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Variation of the components in Cartesian coordinates of the region 2. The permittivity and permeability are, respectively, plotted for the exponential transformation with q 1 1⁄4 1, q 2 1⁄4 À 30, R 1 1⁄4 5 mm, R 2 1⁄4 50 mm, N 1⁄4 2, and a 1⁄4 300 .
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Transformation electromagnetics offers an unconventional approach for the design of novel radiating devices. Here, we propose an electromagnetic metamaterial able to split an isotropic radiation into multiple directive beams. By applying transformations that modify distance and angles, we show how the multiple directive beams can be steered at will...
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Context 1
... parameter a is an angle which can be viewed as the out- put angle of the transformed material. It is introduced in the transformation in order to rotate the beam in the material. The transformations used are general and to validate the possibility of theses transformations, we need to consider different cases. The denominator 1 þ N /2 of the function h 0 have to be an integer meaning that N can only take 0 or positive even values. For example, when N 1⁄4 0, there is no multi- beam creation due to non-segmentation of the space. To calculate the components of the electromagnetic tensors w ij for linear transformation in both regions 1 and 2, the parameters considered are R 1 1⁄4 5 mm, R 2 1⁄4 50 mm, q 1 1⁄4 1/7, N 1⁄4 0, and a 1⁄4 80 . The frequency is set to 20 GHz and Fig. 2 shows the variation of the different components of the permittivity and permeability tensors. The components e xx , e yy , and e zz are always positive but can be negative if the source appearance becomes larger. Only the non-diagonal term e takes both positive and negative values. In order to validate the proposed beam steering concept, we use the commercial software Comsol MULTIPHYSICS to perform numerical simulations of the different transformation cases presented above. All the simulations are run in the microwave domain at 20 GHz. The validation of our design is performed in a two-dimensional configuration in a transverse electric mode (TEz) (E parallel to the z -axis). A current source of dimension d 1⁄4 4 mm placed perpendicular to the xy plane is used as a radiating element. Continuity and matched conditions are applied, respectively, to the boundary of zone 1 and zone 2. To verify our design, we fix R 1 1⁄4 5 mm and R 2 1⁄4 50 mm. The results obtained from linear transformations in both regions 1 and 2 with the material parameters of Fig. 2 are presented in Fig. 3. Since the size of the source is much smaller than the operating wavelength ( k 0 1⁄4 1.5 cm at 20 GHz), the radiation is isotropic in the xy plane, as shown in Fig. 3(a) for free space as surrounding medium. The electric field (Ez-component) distribution of the same linear source placed in both materials calculated by the transformations having parameters N 1⁄4 0, a 1⁄4 80 and q 1 1⁄4 1/7 is presented in Fig. 3(b). The radiating field is equivalent to a new source with dimension d / q 1 1⁄4 2.8 cm producing a 130 beam steering. As stated earlier, the transformation in the second region allows creating multi-beams and steering them. The parameter N enables to vary the number of the radiated beams. However, in Fig. 3, N is considered to be zero. Therefore, we should not be in presence of multi-beam. Such a scenario of two steered directive beams is the result of first the compression region 1, where the source appears as a large aperture radiator with two opposite directive beams as previously reported in, 36 and second to the rotations in region 2 which enables to steer the beams. Figs. 3(c) and 3(d) show the norm of the total electric field in such configuration. If we consider N different from zero, the device does not work properly since the two directive beams emanating from region 1 are not channeled properly into the different space segments. Therefore, for N 1⁄4 0, we need to have an isotropic beam emanating from region 1 in order to input the different space segments. Such a configuration is detailed in the “spiral-like emission of multiple directive beams” section. As stated, when considering N 0, there must be no space compression in region 1. Fig. 4 shows the variation of the tensor for the exponential transformation case in region 2 and when N is different from zero, so as to be in a multi- beam configuration. The parameters are q 1 1⁄4 1, q 2 1⁄4 À 30, N 1⁄4 2, and a 1⁄4 300 . In Fig. 5, the linear transformation of the radial part in region 2 is followed by an exponential one with N different from zero, q 2 1⁄4 À 30 and a 1⁄4 300 , corresponding to the material parameters presented in Fig. 4. Figs. 5(a) and 5(b) show the electric field distribution in the proposed device for N 1⁄4 2. Two steered beams can be clearly observed. The cases for N 1⁄4 4 and N 1⁄4 6 are, respectively, shown in Figs. 5(e) and 5(f) and Figs. 5(g) and 5(h), respectively. In each case, the electromagnetic field is rotated in the material as shown in Figs. 5(c) and 5(d). When a increases, the radiation tends to be more and more tangential to the surface of the material, and the interferences observed in Fig. 5(g) between the emitted beams decrease in Fig. 5(h). This work points out the use of transformation electromagnetics concept to design an artificial shell which allows creating multiple directive beams. The latter concept makes use of two transformations; the first one to compress space and the second one to divide and rotate it. Numerical simulations have confirmed the operating principle of the transformations. We have first shown that a very small source can emit two directive beams comparable to an antenna with a large aperture. These directive beams can then be steered in a desired direction. Furthermore, the concept has also been applied to create more than two directive steered beams. In this case, space has to be divided into different segments so to channel directive beams from an isotropic source. For a possible future prototype fabrication, choosing a polarization in a fixed direction of the electromagnetic field will lead after the parameters reduction procedure to two variations of permittivity or permeability in a spiral-like eigen-base, which can be achieved by common metamaterial structures. This study shows the great possibilities that transformation electromagnetics can offer for the design and synthesis of new devices in both microwave and optical wavelength ...
Context 2
... parameter a is an angle which can be viewed as the out- put angle of the transformed material. It is introduced in the transformation in order to rotate the beam in the material. The transformations used are general and to validate the possibility of theses transformations, we need to consider different cases. The denominator 1 þ N /2 of the function h 0 have to be an integer meaning that N can only take 0 or positive even values. For example, when N 1⁄4 0, there is no multi- beam creation due to non-segmentation of the space. To calculate the components of the electromagnetic tensors w ij for linear transformation in both regions 1 and 2, the parameters considered are R 1 1⁄4 5 mm, R 2 1⁄4 50 mm, q 1 1⁄4 1/7, N 1⁄4 0, and a 1⁄4 80 . The frequency is set to 20 GHz and Fig. 2 shows the variation of the different components of the permittivity and permeability tensors. The components e xx , e yy , and e zz are always positive but can be negative if the source appearance becomes larger. Only the non-diagonal term e takes both positive and negative values. In order to validate the proposed beam steering concept, we use the commercial software Comsol MULTIPHYSICS to perform numerical simulations of the different transformation cases presented above. All the simulations are run in the microwave domain at 20 GHz. The validation of our design is performed in a two-dimensional configuration in a transverse electric mode (TEz) (E parallel to the z -axis). A current source of dimension d 1⁄4 4 mm placed perpendicular to the xy plane is used as a radiating element. Continuity and matched conditions are applied, respectively, to the boundary of zone 1 and zone 2. To verify our design, we fix R 1 1⁄4 5 mm and R 2 1⁄4 50 mm. The results obtained from linear transformations in both regions 1 and 2 with the material parameters of Fig. 2 are presented in Fig. 3. Since the size of the source is much smaller than the operating wavelength ( k 0 1⁄4 1.5 cm at 20 GHz), the radiation is isotropic in the xy plane, as shown in Fig. 3(a) for free space as surrounding medium. The electric field (Ez-component) distribution of the same linear source placed in both materials calculated by the transformations having parameters N 1⁄4 0, a 1⁄4 80 and q 1 1⁄4 1/7 is presented in Fig. 3(b). The radiating field is equivalent to a new source with dimension d / q 1 1⁄4 2.8 cm producing a 130 beam steering. As stated earlier, the transformation in the second region allows creating multi-beams and steering them. The parameter N enables to vary the number of the radiated beams. However, in Fig. 3, N is considered to be zero. Therefore, we should not be in presence of multi-beam. Such a scenario of two steered directive beams is the result of first the compression region 1, where the source appears as a large aperture radiator with two opposite directive beams as previously reported in, 36 and second to the rotations in region 2 which enables to steer the beams. Figs. 3(c) and 3(d) show the norm of the total electric field in such configuration. If we consider N different from zero, the device does not work properly since the two directive beams emanating from region 1 are not channeled properly into the different space segments. Therefore, for N 1⁄4 0, we need to have an isotropic beam emanating from region 1 in order to input the different space segments. Such a configuration is detailed in the “spiral-like emission of multiple directive beams” section. As stated, when considering N 0, there must be no space compression in region 1. Fig. 4 shows the variation of the tensor for the exponential transformation case in region 2 and when N is different from zero, so as to be in a multi- beam configuration. The parameters are q 1 1⁄4 1, q 2 1⁄4 À 30, N 1⁄4 2, and a 1⁄4 300 . In Fig. 5, the linear transformation of the radial part in region 2 is followed by an exponential one with N different from zero, q 2 1⁄4 À 30 and a 1⁄4 300 , corresponding to the material parameters presented in Fig. 4. Figs. 5(a) and 5(b) show the electric field distribution in the proposed device for N 1⁄4 2. Two steered beams can be clearly observed. The cases for N 1⁄4 4 and N 1⁄4 6 are, respectively, shown in Figs. 5(e) and 5(f) and Figs. 5(g) and 5(h), respectively. In each case, the electromagnetic field is rotated in the material as shown in Figs. 5(c) and 5(d). When a increases, the radiation tends to be more and more tangential to the surface of the material, and the interferences observed in Fig. 5(g) between the emitted beams decrease in Fig. 5(h). This work points out the use of transformation electromagnetics concept to design an artificial shell which allows creating multiple directive beams. The latter concept makes use of two transformations; the first one to compress space and the second one to divide and rotate it. Numerical simulations have confirmed the operating principle of the transformations. We have first shown that a very small source can emit two directive beams comparable to an antenna with a large aperture. These directive beams can then be steered in a desired direction. Furthermore, the concept has also been applied to create more than two directive steered beams. In this case, space has to be divided into different segments so to channel directive beams from an isotropic source. For a possible future prototype fabrication, choosing a polarization in a fixed direction of the electromagnetic field will lead after the parameters reduction procedure to two variations of permittivity or permeability in a spiral-like eigen-base, which can be achieved by common metamaterial structures. This study shows the great possibilities that transformation electromagnetics can offer for the design and synthesis of new devices in both microwave and optical wavelength ...
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Citations
... Opt. Express 2009;Li et al. 2011;Kwon 2010;Popa et al. 2009;Kong et al. 2007;Lu et al. 2009;Jiang et al. 2011aJiang et al. , 2012Tichit et al. 2011aTichit et al. , 2014 etc. Similarly, directive antennas have been introduced employing the TO to achieve high-directivity from the smaller-sized antenna (Tichit et al. 2009(Tichit et al. , 2011b(Tichit et al. , 2013Luo et al. 2009;Wang et al. 2014;Segura et al. 2014). ...
Based on transformation optics (TO), this paper uses geometric divisions and linear coordinate transformations to design “shrinking-shifting—and reshaping”, and “amplifying-shifting—and reshaping” devices. The proposed devices can reshape the sizes and locations of the wrapped-objects inside the core-region. The shrinking-shifting device shrinks the larger object into a smaller one and shifts it to different location, whereas the shrinking-reshaping device can generate a smaller-size image with different shape located at different location. In contrast to previously designed shrinking devices, the real object wrapped inside the proposed core-region and the transformed object contains the same material properties, and the location-shifting is another feature. Here, the shifting-region is located inside the physical-space boundaries to achieve the non-negative, homogeneous, and anisotropic material properties of the proposed device, which are easier for real implementations. Thus, we further verified this concept with the amplifying-shifting and -reshaping devices for visually transformation of smaller object into bigger one placed at different location and position. We also applied active scatterer to further validate the working functionality of proposed devices. In addition, the proposed devices behave like the concentrator and (or) rotator effect in the absence of any scatterer. Our findings highlight the role of TO, suggesting directions for future research on bi-functional devices that will be useful for shrinking and amplifying devices, illusion optics, camouflage, and object protection etc.
... As analyzed by Pendry's group, an equivalence between Maxwell's equations described in an initial coordinate system (i.e., a virtual domain) and their counterparts in another arbitrary transformed coordinate system (i.e., a physical domain) results in a direct link between the permittivity and permeability of the materials occupying the spaces and the metric tensor of the transformed space, which has the desired EM properties. Soon after the introduction of TO, many novel devices that seemed impossible * abdolali@iust.ac.ir to achieve with natural materials, such as EM cloaks [8][9][10], multiemission lenses [11][12][13][14][15][16], wave concentrators [17][18][19][20][21][22][23][24], and beam splitters [25][26][27], were proposed. In the same way as the TO methodology, TT has gained much attention due to the intrinsic degree of freedom that it offers. ...
Recently, thermal manipulation has gained the attention of the scientific community due to its several applications. In this paper, based on a transformation-thermodynamics methodology, a special type of material, called a thermal-null medium (TNM), is proposed that leads to the design of various thermal functionalities such as thermal bending devices, arbitrarily shaped heat concentrators, and omnidirectional thermal cloaks. In contrast to the conventional transformation-thermodynamics-based conductivities, which are inhomogeneous and anisotropic, TNMs are homogeneous and easy to realize. It is shown that the TNMs obtained are independent of the desired device shape, meaning that if the geometry of the desired device is changed, there is no need to recalculate the conductivities required. This consequently makes the designed devices suitable for scenarios where reconfigurability is of the utmost importance. Several numerical simulations are carried out to demonstrate the capability of TNMs and their application in directional bending devices, concentrators, and cloaks. In order to validate the concept, by using effective-medium theory, a TNM with an aluminum-air multilayer structure is designed and fabricated. The structure realized is then used to implement an elliptically shaped thermal concentrator. It is observed that the experimental results exhibit good agreement with the results obtained from the numerical simulations, which corroborates the effectiveness of the proposed materials.
... Many novel devices based on TO theory have been proposed and implemented, including cloaks [4][5][6][7][8], concentrators [9][10][11], illusion devices [12][13][14], and lens [15][16][17][18]. Among these devices, multi-beam antennas capable of achieving multi-beam emission show promise for many potential applications in the fields of satellite communications, smart traffic systems, and multiple-input multiple-output (MIMO) systems [19][20][21][22][23][24]. However, the spatially inhomogeneous and highly anisotropic material parameters of the early transformation-based multi-beam antennas represent a considerable challenge for fabrication [19,22]. ...
... Among these devices, multi-beam antennas capable of achieving multi-beam emission show promise for many potential applications in the fields of satellite communications, smart traffic systems, and multiple-input multiple-output (MIMO) systems [19][20][21][22][23][24]. However, the spatially inhomogeneous and highly anisotropic material parameters of the early transformation-based multi-beam antennas represent a considerable challenge for fabrication [19,22]. In an effort to develop devices for practical application, multi-beam antennas with fewer material parameters have been proposed and implemented, such as antennas consisting of anisotropic zero-index materials [20] and uniaxial zero-index materials [21,23]. ...
In this paper, we experimentally demonstrate the performance of a multi-beam antenna based on inductor-capacitor (L-C) transmission line networks. The lumped element parameters of the antenna are derived according to the mapping relations between Maxwell’s equations and the L-C network equations. The simulation results are in good agreement with the measurement results, and the antenna performs well at a wide bandwidth with high directivity. The antenna has potential applications in future communication systems.
... 5 This approach has been extensively used for the design of flat lenses; however, it calls for artificial media having anisotropic electric and magnetic properties (i.e., with both electric permittivity and magnetic permeability in tensor forms) which can be realized, though with difficulty, using metamaterials. 2,[6][7][8][9] The need for anisotropic properties of the medium can be eliminated by applying quasiconformal transformation optics (QCTO), 6 however, at the cost of making the design sensitive to the polarization of light. 11 Nevertheless, this approach does not resolve the need for both gradient permittivity and permeability, even if both were isotropic. ...
... Several studies utilized metamaterials and metasurface; [6][7][8][13][14][15][16][17][18]24,25 however, metamaterials commonly consist of metallic inclusion, which can impose significant losses particularly in high frequencies. Furthermore, the metamaterials based designs (or more broadly, any electrically-small resonators-based design) generally result in narrow bandwidth focusing, dispersion, anisotropy, and sensitivity to the polarization. ...
This work presents two approaches to design and implement three-dimensional (3D) graded index (GRIN) flat lenses consisting of concentric annular segments. Generally, the design of GRIN flat lenses calls for segments with very specific tailored permittivity which makes the realization of the lens challenging. To meet this challenge, each segment of the lens is replaced with a three-layer structure consisting of two materials with a high and a low dielectric constant in such a way that the high permittivity layer is sandwiched between two low permittivity layers. By treating the lens segments as transmission lines and taking the effect of multiple reflections into account, the layer thicknesses are adjusted in such a way that the rays passing through different segments interfere constructively at a focal point. To further improve the focusing performance, a practical design approach is introduced in which each segment of the lens is made of a symmetric seven-layer structure using only two materials (alternating in arrangement) with a high and a low dielectric constant. This design provides the following features: (1) almost all of the incident power passes the lens without considerable reflection, (2) the lens provides a constructive interference of the incident wave at a focal point, and (3) the lens has the potential to be manufactured using available material and technology. Numerical examples are provided in which silica and silicon are utilized as low and high permittivity materials, respectively.
... Fortunately, the combination of transformation optics (TOs) and metamaterial technology has led to an innovative approach to design a new class of electromagnetic (EM) sources that are appropriate for being used in this research area [16][17][18]. The main idea of TO is to create an equivalence between Maxwell's equations described in an initial coordinate system (i.e. ...
... However, it is obvious that for N similar virtual triangles, the radiated power is divided equally in each physical domain section with a relation of P i = P rad /N. It should be mentioned that unlike previous studies [16][17][18] which were focused on the special case of N similar triangles, our approach can be applied for both cases and, as a result, it is more general with respect to customising radiation pattern properties. As previously mentioned, due to the TO methodology, the design approach was made possible by the fact that Maxwell's equations can be written in a form-invariant manner under coordinate transformations where only the permittivity and permeability tensors are changed [28]. ...
Using transformation optics (TO), a general method for tailoring the radiation pattern of a monopole antenna encompassed by a coating layer is proposed. Unlike previous studies, the propounded approach is not restricted to special patterns and can produce arbitrary radiation pattern with customizable beam parameters like number, direction, and directivity in both azimuthal and elevation planes. A linear coordinate transformation is established to simplify the coating layer realization via offering homogeneous materials. As proof-of-principle, two different antennas capable of generating multiple beams and a single-directive beam, are elaborately acquired by a meta-structure consisting of a split-ring resonator (SRR)-wire array composite. It was observed that the experimental results corroborate numerical simulations. The proposed approach is believed to have potential applications in antenna technologies, satellite communication, and multiple-input multiple-output (MIMO) systems.
... It is worthwhile mentioning that due to the importance of highly directive emissions with flexible beam numbers and directions in modern smart communication systems, many works have been carried out for achieving this aim on the basis of the TO methodology [25][26][27]. The earliest designs were based on mapping a circle in virtual space into a square or rectangle in the physical domain in order to change cylindrical waves to planar ones [28][29][30][31]. ...
Multi emission meta-radiating structures have been an important area of focus due to their potential applications in multi-user scenarios. Thereby, based on the concept of multi-folded transformation optics, a general approach for enabling the customizable multiple beam radiation of a planar antenna is propounded. The presented method is competent not only in creating any desired number of beams but also in adjusting their directivities. Antithetical to phased array antennas, with their complex feeding, heavyweight and costly elements, this goal was achieved via utilizing simple homogeneous materials elaborately designed by multiple folding of the virtual spaces to physical ones with an affine transformation. Several illustrative numerical simulations were performed in order to highlight the capability of the presented method in manipulating the planar source radiation. A metantenna structure consisting of a multi-folded layer (MFL) and the patch antenna underneath, was designed, fabricated, and measured to authenticate the concept. The structure divides the radiated wave into two beams with arbitrary directions and different directivities. The experimental results of the realized metantenna structure exhibited good agreement with numerical simulations and theoretical predictions. The proposed approach could find applications in several fields of engineering where multiple emission is of utmost importance such as multi-input-multi-output (MIMO) systems or could be used in waveguide applications like power dividers.
... The complex distribution of electromagnetic parameters is usually achieved using metamaterials with unique electromagnetic properties that can be tailored to alter the normal principles of light and electromagnetic wave propagation [16][17][18]. As such, the TO concept has motivated a series of studies on conceptual and functional devices including waveguiding structures [19][20][21][22][23], lens antennas [24][25][26][27][28] and directive antennas [29][30][31], multi-beam radiating structures [32][33][34], illusion systems [35][36][37], and isotropic emitters [38,39]. We have previously investigated a TO based OAM generation lens [40], the convergence to be solved. ...
Radio waves carrying orbital angular momentum (OAM) may potentially increase spectrum efficiency and channel capacity based on their extra rotational degree of freedom. However, due to their divergence characteristics, vortex waves are not suitable to transmit over a long distance in the radio frequency (RF) and microwave domains. In this paper, a transformation optics (TO) based all-dielectric converging lens is proposed. The beam divergence angle of the vortex wave passing through the lens can be decreased from 25° to 9°. The transformed material parameters of the converging lens are determined by solving Laplace’s equation subject to specific boundary conditions. Far-field antenna radiation patterns as well as near-field helical phase and electric field amplitude distributions obtained from numerical simulations are reported, demonstrating the broadband characteristics of the proposed microwave lens. Moreover, the all-dielectric compact lens design comprised by a graded permittivity profile can be fabricated by additive manufacturing technology, which greatly facilitates the potential development and application of vortex wave based wireless communications.
... The requisite for smart antennas in various applications such as multiple-input multiple-output (MIMO) systems, highresolution sensors and pico-cellular communication systems is escalating expeditiously in telecommunications domain [1]. Beam steering functionalities, which is achieved through different approaches such as phased array antennas (PAA), is of paramount importance for the mentioned applications [2,3]. ...
... TO and folded geometry) has led to many innovative devices, such as external cloak [15], waveguide connection [16], super-scatterer [17], and imaging [18] to name a few. Recently, based on the TO methodology, some works have been performed in order to tilt the radiated beam of an antenna into the desired angle of interest [1,19]. However, these TO-based structures generally require anisotropic and spatial in homogeneous materials, which cause major difficulties for practical implementations in real-life scenarios [20,21]. ...
Based on the folded transformation optics, a methodical approach for manipulating the orientation and directivity of a planar antenna beam is propounded. Unlike the classical phased array antennas, the tilted emitting beam is achieved by a homogeneous metamaterial-based superstrate in a triangle shape configuration, rather than by utilizing complicated and bulky phase shifters. Such a simple coating layer is elaborately designed via folding the virtual space to physical one with an affine transformation. The competency of the proposed method is demonstrated through some examples, indicating the capability of the coating layer in tilting the radiated beam to any desired off-normal directions in both the upper and lower half spaces. To authenticate the concept, a tilted beam metamaterial-based antenna is fabricated and tested; it is observed that the experimental results are in a good agreement with the numerical simulations and theoretical predictions. The presented method paves the way towards steering the beam of a metamaterial-assisted antenna, which has been a topic of great interest in several fields of engineering, such as antenna technologies, satellite communication, and multiple-input multiple-output systems.
... New class of optical and electromagnetic devices has been designed by this approach, including the widely known invisibility cloak [3]. Following this success, TO technique along with metamaterial technology has resulted in the development of other interesting devices, including concentrators [4], waveguide bends and transitions [5], lenses [6][7][8][9][10][11][12] and antennas [13][14][15][16][17][18][19][20][21][22][23]. ...
... The latter concept has led to the design of the now well-known electromagnetic cloak [3] in 2006, and has resulted in the development of conceptual and functional waveguiding [4][5][6][7][8][9] and illusion devices [10][11][12][13][14][15][16][17]. In the field of lenses and antennas, focusing devices [18][19][20][21][22][23][24][25], directive antennas [26][27][28], multibeam [29][30][31] and isotropic emissions [32,33] have been proposed. ...
Quasi-conformal transformation optics is applied to design electromagnetic devices for focusing and collimating applications at microwave frequencies. Two devices are studied and conceived by solving Laplace’s equation that describes the deformation of a medium in a space transformation. As validation examples, material parameters of two different lenses are derived from the analytical solutions of Laplace’s equation. The first lens is applied to produce an overall directive in-phase emission from an array of sources conformed on a cylindrical structure. The second lens allows deflecting a directive beam to an off-normal direction. Full-wave simulations are performed to verify the functionality of the calculated lenses. Prototypes presenting a graded refractive index are fabricated through three-dimensional polyjet printing using solely dielectric materials. Experimental measurements carried out show very good agreement with numerical simulations, thereby validating the proposed lenses. Such easily realizable designs open the way to low-cost all-dielectric microwave lenses for beam forming and collimation.
