Two-dimensional elliptical electromagnetic superscatterer and superabsorber

Xiaofei Zang; Chun Jiang

doi:10.1364/OE.18.006891

1. Introduction

In the past few years, the exciting issue of achieving invisibility of objects has received much attention [1–28]. Based on the coordinate transformation of Maxwell’s equations, Pendry et al theoretically proposed an invisibility cloak that can hide the objects inside the cloak from detection, the cloak having electrical permittivity and magnetic permeability that are both spatially varying and anisotropic [2]. Following this theory, an electromagnetic cloak bends and guides incoming waves smoothly around the cloaked region, so that the fields after having emerged from the cloak are the same as if the incident waves have just passed through the free space. Cummer et al. first reported full-wave electromagnetic simulations of the cylindrical version of this cloaking structure using ideal and nonideal (but physically realizable) electromagnetic parameters [3]. Based on low-index electromagnetic metamaterials, a microwave invisibility cloak was soon realized in experiments [4]. Recently, acoustic transparency phenomenon for a multilayered sphere with acoustic metamaterial has been proposed by Zhou et al., which opened a new possibility to enrich the applications of metamaterials cloaking techniques [12,13]. In addition, electromagnetic cloaks with arbitrary geometries have also been investigated [23,24].

In contrast to concealing an object, another interesting phenomenon such as superscatterer which means that the object looks like a scatterer bigger than its geometric size in electromagnetic wave detection, was proposed by Yang et al. based on the concept of complementary media [29–36]. Such a superscatterer [29] was realized by coating a negative refractive material shell on a perfect electrical conductor cylinder. Based on the mechanism of superscatterer, Jack et al. designed a frequency-selective superabsorber metamaterial [30], which could be constructed by using an absorbing core material coated with a shell of isotropic double negative metamaterial. They found that the absorption cross section of a fixed volume could be made arbitrarily large at one frequency. Chen et al. proposed a general theoretical method to design a superscatterer with an arbitrary cross section [34]. Their theoretical results showed that the PEC reshaper could reshape the PEC boundary arbitrarily. But, they demonstrated the properties of the PEC reshaper with a circular PEC coating with circular metamaterial shell.

In the above investigations [29–33] of superscatterer and superabsorber, all theoretical analyses, numerical simulations and parameter designs were devoted mainly to circular objects(axial and radial symmetries), which were relatively easier to design and analyse. Although Chen et al. [34] used coordinate transformations theory to design a superscatterer with an arbitrary cross section, their numerical simulation is only on a circular object with perfect axial and radial symmetries. In the practical application or in general, most objects don’t have such perfect symmetries, what happen to the arbitrarily shaped superscatterer and superabsorber? An ellipse may have very good representative of arbitrarily shaped objects to a very great extent, because the boundary of the ellipse is a function of angle just as arbitrarily shaped objects’ boundaries versus their corresponding angles (the boundary of a circle is constant versus its angle). Therefore, in this paper, we proposed the designs of two-dimensional elliptical electromagnetic superscatterer and superabsorber with elliptical cross section described by three functions R₁(θ), R₂(θ), and R₃(θ), giving an angle-dependent distance from the origin. Our designs can be extended to the arbitrarily shaped electromagnetic superscatterer and superabsorber. The elliptical electromagnetic superscatterer (or superabsorber) could be realized by coating an elliptical negative refractive material shell. The properties of elliptical electromagnetic superscatterer and superabsorber were investigated by using the finite-element simulation.

2. Elliptical electromagnetic superscatterer

We consider an elliptical-cylindrical object with elliptical cross section embedded in free space, which is covered by elliptical negative refractive material shell as shown in Fig. 1 . The inner and outer boundaries of the shell are described by two functions R₁(θ) and R₂(θ), and their minor semi-axis in the y direction are a and b, respectively. Based on the concept of complementary media, the practical cross section is equal to a bigger ellipsoid with the outer boundary of R₃(θ) as shown in Fig. 1. The geometric coordinate transformation between the new system (f(ρ), θ', z') and the original system (ρ, θ, z) can be expressed as:

f (ρ) = \frac{R_{3} (θ) - R_{2} (θ)}{R_{2} (θ) - R_{1} (θ)} (R {}_{2}{(θ)} - ρ) + R_{2} (θ),

θ^{'} = θ,

z^{'} = z,

where f(ρ) is a monotonic differentiable function. The inner, middle and outer boundaries are:

\begin{array}{l} R_{1} (θ) = \frac{a}{\sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}}} \\ R_{2} (θ) = \frac{b}{\sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}}}, \\ R_{3} (θ) = \frac{c}{\sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}}} \end{array}

where k is the ratio between the major semi-axis and minor semi-axis of the corresponding ellipse.

Fig. 1 (Color online). Computational domain of the elliptical superscatterer and superabsorber. The inner ellipse is a PEC (or an absorber core) with boundary of R₁(θ) and minor semi-axis of a. The middle shell is an elliptical negative refractive material shell with outer boundary of R₂(θ) and minor semi-axis of b. If the inner ellipsoid coating with the elliptical negative refractive shell, its cross section is the same as a large ellipsoid with boundary of R₃(θ) and minor semi-axis of c.

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The corresponding permittivity and permeability tensors of the transformation media in the cylindrical coordinates are,

\frac{\bar{\bar{ε}}}{ε} = \frac{\bar{\bar{μ}}}{μ} = [\begin{matrix} \frac{f [1 + {(\frac{1}{f} \frac{\partial f}{\partial θ})}^{2}]}{ρ \frac{\partial f}{\partial ρ}} & - \frac{1}{f} \frac{\partial f}{\partial θ} & 0 \\ - \frac{1}{f} \frac{\partial f}{\partial θ} & \frac{ρ}{f} \frac{\partial f}{\partial ρ} & 0 \\ 0 & 0 & \frac{f}{ρ} \frac{\partial f}{\partial ρ} \end{matrix}],

Substituting Eqs. (1), (2) into Eq. (3), we obtain

\frac{\bar{\bar{ε}}}{ε} = \frac{\bar{\bar{μ}}}{μ} = ⌊ \begin{matrix} μ_{r r} & μ_{r θ} & 0 \\ μ_{θ r} & μ_{θ θ} & 0 \\ 0 & 0 & μ_{z z} \end{matrix} ⌋ = [\begin{matrix} \frac{f_{0} [1 + \frac{1}{f_{0}^{2}} {(f_{2})}^{2}]}{ρ f_{1}} & - \frac{1}{f_{0}} f_{2} & 0 \\ - \frac{1}{f_{0}} f_{2} & \frac{ρ}{f_{0}} f_{1} & 0 \\ 0 & 0 & \frac{f_{0}}{ρ} f_{1} \end{matrix}],

where

f_{0} = \frac{b - c}{b - a} ρ + \frac{c - a}{b - a} \frac{b}{\sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}}}

,

f_{1} = \frac{b - c}{b - a},

and

f_{2} = \frac{c - a}{b - a} \frac{b (1 / k^{2} - 1) \sin θ \cos θ}{{(\sin^{2} θ + \cos^{2} θ / k^{2})}^{3 / 2}}

.

The corresponding permittivity and permeability tensors of the transformation media in Cartesian coordinates are

[\begin{matrix} μ_{x x} & μ_{x y} \\ μ_{y x} & μ_{y y} \end{matrix}] = [\begin{matrix} \cos θ^{'} & - \sin θ^{'} \\ \sin θ^{'} & \cos θ^{'} \end{matrix}] [\begin{matrix} μ_{r r} & μ_{r θ} \\ μ_{θ r} & μ_{θ θ} \end{matrix}] [\begin{matrix} \cos θ^{'} & \sin θ^{'} \\ - \sin θ^{'} & \cos θ^{'} \end{matrix}]

where

\begin{array}{l} μ_{x x} = (\frac{f_{0}}{ρ f_{1}} + \frac{f_{2}^{2}}{ρ f_{0} f_{1}}) \cos^{2} θ + \frac{ρ f_{1}}{f_{0}} \sin^{2} θ + 2 \frac{f_{2}}{f_{0}} \sin θ \cos θ, \\ μ_{x y} = μ_{y x} = (\frac{f_{0}}{ρ f_{1}} + \frac{f_{2}^{2}}{ρ f_{0} f_{1}} - \frac{ρ f_{1}}{f_{0}}) \sin θ \cos θ - \frac{f_{2}}{f_{0}} (\cos^{2} θ - \sin^{2} θ), \\ μ_{y y} = \frac{ρ f_{1}}{f_{0}} \cos^{2} θ + (\frac{f_{0}}{ρ f_{1}} + \frac{f_{2}^{2}}{ρ f_{0} f_{1}}) \sin^{2} θ - 2 \frac{f_{2}}{f_{0}} \sin θ \cos θ, \\ μ_{z z} = ε_{z z} = \frac{f_{0} f_{1}}{ρ} . \end{array}

In what follows, we perform numerical simulations on elliptical-cylindrical superscatterer by using the finite element method (FEM) to demonstrate the designed formulae of Eq. (6). The computational domain consists of perfectly matched layers (PML) to simulate the absorbing boundary conditions as shown in Fig. 1. The inner ellipse is a perfectly electric conducting (PEC) coating with an elliptical negative refractive material shell. Plane wave propagates from left to right with unit amplitude and frequency of 3GHz.

First, we consider a circular superscatterer with parameters a = 0.1m b = 0.2m, c = 0.3m and k = 1 as shown in Fig. 2(a) (the radius of the PEC is 0.1m). We find that the pattern of electric field (Fig. 2(a)) in the region r>0.3m is equal to Fig. 2 (a). In the interior region, the value of the field exceeds the bounds and appears white flecks. This is the same as Ref [29].

Fig. 2 (Color online) Electric field distribution of PEC (left column) and our designed device (right column) with system parameters (a) c = 0.3m, k = 1, (a’)a = 0.1m, b = 0.2m, k = 1, (b)c = 0.15m, k = 2, (b’)a = 0.05m, b = 0.1m, k = 2, (c) c = 0.15m, k = 2, (c’)a = 0.05m, b = 0.1m, k = 2.The unit is meter.

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Next, we investigate the elliptical superscatterer with parameters a = 0.05m b = 0.1m, c = 0.15m, and k = 2. The elliptical PEC with minor semi-axis c = 0.15m and the corresponding scattering electric field distribution is shown in Fig. 2 (b), where the incident plane wave propagates along the horizontal direction. When a small elliptical PEC with minor semi-axial a = 0.05m is coated by an elliptical negative refractive material shell with outer minor semi-axial b = 0.1m as shown in Fig. 2(b), we can find that scattered field in region r>R₃(θ) is the same as Fig. 2 (b). Figure 2(c) gives the scattering distribution of the electric field when the incident angle of the plane wave is 45°. It is found that scattered field distribution of Fig. 2(c) in region r>R₃(θ) is also the same as Fig. 2(c). Therefore, the elliptical superscatterer is valid for arbitrary angle of incident plane waves. In a word, the elliptical negative refractive material shell makes object look like a scatterer bigger than its geometric size. Such a phenomenon of elliptical superscatterer can be extended to arbitrarily shaped superscatterer.

For the physical realization, one can easily obtain the material parameters of the elliptical superscatterer according to Eq. (6). Figure 3 shows the values of the permeability μ_xx, μ_xy ( = μ_yx), and μ_yy and the permittivity ε_zz( = μ_zz)of the elliptical shell. It is found that the shell of the metamaterial is made of anisotropic and inhomogeneous material similar to the case of circular superscatterer in Ref [29]. The variations of the permeability μ_xx, μ_xy ( = μ_yx), and μ_yy with a fixed radius r ( $r = L / \sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}},$ $a \leq L \leq b$ ) versus angle are shown in Fig. 3 (a), (b), (c). In Fig. 3(d), the value of the permittivity ε_zz increases with the increase of the radius r.

Fig. 3 (Color online) Permittivity and permeability tensor parameters for the superscatterer (a) μ_xx. (b) μ_xy = μ_yx. (c) μ_yy. (d) ε_zz = (μ_zz). The unit is meter.

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3. Elliptical electromagnetic superabsorber

Now, we discuss the elliptical superabsorber. Let us consider geometric coordinate transformation between the new system (f(ρ), θ', z') and the original system (ρ, θ, z):

f (ρ) = \frac{R_{2} {(θ)}^{2}}{ρ},

θ^{'} = θ,

z^{'} = z,

with

R_{1} (θ) = \frac{a}{\sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}}},

R_{2} (θ) = \frac{b}{\sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}}}

(a, b are the inner and outer minor semi-axial of the shell, respectively).

R_{3} (θ) = \frac{b^{2} / a}{\sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}}}

is the outside surface of the practical absorber.

Substituting Eq. (7) into Eq. (4) and using Eq. (5), we obtain

\begin{array}{l} μ_{x x} = (\frac{f_{0}}{ρ f_{1}} + \frac{f_{2}^{2}}{ρ f_{0} f_{1}}) \cos^{2} θ + \frac{ρ f_{1}}{f_{0}} \sin^{2} θ + 2 \frac{f_{2}}{f_{0}} \sin θ \cos θ, \\ μ_{x y} = μ_{y x} = (\frac{f_{0}}{ρ f_{1}} + \frac{f_{2}^{2}}{ρ f_{0} f_{1}} - \frac{ρ f_{1}}{f_{0}}) \sin θ \cos θ - \frac{f_{2}}{f_{0}} (\cos^{2} θ - \sin^{2} θ), \\ μ_{y y} = \frac{ρ f_{1}}{f_{0}} \cos^{2} θ + (\frac{f_{0}}{ρ f_{1}} + \frac{f_{2}^{2}}{ρ f_{0} f_{1}}) \sin^{2} θ - 2 \frac{f_{2}}{f_{0}} \sin θ \cos θ, \\ μ_{z z} = ε_{z z} = \frac{f_{0} f_{1}}{ρ}, \end{array}

where

f_{0} = \frac{b}{\sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}}},

f_{1} = - \frac{b^{2}}{ρ^{2} {(\sin^{2} θ + \cos^{2} θ / k^{2})}^{2}},

and

f_{2} = \frac{b (1 / k^{2} - 1) \sin (2 θ)}{{(\sin^{2} θ + \cos^{2} θ / k^{2})}^{2}}

.

Figure 4(a) illustrates the electric field distribution in the computational domain illuminated by a plane incident wave with frequency 1.1GHz. The circular absorbentmaterial with radius c = 0.4m (k = 1), ε_c = μ_c = 1-i absorbs a large portion of the incident plane wave. Such a large circular absorbentmaterial with radius c = 0.4m can be replaced by a smaller circular absorbentmaterial with radius a = 0.1m coating with a circular negative refractive material shell as shown in Fig. 4(a). As can be seen from Fig. 4(a), when the inner circular has a radius of a = 0.1m, ε_c = 16(1-i), μ_c = 1-i and the shell has an outer radius of b = 0.2m, ε_b = μ_b = −1 and ε_zz = b²/r (0.1m<r<0.2m), the electric field distribution in the region r>0.4m is the same as Fig. 4(a) (similar to the case of Ref [30]).

Fig. 4 (Color online) Electric field distribution of normal absorber(left column) and superabsorber(right column) with system parameters (a) c = 0.4m, k = 1, (a’)c = 0.1m, b = 0.2m, k = 1, (b)c = 0.2m, k = 2, (b’)a = 0.05m, b = 0.1m, k = 2, (c) c = 0.2m, k = 2, (c’)a = 0.05m, b = 0.1m, k = 2. The unit is meter

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Let us now consider the elliptical superabsorber with parameters a = 0.05m, b = 0.1m, c = 0.2m, and k = 2. In Fig. 4(b), we give the electric field distribution of an elliptical absorber with minor semi-axis of 0.2m. For a small elliptical absorbentmaterial with minor semi-axis of 0.05m and coating with an elliptical negative refractive material shell with outer minor semi-axis of 0.1m, we find that the absorption cross section is the same as Fig. 4(b) (as shown in Fig. 4(b)). When the incident angle of the plane wave is 45°, the absorption cross section of the small elliptical absorbentmaterial coating with an elliptical negative refractive material shell shown in Fig. 4(c) is equal to the case of the large elliptical absorbentmaterial as shown in Fig. 4(c) in the region r>R₃(θ). Such a phenomenon of elliptical superabsorber can be also extended to arbitrarily shaped superabsorber.

The corresponding material parameters of elliptical superabsorber are shown in Fig. 5 . The elliptical metamaterial shell is also made of anisotropic and inhomogeneous material different from the case of circular superabsorber with isotropic material in Ref [30]. The variations of the permeability with a fixed radius r ( $r = L / \sqrt{\sin^{2} θ + \cos^{2} θ / k^{2}},$ $a \leq L \leq b$ ) versus angle are given in Fig. 5(a), (b), and (c). As can be seen in Fig. 5(d), the value of the permittivity ε_zz also increases with the increase of the radius r. It can be found that the elliptical superabsorber are highly anisotropic and must be realized with the metamaterial technology.

Fig. 5 (Color online) Permittivity and permeability tensor parameters for the superabsorber (a) μ_xx. (b) μ_xy = μ_yx. (c) μ_yy. (d) ε_zz = (μ_zz). The unit is meter.

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It should be noted that the parameters of μ_rr(ε_rr), μ_rθ(ε_rθ) and μ_θθ(ε_θθ) are no longer constant for the elliptical or arbitrarily shaped superabsorber compared with the Ref [30]. Taking elliptical superabsorber as example, the material parameters of μ_rr μ_rθ and μ_θθ on the outer boundary of the shell versus θ' are shown in Fig. 6 . The parameters of μ_rr(ε_rr) and μ_θθ(ε_θθ) are varied versus θ', while μ_rθ = −1.

Fig. 6 (Color online) Superabsorber parameters of μ_rr, μ_θθ and μ_rθ on the outer boundary (R₂(θ)) of the shell versus θ'.

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4. Conclusion

In summary, we analyzed the electromagnetic response of elliptical superscatterer and superabsorber based on the concept of complementary media. Using coordinate transformation method, the parameters of permittivity and permeability in Cartesian coordinates are deduced. Our designs of elliptical electromagnetic superscatterer and superabsorber are demonstrated by using finite element simulation. The results can be extended to the arbitrarily shaped electromagnetic superscatterer and superabsorber.

Acknowledgment

This work was supported by Ministry of Education Foundation of China (Grant No. 708038).

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Two-dimensional elliptical electromagnetic superscatterer and superabsorber

Abstract

1. Introduction

2. Elliptical electromagnetic superscatterer

3. Elliptical electromagnetic superabsorber

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Equations (14)

Optics Express