Superscatterer: Enhancement of scattering with complementary media

Based on the concept of complementary media, we propose a novel design which can enhance the electromagnetic wave scattering cross section of an object so that it looks like a scatterer bigger than the scale of the device. Such a ``superscatterer'' is realized by coating a negative refractive material shell on a perfect electrical conductor cylinder. The scattering field is analytically obtained by Mie scattering theory, and confirmed by full-wave simulations numerically. Such a device can be regarded as a cylindrical concave mirror for all angles.

nipulating 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 easer for EM wave detection, which we refer hereafter as a superscatterer.
In this Letter, we propose a generalized technique to achieve such a superscatterer. In the quasistatic limit, Nicorovici et al. [15] 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 (NRM) [16,17], Pendry and Ramakrishna [18] 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 [18]. In Fig. 1(a), a slab of metamaterial with ǫ = 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 (−iwt). 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 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 coordinate system (f (r), θ, z), in which f (r) is a continuous and piecewise differentiable function of the original radial coordinate r, the wave equation of fieldẼ z (f (r), θ) = E z (r, θ) is transformed where f ′ (r) denotes df (r)/dr. By taking the components of permittivity and permeability tensors as [14], where ǫ 0 and µ 0 are the vacuum permittivity and permeability, the Eq. 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 θ 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 func- 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 f (R 1 ) = R 3 > R 2 , the material of the shell must be NRM from Eqs. (4), and the fields 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) 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 , 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 [19,20], 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 NRM 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".
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 ǫ and µ are taken as follows: ǫ r , µ r ∈ [−3, −1], ǫ θ , µ θ ∈ [−1, − 1 3 ], and ǫ z , µ z ∈ [−3, −1]. A tiny absorptive imaginary part (∼ 10 −5 ) is added to ǫ and µ due to the inevitable losses of the NRM. 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)-(d) give the snapshots of the Gaussian beam in free space, the total electric field induced by PEC cylinder with radius 0.6m, and 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 there is 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 dimension.