1. Introduction

Transformation optics (TO) is a kind of method for controlling the electromagnetic waves [1, 2]. The basic principle of TO is the form invariance of Maxwells equations under coordinate transformation. As a special case of TO, conformal mapping provides a particular way to control electromagnetic waves. Lots of works have proved that conformal mapping and gradient refractive index (GRIN) materials are effective tools for designing new type of electromagnetic devices. And they have been used in plasmonic devices [3, 4], electromagnetic invisible cloak [5–8 ], various lenses [9–12 ], high performance antenna [13, 14], rotating wave guide [15–17 ] and other devices [18, 19].

In this work we propose and design a semi-elliptic multi-functional lens using the conformal mapping method. We show that waves will propagate along different directions when the source is placed at different positions along the long axis of the elliptic lens. The lens can thus be considered like a flattened Luneburg lens [20], which produces highly-directive electromagnetic waves by adjusting the feed positions along the line connecting the two foci. It also functions like an Eaton lens [21]. When an incoming beam impinges on the same line but outside the two foci, it will be guided through the lens structure and take a U-turn. Besides, if properly shaped, the structure can also be used as a waveguide bend [16]. In addition, as suggested by Luo and coworkers, the structure can also be considered as a perfect lens to break the diffraction limit [22]. The lens can be realized by drilling dielectric materials with holes. It has very good properties like simple structure, large beam steering angle and broad bandwidth. The lens has potential applications in controlling electromagnetic waves and in the antenna field. Moreover, this device can also be applied to make a lampshade which will scatter light to different directions according to actual needs.

2. Design of the lens

The conformal mapping method can accomplish the transformation between the virtual and physical space. Here we adopt sine transformation to transform the complex plane z = x + iy into another one, w = u+iv. Then we have

When x and y are constants, Eq. (2) indicates confocal ellipses and hyperbolas, and c is the focal length. Hence, we achieve the transformation between rectangular grids and elliptic grids by sine transformation.

The schematic diagram of both virtual and physical spaces are shown in Fig. 1. As clearly shown in the figure, vertical and blue lines are mapped into confocal hyperbolas by the transformation, while the horizontal and red lines are transformed into confocal ellipses. As is well known in the conformal mapping theory, the transformations leave Fermats principle intact in both spaces and they correspond to materials with an isotropic refractive index profile, which acts as a transformation medium [23]. If we treat the former lines as light rays in the virtual space with vacuum, then the latter ones will be the corresponding rays in the physical space with appropriate materials. Based on this mapping, the propagate directions of the blue lines and red lines depending on the start positions along the line connecting the two foci. Careful observations suggest that when we put the source at different positions along the long axis, the wave will propagate along different directions. For example, if we consider the blue lines as light paths and put the source between the two foci, the phenomenon of light deflection can be observed. In another case, if we consider the red lines as the light paths, then, a U turn of the incident light is clearly observed. Based on this observation, we propose a multi-functional lens design. The electromagnetic parameters in the lens, as governed by conformal mapping theory, can be described as [2],

 

Fig. 1 The principle diagram of the conformal mapping between rectangular and elliptic grids. (a) the virtual space in the rectangular system; (b) the physical space in the elliptic coordinate system.

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Where n ₀ indicates the refractive index in the rectangle area. Here we set n ₀ = 1, then

And a typical distribution of the refractive index in the elliptic area is shown in Fig. 2(a).

 

Fig. 2 The refractive index distribution in xoy plane and the ray paths in the lens structure. (a) the refractive index in the lens; (b) and (c), ray traces inside the lens.

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3. Numerical simulations

To validate the performance of the lens, we make numerical simulations using the commercial software, COMSOL Multiphysics. A half ellipse with the long half axis a=75mm, short half axis b=65mm, is used in the simulation. We first fill the half elliptic lens with refractive index given in Eq. (4) and give the ray tracing results. Figures 2(b) and 2(c) show the ray paths when a bundle of rays hit the lens between the two foci and outside the two foci, respectively. The simulation results of the light paths in the semi-elliptic lens in Figs. 2(b) and 2(c) agree well with the blue and red lines, respectively, in the principle diagram in Fig. 1(b). It is demonstrated that the lens has obvious effect on changing the propagation direction of the light rays. To further illustrate the performance of the transformation model, full wave simulation is also made using the same software. The simulated electric field distribution on the xoy plane is shown in Fig. 3, where four typical source positions are chosen: (1) the feeding source is located at the lens center (x = 0, y = 0), as shown in Fig. 3(a); (2) 20mm off the lens center (x = 20mm, y = 0), as shown in Fig. 3(b); (3) 29mm off the center (x = 29mm, y = 0), as shown in Fig. 3(c); and (4) 51.5mm off the center (x = 51.5mm, y = 0), as shown in Fig. 3(d), respectively. Figures 3(a)–3(c) indicate that the electromagnetic waves propagation along different directions when the feeding source is placed at different positions, a special property owned by a flattened Luneburg lens [20]. Careful observations suggest that beams in Figs. 3(a)–3(c) propagate like three different blue lines between the two foci in Fig. 1(b). In Fig. 3(d), we illustrate another configuration, where the lens acts like an Eaton lens. Obviously, when the feeding source is placed outside the two foci, the incident wave will be guided inside the lens, take a U-turn and then be released from the lens structure. Using the conformal mapping theory for the explanation, the beam in Fig. 3(d) coincide with the red lines outside the two foci in Fig. 1(b), as a result, it will take a U-turn accordingly. However, we note that the proposed device is neither a real Luneburg lens nor a real Eaton lens due to different refractive index profiles, it owns the similar functions like a Luneburg lens and an Eaton lens. In addition to the near field distribution, we also give the far field radiation patterns in Figs. 4(a)–4(d), which correspond to the near field distribution in Figs. 3(a)–3(d). As can be seen in the figure, the same conclusions can be made. Moreover, the same structure can also be used as a waveguide bend if properly shaped. For example, if we cut the lens in half, the resultant quarter elliptic structure will perform as a 90-degree waveguide, as demonstrated in Figs. 5(a) and 5(b). In the figure, Fig. 5(a) illustrates the propagation characteristics of electric field inside the waveguide bend and Fig. 5(b) gives the radiation pattern for Fig. 5(a). All the simulation results agree well with the theoretical description.

 

Fig. 3 The full wave simulated z components of electric fields at 18GHz in xoy plane, where the feeding source is located at (a) x = 0, y = 0; (b) x = 20mm, y = 0; (c) x = 29mm, y = 0; and (d) x = 51.5mm, y = 0.

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Fig. 4 Far field patterns for the transmitted waves. (a)–(d) corresponds to the radiation pattern of each plot in Fig. 3, respectively.

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Fig. 5 The wave guide bend using a half lens structure. (a) the simulated z component of electric field in the wave guide bend; (b) the far field pattern corresponding to (a).

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4. Realization of the lens

To realize the gradient refractive index, we can adopt the hole-array metamaterials by changing the hole size [24]. The insets in Fig. 6 show the unit structure adopted. It is absolutely true that the hole-array has an anisotropic properties, however, since we focus on the TE polarization, which means the electric field is parallel to the hole (air cylinder) axis, this problem can be naturally resolved. We remark that this fact does not exclude the lenss application for TM polarized waves (magnetic field parallels with the hole axis), however, a totally new design is needed. Since for these two polarizations, the effective refractive index for each cell is different, i.e. the anisotropic property. As far as the effective refractive index is concerned, it is obtained using the standard retrieval process. At first, a unit cell is put inside an ideal waveguide with the plane wave excitation. Then, the S parameters are obtained after full wave simulations. More specific details about these theories can be seen in the appendix. With the help of simulation software (Microwave Studio, CST), we analyze the scattering characteristics of the unit cell. Then the effective electromagnetic parameters are retrieved from the S parameters of the simulation [25]. Here three kinds of dielectric plates and four kinds of unit structures are considered in our design. They are 2×2×1mm ³ with dielectric constant 2.2, 2×2×2mm ³ with dielectric constant 2.2, 2×2×2mm ³ with dielectric constant 4.4, and 2×2×2mm ³ with dielectric constant 12.9. The relationship between the effective refractive index and the hole radius at 15GHz is demonstrated in Figs. 6(a) and 6(b). In the figure, subplot Fig. 6(a) indicates the refractive index of the unit cell with a 2 mm thickness and the dielectric constants are 2.2, 4.4 and 12.9, respectively. Subplot Fig. 6(b) shows the refractive index of the unit cell with a 1mm thickness whose dielectric constant is 2.2. Based on the above curves, we can get the gradient refractive index ranging from 1.069 to 3.59, which covers almost all the required data.

 

Fig. 6 Relations between refraction index and the hole’s radius. (a) The refractive index analysis of the unit cell with 2 mm thickness when the dielectric constant is 2.2, 4. 4 and 12.9 respectively; (b) The refractive index analysis of the unit cell with 1mm thickness when the dielectric constant is 2.2.

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According to Fig. 2(a), larger refractive index concentrates near the two foci, and most part of the lens is filled with refractive index less than 1, which will be treated as 1 approximately. Based on the above observation, we design a quarter of the ellipse and fill it with the above four kinds of drilled materials. Then we make several full wave simulations for the whole structure, where the feeding source is put at different positions along the long axis of the lens. As predicted, the electromagnetic waves propagate along different directions which are perpendicular to the edge of lens, as shown in Fig. 7. Meanwhile, we confirm this phenomenon at different frequencies (12, 15, 20GHz). Figures 7(a)–7(c) illustrate the results where the feeding source is located at x = 20mm, y = 0 for frequencies at 12, 15 and 20GHz, respectively, and Figs. 7(d)–7(f) show the case where feeding source is located at x = 29mm, y = 0 for frequencies at 12, 15 and 20GHz, respectively. The whole structure simulation results agree well with theoretical analysis. Here, the Eaton lens and wave guide bending can not be directly achieved when they work in the vacuum, because the refractive index contributing to the two devices is mostly less than 1, while the refractive index of the drilling unit is always larger than 1. However, If the lenses are working in a background medium with refractive index n ₀ lager than 1, the material parameters in the elliptic lens will be multiplied by a constant n ₀ accordingly (See Eq. (3)). As a result, the original refractive index in the lens which are less than 1 will become bigger than 1. Hence the same approach can be utilized when the lenses are working in a background medium, similar method has been used by Hunt and coworkers in their Luneburg lens implementation at the infrared band [20].

 

Fig. 7 the whole structure simulation diagram. (a–c) The distribution of electric field at the 12, 15 and 20GHz respectively when the feeding source is located at x = 20mm, y = 0; (d–f) The distribution of electric field at the 12, 15 and 20GHz respectively when the feeding source is located at x = 29mm, y = 0.

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5. Discussion

The whole structure simulation results agree well with theoretical predictions. However, there are still some discrepancies. The reason is as follows: (1) we give up a lot of refractive index which are less than unit in the model; (2) the feeding source is not narrow enough and its directivity is not high. The first problem can be eliminated by using other metamaterials, e.g. H-fractal structures [26], which can give refractive index lower than one, while the second point can be solved by using a collimated beam. By done so, a much better multi-functional lens will be realized.

In addition, we have to say that there exist some defects in our design. Since the refractive index is not matched to the environment on the edge of the lens, there will be reflections at this part. So the gain will be reduced accordingly. This can be partially solved by using matching layers at the boundary to reduce reflections. Moreover, the lens structure works like a flattened Luneburg lens, but it is not a true one. According to ray analysis, a Luneburg lens can change a point source radiation into a collimated beam and realize the beam steering by changing its position, while our lens can only change the propagation direction of a collimated beam through positions. Theoretically speaking, the lens works much better at optics band, however, its realization highly depends on other gradient index materials.

6. Conclusion

Based on conformal mapping, we propose and design a multi-functional lens by dielectric elements which are drilled with inhomogeneous holes. And the correctness of our theory is verified by full wave simulations. According to the simulation, when the feeding source is placed at different positions along the long axis of the elliptic lens, electromagnetic waves will propagate along different directions, which functions like a flattened Luneburg lens, Eaton lens or a waveguide bend for a wide bandwidth (12 to 20GHz). The simulation results agree well with theoretical analysis. The lens has good properties like simple structure, large beam steering angle (up to 45 degree bending and 180 degree rotating), and broad bandwidth (from 12 to 20GHz). The device we designed can also be applied to make a lampshade which is able to adjust the propagation direction of the light according to actual needs. This theory also works to higher frequencies and has potential applications at optics band.

Appendix

In this appendix, we give the detailed analysis for why different thickness produces different effective index.

By treating the unit cell as an effective material, the equivalent material parameters can be given in Smith and coworkers work (D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Determination of effective permittivity and permeability of metamaterials from reflections and transmission coefficient, Phy. Rev. B 65, 195104 (2002)), which is

(5)$$z=\pm \sqrt{\frac{{\left(1+r\right)}^2-{t^{\prime }}^2}{{\left(1-r\right)}^2-{t^{\prime }}^2}}$$

(6)$$\mathrm{Im}(n)=\pm \mathrm{Im}\left(\frac{{\cos }^{-1}\left(\frac{1}{2t^{\prime }}\left[1-\left(r^2-{t^{\prime }}^2\right)\right]\right)}{kd}\right)$$

(7)$$\mathrm{Re}(n)=\pm \mathrm{Re}\left(\frac{{\cos }^{-1}\left(\frac{1}{2t^{\prime }}\right)\left[1-\left(r^2-{t^{\prime }}^2\right)\right]}{kd}\right)+\frac{2\pi m}{kd}$$

where t′ = exp(ikd)t.

When the electric size of the unit cell is very small, the effective medium theory can also be utilized for obtaining the material parameters. In this regard, the ”capacitor model” will make things much clearer. In the effective medium theory, since the working wavelength is larger compared with the hole dimensions, the problem can be approximately treated as a quasi-static problem. Then, the total capacitor between the two conducting plates (imagined ones, which are perpendicular to the electric fields) can be expressed as C = εS/d, where S is the area for one plate, ε is the effective material permittivity between the two conducting plates, and d is their distance. However, if we treat the capacitor as the parallel connection of many small capacitors, as shown in Fig. 8. It is quite clear that the following equation holds C = ∑C_i, which can be further expressed as εS/d = ∑ε_iS_i/d. Then, it is easily proved that the effective material parameter can be represented as ε = ∑ε_i f_i, where f_i = S_i/S is the filling ratio for each constitutive material. By adjusting the filling ratio, which is easily realizable in practice, the effective material parameters can be flexibly adjusted. In most cases, two materials are used to get the required effective parameters, and for most of the time, one of the two materials is air. Then, it is easy to get the following equation: ε = f +ε_d(1−f), where ε_d is the dielectric constant of the material (substrate). Our earlier work (Z. L. Mei, J. Bai, and T. J. Cui, “Gradient index metamaterials realized by drilling hole arrays,” J. Phys. D: Appl. Phys. 43, 055404 (2010).) shows that the above expression works well for the quasi-static conditions.

 

Fig. 8 The parallel connection capacitor model.

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Now, let’s move to another problem, why different thickness produces different effective index. In this case, the unit cell can be considered to include 3 parts, or 3 capacitors in our term, two air-filled capacitors, and one dielectric filled capacitor (but with an air hole in the middle), as shown in Fig. 9(b). The former capacitors are well known, while the latter one can be obtained using the above theory. However, the total capacitor is the serial connection of the three 1/C = ∑1/C_i, i.e. d/(εS) = ∑d_i/(ε_iS). Since the area for each plate is the same in this case, we have 1/ε = ∑(d_i/d)/ε_i. Then, it is quite clear that the change of height will change the capacitor of the latter one, and hence, the permittivity, which is clearly related to the effective refractive index.

 

Fig. 9 The series connection capacitor. (a) The unit we adopt, (b) The equivalent series connection capacitor model.

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Acknowledgments

Z.L.M. Acknowledges the Open Research Funds of State Key Laboratory of Millimeter Waves (Grant No. K201409), and Fundamental Research Funds for the Central Universities (Grant Nos. LZUJBKY-2015-k07, LZUJBKY-2014-43 and LZUJBKY-2014-237).

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