1. Introduction

In the past eight years, there has been a surge of interest in electromagnetic (EM) metamaterials deduced from the coordinate transformation approach proposed by Leonhardt [1] and Pendry et al. [2], such as invisibility cloaks designed through the blowup of a point [1, 2], or space folding [3, 4, 5, 6, 7, 8] – the latter being based upon the powerful concept of complementary media introduced by Pendry and Ramakrishna ten years ago [9] – or even superscatterers [10]. Transformation optics is a useful mathematical tool enabling a deeper analytical insight into the scattering properties of EM fields in metamaterials. Geometric transforms [11, 12, 13] can be chosen properly to design the metamaterials.

In this work, we make use of complementary media [9] and geometric transforms in order to design an heterogeneous bianisotropic shell behaving either as a superscatterer or an external cloak, depending upon whether its core is a perfect electric conductor (PEC) or certain bianisotropic medium. Indeed, the twofold functionality of the bianisotropic cylindrical shell which we propose displays the similar homeopathic effect to the dielectric shell studied in [14]. Surface plasmon type resonances are visible on its interfaces when a set of electric antennas appears to be in its close neighbourhood, and this leads to the similar cloaking to [7], although in the present case this cloaking occurs at any frequency. It is also interesting to make large objects of negatively refracting media invisible when they are located closeby a slab or a cylindrical lens of opposite permittivity, permeability and magnetoelectric coupling parameters with a hole of same shape as the object to hide. All these aforementioned bianisotropic cloaks work at finite frequency, but it is interesting to simplify their optical parameters and check to which extent invisibility is preserved: We argue one can achieve an ostrich effect [15] whereby it is virtually impossible to detect the scattering of a set of small electric antennas located nearby a (much visible) slab or cylindrical perfect lens at quasi-static frequencies.

2. Superscatterer through complementary bianisotropic media

The source-free Maxwell-Tellegen’s equations for a time-harmonic plane wave in a bian-isotropic medium can be expressed as

Considering a map ϕ : x → x′ (x, x′ ∈ ℝ³) described by x′(x) (i.e. x′ is given as a function of x), the electromagnetic fields in the two coordinate systems satisfy E′(x′) = J^TE(x), H′(x′) = J^TH(x) [19, 16, 11], and the parameter tensors satisfy

Let us consider a cylindrical lens consisting of three regions: A core (r ≤ r_c), a shell (r_c < r ≤ r_s) and a matrix (r > r_s), which are filled with bianisotropic media as shown in Fig. 1(a) in Cartesian coordinates. The parameter in region p (p = 1,2,3) is denoted by ${\underset{̳}{v}}^{(p)}=v_0\mathbf{diag}\left(v_x^{(p)},v_y^{(p)},v_z^{(p)}\right)$ (v = ε, μ, ξ), region 3 is isotropic: v̳⁽³⁾ = v₀I with I the identity matrix.

 

Fig. 1 (a) Schematic diagram of cylindrical lens consisting of a negatively refracting bianisotropic ring (grey color) r_c < r ≤ r_s which optically cancels out the positively refracting ring r_s < r ≤ r_* (white color). (b) Mapping of the three axisymmetric regions of panel (a) into layers in the new coordinate system (r, θ, z). (c) The enlarged region 1 in final Cartesian coordinates via transformation optics (TO).

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To design a superscatterer with an enhanced optical scattering cross section, i.e. the region 1 appears to be optically enlarged up to the boundary of region 3, the sum of optical paths 2+3 should be zero, which can be achieved by a pair of complementary bianisotropic media [17] according to the generalized lens theorem [9].

Firstly, we introduce a map from Cartesian to cylindrical coordinates (r, θ, z) defined by [9, 20]

Considering a transparent superscatterer with v̳_eff/v₀ = I, then the relative permittivity, permeability and magnetoelectric tensors of those three regions are

It should be pointed out that the parameters of the background could be anisotropic 3 × 3 matrices, which allows us to have a desired bianisotropic background. Here we have assumed a bianisotropic background with isotropic parameters ε = ε₀I, μ = μ₀I, ξ = ξ₀I to simplify the design of superscatterer. Numerical illustration is carried out with COMSOL Multiphysics, the finite element method (FEM) result of a PEC core surrounded by a cylindrical shell consisting of bianisotropic media is shown in Fig. 2(a), while the equivalent PEC cylinder with r_* is shown for comparison in Fig. 2(b); in these computations, an s-polarized (the electric field is perpendicular to the x–y plane) plane wave with frequency 8.7GHz is incident from above. The white regions at r_s in Fig. 2(a) are for values outside the color scale. In these three regions, v₀ = ε₀, μ₀ are the permittivity, permeability of the vacuum, while |ξ₀| ≠ 1/c₀ with c₀ the velocity of light in vacuum should be satisfied to ensure convergence of the numerical algorithm. Here we take ξ₀ = 0.99/c₀, and the radii are r_c = 0.02m, r_s = 0.04m. It can be seen that the scattered fields in Figs. 2(a) and 2(b) are quite similar outside the disc of radius $r_{*}=r_s^2/r_c$ (equivalent for an external observer to a disc of radius r_* shown in Fig. 2). Moreover, the scattering by a PEC core is shown for comparison in Fig. 2(c). The profiles of Re(E_z) of the scattered fields along the dashed black line depicted in Figs. 2(a)–2(c) are drawn in Fig. 2(d) with solid black, blue crosses and dotted black curves, respectively. The solid black curve and blue crosses are nearly superimposed, unlike for the dotted black curve, which proves the super scattering effect for a cylindrical lens with complementary bianisotropic media, similarly to the achiral case [10].

 

Fig. 2 Plots of Re(E_z) under an s-polarized incidence with frequency 8.7GHz: (a) Cylindrical lens made of bianisotropic media with parameters as in (7), r_s = 0.04m (shell radius) and a PEC boundary at r = r_c = 0.02m; (b) Enlarged PEC boundary at $r_{*}=r_s^2/r_c=0.08\text{m}$ in bianisotropic background; (c) Core with PEC boundary at r = r_c in bianisotropic background; (d) Comparison of Re(E_z) on the intercepting line for (a) (solid black curve), (b) (blue crosses) and (c) (dotted black curve).

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3. Bianisotropic superscatterer through the space folding technique

To understand how the superscatterer works, we introduce a space folding technique [3]. We consider the geometric transform which includes first a map from the Cartesian system to cylindrical coordinates through x = r cos θ, y = r sin θ, z = z. Then a stretched cylindrical coordinates (r′, θ′, z′) is introduced through a radial transform r′ = f (r) while θ′ = θ and z′ = z. Finally we go back to Cartesian coordinates (x′, y′, z′). This compound transform leads to a Jacobian matrix

For any such bianisotropic media with a translational invariance along the z′ = z axis, we can write the electromagnetic field in stretched cylindrical coordinates, wherein E = (E_r′, E_θ′, E_z′)^T, H = (H_r′, H_θ′, H_z′)^T. Equation (1) can be rewritten as:

According to (10), the permittivity, permeability and magnetoelectric coupling tensors can be expressed in the polar eigenbasis of the metric tensor as

Figure 3(a) shows the mapping from unfolded system (r, θ, z) to folded system (r′, θ′, z′), wherein the space between r′_s and r′_c overlaps itself but without intersection [8]: Starting from the origin, one first moves in the core with increasing radius until one encounters the core radius r′_c, then one moves into the shell with decreasing radius until one reaches the shell radius r′_s < r′_c, at which point one moves into the matrix with increasing radius again. By choosing a proper function g, folded space can be achieved through a negative slope in region r′_s < r′ < r′_c, hence a negatively refracting index medium in the shell: The electromagnetic field inside the shell (folded region) is then equal to that in the matrix in this region. Continuity of the radial mapping function is required in order to achieve impedance-matched material interfaces.

 

Fig. 3 (a) Graph of g(r′) versus radius r′, where parameters in vertical are r_c = 0.02m (core radius), r_s = 0.04m (shell radius), and $r_{*}=r_s^2/r_c=0.08\text{m}$, r′_c = r_*. (b) Curves of parameters v_r′, v_θ′, v_z′ defined in (12) versus r in the three regions, where the function g(r′) is defined in (15). (c) Curves of parameters v_x, v_y, v_z defined in (7) versus r. In the matrix, we have v_r′ = v_θ′ = v_z′ in panel (b), and $v_x = v_y = v_z in panel (c): the solid and dashed lines are overlapped..

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To design a superscatterer as in (7), we take

4. External cloaking at all frequency

Milton and Nicorovici pointed out in 2006 [7] that anomalous resonance occurring near a lens consisting of complementary media will lead to cloaking effect: When there is a finite collection of polarizable line dipoles near the lens, the resonant field generated by the dipoles acts back onto the dipoles and cancels the field acting on them from outside sources. In actuality, the first cloaking device of this type (now known as external cloaking) was introduced by the former authors with McPhedran back in 1994 [14], but for a different purpose. The first numerical study of this quasi-static cloaking appeared in [8], and it was subsequently realized that this is associated with some optical space folding [6].

More precisely, for a Veselago lens [22] consisting of a slab of thickness d with ε = −1 and μ = −1, surrounded by a medium with ε = 1 and μ = 1, Milton et al [6] showed that: For a polarizable dipolar line source or single constant energy line source located at a distance d₀ in front of the slab, when d₀ < d/2, then the source will be cloaked. However, for a cylindrical lens, the proof of the cloaking phenomenon in the quasi-static limit relies upon the fact that when the permittivities in the core (ε_c), shell (ε_s) and surrounding matrix (ε_m) satisfy ε_s ≈ −ε_m ≈ −ε_c, the collection of polarizable line dipoles lying within a specific distance from the cylindrical lens will be cloaked, with various examples being analyzed in [7].

Similar ideas are investigated here for both the slab lens and cylindrical lens, which are all made of bianisotropic media. However, we stress that in our case we would like to achieve external cloaking at finite frequencies. For this, we first consider a slab lens as shown in Fig. 4(a), the thickness of the slab is d = 0.1m. The parameters in the slab are defined as ε = (−ε₀ + iδ)I, μ = −μ₀I and ξ = −0.99/c₀I, while the upper and lower regions are the complementary bianisotropic medium with positive parameters. Assuming an s-polarized plane wave from above, the frequency is 8.7GHz, the scattering pattern of the slab lens is depicted in Fig. 4(b), a small absorption δ = 10⁻¹⁸ has been introduced as the imaginary part of the permittivity of the slab to improve the convergence of the simulation. Figure 4(c) shows the scattering pattern when a small electric antenna (the radius is 0.002m) locates in the bianisotropic background, a significant perturbation of the field can be observed. However, if we place a slab as designed in Fig. 4(a) below such kind of antenna at a distance d₀ = 0.02m, we can see that the antenna is indeed cloaked as shown in Fig. 4(d), where the white regions on the interfaces are for values outside the color scale. As discussed in [17], the extension of Pendry-Ramakrishna’s generalized lens theorem [9] to bianisotropic media shows that: A pair of complementary media makes a vanishing optical path, i.e. the region between the two dashed lines in Fig. 4(a) behaves as though it had zero thickness. This is checked again by comparing the distribution of the fields along the dashed lines in the regions above and below of Figs. 4(b) and 4(d).

 

Fig. 4 (a) Schematic diagram of a slab lens (d = 0.1m), and a small electric antenna (radius of 0.002m), which is located at a distance d₀ = 0.02m above the slab lens in the cloaking region highlighted by dashed line. The parameters in the upper and lower regions are ε = ε₀I, μ = μ₀I and ξ = 0.99/c₀I, while for the slab they are ε = (−ε₀ + iδ)I, μ = −μ₀I and ξ = −0.99/c₀I, I is the 3 × 3 identity matrix. Plots of Re(E_z) for: (b) A transparent slab lens; (c) An electric antenna lies in a background with ε = ε₀I, μ = μ₀I and ξ = 0.99/c₀I; (d) A small electric antenna located at a distance d₀ (in the cloaking region) from the slab. An s-polarized plane wave is assumed to be incident from above, with a frequency of 8.7GHz, and δ = 10⁻¹⁸.

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Furthermore, we consider a system consisting of a bianisotropic cylindrical lens and a small electric antenna (radius 0.002m) lying at a distance r from the center point. According to (12) along with (15), the parameters for a transparent superscatterer are defined by $\epsilon ={\epsilon }_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r_c^4\right)$, $\mu ={\mu }_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r_c^4\right)$, $\xi =0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r_c^4\right)$ in the core; $\epsilon =(-{\epsilon }_0+i\delta )\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\mu =-{\mu }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\xi =-0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r^4\right)$ in the shell; and ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I in the background. A small absorption δ = 10⁻¹⁴ is introduced as the imaginary part of permittivity of the shell to ensure numerical convergence of the finite element algorithm. The radii of the cylindrical lens are r_c = 0.02m, r_s = 0.04m, respectively. Assuming an s-polarized plane wave with frequency 8.7GHz from above, the scattering pattern of the cylindrical lens is shown in Fig. 5(b), while Figs. 5(c)–5(d) describe the phenomena when the antenna moves towards the cylindrical lens from r > r_# to r < r_# with the cloaking radius $r_{\#}=\sqrt{r_s^3/r_c}=0.0566\text{m}$: Cloaking can be observed.

 

Fig. 5 (a) Diagram of a bianisotropic cylindrical lens (r_c = 0.02m, r_s = 0.04m) and a small electric antenna (radius of 0.002m). Plots of Re(E_z) for: (b) A transparent cylindrical lens without the electric antenna, the parameters are $\epsilon ={\epsilon }_0\mathbf{diag}\left(1,1,r_s^4/r_c^4\right)$, $\mu ={\mu }_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r_c^4\right)$, $\xi =0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r_c^4\right)$ in the core, $\epsilon =(-{\epsilon }_0+i\delta )\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\mu =-{\mu }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\xi =-0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r^4\right)$ in the shell, and ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I in the background; (c) An antenna located at r = 0.07m which is outside the cloaking region, a significant perturbation of the field can be observed; (d) An antenna located at r = 0.045m which is inside the cloaking region becomes virtually invisible to the incident field. An s-polarized plane wave incidence with frequency 8.7GHz is assumed from above, and δ = 10⁻¹⁴.

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Moreover, an interplay between a triangular set of small electric antennas with the cylindrical lens is shown in Fig. 6, the radius of each antenna is 0.001m, the center-to-center spacing between the upper two antennas is 0.006m and the curves connecting their centers form an isosceles right triangle. Figure 6(b) shows what happens when all three antennas are outside the cloaking region; when the lower antenna in the triangle is moving into the cloaking region, while the upper two are outside, the distribution of the electric field is depicted in Fig. 6(c); external cloaking is more pronounced when all antennas enter the cloaking region, see Fig. 6(d). However, cloaking deteriorates with an increasing number of antennas, which suggests it would not hold for finite bodies [23], which is reminiscent of the ostrich effect [15]. We will come back to the ostrich effect in the next section.

 

Fig. 6 (a) Diagram of a bianisotropic cylindrical lens as in Fig. 5 and a triangular set of small electric antennas, the radius of each antenna 0.001m, the center-to-center spacing between the upper two antennas is 0.006m and the curves connecting their centers form an isosceles right triangle. Plots of Re(E_z) for: (b) Triangular set of antennas are totally outside the cloaking region of radius $r_{\#}=\sqrt{r_s^3/r_c}$; (c) The upper two antennas of triangular set are outside while the lower one is inside the cloaking region; (d) All antennas sit inside the disc of radius r_#. An s-polarized plane wave incidence with frequency 8.7GHz is assumed from above, all the other parameters for the cylindrical lens are the same as in Fig. 5.

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As a quantitative illustration, Fig. 7(a) shows the distribution of electric field along the intercepting lines in Figs. 5(b)–5(d) by solid black, dotted-dashed blue and dashed red curves, respectively. Comparing with the black crosses, which is the distribution of electric field in a fully bianisotropic background, the solid black one totally matches it, i.e. the cylindrical lens is transparent with respect to the incident wave. Although the dashed red curve does not coincide with the solid black one, it somehow achieves the cloaking effect when the electric antenna lies inside the cloaking region, by comparison with the dotted-dashed blue curve when the antenna is outside the cloaking region. Similarly, Fig. 7(b) shows the distribution of electric field along the intercepting line in Figs. 6(b)–6(d) by dotted black curve, dotted-dashed blue curve and dashed red curve, respectively; while the solid black curve for a transparent cylindrical lens without electric antennas is the benchmark. Again, an improved scattering electric field can be achieved by moving the antennas into the cloaking region.

 

Fig. 7 (a) Comparison of Re(E_z) on the intercepting line for Figs. 5(b)–5(d) in solid black curve, dotted-dashed blue curve and dashed red curve, respectively. The black crosses is the distribution of electric field in a fully bianisotropic background. (b) Comparison of Re(E_z) on the intercepting line for Figs. 6(b)–6(d) in dotted black curve, dotted-dashed blue curve and dashed red curve, respectively. The solid black one corresponds to the distribution of electric field along the intercepting line for a transparent cylindrical lens without electric antennas.

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However, this type of external cloaking only works for small polarizable objects (compared to the wavelength) [23]. If one wishes to cloak a large obstacle, it is possible to resort to an optical paradox put forward by Pendry and Smith [24] in conjunction with the theory of complementary media developed by Pendry and Ramakrishna [9]. In the context of complementary bianisotropic media [17], a circular inclusion consisting of a material with optical parameters ε = (−ε ₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I and of radius r₀, which is located in a medium with parameters ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I is optically canceled out by an inclusion with ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I of same diameter r₀ in a slab lens of medium ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I, see Fig. 8 for r₀ = 0.025m and δ = 10⁻¹⁷. The physical interpretation of this striking phenomenon is that some resonances building up in this optical system make possible some tunneling of the electric field through the slab lens and inclusions. One can see in Fig. 8 that the transmission is nevertheless not perfect, but the forward scattering is much reduced in panel (c), compared to panel (b), and we numerically checked that this kind of optical cancellation breaks down at higher frequencies.

 

Fig. 8 (a) A mirror system made of complementary bianisotropic media, an inclusion with ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I is placed inside the slab lens with ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I, while a mirror inclusion with ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I locates in the media with ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I; (b) Plot of Re(E_z) for the system wherein only an inclusion with ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I in the background, an s-polarized plane wave comes from above, the negative inclusion interrupts the propagation of the incident wave; (c) Plot of Re(E_z) for the system (a): waves propagate from top to bottom undisturbed. Absorption is δ = 10⁻¹⁷ and the frequency is 1GHz, the thickness of slab lens is d = 0.1m, and the radius of the inclusion is r₀ = 0.025m.

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Similarly for the cylindrical lens in ^{Figs. 5–6}, an annulus with parameters ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I and radii 0.05m–0.064m can be cloaked by an annulus with parameters $\epsilon ={\epsilon }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\mu ={\mu }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\xi =-0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r^4\right)$ and of radii 0.025m–0.032m, which is located in the shell with $\epsilon =(-{\epsilon }_0+i\delta )\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\mu =-{\mu }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\xi =-0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r^4\right)$, as shown in Fig. 9(a). Note that the ratios for each component of the parameters of these two complementary annuli are equal to diag(g/g′r′, g′r′/g, g′g/r′) with coordinate transformation function defined in (15); while their radii are also satisfying the relation (15). For an s-polarized plane wave with frequency 8.7GHz coming from above, the distribution of electric field of the optical system with a negative annulus in the background and that of the system in Fig. 9(a) are depicted in Figs. 9(b) and 9(c), respectively. The negative inclusion in the background is cloaked by introducing the negative cylindrical lens with a complementary annulus.

 

Fig. 9 (a) An annulus with $\epsilon ={\epsilon }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\mu ={\mu }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\xi =-0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r^4\right)$ and of radii 0.025m–0.032m is placed inside the shell of a cylindrical lens as shown in Figs. 5–6, while a mirror inclusion with ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I and of radii 0.05m–0.064m is placed in the background; (b) Plot of Re(E_z) for the optical system with only an inclusion with ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I in the background; (c) Plot of Re(E_z) for the optical system (a): waves propagate from top to bottom undisturbed. An s-polarized plane wave comes from above, absorption is δ = 10⁻¹⁴ and the frequency is 8.7GHz.

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We then replace the annuli by curved sheets, which are pieces of the two annuli in Fig. 9(a), as shown in Fig. 10(a). The parameters for the structure are the same as before, the distributions of electric field are depicted in Figs. 10(b)–10(c). When there is a negative curved sheet located in the background, a scattering effect can be observed in Fig. 10(b), if we introduce the negative shell with a positive sheet to cancel the negative sheet, the scattering arising from the negative sheet is improved (the left side), although one can nevertheless observe some side-scattering effects. Note also that we numerically checked cloaking worsens with higher-frequencies, and improves in the quasi-static limit. We also looked at sheets of more complex shapes, with similar results.

 

Fig. 10 (a) A curved sheet with $\epsilon ={\epsilon }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\mu ={\mu }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\xi =0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r^4\right)$ and of radii 0.025m–0.032m is placed inside the shell with $\epsilon =(-{\epsilon }_0+i\delta )\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\mu =-{\mu }_0\mathbf{diag}\left(1,1,r_s^4/r^4\right)$, $\xi =-0.99/c_0\mathbf{diag}\left(1,\hspace{0.17em}1,\hspace{0.17em}{r}_s^4/r^4\right)$, while a mirror sheet with ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I and of radii 0.05m–0.064m locates in the background with ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I; (b) Plot of Re(E_z) for a curved sheet with ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I in the background; (c) Plot of Re(E_z) for the optical system shown in (a): propagation of the incident plane wave from top to bottom is less perturbed than in (b). Absorption is δ = 10⁻¹⁴ and the frequency is 8.7GHz.

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5. Ostrich effect at low frequency

Finally, we numerically checked that if one increases the wavelength of the incident wave to allow the bianisotropic slab lens to become visible, then an ostrich effect can be observed in the system as shown in Fig. 4(a). First, we take the frequency as f = 3GHz, Fig. 11(a) shows the scattering pattern of a slab lens with thickness d = 0.1m: The slab lens becoming more visible than Fig. 4(b) by comparing the forward scattering fields in the lower space of the lens; furthermore, if we put a small electric antenna (radius 0.002m) at a distance d₀ < d/2 (d₀ = 0.02m), the distribution of the fields is shown in Fig. 11(b), where Fig. 11(c) shows the scattering fields for an electric antenna in the bianisotropic background. Comparing Figs. 11(a) and 11(b), we can see that the antenna is cloaked, i.e. external cloaking leads to the ostrich effect [15]. Similar effects can be observed by increasing the wavelength of the incidence as shown in Figs. 11 (d)–11(f) with f = 1.5GHz.

 

Fig. 11 Plots of Re(E_z) for the same system as in Fig. 4(a), an s-polarized plane wave incidence with frequency 3GHz: (a) A bianisotropic slab lens with the thickness d = 0.1m; (b) A small electric antenna (radius of 0.002m) is inside the cloaking region with d₀ < d/2 (d₀ = 0.02m); (c) The antenna sits inside a bianisotropic background. Similar illustration is indicated in (d)–(f) when the frequency becomes 1.5GHz.

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Meanwhile, we also numerically checked that if one considers a bianisotropic shell with ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I, and a small absorption δ = 10⁻²⁰ has been introduced as the imaginary part of the permittivity of the shell to improve the convergence of the package COMSOL; while the parameters in the bianisotropic background and core are ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I. The frequency of the incidence is assumed to be f = 2.5GHz allowing a wavelength comparable with the size of the cylindrical lens (same radii as Fig. 5). The numerical illustration for such a lens is shown in Fig. 12(a), where the cylindrical lens becomes visible. When we place an electric antenna (radius of 0.002m) inside the cloaking region r < r_# (r = 0.045m), similar distribution of electric field as Fig. 12(a) can be observed, i.e. an external cloaking can still be observed, which is the ostrich effect [15]; Fig. 12(c) shows the case when there is only an electric antenna located at the bianisotropic background as a comparison. If one increases the frequency to f = 3.5GHz even f = 5GHz, the ostrich effect becomes weak, as shown in Figs. 12(d)–12(f), Figs. 12(g)–12(i).

 

Fig. 12 Plots of Re(E_z) under an s-polarized incidence with frequency 2.5GHz: (a) Cylindrical lens with parameters ε = (−ε₀ + iδ)I, μ = −μ₀I, ξ = −0.99/c₀I in the shell, and ε = ε₀I, μ = μ₀I, ξ = 0.99/c₀I in both the background and core; a small absorption δ = 10⁻²⁰ is introduced as the imaginary part of permittivity in the shell to improve the convergence of the package COMSOL; (b) An electric antenna (radius of 0.002m) locates inside the cloaking region r < r_# (r = 0.045m); (c) An electric antenna in the bianisotropic background. The ostrich effect becomes weaker along with an increasing frequency, e.g. panels (d)–(f) for f = 3.5GHz, and (g)–(i) for f = 5GHz.

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Again, we compare the distribution of the electric field along the upper intercepting line in the three systems of each line in Fig. 12, they are denoted respectively in solid black, dashed red and dotted-dashed blue curves as shown in Fig. 13. At low frequency f = 2.5GHz, the electric field along the intercepting line in bianisotropic background without cylindrical lens and antenna is denoted in dotted black curve as a comparison, the mismatch between it and the solid black curve indicates that the cylindrical lens becomes visible; if we place an electric antenna in the cloaking region as shown in Fig. 12(b), the distribution of electric field of which is marked by dashed red curve, a relatively small difference can be observed from the solid black line; however, for an electric antenna located in the background, we have a quite different dotted-dashed blue line for the distribution of electric field; in other words, an ostrich effect can be achieved. Note that, if we increase the frequency, the phenomenon collapses, see Figs. 13(b) and 13(c) for the frequencies 3.5GHz and 5GHz.

 

Fig. 13 Comparison of Re(E_z) on the intercepting line for Fig. 12; solid black, dashed red and dotted-dashed blue curves are respectively representing the distribution of electric field along the upper intercepting line of the cylindrical lens, a lens with an electric antenna located in the cloaking region and a single electric antenna located at the bianisotropic background: (a) f = 2.5GHz, as a benchmark, the electric field along the intercepting line in a bianisotropic background without cylindrical lens and antenna is denoted in dotted black curve; (b) f = 3.5GHz; (c) f = 5GHz.

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6. External cloaking in air with chiral slab lens

The previous designs involve fairly complex fully bianisotropic media. It is interesting to simplify these parameters in order to foster experimental efforts in this emergent area. We explore the external cloaking effect in air with a chiral slab lens, wherein the slab lens is exactly the same as pointed out by Jin and He [25], since the chiral slab lens possess a negative refractive index, where the anomalous resonance occurs. Figure 14(a) shows the chiral slab lens with the upper and lower regions being air, the parameters in the slab are ε = (ε₀ + iδ)I with δ = 10⁻¹⁶, μ = μ₀I, ξ = 1.975/c₀I, and the thickness of the slab is d = 0.1m. We consider an s-polarized plane wave from above, the wavelength of which is equal to the thickness of the slab as defined in [25]. Figure 14(b) is the scattering pattern of this chiral slab lens; while Fig. 14(c) shows the distribution of electric field in a system, wherein an electric antenna with radius 0.002m located in the air. Since the refractive index of the chiral slab satisfies n ≈ −1, anomalous resonance can occur at the interfaces between the air and the slab, if we put the electric antenna quite close to the slab at a distance d₀ = 0.01m, then a quasi-cloaking effect can be observed as shown in Fig. 14(d).

 

Fig. 14 (a) Diagram of a chiral slab lens with ε = (ε₀ + iδ)I, μ = μ₀I, ξ = 1.975/c₀I, and the upper and lower regions are air. An s-polarized plane wave with wavelength λ = d = 0.1m is incident from above, and δ = 10⁻¹⁶. Plots of Re(E_z) for: (b) A chiral slab lens [25] when the wavelength of radiation is comparable with the size of the structure; (c) An electric antenna is placed in air; (d) When the antenna is placed very close to the chiral slab lens, partial external cloaking (i.e. for the forward scattering) can be observed comparing with panel (c).

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To cloak a large obstacle, we implement the similar idea as Fig. 8(a), a circular inclusion with parameters ε = (ε₀ + iδ)I, μ = μ₀I, ξ = 1.975/c₀I and of radius 0.025m is placed in air, is partly canceled out by an inclusion of air in a chiral slab lens with parameters ε = (ε₀ + iδ)I, μ = μ₀I, ξ = 1.975/c₀I, since the chiral medium possess a negative index n ≈ −1 which is opposite to the index of air. For an s-polarized incidence, the distribution of the electric field is shown in Fig. 15, panel (b) is the case when there is only a chiral inclusion located in air, while panel (c) shows the reduced scattering (in forward scattering) for the chiral inclusion in air, when a chiral slab with a mirror inclusion of air is added nearby. The frequency is 1.5GHz, the thickness of slab lens is d = 0.1m, and the radius of the inclusion is 0.025m. A small absorption δ = 10⁻¹⁶ is introduced as the imaginary part of the permittivity in chiral medium. Although the transmission of the incident wave is not perfect, it opens us a possible route to the application of the bianisotropic media. Importantly, bianisotropic media can be achieved from dielectric periodic structures as proved using the mathematical tool of high-order homogenization in [26].

 

Fig. 15 (a) A mirror system made by air and bianisotropic media, a circular inclusion with ε = (ε₀ + iδ)I, μ = μ₀I, ξ = 1.975/c₀I is placed in the air, while a mirror inclusion of air locates in the media with ε = (ε₀ + iδ)I, μ = μ₀I, ξ = 1.975/c₀I; (b) Plot of Re(E_z) for the system wherein only an inclusion with ε = (ε₀ + iδ)I, μ = μ₀I, ξ = 1.975/c₀I in air, an s-polarized plane wave comes from above, the inclusion interrupt the propagation of the incidence; (c) Plot of Re(E_z) for the system (a): waves are transmitted. A small absorption δ = 10⁻¹⁶ is introduced and the frequency is 1.5GHz, the thickness of slab lens is d = 0.1m, and the radius of the inclusion is 0.025m.

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7. Concluding remarks

In conclusion, we have studied numerically the EM scattering properties of a cylindrical lens. Coordinates transformation can be used to realize a superscatterer with negatively refracting heterogeneous bianisotropic media, wherein a core with PEC boundary acts like a magnified PEC with radius r_*. Moreover, if the core is filled with certain bianisotropic media, a cloaking effect can be observed for a set of small electric antennas lying at a specific distance from the shell, which can be attributed to the anomalous resonances of such kind of complementary media. Similarly, we explore the external cloaking effect in bianisotropic lenses; at low frequencies, an ostrich effect induced by the external cloaking can be observed with both the cylindrical lens and slab lens. Finally, it is possible to cloak finite size objects made of negatively refracting bianisotropic material at finite frequencies with slab and cylindrical lenses having a hole of same shape as the object, and opposite refractive index. Finally, we note that we have numerically checked that all our computations remain valid for smaller values of magnetoelectric coupling. We point out that finite frequency external cloaking, perfect lensing and superscattering for usual media without magnetoelectric coupling is a limiting case of this paper.