# Parametric Analysis of Negative and Positive Refractive Index Lens Antenna by ANSYS HFSS

Authors: Phan Van Hung; Nguyen Quoc Dinh; Yoshihide Yamada; Naobumi Michishita; Mohammad Tariqul Islam
Journal: International Journal of Antennas and Propagation
 (2020)

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## Abstract

Lens antennas with multibeam, high gain, and low sidelobe level are potential candidates for base station antennas in 5G mobile communication. In this paper, the authors perform simulation and parametric analysis of a lens antenna with positive and negative refractive indexes (NRI) using the modern electromagnetic field simulation software ANSYS HFSS. The simulation results of structures and theoretical calculations are analyzed and compared. The simulation results show the effectiveness of using negative refractive index lens antennas to minimize the dimension. The lens thickness with a negative refractive index decreased from 24.5 mm to 6.1 mm compared to the positive refractive index lens’s thickness. The results also indicate the similarities in gain, sidelobe level, amplitude, and electric field distribution on the aperture plane of the negative and positive refractive indexes (PRI) lens antennas compared to the theoretical calculation. In addition, the authors simulate a lens structure with additional quarter wavelength matching layers (MLs) to estimate the antireflection performance.


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## Body

## 1. Introduction

The fifth generation mobile system is now under development in the world [1–3]. The features of 5G in radio wave technology are millimeter wave frequency, small-size cells, and multiple input and multiple output (MIMO) systems [4, 5]. The new technology will allow a 5G antenna to low latency, low path loss, multibeam, beam scanning ability, broadband, and stable radiation pattern. In the massive MIMO, the base station antenna should assign one access line corresponding to one radio beam to one user [5, 6]. Thus, the base station antenna requires a multibeam radiation pattern like Figure 1. By using millimeter wave, the size of the base station antenna is reduced to about 30 cm; hence, types of antennas such as reflector, phased arrays, and dielectric lens antennas are proposed. As for a reflector antenna, a bifocal dual reflector and a reflectarray are proposed for multibeam application [7–9]. A phased array antenna is combined with a metasurface lens for extending the angular scan range, broadband, and multibeam [10–17]. As for a dielectric lens antenna, it has better multibeam radiation patterns than those of a dual reflector antenna, and this type of antenna also has structural features such as no blockage and having the same size beam forming surfaces of the front and rear surfaces, and stable radiation pattern [18–22]. However, on the downside, the thickness of dielectric lens antenna becomes larger. Meanwhile, when the refractive index is changed from the plus value to the minus value, the lens thickness can be remarkably reduced [23, 24]. In the study [25], the authors have designed simulated electromagnetic fields and measured the structure of shaped dielectric lens antennas with refractive index n=1.96 (Teflon) operating at 60 GHz using FEKO software. The results demonstrate very good agreements between the simulation results and the measured results.Figure 1
A multibeam lens antenna model for the base station.However, there have been no studies of the simulation and comparison of negative and positive refractive index lens antennas in the electromagnetic environment. Therefore, in this paper, the authors research and simulate negative and positive refractive index lens antennas using ANSYS HFSS electromagnetic simulator. The value of the positive refractive index (n=2) is approximately equal to that of Teflon refractive index (n=1.96), which is studied and experimented in reference [25]. The comparison and analysis of lens size, gain, sidelobe level, and electric field distribution on the antenna planes and the effects of multiple reflections are performed. Also, the results are compared with the theoretical calculation of lens antennas. This paper consists of 5 sections. Section 2 shows the structures of lens antennas with negative and positive refractive indexes and fundamental expressions in determining the curve surface and the theoretical calculation of the lens antennas. Simulation conditions, parameters, and the matching layers structure are presented in section 3. Section 4 illustrates the simulation results, the comparison, and the discussion. Finally, the conclusions are summarized in section 5.


## 2. Antenna Structures

### 2.1. Lens Antenna Features and Structures

The geometries of the negative and positive refractive index lens antennas are shown in Figure2. The lenses are rotationally symmetric around the Z-axis. The conical horn antenna is used as a feed. The lens materials have the refractive indexes of n=−2 and n=2. The lens has a basic structure, and its inner surface is curved and the outer surface is planar. The inner surface is generated by [20, 26, 27](1)r=n−1Fncosθ−1,where r is the distance from the focal point to the curve lens surface. The focal point is located at the origin of the coordinates axis. F is the distance between the focal point and the lens vertex. θ is the angle from the focal point to the inner lens surface and the Z-axis. n is a refractive index. εr is relative permittivity. μr is relative permeability. θi and θr indicate an incident angle and a refracted angle at a curve lens surface, respectively. These angles satisfy Snell’s law.(2)n=sinθisinθr,(3)n=εrμr.Figure 2
Lens antenna structures. (a) Horn with PRI lens (n=2), (b) Horn with NRI lens (n=−2).
(a)(b)The distance from the origin to the lens vertex and the diameter of the lens areF = D = 100 mm. T indicates the lens thickness. The rays from the feed horn are refracted at the curve lens surface, which makes outgoing rays parallel to the Z-axis.For a symmetrical axis lens with a feed horn at focal point, in Figure2, Ep2θdθ is the power radiated per unit length by the feed horn between the angles θ and θ+dθ. Then, if Ed2x is the power per unit length in the corresponding aperture interval between x and x+dx, where x is the distance (x=rsinθ) from a point on the aperture plane to the oz axis, from equation (1), x distance is given by the following equation:(4)x=Fn−1sinθncosθ−1,(5)dxdθ=n−1Fn−cosθncosθ−12.Reflection from the lens surface is neglected. For the hyperbolic surface, the relationship between the electric field distribution on the aperture plane and the electric field of the feed horn is given by [27](6)Ed2xEp2θ=dθdx.From equations (5) and (6), the electric field distribution on the aperture plane is determined by:(7)Ed2x=dθdxEp2θ=ncosθ−13Ep2θn−12F2n−cosθ.Applying equation (1) to determine the points on the upper curve lens surface on the xz plane with the negative and positive refractive indexes is shown in Table 1.Table 1
Points on curve lens surface (F=D=100 mm).
θn=2n=−2xzxz0°010001005°8.83100.98.799.810°18.3103.917.599.415°29.3109.326.498.620°43.1118.335.597.422.04°50123.539.296.825°44.795.927.8°5094.9


### 2.2. Conical Horn Antenna Structures

The structure, fundamental dimensions, and the electric field of the conical horn antenna are illustrated in Figure3(a). The feed horn is used to illuminate a wide angle for the lens. The radiation pattern of the conical horn antenna is shown in Figure 3(b). The horn has the maximum gain of 15.15 dBi. The lens area is stationed in the horn radiation angle (θ) from −27.8° to 27.8°, where the horn electromagnetic radiation obtains −10 dB compared to the maximum value. The focus of lens is set at center of the horn throat.Figure 3
Parameters and the radiation pattern of the conical horn. (a) Conical horn antenna structure, (b) horn radiation patterns.
(a)(b)


## 2.1. Lens Antenna Features and Structures

The geometries of the negative and positive refractive index lens antennas are shown in Figure2. The lenses are rotationally symmetric around the Z-axis. The conical horn antenna is used as a feed. The lens materials have the refractive indexes of n=−2 and n=2. The lens has a basic structure, and its inner surface is curved and the outer surface is planar. The inner surface is generated by [20, 26, 27](1)r=n−1Fncosθ−1,where r is the distance from the focal point to the curve lens surface. The focal point is located at the origin of the coordinates axis. F is the distance between the focal point and the lens vertex. θ is the angle from the focal point to the inner lens surface and the Z-axis. n is a refractive index. εr is relative permittivity. μr is relative permeability. θi and θr indicate an incident angle and a refracted angle at a curve lens surface, respectively. These angles satisfy Snell’s law.(2)n=sinθisinθr,(3)n=εrμr.Figure 2
Lens antenna structures. (a) Horn with PRI lens (n=2), (b) Horn with NRI lens (n=−2).
(a)(b)The distance from the origin to the lens vertex and the diameter of the lens areF = D = 100 mm. T indicates the lens thickness. The rays from the feed horn are refracted at the curve lens surface, which makes outgoing rays parallel to the Z-axis.For a symmetrical axis lens with a feed horn at focal point, in Figure2, Ep2θdθ is the power radiated per unit length by the feed horn between the angles θ and θ+dθ. Then, if Ed2x is the power per unit length in the corresponding aperture interval between x and x+dx, where x is the distance (x=rsinθ) from a point on the aperture plane to the oz axis, from equation (1), x distance is given by the following equation:(4)x=Fn−1sinθncosθ−1,(5)dxdθ=n−1Fn−cosθncosθ−12.Reflection from the lens surface is neglected. For the hyperbolic surface, the relationship between the electric field distribution on the aperture plane and the electric field of the feed horn is given by [27](6)Ed2xEp2θ=dθdx.From equations (5) and (6), the electric field distribution on the aperture plane is determined by:(7)Ed2x=dθdxEp2θ=ncosθ−13Ep2θn−12F2n−cosθ.Applying equation (1) to determine the points on the upper curve lens surface on the xz plane with the negative and positive refractive indexes is shown in Table 1.Table 1
Points on curve lens surface (F=D=100 mm).
θn=2n=−2xzxz0°010001005°8.83100.98.799.810°18.3103.917.599.415°29.3109.326.498.620°43.1118.335.597.422.04°50123.539.296.825°44.795.927.8°5094.9


## 2.2. Conical Horn Antenna Structures

The structure, fundamental dimensions, and the electric field of the conical horn antenna are illustrated in Figure3(a). The feed horn is used to illuminate a wide angle for the lens. The radiation pattern of the conical horn antenna is shown in Figure 3(b). The horn has the maximum gain of 15.15 dBi. The lens area is stationed in the horn radiation angle (θ) from −27.8° to 27.8°, where the horn electromagnetic radiation obtains −10 dB compared to the maximum value. The focus of lens is set at center of the horn throat.Figure 3
Parameters and the radiation pattern of the conical horn. (a) Conical horn antenna structure, (b) horn radiation patterns.
(a)(b)


## 3. Designing Matching Layers and Simulation Conditions

### 3.1. Designing Matching Layers

A lens antenna with high-density dielectric material allows improving the energy transmission efficiency. However, using a lens whose material is a dielectric causes the worsening, which affects the radiation properties of the antenna. These effects are due to multiple reflections on the curve surface and inside the lens. This can be overcome by covering the lens with matching layers. Matching layers thickness is limited to a quarter wavelength [28–32].The designed lens structure with a quarter wavelength matching layers at the front and the rear of the lens and some layers parameters are shown in Figure4, where εr is relative permittivity of the lens. εML indicates permittivity of the matching layers (εML=εr). DML is the matching layer thickness (DML=λg/4). λg is the internal MLs wavelength.Figure 4
Lens structures with matching layers. (a) PRI lens with MLs. (b) NRI lens with MLs.
(a)(b)


### 3.2. Simulation Conditions and Parameters

The simulation parameters and the personal computer configurations are indicated in Table2. The ANSYS HFSS and FEKO electromagnetic simulators are used to calculate the electrical field on the planes and the electric field distribution on the aperture plane based on designed structures. The Multilevel Fast Multipole Method (MLFMM) is applied simultaneously in order to optimize the calculation ability, save the computer memory, and minimize the calculation time. The lenses are set with the relative permeability μr=1 and μr=−1 and the relative permittivity εr=2 and εr=−2 corresponding to the refractive index of n=2 and n=−2 , respectively. For the lenses with matching layers, the parameters are set: εML=2 , εML=−2 and DML≈2.25 . The distance between the focal point and lens vertex equals the diameter of the lens aperture (F=D=100 mm). The survey frequency is 28 GHz.Table 2
Simulation parameters.
Computer specificationCPUIntel xeon 5ELoaded memory128 GBSoftwareHFSS and FEKOSimulation methodMLFMMPositive and negative refractive index lens parametersLens diameter (mm)D100Distance from the origin to the lens vertex (mm)F100Relative permeabilityμr1−1Relative permittivityεr2−2Relative permeability MLsεML2−2Matching layer thickness (mm)DML2.25Survey frequency28 GHz


## 3.1. Designing Matching Layers

A lens antenna with high-density dielectric material allows improving the energy transmission efficiency. However, using a lens whose material is a dielectric causes the worsening, which affects the radiation properties of the antenna. These effects are due to multiple reflections on the curve surface and inside the lens. This can be overcome by covering the lens with matching layers. Matching layers thickness is limited to a quarter wavelength [28–32].The designed lens structure with a quarter wavelength matching layers at the front and the rear of the lens and some layers parameters are shown in Figure4, where εr is relative permittivity of the lens. εML indicates permittivity of the matching layers (εML=εr). DML is the matching layer thickness (DML=λg/4). λg is the internal MLs wavelength.Figure 4
Lens structures with matching layers. (a) PRI lens with MLs. (b) NRI lens with MLs.
(a)(b)


## 3.2. Simulation Conditions and Parameters

The simulation parameters and the personal computer configurations are indicated in Table2. The ANSYS HFSS and FEKO electromagnetic simulators are used to calculate the electrical field on the planes and the electric field distribution on the aperture plane based on designed structures. The Multilevel Fast Multipole Method (MLFMM) is applied simultaneously in order to optimize the calculation ability, save the computer memory, and minimize the calculation time. The lenses are set with the relative permeability μr=1 and μr=−1 and the relative permittivity εr=2 and εr=−2 corresponding to the refractive index of n=2 and n=−2 , respectively. For the lenses with matching layers, the parameters are set: εML=2 , εML=−2 and DML≈2.25 . The distance between the focal point and lens vertex equals the diameter of the lens aperture (F=D=100 mm). The survey frequency is 28 GHz.Table 2
Simulation parameters.
Computer specificationCPUIntel xeon 5ELoaded memory128 GBSoftwareHFSS and FEKOSimulation methodMLFMMPositive and negative refractive index lens parametersLens diameter (mm)D100Distance from the origin to the lens vertex (mm)F100Relative permeabilityμr1−1Relative permittivityεr2−2Relative permeability MLsεML2−2Matching layer thickness (mm)DML2.25Survey frequency28 GHz


## 4. Performance Results

### 4.1. The Electric Field Distribution on the yz Plane

Electric field distributions of a lens antenna with the refractive indexes ofn=2 and n=−2 are shown in Figure 5. It is clearly observed that the radiation rays from the conical horn antenna are spherical waves, which are refracted and reformed when reaching the curve inner lens surface, and the waveforms at the outer surface are planar. This is correct in both cases of n=2 and n=−2, and this meets the theoretical conditions of the lens antennas. In order to ensure the simulation accuracy of HFSS software, a comparison with FEKO software is performed. In the study [25], the authors have made the design and simulation using FEKO and conducted experimental measurements of the shaped dielectric lens antenna structure with a positive refractive index (n=1.96) operating on 60 GHz. The results in the study [25] show a similarity between simulation by FEKO and measured results. This demonstrates the simulation calculation accuracy of the FEKO software. Figure 5(c) shows the electromagnetic field distribution of a lens antenna with a positive refractive index (n=2) when simulated by the FEKO software. Comparing the results between Figures 5(a) and 5(c), it can be seen that the antenna's field distribution with the positive refractive index is uniform when simulated by two different software tools, HFSS and FEKO. This proves the precision in constructing the structure and setting the methods and parameter calculations.Figure 5
The electric field distributions on the yz plane. (a) PRI lens antenna (n=2). (b) NRI lens antenna (n=−2), (c) PRI lens antenna (n=2) by FEKO.
(a)(b)(c)


### 4.2. Radiation Patterns

The radiation patterns of the conical horn antenna and the lens antennas with different refractive indexes are illustrated in Figure6. The green dashed dotted line shows the radiation pattern when n=2, the blue dashed one represents the radiation pattern when n=−2, and the black dashed squared line indicates the radiation pattern of conical horn antenna without lens. Clearly, the gain of the lens antenna when n=−2 reaches 27.34 dBi while that of the lens antenna when n=2 is 27.20 dBi, which are significantly higher than the gain of the conical horn antenna without lens, at 15.15 dBi. This demonstrates the effectiveness in improving the ability of the radiation properties of the lens antenna.Figure 6
Radiation patterns of the horn antenna and the horn with PRI lens and NRI lens.The red solid dotted line represents the PRI antenna’s radiation pattern of the designed lens antenna with the same size, simulated by FEKO electromagnetic field software. The results show that the red solid dotted line and the green dashed dotted line are in good agreement. This shows the accuracy of the result in the simulation calculation using both software tools. Consequently, it can be concluded that the simulation results using HFSS software in the proposed model are accurate because the simulation process done by HFSS and FEKO software tools in the study [25] used the same calculation method, the multilevel fast multipole method (MLFMM). Thus, from the results of the electric field distribution, from Figures 5(a) and 5(c) and the radiation pattern of the lens antenna with a positive refractive index, we can see that the correlation results between the HFSS and FEKO software tools are done with the same calculation method to confirm the results' accuracy based on our proposed model.The simulation results are indicated in Table3. The sidelobe level in H-plane reaches the lowest value of -21.80 dB for a positive refractive index lens n=2, and −20.17 dB for a negative refractive index lens n=−2, while the sidelobe level of the conical horn antenna just gets −18.00 dB. Besides, the half power beam-widths (HPBW) of negative and positive refractive index lens antennas are fairly similar, at 7.78° and 7.13°, respectively. By contrast, the HPBW of a conical horn antenna without lens is 30.50°. The dimension of the lens is determined based on equation (1) and the structure-constructing method of using the simulator. From equation (1), we can see that when the refractive index of the lens is a positive value (n=2), the inner curved surface structure of the lens is convex and directed towards the feed horn. The lens thickness at the center is a maximum, 24.5 mm. However, when the lens's refractive index has a negative value (n=−2), the inner curved surface structure of the lens is concave, and the lens dimension has the thickest at the lens edge and is equal to 6.1 mm, which is significantly thinner than that of positive refractive index lens. As a result, negative refractive index lens antennas for base stations could be designed with smaller sizes reducing outer impacts.Table 3
Comparison results.
Antenna structuresnGain [dBi]SLL (dB)θB (deg.)Lens thickness (mm)Horn without lens15.15−18.0030.50Horn with PRI lens227.20−21.807.1324.5Horn with NRI lens−227.34−20.177.786.1Horn with PRI lens (FEKO)226.83−21.737.2324.5PRI lens with MLs2427.80−22.496.7029.1NRI lens with MLs−2427.30−18.527.4910.7


### 4.3. The Electric Field Amplitude Distribution on the Aperture Plane

The electric field amplitude distributions on the aperture plane of the lens antennas forn=2 and n=−2 are shown in Figures 7 and 8. In Figure 7, it is obvious that electric field distributions in both cases are uniform and symmetric around the Z-axis. In addition, Figure 8 shows that the electric field amplitude distribution of the lens antenna is theoretically calculated according to equation (7) and simulated with positive and negative refraction indexes. Thus, electric field amplitudes on the aperture plane distributed along the lens diameter line are high and tend to decrease gradually to the sides. The simulation results and theoretical calculations are quite similar. This shows the accuracy in the design and theoretical calculations. The ripples on the amplitude distribution lines in case of the simulation occur due to the reflection from the lens surface.Figure 7
The electric field distributions on the aperture plane. (a) PRI lens (n=2). (b) NRI lens (n=−2).
(a)(b)Figure 8
The comparison of electric field amplitude distributions on the aperture plane for PRI and NRI lens antennas based on theoretical and simulation.


### 4.4. The Effects of Matching Layers

The comparisons of the radiation patterns of the different refractive index lens antennas with and without matching layers on the yz plane are illustrated in Figure9. It can be observed that when utilizing quarter wavelength matching layers for the positive refractive index lens, the antenna gain increases by 0.60 dBi compared to the one without quarter wavelength matching layers, whereas the antenna gain is almost unchanged when using MLs for negative refractive index lens. The sidelobe level in H-plane also decreases from −21.80 dB to −22.49 dB with positive refractive index lens antennas and increases from −20.17 dB to −18.52 dB with negative refractive index lens antennas. The results show the efficiency of using a quarter wavelength matching layers for a positive refractive index lens antenna.Figure 9
The comparison of radiation patterns of lens antennas on the yz plane. (a) PRI lens with and without MLs. (b) NRI lens with and without MLs.
(a)(b)The investigation into the electric field distribution of lens antennas with and without quarter wavelength matching layers is presented in Figure10. Spillover and diffraction occur more clearly at the lens edge for a positive refractive index. In Figure 11, the electric field amplitude distributions on the aperture planes between the antenna lens with and without the quarter wavelength matching layers are compared with each other. The results show that, in the case of the positive refractive index lens antennas, the black solid line, which illustrates the field amplitude distribution of the lens antenna with matching layers, is smoother and less jagged than the red dashed one presenting the case without matching layers. In contrast, the field amplitude distribution line of the negative refractive index lens antenna with matching layers is even more rugged than without MLs. This shows the more uniform electric field distribution on the aperture plane of the positive refractive index lens antennas when using the quarter wavelength matching layers.Figure 10
The electric field distributions of lens antennas with and without the quarter wavelength matching layers. (a) PRI lens. (b) NRI lens.
(a)(b)Figure 11
Electric field amplitude distributions on the aperture planes. (a) PRI lens with and without MLs. (b) NRI lens with and without MLs.
(a)(b)The authors use the ray tracing method to investigate the effects of multiple reflection from the curve lens surface, as shown in Figure12. Accordingly, black solid lines are the incident rays that travel from the focal point to the inner surface of the lens where the refraction and reflection phenomena occur. Most of the parts of the incident ray power are refracted, which makes outgoing rays parallel to the Z-axis, the solid red lines, while the rest are reflected at the curve surface, the blue dashed lines. This phenomenon causes spillover and diffraction. As to the positive refractive index lens, the lens structure is convex. Thus, reflected rays from the curve lens surface tend towards diffraction at the lens edge. Meanwhile, reflected rays from negative refractive index lens surface tend to bounce back because of the concave lens structure. Spillover and diffraction can be solved by covering the lens with an electromagnetic absorber, shown as in the third picture of Figures 10(a) and 10(b).Figure 12
Multiple reflected rays on the curve lens antenna surface. (a) PRI lens. (b) NRI lens.
(a)(b)


## 4.1. The Electric Field Distribution on the yz Plane

Electric field distributions of a lens antenna with the refractive indexes ofn=2 and n=−2 are shown in Figure 5. It is clearly observed that the radiation rays from the conical horn antenna are spherical waves, which are refracted and reformed when reaching the curve inner lens surface, and the waveforms at the outer surface are planar. This is correct in both cases of n=2 and n=−2, and this meets the theoretical conditions of the lens antennas. In order to ensure the simulation accuracy of HFSS software, a comparison with FEKO software is performed. In the study [25], the authors have made the design and simulation using FEKO and conducted experimental measurements of the shaped dielectric lens antenna structure with a positive refractive index (n=1.96) operating on 60 GHz. The results in the study [25] show a similarity between simulation by FEKO and measured results. This demonstrates the simulation calculation accuracy of the FEKO software. Figure 5(c) shows the electromagnetic field distribution of a lens antenna with a positive refractive index (n=2) when simulated by the FEKO software. Comparing the results between Figures 5(a) and 5(c), it can be seen that the antenna's field distribution with the positive refractive index is uniform when simulated by two different software tools, HFSS and FEKO. This proves the precision in constructing the structure and setting the methods and parameter calculations.Figure 5
The electric field distributions on the yz plane. (a) PRI lens antenna (n=2). (b) NRI lens antenna (n=−2), (c) PRI lens antenna (n=2) by FEKO.
(a)(b)(c)


## 4.2. Radiation Patterns

The radiation patterns of the conical horn antenna and the lens antennas with different refractive indexes are illustrated in Figure6. The green dashed dotted line shows the radiation pattern when n=2, the blue dashed one represents the radiation pattern when n=−2, and the black dashed squared line indicates the radiation pattern of conical horn antenna without lens. Clearly, the gain of the lens antenna when n=−2 reaches 27.34 dBi while that of the lens antenna when n=2 is 27.20 dBi, which are significantly higher than the gain of the conical horn antenna without lens, at 15.15 dBi. This demonstrates the effectiveness in improving the ability of the radiation properties of the lens antenna.Figure 6
Radiation patterns of the horn antenna and the horn with PRI lens and NRI lens.The red solid dotted line represents the PRI antenna’s radiation pattern of the designed lens antenna with the same size, simulated by FEKO electromagnetic field software. The results show that the red solid dotted line and the green dashed dotted line are in good agreement. This shows the accuracy of the result in the simulation calculation using both software tools. Consequently, it can be concluded that the simulation results using HFSS software in the proposed model are accurate because the simulation process done by HFSS and FEKO software tools in the study [25] used the same calculation method, the multilevel fast multipole method (MLFMM). Thus, from the results of the electric field distribution, from Figures 5(a) and 5(c) and the radiation pattern of the lens antenna with a positive refractive index, we can see that the correlation results between the HFSS and FEKO software tools are done with the same calculation method to confirm the results' accuracy based on our proposed model.The simulation results are indicated in Table3. The sidelobe level in H-plane reaches the lowest value of -21.80 dB for a positive refractive index lens n=2, and −20.17 dB for a negative refractive index lens n=−2, while the sidelobe level of the conical horn antenna just gets −18.00 dB. Besides, the half power beam-widths (HPBW) of negative and positive refractive index lens antennas are fairly similar, at 7.78° and 7.13°, respectively. By contrast, the HPBW of a conical horn antenna without lens is 30.50°. The dimension of the lens is determined based on equation (1) and the structure-constructing method of using the simulator. From equation (1), we can see that when the refractive index of the lens is a positive value (n=2), the inner curved surface structure of the lens is convex and directed towards the feed horn. The lens thickness at the center is a maximum, 24.5 mm. However, when the lens's refractive index has a negative value (n=−2), the inner curved surface structure of the lens is concave, and the lens dimension has the thickest at the lens edge and is equal to 6.1 mm, which is significantly thinner than that of positive refractive index lens. As a result, negative refractive index lens antennas for base stations could be designed with smaller sizes reducing outer impacts.Table 3
Comparison results.
Antenna structuresnGain [dBi]SLL (dB)θB (deg.)Lens thickness (mm)Horn without lens15.15−18.0030.50Horn with PRI lens227.20−21.807.1324.5Horn with NRI lens−227.34−20.177.786.1Horn with PRI lens (FEKO)226.83−21.737.2324.5PRI lens with MLs2427.80−22.496.7029.1NRI lens with MLs−2427.30−18.527.4910.7


## 4.3. The Electric Field Amplitude Distribution on the Aperture Plane

The electric field amplitude distributions on the aperture plane of the lens antennas forn=2 and n=−2 are shown in Figures 7 and 8. In Figure 7, it is obvious that electric field distributions in both cases are uniform and symmetric around the Z-axis. In addition, Figure 8 shows that the electric field amplitude distribution of the lens antenna is theoretically calculated according to equation (7) and simulated with positive and negative refraction indexes. Thus, electric field amplitudes on the aperture plane distributed along the lens diameter line are high and tend to decrease gradually to the sides. The simulation results and theoretical calculations are quite similar. This shows the accuracy in the design and theoretical calculations. The ripples on the amplitude distribution lines in case of the simulation occur due to the reflection from the lens surface.Figure 7
The electric field distributions on the aperture plane. (a) PRI lens (n=2). (b) NRI lens (n=−2).
(a)(b)Figure 8
The comparison of electric field amplitude distributions on the aperture plane for PRI and NRI lens antennas based on theoretical and simulation.


## 4.4. The Effects of Matching Layers

The comparisons of the radiation patterns of the different refractive index lens antennas with and without matching layers on the yz plane are illustrated in Figure9. It can be observed that when utilizing quarter wavelength matching layers for the positive refractive index lens, the antenna gain increases by 0.60 dBi compared to the one without quarter wavelength matching layers, whereas the antenna gain is almost unchanged when using MLs for negative refractive index lens. The sidelobe level in H-plane also decreases from −21.80 dB to −22.49 dB with positive refractive index lens antennas and increases from −20.17 dB to −18.52 dB with negative refractive index lens antennas. The results show the efficiency of using a quarter wavelength matching layers for a positive refractive index lens antenna.Figure 9
The comparison of radiation patterns of lens antennas on the yz plane. (a) PRI lens with and without MLs. (b) NRI lens with and without MLs.
(a)(b)The investigation into the electric field distribution of lens antennas with and without quarter wavelength matching layers is presented in Figure10. Spillover and diffraction occur more clearly at the lens edge for a positive refractive index. In Figure 11, the electric field amplitude distributions on the aperture planes between the antenna lens with and without the quarter wavelength matching layers are compared with each other. The results show that, in the case of the positive refractive index lens antennas, the black solid line, which illustrates the field amplitude distribution of the lens antenna with matching layers, is smoother and less jagged than the red dashed one presenting the case without matching layers. In contrast, the field amplitude distribution line of the negative refractive index lens antenna with matching layers is even more rugged than without MLs. This shows the more uniform electric field distribution on the aperture plane of the positive refractive index lens antennas when using the quarter wavelength matching layers.Figure 10
The electric field distributions of lens antennas with and without the quarter wavelength matching layers. (a) PRI lens. (b) NRI lens.
(a)(b)Figure 11
Electric field amplitude distributions on the aperture planes. (a) PRI lens with and without MLs. (b) NRI lens with and without MLs.
(a)(b)The authors use the ray tracing method to investigate the effects of multiple reflection from the curve lens surface, as shown in Figure12. Accordingly, black solid lines are the incident rays that travel from the focal point to the inner surface of the lens where the refraction and reflection phenomena occur. Most of the parts of the incident ray power are refracted, which makes outgoing rays parallel to the Z-axis, the solid red lines, while the rest are reflected at the curve surface, the blue dashed lines. This phenomenon causes spillover and diffraction. As to the positive refractive index lens, the lens structure is convex. Thus, reflected rays from the curve lens surface tend towards diffraction at the lens edge. Meanwhile, reflected rays from negative refractive index lens surface tend to bounce back because of the concave lens structure. Spillover and diffraction can be solved by covering the lens with an electromagnetic absorber, shown as in the third picture of Figures 10(a) and 10(b).Figure 12
Multiple reflected rays on the curve lens antenna surface. (a) PRI lens. (b) NRI lens.
(a)(b)


## 5. Conclusions

Two types of antennas with the positive and negative refractive index lens are simulated by using the electromagnetic simulator ANSYS HFSS. The result accuracy is verified by comparing the obtained antenna parameters with the theoretical calculations. The findings show that when the negative refractive index material is used, the lens thickness is 6.1 mm, which is significantly smaller than that of the positive refractive index material, at 24.5 mm. Further, sidelobe levels are also maintained at a low level, as shown in Table3. Those results show the effectiveness of using negative refractive index lens antennas in constructing base station antennas. In addition, when investigating the antenna structure, quarter wavelength matching layers are added to the lens to reduce reflection from the curve surface. The results show that the matching layers are an effective choice for improving the radiation properties of a positive refractive index lens antenna.



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*Source: 9128921-2020-11-25.xml*