Multi Beam Dielectric Lens Antenna for 5G Base Station

In the 5G mobile system, new features such as millimetre wave operation, small cell size and multi beam are requested at base stations. At millimetre wave, the base station antennas become very small in size, which is about 30 cm; thus, dielectric lens antennas that have excellent multi beam radiation pattern performance are suitable candidates. For base station application, the lens antennas with small thickness and small curvature are requested for light weight and ease of installation. In this paper, a new lens shaping method for thin and small lens curvature is proposed. In order to develop the thin lens antenna, comparisons of antenna structures with conventional aperture distribution lens and Abbe’s sine lens are made. Moreover, multi beam radiation pattern of three types of lenses are compared. As a result, the thin and small curvature of the proposed lens and an excellent multi beam radiation pattern are ensured.


Introduction
Nowadays, the 5G mobile system is rapidly developing to achieve fast rate transmission, low latency, extremely high traffic volume density, super-dense connections and improved spectral energy, as well as cost efficiencies [1][2][3]. With the introduction of 5G, the mobile technology has new features such as millimetre wave operation, small cell size and multi beam base station antenna to meet massive multiple-input multiple-output (MIMO) requirements [4][5][6]. At millimetre wave, the base station antenna size is expected to be less than 30 cm, and due to the massive MIMO usage in 5G technology, the antenna system shall have excellent multi beam radiation patterns. Aperture antennas such as a dielectric lens and reflector can be among the alternatives proposed to replace the present array antenna system. Based on recent studies [7,8], the dielectric lens antenna is known to produce excellent multi beam patterns as compared to the reflector antenna. A Luneburg lens that is composed of spherical layered material has achieved a good multi beam radiation pattern in very wide-angle region [9]. However, having a Luneburg lens with continuously varying material permittivity as a function of the lens radius is quite difficult to accomplish in practice. As a result, many methods for these non-uniform geometries have been studied to simplify the lens geometry for various communication applications as well as its feeding network [10][11][12]. However, in those works, the capabilities of performing wide angular scanning for multi beam applications were not discussed in detail.

Lens Surface Design Equations
In order to clarify the proposed lens design method, conventional design methods such as aperture distribution condition and Abbe's sine condition are explained. This section shows the equations used in designing the lens shape based on all three methods.

Fundamental Ray Equations
There are three important expressions to represent the ray tracing principle, which are derived based on the Snell's Law [24]. The Snell's Law on the inner surface (S1) is given by Equation (1): The Snell's Law on the outer surface (S2), is given by Equation (2), where n is the refractive index of the lens and ɸ is the refracted angle of the lens.
The expression for slope can be derived from the condition that all exit rays after refraction are parallel to the z-axis. The and shown in Equation (2) for variable change from dx to dθ.
In the ray tracing calculation, the constant condition of the total electric path length, Lt, is given by expression (3): where Zo is the aperture plane position and Z is the distance from inner surface at the edge to aperture plane, respectively. Equation (4) determines the ɸ value for a given θ value. Then, Equations (1) and (2) can be solved for the variable θ, if is known, which can be calculated from Equation (5).

Aperture Distribution Condition
The following electric power conservation condition is given to obtain the differential equation.
Here, the total horn power, P, is given by Equation (6).

Fundamental Ray Equations
There are three important expressions to represent the ray tracing principle, which are derived based on the Snell's Law [24]. The Snell's Law on the inner surface (S1) is given by Equation (1): The Snell's Law on the outer surface (S2), is given by Equation (2), where n is the refractive index of the lens and φ is the refracted angle of the lens.
The expression for slope dz dx can be derived from the condition that all exit rays after refraction are parallel to the z-axis. The dz dx and dx dθ shown in Equation (2) for variable change from dx to dθ. dz dx = n sin φ 1 − n cos φ , dz dθ = n sin φ 1 − n cos φ dx dθ (2) In the ray tracing calculation, the constant condition of the total electric path length, L t , is given by expression (3): where Z o is the aperture plane position and Z is the distance from inner surface at the edge to aperture plane, respectively. Equation (4) determines the φ value for a given θ value. Then, Equations (1) and (2) can be solved for the variable θ, if dx dθ is known, which can be calculated from Equation (5).

Aperture Distribution Condition
The following electric power conservation condition is given to obtain the differential dx dθ equation.
Here, the total horn power, P, is given by Equation (6).
The fundamental parameters in lens shaping are the feed radiation, E p 2 (θ), and aperture distribution, E d 2 (x). The values of E p 2 (θ) and E d 2 (x) are given as follows: where C is the edge illumination of an aperture distribution. The value p determines the aperture distribution taper and X m is the maximum radius of the aperture. Differential Equations (1), (2) and (5) can be solved based on the constant path length condition of Equations (3) and (4). The performance of this method is determined based on the calculated or designed aperture distribution, E d 2 (x), for a given horn radiation pattern, E p 2 (θ). An example of a designed lens shape is shown in Figure 2. The newly developed MATLAB program will be discussed in Section 2.3, and the antenna parameters will be described in Section 2.4. The shapes of the inner and outer lens surfaces have special curvatures. During transmitting mode, all rays go into the aperture plane and become parallel to the horizontal axis. The spacing of rays is observed to be gradually increased towards the lens edge in order to achieve the aperture distribution taper.
The total aperture power, D, is given by Equation (7).
The fundamental parameters in lens shaping are the feed radiation, Ep 2 (θ), and aperture distribution, Ed 2 (x). The values of Ep 2 (θ) and Ed 2 (x) are given as follows: where C is the edge illumination of an aperture distribution. The value p determines the aperture distribution taper and Xm is the maximum radius of the aperture. Differential Equations (1), (2) and (5) can be solved based on the constant path length condition of Equations (3) and (4). The performance of this method is determined based on the calculated or designed aperture distribution, Ed 2 (x), for a given horn radiation pattern, Ep 2 (θ). An example of a designed lens shape is shown in Figure 2. The newly developed MATLAB program will be discussed in Section 2.3, and the antenna parameters will be described in Section 2.4. The shapes of the inner and outer lens surfaces have special curvatures. During transmitting mode, all rays go into the aperture plane and become parallel to the horizontal axis. The spacing of rays is observed to be gradually increased towards the lens edge in order to achieve the aperture distribution taper. The aperture distribution calculated by Equation (9) shows that the maximum radius of outer surface, Xm = 51.79 mm, for tapered must be larger than 1 (C > 1). Here, C is the edge illumination (C = 6), and the edge level of the aperture electric field is −8.63 dB as shown in Figure 3.  The aperture distribution calculated by Equation (9) shows that the maximum radius of outer surface, X m = 51.79 mm, for tapered must be larger than 1 (C > 1). Here, C is the edge illumination (C = 6), and the edge level of the aperture electric field is −8.63 dB as shown in Figure 3.
The total aperture power, D, is given by Equation (7).
The fundamental parameters in lens shaping are the feed radiation, Ep 2 (θ), and aperture distribution, Ed 2 (x). The values of Ep 2 (θ) and Ed 2 (x) are given as follows: where C is the edge illumination of an aperture distribution. The value p determines the aperture distribution taper and Xm is the maximum radius of the aperture. Differential Equations (1), (2) and (5) can be solved based on the constant path length condition of Equations (3) and (4). The performance of this method is determined based on the calculated or designed aperture distribution, Ed 2 (x), for a given horn radiation pattern, Ep 2 (θ). An example of a designed lens shape is shown in Figure 2. The newly developed MATLAB program will be discussed in Section 2.3, and the antenna parameters will be described in Section 2.4. The shapes of the inner and outer lens surfaces have special curvatures. During transmitting mode, all rays go into the aperture plane and become parallel to the horizontal axis. The spacing of rays is observed to be gradually increased towards the lens edge in order to achieve the aperture distribution taper. The aperture distribution calculated by Equation (9) shows that the maximum radius of outer surface, Xm = 51.79 mm, for tapered must be larger than 1 (C > 1). Here, C is the edge illumination (C = 6), and the edge level of the aperture electric field is −8.63 dB as shown in Figure 3.

Abbe's Sine Condition
Another method of obtaining the dx dθ expression is through the Abbe's sine condition as explained in [18], where the coma free condition is found for a limited scan. The condition, as expressed by Equation (10), is called Abbe's sine condition.
The meaning of this equation is shown in Figure 4. Based on this figure and by employing the Abbe's sine condition, the crossing point of the incoming and the refracted rays should exist on the circle of radius, Fs. The differential form of Equation (10) if given by the next equation. The lens shape shown in Figure 4 is solved by Equations (1), (2) and (11), respectively.

Abbe's Sine Condition
Another method of obtaining the expression is through the Abbe's sine condition as explained in [18], where the coma free condition is found for a limited scan. The condition, as expressed by Equation (10), is called Abbe's sine condition. = The meaning of this equation is shown in Figure 4. Based on this figure and by employing the Abbe's sine condition, the crossing point of the incoming and the refracted rays should exist on the circle of radius, Fs. The differential form of Equation (10) if given by the next equation. The lens shape shown in Figure 3 is solved by Equations (1), (2) and (11), respectively.

Proposed Lens by Straight-Line Condition
As explained in Section 2.2.3, the differential equation of Abbe's sine is derived from the condition that the refraction point is on the circle. As for the straight-line condition, small lens curvature and thin lens thickness are required; thus, the refraction point is expected to be better on the straight-line for a wide scanning beam.
By taking into account the radius of the circle in obtaining in the Abbe's sine condition, the straight-line condition can also be obtained by the equation. The straight-line condition of the lens shape is shown in Figure 5, and the equation is given by Equation (12).
The differential equation form is expressed by Equation (13), where L is the distance from the feed to the straight-line curve on the lens.

=
From the developed MATLAB program, Equations (1), (2) and (13) have been solved, which has resulted in smaller lens thickness, T, and smaller curvature. Thus, this shape is suitable for a base station application.

Proposed Lens by Straight-Line Condition
As explained in Section 2.2.3, the differential equation dx dθ of Abbe's sine is derived from the condition that the refraction point is on the circle. As for the straight-line condition, small lens curvature and thin lens thickness are required; thus, the refraction point is expected to be better on the straight-line for a wide scanning beam.
By taking into account the radius of the circle in obtaining dx dθ in the Abbe's sine condition, the straight-line condition can also be obtained by the dx dθ equation. The straight-line condition of the lens shape is shown in Figure 5, and the equation is given by Equation (12).
The differential equation form is expressed by Equation (13), where L is the distance from the feed to the straight-line curve on the lens.
From the developed MATLAB program, Equations (1), (2) and (13) have been solved, which has resulted in smaller lens thickness, T, and smaller curvature. Thus, this shape is suitable for a base station application.

MATLAB Program for Solving Differential Equations
In order to solve the three differential equations, a MATLAB program is developed. The flow of the developed program is explained by the flow chart shown in Figure 6. The parameter of the initial condition as shown in Figure 7 is determined. Here, θo is an important parameter influencing the lens thickness. In accordance with the Δθ, changes of r and z are given by Equations (1) and (2), respectively. The change of x is determined by Equations (5), (11) and (13) for the design method of aperture distribution, Abbe's sine condition and straight-line condition, respectively. As a result, the inner surface (θ,r) and the outer surface (z,x) are determined to design the lens surfaces and rays are plotted on the structure.

MATLAB Program for Solving Differential Equations
In order to solve the three differential equations, a MATLAB program is developed. The flow of the developed program is explained by the flow chart shown in Figure 6. The parameter of the initial condition as shown in Figure 7 is determined. Here, θ o is an important parameter influencing the lens thickness. In accordance with the ∆θ, changes of r and z are given by Equations (1) and (2), respectively. The change of x is determined by Equations (5), (11) and (13) for the design method of aperture distribution, Abbe's sine condition and straight-line condition, respectively. As a result, the inner surface (θ,r) and the outer surface (z,x) are determined to design the lens surfaces and rays are plotted on the structure.

MATLAB Program for Solving Differential Equations
In order to solve the three differential equations, a MATLAB program is developed. The flow of the developed program is explained by the flow chart shown in Figure 6. The parameter of the initial condition as shown in Figure 7 is determined. Here, θo is an important parameter influencing the lens thickness. In accordance with the Δθ, changes of r and z are given by Equations (1) and (2), respectively. The change of x is determined by Equations (5), (11) and (13) for the design method of aperture distribution, Abbe's sine condition and straight-line condition, respectively. As a result, the inner surface (θ,r) and the outer surface (z,x) are determined to design the lens surfaces and rays are plotted on the structure.

Structural Parameters for Lens Designing
In designing lens shapes shown in Figures 2, 4 and 5, the optimum antenna parameters used are shown in Table 1. As for common parameters, the lens diameter, D, is set to 100 mm and the feed radiation pattern is fixed. Other structural parameters of aperture distribution condition (ADC), Abbe's sine condition (ASC) and straight-line condition (SLC) are independently determined based on the different design methods. In order to clarify the feature of SLC, smaller focal length structures are shown in Figure 8. Calculation parameters are summarised in Table 2. From the figure, the lens thickness and area ratio of SLC become the smallest. In order to make clear the difference, the lens area and thickness are shown in Figure 9. It is clarified that SLC achieves the smallest lens area and thickness.

Structural Parameters for Lens Designing
In designing lens shapes shown in Figures 2, 4 and 5, the optimum antenna parameters used are shown in Table 1. As for common parameters, the lens diameter, D, is set to 100 mm and the feed radiation pattern is fixed. Other structural parameters of aperture distribution condition (ADC), Abbe's sine condition (ASC) and straight-line condition (SLC) are independently determined based on the different design methods. In order to clarify the feature of SLC, smaller focal length structures are shown in Figure 8. Calculation parameters are summarised in Table 2. From the figure, the lens thickness and area ratio of SLC become the smallest. In order to make clear the difference, the lens area and thickness are shown in Figure 9. It is clarified that SLC achieves the smallest lens area and thickness.    Figure 9. Lens area ratio and thickness at focal length F = 40 mm, F = 60 mm and F = 100 mm.

Radiation Pattern of a Feed Horn
The feed horn radiation, Ep 2 (θ), employed in MATLAB is expressed by Equation (8) with radiation coefficient, m = 17, and the maximum angle from the feed horn is according to the lens shapes as tabulated in Table 1. The maximum power for lens illumination is about −10.17 dB, as shown in Figure 10. Based on the focal length to lens diameter ratio, F/D, the maximum angle from feed to lens edge is calculated to be about θm = 26.56°.    Figure 9. Lens area ratio and thickness at focal length F = 40 mm, F = 60 mm and F = 100 mm.

Radiation Pattern of a Feed Horn
The feed horn radiation, Ep 2 (θ), employed in MATLAB is expressed by Equation (8) with radiation coefficient, m = 17, and the maximum angle from the feed horn is according to the lens shapes as tabulated in Table 1. The maximum power for lens illumination is about −10.17 dB, as shown in Figure 10. Based on the focal length to lens diameter ratio, F/D, the maximum angle from feed to lens edge is calculated to be about θm = 26.56°.

Radiation Pattern of a Feed Horn
The feed horn radiation, E p 2 (θ), employed in MATLAB is expressed by Equation (8) with radiation coefficient, m = 17, and the maximum angle from the feed horn is according to the lens shapes as tabulated in Table 1. The maximum power for lens illumination is about −10.17 dB, as shown in Figure 10. Based on the focal length to lens diameter ratio, F/D, the maximum angle from feed to lens edge is calculated to be about θ m = 26.56 • .

Parameters of Electromagnetic Simulations
For validation, the shaped lenses developed in MATLAB are simulated by using an electromagnetic tool called FEKO to recognize its electrical performance. The numerical analysis in FEKO software is based on the Method of Moment (MoM) technique [25], and the antenna system consists of a horn antenna as the feeding element and a dielectric lens antenna. The use of the multilevel fast multipole method (MLFMM) becomes inevitable; thus, the surface equivalent principle (SEP) is employed for the feed horn and the dielectric lens body. The details of the simulation parameters are shown in Table 3. MoM is an accurate solver because it performs full wave analysis to derive rigorous solution for the complex model.

Feed Horn Design
In order to achieve the Ep 2 (θ) radiation pattern as shown in Figure 10, a pyramidal horn antenna is employed in FEKO. The horn structure and size are shown in Figure 11a. The simulated radiation pattern for the E-plane and H-plane is shown in Figure 11b, which shows the obtained gain of 15.13 dBi with an edge level of −9.11 dB and −7.23 dB at the E-plane and H-plane, respectively.

Parameters of Electromagnetic Simulations
For validation, the shaped lenses developed in MATLAB are simulated by using an electromagnetic tool called FEKO to recognize its electrical performance. The numerical analysis in FEKO software is based on the Method of Moment (MoM) technique [25], and the antenna system consists of a horn antenna as the feeding element and a dielectric lens antenna. The use of the multilevel fast multipole method (MLFMM) becomes inevitable; thus, the surface equivalent principle (SEP) is employed for the feed horn and the dielectric lens body. The details of the simulation parameters are shown in Table 3. MoM is an accurate solver because it performs full wave analysis to derive rigorous solution for the complex model.

Feed Horn Design
In order to achieve the E p 2 (θ) radiation pattern as shown in Figure 10

Radiation Characteristics of the Centre Beam
The radiation patterns of all three lens antennas (ADC, ASC, SLC) are shown in Figure 13. All lenses have achieved the same main beam patterns and almost similar sidelobe levels. In order to examine the accuracy of the radiation patterns shown in Figure 10, the theoretical values of antenna gain and the beamwidth of uniform aperture distribution, as shown by Equations (14) and (15) Table 4. Based on the data, the simulated beam widths are slightly increased for all lenses as compared to the theoretical beam width, θBT = 6.25°, due to the tapered aperture distributions of the designed lens. The increased value of the taper aperture distribution has resulted in an increase of the beam width. In this paper, the taper aperture distribution, p, is 1; thus, the values of the beam width for all lenses are increased as compared to the beam width of uniform aperture distribution. In addition to that, the gain reduction from the theoretical value GT = 29.34 dBi is produced by the tapered aperture distributions when ADC, ASC and SLC gain are 27.56 dBi, 27.74 dBi and 27.69 dBi, respectively. Furthermore, the aperture efficiencies of 66.37% to 69.20% are also achieved.

Radiation Characteristics of the Centre Beam
The radiation patterns of all three lens antennas (ADC, ASC, SLC) are shown in Figure 13. All lenses have achieved the same main beam patterns and almost similar sidelobe levels. In order to examine the accuracy of the radiation patterns shown in Figure 10, the theoretical values of antenna gain and the beamwidth of uniform aperture distribution, as shown by Equations (14) and (15) Table 4. Based on the data, the simulated beam widths are slightly increased for all lenses as compared to the theoretical beam width, θBT = 6.25°, due to the tapered aperture distributions of the designed lens. The increased value of the taper aperture distribution has resulted in an increase of the beam width. In this paper, the taper aperture distribution, p, is 1; thus, the values of the beam width for all lenses are increased as compared to the beam width of uniform aperture distribution. In addition to that, the gain reduction from the theoretical value GT = 29.34 dBi is produced by the tapered aperture distributions when ADC, ASC and SLC gain are 27.56 dBi, 27.74 dBi and 27.69 dBi, respectively. Furthermore, the aperture efficiencies of 66.37% to 69.20% are also achieved.

Radiation Characteristics of the Centre Beam
The radiation patterns of all three lens antennas (ADC, ASC, SLC) are shown in Figure 13. All lenses have achieved the same main beam patterns and almost similar sidelobe levels. In order to examine the accuracy of the radiation patterns shown in Figure 10, the theoretical values of antenna gain and the beamwidth of uniform aperture distribution, as shown by Equations (14) and (15) [26], are calculated and compared. Here, D indicates the antenna diameter and θ BT is the half-power beamwidth (HPBW). The comparisons of the theoretical values and the simulation results are summarised in Table 4. Based on the data, the simulated beam widths are slightly increased for all lenses as compared to the theoretical beam width, θ BT = 6.25 • , due to the tapered aperture distributions of the designed lens. The increased value of the taper aperture distribution has resulted in an increase of the beam width. In this paper, the taper aperture distribution, p, is 1; thus, the values of the beam width for all lenses are increased as compared to the beam width of uniform aperture distribution. In addition to that, the gain reduction from the theoretical value G T = 29.34 dBi is produced by the tapered aperture distributions when ADC, ASC and SLC gain are 27.56 dBi, 27.74 dBi and 27.69 dBi, respectively. Furthermore, the aperture efficiencies of 66.37% to 69.20% are also achieved.

Radiation Characteristics of Multi Beam
The multi beam antennas are capable of generating a number of synchronised and independent directive beams to cover the predefined angular and to provide a solution to overcome the shortcomings of the antenna with a single-directive beam. In this section, the radiation mode is explained for determining each of the feed coordinates at a specific angular range.

Feed Positions for Multi Beams
The off-focus radiation patterns are investigated in the y-z plane. The relation of the feed positions with respect to the shift angles is shown in Figure 14. The feed positions of F4 and F5 seem to be approaching the inner surface of the lens. The locus of the feed position, F(y,z), is determined below with R = 100 [27,28].
The feed coordinates are expressed by Equations (17) and (18).
Based on the equations, the feed coordinates are calculated and shown in Table 5 for θF of the 10° step angle.

Radiation Characteristics of Multi Beam
The multi beam antennas are capable of generating a number of synchronised and independent directive beams to cover the predefined angular and to provide a solution to overcome the shortcomings of the antenna with a single-directive beam. In this section, the radiation mode is explained for determining each of the feed coordinates at a specific angular range.

Feed Positions for Multi Beams
The off-focus radiation patterns are investigated in the y-z plane. The relation of the feed positions with respect to the shift angles is shown in Figure 14. The feed positions of F4 and F5 seem to be approaching the inner surface of the lens. The locus of the feed position, F(y,z), is determined below with R = 100 [27,28].
The feed coordinates are expressed by Equations (17) and (18).
Based on the equations, the feed coordinates are calculated and shown in Table 5 for θ F of the 10 • step angle.

Aperture Distribution Condition
The design parameters of the aperture distribution condition (ADC) are tabulated in Table 1, which is shown in Section 2.4. The ADC lens structure and multi beam radiation characteristic are shown in Figure 15a,b, respectively. The ADC lens produces a good multi beam pattern in the angle range of 40° (F1 to F4). It shows a slight drop in gain at the scanning angle θF of 50° (F5). Moreover, the deterioration of the half power beam width is not significant with a difference of about 1.43° as compared to the on-focus feed (F0).  Table 6 summarizes the multi beam radiation characteristics and feed angles. The relation between the feed angle, θF, and the beam shift angle, θS, can be expressed as θF = aθS. The value of a is changed from 1.14 to 1.33 depending on the feed angles of 10° to 50°.

Aperture Distribution Condition
The design parameters of the aperture distribution condition (ADC) are tabulated in Table 1, which is shown in Section 2.4. The ADC lens structure and multi beam radiation characteristic are shown in Figure 15a,b, respectively. The ADC lens produces a good multi beam pattern in the angle range of 40 • (F1 to F4). It shows a slight drop in gain at the scanning angle θ F of 50 • (F5). Moreover, the deterioration of the half power beam width is not significant with a difference of about 1.43 • as compared to the on-focus feed (F0).

Aperture Distribution Condition
The design parameters of the aperture distribution condition (ADC) are tabulated in Table 1, which is shown in Section 2.4. The ADC lens structure and multi beam radiation characteristic are shown in Figure 15a,b, respectively. The ADC lens produces a good multi beam pattern in the angle range of 40° (F1 to F4). It shows a slight drop in gain at the scanning angle θF of 50° (F5). Moreover, the deterioration of the half power beam width is not significant with a difference of about 1.43° as compared to the on-focus feed (F0).  Table 6 summarizes the multi beam radiation characteristics and feed angles. The relation between the feed angle, θF, and the beam shift angle, θS, can be expressed as θF = aθS. The value of a is changed from 1.14 to 1.33 depending on the feed angles of 10° to 50°.  Table 6 summarizes the multi beam radiation characteristics and feed angles. The relation between the feed angle, θ F , and the beam shift angle, θ S , can be expressed as θ F = aθ S. The value of a is changed from 1.14 to 1.33 depending on the feed angles of 10 • to 50 • .

Abbe's Sine Condition
In optics, Abbe's sine condition is well known for designing a collimating lens for a limited scan. This ASC lens structure and multi beam radiation characteristic for on-focus (F0) and off-focus feeds (F1-F5) are shown in Figure 16a,b, respectively. A similar trend with the ADC lens has been discussed in Section 3.3.2, which shows a good radiation pattern produced by the ASC lens for angle range of 40 • .

Abbe's Sine Condition
In optics, Abbe's sine condition is well known for designing a collimating lens for a limited scan. This ASC lens structure and multi beam radiation characteristic for on-focus (F0) and off-focus feeds (F1-F5) are shown in Figure 16a,b, respectively. A similar trend with the ADC lens has been discussed in Section 3.3.2, which shows a good radiation pattern produced by the ASC lens for angle range of 40°. The multi beam characteristics of ASC are tabulated in Table 7, which specifies the shifted beam direction (θS), HPBW(θBS) and gain for each feed angle. Based on the table, a clear correlation between the feed angle, θF, and shifted beam direction, θS, is observed and it can be expressed as θF = bθS, where the b value is determined to be from 1.11 to 1.27. It can be clearly seen that there are no changes in gain for feed position, F1, but a slight decrement is observed from F2 to F4. Slight deterioration occurs in the main beam for F5 as the gain is dropped to 23.03 dBi from F0 with a gain difference of about −4.68 dBi. The maximum shifted beam direction is 39.3° when the feed angle is at 50°.  The multi beam characteristics of ASC are tabulated in Table 7, which specifies the shifted beam direction (θ S ), HPBW(θ BS ) and gain for each feed angle. Based on the table, a clear correlation between the feed angle, θ F , and shifted beam direction, θ S , is observed and it can be expressed as θ F = bθ S , where the b value is determined to be from 1.11 to 1.27. It can be clearly seen that there are no changes in gain for feed position, F1, but a slight decrement is observed from F2 to F4. Slight deterioration occurs in the main beam for F5 as the gain is dropped to 23.03 dBi from F0 with a gain difference of about −4.68 dBi. The maximum shifted beam direction is 39.3 • when the feed angle is at 50 • .

Proposed Lens by Straight-Line Condition
The detailed lens parameters and structures calculated based on the newly proposed SLC method are tabulated in Table 1 and are shown in Figure 17a. This lens shape provides the lowest thickness as compared to the ADC and ASC lenses. The multi beam radiation characteristic for SLC is shown in Figure 17b, which is calculated for feed angle of θ F = 0 • to 50 • . From the figure, it is clearly seen that the tilt angle of 40 • can also produce a good multi beam radiation pattern. At the beam direction of θs = 42.6 • , the beam deformation becomes large.

Proposed Lens by Straight-Line Condition
The detailed lens parameters and structures calculated based on the newly proposed SLC method are tabulated in Table 1 and are shown in Figure 17a. This lens shape provides the lowest thickness as compared to the ADC and ASC lenses. The multi beam radiation characteristic for SLC is shown in Figure 17b, which is calculated for feed angle of θF = 0° to 50°. From the figure, it is clearly seen that the tilt angle of 40° can also produce a good multi beam radiation pattern. At the beam direction of θs = 42.6°, the beam deformation becomes large. The details of the SLC multi beam characteristics are shown in Table 8. It is found that F1 is similar to the ASC lens where there is no reduction in gain but it shows a slightly wider HPBW. It also has an almost similar relationship to the ADC and ASC, where the feed angle, θF, and shifted beam direction, θS, can be expressed as θF = cθS. The range value of c varies from 1.07 to 1.17. It can be seen that the feed angle, θF, is larger than the shifted beam direction, θS, for all lenses (ADC, ASC and SLC). This condition is demonstrated based on the principle of Snell's Law when the waves are passing through between two boundaries from free space to a dielectric material. However, gain reduction becomes large for both feeds at F4 and F5, respectively. In order to understand the beam shapes more in detail, 2D radiation patterns are shown in Figure 18. At a feed angle of θF = 0° and 20°, the symmetrical main beam shapes in the theta (θ) and phi (ɸ) directions are achieved. In θF = 40°, the main beam shape is not symmetrical. This beam shape deformation is caused by the aperture phase aberration in the off-focus feed. The details of the SLC multi beam characteristics are shown in Table 8. It is found that F1 is similar to the ASC lens where there is no reduction in gain but it shows a slightly wider HPBW. It also has an almost similar relationship to the ADC and ASC, where the feed angle, θ F , and shifted beam direction, θ S , can be expressed as θ F = cθ S. The range value of c varies from 1.07 to 1.17. It can be seen that the feed angle, θ F , is larger than the shifted beam direction, θ S , for all lenses (ADC, ASC and SLC). This condition is demonstrated based on the principle of Snell's Law when the waves are passing through between two boundaries from free space to a dielectric material. However, gain reduction becomes large for both feeds at F4 and F5, respectively. In order to understand the beam shapes more in detail, 2D radiation patterns are shown in Figure 18. At a feed angle of θ F = 0 • and 20 • , the symmetrical main beam shapes in the theta (θ) and phi (φ) directions are achieved. In θ F = 40 • , the main beam shape is not symmetrical. This beam shape deformation is caused by the aperture phase aberration in the off-focus feed. According to Tables 6-8, the beam directions of ADC, ASC and SLC from the feed angles 0° to 50° are shown in Table 9. The usefulness of the SLC design method is clarified, as maximum beam direction obtained is at θs = 42.6° as compared to ADC and ASC, where the beam directions are θs = 37.5° and θs = 39.3°, respectively. The correlation between the feed angle and the beam direction for all lenses is shown in Figure 19. Table 9. Beam direction of ADC, ASC and SLC lenses. According to Tables 6-8, the beam directions of ADC, ASC and SLC from the feed angles 0 • to 50 • are shown in Table 9. The usefulness of the SLC design method is clarified, as maximum beam direction obtained is at θs = 42.6 • as compared to ADC and ASC, where the beam directions are θs = 37.5 • and θs = 39.3 • , respectively. The correlation between the feed angle and the beam direction for all lenses is shown in Figure 19. (a) θF = 0° (b) θF = 20° (c) θF = 40° According to Tables 6-8, the beam directions of ADC, ASC and SLC from the feed angles 0° to 50° are shown in Table 9. The usefulness of the SLC design method is clarified, as maximum beam direction obtained is at θs = 42.6° as compared to ADC and ASC, where the beam directions are θs = 37.5° and θs = 39.3°, respectively. The correlation between the feed angle and the beam direction for all lenses is shown in Figure 19.  The correlation between the gain reduction and the shifted beam directions is shown in Figure  20. At beam directions of less than θS = 40°, the gain reduction of ADC, ASC and SLC become similar. At θS = 42.6°, the gain reduction increases to −6.69 dBi. From this figure, it is concluded that the proposed SLC achieves good multi beam radiation patterns as compared to the ADC and ASC lenses. The correlation between the gain reduction and the shifted beam directions is shown in Figure 20. At beam directions of less than θ S = 40 • , the gain reduction of ADC, ASC and SLC become similar. At θ S = 42.6 • , the gain reduction increases to −6.69 dBi. From this figure, it is concluded that the proposed SLC achieves good multi beam radiation patterns as compared to the ADC and ASC lenses.

Conclusions
The new thin lens antenna has been designed by applying a straight-line condition. Multi beam radiation patterns are achieved and, as compared to the conventional lens antenna designed based on Abbe's sine condition (ASC) and aperture distribution condition (ADC), this method also produced good results. For wide-angle beam scanning operation of up to 30° from the centre beam, good multi beam radiation patterns with small beam shape distortion are achieved. The usefulness of the newly developed shaped lens is ensured for multi beam radiation pattern characteristics in which an aperture efficiency of approximately 68.39% is achieved.
Author Contributions: F.A. made an extensive contribution of the conceptualization, design, analysis and

Conclusions
The new thin lens antenna has been designed by applying a straight-line condition. Multi beam radiation patterns are achieved and, as compared to the conventional lens antenna designed based on Abbe's sine condition (ASC) and aperture distribution condition (ADC), this method also produced good results. For wide-angle beam scanning operation of up to 30 • from the centre beam, good multi Sensors 2020, 20, 5849 16 of 17 beam radiation patterns with small beam shape distortion are achieved. The usefulness of the newly developed shaped lens is ensured for multi beam radiation pattern characteristics in which an aperture efficiency of approximately 68.39% is achieved.