1. Introduction

The use of Transformation Electromagnetics (TE) [1,2] to produce material with specific specifications that control electromagnetic fields in interesting and useful ways [3,4] has been discussed. Such unprecedented flexibility in controlling EM waves by various optical elements via coordinate transformation method has inspired considerable research interests in the design of new class of optical and electromagnetic devices. The best known design by this approach is the invisibility cloak [4]. Following this success, TE technique along with metamaterial technology has resulted in the development of interesting devices, including concentrators [5], waveguide bends and transitions [6–9], illusion systems [10–15], lenses [16–20] and antennas [21–30]. However, both permittivity and permeability values tailored by traditional transformation are generally inhomogeneous and anisotropic, thus entailing a huge challenge for practical implementations. Besides that, transformation medium often needs the resonant artificial electromagnetic materials, which largely limits the frequency bandwidth and performance of devices.

In order to use the full potential of TE for practical design of optical devices, it is important to relax the exact required material parameters. Obviously this would deteriorate the performance of the device. Therefore, the mathematics of the transformation should be improved to find a trade-off between the exact required material parameters and the desired performance of the device. In order to design arbitrary shaped cloaks, Hu et al. [31] proposed an equivalence between coordinate transformation and spatial deformation by using Laplace’s equation to determine the deformation of coordinate grids during the transformation. In addition, the singularity of arbitrary 2D cloaks [32] can be easily avoided by such method. In this study, it is interesting to examine Laplace’s equation with sliding boundary for quasi-isotropic material design for transformation electromagnetics, since the mapping based on Laplace’s equation together with Dirichlet-Neumann (sliding) boundary is quasi-conformal [33]. It has also been shown that quasi-conformal mapping minimizes material anisotropy.

Quasi-Conformal Transformation Optics (QCTO) was proposed and have been widely applied to design quasi-isotropic devices, such as carpet cloaks [34] and bend waveguides [35,36]. Thus, some optical devices were realized simply with dielectric materials or gradient-index metamaterials [37,38]. Therefore, QCTO opens up new horizons for antenna designs where operation bandwidth is always a key design consideration. The method has been employed to conceive high-gain and multi-beam lens antennas [39,40], to achieve high directivity through a flattened lens [41], and to design conformal devices [42]. Though confined to two-dimensional (2D) geometries with TE polarized waves, this technique can be extended to three-dimensional (3D) models via the use of suitable mathematical method.

In this paper, we use Laplace’s equation to determine the transformation matrix in a unified manner. It is shown that the function of the transformation media corresponds in fact to specific boundary conditions. By applying two different boundary conditions, two different designs of a bean steering lens antenna with different properties, which support the same functionality, are proposed. Two-dimensional full wave simulations based on finite element method are used to validate the design method. Through the two lens designs, we show the effect of ignoring anisotropy in the material parameters on the beam steering performances of the lenses.

2. Theoretical design of the lens

The scalar two-dimensional Helmholtz equation is form invariant with respect to coordinate transformations which are equivalent to a conformal mapping. Thus QCTO is an approximate solution of minimizing the anisotropy for general boundary conditions. Here, we use QCTO to propose the design of a lens capable of deflecting electromagnetic waves. To transform the angle between the wave vector k and y-axis from 0° to 45°, we propose two designs based on Laplace’s equation, as illustrated by the schematics in in Fig. 1. For the first design, the virtual space which is free space filled with air and the physical space which is the transformed medium lens are respectively presented in Figs. 1(a) and 1(b). The points B’, C’ and D’ in the physical space share the same location as B, C and D in the virtual space. We consider the length of the segment CD to equal to W and that of BC to be L. Therefore the segment AB has a length $W/\cos \left(\pi /4\right)$ and DA has a length W + L. The second design is illustrated by the virtual and physical spaces shown in Figs. 1(c) and 1(d). The rectangle E’F’G’H’ is mapped from the quadrilateral EFGH, where E’F’ = W, F’G’ = L. In the quadrilateral EFGH, the different dimensions are: $EF=W/\cos \left(\pi /4\right)$, $FG=\frac{2}{5}L$, GH = W and $EH=W+\frac{2}{5}L$. The determination of the mapping is introduced by solving Laplace’s equations subject to predefined boundary conditions by Comsol Multiphysics Partial Differential Equation (PDE) solver [43].

 

Fig. 1 Illustration of conformal mapping from the virtual space to the physical space for the two different designs of the beam steering lens.

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In Fig. 1, a color plot of the contour of the two designs is shown. For simplicity, the transformation deals with a 2D model with incident Transverse Electric (TE) polarized wave. In this case, the electric field only has the z-directed component. Suppose that the coordinate transformation between the physical space (x, y) and the virtual space (x’, y’) is x’ = f(x, y), y’ = g(x, y). The mathematical equivalence of this mapping can be expressed by a Jacobian matrix J whose elements are defined by $J=\partial \left(x\text{'},y\text{'}\right)/\partial \left(x,y\right)$. By solving Laplace’s equations in the virtual space with respect to specific boundary conditions, the Jacobian matrix J of the mapping can be obtained:

The physical space performs an inverse function of the virtual space. Thus the Jacobian matrix of this inverse transformation from (x, y) to (x’, y’) can be represented by J⁻¹. Here we assume that the conformal module of the virtual space is 1 while the conformal module of the physical space is M. Once J⁻¹ is known, the properties of the intermediate medium can be calculated. In terms of fields’ equivalence with the virtual space upon the outer boundaries, Neumann and Dirichlet boundary conditions are set at the edges of the lens. For the lens design 1, the boundary conditions are:

The properties of both lens designs shown in Fig. 2, share similar parameter variation but different range of values. It can be observed that the effective property tensors obtained from Laplace’s equation are not isotropic in the x-y plane anymore. But if the conformal module M of the physical space is not quite different with the conformal module of the virtual space, which is 1 in this case, Li and Pendry suggested that the small anisotropy can be ignored in this case [34]. Considering the polarization of the excitation, the properties of the intermediate medium can be further simplified as:

 

Fig. 2 Material parameter values for the two different designs of the beam steering lens.

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It is clear that for the design 2 the conformal module difference between the virtual space and physical space is smaller than in design 1. In summary, the anisotropy in design 2 is smaller than in design 1.

3. Numerical validation

In this section, finite element method based numerical simulations are used to design and characterize the proposed transformed beam steering lens. In the models using Comsol Multiphysics, scattering boundary conditions are set around the computational domain. In the 2D models, a current line of length L_s = 0.1 m is used as source in the numerical simulations. The electric field of the source is polarized along the z-direction. The operation frequency is set to 10 GHz. As shown in Fig. 3, we consider that the radiation emitted by the source propagates into the lens from boundary A’B’ or E’F’ and exits at boundary C’D’ or G’H’. Both the electric field distribution and the norm of the electric field are presented in Fig. 3.

 

Fig. 3 (a)-(b) Electric field distribution at 10 GHz illustrating the propagation of the radiated beam of a linear source through and out of the lens. (c)-(d) Norm of the electric field. A 45° beam deflection is observed in both designs.

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By applying the material properties obtained from calculations for the two different designs based on Laplace’s equation, the radiated beam is deflected by 45° after propagating through the lens, as it can be clearly observed in Fig. 3. The beam steering performances are also clearly shown by the antenna radiation patterns in Fig. 4.

 

Fig. 4 Normalized antenna radiation patterns showing the beam steering performances of the two proposed lenses.

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In the x-y plane, the wavefronts in design 2 are less distorted compared to design 1. Out of plane, along z-direction, the distribution of ψ_zz also contributes in the beam steering mechanism. The anisotropy of the physical domain can be ignored since it is small enough [34]. Besides, if the in-plane anisotropy can be ignored, the proposed lenses can be further physically realized by isotropic graded refractive index (GRIN) materials. Such isotropic materials suggest a broadband frequency operation. So considering the polarization of the electric field, we neglect the permeability tensor, therefore assuming it to be the same as the virtual space (vacuum).

As shown in Fig. 5, the lenses are now assigned only a permittivity parameter along z-direction, while the permeability is isotropic. It is clear that, since the physical space of lens design 2 is less distorted and the conformal module is close to 1, the lens still steers the beam by 45° when we ignore the in-plane property tensors. On the other hand, when we assume the permeability to be isotropic in the lens design 1, the beam is steered by an angle of 28°, smaller than the expected 45°. This is due to the fact that the conformal module is much larger than 1. So the anisotropy cannot be ignored if a deflection of 45° is still desired. Therefore, the beam steering functionality is weakened when the anisotropy is ignored. If we consider the ε_zz distribution for both lens designs in Fig. 2, it can be noted that for design 1, ε_zz value ranges from 1 to 6.4. Such values can be achieved using nano- and micro-sized titanates dispersed in a polymeric host material [44] or by drilling holes in a high constant dielectric [19,20]. However for lens design 2, where 45° beam deflection is retained, ε_zz value ranges from 0.4 to 5.8. Though values higher than unity can be easily obtained from dielectrics, it is not the case for values less than 1. Such design 2 solution then implies the use of resonant metamaterials such as electric-LC (ELC) resonators [45] or cut wires [46,47] that present an electric resonance where the permittivity ranges from positive to negative values. However, such solution implies a limitation on the operating frequency bandwidth.

 

Fig. 5 Permeability is assumed to be isotropic. (a)-(b) Electric field distribution at 10 GHz illustrating the propagation of the radiated beam of a linear source through and out of the lens. (c)-(d) Norm of the electric field. In the lens design 1, the beam is steered by only 28° since the conformal module is much larger than 1 and hence, beam steering capability is weakened when anisotropy is ignored. However, in lens design 2, 45° beam deflection is maintained even when ignoring material anisotropy since the conformal module is close to 1.

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Furthermore, we have designed a beam splitter by using four adjacent lenses based on the design 1. The line source is enlarged to L_s = 0.2 m and is placed in the middle of the four lenses as illustrated in Fig. 6. The electric field distribution and the norm of the electric field show clearly the transmission of four splitted beams through the lenses. The angle of deflection of the beams with respect to the normal is 28° as for the lens design 1.

 

Fig. 6 Beam splitting device. (a) Electric field distribution at 10 GHz illustrating the transmission of four splitted beams. (b) Norm of the electric field.

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4. Conclusion

In conclusion, using transformation electromagnetics, we have proposed a concept to manipulate electromagnetic waves and design a beam steering lens. The latter manipulation is enabled by using quasi-conformal transformation and Laplace’s equation is utilized to construct the mapping between the virtual space and the physical space. Numerical verifications have been performed on two different designs; one where the conformal module of the mapping is close to 1 and another one were the conformal module is far from 1. The radiation of a current line source has been steered off the normal in both designs. It has also been observed that when ignoring material anisotropy in the design where the conformal module is far from 1, the beam steering functionality is weakened and hence, beam steering angle is less compared to the case where anisotropic parameters are used. However, when the conformal module is close to 1, material anisotropy can be ignored without deterioration of the steering performances. Such concept shows the potentials of transformation electromagnetics in both microwave and optical applications.