## Abstract

A three-dimensional cone-shaped concentrator was designed and analyzed through an approach of coordinate transformation theory. The device can provide varying performances for concentrating along the symmetric axis. The physical picture regarding concentrating ability of this structure was revealed and quantitative analyses were performed for the purpose of investigating the dependence of the concentrating properties on the structural parameters. Moreover, reduced material parameters were theoretically derived and the corresponding mismatched impedance at boundaries was analyzed. Finite element method-based numerical simulations results of the device were further presented to verify our theoretical design.

© 2008 Optical Society of America

## 1. Introduction

Since control of electromagnetic waves using the coordinate transformation was proposed by Ward, *et al.*, [1], various types of devices, recently termed transformation media, have been suggested for the purpose of realization of some specific electromagnetic functions. The cylindrical and sphere cloaks were put forth firstly based on this theory [2, 3]. Other optical transformation media such as magnifying perfect lens and superlens were designed by applying this method [4]. This approach was confirmed by interpreting the coordinate transformation in terms of Jacobean matrix called form-invariant transformation of Maxwell’s equation by D.Schurig later [5]. Consequently, an electromagnetic fields rotator, a square-shaped cloak were designed using this method [6, 7]. Most recently, the appearance of metamaterials provides a way to realize these anisotropic and inhomogeneous materials, and a cloak in microwave range has been experimentally demonstrated [3, 8]. Since the material parameters are complex, a way to derive an imperfect but simpler version of the materials parameters was offered to make the fabrication easier [3, 9].

A cylindrical electromagnetic concentrator that makes the incident wave power flow concentrate within the inner region was proposed [7]. It is expected to be of potential use in the applications where high field intensities are required. However, the constant enhancement of the energy along the symmetric axis in the cylindrical concentrator can not satisfy some requirements where various enhancement of energy are needed. In this work, a cone-shaped concentrator was proposed to provide varying energy distribution along the symmetry axis. The design method and approach to derive the material parameters based on basic theory of Ward’s coordinate transformation are given in the following section. Then the numerical simulation are presented and analyzed. Finally, several sets of reduced material parameters are discussed.

## 2. Principle

Concentrator is a device to collect the energy into a small region and the schematic diagram of the cone-shaped concentrator is represented in Fig. 1(a). When the electromagnetic wave propagates into the region enclosed by the dashed circle, the energy is concentrated into the yellow region. For this purpose, anisotropic and inhomogeneous material is needed which can be designed by the approach of coordinate transformation according to the change of the space.

To redistribute the field, a two-steps space deformation is necessary as suggested in Ref. [7]: the first one is compressing the space within (0≤r<*R*
_{2}) into a circle with a radius of *R*
_{1}, named as region I hereinafter; the second one is expanding the space between *R _{2}* and

*R*to a region between

_{3}*R*and

_{1}*R*so as to make the space continuous, named region II hereinafter. Figure 1(b) is the schematic diagram of the space transformation. Note that

_{3}*R*(

_{i}*i*=1, 2, 3) is a function of

*z*in this device. The process can be mathematically described as:

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\theta \prime =\theta $$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}z\prime =z$$

According to the theory of coordinate transformation in Ref. [1], we firstly calculate the transformation coefficients with respect to different regions in the cylindrical system:

In region I :

In region II :

$${Q}_{\theta \prime}=\left[1+\frac{{R}_{3}\left({R}_{2}-{R}_{1}\right)}{r\prime \left({R}_{3}-{R}_{2}\right)}\right]\frac{{R}_{3}-{R}_{2}}{{R}_{3}-{R}_{1}}$$

$${Q}_{z\prime}=1$$

So the material parameters in the cylinder coordinate are

Here, we designed a structure with parameters of *R _{1}*=

*R*-

_{10}*zt*,

*R*=

_{2}*R*-

_{20}*zt*,

*R*=

_{3}*R*-

_{30}*zt*, where

*z*is the symmetric axis,

*R*(

_{i0}*i*=1, 2, 3) is the radius at

*z*=0, and

*t*is the slope of the cone. The ratio

*R*/

_{2}*R*that affects the concentrating performance greatly can be expressed as

_{1}It can be seen from Eq. (5) that the parameters in region II are anisotropic and inhomogeneous which is difficult to fabricate using current technologies, so reduced parameters are required. For a complete EM mode concentrator, we have 6 material parameters taken into account, including both the permittivity and permeability in the radial (*r*), azimuthally (*φ*) and *z* directions which are hard to satisfy at the same time. Here a TE mode design is considered and the TM mode design follows the same principle by making *ε*→*µ* and *µ*→*ε* substitutions. In the TE mode design case, only *ε _{z}*,

*µ*and

_{φ}*µ*enter into the Maxwell’s equations. Therefore, we just consider the

_{r}*ε*,

_{z}*µ*and

_{φ}*µ*, and assume that the other three parameters,

_{r}*µ*,

_{z}*ε*and

_{φ}*ε*, are units which means the TM mode energy will remain the same as in free space. The reduced parameters that make the same solutions for the wave equations are obtained as long as the product of

_{r}*µ*and

_{r}ε_{z}*µ*remain the same value [3, 9]. When

_{φ}ε_{z}*ε*and

_{z}*µ*are constants, only

_{φ}*µ*is dependent on the coordinate; and there are numerous sets of the parameters, theoretically. Considering the convenience of fabrication and the matching condition at the boundary, three sets of the reduced material parameters are derived. For the first set,

_{r}*ε*=1, which is easy to realize; for the second set, the parameters in

_{z}*φ*and

*z*directions are the same, and the impedance matches at the boundary of

*r*=

*R*; and the third set has an impedance matched at the boundary of

_{3}*r*=

*R*

_{1}. They can be written in the following equations respectively.

## 3. Ideal material parameters of the cone-shaped concentrator

Numerical simulation results for the ideal concentrator using finite element method are given in Fig. 2(a). The simulation area is 30 cm×30 cm×15 cm. Origin of the coordinate is at the center of the cone, and *z* axis is the symmetric axis. The incident wave is TE polarized with the frequency of 4GHz, which propagates along the y axis with parameters of *R _{10}*=1.5 cm,

*R*=3.5 cm,

_{20}*R*=5.5 cm and

_{30}*t*=1/15. Figure 2(a) shows the distribution of

*E*in

_{z}*x*-

*y*plane and

*y*-

*z*plane respectively. And the distribution of

*E*in a cross-section of the cone parallel to the

_{z}*x*-

*y*plane at

*z*=6 cm with

*R*=1.1 cm,

_{1}*R*=3.1 cm and

_{2}*R*=5.1 cm is presented in Fig. 2(b). The electric field is concentrated to the region with a radius of

_{3}*R*. The wavelength in free space is about

_{1}*R*/

_{2}*R*times of that in region I.

_{1}It can be seen from Fig. 2 that the phase velocity in region II which is caused by the distribution of the inhomogeneous refraction index defined as

Refraction index in the region II decrease with the increase of the radius, so a smaller *r* will lead to a larger wave number and a smaller velocity.

Considering the space deformation, the enhancement of energy in region I is proportion to the square of *R _{2}*/

*R*, because we have compressed the space with radius of

_{1}*R*into a region with radius of

_{2}*R*in the first step of the transformation. Figure 3(a) shows the energy distribution along the

_{1}*z*axis at

*x*=

*y*=0. It can be seen that the energy varies with different position of

*z*, which means that the ability of concentrating is not constant along the symmetrical axis. Figure 3(b) represents the relationship between

*R*/

_{2}*R*and the concentrating ability. The energy distribution in free space is marked by the red line, and the average value along the line is 4.5×10

_{1}^{-12}J/m

^{2}. The green curve represent energy distribution along the

*y*axis of the concentrator at site of

*x*=0 and

*z*=7.5 cm with

*R*=1 cm,

_{1}*R*=3 cm and

_{2}*R*=5 cm, so

_{3}*R*/

_{2}*R*=3. Obviously, the energy in the center, whose average value is 4.25×10

_{1}^{-11}J/m

^{2}, is about 9 times larger than that of the value in free space. A similar result can be obtained from the yellow curve which represents energy distribution along

*y*axis at

*x*=0 and

*z*=0 with

*R*=1.5 cm,

_{1}*R*=3.5 cm and

_{2}*R*=5.5 cm. In this case

_{3}*R*/

_{2}*R*=2.3, the averaged energy value in region I is 2.4×10e

_{1}^{-11}J/m

^{2}, so the enhancement obtained from Fig. 3(b) is ~5.3. The averaged energy in region I of the blue line at

*x*=0 and

*z*=-7.5 cm with

*R*=2 cm,

_{1}*R*=4 cm and

_{2}*R*=6 cm is 1.68×10

_{3}^{-11}J/m

^{2}which is almost 4 times larger than the case in free space. The coarseness of the lines is caused by the finite number from the finite elements.

## 4. Reduced parameters of the cone-shaped concentrator

The designed parameters are all complicated and position-relevant, and difficult to realize in practice. Therefore, sets of reduced parameters are proposed here based on the Maxwell’s equations and wave equations. The physical insight of this change is that the reduced parameters have the same refraction index which controls the trace of the light as the ideal ones with the penalty of causing the scattering due to the impedance mismatch at the interfaces. In order to keep the product of *µ _{r}ε_{z}* and

*µ*to be the same, three sets of reduced parameters have been derived in section2 for their specificity. Figure 4 shows the impedance of four conditions at the site of

_{φ}ε_{z}*x*=

*z*=0. For the ideal case, the impedance is

*Z*=(

*R*-

_{3}*R*)/[(

_{1}*R*-

_{3}*R*)

_{2}*k*], which is matched at both boundaries with

*r*=

*R*and

_{3}*r*=

*R*. The impedance of the reduced parameters is

_{1}*Z*=(

*R*-

_{3}*R*)/(

_{2}*R*-

_{3}*R*) for set 1 (green line), and it does not match the impedance at both boundaries of

_{1}*r*=

*R*and

_{1}*r*=

*R*in this case; it can be matched at the inner boundary at

_{3}*r*=

*R*only in the case of

_{1}*R*=

_{3}*R*+

_{2}*R*. The impedance is

_{1}*Z*=1 for set 2 which is matched at

*r*=

*R*and

_{3}*Z*=

*R*/

_{1}*R*for set 3 matched at

_{2}*r*=

*R*. Hence, we can choose the proper set of the reduced material parameters as desired.

_{1}## 5. Conclusion

We have presented a cone-shaped concentrator with various abilities for concentrating along the symmetric axis based on the coordinate transformation in this paper. Both cases of ideal and reduced material parameters are given respectively. Electromagnetic behavior of the device was simulated and analyzed by means of a finite element method. Our numerical simulation results demonstrated that the proposed configuration is an appropriate structure for obtaining varying energy enhancements. This structure can be extended to design a concentrator with various functioning effects along the symmetric axis as we needed by specifying the generatrix of the structure.

## Acknowledgments

This work was supported by 973 Program of China (No.2006CB302900) and 863 Program of China (2006AA04Z310).

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