Recently great progress has been made[1, 2, 3, 4, 5, 6, 9, 7, 8, 10, 11, 12, 13] in manipulating the electromagnetic (EM) fields by means of metamaterials. By employing the coordinate transformation approach proposed by Leonhardt [1] and Pendry et al. [2], various exciting functional EM devices have been reported [4, 9, 10]. This methodology provides a clear geometric picture of those designed devices, and the permittivity and permeability tensors of designed functional materials can be derived from coordinate transformations directly. On the other hand, Mie scattering theory [11, 12, 14] provided an analytic approach to quantitatively analyze the scattering properties of EM fields, and the aforementioned functional devices can also be realized by choosing different scalar transformation functions. The combination of the geometric and the analytic approaches may give novel and surprising results. For example, in contrast to invisibility cloak, we may design an EM transformation media device to enlarge the scattering cross-section of a small object. In this way the object is effectively magnified to a size larger than the object plus the device so that it is much easier for EM wave detection, which we refer hereafter as a superscatterer.

 

Fig. 1. (a) The schematic demonstration of the behavior of a beam propagates in a complementary media and PEC boundary. (b) The behavior is extended to the 2D case, where a superscatterer is formed with its effective size shown by the dashed line.

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In this paper, we propose a generalized technique to achieve such a superscatterer. In the quasistatic limit, Nicorovici et al. [15, 16] had demonstrated that the properties of a coated core can be extended beyond the shell into the matrix. It is called a partially-resonant system in which at least one dielectric constant of three-phase composite structure is negative. Moreover, by using the negative refracting material (NIM) [17, 18], Pendry and Ramakrishna [19] showed how to image an object by perfect cylindrical lens based on the concept of complementary media. Here we limit our discussion on the 2D case and prove that a special magnified image acts as a real object for EM wave detection.

Figure 1 shows a heuristic model stemmed from the concept of complementary media [19]. In Fig. 1(a), a slab of metamaterial with ε=µ=-1 images point source S at F ₂ inside and F ₁ outside the metamaterial if the distance between source and slab is less than the slab thickness. Now a perfect electrical conductor(PEC) boundary is set at R ₁, so the light propagating in the slab is completely reflected by it and focuses at I ₂ inside and I ₁ outside the slab. Since the slab and the vacuum with the same thickness are complementary to each other, it looks as if the medium between R ₃ and R ₁ is moved, and the PEC at R ₁ is virtually shifted to R ₃ and takes effect as a real PEC boundary. The picture can be extended to the 2D case as shown in Fig. 1(b), in which z axis is perpendicular to the paper. In the 2D case, the R ₁, R ₂ and R ₃ represent the radius of the inner region, the outer radius of the cylindrical annulus (metamaterial shell) and the effective radius of the virtual cylinder, respectively. In order to move effectively the inner PEC boundary at r=R ₁ to the surface of the virtual cylinder, where the radius is r=R ₃>R ₂, both permittivity and permeability tensors in the metamaterial shell should be selected properly. The Mie scattering theory is a powerful tool to accomplish the goal.

We consider a transverse-electric (TE) polarized EM incident field with harmonic time dependence exp(-iωt). The coordinate system is the cylindrical coordinate coinciding with the cylinder we considered. In this coordinate system the permittivity and permeability tensors can be put in the following general form

where ε ₀ and µ₀ are the vacuum permittivity and permeability. The EM fields in the homogeneous material region (r>R ₂) are well known and can be expressed by the superposition of cylindrical functions. The wave equation of E _z in the shell is written as

where k ₀ is the wave vector of the EM wave in vacuum. Now we introduce a new variable u=f(r), in which f(r) is a continuous and piecewise differentiable function of the original radial coordinate r. In the new coordinate system (u,θ,z), the wave equation of Eq.(2) is transformed to

where F(u,θ)=E _z(r,θ) and u′ denotes du(r)/dr. By taking the components of permittivity and permeability tensors as [14],

The Eq. (3) can be solved by separation of variables F=R(u)Θ(θ). The solution of R(u) and Θ(θ) are just the mth-order Bessel functions and exp(imθ), in which m are integers.

With the above analysis, the electric fields in each domain are expressed as

where J _m and H ⁽¹⁾ _m are the mth-order Bessel function and Hankel function of the first kind, respectively. By means of the orthogonality of exp(imθ) and the continuity of E _z and $H_{\theta }=\frac{i}{\omega {\mu }_{\theta }}{f}^{\text{'}}\frac{\partial E_z}{\partial f}$ at two interfaces (r=R ₁ and r=R ₂), we obtain the linear relationships between the coefficients as follows,

where the prime denotes differentiation with respect to the entire argument of Bessel functions. By imposing the boundary condition of f(r), f(R ₂)=R ₂, we get the solutions of Eqs.(6) as follows,

This is the exact solution for scattering matrix.

We can draw some interesting conclusions from Eqs. (7). If f(r) is a monotonic function and R ₃=f(R ₁)>R ₂, the material of the shell must be the NIM from Eqs.(4), and the E _z field at point (r,θ,z) in inner annulus R ₁<r<R ₂ are equal to those at point (f (r),θ,z) in outer region R ₂<r<R ₃. Moreover, in the region r>R ₃, the scattering fields are exactly equal to those scattered by a PEC cylinder with radius R ₃, as if whole cylindrical annulus of R ₁<r<R ₃ is moved and the PEC boundary at r=R ₁ is magnified and shifted to r=R ₃. Here, the cylinder with radius f(R ₁) is called a virtual cylinder. It is just the expected result in Fig. 1(b).

 

Fig. 2. A simple function f(r) satisfies the condition of Fig.1(b)

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For simplicity, we choose a linear function for f(r),

Here b ₀ is an important parameter to determine the magnification factor. It should be noted that both cloak and concentrator can also be obtained in the same manner. For example, it is a perfect cloak when b ₀=-R ₂, and it becomes an imperfect cloak when R ₁-R ₂>b ₀>-R ₂, or a concentrator when b ₀>R ₁-R ₂ and f(r)=rf(R ₁)/R ₁ in the region r<R ₁. However, the case b ₀>0, which provides a folded geometry [20, 21], is less discussed. In fact, with the help of the geometric picture of the concentrator, it means a big cylinder with radius b ₀+R ₂ is compressed into a small cylinder with radius R ₁, and the gap between r=R ₁ and r=b ₀+R ₂ has to be filled by a pair of complementary media: One is the vacuum in R ₂<r<b ₀+R ₂ and the other is the NIM shell in R ₁<r<R ₂. This device scatters the same fields as the uncompressed cylinder, which extends beyond the shell, so we call it “superscatterer”.

It should be remarked that Fig. 1 only demonstrates the special case that has the biggest virtual PEC surface. For the general case, if there are various objects in the big cylindrical region r≤R ₃, and both objects and vacuum in this region are compressed into the small cylindrical region r≤R ₁, all material parameters in the small region can be obtained by the corresponding coordinate transformation. As soon as we impose the pair of complementary media on the region R ₁<r<R ₃, all should be restored in the EM detection as if nothing has happened. On the other hand, if we know the material parameters in the region r≤R ₁, the restored results can be easily obtained by the inverse coordinate transformation. In this way, using a small device in EM detection, we can get any detectable virtual object bounded in the virtual cylinder which is bigger than the device, although it is vacuum outside the small device.

 

Fig. 3. Snapshot of the total electric field. (a) is the total electric field induced by PEC cylinder with radius R ₃=0.3m and (b) is the total electric field induced by the designed device (the radius of virtual cylinder is 0.3m)

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Next, we present the patterns of electric field calculated by finite element solver of the Comsol Multiphysics software package. In Fig. 3, the plane wave is normal incident from left to right with frequency 3 GHz and unit amplitude, and the inner and outer radii of the shell are R ₁=0.1m and R ₂=0.2m, respectively. When b ₀ is equal to 0.1m and the function f(r) is taken as in Eq. (8), the radius of the virtual cylinder becomes f(R ₁)=b ₀+R ₂=0.3m. The ranges of the components of ε̿ and µ̿ are taken as follows: ${\epsilon }_r,{\mu }_r\in [-3,-1],{\epsilon }_{\theta },{\mu }_{\theta }\in [-1,-\frac{1}{3}],$, and ε _z,µ _z ∈ [-3,-1]. A tiny absorptive imaginary part (~10^-5) is added to ε̿ and µ̿ due to the inevitable losses of the NIM.

Figure 3(a) is a snapshot of the total electric field induced by a PEC cylinder with radius R ₃=0.3m; Fig. 3(b) is the corresponding field induced by the superscatterer. Comparing the patterns of electric field in the region r>R ₃, one can find they are almost equivalent. Here, the bounds of the amplitude of electric field in Fig. 3 set from -2 to 2 for clarity. The white flecks in the region of the complementary media show the regions where the values of fields exceed the bounds. The highest value of the field in the flecks is about 10 ², which comes from the dominative high-m modes with the factor J _m(k ₀ R ₃)H ⁽¹⁾ _m(k ₀ f(r))/H ⁽¹⁾ m(k ₀ R ₃) in the scattering field.

We define the magnification factor ηas f(R ₁)/R ₂, which is the ratio of the radius of the virtual cylinder to the real size of this device. The components of permittivity and permeability tensors are negative for the case of η>1. There is no singularity for any finite η as long as both f(r) and f′(r) are nonzero in R ₁≤r≤R ₂. For example, when η=20, the ranges of the parameters are: ε _r,µ _r ∈ [-1.1,-0.052], ε _θ,µ _θ ∈ [-19,-0.90], and ε _z,µ _z ∈ [-399,-19]. From the point of view of fabrication, one can choose a more proper function than that of Eq. (8). For the TE wave, one simple choice is f(r)=R ² ₂/r in region R ₁<r<R ₂, the corresponding parameters are µ _r=µθ=-1 and ε _z=-R ⁴ ₂/r ⁴, which is easier in fabrication. However, in this case, the magnification η=f(R ₁)/R ₂=R ₂/R ₁ is completely fixed by the inner and outer radius of the NIM shell, so it might be unsuitable for large magnification. As a comparison, when η=20, there is a large value ε _z=-(R ₂/R ₁)⁴=-1.6×10⁵ at the inner surface r=R ₁.

 

Fig. 4. A Gaussian beam in free space is shown in (a). (b) and (c) are the scattering and total fields induced by the superscatterer (the radius of the virtual cylinder is 0.6m), respectively.

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Moreover, the device in Fig. 1(b) can be regarded as a cylindrical concave mirror for all angles, which can not be realized by ordinary media. From the theory of geometrical optics, when a plane wave incidents on it, the paraxial beams focus at r=f(R ₁)/2=(b ₀+R ₂)/2. Here, a full-wave simulations with COMSOL Multiphysics are used to test the focus behavior of a Gaussian beam propagated from left to right with unit amplitude. Fig. 4(a)–(c) give the snapshots of the Gaussian beam in free space, the scattering and total electric fields induced by the superscatterer (here R ₁=0.1m, R ₂=0.2m and b ₀=0.4m), respectively. The scattering electric field in Fig. 4(c) shows the focus is approximately at r=0.3m since Gaussian beam is close to a paraxial beam. Here, it is worthy noting that the superscatterer becomes a cylindrical concave mirror for all angles if and only if b ₀>R ₂. In principle, if we make the radius f(R ₁) of virtual cylinder to infinity by adjusting the function f(r), the cylindrical concave mirror becomes a plane mirror for all angles, so that any parallel light incidents to the superscatterer will be reflected straight back along the incident path.

In conclusion, we demonstrated the properties of a “superscatterer” in terms of PEC boundary and properly complementary media. This kind of functional devices might be important in EM detection. Similar concept can be extended to the case of non PEC boundary and three dimensional.