# Planar Lens Antenna for High Data Rate Applications

Authors: Daniel Santillán-Haro; Daniel Sánchez-Escuderos; Eva Antonino-Daviu; Miguel Ferrando-Bataller
Journal: Wireless Communications and Mobile Computing
 (2019)

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## Abstract

A low-profile lens antenna formed by 2 metallic rings with strips short-circuiting both rings is presented. The theory of characteristic modes is used to facilitate the design, optimization, and analysis of the structure. Simulations and measurements are presented for the optimized single-layer metallic lens antenna. Measured results show a large operating bandwidth (14.7% relative -14 dB impedance bandwidth) with a maximum directivity above 13.70 dBi and a return loss better than 14 dB.


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## Body

## 1. Introduction

The increase of small cell base stations, wireless access points, laptops, tablets, and smartphones, each with a computing capacity comparable to that of a computer server a decade ago, and the growing demand for multimedia services are leading to an increase in the traffic of associated broadband connections, with high access speed and low latency [1]. From the perspective of the system, these needs require the use of low-profile directive antennas that maximize the gain, and exhibit energy-focusing capability.Over the past few years, metal lenses have attracted an increasing interest as high-gain antennas in multiple applications. Several types of lenses have been proposed to improve the gain and efficiency of simple radiators. These lenses are commonly designed using periodic structures, such as EBG (Electromagnetic Band Gap) or FSS (Frequency Selective Surfaces). In [2], a nonuniform three-layer FSS lens is presented, whereas, in [3], a metallic luneburg lens is proposed with pin-loaded holes with glide symmetry to increase the equivalent refractive index of the structure.Periodic structures have also been used to design transmitarrays [4, 5], and reflectarrays [6], based on the analysis of the unit cell with infinitely periodic boundary conditions. In [7, 8], thin planar lenses are proposed for massive MIMO applications, and, in [9], gradient metasurfaces are used to construct a planar lens antenna system. In all cases, the truncation to a finite number of cells from the theoretical infinite number leads to inaccurate results, thereby demanding a costly optimization process.Alternatively, the design might be carried out using the Theory of Characteristic Modes (TCM). The TCM facilitates the design process because it provides a very clear understanding of the radiation behavior of metal and dielectric bodies [10]. This theory was initially formulated by Garbacz [11] and improved by Harrington and Mautz in [12], where they explained how to compute the characteristics eigenvalues and eigencurrents for a conducting body.For more than a decade, the TCM has been used to design mobile antennas [13], analyze slotted planar structures [14, 15], study some canonical structures such as spheres or cylinders [16], design wideband omnidirectional antennas [17], or provide design guidelines for open slot antennas embedded in finite platforms [18]. The TCM has also been used to analyze metallic lenses, such as a lens formed by twelve metallic rings distributed in a single-layer in [19], a low-profile lens made of metallic cylinders [20], or a metallic lens designed with a central circular metallic ring surrounded by a set of eight metallic rings.In this paper, we propose the use of the TCM not only to analyze but also to design the lens antenna. In particular, the TCM is used to design a single-layer metallic lens antenna formed by two short-circuited metallic rings and fed by a rectangular waveguide. A similar structure was developed in [21], where the results of the simulation of the lens were presented, but in this work we demonstrate the usefulness of the TCM for the design of the lens, presenting measured results as well. Characteristic modes (CM) have been calculated using FEKO [22], and the design of the lens has been performed with CST Microwave Studio [23].


## 2. Analysis of Characteristic Modes of the Proposed Metallic Structure

### 2.1. Structure

Considering that the CM analysis is based on the structure without feeder, for the initial dimensions, the lens is modeled as a PEC object identical to that shown in Figure1(a), with a thickness of 0.35 mm, a width of w=1.37 mm, an inner diameter Di=29.32 mm (2λ0), a shorting strip w1=4.04 mm, and an external diameter Do=42.89 mm (3λ0), where λ0 is the free-space wavelength at the central operating frequency (20.50 GHz). The lens antenna consists of two short-circuited metallic rings, whose sectors are no larger than λ at the design frequency, distributed in twenty-four sectors as shown in Figure 1(a), intended to operate in the 19−22 GHz band. The lens is placed at a distance f from the feeder.Figure 1
Geometry of the proposed antenna: (a) metallic lens antenna; (b) lateral view.
(a) (b)


### 2.2. Analysis of Characteristic Modes

As discussed in [10–12], the TCM provides a physical interpretation of the radiation by a given structure, which is of great help in antenna design. The CM are real current modes (Jn) that are extracted at each frequency from the generalized impedance matrix of the antenna. These modes form a set of orthogonal functions that can be used to expand the total current in the surface of the structure, as described in the following equations:(1)J=∑nVniJn1+jλn(2)Vni=∮sJn•Eids,where Jn are the characteristic currents on the conducting body, λn are the eigenvalues, and Vni is the modal excitation coefficient. Despite Jn and λn being the only two quantities solved in the base equation, there are other physical interpretations of the eigenvalue, such as the characteristic angle (αn) and the modal significance (MSn). The characteristic angle (αn) is associated with each eigenvalue as(3)αn=180°-arctan⁡λn.The characteristic angle (αn) indicates how the phase of a single current changes over the frequency, and can be used to design antennas with a specific phase distribution [10]. The modal significance (MSn) is an intrinsic property of each mode and is independent of any specific external source. The MSn provides a convenient way to measure the bandwidth of each characteristic mode, mathematically expressed as(4)MSn=11+jλn.The mode is at resonance when the characteristic angle is 180° (αn=180°) or the modal significance is one (MSn=1).The half-power radiating bandwidth of a mode (BWn) constitutes an important figure of merit to determine the radiation performance of characteristic modes [10]. It can be expressed as BWn=(fU-fL)/fres, where fres is the resonance frequency of the mode and fL and fU are the lower and upper frequencies, respectively, for a MSn threshold of 1/2. Furthermore, a modal quality factor can be defined from the fractional bandwidth as Qrad,n=1/BWn [10].The modal currents of the first four modes of the proposed metallic structure are illustrated in Figure2. Black lines have been added to indicate the direction of the currents.Figure 2
Modal currents of the lens proposed at 20.5 GHz: (a) Mode 1; (b) Mode 1’; (c) Mode 2; (d) Mode 3.
(a) (b) (c) (d)Figure2(a) shows a very intense current distribution in the short-circuiting strips between the inner and outer rings for Mode 1. The orthogonal modal current (Mode 1’) is included in Figure 2(b). Mode 2 presents a vertical current distribution in the two metal rings (see Figure 2(c)), and Mode 3 (see Figure 2(d)) presents a loop inside and outside the total structure. Each modal current maintains this current distribution in the metallic structure over the whole frequency range (19−22 GHz).The radiated far field of the first four modes is shown in Figure3. From these radiation patterns it is clear that Mode 1, Mode 1’, and Mode 2 are the desirable modes in order to obtain a maximum radiation in the broadside direction. However, if a vertically polarized plane wave incides in the lens, only Mode 1 and Mode 2 will be excited.Figure 3
Modal radiation pattern of the proposed lens at 20.5 GHz: (a) Mode 1; (b) Mode 1’; (c) Mode 2; (d) Mode 3.
(a) (b) (c) (d)


### 2.3. Design of the Lens Antenna

As seen, fundamental Mode 1 presents a broadside radiation pattern at the central frequency. In order to excite this fundamental Mode 1, a parametric study ofαn and MS for different lens sizes is performed. The aim is to have a resonance for Mode 1 at the central frequency, together with a large bandwidth.As a starting point for the design, the structure considered is the one presented in Section2.1. The two variables of the design are Di and w1. The number of sectors in the structure is left fixed, in order to keep the symmetry. Note that, considering w1=4.04 mm and a variable internal diameter Di, the value of the external diameter of the structure can be found by the relationship Do=w1+Di. With this in mind, we first present the results for a fixed w1 value (w1=4.04 mm) and Di ranging between 22 and 32 mm.Figure4(a) plots the phase values of Mode 1 for different Di values. As seen in the band of interest (19−22 GHz), Di values for the resonance of the structure (αn=180°) are within the range 26−30 mm. For Di≤24 mm, the structure has a capacitive behavior, and for Di≥32 mm it has an inductive behavior (αn<180°).Figure 4
Modal parameters of the proposed lens by varying the internal diameterDi with w1=4.04 mm: (a) characteristic angle Mode 1; (b) modal significance Mode1.
(a) (b)Figure4(b) shows the values of modal significance for Mode 1. As can be observed, for Di=28 mm, we have values higher than 0.98 in the whole band, which is a very good value above the commonly adopted 0.7 threshold to guarantee a large bandwidth.From the above results, it can be easily deduced that forDi=28 mm there is a resonance at the central frequency 20.5 GHz and a high modal significance for Mode 1.Figure5 presents the characteristic angle αn of the proposed metallic structure by varying Di and w1 for Mode 1, at 20.5 GHz. As can be seen, Mode 1 resonates for Di=27.87 mm with w1=4.04 mm. There exists other possible values, but these lead to the smallest lens.Figure 5
Characteristic angle of the proposed lens by varyingDi and w1 at 20.5 GHz for Mode 1.Figure6 represents the contribution of each mode to the total radiated power for the proposed lens with Di = 27.87 mm and w1=4.04 mm, when excited with a vertically polarized plane wave with normal incidence. As can be seen, at the central frequency, a 60% of Mode 1 and a 40% of Mode 2 generate the total radiated power. Since these two modes have a maximum at broadside direction, the total power field will also have a maximum at this direction.Figure 6
Contribution of the each mode to the total radiated power for the proposed lens withDi=27.87 mm and w1=4.04 mm considering plane wave with normal incidence.


## 2.1. Structure

Considering that the CM analysis is based on the structure without feeder, for the initial dimensions, the lens is modeled as a PEC object identical to that shown in Figure1(a), with a thickness of 0.35 mm, a width of w=1.37 mm, an inner diameter Di=29.32 mm (2λ0), a shorting strip w1=4.04 mm, and an external diameter Do=42.89 mm (3λ0), where λ0 is the free-space wavelength at the central operating frequency (20.50 GHz). The lens antenna consists of two short-circuited metallic rings, whose sectors are no larger than λ at the design frequency, distributed in twenty-four sectors as shown in Figure 1(a), intended to operate in the 19−22 GHz band. The lens is placed at a distance f from the feeder.Figure 1
Geometry of the proposed antenna: (a) metallic lens antenna; (b) lateral view.
(a) (b)


## 2.2. Analysis of Characteristic Modes

As discussed in [10–12], the TCM provides a physical interpretation of the radiation by a given structure, which is of great help in antenna design. The CM are real current modes (Jn) that are extracted at each frequency from the generalized impedance matrix of the antenna. These modes form a set of orthogonal functions that can be used to expand the total current in the surface of the structure, as described in the following equations:(1)J=∑nVniJn1+jλn(2)Vni=∮sJn•Eids,where Jn are the characteristic currents on the conducting body, λn are the eigenvalues, and Vni is the modal excitation coefficient. Despite Jn and λn being the only two quantities solved in the base equation, there are other physical interpretations of the eigenvalue, such as the characteristic angle (αn) and the modal significance (MSn). The characteristic angle (αn) is associated with each eigenvalue as(3)αn=180°-arctan⁡λn.The characteristic angle (αn) indicates how the phase of a single current changes over the frequency, and can be used to design antennas with a specific phase distribution [10]. The modal significance (MSn) is an intrinsic property of each mode and is independent of any specific external source. The MSn provides a convenient way to measure the bandwidth of each characteristic mode, mathematically expressed as(4)MSn=11+jλn.The mode is at resonance when the characteristic angle is 180° (αn=180°) or the modal significance is one (MSn=1).The half-power radiating bandwidth of a mode (BWn) constitutes an important figure of merit to determine the radiation performance of characteristic modes [10]. It can be expressed as BWn=(fU-fL)/fres, where fres is the resonance frequency of the mode and fL and fU are the lower and upper frequencies, respectively, for a MSn threshold of 1/2. Furthermore, a modal quality factor can be defined from the fractional bandwidth as Qrad,n=1/BWn [10].The modal currents of the first four modes of the proposed metallic structure are illustrated in Figure2. Black lines have been added to indicate the direction of the currents.Figure 2
Modal currents of the lens proposed at 20.5 GHz: (a) Mode 1; (b) Mode 1’; (c) Mode 2; (d) Mode 3.
(a) (b) (c) (d)Figure2(a) shows a very intense current distribution in the short-circuiting strips between the inner and outer rings for Mode 1. The orthogonal modal current (Mode 1’) is included in Figure 2(b). Mode 2 presents a vertical current distribution in the two metal rings (see Figure 2(c)), and Mode 3 (see Figure 2(d)) presents a loop inside and outside the total structure. Each modal current maintains this current distribution in the metallic structure over the whole frequency range (19−22 GHz).The radiated far field of the first four modes is shown in Figure3. From these radiation patterns it is clear that Mode 1, Mode 1’, and Mode 2 are the desirable modes in order to obtain a maximum radiation in the broadside direction. However, if a vertically polarized plane wave incides in the lens, only Mode 1 and Mode 2 will be excited.Figure 3
Modal radiation pattern of the proposed lens at 20.5 GHz: (a) Mode 1; (b) Mode 1’; (c) Mode 2; (d) Mode 3.
(a) (b) (c) (d)


## 2.3. Design of the Lens Antenna

As seen, fundamental Mode 1 presents a broadside radiation pattern at the central frequency. In order to excite this fundamental Mode 1, a parametric study ofαn and MS for different lens sizes is performed. The aim is to have a resonance for Mode 1 at the central frequency, together with a large bandwidth.As a starting point for the design, the structure considered is the one presented in Section2.1. The two variables of the design are Di and w1. The number of sectors in the structure is left fixed, in order to keep the symmetry. Note that, considering w1=4.04 mm and a variable internal diameter Di, the value of the external diameter of the structure can be found by the relationship Do=w1+Di. With this in mind, we first present the results for a fixed w1 value (w1=4.04 mm) and Di ranging between 22 and 32 mm.Figure4(a) plots the phase values of Mode 1 for different Di values. As seen in the band of interest (19−22 GHz), Di values for the resonance of the structure (αn=180°) are within the range 26−30 mm. For Di≤24 mm, the structure has a capacitive behavior, and for Di≥32 mm it has an inductive behavior (αn<180°).Figure 4
Modal parameters of the proposed lens by varying the internal diameterDi with w1=4.04 mm: (a) characteristic angle Mode 1; (b) modal significance Mode1.
(a) (b)Figure4(b) shows the values of modal significance for Mode 1. As can be observed, for Di=28 mm, we have values higher than 0.98 in the whole band, which is a very good value above the commonly adopted 0.7 threshold to guarantee a large bandwidth.From the above results, it can be easily deduced that forDi=28 mm there is a resonance at the central frequency 20.5 GHz and a high modal significance for Mode 1.Figure5 presents the characteristic angle αn of the proposed metallic structure by varying Di and w1 for Mode 1, at 20.5 GHz. As can be seen, Mode 1 resonates for Di=27.87 mm with w1=4.04 mm. There exists other possible values, but these lead to the smallest lens.Figure 5
Characteristic angle of the proposed lens by varyingDi and w1 at 20.5 GHz for Mode 1.Figure6 represents the contribution of each mode to the total radiated power for the proposed lens with Di = 27.87 mm and w1=4.04 mm, when excited with a vertically polarized plane wave with normal incidence. As can be seen, at the central frequency, a 60% of Mode 1 and a 40% of Mode 2 generate the total radiated power. Since these two modes have a maximum at broadside direction, the total power field will also have a maximum at this direction.Figure 6
Contribution of the each mode to the total radiated power for the proposed lens withDi=27.87 mm and w1=4.04 mm considering plane wave with normal incidence.


## 3. Lens Design

Considering the results obtained in previous section, the optimum values for the metallic lens in order to resonate at 20.5 GHz areDi=27.87 mm and w1=4.04 mm, which means that the external diameter Do is 41.43 mm. Note that the width and thickness are the same as in the structure presented in Section 2.1.The next step is to find the optimal feeding point of the lens. This point corresponds to the distance between the lens and the feeding aperture, where the resulting radiated electromagnetic waves have a plane wave front at the upper level of the metallic structure.


### 3.1. Plane Wave Analysis (Focal Point)

The focal point can be determined using several methods such as geometrical optics used in the Fresnel zone plate [24], or the plane wave incidence described in [19], where a plane wave impinges on the metallic structure. In this work, we use the second method, considering an incident field produced by a plane wave propagating in the z direction, the lens being located in the x−y plane. Figure 7 shows the electric field with this configuration at 20.5 GHz. It can be seen that the field is focused on the other side of the metallic structure, at f=9.54 mm from the lens. This is the focus of the lens and is where the feeder must be placed.Figure 7
Simulation of the plane wave at 20.5 GHz in the proposed lens.The feeder is a square aperture (10.66 mm × 10.66 mm). This size guarantees a −10 dB taper on the edge of the lens. The aperture is placed on a ground plane (90 mm × 90 mm) which only introduces a small ripple in the radiation pattern of the lens. The aperture is fed by a transition of lengthl=10.66 mm to allow the feeding of the structure by means of a standard rectangular waveguide (WR-42). Figure 8 shows a picture of the prototype. The feeder has been fabricated in aluminum (see Figure 8(a)), while the single-layer lens antenna has been fabricated in a copper sheet, as shown in Figure 8(b), and attached to a foam layer with the same height as the focal distance f (see Figure 8(c)), to guarantee the correct separation between the lens and the feeder.Figure 8
Photograph of the manufactured prototype of the proposed lens antenna: (a) feeder, (b) single-layer lens antenna, and (c) lens with feeder.
(a) (b) (c)


## 3.1. Plane Wave Analysis (Focal Point)

The focal point can be determined using several methods such as geometrical optics used in the Fresnel zone plate [24], or the plane wave incidence described in [19], where a plane wave impinges on the metallic structure. In this work, we use the second method, considering an incident field produced by a plane wave propagating in the z direction, the lens being located in the x−y plane. Figure 7 shows the electric field with this configuration at 20.5 GHz. It can be seen that the field is focused on the other side of the metallic structure, at f=9.54 mm from the lens. This is the focus of the lens and is where the feeder must be placed.Figure 7
Simulation of the plane wave at 20.5 GHz in the proposed lens.The feeder is a square aperture (10.66 mm × 10.66 mm). This size guarantees a −10 dB taper on the edge of the lens. The aperture is placed on a ground plane (90 mm × 90 mm) which only introduces a small ripple in the radiation pattern of the lens. The aperture is fed by a transition of lengthl=10.66 mm to allow the feeding of the structure by means of a standard rectangular waveguide (WR-42). Figure 8 shows a picture of the prototype. The feeder has been fabricated in aluminum (see Figure 8(a)), while the single-layer lens antenna has been fabricated in a copper sheet, as shown in Figure 8(b), and attached to a foam layer with the same height as the focal distance f (see Figure 8(c)), to guarantee the correct separation between the lens and the feeder.Figure 8
Photograph of the manufactured prototype of the proposed lens antenna: (a) feeder, (b) single-layer lens antenna, and (c) lens with feeder.
(a) (b) (c)


## 4. Experimental Results of the Single-Layer Lens Antenna

To validate the performance of the proposed single-layer lens antenna, theS11 parameter has been measured. Figure 9 compares measured and simulated results. As can be observed, both results are quite similar, with a good matching (S11<-14 dB) from 19 to 22 GHz.Figure 9
Simulated and measuredS11 parameter of the proposed metallic lens antenna.The measured and simulated copolar and crosspolar components of the radiated field are shown in Figure10. A good measured crosspolar level (below -27 dB) is observed within the main beam of the pattern. The differences between the measured and simulated patterns are caused by an inaccurate characterization of the foam material between the lens and the feeder and, also, by unexpected manufacturing errors.Figure 10
Measured (MEAS) and simulated (SIM) copolar and crosspolar components of the radiation pattern at 20.5 GHz.Figure11 shows that the maximum measured gain is 14 dBi at 19 GHz, which corresponds to an increase in directivity of 5.23 dB with respect to the measurement of an open-ended square waveguide.Figure 11
Comparison of the maximum gain, measured (MEAS) and simulated (SIM), of the proposed lens.It can be clearly observed that the proposed antenna has an operational bandwidth of 14.7% forS11<-14 dB, and a radiation efficiency above 90%.


## 5. Conclusion

A low profile metallic lens antenna formed by 2 short-circuited metallic rings has been proposed. The single-layer lens antenna has been designed using the Theory of Characteristic Modes to facilitate the design process and evaluate the performance of currents in the metallic structure. Note that a final optimization has not been required. An open-ended square waveguide is used as the primary feed for the metallic lens. The resulting structure improves the maximum directivity with respect to the feeder along a large bandwidth (more than 5 dB) and provides a good crosspolar level (better than -27 dB), with a measured directivity above 13.70 dB.



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*Source: 5125287-2019-02-04.xml*