We investigate the twofold functionality of a cylindrical shell consisting of a negatively refracting heterogeneous bianisotropic (NRHB) medium deduced from geometric transforms. The numerical simulations indicate that the shell enhances their scattering by a perfect electric conducting (PEC) core, whereas it considerably reduces the scattering of electromagnetic waves by closely located objects when the shell surrounds a bianisotropic core. The former can be attributed to a homeopathic effect, whereby a small PEC object scatters like a large one as confirmed by numerics, while the latter can be attributed to space cancellation of complementary bianisotropic media underpinning anomalous resonances counteracting the field emitted by small objects (external cloaking). Space cancellation is further used to cloak a NRHB finite size object located nearby a slab of NRHB with a hole of same shape and opposite refracting index. Such a finite frequency external cloaking is also achieved with a NRHB cylindrical lens. Finally, we investigate an ostrich effect whereby the scattering of NRHB slabs and cylindrical lenses with simplified parameters hide the presence of small electric antennas in the quasi-static limit.
© 2014 Optical Society of America
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  and Pendry et al. , 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  – or even superscatterers . 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  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 . 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 , 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  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 as16, 17], which can be derived as an extension of Ward and Pendry’s result .
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′) = JTE(x), H′(x′) = JTH(x) [19, 16, 11], and the parameter tensors satisfy
Let us consider a cylindrical lens consisting of three regions: A core (r ≤ rc), a shell (rc < r ≤ rs) and a matrix (r > rs), 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 (v = ε, μ, ξ), region 3 is isotropic: v̳(3) = v0I with I the identity matrix.
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  according to the generalized lens theorem .(2) as v̳′(3) = v0diag(1, 1, exp(2r)). Furthermore, according to the generalized perfect lens theorem, the region 2’ should be designed as the complementary medium of region 3’, i.e. region 2’(respectively 3’) is mirror imaged onto region 3’(respectively 2’) along the axis r = ln rs, see Fig. 1(b). More precisely, we have Fig. 1(c), and suppose the new parameter of the enlarged region is v̳eff, then we can fix v̳(1) from the reverse of (2). If the scaling factor is γ in the x-, y-directions while it is equal to 1 in z-direction for r ≤ rc, i.e. (x′, y′, z′) = F(x, y, z) = (γx, γy, γz), then we have v̳(1) = v̳effdiag(1, 1, γ2). On the other hand, the boundary rc of region 1 is enlarged to r* as discussed above, hence .
Considering a transparent superscatterer with v̳eff/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 rs 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 rc = 0.02m, rs = 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 (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(Ez) 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 .
3. Bianisotropic superscatterer through the space folding technique
To understand how the superscatterer works, we introduce a space folding technique . 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(2) .
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 = (Er′, Eθ′, Ez′)T, H = (Hr′, Hθ′, Hz′)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(11), we obtain: (13) can be expressed in the form of a combination of cylindrical Bessel and Hankel functions Jm and 
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 : 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.
To design a superscatterer as in (7), we take(12) and illustrated in Fig. 3(b). Figure 3(c) shows the curves of parameters vx, vy, vz defined in (7) versus r.
4. External cloaking at all frequency
Milton and Nicorovici pointed out in 2006  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 , but for a different purpose. The first numerical study of this quasi-static cloaking appeared in , and it was subsequently realized that this is associated with some optical space folding .
More precisely, for a Veselago lens  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  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 (εc), shell (εs) and surrounding matrix (εm) satisfy εs ≈ −εm ≈ −εc, the collection of polarizable line dipoles lying within a specific distance from the cylindrical lens will be cloaked, with various examples being analyzed in .
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 , the extension of Pendry-Ramakrishna’s generalized lens theorem  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).
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 , , in the core; , , 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 rc = 0.02m, rs = 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 : Cloaking can be observed.
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 , which is reminiscent of the ostrich effect . We will come back to the ostrich effect in the next section.
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.
However, this type of external cloaking only works for small polarizable objects (compared to the wavelength) . If one wishes to cloak a large obstacle, it is possible to resort to an optical paradox put forward by Pendry and Smith  in conjunction with the theory of complementary media developed by Pendry and Ramakrishna . In the context of complementary bianisotropic media , 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.
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 , , and of radii 0.025m–0.032m, which is located in the shell with , , , 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.
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.
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 . Similar effects can be observed by increasing the wavelength of the incidence as shown in Figs. 11 (d)–11(f) with f = 1.5GHz.
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 ; 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).
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.
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 , 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 . 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).
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 .
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.
Y. Liu acknowledges a PhD funding from China Scholarship Council and groupe Ecole Centrale. S. Guenneau is thankful for a European funding through ERC Starting Grant ANAMORPHISM. R. McPhedran acknowledges support from the Australian Research Council ( CUDOS Centre of Excellence CE110001018 and Discovery Projects).
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