laplace’s equation in the virtual space for the variables
The following equations (2) represent the underlying differential equations and the Dirichlet and Neumann boundary conditions involved in the process.\begin{align*} \begin{cases} \displaystyle \nabla ^{2}x=0 \\[4pt] \displaystyle \left.{ x }\right |_{\textrm {AB}} =-60 \\[4pt] \displaystyle \left.{ x }\right |_{\textrm {FG}} =60 \\[4pt] \displaystyle \left.{ {\partial x \mathord {\left /{ {\vphantom {\partial x {\partial N}}} }\right. } {\partial N}} }\right |_{\textrm {AG,BC,CD,DE,EF}} =0, \\[4pt] \displaystyle \end{cases}\quad \begin{cases} \displaystyle \nabla ^{2}y=0 \\[4pt] \displaystyle \left.{ y }\right |_{\textrm {AG}} =60 \\[4pt] \displaystyle \left.{ y }\right |_{\textrm {BC,CD,DE,EF}} =0 \\[4pt] \displaystyle \left.{ {\partial y \mathord {\left /{ {\vphantom {\partial y {\partial N}}} }\right. } {\partial N}} }\right |_{\textrm {AB,FG}} =0, \\[4pt] \displaystyle \end{cases}\!\!\!\!\!\!\!\!\!\! \\[4pt]\tag{2}\end{align*}
\begin{equation*} {\varepsilon }'_{\textrm {r}} =\frac {\varepsilon _{\textrm {r}}}{x_{u} y_{v} -x_{v} y_{u}},\tag{3}\end{equation*}
The permittivity profile of the physical space and the corresponding mapped
The original untransformed spherical Luneburg lens has a diameter of 60 mm, while the transformed flattened Luneburg lens has a bottom diameter of 34 mm and the center diameter of 52 mm while the minimum relative permittivity is limited to the value of 1. The height of the lens is then limited by the height of the physical space (60 mm) with the minimum relative permittivity of 1.11. The relative permittivity values below 1 are omitted because we focus on the implementation involving only dielectric materials. In addition, due to minimum achievable effective relative permittivity of the cross unit cell used for the alumina ceramic 3D printing fabrication process, which is about
The calculated maximum and minimum relative permittivity of the QCTO Luneburg lens is then 8.89 and 1.25, respectively. The relation between the maximum relative permittivity of the transformed lens and the half beam steering angle
B. QCTO Luneburg Lens Performance
To evaluate the designed QCTO Luneburg lens, it is assumed that the excitation with the WR-28 waveguide is placed on the lens bottom surface. The numerical simulations in CST Studio Suite were performed in the Ka-band (26.5 GHz to 40 GHz) while the waveguide was shifted from the lens center to the lens edge in 2 mm steps to obtain the radiation patterns in the H-plane (azimuth) (Fig. 5) and the reflection coefficient responses (Fig. 6). The lens antenna realized gain achieves 16.12 dBi at the lens center and 17.31 dBi at the lens edge, steering the beam in the range of ±75°. The maximum reflection coefficient varies between −3.57 dB and −9.06 dB while shifting the feed position. In the next section, we analyze several impedance transformer profiles to improve the antenna impedance matching.
C. Impedance Matching Layer
The effect of adding IML as an antireflective layer on the lens bottom surface was investigated in [52] with the conclusion that the Klopfenstein impedance matching profile with \begin{align*} \varepsilon _{\textrm {AR}\_{}\textrm {Klop}}=&\left ({{\sqrt {\varepsilon _{\textrm {i}} \varepsilon _{\textrm {l}} (x,y)} \exp \left [{ {\Gamma _{\textrm {m}} A^{2}\phi \left ({{2\frac {y}{\textrm {L}}-1,A} }\right)} }\right]} }\right)^{2}, \\ \tag{4}\\ \Gamma _{\textrm {m}}=&\frac {\Gamma _{0}}{\cosh A};\quad \Gamma _{0} =\frac {1}{2}\ln \left ({{\frac {\sqrt {\varepsilon _{\textrm {l}}}}{\sqrt {\varepsilon _{\textrm {i}}}}} }\right), \tag{5}\\ \phi \left ({{x,A} }\right)=&\frac {1}{2}\int \limits _{0}^{x} {\sum \limits _{m=0}^\infty {\frac {\left ({{\frac {A^{2}}{4}} }\right)^{m}\left ({{1-y^{2}} }\right)^{m}}{m!\left ({{m+1} }\right)!}}} dy, \tag{6}\\ \varepsilon _{\textrm {AR}\_{}\textrm {exp}}=&\left ({{\sqrt {\varepsilon _{\textrm {i}}} \exp \left [{ {\frac {y}{\textrm {L}}\ln \left ({{\frac {\sqrt {\varepsilon _{\textrm {l}} \left ({{x,y} }\right)}}{\sqrt {\varepsilon _{\textrm {i}}}}} }\right)} }\right]} }\right)^{2}, \tag{7}\\ \varepsilon _{\textrm {AR}\_{}\textrm {Gauss}}=&\left ({{\sqrt {\varepsilon _{\textrm {i}} } \exp \left [{ {2\left ({{\frac {y}{\textrm {L}}} }\right)^{2}\ln \left ({{\frac {\sqrt {\varepsilon _{\textrm {l}} \left ({{x,y} }\right)}}{\sqrt {\varepsilon _{\textrm {i}}}}} }\right)} }\right]} }\right)^{2} \\&\quad \textrm {for } 0\le y\;\le \frac {\textrm {L}}{2} \tag{8}\\ \varepsilon _{\textrm {AR}\_{}\textrm {Gauss}}=&\left ({{\sqrt {\varepsilon _{\textrm {i}}} \textrm {exp}\left \{{{\begin{array}{l} 4\left ({{\frac {\textrm {y}}{\textrm {L}}} }\right)-2\left [{ {\left ({{\frac {\textrm {y}}{\textrm {L}}} }\right)^{2}-1} }\right]\times \\ \textrm {ln}\left ({{\frac {\sqrt {\varepsilon _{\textrm {l}} \left ({{\textrm {x,y}} }\right)}}{\sqrt {\varepsilon _{\textrm {i}}}}} }\right) \\ \end{array}} }\right \}} }\right)^{2}, \\&\quad \textrm {for } \frac {\textrm {L}}{2}\le y\;\le \textrm {L}\tag{9}\end{align*}
The discretized impedance matching profiles were implemented by MATLAB codes and are shown in Fig. 7. In our case, the thickness of the broadband IML is chosen to be
From the comparison of the broadband IML, the exponential profile provides the largest improvement of the antenna gain while maintaining the maximum beam steering angle of ±75° and the reflection coefficient better than −11.7 dB in the whole Ka-band. Further, the Klopfenstein profile provides a slightly better impedance matching with a reflection coefficient better than −15.4 dB. However, the antenna gain is 0.48 dB less than for the exponential profile at the lens center. In addition, compared to the exponential profile, the Gaussian profile gives 0.95 dB lower antenna gain, and for lens without the IML, the gain improvement of 2.7 dB is achieved at the lens center. The comparison of selected IML and corresponding lens antenna parameters at a frequency of 40 GHz is summarized in Table 2. The higher antenna gain at the lens positions (8 mm; 15 mm) for the case without the IML is caused by a properly located focal point of an excitation waveguide. While for the case of the IML, the focal point is slightly shifted, which could be partially mitigated by creating a multi-sectional IML of different thicknesses as proposed in [57]. Due to those facts, the exponential IML is chosen for manufacturing.
Fabrication and Measurements
A. Lens Models Involving Fabrication Limits
The designed flattened QCTO Luneburg lens must be discretized into unit cells which can be fabricated by the ceramic 3D printing process applied for this study [25], [26]. These unit cell types should be ideally isotropic and easily manufacturable, such as crosses [59] and cubes with rods [60], [61] suitable for 3D objects, or square hole blocks [62] and cylindrical hole blocks [63] applicable for 2D objects.
In our design, we first evaluated the effective relative permittivity of the following unit cell types: crosses, cubes with rods and square hole blocks depicted in Fig. 10 by the numerical simulation of dispersion diagrams [64]. The effective relative permittivity and corresponding eigen-frequencies for which the permittivity value is valid are shown in Fig. 11. A 90° phase shift with E-field oriented in the vertical direction across the unit cell is assumed and the inner structure’s dimensions are varied (squared hole size, cross’s rod thickness, cube size for 100
The alumina relative permittivity used for our design is 9.5 [25], leading to the effective relative permittivity values between 1.27 and 8.81, 1.27 and 5.77, and 2.23 and 9.13 in the cases of cross, cube with rods and square hole unit cells, respectively. According to this comparison, it is clear that the cross unit cell provides the best ability to vary the effective relative permittivity, which follows a nearly linear slope, while for the cube with rods, the effective relative permittivity increases exponentially and is limited to the maximum cube size of 0.65 mm. The squared hole unit cell shows a logarithmic increase of the effective relative permittivity, and its minimum achievable value is limited by the minimum wall thickness. We choose the cross unit cells for the 3D lens fabrication models based on this analysis. The ideal lens permittivity profile will differ from the actual one due to the limited number of feasible effective relative permittivity values for each unit cell type. The absolute difference of the spatial effective relative permittivity distribution between the ideal and the fabricated QCTO Luneburg lens is plotted in Fig. 12.
The mean absolute relative permittivity difference is 0.1465, with the greatest absolute relative permittivity differences concentrated at the lens’s outer surface areas. The 3D printer limits the size of the green body to 64 mm
Simulated radiation patterns of the QCTO Luneburg lens involving non-ideal spatial permittivity distribution [fab] and the final fabrication dimensions [after] at a frequency of 40 GHz.
Simulated reflection coefficients of the QCTO Luneburg lens involving non-ideal spatial permittivity distribution [fab] and the final fabrication dimensions [after].
The non-ideal lens permittivity distribution leads to slightly reduced antenna gain at the lens center position (1.2 dB), but increased gain and decreased beam steering angle at the lens edge positions (up to 2.9 dB and 7 degrees). Due to the lens size reduction, the antenna gain and maximum beam steering capability are further reduced (up to 2.4 dB and 3 degrees), so the maximum beam steering angle of the designed lens is ±65°. By adding the IML, the antenna gain is further increased by 4.4 dB to the value of 16.23 dBi for the boresight direction. The 3D radiation patterns of the 3D QCTO Luneburg lens, including the full measurement setup (the lens with a holder and a metallic waveguide probe) are shown in Fig. 16. It can be observed that the designed lens holder has a minor effect on the lenses’ radiation patterns since it is designed as an open structure in the lenses’ radiation directions.
3D radiation patterns of the 3D QCTO Luneburg lens without exponential IML (realized gain) using solid fabrication models, (a) feed at the lens center; (b) feed at the lens edge.
To evaluate the fabrication version of the QCTO Luneburg lens as a retroreflector, a circular metallic part with a diameter of 25.75 mm and thickness of 2.20 mm is placed on the lens’s bottom, simulating a 2 EUR coin employed as the reflective metallic layer. The simulated angular RCS response at a frequency of 40 GHz for the metallic coin and the lens with and without the exponential IML is depicted in Fig. 17. From the comparison, it is obvious that the IML improves the overall RCS and achieves a more stable response over a large angular range. It is shown that the QCTO retroreflective lens presents a maximum RCS of −24.3 dBsqm, while the QCTO retroreflective lens with the IML has a maximum RCS of −17.16 dBsqm at boresight. The abrupt RCS drop near angle of ±45° is caused by a destructive interference of the incoming and reflected EM wave.
Simulated RCS of the QCTO Luneburg lens fabrication version with and without exponential IML.
B. QCTO Luneburg Lens Fabrication
The lithography-based ceramic manufacturing (LCM) technology was employed for the realization of the discussed lens structures. With this 3D printing method, lens’s parts are created layer-by-layer via DLP-controlled polymerization of a photosensitive slurry [65], [66]. The utilized printer was a Lithoz CeraFab 7500 [67] with a printing resolution of 25
Lens’s parts were printed using a newly formulated slurry (i. e., LithaLox 360 [67]) and with an illumination energy of 450 mJ/cm2/layer. LithaLox 360 was intentionally developed for the fabrication of delicate samples, such as the Luneburg lens structures and contains 49 vol% of high purity Al2O3 powder as well as 51 vol% of UV-curable polymers. Directly after printing, the parts were cleaned to wash out the residual unpolymerized slurry. This was done by dipping the samples in LithaSol 20 [67] cleaning fluid, letting them soak for 5 minutes, and then ultrasonication of the system for 2 minutes. To ensure the cleanness of the samples, especially inside the lens’s narrow holes and channels, the cleaning process was repeated 10 times for each sample.
Afterward, the following thermal processing steps were performed to convert the cleaned green samples to dense ceramics parts. First, samples were slowly dried in an electrical laboratory dryer. The maximum temperature and duration of the drying step were 140°C and 6 days, respectively. The dried samples were then sintered in another electric furnace and under ambient atmosphere. Since the dried parts still consisted of 51 vol% of cured polymers, heating rates were low up to temperatures of 430°C (i.e., the temperature by which polymer compounds will be fully decomposed into volatile components) to guarantee the fabrication of crack-free samples. In this study, the lens’s parts were sintered at 1600°C, and the sintering duration was 4 days. The fabricated lenses are illustrated in Fig. 18 where the lenses are illuminated with a light source to appreciate their solid core through the whole grid array structure.
(a) Fabricated samples of the 3D QCTO Luneburg lens (right) and the lens with exponential IML (left), (b) and (c) manufactured lens with and without IML, respectively, illuminated by a torch to appreciate their solid core.
C. Lens Antenna Characterization
The measurement of the radiation pattern was performed in an anechoic chamber with the antenna scanner NSI 700S-30 and a Vector Network Analyzer (VNA) R&S ZVA67 with a 23 dBi horn antenna used as a transmitting antenna. A standard WR-28 waveguide probe was used with the lens as a receiving antenna. The lens was sequentially placed into individual 3D printed holders with a gradually shifted excitation position by 3 mm to precisely control the excitation position. The arm with lens was then rotated in 1-degree steps by a computer controller. The lens with holder in the anechoic chamber is shown in Fig. 19. The measured gain of the QCTO Luneburg lens is shown in Fig. 20.
Measurement setup of QCTO Luneburg lens antenna with a lens holder in the anechoic chamber.
Measured radiation patterns of the fabricated 3D QCTO Luneburg lens: (a) without the IML, (b) with the IML. The cross-polarization component is shown as dotted (30 GHz) and dash-dotted (40 GHz) lines.
For the lens without the IML, the gain is 12.62 dBi at 40 GHz at the lens center. Due to the radiation pattern flatness, the measured gain at the lens edge achieves 8.75 dBi and 8.25 dBi with the maximum beam steering angle of 45° and 90°, respectively. The measured gain of the lens with the exponential IML at 40 GHz is 16.51 dBi at the lens center and 10.49 dBi at the lens edge with the maximum beam steering angle of 70°. It is obvious that IML improves the gain of the lens by nearly 4 dB. The input reflection coefficient responses of the lens for different positions with and without the IML is shown in Figs. 21 and 22. The improvement caused by the IML is significant mainly for the lens center position. The input reflection coefficient is lower than −10 dB in the whole Ka-band. Those values are in good agreement with the simulation results. The discrepancy of the measured gain for higher beam steering directions can be caused by the excessive alumina material trapped inside the lens which could not be removed during fabrication, and the interaction of the wave with the lens holder.
Measured and simulated reflection coefficient of the fabricated 3D QCTO Luneburg lens without the IML.
Measured and simulated reflection coefficient of the fabricated 3D QCTO Luneburg lens with the IML.
D. Retroreflective Lens Characterization
The characterization of the lens with reflective layer was performed by employing the Vector Network Analyzer PNA-X N5247A from Agilent Technologies, with its bandwidth set between 30 GHz to 40 GHz and 10001 frequency points, as well as IF bandwidth of 5 kHz. The measurements were performed with an 18 dBi Ka-band horn antenna, connected to a right angle (90°) WR-28 waveguide to coaxial adapter, and then to the VNA via a K-V transition and a V(m)-V(f) cable. Moreover, the structures are located within the transmitting/receiving horn antenna’s far field, at a distance of 0.3 m. The corresponding measurement setup is displayed in Fig. 23. In this case, the reflective layer was implemented by a metallic 2 EUR coin, whereas an electronically controlled turntable was used to perform angular measurements with 1° resolution. For each angle, a reference measurement without the tag was taken and subtracted from the raw data, to remove the influence of reflections from the horn antenna, as well as reflections from the surrounding environment. The measured reflection coefficients for the employed coin, as well as the two fabricated lenses with and without the coin as the reflective layer, are presented in Fig. 24.
(a) Measurement setup for the retroreflective lens characterization, and (b) front view of the combination of lens and coin.
Measured reflection coefficient for the frontal incidence in the frequency-domain.
The results in Fig. 24 show that the backscattered power is higher when the coin is added at the bottom of each lens. Concretely, in the case of lens with matching layer, the backscattered power is an average 7.5 dB larger than when no metal coin is employed as reflective layer. In the case of the lens without IML, the backscattered power when the coin is added is less than its coin-less counterpart between 31.5 GHz and 36.5 GHz. This is attributed to an interaction between the high-
The monostatic radar cross-section (RCS) is a normalized measurement of the backscattered power of a structure, independent of the distance between the structure and the monostatic radar antenna. When employing a radar target with known RCS, the unknown RCS of a reflective structure is calculated by (10):\begin{equation*} RCS_{\textrm {tag}} =RCS_{\textrm {ref}} \cdot \frac {\left |{ {S_{\textrm {11,tag}}} }\right |^{2}}{\left |{ {S_{\textrm {11,ref}}} }\right |^{2}},\tag{10}\end{equation*}
A corner reflector (CR) with an edge of 8.3 cm is employed as a reference radar target, with an analytical RCS in boresight at 40 GHz of 5.48 dBsqm, according to (11) [69]:\begin{equation*} RCS_{\textrm {CR(ref)}} =10\log \left ({{\frac {4\pi }{3}\cdot \frac {a^{4}}{\lambda _{0}^{2}}} }\right),\tag{11}\end{equation*}
When the CR is placed at a distance of 0.3 m from the horn antenna, the latter is not located in the far field of the former, which starts at 3.67 m. Therefore, the RCS cannot be computed accurately with (10) and (11). Instead, the CR is measured when placed 4 m away from the horn antenna. Then, the magnitude of the received \begin{equation*} P_{\text {r}} =\frac {P_{\textrm {t}} G_{\textrm {t}}^{2} \lambda _{0}^{2} RCS_{\textrm {target}}}{\left ({{4\pi } }\right)^{3}}\cdot \frac {1}{d^{4}},\tag{12}\end{equation*}
\begin{equation*} \left |{ {S_{\textrm {11, 0.3 m}}} }\right |^{2}=\left |{ {S_{\textrm {11,4 m}}} }\right |^{2}\cdot \left ({{\frac {4\;\textrm {m}}{0.3\;\textrm {m}}} }\right)^{4},\tag{13}\end{equation*}
It should be pointed out that (13) is only accurate when (i) the measurement of the corner reflector is in line-of-sight, (ii) the distance at which the
The corresponding measurement setup for the corner reflector is shown in Fig. 25, whereas the received backscattered signals of the corner reflector are presented in Fig. 26. Empty room subtraction and time gating between 26 ns to 28 ns are employed to minimize the influence of the surrounding environment.
Measurement setup for the corner reflector with an 8.3 cm edge employed as a known radar target for RCS computation. The analytical RCS in boresight at 40 GHz is
The calculated RCSs at 40 GHz for the QCTO-based retroreflective lens with and without IML are presented in Fig. 27. It is shown that the QCTO retroreflective lens presents a maximum RCS of −20.05 dBsqm at the angle of −7°, whereas the QCTO retroreflective lens with IML has a maximum RCS of −15.78 dBsqm for the frontal incidence. Both values are 6.85 dB and 2.67 dB below the single coin’s RCS for the frontal incidence, respectively.
Furthermore, the retroreflective lenses also present a wide angular range where the backscattered power remains with relatively high magnitude. The operating angular range, defined between the angles in which the RCS decreases by 10 dB regarding the maximum value, is ±7° for the coin, ±37° for the lens and ±50° for the lens with IML. The retroreflective structures present less angular range than the maximum steerable angle of the designed lens. This is owing to the finite size of the metallic coin used as reflective layer. As the lens is rotated, the effective size of the coin is smaller, which in turn reflects less power.
The radar range equation (12) is used to calculate the maximum read-out range with our setup by solving it for the distance. In our measurements,
Conclusion
In this paper, we have proposed a novel 3D QCTO based Luneburg lens for Ka-band mm-wave self-localization systems, which has been manufactured from high permittivity alumina with the LCM 3D printing process. The maximum realized gain of the lens without an IML is 12.62 dBi, and the lens with the exponential IML led to the maximum realized gain of 16.51 dBi at 40 GHz with the maximum beam steering angles of 90° and 70°, respectively. The comparison with other related works is summarized in Table 3. The maximum RCS of the lens backed by a reflective layer, which is firstly evaluated for this kind of retroreflector, without and with the exponential IML is −20.05 dBsqm and −15.78 dBsqm at 40 GHz, respectively.
To the best of the authors’ knowledge, this is the first demonstration of a 3D and retroreflective ceramic QCTO Luneburg lens. The employment of ceramics such as alumina for the lens allows for the realization of wide-angle frequency-coded retroreflectors that can be used for mm-wave indoor localization in dynamic cluttered harsh and high-temperature environments, e.g., withstanding a fire. Future development will be focused on the integration of the lens with 3D photonic crystal-based frequency-coding particles into a monolithic block, and their placement along the lens’s bottom, with the objective of achieving angle-of-arrival identification.
ACKNOWLEDGMENT
(Petr Kaděra and Jesús Sánchez-Pastor contributed equally to this work.)