Wide-Angle Ceramic Retroreflective Luneburg Lens based on Quasi-Conformal Transformation Optics for mm-Wave Indoor Localization

This paper presents a quasi-conformal transformation optics (QCTO) based three-dimensional (3D) retroreflective flattened Luneburg lens for wide-angle millimeter-wave radio-frequency indoor localization. The maximum detection angle and radar cross-section (RCS) are investigated, including an impedance matching layer (IML) between the lens antenna and the free-space environment. The 3D QCTO Luneburg lenses are fabricated in alumina by lithography-based ceramic manufacturing, a 3D printing process. The manufactured structures have a diameter of 29.9 mm (4 λ0), showing a maximum realized gain of 16.51 dBi and beam steering angle of ±70° at 40 GHz. The proposed QCTO Luneburg lens with a metallic reflective layer achieves a maximum RCS of -20.05 dBsqm at 40 GHz with a wide-angle response over ±37°, while the structure with an IML between the lens and air improves these values to a maximum RCS of -15.78 dBsqm and operating angular response between ±50°.


I. INTRODUCTION
The area of passive retroreflective devices for mm-wave  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 crosssection (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 amount 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 OLUME XX, 2017 1 omnidirectional retroreflector when it is surrounded by slant polarizers [14]. Most of the aforementioned references comprise frequency coding, enabling their employment in different areas. For instance, in chipless indoor selflocalization 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 on 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.  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 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 large angular separation needed 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 (εr = 17.5) does not allow the usage of the same material as for the PhC coding part (εr = 9.5). Therefore, including the manufacturing and integration options enabled by ceramic 3D printing [25]- [26], a new design addressing this problem is required. 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. 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], 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].

II. QUASI-CONFORMAL TRANSFORMATION OPTICS ENABLED LUNEBURG LENS
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 more 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: 1) 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). 2) 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). 3) Creating the discretized 3D model (solid cubes in Fig. 3b) by using customized MATLAB code for the full-wave simulations in CST Studio Suite.

4)
Assigning the targeted 3D permittivity distribution of the QCTO lens to the closest available permittivity values for the selected unit cell (Fig. 11). 5) Replacing the solid cubes by the crosses with corresponding dimensions for creation of the suitable 3D printing fabrication model (Fig. 13).

A. QUASI-CONFORMAL MAPPING OF THE LUNEBURG LENS
We assume that the Cartesian coordinates that define the virtual and physical spaces are (u,v) and (x,y), respectively. The virtual space is a rectangle with a height of 60 mm and a width of 120 mm that is attached to the Luneburg lens with a diameter of 60 mm. The bottom-left corner of the rectangle is at (-60 mm, -9.33 mm). The relative permittivity of the virtual space follows the below equation (1): where R is the radius of the lens (30 mm) that is centered at the coordinate's origin. The virtual space and the vertical u-constant and horizontal v-constant lines are depicted in Fig. 1 as well as the relative permittivity of the virtual space. The θ ≈ 144° angle defines the lens steering angle.
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 quasiconformal transformation between the virtual and physical spaces is calculated by solving the Laplace's equation in the virtual space for the variables x and y. The following equations (2) represent the underlying differential equations and the Dirichlet and Neumann boundary conditions involved in the process.
where N is the normal vector of the boundary. The above equations can be solved using MATLAB [55] or COMSOL [56] PDE solvers. After solving the Laplace's equations, the x(u,v) and y(u,v) functions and their partial derivatives are derived. The permittivity distribution in the physical space is calculated by the following equation (3): where the sub-scripts denote partial derivatives.
The permittivity profile of the physical space and the corresponding mapped u-constant and v-constant lines are depicted in Fig. 2a. Note that the corresponding image points are labeled with a prime sign. The images of physical space's x-constant and y-constant lines in the virtual space are depicted in Fig. 2b. The mapped physical space's relative permittivity is included as well.
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 εr,eff = 1.27 (see Section V), the transformed QCTO Luneburg lens is further evaluated in a manufacturable region with a bottom diameter of 34 mm, a center diameter of 44 mm and a height of 32 mm. After   calculations of the lens relative permittivity distribution in COMSOL Multiphysics, the in-house MATLAB codes were created and exploited for generating a discretized lens for CST Studio Suite (Fig. 3). 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 θ/2, is shown in Fig. 4. Our numerical calculation corresponds to the approach presented by Kundtz et al. in [40]. However, the calculation done by Biswas et al. in [52], [57] leads to the maximum relative permittivity of 4, that is caused by imposing a different Dirichlet boundary conditions. To verify the COMSOL calculations, we used a simplified lens model discretized into ringed cylinders with the CST time-domain solver in the Ka-band (40 GHz) and the WR-28 waveguide excitation placed at the lens edge. Note that in Fig. 4 we added available results from the literature. Generally, we can see good agreement with our results.

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 0.5 λ0 layer thickness at the center frequency provides the largest beam steering angle while maintaining a sufficient level of impedance matching. Further, in [57], the exponential and Gaussian impedance matching profiles were evaluated with slightly worse beam steering capabilities. However, these evaluations were carried out on the low maximum relative permittivity (εr = 2.89) QCTO Luneburg lens. Since our designed lens has a much higher maximum relative permittivity (εr = 8.89), we investigate the combination of the lens and IML on the overall antenna performance for a broadband Klopfenstein, exponential, and Gaussian profiles described by the following equations (7)-(11), respectively [57]- [58].
l 0 where εi represents the relative permittivity of free space, εl is the relative permittivity of the transformed lens gradient-index profile, L represents the antireflective layer thickness, Γm is the maximum passband ripple (0.001 in our case), the function ϕ (x, y) describes expanded power series of the first kind Bessel function of the order one, λ0 is the free-space wavelength at the center frequency.
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 0.5 λ0 (3.5 mm at 40 GHz) to accommodate 5 cells in the longitudinal direction to achieve better manufacturing mechanical stability and the requirement of the integer count of cells (assumed building unit cells size is 0.7 mm x 0.7 mm x 0.7 mm). The influence of the broadband IML on the antenna radiation patterns and the reflection coefficient are depicted in Figs. 8 and 9.
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 slightly better impedance matching with 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.

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 eigenfrequencies 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 µm thick connecting rods). To ensure the success of the ceramic 3D printing process, we need to consider the effect of over polymerization (i.e., unwanted polymerization in vicinity of illuminated pixels, due to the light scattering during printing) and set the minimum inner dimensions to 100 µm (cross, cube with rods), or 50 µm (squared hole block) which results in a 100 µm wall thickness between adjacent cells. Furthermore, the minimum separation between each wall is 150 µm, and the overall unit cell dimensions must be a multiple of 25 µm due to the minimum 3D printer´s pixel size. To comply with the trade-off between those limits and the maximum operational frequency of the lens, the overall selected unit cell size is 0.7 mm x 0.7 mm x 0.7 mm.
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 x 40 mm x 100 mm. Further, the expected shrinkages caused by sintering are 1.22 and 1.28 in the xy plane and z direction (i.e., height), respectively. As a result, the maximum dimensions after sintering are restricted to 52 mm x 32 mm x 78 mm. Therefore, the designed lens was recalculated to comply with the green body requirements, reaching a middle diameter of 36.4 mm, a bottom diameter of 28 mm, and a height of 25.9 mm. The example of the fabricated model including the exponential IML is depicted in Fig. 13. The comparison of lens antenna radiation patterns and reflection coefficient involving non-ideal permittivity distribution and lens size after final fabrication (including all size reductions) are shown in Figs. 14 and 15. The focal points of the reduced size lens were shifted accordingly to match the beam steering direction of the original size of the lens.
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.  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 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.

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 µm and a UV light source (wavelength: ~450 nm) for slurry polymerization.
Lens´s parts were printed using a newly formulated slurry (i. e., LithaLox 360 [67]) and with an illumination energy of 450 mJ/cm 2 /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.

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.

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 to the raw data, to remove the influence of reflections from the horn antenna, as well 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.
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-lens's core and the metallic coin, but more research needs to be conducted to ascertain the concrete reason. Regardless, when the coin is added, the lens without the IML increases its backscattered power by an average of 1.3 dB.
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 (12): where |S11| is the measured reflection coefficient, tag corresponds to the retroreflective lens-based structure and ref is the radar target employed as reference for calculating the RCS.
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 (13) [69]: where a denotes the corner reflector´s edge and λ0 is the freespace wavelength. 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 (12) and (13). Instead, the CR is measured when placed 4 m away from the horn antenna. Then, the magnitude of the received S11 in frequency-domain is predicted at a distance of 0.3 m by considering the received power Pr in a monostatic radar system (14): where Pt is the transmitted power, Gt is the horn antenna gain, and d the distance between radar and target. Assuming the same measurement setup is employed for different distances, Pt, Gt and RCStarget remain constant. Thus, by dividing Eq. (14) for the two different distances of 0.3 m and 4 m, the following relation between the corresponding |S11| is obtained: It should be pointed out that (15) is only accurate when (i) the measurement of the corner reflector is in line-of-sight , (ii) the distance at which the |S11| is predicted is within the far field of the transmitting horn antenna, and (iii) the same monostatic radar setup is employed for both retroreflective lens and corner reflector measurements, which is our case.  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.
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 (14) is used to calculate the maximum read-out range with our setup by solving it for the distance. In our measurements, Pt is -5 dBm, Gt is 18 dBi, 0 is set to the vacuum wavelength at 40 GHz and Pr is set to the minimum detectable power at the receiver, Pr,min, which is -70 dBm. With these values, the maximum read-out ranges are 48.63 cm and 62.18 cm for the retroreflective lens without and with IML, respectively.

IV. 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 frequencycoded 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.