## Abstract

The electromagnetic concentrators play an important role in the harnessing of light in solar cells or similar devices, where high field intensities are required. The material parameters for two-dimensional (2D) metamaterial-assisted electromagnetic concentrators with arbitrary geometries are derived based on transformation-optical approach. Enhancements in field intensities of the 2D concentrator have been shown by full-wave simulation. All theoretical and numerical results validate the material parameters for the 2D concentrator with irregular cross section we developed.

©2009 Optical Society of America

## 1. Introduction

Control of electromagnetic wave with metamaterials is of great topical interest, and is fuelled by rapid progress in electromagnetic cloaks [1–6]. Cloaking techniques rely on the transformation of coordinates, e.g., a point in the electromagnetic space is transformed into a special volume in the physical space, thus leading to the creation of the volume where electromagnetic fields do not exist, but are instead guided around this volume [7]. Based on the coordinate transformation idea, some interesting optical and microwave applications, such as illusion device [8], concentrators [9], field shifters [10], anti-cloaks [11], super-scatterers [12], superabsorbers [13], remote cloaks [14] and cloaking sensor [15] have been proposed. Among various novel applications, the phenomenon of near-field concentration of light plays an important role in the harnessing of light in solar cells or similar devices, where high field intensities are required [9,16]. Recently, Rahm et al [9] derived the material parameters for circular cylindrical metamaterial electromagnetic concentrator, and confirmed by numerical simulation. However, to the best of our knowledge, there is no report about the metamaterial electromagnetic concentrators with any other geometry.

Inspired by the work of Li et al [17], we develop the material parameters for 2D concentrators with arbitrary geometries, and validate them by numerical simulation. This work has greatly improved the designing flexibility of concentrators, since material parameters for the concentrator with arbitrary geometries can be easily obtained for the given contour equations. We show that the material parameters developed in this paper can be also specialized to the 2D concentrator with conformal inner and outer boundaries or the other regular shapes, such as circular, elliptical and square, which represents an important progress towards the realization of arbitrary shaped concentrator.

## 2. Theoretical model

The schematic diagram of the space transformation is shown in Fig. 1
, where three cylinders with arbitrary cross section enclosed by contours ${R}_{1}(\theta )$, ${R}_{2}(\theta )$and ${R}_{3}(\theta )$divide the space into three regions S_{1}, S_{2} and S_{3}. The space is compressed into S_{1} at the expense of an expansion of space in regions S_{2} and S_{3}. Here,${R}_{n}(\theta )$
$(n=1,2,3)$is an arbitrary continuous function with period $2\pi $ [17]. According to the coordinate transformation method, the permittivity ${\epsilon}^{{i}^{\prime}{j}^{\prime}}$ and permeability ${\mu}^{{i}^{\prime}{j}^{\prime}}$ tensors of the transformation media can be written as [18]

In the compressive region ($r<{R}_{1}(\theta )$), with the transformation${r}^{\prime}={k}_{3}r$,${\theta}^{\prime}=\theta $, ${z}^{\prime}=z$, where ${k}_{3}={R}_{1}(\theta )/{R}_{2}(\theta )$,we obtain the Jacobi matrix ${\Lambda}_{i}^{{i}^{\prime}}=[{C}_{1},{C}_{2},0;{D}_{1},{D}_{2},0;0,0,1]$and its determinant $\mathrm{det}({\Lambda}_{i}^{{i}^{\prime}})={C}_{1}{D}_{2}-{C}_{2}{D}_{1}$, where${C}_{1}={k}_{3}-{k}_{4}\mathrm{sin}\theta \mathrm{cos}\theta $, ${C}_{2}={k}_{4}{\mathrm{cos}}^{2}\theta $, ${D}_{1}=$ $-{k}_{4}{\mathrm{sin}}^{2}\theta $,${D}_{2}={k}_{3}+{k}_{4}\mathrm{sin}\theta \mathrm{cos}\theta $, ${k}_{4}=[{R}_{2}(\theta )d{R}_{1}(\theta )/d\theta -{R}_{1}(\theta )d{R}_{2}(\theta )/d\theta ]/{R}_{2}{}^{2}(\theta )$. Substituting ${\Lambda}_{i}^{{i}^{\prime}}$ and $\mathrm{det}({\Lambda}_{i}^{{i}^{\prime}})$into Eq. (1), we can obtain the relative permittivity and permeability tensors for compressive region as

In the stretching region (${R}_{1}(\theta )<r<{R}_{3}(\theta )$), with the transformation ${r}^{\prime}={k}_{1}r+{k}_{2}$, ${\theta}^{\prime}=\theta $, ${z}^{\prime}=z$, where ${k}_{1}=[{R}_{3}(\theta )-{R}_{1}(\theta )]/[{R}_{3}(\theta )-{R}_{2}(\theta )]$,${k}_{2}={R}_{3}(\theta )(1-{k}_{1})$, the coordinate transformation equations are expressed by ${x}^{\prime}={r}^{\prime}\mathrm{cos}({\theta}^{\prime})={k}_{1}x+{k}_{2}x/\sqrt{{x}^{2}+{y}^{2}}$, ${y}^{\prime}={r}^{\prime}\mathrm{sin}({\theta}^{\prime})={k}_{1}y+{k}_{2}y/\sqrt{{x}^{2}+{y}^{2}}$,${z}^{\prime}=z$with the Jacobi matrix ${\Lambda}_{i}^{{i}^{\prime}}=[{A}_{1},{A}_{2},0;{B}_{1},{B}_{2},$ $0;0,0,1]$and its determinant $\mathrm{det}({\Lambda}_{i}^{{i}^{\prime}})=$ ${A}_{1}{B}_{2}-{A}_{2}{B}_{1}$, where

Substituting ${\Lambda}_{i}^{{i}^{\prime}}$ and $\mathrm{det}({\Lambda}_{i}^{{i}^{\prime}})$ into Eq. (1), we can obtain the relative permittivity and permeability tensors for the stretching region as

Equations (2) and (3) give the general expressions of the material parameters for the 2D concentrators with arbitrary shapes. They can be used not only in the design of concentrator with non-conformal boundaries, but also conformal boundaries with ${R}_{1}(\theta )={t}_{1}R(\theta )$, ${R}_{2}(\theta )={t}_{2}R(\theta )$, ${R}_{3}(\theta )={t}_{3}R(\theta )$, where ${t}_{i}\le 1$ $(i=1,2,3)$represents the linear compression ratio between the inner and outer boundaries. For the cylindrical concentrator with regular geometric shapes such as circular, elliptical and square, the contour equation ${R}_{i}(\theta )$can be simplified by the procedure illustrated in [19] to obtain the corresponding material parameters. It means that the material parameters developed in this paper can be specialized to the formally designed 2D concentrators. It will be confirmed by full-wave simulation based on finite element software COMSOL Multiphysics in the next section.

## 3. Simulation results and discussion

Figure 2(a) displays the electric field distribution in the vicinity of the concentrator with non-conformal boundaries under TE wave irradiation. The frequency of the TE wave is 4GHz. The contour equations used in the simulation are as follows

As can be seen in Fig. 2(a), the waves are focused into the compressive region with slight distortion. Figure 2(b) is the corresponding power flow distribution. It is calculated according to the Poynting vector $\stackrel{\rightharpoonup}{S}=\stackrel{\rightharpoonup}{E}\times \stackrel{\rightharpoonup}{H}$, where $\stackrel{\rightharpoonup}{E}$ and$\stackrel{\rightharpoonup}{H}$are the electric field and magnetic field in the computational domain. Red lines indicate the direction of the power flow. It can be seen that although the power flow is enhanced in the compressive region, it exhibits strong spatial fluctuations as a consequence of the non-conformal inner and outer boundaries. Figure 2(c) and 2(d) show the simulation results for the concentrator with conformal boundaries, of which the geometry parameters are chosen as$R(\theta )\text{=}$ ${R}_{1}(\theta )$,${t}_{1}=0.3$,${t}_{2}=0.8$, ${t}_{3}=1$. Form Fig. 2(c), it can be clearly seen that the waves are focused by the concentrator into the compressive region without any distortion compared with Fig. 2(a). Comparing Fig. 2(d) with 2(b), we can observe that the power flow distribution in the compressive region is spatially uniform and the intensity of the power flow is strongly enhanced. Significantly stronger enhancements can be achieved by increasing the ratio of${R}_{2}/{R}_{1}$.

Since the proposed concentrator has no symmetry in any direction, it’s necessary to study its interaction with electromagnetic waves from different orientations. Figure 3 shows the power flow distribution for the concentrator with conformal boundaries under cylindrical wave irradiation. The line source with a current of 0.001A/m is located at (−0.45, 0), (0, 0.45) and (−0.32, −0.32) for panels (a), (b) and (c), respectively. It can be clearly seen that the power flow is strongly enhanced in the compressive region, and the focusing effects are independent on the location of the line source.

Since metamaterials are always lossy in real applications, it does make sense to investigate the effects of loss on the performance of the concentrator. Figure 4(a) shows the power flow distribution in the vicinity of the concentrator with electric and magnetic-loss tangents of 0.01. It can be seen that the power flow distribution in the compressive region is greatly fluctuated with the influence of loss. The electric field distributions along the x axis of the concentrator with different electric and magnetic-loss are shown in Fig. 4(b). It can be seen that the power flow distribution is basically undisturbed when loss tangent of 0.001 is added to both permittivity and permeability tensors of the anisotropic and inhomogeneous materials. For the back-scattering region, performance of the concentrator is independent on the loss of the metamaterials. The increase of loss mainly deteriorates the performance of the concentrator in the transformation region and the forward-scattering region of the near field.

Power flow distributions for the concentrator with circular, elliptical and square cross sections under TE wave irradiation are shown in Fig. 5 . The TE wave is irradiated along the x axis. In the simulation, the linear compression ratios between the inner and outer boundaries are chosen as ${t}_{1}=0.3$,${t}_{2}=0.8$, ${t}_{3}=1$. The radius for the out boundary of the circular concentrator is 0.2m; the semi-major axis and the semi-minor axis for the out boundary of the elliptical concentrator is 0.22m and 0.11m, respectively; the side length for the outer boundary of the square concentrator is 0.3m. From Fig. 5, we can observe that the power flow is strongly enhanced in the compressive region. Besides, under the irradiation of cylindrical wave, the concentration phenomenon can also be observed, as shown in Fig. 6 .The simulation results for the circular concentrator are in good agreement with Ref [9], which further confirms the effectiveness and the generality of the material parameters we developed.

## 6. Conclusion

The material parameters for 2D concentrator with arbitrary geometries are developed, which can be specialized to the formally designed concentrators with conformal inner and outer boundaries. All theoretical and numerical results validate the effectiveness and the generality of the material parameter for the 2D concentrator with arbitrary cross section we deduced. Besides, we have investigated the influence of metamaterial loss on the performance of the device, and found that the power flow distribution of the concentrator is basically undisturbed when loss tangent of metamaterials is less than 0.001. The phenomenon of field concentration of light plays an important role in the harnessing of light in solar cells similar devices. It is expected that our works are helpful for designing concentrators for electromagnetic and optical fields, where high field intensities are required.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant no. 60861002), Training Program of Yunnan Province for Middle-aged and Young Leaders of Disciplines in Science and Technology (Grant No. 2008PY031), the Research Foundation from Ministry of Education of China (grant no. 208133), the Natural Science Foundation of Yunnan Province (grant no.2007F005M), Research Foundation of Education Bureau of Yunnan Province (grant no. 07Z10875), and the National Basic Research Program of China (973 Program) (grant no. 2007CB613606).

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