Manipulating electromagnetic radiation through metamaterial structures designed by coordinate transformation

Yijun Feng*, Xiaofei Xu, Zhenzhong Yu, and Tian Jiang

Department of Electronic Science and Engineering, Nanjing University, Najing, 210093, China.

*Email:yjfeng@nju.edu.cn

ABSTRACT: In this presentation, we will demonstrate some examples of applying the transformation optics to the manipulation of electromagnetic radiations from antenna or antenna array. By designing through different kinds of coordinate transformations, the radiation direction of an omni-directional antenna could be modulated through a metamaterial cylindrical shell, and conformal antenna array located on an arbitrary curved surface could be modulated to a planar array by covering with properly designed metamaterial structure. The validity of the proposed wave manipulations have been verified through two dimensional full-wave numerical simulations based on the finite element method. The electromagnetic wave manipulation through metamaterials provides alternative ways to steer electromagnetic irradiation and may find novel applications in microwave antenna technology. INTRODUCTION Recent development on metamaterial has expanded the electromagnetic (EM) wave propagation phenomenon such as negative refraction [1], perfect imaging [2], and the EM invisibility cloaking [3-8], since the electromagnetic parameters could now be arbitrary designed as desired. These extraordinary properties are directly determined by media parameters, the permittivity and the permeability. From the application point of view, it requires a certain method to regulate the material parameters to obtain the desired EM device properties. Recently based on the form-invariant of Maxwell’s equations under certain coordinate transformations, the transformation optics proposed in [3-4] for controlling the EM fields has been proved to be an effective approach for manipulating the material properties to satisfy the desired ray traces of the EM waves [5]. The successful application of the transformation optics to the invisibility cloaks has triggered more intensive explorations of this idea theoretically and experimentally, and many interesting theoretical results and practical approaches have been obtained [6-8]. Besides the invisibility cloak, EM beam shifter, splitter, rotators and other interesting devices have also been proposed with exotic EM behaviors by utilizing the transformation optics [9-10]. Particularly, M. Rahm et. al., have expanded the transformation optics by the use of finite-embedded coordinate transformation which has more flexibility to the transformation design of complex materials and enabling the transfer of field manipulations from the transformation optical structures to the surrounding normal medium [10]. In this presentation, we will demonstrate some examples of applying the finite-embedded coordinate transformation method to the manipulation of EM propagations and radiations. By designing the material constitutive tensors through different kinds of coordinate transformations, the wave radiation direction of an omni-directional line source antenna could be modulated through a metamaterial cylindrical shell. Moreover, antenna array with its location limited to a curved surface could behave as a planar array by covering it with a properly designed metamaterial structure through certain coordinate transformation. The performances of the modulation for the omni-directional antenna and the directional antenna array have been verified through two dimensional (2D) full-wave numerical simulations based on the finite element method. Such EM wave manipulating method provides alternative ways to steer EM irradiation and may find novel applications in microwave antenna technology. BEAM MODULATION OF OMNI-DIRECTIONAL ANTENNA The beam modulating device we proposed here is composed of a region with finite size made of transformation designed anisotropic metamaterial in which the electromagnetic waves are transformed as required. This functional material is embedded in free space with the size of only several wavelengths. To design the functional metamaterial, we utilize the so-called finite-embedded coordinate transformation method as proposed in [10]. The embedded transformations add more flexibility to the design of transformation optical structures. It enables the transfer of EM field manipulations from the transformation optical structure to the surrounding medium. The first example is to show how we can use the coordinate transformation to design metamaterial that manipulate the radiation from an EM source. We consider a 2D case with the cylindrical coordinate transformation and modulate the

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EM beam radiation from an omni-directional line source in the circumferential direction. By filling a shell region with transformation designed functional material that surrounding the source, such beam modulation in the circumferential direction could be used to squeeze the omni-directional cylindrical wave to some special directions, either to increase the radiation directivity of the source or to render the original source confused to the outsider observer. We use a coordinate transformation between two cylindrical coordinates, under which the uniformly distributed radial traces in the original cylindrical system could be squeezed in the circumferential direction to redistribute within certain specified directions in the transformed cylindrical system. Different mappings between the two coordinates will yield different beam modulating effects and result in different field distributions. Fig. 1 describes the 2D mappings from a cylindrical shell region (gray region in the left) in the original virtual space to a shell region (gray region in the right) in the coordinate transformed space (only the right half space is illustrated due to the symmetry). Such transformation ensures that the uniformly distributed radial traces within the original shell have been squeezed towards the x-axis within the shell. One simple example of the transformation is given in the following,

'' /a

z z

ρ ρθ θ ρ

=⎧⎪ =⎨⎪ ′ =⎩

(within the shell region), and ''

z z

ρ ρθ θ

=⎧⎪ =⎨⎪ ′ =⎩

(outside the shell region). (1)

If b = 2a, the above transformation indicates that the uniformly distributed radial traces within -π/2 ≤ θ ≤ π/2 in the original shell have been squeezed into -π/4 ≤ θ ≤ π/4 upon the outer boundary of the transformed shell. The material constitutive parameters of the shell in the transformed space are derived through the procedure of transformation optics based on the above cylindrical coordinates transformation [4-5]. The associated relative permittivity and permeability tensors in the transformed space could be calculated as

'det( )

Tεε = J JJ

, and 'det( )

Tμμ = J JJ

, (2)

where, ε and μ tensors are those for the medium in the original virtual space and J denotes the Jacobian transformation matrix, respectively. For the particular coordinate transformation described in (1), the resulted permittivity tensor is

2 2

cos ' sin ' 0 1 / 0 cos ' sin ' 01' sin ' cos ' 0 / ( / ) ( / ) 0 sin ' cos ' 0/

0 0 1 0 0 1 0 0 1

aa a a

a

θ θ θ ρ θ θε θ θ θ ρ θ ρ ρ θ θ

ρ

− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − − + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (3)

The permeability tensor is obtained through (2) and is of the same form as that of the permittivity. The performance of the beam modulating in the circumferential direction has been analyzed through full-wave simulations based on finite element method, and have been displayed in Fig. 2. A linear line source is located in the origin which radiates cylindrical waves omni-directionally in the free space (Fig. 2a). To modulate the beam, the line source is surrounded by a shell with the outer radius b = 3.33λ0 and inner radius a = b/2 =1.67λ0, which is filled with the transformed medium. As shown in Fig. 2b, the omni-directional cylindrical waves radiated from the line source are condensed to left and right directions, resulting in a directive radiation to the broadsides. We also demonstrated that by using metamaterial shell designed through other cylindrical coordinate transformations, treble (Fig. 2c) or even

Fig. 1 Coordinate mapping from a cylindrical shell region (gray region in the left) in the original virtual space to a shell region (gray region in the right) in the coordinate transformed space with the outer radius b and inner radius a. Radial traces have been squeezed through the transformation.

quadruple beam splitter could be obtained. Both the EM field distributions and the power density flows have confirmed the performance of the circumferential beam modulating. We believe such EM wave manipulating devices and the design method could also find applications in other potential EM devices. MANIPULATING ANTENNA ARRAY THROUGH COORDIANTE TRANSFORMATION As a second example, we consider manipulating the performance of antenna array by metamaterial designed through coordinate transformation. We will show that by covering properly designed metamaterial structure, antenna array with its location limited to an arbitrary curved surface could behave as a planar array on a horizontal surface. This could be useful for the design of conformal antenna array where the array should be set at the boundary of an object. For simplicity, we assume a 2D problem as indicated in Fig. 3, where an antenna array is located at the surface of a curved object. We can utilize a particularly designed coordinate transformation as shown in Fig. 3, through which the curved surface could be transformed to planar surface. Therefore by designing a metamaterial in the grey region that covers the curved surface, the antenna array in the right part of Fig. 3 will behaves like a planar array located on the horizontal surface in the free-space and enables us to easily design the antenna performance through the planar array. To demonstrate this method, we show one simple example as described in Fig. 4. A five-element antenna array is located on the curved boundary of a triangle shape. By covering the boundary with a metamaterial structure (two back to back triangle structures as denoted in Fig. 4b) that has been designed through a particular coordinate transformation, the whole structure is equivalent to an antenna array just set on the horizontal boundary in free-space. The detail design of the metamaterial structure will be discussed in the presentation. To verify the equivalence, full wave simulations have been carried out to calculate both the near-field EM field distributions and the far-field antenna radiation patterns.

Fig. 3 Coordinate transformation between the two grey region in the left and right. Through this coordinate transformation the antenna array (denoted by the white dots) on an arbitrary curved surface (right) is transformed to a planar array (left).

Fig. 2 Transverse electric field distribution, power flow lines (white lines) [the lower row] and the power density distribution [the upper row] for a line source at the origin radiating in the free space (a), a double beam splitter (b), a treble beam splitter (c) and a quadruple beam splitter (d) (the black lines denote the boundary of the shell filled with the metamaterial).

The antenna array is assumed to be a five-element Doph-Tchebyscheff array with phase difference of π/3, element separation of half wavelength, and an amplitude distribution of 1.0, 1.6, 1.9, 1.6, and 1.0. As shown in Fig. 4, both the near-field distributions and the far-field radiation patterns reveal a highly directive radiation along the direction of 108° for either of the two array systems. The equivalence indicates that we can change an arbitrary conformal antenna array to a simple planar array by covering it with specially designed metamaterial, which simplifies the performance design of the conformal antenna array. To summary, we have proposed the application of transformation optics through specially designed coordinate transformation to the manipulation of the EM wave radiation from either single antenna or antenna array. We believe such method provides us alternative ways and more freedom in design and optimization of antenna systems. This work is supported by the National Basic Research Program of China (2004CB719800) and the National Nature Science Foundation of China (60990320, 60990322, 60801001, and 60671002). REFERENCES [1] R.A. Shelby, D.R. Smith and S. Schultz, “Experimental verification of a negative index of refraction,” Science,

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(a) (b) (c)

Fig. 4 The near-field electric field distributions [(a), (b)], and the far-field radiation patterns (c) for both the five element (denoted by white dots) antenna array on the curved boundary covered by metamaterial and its equivalent planar array in the free-space. Perfect magnetic conducting (PMC) boundary is assumed as the surface to insure radiation to the upper half space.