Improving the scanning coverage of array antennas with multilayer lenses designed with a ray tracing

IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. XX, NO. XX, MONTH 202X 1 Improving the scanning coverage of array antennas with multilayer lenses designed with a ray tracing Pilar Castillo-Tapia, Graduate Student Member, IEEE, N´ uria Flores-Espinosa, Graduate Student Member, IEEE, Francisco Mesa, Fellow, IEEE Maria Carolina Vigan´ o, Member, IEEE, and Oscar Quevedo-Teruel, Fellow, IEEE Abstract—Integrating dielectric lenses with phased array antennas can be beneficial in numerous applications, yet the design procedure typically requires significant computational effort. In this work, we employ a streamlined in-house two-dimensional ray-tracing model to design dielectric lenses, with the goal of enhancing the gain of an array antenna at large scanning angles. The ray-tracing model also accounts for losses from material absorption and reflections. The interfaces between the dielectric layers of these lenses devised in this study are defined using splines to allow a large flexibility of their shape. First, dielectric lenses consisting of a core layer supplemented by two matching layers are investigated. The results show better performance compared to the lens constructed using the conic equation. In addition, multilayer dielectric lenses also provide more flexibility, as proved by the proposed configuration with five layers which also minimizes reflections in the optimized direction. Finally, a nonuniform multilayer lens, featuring two layers on the sides and a single layer near to broadside can improve gain at 60◦while maintaining efficiency at smaller angles. Index Terms—Array antennas, dielectric lens, multilayer lens, ray tracing. I. INTRODUCTION Certain applications, such as satellite and terrestrial communications, are moving to higher frequencies to increase capacity. To overcome high path losses, these applications need highly directive antennas capable of steering the beam within a specific angular range [1]. A well suited option for this new generation of terminals is the phased array antenna (PAA) due to its scanning capabilities, simplicity of design, and compactness [2]. However, at high frequencies, the feeding network of these antennas can become complex and lossy [3]. In addition, because of the planar implementation of these antennas, scan losses follow the cosine function. Recently, there has been renewed interest in the use of lenses to design antennas operating in the millimeter-wave frequency range [4]. At these frequencies, the physical size of the lens can meet the size requirements of certain applications. In contrast to planar array antennas, lens antennas can have a more simplified feeding network and can reduce losses The work of P. Castillo-Tapia, N. Flores-Espinosa and O. Quevedo-Teruel has been funded by the strategic innovation program Smarter Electronics System –a joint venture of Vinnova, Formas and the Swedish Energy Agency, under project 2023-00648. The work of F. Mesa has been partially supported by Grant PID2020-116739GB-I00 funded by MICIU/AEI/10.13039/501100011033 P. Castillo-Tapia, N. Flores-Espinosa and O. Quevedo-Teruel are with the Division of Electromagnetic Engineering and Fusion Science, School of Electrical Engineering, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden. F. Mesa is with the Department of Applied Physics 1, Universidad de Sevilla, Sevilla, Spain M.C. Vigan´ o is with Viasat Antenna Systems S.A., 1015 Lausanne, Switzerland. Manuscript received Month XX, 202X; revised Month XX, 202X. due to scanning. Graded-index lenses have been proposed to produce highly directive beams in the mm-wave band [5], [6]. A significant example is the Luneburg lens antenna, which can produce highly directive beams with minimal scan losses due to its specific graded refractive index and rotational symmetry [7], [8]. For different applications, the combination of PAA and lenses can benefit from the advantages of both solutions [9]. Dielectric lenses can increase the gain at high scanning angles, as proposed in [10], [11]. This can also be accomplished through the use of metadomes, although their operational bandwidth may be more restricted [12], [13]. Dielectric lenses can also increase the gain in a wide range of scanning to reduce the power consumption of amplifiers in the PAA [14]. Another interesting solution is the use of dielectric lenses to mitigate grating-lobe levels in array antennas with element spacings greater than λ/2[15]. The entire process of designing PAA combined with lenses requires significant computational resources. As an example, a single three-dimensional fullwave simulation took approximately 1 h in [14]. This makes any optimization procedure lengthy and challenging. More efficient solutions based on geometric optics can speed up the design of these antennas [16], [17]. Recently, a ray tracing (RT) model has been proposed to calculate the radiation patterns of geodesic lens antennas [18], [19]. This model can also be complemented with physical optics (PO) to calculate directivity and gain [20]. Based on this previous experience, in [21] a ray-tracing model was proposed to efficiently calculate the nonlinear phased distribution and radiation patterns of a dielectric lens placed on a PAA. Furthermore, [21] presents a method for determining the losses due to material absorption and reflection. The method in [21] is here adapted to design three different lenses to increase gain at high scanning angles. To gain more flexibility in design, we now define the dielectric layer profiles using splines instead of conic equations. The first lens designed is made up of a core layer and two matching layers to minimize reflections. The second is a lens composed of five different dielectric layers, and the last design is a nonuniform multilayer lens, specifically designed to enhance the gain at 60◦without significantly decreasing the efficiency at smaller angles. II. DESIGN PROCEDURE A two-dimensional RT-PO model is employed for the design of multilayer dielectric lenses. This RT-PO model can efficiently calculate the nonlinear phase distribution and radiation patterns of a PAA in combination with a dielectric lens. In addition, it can calculate the losses due to material absorption This article has been accepted for publication in IEEE Antennas and Wireless Propagation Letters. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/LAWP.2024.3507178 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. XX, NO. XX, MONTH 202X 2 and reflections between different dielectric layers, as indicated in [21]. Radiation patterns are calculated using geometric optics, the ray tube theory, and the Kirchhoff diffraction formula. Geometric optics is used to calculate the ray paths. First, we trace the rays from their aperture plane to the array plane. This is done to calculate the nonlinear phase distribution needed in the array antenna to maximize the directivity [22]. Second, we trace the rays from the array plane to the aperture plane using the phase distribution previously calculated. From this step, we obtain the path length and spatial distribution of the rays. The path length gives us the phase of the electric field in the aperture while the distance between rays is used in the ray tube theory to calculate the amplitude distribution of this electric field. Finally, the Kirchhoff diffraction formula is used to calculate the electric far field from these values of phase and amplitude [20]. In the Kirchhoff diffraction formula, we can take into account the losses from material absorption assuming a complex wavenumber [19] and the losses from reflections by means of a multiplicative transmission coefficient of each ray. The transmission coefficient is determined by modifying the reflection and transmission coefficients in a multilayer planar dielectric medium [21]. Due to the 2D nature of the present approach, determining the antenna’s directivity is not feasible. Instead, we evaluated the variation in peak amplitude by comparing the 3-dB beamwidth of the radiation patterns at different scanning angles with and without the presence of the dielectric lens. If the 3-dB beamwidth of the PAA with the antenna is narrower than that of the PAA alone at a specific angle, the antenna gains in directivity. In addition, the efficiency of the lens antenna is calculated by comparing the peak magnitude with and without losses. In the authors’ experience, this simplified and fast procedure provides accurate enough results when compared to 3-D full-wave simulations. In previous research, the surfaces that defined the dielectric lens followed conic equations [11], [14], [21]. Here, we use splines to define the lenses, as they provide more degrees of freedom. We define the spline curves with the coordinates ρand zof four points (see Fig. 1), as this is found to give enough degrees of freedom while keeping a simple design. Rotational symmetry is assumed with respect to the z-axis. This gives a total of 6 degrees of freedom per curve (ρ0= 0, z3= 0) compared to the three degrees of freedom that the conic equation has. No optimization algorithm has been used to design lens profiles. Instead, a user-controlled procedure similar to that proposed in [14] was carried out. It consists Fig. 1: Definition of a spline curve. 0 30 60 90 3 (deg) -2 0 2 4 6 8 10 TSL (dB) Standalone array h=L/10 h=L/3 h=L/2 h=2L/3 h=L Fig. 2: Theoretical scan losses (TSL) of different radome heights, with Lthe array length. of parameter sweeps where one of the ρor zcoordinates is changed at a time until optimal results are achieved. Since the ray-tracing model is significantly faster than full-wave simulations (8 different scanning angles are simulated in a few seconds), numerous iterations can be quickly carried out to achieve an optimal shape. A performance metric is defined to predict the maximum achievable gain of a dielectric lens based on its geometric size. This metric, termed “theoretical scan losses” (TSL), is given in terms of the surface projection as TSL = cos θ+h Lsin θ(1) where θis the scan angle, his the lens height, and Lis the array length. For a standalone array (h= 0), the TSL follows the cosine function, as clearly expected for planar arrays. Fig. 2 shows the TSL vs. the scan angle for different lens heights. Note that a practical lens can provide a gain close to that shown in Fig. 2only within a limited scanning range. In this work, we will consider a lens height of h=L/2and try to maximize the gain at 60◦. From Fig. 2, we observe that the maximum achievable gain with these parameters is 2.7 dB. This value will be used as a physical limit for the last design presented in Sec. V. III. DIELECTRIC LENS DESIGN BASED ON SPLINE CURVES As a first assessment of the use of splines to define the shape of the dielectric lens, we design a lens with one core layer. Matching layers are added on each side of the lens to reduce reflections. The spline parameters that define the TABLE I: Spline parameters for the lens with 3 dielectric layers (one core layer with n=√2.6and two thin matching layers with nML =√1.7). Surface 1 z10 ρ11 z11 ρ12 z12 ρ13 59 23 46 69 11 80 Surface 2 z20 ρ21 z21 ρ22 z22 ρ23 71 70 54 109 18 125 Data in mm. This article has been accepted for publication in IEEE Antennas and Wireless Propagation Letters. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/LAWP.2024.3507178 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. XX, NO. XX, MONTH 202X 3 0 20 40 60 80 3 (deg) -20 -15 -10 -5 0 Norm. Radiation Pattern (dB) Comsol (2D) CST (3D) RT-PO (2D) (a) (b) 0 10 20 30 40 50 60 70 3 (deg) -3 -2 -1 0 1 2 3 Peak Magnitude Difference (dB) 0r=2.4 0r=2.6 0r=2.8 (c) Fig. 3: (a) Normalized radiation patterns for ϵr= 2.6. (b) Normalized peak difference for the dielectric lens shown in Table Icompared to that of a conic lens. (c) Normalized peak difference vs scanning angle for different values of ϵrfor the dielectric lens shown in Table I. All results are at 29 GHz. lower surface (surface 1) and the upper surface (surface 2) of the core layer of the lens are given in Table Itogether with a graphical illustration of this dielectric lens. The matching layers are defined by two parallel coatings of thickness 2 mm. The relative permittivity of the core layer is ϵr= 2.6and in the matching layers ϵr,ML = 1.7. The absorption losses are set to tan δ= 10−3n, mimicking the losses of the dielectric metamaterials [ϵ=ϵ0ϵr(1 −j tan δ)]. The central frequency of operation is 29 GHz. The upper surface is designed to have a flat shape near the broadside direction, as this configuration may be more appropriate for applications where flat radomes are desired. This characteristic is easily achieved using spline curves due to their greater versatility compared to conic equations. This versatility is also key to designing the shape of the other two lenses presented in Secs. IV and V. The normalized radiation patterns of the above dielectric lens for different scanning angles are plotted in Fig. 3(a). In this figure, we compare the results calculated using a 2D model in COMSOL (solid lines), a 3D model in CST (dotted lines), and our in-house RT-PO model (dashed lines). These simulations use the non-linear phase distribution previously obtained by the inverse RT approach [21], [22]. The three sets of radiation patterns agree well in the beamwidth and side lobe levels (SLL), with only relevant discrepancies in the SLL at 70◦. This comparison is therefore taken as a validation of our RT-PO model for shapes defined with splines. As radiation patterns maintain a narrow beamwidth throughout the scanning range (0 -70◦), very low scan losses are expected. Fig. 3(b) shows the difference in the peak magnitude for the lens considered. The results are compared with a lens defined with conic surfaces, which follows the equation proposed in [11] with the following parameters: h1=51 mm, c1=−6.5,k1=2, h2=0.06,c2=−4and k2= 3. To ensure a fair comparison, identical size constraints were applied to both lenses. Under these constraints, the conic-shaped lens has a very small impact on the gain for all scanning angles. Conversely, the spline lens shows an increase in gain starting from 50◦. The variation in the peak magnitudes for various values of ϵris depicted in Fig. 3(c), including the material variations in the matching layers. The results of the three cases are similar, indicating that the lens can tolerate manufacturing deviations. The differences observed near broadside for ϵr= 2.8can be attributed to the considerable reduction in the effective aperture caused by the lens shape. Fig. 3(c) shows that the antenna loses 0.15 dB in the broadside direction and starts to increase the directivity at 50◦with an increase of 1.23 dB at 60◦. IV. UNIFORM MULTILAYER LENS DESIGN The second configuration is a multilayer lens composed of five dielectric layers of three different materials as illustrated TABLE II: Surface parameters of the multilayer lens with 5 dielectric layers, n1=√1.3,n2=√1.7and n3=√2.6. Data in mm. Surface 1 Surface 2 z10 ρ11 z11 ρ12 z12 ρ13 z20 ρ21 z21 ρ22 z22 ρ23 50 15 47 72 7 80 53 25 51 77 12 92 Surface 3 Surface 4 z30 ρ31 z31 ρ32 z32 ρ33 z40 ρ41 z41 ρ42 z42 ρ43 58 33 52 86 16 99 61 43 58 103 10 107 Surface 5 Surface 6 z50 ρ51 z51 ρ52 z52 ρ53 z60 ρ61 z61 ρ62 z62 ρ63 66 51 63 106 23 115 71 72 64 114 22 125 Data in mm. This article has been accepted for publication in IEEE Antennas and Wireless Propagation Letters. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/LAWP.2024.3507178 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. XX, NO. XX, MONTH 202X 4 0 20 40 60 80 3 (deg) -20 -15 -10 -5 0 Norm. Radiation Pattern (dB) Comsol RT-PO (a) 0 10 20 30 40 50 60 3 (deg) -3 -2 -1 0 1 2 3 Peak Magnitude Difference (dB) Ideal Absorption losses Absorption+Reflections (b) Fig. 4: (a) Normalized radiation patterns, and (b) normalized peak difference vs scanning angle of the multilayered lens in Table II. The results are at 29 GHz. in Table II. This design incorporates refractive indices of n1=√1.3,n2=√1.7, and n3=√2.6. The absorption losses are again set to tan δ= 10−3n. The parameters of the six surfaces of the lens are given in Table II. The normalized radiation patterns calculated with Comsol and the RT-PO model are shown in Fig. 4(a). The results agree well in both beamwidth and SLL, except for some disagreement in SLL at 60◦, likely due to diffraction. This shows the ability of the proposed RT-PO model to efficiently calculate the radiation patterns of dielectric lenses with several non-parallel layers. Interestingly, the 3dB-beamwidth at 60◦is narrower than at lower scanning angles, which shows the good behavior of this lens. Figure 4(b) shows the maximum magnitude difference calculated with the RT-PO model for three cases: the ideal case without losses, with only absorption losses, and with both absorption and reflections. The absorption losses remain constant around 0.18 dB for all scanning angles. However, the reflections losses decrease for higher scanning angles. At 0◦ the peak magnitude difference with respect to the standalone array is −1.5dB. At 60◦, the reflection losses are almost null since the design was made to maximize the gain at this angle taking into account the reflection losses. Moreover, the peak difference is 1.7 dB. V. NON-UNIFORM MULTILAYER LENS DESIGN The third design considered in this work consists of an optimal lens shape to increase the gain at 60◦. The inset of Fig. 5illustrates the geometric configuration of this lens z ρ n1 n2 Physical limits Fig. 5: Scan loss of the non-uniform multilayer lens shown as inset. The results are at 29 GHz. with refractive indices of n1=√1.7and n2=√2.6. A key aspect of this design is that it has one layer that does not touch the zand ρaxes. Fig. 5shows the scan loss of the standalone array compared to the designed lens (these results do not account for the losses due to reflections). As in previous cases, absorption losses are taken into account in the RT-PO considering tan δ= 10−3n. The lens starts to increase the gain from 40◦and, at 60º, the lens increases the gain 2.29 dB; getting very close to the physical limits. Compared to the uniform multilayer lens, the maximum loss in gain close to broadside (0º≤θ≤30º) is 0.48 dB. The previous results demonstrate that achieving a high gain (close to the physical limits) for a specific angle is feasible if the lens’s shape is appropriately selected. VI. CONCLUSION We proposed a RT-PO model to efficiently design multilayer dielectric lenses in combination with array antennas. The RTPO model can calculate the radiation patterns using a nonlinear phase distribution at the array and taking into account the reflections and the losses due to material absorption. Three different designs have been made: one lens with a core layer and two matching layers, one lens with five dielectric layers, and a non-uniform multilayer lens. The goal of these lenses is to increase the gain of the array antenna at 60◦. With the first lens, we show that using spline curves to define the dielectric interfaces, we can obtain more flexibility in the lens shape compared to the conic shape. We also showed that we can increase the gain in the angular range of 50º to 70º. To increase even more the flexibility of the design and get closer to the physical limits of the lens, multilayer dielectric lenses are also investigated. 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Martini, S. Monni, M. C. Vigan´ o, F. Silvestri, E. Gandini, G. Gerini, G. Toso, and S. Maci, “Phase-gradient meta-dome for increasing grating-lobe-free scan range in phased arrays,” IEEE Trans. Antennas Propag., vol. 66, no. 8, pp. 3973–3982, 2018. This article has been accepted for publication in IEEE Antennas and Wireless Propagation Letters. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/LAWP.2024.3507178 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/