## Abstract

In this paper, based on quasi-conformal transformation optics, a 3D conformal lens made of isotropic and non-resonant metamaterial is designed, which can make a cylindrical conformal array behave similarly to a uniform linear array. After discussion and simplification in the two-dimensional model, we realize the proposed lens by utilizing drilling-hole material in the three-dimensional structure. The ring-like shape and forward-only radiation make it possible to equip the lens on a cylindrical device.

© 2014 Optical Society of America

## 1. Introduction

Transformation optics (TO) [1] have provided a new approach to design electromagnetic device, with the unprecedented capability of manipulating electromagnetic field at will [2]. With significant developments of recent years, it has become a research hotspot in optics and electromagnetic society. The best known design by this approach must be the invisible cloak [3]. In addition, a variety of transformation-based designs have been put forward, including EM field concentrators [4], rotators [5], dual-function device [6], and so on [7]. More recently, new types of antennas have been designed using this powerful theory [8–13].

However, both permittivity and permeability designed by the traditional transformation are generally inhomogeneous and anisotropic, thus entailing a huge challenge for practical implementation. Besides that, transformation medium often needs the resonant artificial electromagnetic materials, which largely limits the bandwidth and performance of devices [14–16], although the cloak with anisotropic material has been validated by experiment [3]. Therefore, an improved transformation method is urgently needed. In order to overcome these issues, especially for removing the need of magnetic constituent materials, quasi-conformal transformation optics (QCTO) was proposed and have been widely applied to design quasi-isotropic devices, such as carpet cloaks [17] and bend waveguides [18, 19]. With that, some optical devices were realized simply with dielectric materials or gradient-index metamaterial [20, 21]. Therefore, the QCTO opens up new horizons for antenna designs where the bandwidth is always a key design consideration. They have been employed to make high-gain and multibeam lens antennas [22, 23], to improve a surface wave antenna [24] and to design conformal devices [25]. Although confined to two-dimensional (2D) geometries with TE polarized waves, this technique can be extended to some three-dimensional (3D) models which are not so complicated, and its development depends on the mathematical method.

The concept of source transformation has provided a new idea for the design of conformal antenna which is widely used in the satellite or missile where the design of antenna is needed to conform or follow some prescribed shape. As known to antenna designers, the research on linear antenna array has been developed into a mature theory, compared to that for conformal antenna, especially for that with the irregular configurations. Therefore, if it has a possibility of transforming the design of linear array with desired radiation pattern into a conformal one, it will be a powerful tool for solving the difficulties in the design of conformal antenna. The idea of designing conformal antennas using the concept of transformation optics can be found in [13, 26, 27]. However, these works are mainly theoretical simulations dealing with a two-dimensional ideal model [13]. In this article, by applying suitable transformation, a 3D conformal lens made of isotropic dielectric materials is designed, which can make a cylindrical conformal array perform as a linear one [26]. Firstly, through comparing with the linear array, we analyzed the performance of the lens in two-dimensional configurations, where antennas are temporarily simplified as point sources. Then, by trying several times, we found that the thickness of the lens had an influence on the result. Finally, we chose a proper size and finished the lens with drilling holes material [28–30]. Different from [27], the shape of this lens is similar to a part of the circular ring and a reflector is placed behind the arrays to prohibit radiation in the back, so that it is possible to embed the lens into the surface of a cylindrical device without interference. The main goal of this work is to design a conformal lens from practical point of view, and the result indicates that the idea in this paper is feasible.

## 2. Design of the conformal lens based on QCTO

For transforming a linear array to a cylindrical conformal array, a conformal transformation is presented in Fig. 1, where the virtual space represents the free space filled with air (in the left) and the transformed physical space represents the lens filled with transform medium (in the right). The points *A*, *B*, *C*, *D* have same locations as *A'*, *B'*, *C'*, *D'*, which means that only the line *CD* in the virtual space is transformed to the lines *C'F'*, *F'E'* (arc), *E'D'* and other boundaries remain unchanged. As shown in Fig. 1, the generated grid lines are approximately orthogonal and the mappings between two spaces are nearly conformal, especially the lines near the three upside boundaries have a tendency to overlap. With this transformation, the far field radiation of conformal array with lens is equivalent to the one of linear array.

The conformal mapping in the physical space is obtained by solving Laplace’s equation with predefined boundary conditions. For the sake of simplicity, the transformation deals with a two-dimensional model with TE polarized wave, in which the electric field only has the *z*-directed component. Suppose that the coordinate transformation between the physical space and the virtual space is$x\text{'}=(x,y),\text{\hspace{0.17em}}y\text{'}=(x,y)$, where$(x,y)$and $(x\text{'},y\text{'})$are the local coordinate in the physical and virtual space respectively. The mathematical significance of this mapping can be described by a Jacobian matrix $J$ whose elements are defined by$J=\partial (x\text{'},y\text{'})/\partial (x,y)$. The solution of matrix $J$ and $(x\text{'},y\text{'})$are subjected to the following Laplace’s equations:

## 3. Numerical simulation

This section is devoted to analyzing the performance of the conformal lens antennas, and these works are done with COMSOL, a commercial finite element simulator. The operating frequency is 6 GHz, with a wavelength of *λ* = 50 mm in the free space. As shown in Fig. 2(a), the lens for design can be fully parameterized with *a* = 25 mm, *d* = 30 mm, *L* = 250 mm, *R* = 275.42 mm and a variable *T,* the thickness of the lens which will be discussed later with different values. In this paper, we aim to design a lens equipped upon a cylindrical device, for simplicity, its two curve boundaries have an equal curvature. As shown in Fig. 2(b), the linear array in virtual space consists of eight *z*-directed electric currents, and the corresponding array in the physical space is circumferentially located along the lens as illustrated in Fig. 2(c). In 3D model, the array element is chosen as dipole antenna and the interval between two adjacent elements is *λ*/2 in linear array. In order to eliminate the possible electromagnetic interference with the other devices, a PEC reflector is placed behind the array to prohibit the backward radiation. Considering the mutual coupling, there should be a distance between the reflector and antennas. Therefore, a foam material, with a permittivity of *ε _{r}* = 1.2, is put between them for supporting. According to the image theory [31], we set the thickness of the foam material as 10 mm (about

*λ*/4) for maximal radiation efficiency. The antennas and reflector are treated as a whole in this work.

The thickness, *T* is an important factor of having significant effect on the distribution of material. For investigating its influence, simulations are made with four different values: *T* = 80 mm, 65 mm, 50 mm and 35 mm. The distributions of permittivity, calculated by QCTO, are displayed in Figs. 2(d)-2(g), in which the values less than 1 are set to 1. It will have little influence on the performance of the transformed lens because the waves are mainly limited in the region with high permittivity. As *T* decreases, the values of the permittivity increase, which implies that the materials with higher dielectric constant are needed. In order to examine the performance of the proposed lens, comparisons are made with the linear array and the conformal array without lens. In the following cases, perfectly matched layers surrounding the simulation domain are chosen as the absorbing boundary conditions. The linear sources are excited in equal amplitude and phase. However, it is necessary to explain that the conformal transformation moves these point-like current sources to new locations without stretching or compressing [32], so the conformal array has an identical excitation with the linear one, which also conforms to the law of energy conservation.

The electric field (*E _{z}*) distribution of the linear array is shown in Fig. 3(a), and Fig. 3(b) illustrates the field of the conformal array without lens. The performances of the lenses with different thicknesses are presented in Figs. 3(c)-3(f), respectively. It can be easily seen that the outgoing waves in Fig. 3(b) exhibit an obvious diffusion, which has been effectively corrected with the designed lens, as shown in Figs. 3(c)-3(f). From the far-field radiation patterns shown in Fig. 4, the results of the conformal arrays with lens agree well with that of the linear array, especially for the thicker one. This means that the lenses can largely improve the performance of conformal antennas, although a little loss (1-2 dB) is resulted. As the thickness decreases, the improvement of the lens has been weakened. The explanation for this phenomenon is that the grid lines of the conformal mapping tend to be more divergent at the upper boundary as shown in Fig. 5. This results in more reflections at the boundary between the transformed space and free space. But for this case, the results with

*T*= 35 mm is still acceptable in some senses. Considering practical application, the thickness should be as small as possible. Therefore, we choose the lens of

*T*= 35 mm for further research in three-dimensional model.

## 4. Simplification and implementation

After discussions, the lens whose thickness is 35 mm is selected for practical implementation. The permittivity profile of the designed lens is shown in Fig. 2(g). The profile is divided into two regions by two lines pointing to the center of the circle in Fig. 2(a). Because the values are very close to unity in region I, the permittivity there is set to 1, therefore, this part is omitted in the practical realization. Region ΙΙ is discretized into several unit cells as shown in Fig. 6(b). The arc length of the internal boundary is 202 mm, which is divided into twenty parts with equal size. The thickness (*T* = 35 mm) is also cut into four segments: 10 mm, 10 mm, 10 mm and 5 mm. Therefore, the lens is composed of sixty square cells (about 10 mm × 10 mm) and twenty rectangle cells (double 5 mm × 5 mm). The discrete permittivity of each cell is constant and is equal to the average permittivity within the cell. In addition, the maximum is 2.2, and the minimum is 1.2. The results in Fig. 7 show that the process of discretization has little influence on the performance of the conformal lens, as compared to that with continuous profile.

In order to realize the lens with non-resonant materials, drilling holes on dielectric medium [28–30] is applied in this article. The method is based on the idea of composite materials, in which the property can be changed by adjusting the volume fraction of each component. According to the effective medium theory, if the operating wavelength is enough large (relative to the actual size), the composite material can be considered to be isotropic and homogenous.

Suppose that two dielectric materials are mixed together, whose permittivities are *ε*_{1} and *ε*_{2}, respectively. The effective parameter is calculated by the following equation:

*f*

_{1}and

*f*

_{2}are the volume fraction of the two materials, and

*f*

_{1}+

*f*

_{2}= 1. In this work, several holes are drilled on conventional dielectric material, which simplifies the equation as follow:where

*ε*is the permittivity of the dielectric material and

_{c}*f*is the volume fraction of air. Therefore, by changing the size of the hole, the value of permittivity can be adjusted. Due to the discrete permittivity of the proposed lens ranging from 1.2 to 2.2, the plastic material PTFE is applied as the dielectric substrate, which has a permittivity of

*ε*= 2.2 (with loss tangent 0.0002). Circular holes can be drilled to achieve the required permittivity. As demonstrated in Fig. 8(a), the dielectric constant of circular hole ranges from 1.3 to 2.2. Considering the machining accuracy (generally 0.1 mm) and difficulty, the discrete dielectrics from 1.2 to 1.35 are made with foam material (

_{c}*ε*= 1.2), and others are realized by circular holes. As shown in Fig. 8(b), the proposed lens is implemented utilizing about 10 mm × 10 mm and 5 mm × 5 mm PTFE unit cells with different-sized holes. The performance of the lens with holes in the 2D model is shown in Fig. 9, which is very similar to the discrete lens.

_{r}The 2D configuration can be seen as a cross-section of the 3D model with an infinite length in the *z*-direction. If the finite length is considered, the 2D model is extended to a 3D one, in which the electric current sources (2D) are replaced with half-wave dipole antennas. In the design of an array antenna, the spatial distribution and radiation gains of beams are always the major considerations, which are well described in the radiation patterns of E-plane and H-plane, especially for a linearly polarized antenna. It is well known that the electrical field of dipole only has *z*-directed component at the H-plane. Therefore, for 3D model, it can be approximated that only TE polarized wave is radiated from dipoles so that the transformation medium can be implemented simply by isotropic material, just as the 2D model discussed above. The simulation results given later will verify this point.

To illustrate the 3D performance of the proposed lens, simulations are made using the full-wave software (Microwave Studio CST). The dipole antenna is designed to be operated at 6 GHz (C-band) with parameters: *L _{a}* = 20 mm,

*s*= 2 mm and

*w*= 2 mm. The heights of the lens and reflector are 80 mm (

*H*= 80 mm) and 120 mm, respectively, which are high enough to cover the antennas. As shown in Fig. 10(a), the elements are symmetrically arrayed and each pair is numbered from 1 to 4. Their return losses are illustrated in Figs. 10(d)-10(f). The curves in Figs. 10(d) and 10(e) are similar, while the one in Fig. 10(f) has an obvious rise (about 10 dB). This suggests that covering the arrays with lens makes the antennas’ impedances mismatch, a major reason for that is the truncated length which results in the reflection at the interface with the air. Despite that, a high level of radiation efficiency is still maintained. The total radiation efficiencies of these three cases are −0.1098 dB, −0.1013 dB and −0.2163 dB, respectively. The distributions of the electric field (absolute value) at the H-plane of the three situations are shown in Figs. 11(a)-11(c), which present a similar trend to the 2D simulations (Fig. 3). The gain pattern of E-plane and H-plane are illustrated in Fig. 12. We can clearly see that the performance of the conformal lens in the H-plane is very good (taking no account of the radiation in the back). The peak gains of the three cases are 15.8 dB, 13.5 dB and 15.5 dB, respectively, which means that the proposed lens can improve the gain by 2 dB. However, due to the approximate 3D model, the distribution of the permittivity in other planes has not been carefully considered. Therefore, small fluctuations appear in the E-plane pattern of the conformal lens array when compared to others. The 3D patterns of the gain are displayed in Figs. 12(c)-12(f), which shows the overall performance of the proposed lens.

The performance also depends on the height (*H*) of the lens, especially the radiation pattern of E-plane. We tune the height to other three values: 60 mm, 100 mm and 120 mm, and their comparisons are displayed in Fig. 13. We can see that the E-plane pattern of *H* = 60 mm has a large fluctuation, and the ones of *H* = 100 mm and *H* = 120 mm appear two peaks which are not in the direction of 0°. These will change the performance of radiation largely, thus resulting in a big difference from the linear array. The reason for this phenomenon is that the irregular distribution of the permittivity at the E-plane affects the phase of electromagnetic wave, so the directivity will be also altered. By contrast, the lens with a medium height of 80 mm performs better, which means that the value should be neither too large nor too small. In generally, these results verify our designs and allow the practical realization of the conformal lens with isotropic material.

## 5. Conclusion

In this article, quasi-conformal mapping is applied to design a conformal lens, which can make a cylindrical array radiate like a linear one. Through approximately extruding along the *z*-axis, we realize the lens in the 3D structure using drilling holes materials. The numerical simulation results have validated a good overall performance, although small fluctuations exist in the E-plane of the array. It is shown that by covering the array with the proposed lens, the radiation of the conformal array has been improved to some extent. The property of the lens depends on its thickness, height and curvature, which means that it is possible to equip the lens on a cylindrical device by properly adjusting. In summary, the lens based on the method of conformal transformation can make a contribution to the development of the conformal devices. Moreover, not limited to the proposed lens in this work, designs by QCTO technology have great potential in both microwave and optical domains.

## Acknowledgments

This work is supported by the Nation Natural Science Foundation of China under Grant No. 61101063 and 61331002.

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