1. Introduction

Pendry et al. [1] has proposed an interesting idea of using geometrical transformation combined with Maxwell’s equations to yield unique electromagnetic devices, recently termed “invisibility cloaks” with the electrical permittivity and magnetic permeability both anisotropic and spatially varying, in which the electromagnetic wave can be smoothly guided around a “cloaked” region without disturbing the exterior fields. This new method of controlling electromagnetic fields have been theoretically verified soon by D. Schurig [2] using Cartesian tensors directly to calculate material properties of the transformed medium and by U. Leonhardt [3] who transformed the two dimensional Helmholtz equation to produce similar cloaking effects in the geometrical optic limit. Based on coordinate transformation, other electromagnetic structures with special electromagnetic functionality have been reported such as rotation coatings [4], magnifying perfect lens [5, 6], beam refactors [7], square cloaks and concentrators [8, 9].

Although coordinate transformation associated with Jacobian matrix enables us easily to obtain the material parameters and the development of artificially structured metamaterials provides the flexibility of tuning the electrical permittivity and magnetic permeability independently and arbitrarily covering positive and negative values as desired [10–13], obtaining the tensor components that are generally inhomogeneous and spatially dependent is still challenging for practical fabrication. Most papers reported to date focus on theoretical structural design and numerical analysis [14–18]. Fortunately, transformed material parameters can be simplified by applying EM wave with specified polarization direction, which significantly facilitates the implementation of designed structures in reality. A 2D cloak device with reduced permittivity and permeability tensors have been experimentally fabricated by assembling split ring resonators (SRRs) in desired way [19].

In this paper, using the traditional geometrical transformation approach as first proposed by Pendry for its convenience and intuitiveness in handling co-ordinate systems that are orthogonal, such as cylindrical and spherical, we focus on the concentrator first proposed in Reference [8]. It is noted that the concentrator in the ideal case has the capability of perfectly concentrating the incident EM power without any refection and absorption, but the spatially varying material tensor components are difficult for practical realization. So we derive different sets of reduced material specifications for cylindrical concentrators by taking advantage of TE incident wave and systematically analyze their concentrating performance as well as the scattering property. A set of simpler reduced material tensors is determined to make both the inner and the outer interface perfectly matched, 2D numerical simulations are also performed to confirm our theoretical analysis and conclusions.

2. The ideal concentrator

In cylindrical coordinate system, the concentrator experiences a two-step space deformation as Marco Rahm suggested in Reference [8]:

 

Fig. 1. Schematic diagram of 2D concentrator, including three rigions separated by three circles with radius being R1, R2 and R3 respectively from the inner circle to the outer one.

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Space in the region of 0 ≤ r ≤ R ₂ is compressed into the inner circle, and simultaneously space between R ₂ and R ₃ is stretched to fill the R ₁ ≤ r ≤ R ₃ area, the transformation can be mathematically denoted in the following piecewise equation:

According to Pendry’s geometrical transformation method, the transformed material tensor components from Cartesian system to the cylindrical have the following form:

Where $Q_i=\sqrt{{\left(\frac{\partial x}{\partial q_i}\right)}^2+{\left(\frac{\partial y}{\partial q_i}\right)}^2+{\left(\frac{\partial z}{\partial q_i}\right)}^2,}\left(i=\mathrm{1,2,3}\right)$, denotes Lamé coefficient with q_i representing three coordinate curves of cylindrical system. We should calculate the corresponding Lamé coefficients for the two regions in Equation(1) respectively:

for 0 ≤ r ≤ R ₁, we have

and for R ₁ < r ≤ R ₃ region,

Combining Eqs. (4) and (5) with Eqs. (2) and (3), the diagonal material tensor of the permittivity and permeability for the two Cartesian regions are:

here, M = (R ₂ - R ₁)R ₃/(R ₃ - R ₂) and N = [(R ₃ - R ₂)/(R ₃ - R ₁)]² .The results are exactly equivalent to those derived by Marco Rahm.

3. Concentrators with reduced parameters

It is noted that in 0 ≤ r ≤ R ₁ region material tensor components are all constants and only the z-component need to be considered for the field polarization oriented along the z-axis for future experimental implementation, but the tensor for R ₁ ≤ r ≤ R ₃ region shows its components that are all dependent on the radial position, indicating tremendous trouble in satisfying them all at once. Here we use TE incidence as first proposed by Pendy et al. for material simplification in practical realization of cylindrical cloaking structure [19] to reduce material specification in the concentrator. Given the incident plane wave polarized along z-direction propagates in positive x axis, electric field has only one component, which can be expressed in terms of (r,ϕ,z) basis as Ē = ẑE_z. Combining with Maxwell’s equations the wave equation determining the propagation property of EM wave inside the concentrating structure can be derived in the form of cylindrical system:

We note from Eq. (7) that only the radial permeability component μ_ϕ, azimuthal permeability component μ_ϕ and z-component ε_z are involved in the wave equation. We choose a set of material parameters for cylindrical concentrators that can maintain the same wave propagation behavior in the region of R ₁ < r ≤ R ₃ , e.g., the reduced parameters:

in which we realize that ε_z and μ_ϕ are greatly simplified with the radial permeability specification being position-relevant. Parameters presented in Equation (8) minimize the most challenging part of experimental design. But this set of parameter specification leads to the impedance mismatch at the outer boundary of the concentrator. The ideal parameters in the region of R ₁ < r ≤ R ₃ in Equation (6) have the impedance of $Z_{r=R_3}=\sqrt{\frac{{\mu }_{\varphi }}{{\epsilon }_z}}=1$ at the outer surface r = R ₃ surface and Z _r=R1 = R ₁/R ₂ = at the inner surface r = R ₁ that are both matched at these two interfaces, while the simplified specification corresponds to the mismatched impedance of $Z_{r=R_3}=Z_{r=R_1}=\sqrt{\frac{1}{N}}=\frac{\left(R_3-R_1\right)}{\left(R_3-R_2\right)}$ at the two interfaces, evidently the energy loss due to these impedance mismatch will degenerate the concentrating performance. To improve the concentrating efficiency, we derive another set of simplified material tensors, which also have simple tensor components and can make the outer boundary perfectly matched to the free space. With TE wave mode, the wave equation still remain unchanged if we choose the following material components for the R ₁ < r ≤ R ₃ region

In this set of parameters, only the radial permeability is spatially varying, and the region of R ₁ < r ≤ R ₃ becomes a perfectly matched layer with a constant impedance of $Z_{R_1\sim R_3}=\sqrt{\frac{{\mu }_{\varphi }}{{\epsilon }_z}}=1$. But at the inner boundary the impedance mismatch is not eliminated. So we can further reduce the parameters for the 0 ≤ r ≤ R ₁ region to obtain a zero-reflectance inner interface in the TE mode case, the parameters experience the following simplification:

Actually, in the case of z-polarized wave the simplified material parameters indicate an isotropic material within the inner circular cylinder with the permittivity and permeability being both constants, and the inner surface as well as the outer one are both matched interface without reflection. Now we have proposed three kinds of reduced material specification for the concentrator, their simple forms are expected to provide the ease for the future practical realization. The different impendence patterns at the interfaces give rise to the variation of their concentrating capabilities, which will later be demonstrated by 2D simulations.

4. Simulation and discussion

In this section, finite element method is used to simulate the designed concentrators. We use circular concentrators that have the same size as used in Reference [8], where R ₁ = 2cm, R ₂ = 4cm and R ₃ = 6cm. TE wave with working frequency of f = 15GHz is incident along x axis throughout the simulation area from left to right. All computational domain boundaries are perfectly matched layers (PML) so that the scattered fields can be absorbed. We first simulate the 2D concentrator with ideal parameters to confirm our previous analyses, then we illustrate the simulated results of those with different set of reduced parameters, their scattering properties and concentrating abilities are discussed.

Although the concentrators with ideal material parameters have excellent concentrating performance, it is actually difficult to realize the complicated tensor components in practice. Here we numerically demonstrate three kinds of concentrators having reduced material specifications. The first kind is the one with material tensors presented in Eq. (8), which has simplest tensor components but non-zero reflection occurs at both the inner and the outer interface, we denote this kind of concentrator as case (1), the second kind, denoted as case (2) is the one possessing the material parameters given by Eq. (9), in this case only the outer boundary is perfectly matched. The third kind of concentrator is case (3), where Equation (9) and (10) is combined to specify the simplified material in region of R ₁ < r ≤ R ₃ and 0 ≤ r ≤ R ₁ respectively, eliminating the impendence mismatches at the two boundaries. The simulation results are shown in Fig. (2).

 

Fig. 2. Electric field distributions in the vicinity of the concentrators with reduced parameters. Three graphs from left to right denote the concentrator of the first kind (a), the second kind (b) and the third kind (c). All the concentrators have the same size with R1=0.02m and R2=0.04m and R3=0.06m.

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The E field distributions in Fig. (2) clearly demonstrate the scattering properties of these reduced concentrators. In case (1), the field outside the concentrator is disturbed by the scattered wave from the outer boundary, and inside the inner circle, the wave front that is flat in the ideal cases deforms in shape due to perturbation of the scattering occurring at the inner interface. The energy coming into the structure from the left side is no longer equals to that flowing out from the right side, which surely decreases the concentrating efficiency of the concentrator. The same situation appears in case (2), but the scattering wave amount decreases as the simplified material parameters have made the outer boundary perfectly matched, we can see from the middle graph in Fig. (2) that the field distributed in the region between R ₁ and R ₃ is only slightly disturbed. This kind of structure is expected to have better performance for its less energy loss. In case (3), where reflections at both interfaces are eliminated, the field almost has the same pattern when compared with that in the ideal concentrators. We can expect great improvement in the concentrating functionality for its least scattering. The following time-averaged energy densities give a more intuitive illustration of the concentrating property for each case.

 

Fig. 3. Time-averaged total energy density around the concentrators for the first case (a), the second (b) and the third (c)

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As shown in Fig. (3), the energy density is inhomogeneously distributed within the interior circular region in case (1) and (2), the combination of the incident wave with the scattering part strengthens the field energy at some places, and weaken the power at other locations, while the energy intensity in the third case is almost invariant with the positions inside the inner circle in spite of slightly energy loss in the area between the two interfaces. This kind of concentrator with simpler material specification and good concentrating ability is believed to be preferable in the future practical implementation.

5. Conclusion

In summary, cylindrical concentrators with different sets of simplified material parameters have been proposed using TE polarized wave incidence. Spatial dependence of the material tensor components was eliminated, dramatically reduced the difficulty in future practical implementation. Their abilities to concentrate incident field energy in the inner circle and the scattering properties at the inner and outer boundary have also been theoretically and numerically analyzed. We believe that the reduced concentrator with matched interior and exterior interfaces would provide the convenience for the realization in practice.