# Practical Design Considerations for Compact Array-Fed Huygens’ Dielectric Lens Antennas

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## Abstract

**:**

_{0}/2) from a feed antenna to overcome the limitation of the conventional design method. It is suggested that certain ranges of dielectric thickness values are not considered to exclude undesired resonant effects that hamper the effectiveness of Huygens’ lens which relies on phase shifting elements. In the proposed HP-based design method, phase distributions are captured at the target distance away from the feed array for the two cases of 2 × 2 and 1 × 4 array antennas and based on these, the proposed lens topology is designed to compensate the phase distributions for gain enhancement. A case study shows that the proposed HP-based design approach considering the actual phase information and undesired dielectric resonant phenomenology can achieve a gain enhancement of up to 5.34 dB compared to the conventional dielectric lens, depending on the feed array arrangement that can render circular or elliptic shapes of phase distributions for radiated fields.

## 1. Introduction

## 2. Lens Design Method

#### 2.1. PE Method

_{max}are first chosen as input parameters. F is the distance between the lens and the antenna and T

_{max}is the thickness of the thickest part of the dielectric lens. The bottom surface of the lens is selected to be planar as a conventional reference geometry and the upper surface is determined by the simultaneous equations above. The radius at the bottom of the lens, R

_{b}is determined by the total reflection condition and the corresponding incidence critical angle is given as follows:

_{c}is the ray’s critical incidence angle measured with respect to the local normal vector of the lens, $\widehat{n}$ [1]. It should be noted that once F, T

_{max}and n in Figure 1 are chosen maximum available radius of the lens is decided by the total reflection condition at the top surface of the lens. In more detail, once a design point on the lens is chosen by η to decide the dielectric thickness at that point, F, T

_{max}, n and η become known. Accordingly, other five variables r, γ, l, α and s can be solved by five Equations (1)–(5). As α goes over α

_{c}, reflections on the top surface of the lens goes into the total reflection condition and thus no solutions exist for the five variables related to the lateral size of the lens by R

_{b}= r * sin(η

_{max}). Therefore, if one want to further increase the lateral size of the dielectric lens, the fixed values for F, T

_{max}and n must be changed. For example, the increase of T

_{max}allows for the increase of the lateral dimension of the lens. Unlike the PE method assuming a point source, HP method that can consider multiple sources practically allows for design of the wider dielectric lens. This is why once F, T

_{max}and n are fixed for fair comparisons, the lateral dimensions of the two different method-based lenses can be different, suggesting practical advantages of HP method.

#### 2.2. HP Method

_{max}is the input parameter indicating the maximum thickness of the dielectric lens, λ

_{0}is the free space wavelength and n is the refraction index of the lens material. Since s = T

_{max}– l, the required dielectric thickness l can be expressed as

_{max}is chosen.

## 3. Design Condition for Validation

_{0}/2, based on the fact that microwave impedance matching features are repeated every λ

_{0}/2 and the width of the cylindrical dielectric is wide enough to cover the antennas. A patch element in the antenna arrays was designed to resonate at a centre frequency of 28 GHz and the dielectric constant of the cylindrical dielectric medium was 2.08. The gain of the 1 × 4 antenna array was 12.04 dBi without the dielectric medium. In the 1 × 4 antenna array, the width, P

_{w}and length, P

_{l}, of the patch were both 2.7 mm and the radius of the cylindrical dielectric medium, R, was 20 mm, as shown in Figure 3a. The centre distance between patches of the antenna array, d, was λ

_{0}/2. The total gain of the 1 × 4 antenna array-fed dielectric medium was simulated by using Ansys HFSS in the range of 1 to 30 mm for the thickness of the cylindrical dielectric medium, h. Figure 3b shows that the gain varied periodically at every λ

_{0}/2, as can be predicted from Equation (9). However, the resonant condition at the specific thickness made it difficult to compare the PE method with the HP method because the lens gain is critically affected by the thickness as well as the surface shape of the lens. It was observed that the antenna gain of the 1 × 4 antenna array could be increased up to 16.04 dBi depending on the thickness of the dielectric medium. It should be noted that as the thickness is adjusted to approach the resonant condition, the total antenna gain may be drastically lowered with beam steering compared to an ordinary lens designed based on the phase shifting condition. In this sense, gain enhancement from phase shifting collimation by engineering the shape of the dielectric lens is prohibited at some thickness values of the cylindrical dielectric medium, causing the aforementioned resonant condition and thus, a loss in beam steering capability. Therefore, the appropriate thickness of the dielectric lens should be selected by avoiding thicknesses related to the undesired resonance. It is obvious that the selected thickness becomes the aforementioned T

_{max}in Figure 1 and Figure 2. In this study, T

_{max}was chosen to be 19 and 21 mm for the 1 × 4 antenna array. When the thicknesses of the cylindrical dielectric medium were 19 mm and 21 mm, the antenna gains were 7.39 dBi and 12.32 dBi, respectively. In one case of selecting 7.39 dBi, the antenna beam is most widely spread and thus, this case has the greatest capacity to improve the total gain by using the dielectric lens. In the other case of selecting 12.32 dBi, the gain value is similar to the antenna gain without any dielectric medium and thus, this case has a minimum effect from the resonance of the dielectric medium.

_{max}was chosen to be 4 and 26 mm. When the thicknesses of the cylindrical dielectric were 4 mm and 26 mm, the simulated antenna gains were 13.88 dBi and 14.64 dBi, respectively. Similar to the case of 1 × 4 antenna array, The 4 mm and 26 mm are chosen for total gain of the cylindrical dielectric-medium combined antenna array to be similar to the gain of the antenna array without any dielectric medium and thus gain enhancement effectiveness by phase compensation of the lens can be realized. The reason why 1 or 2 mm is not selected is that too small T

_{max}limits in maximum available lateral dimension (R

_{b}) of the lens due to total reflection condition on the top surface of the dielectric lens as stated in Section 2.1.

## 4. Bench Marketing

#### 4.1. 2 × 2 Patch Antenna Array

_{max}, as shown in Figure 1. The distance between the lens and the antenna, F, was set to be λ

_{0}/2 and the maximum thickness of the lens, T

_{max}, was set to be the selected value, as described in Section 3. The selected T

_{max}values for the 2 × 2 antenna array were 4 mm and 26 mm and two different lenses were designed for the two thicknesses. Figure 5a shows the simulation model of the dielectric lens designed based on the PE method for the 2 × 2 antenna array. Since the conventional method assumes a point source, it can be seen that the lens is symmetrical with respect to the z-axis direction. The dimension parameters of the lenses in Figure 5a are listed in Table 1. When T

_{max}is 4 mm, the R

_{b}is 4.6 mm where R

_{b}is radius at the bottom of the lens. When T

_{max}is 26 mm, the R

_{b}is 16 mm. In Figure 1, R

_{b}is given by r × sin(η

_{max}). In more detail, total reflection condition on the top surface of the lens, α = α

_{c}, gives r and η

_{max}by solving Equations (1)–(5). It should be noted that as α goes over α

_{c}, reflections on the top surface of the lens goes into the total reflection condition and thus no solutions exist for R

_{b}. This limits in maximum available lateral dimension of the lens for the fixed F, T

_{max}and n. Through the aforementioned design procedure, when F, T

_{max}and n are 5.3 mm, 4 mm and 1.44, R

_{b}is decided to be 4.6 mm.

_{0}/2 from the 2 × 2 antenna array when there is no dielectric lens above the feed array. Since antenna elements in the array are arranged in 2 × 2 dimensions, a symmetrical arrangement, the shape rendered by the phase distributions is circular and symmetrical. Figure 5b shows the lens designed based on the HP method for the 2 × 2 antenna array and the shape of the lens follows the shape of the phase distributions. The dimension parameters of the lenses in Figure 5b are listed in Table 2. a is a major axis in the bottom of the lens and b is a minor axis. When T

_{max}is 4 mm, a is 8.6 mm and b is 7.7 mm. When T

_{max}is 26 mm, a is 21.3 mm and b is 19.5 mm.

_{max}are shown by comparing the PE method-based lens (PE lens) with the HP method-based lens (HP lens). The gains of the lens antennas are given in Table 3. The gain of the 2 × 2 antenna array was 12.04 dBi without the lens. When T

_{max}is 4 mm, the antenna with the PE lens has a gain of 13.07 dBi and the antenna with the HP lens has a gain of 13.90 dBi. When T

_{max}is 26 mm, the antenna with the PE lens has a gain of 19.02 dBi and the antenna with the HP lens has a gain of 19.40 dBi. When the T

_{max}values are 4 mm and 26 mm, the HP lens achieves 0.83 dB and 0.38 dB higher gain enhancements than the PE lens, respectively. Since the 2 × 2 antenna array does not deviate significantly from the assumption of a point source in PE method, the shapes of the PE lens and HP lens are not so different and thus there is no big difference in total antenna gain. However, it was observed that there is still a benefit for improving the gain by using the HP lens.

#### 4.2. 1 × 4 Patch Antenna Array

_{max}values for the 1 × 4 antenna array were 19 mm and 21 mm and two different lenses were designed for the two thicknesses. Figure 8a shows the simulation model of the dielectric lens based on the PE method for the 1 × 4 antenna array. As in the PE method, the shape of the lens for the 1 × 4 antenna array is still symmetrical with respect to the z-axis direction. It is obvious that the PE method, which assumes a single point source, cannot fully reflect the asymmetrical features of the 1 × 4 feed array. The dimension parameters of the 1 × 4 antenna array-fed PE lens in Figure 8a are listed in Table 4. When T

_{max}is 19 mm or 21 mm, the R

_{b}is 12.7 mm or 13.6 mm, respectively.

_{max}is 19 mm, a is 22 mm and b is 12 mm. When T

_{max}is 21 mm, a is 23 mm and b is 12.7 mm. Figure 10 shows the phase distributions captured to design the 1 × 4 antenna array-fed dielectric lens at a distance of λ

_{0}/2 from the feed array. It is observed that the phase distributions draw a long oval shape.

_{max}is 19 mm, the 1 × 4 antenna array-fed PE lens has a gain of 13.53 dBi and the 1 x 4 antenna array-fed HP lens has a gain of 17.97 dBi. When T

_{max}is 21 mm, the antenna with the PE lens has a gain of 12.68 dBi and the antenna with the HP lens has a gain of 18.11 dBi. When T

_{max}is 19 mm or 21 mm, the HP lens achieves a 4.14 or 5.34 dB higher gain enhancement than the PE lens, respectively. Since the 1 × 4 antenna array deviates significantly from the assumption of a single point source in the PE method, the levels of gain enhancement of the PE lens and HP lens are very different. The PE method assumes that the radiation pattern or the phase distribution of radiated fields from the feed array are symmetrical but for the 1 × 4 feed array, they are asymmetric, as shown in Figure 10. Finally, it was found that this discrepancy can be fixed by employing the HP lens, leading to significant gain enhancement.

## 5. Conclusions

_{0}/2) away from the feed array was presented. It was demonstrated that employment of the lens designed based on the proposed practical considerations enables a gain enhancement of up to 5.34 dB for the 28 GHz antenna array whose radiated fields render elliptic shapes of phase distributions. The proposed design method that can consider actual field profiles radiated from an arbitrary source arrangement and the undesired resonant conditions of the dielectric medium overcome the limitations of conventional design methods based on the parabolic equation. It is expected that the proposed approach will be very useful for many different integration scenarios of dielectric lenses, which is essentially required for recent industrial systems such as repeaters and base stations in massive MIMO and 5G. In addition, recently evolving 3D/4D printing technology will allow for fabrication of smoothly varying HP dielectric lenses with high-resolution in the near future.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Fernandes, C.A.; Costa, J.R.; Lima, E.B. Dielectric Lens Antennas. In Handbook of Antenna Technologies; Chen, Z.N., Ed.; Springer: Berlin/Heidelberg, Germany, 2016; pp. 1001–1064. [Google Scholar][Green Version]
- Piksa, P.; Zvanovec, S.; Cerny, P. Elliptic and hyperbolic dielectric lens antennas in mmwaves. Radioengineering
**2011**, 20, 270–275. [Google Scholar] - Milligan, T.A. Modern Antenna Design, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2005; pp. 56–451. [Google Scholar]
- Pohl, N. A dielectric lens antenna with enhanced aperture efficiency for industrial radar applications. In Proceedings of the IEEE Middle East Conference on Antennas and Propagation (MECAP 2010), Cairo, Egypt, 20–22 October 2010; pp. 1–5. [Google Scholar]
- Neto, A.; Maci, S.; De Maagt, P.J.I. Reflections inside an elliptical dielectric lens antenna. IEE Proc. Microw. Antennas Propag.
**1998**, 145, 243–247. [Google Scholar] [CrossRef] - Lee, J. Dielectric lens shaping and coma-correction zoning, part I: Analysis. IEEE Trans. Antennas Propag.
**1983**, 31, 211–216. [Google Scholar] [CrossRef] - Yuan, Y.; Zhang, K.; Ding, X.; Ratni, B.; Burokur, S.N.; Wu, Q. Complementary transmissive ultra-thin meta-deflectors for broadband polarization-independent refractions in the microwave region. Photonics Res.
**2019**, 7, 80–88. [Google Scholar] [CrossRef] - Yu, S.G.; Yeon, D.M.; Kim, Y.H. Design of plano-convex lens antenna fed by microstrip patch considering integration with microwave planar circuits. J. Electromagn. Eng. Sci.
**2001**, 1, 67–72. [Google Scholar] - Oh, J. Millimeter Wave Short-Focus Thin Lens Employing Disparate Filter Arrays. IEEE Antennas Wirel. Propag. Lett.
**2016**, 15, 1446–1449. [Google Scholar] [CrossRef] - Enders, P. Huygens’ Principle as Universal Model of Propagation. Lat.-Am. J. Phys. Educ.
**2009**, 3, 4. [Google Scholar] - Zeb, B.A.; Hashmi, R.M.; Esselle, K.P. Wideband gain enhancement of slot antenna using one unprinted dielectric superstrate. Electron. Lett.
**2015**, 51, 1146–1148. [Google Scholar] [CrossRef] - Attia, H.; Abdelghani, M.L.; Denidni, T.A. Wideband and High-Gain Millimeter-Wave Antenna Based on FSS Fabry–Perot Cavity. IEEE Trans. Antennas Propag.
**2017**, 65, 5589–5594. [Google Scholar] [CrossRef] - Ta, S.X.; Nguyen, T.K. AR bandwidth and gain enhancements of patch antenna using single dielectric superstrate. Electron. Lett.
**2017**, 53, 1015–1017. [Google Scholar] [CrossRef] - Feresidis, A.P.; Vardaxoglou, J.C. High gain planar antenna using optimised partially reflective surfaces. IEE Proc. Microw. Antennas Propag.
**2001**, 148, 345–350. [Google Scholar] [CrossRef]

**Figure 1.**Geometry and dimension parameters of the dielectric lens based on the parabolic equation (PE) design method.

**Figure 2.**Geometry and dimension parameters of the dielectric lens based on the Huygens’ principle (HP) design method.

**Figure 3.**(

**a**) Simulation model for the cylindrical dielectric medium fed by a 1 × 4 feed antenna array and (

**b**) total gain of the cylindrical dielectric medium fed by the 1 × 4 antenna array as a function of the thickness of the medium.

**Figure 4.**(

**a**) Simulation model for the cylindrical dielectric medium fed by a 2 × 2 feed antenna array and (

**b**) total gain of the cylindrical dielectric medium fed by the 2 × 2 antenna array as a function of the thickness of the medium.

**Figure 5.**2 × 2 antenna array-fed dielectric lenses designed by (

**a**) the PE method and (

**b**) the HP method.

**Figure 7.**Simulated radiation patterns of the 2 × 2 antenna array with and without PE lens and HP lens at 28 GHz when (

**a**) T

_{max}is 4 mm and (

**b**) T

_{max}is 26 mm.

**Figure 9.**Cross sections of the 1 × 4 antenna array-fed HP lens in (

**a**) the XZ plane and (

**b**) the YZ plane.

**Figure 11.**Simulated radiation pattern of the 1 × 4 antenna array with and without PE lens and HP lens at 28 GHz when (

**a**) T

_{max}is 19 mm and (

**b**) T

_{max}is 21 mm.

**Figure 12.**Comparison between with the PE lens and the HP lens in terms of beam steering when (

**a**) T

_{max}is 4 mm and (

**b**) T

_{max}is 26 mm for the 2 × 2 antenna array.

**Figure 13.**Comparison between with the PE lens and the HP lens in terms of beam steering when (

**a**) T

_{max}is 4 mm and (

**b**) T

_{max}is 26 mm for the 1 × 4 antenna array.

**Table 1.**Dimension parameters of the 2 × 2 antenna array-fed dielectric lens based on the PE method.

T_{max} (mm) | R_{b} (mm) |
---|---|

4 | 4.6 |

26 | 16 |

**Table 2.**Dimension parameters of the 2 × 2 antenna array-fed dielectric lens based on the HP method.

T_{max} (mm) | a (mm) | b (mm) |
---|---|---|

4 | 8.6 | 7.7 |

26 | 21.3 | 19.5 |

T_{max} (mm) | PE Lens (dBi) | HP Lens (dBi) |
---|---|---|

4 | 13.07 | 13.90 |

26 | 19.02 | 19.40 |

**Table 4.**Dimension parameters of the 1 × 4 antenna array-fed dielectric lens based on the PE method.

T_{max} (mm) | R_{b} (mm) |
---|---|

19 | 12.7 |

21 | 13.6 |

**Table 5.**Dimension parameters of the 1 × 4 antenna array-fed dielectric lens based on the HP method.

T_{max} (mm) | a (mm) | b (mm) |
---|---|---|

19 | 22 | 12 |

21 | 23 | 12.7 |

T_{max} (mm) | PE Lens (dBi) | HP Lens (dBi) |
---|---|---|

19 | 13.53 | 17.67 |

21 | 12.68 | 18.11 |

**Table 7.**Comparison of simulated gains between with the PE lens and the HP lens when T

_{max}is 4 mm for the 2 × 2 antenna array.

Lens | PO (deg) | Tilted Angle (deg) | Gain (dBi) |
---|---|---|---|

0 | 0 | 13.07 | |

PE | 45 | 10.53 | 13.07 |

90 | 15.32 | 12.67 | |

0 | 0 | 13.90 | |

HP | 45 | 9.52 | 13.90 |

90 | 16.43 | 13.75 |

**Table 8.**Comparison of simulated gains between with the PE lens and the HP lens when T

_{max}is 26 mm for the 2 × 2 antenna array.

Lens | PO (deg) | Tilted Angle (deg) | Gain (dBi) |
---|---|---|---|

0 | 0 | 19.02 | |

PE | 45 | 3.45 | 18.54 |

90 | 8.14 | 17.64 | |

0 | 0 | 19.40 | |

HP | 45 | 4 | 18.96 |

90 | 7.87 | 18.14 |

**Table 9.**Comparison of simulated gains between with the PE lens and the HP lens when T

_{max}is 19 mm for the 1 × 4 antenna array.

Lens | PO (deg) | Tilted Angle (deg) | Gain (dBi) |
---|---|---|---|

0 | 0 | 13.53 | |

PE | 45 | −7.04 | 14.02 |

90 | −13 | 14.47 | |

0 | 0 | 17.67 | |

HP | 45 | −8.70 | 17.24 |

90 | −18.31 | 16.19 |

**Table 10.**Comparison of simulated gains between with the PE lens and the HP lens when T

_{max}is 21 mm for the 1 × 4 antenna array.

Lens | PO (deg) | Tilted Angle (deg) | Gain (dBi) |
---|---|---|---|

0 | 0 | 12.68 | |

PE | 45 | −5.94 | 13.33 |

90 | −13 | 13.98 | |

0 | 0 | 18.11 | |

HP | 45 | −9.8 | 17.52 |

90 | −16.91 | 16.87 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Seo, B.; Yoon, I.; Oh, J. Practical Design Considerations for Compact Array-Fed Huygens’ Dielectric Lens Antennas. *Sensors* **2019**, *19*, 538.
https://doi.org/10.3390/s19030538

**AMA Style**

Seo B, Yoon I, Oh J. Practical Design Considerations for Compact Array-Fed Huygens’ Dielectric Lens Antennas. *Sensors*. 2019; 19(3):538.
https://doi.org/10.3390/s19030538

**Chicago/Turabian Style**

Seo, Bora, Inseop Yoon, and Jungsuek Oh. 2019. "Practical Design Considerations for Compact Array-Fed Huygens’ Dielectric Lens Antennas" *Sensors* 19, no. 3: 538.
https://doi.org/10.3390/s19030538