Abstract

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

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

×E=ωξ̳E+iωμ̳H×H=iωε̳E+ωξ̳H
with ω the wave angular frequency, ε̳ the permittivity, μ̳ the permeability and ξ̳ the tensor of magnetoelectric coupling. These equations retain their form under geometric changes [16, 17], which can be derived as an extension of Ward and Pendry’s result [18].

Considering a map ϕ : xx′ (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

v̳=J1v̳JTdet(J),v=ε,μ,ξ
where J is the Jacobian matrix of the coordinate transformation: J = ∂x/∂x′ with elements Jij = ∂xi/∂x′j, the inverse J−1 = ∂x′/∂x with {J−1}kl = ∂x′k/∂xl. Moreover, J−T denotes the inverse transposed Jacobian.

Let us consider a cylindrical lens consisting of three regions: A core (rrc), a shell (rc < rrs) 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 v̳(p)=v0diag(vx(p),vy(p),vz(p)) (v = ε, μ, ξ), region 3 is isotropic: (3) = v0I with I the identity matrix.

 figure: Fig. 1

Fig. 1 (a) Schematic diagram of cylindrical lens consisting of a negatively refracting bianisotropic ring (grey color) rc < rrs which optically cancels out the positively refracting ring rs < rr* (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).

Download Full Size | PPT Slide | PDF

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]

x=exp(r)cosθ,y=exp(r)sinθ,z=z
and the Jacobian matrix is
Jxr=(x,y,z)(r,θ,z)=[exp(r)cosθexp(r)sinθ0exp(r)sinθexp(r)cosθ0001].
Region p is mapped to region p′ in the new coordinates. The transformed tensors of region 3’ can be derived from (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
v̳(2)(a)=v̳(3)(b)=v0diag(1,1,exp(2lnrb)),
where rb=rs2/ra is derived from ln rb − ln rs = ln rs − ln ra. Moreover, the boundary r* of region 3 can also be fixed by ln r* − ln rs = ln rs − ln rc. Finally, we go back to the Cartesian coordinates through an inverse transformation with Jxr=Jrx1, and we obtain
v̳(2)(a)=v0diag(1,1,exp(2lnrb)exp(2lnra))=v0diag(1,1,rs4/ra4).
Regarding the tensors (1) in region 1, if we define a function F(x) which enlarges the region 1 to fill up the optically cancelled region (regions 2+3) as shown in Fig. 1(c), and suppose the new parameter of the enlarged region is eff, then we can fix (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 rrc, i.e. (x′, y′, z′) = F(x, y, z) = (γx, γy, γz), then we have (1) = effdiag(1, 1, γ2). On the other hand, the boundary rc of region 1 is enlarged to r* as discussed above, hence γ=r*/rc=rs2/rc2.

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

vx(1)=+1,vy(1)=+1,vz(1)=+rs4/rc4,rrcvx(2)=1,vy(2)=1,vz(2)=rs4/r4,rc<rrsvx(3)=+1,vy(3)=+1,vz(3)=+1,rs<r.

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 xy 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 r*=rs2/rc (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 [10].

 figure: Fig. 2

Fig. 2 Plots of Re(Ez) under an s-polarized incidence with frequency 8.7GHz: (a) Cylindrical lens made of bianisotropic media with parameters as in (7), rs = 0.04m (shell radius) and a PEC boundary at r = rc = 0.02m; (b) Enlarged PEC boundary at r*=rs2/rc=0.08m in bianisotropic background; (c) Core with PEC boundary at r = rc in bianisotropic background; (d) Comparison of Re(Ez) on the intercepting line for (a) (solid black curve), (b) (blue crosses) and (c) (dotted black curve).

Download Full Size | PPT Slide | PDF

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

Jxx=JxrJrrJrx=R(θ)diag(1,r,1)diag(g,1,1)diag(1,1r,1)R1(θ)=R(θ)diag(g,gr,1)R(θ)
where g(r′) = r is the inverse function of f, and
R(θ)=[cosθsinθ0sinθcosθ0001].
The transformation matrix (representation of metric tensor) reads
Txx1=(JxxTJxx/det(Jxx))1=R(θ)diag(ggr,grg,ggr)R(θ).
If the parameter in the coordinates (x, y, z) is isotropic, then the transformed parameter in new coordinates (x′, y′, z′) is v̳=v̳Txx1 according to (2) [19].

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:

1rr(rεθ(1)Ezr)+1r2θ(εr(1)Ezθ)irr(rξθ(1)Hzr)ir2θ(ξr(1)Hzθ)=ω2εzEziω2ξzHz,1rr(rξθ(1)Ezr)+1r2θ(ξr(1)Ezθ)+irr(rμθ(1)Hzr)+ir2θ(μr(1)Hzθ)=iω2ξzEzω2μzHz,
with vr(1)=vr/(εrμrξr2), vθ(1)=vθ/(εθμθξθ2).

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

diag(vr,vθ,vz)=v0diag(ggr,grg,ggr).
Let us now substitute this formula into (11), we obtain:
ε0[r(ggEzr)+gg2Ezθ2+ω2aggEz]iξ0[r(ggHzr)+gg2Hzθ2ω2aggHz]=0,iξ0[r(ggEzr)+gg2Ezθ2ω2aggEz]+μ0[r(ggHzr)+gg2Hzθ2+ω2aggHz]=0,
with a=ε0μ0ξ02. The wave solution for (13) can be expressed in the form of a combination of cylindrical Bessel and Hankel functions Jm and Hm(1) [21]
Ez(p)(r,θ)=m[am±(p)Jm(k±g(r))+bm±(p)Hm(1)(k±g(r))]eimθ.
where the superscript p represents different regions (core, shell and matrix) in a superscatterer, k±=ω(ε0μ0±ξ0) is the wave number where the subscript ± stands for the right and left polarized wave in bianisotropic medium. The coefficients am±(p) and bm±(p) can be fixed according to the boundary conditions: Tangential electric and magnetic fields should be continuous across each interface. Similar results hold true for the magnetic field.

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.

 figure: Fig. 3

Fig. 3 (a) Graph of g(r′) versus radius r′, where parameters in vertical are rc = 0.02m (core radius), rs = 0.04m (shell radius), and r*=rs2/rc=0.08m, r′c = r*. (b) Curves of parameters vr′, vθ′, vz′ defined in (12) versus r in the three regions, where the function g(r′) is defined in (15). (c) Curves of parameters vx, vy, vz defined in (7) versus r. In the matrix, we have vr′ = vθ′ = vz′ in panel (b), and $vx = vy = vz in panel (c): the solid and dashed lines are overlapped..

Download Full Size | PPT Slide | PDF

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

r=g(r)={rrc2/rs2,rrcrs2/r,rc<rrsr,r>rs
which leads to parameters whose expressions are given in (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 [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 (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 [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 + )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).

 figure: Fig. 4

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 + )I, μ = −μ0I and ξ = −0.99/c0I, I is the 3 × 3 identity matrix. Plots of Re(Ez) 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.

Download Full Size | PPT Slide | PDF

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 ε=ε0diag(1,1,rs4/rc4), μ=μ0diag(1,1,rs4/rc4), ξ=0.99/c0diag(1,1,rs4/rc4) in the core; ε=(ε0+iδ)diag(1,1,rs4/r4), μ=μ0diag(1,1,rs4/r4), ξ=0.99/c0diag(1,1,rs4/r4) 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 r#=rs3/rc=0.0566m: Cloaking can be observed.

 figure: Fig. 5

Fig. 5 (a) Diagram of a bianisotropic cylindrical lens (rc = 0.02m, rs = 0.04m) and a small electric antenna (radius of 0.002m). Plots of Re(Ez) for: (b) A transparent cylindrical lens without the electric antenna, the parameters are ε=ε0diag(1,1,rs4/rc4), μ=μ0diag(1,1,rs4/rc4), ξ=0.99/c0diag(1,1,rs4/rc4) in the core, ε=(ε0+iδ)diag(1,1,rs4/r4), μ=μ0diag(1,1,rs4/r4), ξ=0.99/c0diag(1,1,rs4/r4) 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.

Download Full Size | PPT Slide | PDF

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.

 figure: Fig. 6

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(Ez) for: (b) Triangular set of antennas are totally outside the cloaking region of radius r#=rs3/rc; (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.

Download Full Size | PPT Slide | PDF

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.

 figure: Fig. 7

Fig. 7 (a) Comparison of Re(Ez) 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(Ez) 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.

Download Full Size | PPT Slide | PDF

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 + )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 + )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.

 figure: Fig. 8

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 + )I, μ = −μ0I, ξ = −0.99/c0I, while a mirror inclusion with ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I locates in the media with ε = ε0I, μ = μ0I, ξ = 0.99/c0I; (b) Plot of Re(Ez) for the system wherein only an inclusion with ε = (−ε0 + )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(Ez) 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.

Download Full Size | PPT Slide | PDF

Similarly for the cylindrical lens in Figs. 56, an annulus with parameters ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I and radii 0.05m–0.064m can be cloaked by an annulus with parameters ε=ε0diag(1,1,rs4/r4), μ=μ0diag(1,1,rs4/r4), ξ=0.99/c0diag(1,1,rs4/r4) and of radii 0.025m–0.032m, which is located in the shell with ε=(ε0+iδ)diag(1,1,rs4/r4), μ=μ0diag(1,1,rs4/r4), ξ=0.99/c0diag(1,1,rs4/r4), 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.

 figure: Fig. 9

Fig. 9 (a) An annulus with ε=ε0diag(1,1,rs4/r4), μ=μ0diag(1,1,rs4/r4), ξ=0.99/c0diag(1,1,rs4/r4) and of radii 0.025m–0.032m is placed inside the shell of a cylindrical lens as shown in Figs. 56, while a mirror inclusion with ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I and of radii 0.05m–0.064m is placed in the background; (b) Plot of Re(Ez) for the optical system with only an inclusion with ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I in the background; (c) Plot of Re(Ez) 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.

Download Full Size | PPT Slide | PDF

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.

 figure: Fig. 10

Fig. 10 (a) A curved sheet with ε=ε0diag(1,1,rs4/r4), μ=μ0diag(1,1,rs4/r4), ξ=0.99/c0diag(1,1,rs4/r4) and of radii 0.025m–0.032m is placed inside the shell with ε=(ε0+iδ)diag(1,1,rs4/r4), μ=μ0diag(1,1,rs4/r4), ξ=0.99/c0diag(1,1,rs4/r4), while a mirror sheet with ε = (−ε0 + )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(Ez) for a curved sheet with ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I in the background; (c) Plot of Re(Ez) 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.

Download Full Size | PPT Slide | PDF

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.

 figure: Fig. 11

Fig. 11 Plots of Re(Ez) 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.

Download Full Size | PPT Slide | PDF

Meanwhile, we also numerically checked that if one considers a bianisotropic shell with ε = (−ε0 + )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).

 figure: Fig. 12

Fig. 12 Plots of Re(Ez) under an s-polarized incidence with frequency 2.5GHz: (a) Cylindrical lens with parameters ε = (−ε0 + )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.

Download Full Size | PPT Slide | PDF

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.

 figure: Fig. 13

Fig. 13 Comparison of Re(Ez) 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.

Download Full Size | PPT Slide | PDF

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 + )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).

 figure: Fig. 14

Fig. 14 (a) Diagram of a chiral slab lens with ε = (ε0 + )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(Ez) 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).

Download Full Size | PPT Slide | PDF

To cloak a large obstacle, we implement the similar idea as Fig. 8(a), a circular inclusion with parameters ε = (ε0 + )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 + )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].

 figure: Fig. 15

Fig. 15 (a) A mirror system made by air and bianisotropic media, a circular inclusion with ε = (ε0 + )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 + )I, μ = μ0I, ξ = 1.975/c0I; (b) Plot of Re(Ez) for the system wherein only an inclusion with ε = (ε0 + )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(Ez) 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.

Download Full Size | PPT Slide | PDF

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.

Acknowledgments

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).

References and links

1. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef]   [PubMed]  

2. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef]   [PubMed]  

3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006). [CrossRef]  

4. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387–396 (2010). [CrossRef]   [PubMed]  

5. M. Kadic, S. Guenneau, S. Enoch, and S. A. Ramakrishna, “Plasmonic space folding : focussing surface plasmons via negative refraction in complementary media,” ACS Nano 5, 6819–6825 (2011). [CrossRef]   [PubMed]  

6. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. A 461, 3999–4034 (2005). [CrossRef]  

7. G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. A 462, 3027–3059 (2006). [CrossRef]  

8. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008). [CrossRef]  

9. J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).

10. T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: Enhancement of scattering with complementary media,” Opt. Express 16, 18545 (2008). [CrossRef]   [PubMed]  

11. X. X. Cheng, H. S. Shen, B.-I. Wu, and J. A. Kong, “Cloak for bianisotropic and moving media,” Prog. Electromagn. Res. 89, 199–212 (2009). [CrossRef]  

12. Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transofrmation optics,” Opt. Express. 18, 21419–21426 (2010). [CrossRef]   [PubMed]  

13. F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Phys. Rev. Lett. 111, 033901 (2013). [CrossRef]   [PubMed]  

14. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994). [CrossRef]  

15. N. A. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008). [CrossRef]  

16. A. V. Novitsky, S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, “Field approach in the transformation optics concept,” PIER 129, 485–515 (2012). [CrossRef]  

17. Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

18. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996). [CrossRef]  

19. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effect,” Opt. Lett. 32, 1069–1071 (2007). [CrossRef]   [PubMed]  

20. S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005). [CrossRef]   [PubMed]  

21. M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas. Propag. 39, 91–96 (1991). [CrossRef]  

22. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. sp. 10, 509–514 (1968). [CrossRef]  

23. O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. 102, 124502 (2007). [CrossRef]  

24. J. B. Pendry and D. R. Smith, “Reversing light: negative refraction,” Physics Today (2003).

25. H. Jin and S. L. He, “Focusing by a slab of chiral medium,” Opt. Express 13, 4974–4979 (2005). [CrossRef]   [PubMed]  

26. Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: Magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013). [CrossRef]   [PubMed]  

References

  • View by:

  1. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref] [PubMed]
  2. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
    [Crossref]
  4. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387–396 (2010).
    [Crossref] [PubMed]
  5. M. Kadic, S. Guenneau, S. Enoch, and S. A. Ramakrishna, “Plasmonic space folding : focussing surface plasmons via negative refraction in complementary media,” ACS Nano 5, 6819–6825 (2011).
    [Crossref] [PubMed]
  6. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. A 461, 3999–4034 (2005).
    [Crossref]
  7. G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. A 462, 3027–3059 (2006).
    [Crossref]
  8. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008).
    [Crossref]
  9. J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).
  10. T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: Enhancement of scattering with complementary media,” Opt. Express 16, 18545 (2008).
    [Crossref] [PubMed]
  11. X. X. Cheng, H. S. Shen, B.-I. Wu, and J. A. Kong, “Cloak for bianisotropic and moving media,” Prog. Electromagn. Res. 89, 199–212 (2009).
    [Crossref]
  12. Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transofrmation optics,” Opt. Express. 18, 21419–21426 (2010).
    [Crossref] [PubMed]
  13. F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Phys. Rev. Lett. 111, 033901 (2013).
    [Crossref] [PubMed]
  14. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
    [Crossref]
  15. N. A. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008).
    [Crossref]
  16. A. V. Novitsky, S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, “Field approach in the transformation optics concept,” PIER 129, 485–515 (2012).
    [Crossref]
  17. Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).
  18. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
    [Crossref]
  19. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
    [Crossref] [PubMed]
  20. S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
    [Crossref] [PubMed]
  21. M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas. Propag. 39, 91–96 (1991).
    [Crossref]
  22. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. sp. 10, 509–514 (1968).
    [Crossref]
  23. O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. 102, 124502 (2007).
    [Crossref]
  24. J. B. Pendry and D. R. Smith, “Reversing light: negative refraction,” Physics Today (2003).
  25. H. Jin and S. L. He, “Focusing by a slab of chiral medium,” Opt. Express 13, 4974–4979 (2005).
    [Crossref] [PubMed]
  26. Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: Magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
    [Crossref] [PubMed]

2013 (3)

F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Phys. Rev. Lett. 111, 033901 (2013).
[Crossref] [PubMed]

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: Magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

2012 (1)

A. V. Novitsky, S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, “Field approach in the transformation optics concept,” PIER 129, 485–515 (2012).
[Crossref]

2011 (1)

M. Kadic, S. Guenneau, S. Enoch, and S. A. Ramakrishna, “Plasmonic space folding : focussing surface plasmons via negative refraction in complementary media,” ACS Nano 5, 6819–6825 (2011).
[Crossref] [PubMed]

2010 (2)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387–396 (2010).
[Crossref] [PubMed]

Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transofrmation optics,” Opt. Express. 18, 21419–21426 (2010).
[Crossref] [PubMed]

2009 (1)

X. X. Cheng, H. S. Shen, B.-I. Wu, and J. A. Kong, “Cloak for bianisotropic and moving media,” Prog. Electromagn. Res. 89, 199–212 (2009).
[Crossref]

2008 (3)

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: Enhancement of scattering with complementary media,” Opt. Express 16, 18545 (2008).
[Crossref] [PubMed]

N. A. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008).
[Crossref]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008).
[Crossref]

2007 (2)

O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. 102, 124502 (2007).
[Crossref]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[Crossref] [PubMed]

2006 (4)

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. A 462, 3027–3059 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

2005 (3)

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. A 461, 3999–4034 (2005).
[Crossref]

S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
[Crossref] [PubMed]

H. Jin and S. L. He, “Focusing by a slab of chiral medium,” Opt. Express 13, 4974–4979 (2005).
[Crossref] [PubMed]

2003 (1)

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).

1996 (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

1994 (1)

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[Crossref]

1991 (1)

M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas. Propag. 39, 91–96 (1991).
[Crossref]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. sp. 10, 509–514 (1968).
[Crossref]

Barkovsky, L. M.

A. V. Novitsky, S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, “Field approach in the transformation optics concept,” PIER 129, 485–515 (2012).
[Crossref]

Bruno, O. P.

O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. 102, 124502 (2007).
[Crossref]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387–396 (2010).
[Crossref] [PubMed]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387–396 (2010).
[Crossref] [PubMed]

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: Enhancement of scattering with complementary media,” Opt. Express 16, 18545 (2008).
[Crossref] [PubMed]

Cheng, X. X.

X. X. Cheng, H. S. Shen, B.-I. Wu, and J. A. Kong, “Cloak for bianisotropic and moving media,” Prog. Electromagn. Res. 89, 199–212 (2009).
[Crossref]

Cherednichenko, K.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008).
[Crossref]

Ding, K.

Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transofrmation optics,” Opt. Express. 18, 21419–21426 (2010).
[Crossref] [PubMed]

Enoch, S.

M. Kadic, S. Guenneau, S. Enoch, and S. A. Ramakrishna, “Plasmonic space folding : focussing surface plasmons via negative refraction in complementary media,” ACS Nano 5, 6819–6825 (2011).
[Crossref] [PubMed]

N. A. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008).
[Crossref]

Gralak, B.

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: Magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
[Crossref] [PubMed]

Guenneau, S.

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: Magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

M. Kadic, S. Guenneau, S. Enoch, and S. A. Ramakrishna, “Plasmonic space folding : focussing surface plasmons via negative refraction in complementary media,” ACS Nano 5, 6819–6825 (2011).
[Crossref] [PubMed]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[Crossref] [PubMed]

S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
[Crossref] [PubMed]

He, S. L.

Jacob, Z.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008).
[Crossref]

Jin, H.

Kadic, M.

M. Kadic, S. Guenneau, S. Enoch, and S. A. Ramakrishna, “Plasmonic space folding : focussing surface plasmons via negative refraction in complementary media,” ACS Nano 5, 6819–6825 (2011).
[Crossref] [PubMed]

Kluskens, M. S.

M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas. Propag. 39, 91–96 (1991).
[Crossref]

Kong, J. A.

X. X. Cheng, H. S. Shen, B.-I. Wu, and J. A. Kong, “Cloak for bianisotropic and moving media,” Prog. Electromagn. Res. 89, 199–212 (2009).
[Crossref]

Lavrinenko, A. V.

A. V. Novitsky, S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, “Field approach in the transformation optics concept,” PIER 129, 485–515 (2012).
[Crossref]

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

Li, J.

F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Phys. Rev. Lett. 111, 033901 (2013).
[Crossref] [PubMed]

Liang, Z.

F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Phys. Rev. Lett. 111, 033901 (2013).
[Crossref] [PubMed]

Lintner, S.

O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. 102, 124502 (2007).
[Crossref]

Liu, F.

F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Phys. Rev. Lett. 111, 033901 (2013).
[Crossref] [PubMed]

Liu, Y.

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: Magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

Luo, X.

Ma, H.

McPhedran, R. C.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008).
[Crossref]

N. A. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008).
[Crossref]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. A 461, 3999–4034 (2005).
[Crossref]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[Crossref]

Milton, G. W.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008).
[Crossref]

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. A 462, 3027–3059 (2006).
[Crossref]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. A 461, 3999–4034 (2005).
[Crossref]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[Crossref]

Newman, E. H.

M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas. Propag. 39, 91–96 (1991).
[Crossref]

Nicolet, A.

Nicorovici, N. A.

N. A. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008).
[Crossref]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[Crossref]

Nicorovici, N.-A. P.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008).
[Crossref]

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. A 462, 3027–3059 (2006).
[Crossref]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. A 461, 3999–4034 (2005).
[Crossref]

Novitsky, A. V.

A. V. Novitsky, S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, “Field approach in the transformation optics concept,” PIER 129, 485–515 (2012).
[Crossref]

Pendry, J. B.

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[Crossref] [PubMed]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
[Crossref] [PubMed]

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

J. B. Pendry and D. R. Smith, “Reversing light: negative refraction,” Physics Today (2003).

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

Podolskiy, V. A.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. A 461, 3999–4034 (2005).
[Crossref]

Ramakrishna, S. A.

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

M. Kadic, S. Guenneau, S. Enoch, and S. A. Ramakrishna, “Plasmonic space folding : focussing surface plasmons via negative refraction in complementary media,” ACS Nano 5, 6819–6825 (2011).
[Crossref] [PubMed]

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).

Shen, H. S.

X. X. Cheng, H. S. Shen, B.-I. Wu, and J. A. Kong, “Cloak for bianisotropic and moving media,” Prog. Electromagn. Res. 89, 199–212 (2009).
[Crossref]

Shen, Y.

Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transofrmation optics,” Opt. Express. 18, 21419–21426 (2010).
[Crossref] [PubMed]

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387–396 (2010).
[Crossref] [PubMed]

Shurig, D.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Smith, D. R.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

J. B. Pendry and D. R. Smith, “Reversing light: negative refraction,” Physics Today (2003).

Sun, W.

Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transofrmation optics,” Opt. Express. 18, 21419–21426 (2010).
[Crossref] [PubMed]

Tayeb, G.

N. A. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008).
[Crossref]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. sp. 10, 509–514 (1968).
[Crossref]

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

Wu, B.-I.

X. X. Cheng, H. S. Shen, B.-I. Wu, and J. A. Kong, “Cloak for bianisotropic and moving media,” Prog. Electromagn. Res. 89, 199–212 (2009).
[Crossref]

Yang, T.

Zhou, L.

Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transofrmation optics,” Opt. Express. 18, 21419–21426 (2010).
[Crossref] [PubMed]

Zhukovsky, S. V.

A. V. Novitsky, S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, “Field approach in the transformation optics concept,” PIER 129, 485–515 (2012).
[Crossref]

Zolla, F.

ACS Nano (1)

M. Kadic, S. Guenneau, S. Enoch, and S. A. Ramakrishna, “Plasmonic space folding : focussing surface plasmons via negative refraction in complementary media,” ACS Nano 5, 6819–6825 (2011).
[Crossref] [PubMed]

IEEE Trans. Antennas. Propag. (1)

M. S. Kluskens and E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas. Propag. 39, 91–96 (1991).
[Crossref]

J. Appl. Phys. (1)

O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. 102, 124502 (2007).
[Crossref]

J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

J. Phys.: Condens. Matter (2)

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).

Y. Liu, S. Guenneau, B. Gralak, and S. A. Ramakrishna, “Focussing light in a bianisotropic slab with negatively refracting materials,” J. Phys.: Condens. Matter 25, 135901 (2013).

Nature Materials (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Materials 9, 387–396 (2010).
[Crossref] [PubMed]

New J. Phys. (3)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[Crossref]

N. A. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008).
[Crossref]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008).
[Crossref]

Opt. Express (2)

Opt. Express. (1)

Y. Shen, K. Ding, W. Sun, and L. Zhou, “A chirality switching device designed with transofrmation optics,” Opt. Express. 18, 21419–21426 (2010).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Rev. B (1)

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[Crossref]

Phys. Rev. Lett. (1)

F. Liu, Z. Liang, and J. Li, “Manipulating polarization and impedance signature: A reciprocal field transformation approach,” Phys. Rev. Lett. 111, 033901 (2013).
[Crossref] [PubMed]

PIER (1)

A. V. Novitsky, S. V. Zhukovsky, L. M. Barkovsky, and A. V. Lavrinenko, “Field approach in the transformation optics concept,” PIER 129, 485–515 (2012).
[Crossref]

Proc. R. Soc. A (3)

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. R. Soc. A 461, 3999–4034 (2005).
[Crossref]

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. A 462, 3027–3059 (2006).
[Crossref]

Y. Liu, S. Guenneau, and B. Gralak, “Artificial dispersion via high-order homogenization: Magnetoelectric coupling and magnetism from dielectric layers,” Proc. R. Soc. A 469, 20130240 (2013).
[Crossref] [PubMed]

Prog. Electromagn. Res. (1)

X. X. Cheng, H. S. Shen, B.-I. Wu, and J. A. Kong, “Cloak for bianisotropic and moving media,” Prog. Electromagn. Res. 89, 199–212 (2009).
[Crossref]

Science (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Sov. Phys. sp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. sp. 10, 509–514 (1968).
[Crossref]

Other (1)

J. B. Pendry and D. R. Smith, “Reversing light: negative refraction,” Physics Today (2003).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1 (a) Schematic diagram of cylindrical lens consisting of a negatively refracting bianisotropic ring (grey color) rc < rrs which optically cancels out the positively refracting ring rs < rr* (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).
Fig. 2
Fig. 2 Plots of Re(Ez) under an s-polarized incidence with frequency 8.7GHz: (a) Cylindrical lens made of bianisotropic media with parameters as in (7), rs = 0.04m (shell radius) and a PEC boundary at r = rc = 0.02m; (b) Enlarged PEC boundary at r * = r s 2 / r c = 0.08 m in bianisotropic background; (c) Core with PEC boundary at r = rc in bianisotropic background; (d) Comparison of Re(Ez) on the intercepting line for (a) (solid black curve), (b) (blue crosses) and (c) (dotted black curve).
Fig. 3
Fig. 3 (a) Graph of g(r′) versus radius r′, where parameters in vertical are rc = 0.02m (core radius), rs = 0.04m (shell radius), and r * = r s 2 / r c = 0.08 m , r′c = r*. (b) Curves of parameters vr′, vθ′, vz′ defined in (12) versus r in the three regions, where the function g(r′) is defined in (15). (c) Curves of parameters vx, vy, vz defined in (7) versus r. In the matrix, we have vr′ = vθ′ = vz′ in panel (b), and $vx = vy = vz in panel (c): the solid and dashed lines are overlapped..
Fig. 4
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 + )I, μ = −μ0I and ξ = −0.99/c0I, I is the 3 × 3 identity matrix. Plots of Re(Ez) 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.
Fig. 5
Fig. 5 (a) Diagram of a bianisotropic cylindrical lens (rc = 0.02m, rs = 0.04m) and a small electric antenna (radius of 0.002m). Plots of Re(Ez) for: (b) A transparent cylindrical lens without the electric antenna, the parameters are ε = ε 0 diag ( 1 , 1 , r s 4 / r c 4 ), μ = μ 0 diag ( 1 , 1 , r s 4 / r c 4 ), ξ = 0.99 / c 0 diag ( 1 , 1 , r s 4 / r c 4 ) in the core, ε = ( ε 0 + i δ ) diag ( 1 , 1 , r s 4 / r 4 ), μ = μ 0 diag ( 1 , 1 , r s 4 / r 4 ), ξ = 0.99 / c 0 diag ( 1 , 1 , r s 4 / r 4 ) 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.
Fig. 6
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(Ez) for: (b) Triangular set of antennas are totally outside the cloaking region of radius r # = 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.
Fig. 7
Fig. 7 (a) Comparison of Re(Ez) 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(Ez) 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.
Fig. 8
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 + )I, μ = −μ0I, ξ = −0.99/c0I, while a mirror inclusion with ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I locates in the media with ε = ε0I, μ = μ0I, ξ = 0.99/c0I; (b) Plot of Re(Ez) for the system wherein only an inclusion with ε = (−ε0 + )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(Ez) 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.
Fig. 9
Fig. 9 (a) An annulus with ε = ε 0 diag ( 1 , 1 , r s 4 / r 4 ), μ = μ 0 diag ( 1 , 1 , r s 4 / r 4 ), ξ = 0.99 / c 0 diag ( 1 , 1 , r s 4 / r 4 ) and of radii 0.025m–0.032m is placed inside the shell of a cylindrical lens as shown in Figs. 56, while a mirror inclusion with ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I and of radii 0.05m–0.064m is placed in the background; (b) Plot of Re(Ez) for the optical system with only an inclusion with ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I in the background; (c) Plot of Re(Ez) 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.
Fig. 10
Fig. 10 (a) A curved sheet with ε = ε 0 diag ( 1 , 1 , r s 4 / r 4 ), μ = μ 0 diag ( 1 , 1 , r s 4 / r 4 ), ξ = 0.99 / c 0 diag ( 1 , 1 , r s 4 / r 4 ) and of radii 0.025m–0.032m is placed inside the shell with ε = ( ε 0 + i δ ) diag ( 1 , 1 , r s 4 / r 4 ), μ = μ 0 diag ( 1 , 1 , r s 4 / r 4 ), ξ = 0.99 / c 0 diag ( 1 , 1 , r s 4 / r 4 ), while a mirror sheet with ε = (−ε0 + )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(Ez) for a curved sheet with ε = (−ε0 + )I, μ = −μ0I, ξ = −0.99/c0I in the background; (c) Plot of Re(Ez) 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.
Fig. 11
Fig. 11 Plots of Re(Ez) 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.
Fig. 12
Fig. 12 Plots of Re(Ez) under an s-polarized incidence with frequency 2.5GHz: (a) Cylindrical lens with parameters ε = (−ε0 + )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.
Fig. 13
Fig. 13 Comparison of Re(Ez) 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.
Fig. 14
Fig. 14 (a) Diagram of a chiral slab lens with ε = (ε0 + )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(Ez) 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).
Fig. 15
Fig. 15 (a) A mirror system made by air and bianisotropic media, a circular inclusion with ε = (ε0 + )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 + )I, μ = μ0I, ξ = 1.975/c0I; (b) Plot of Re(Ez) for the system wherein only an inclusion with ε = (ε0 + )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(Ez) 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.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

× E = ω ξ ̳ E + i ω μ ̳ H × H = i ω ε ̳ E + ω ξ ̳ H
v ̳ = J 1 v ̳ J T det ( J ) , v = ε , μ , ξ
x = exp ( r ) cos θ , y = exp ( r ) sin θ , z = z
J x r = ( x , y , z ) ( r , θ , z ) = [ exp ( r ) cos θ exp ( r ) sin θ 0 exp ( r ) sin θ exp ( r ) cos θ 0 0 0 1 ] .
v ̳ ( 2 ) ( a ) = v ̳ ( 3 ) ( b ) = v 0 diag ( 1 , 1 , exp ( 2 ln r b ) ) ,
v ̳ ( 2 ) ( a ) = v 0 diag ( 1 , 1 , exp ( 2 ln r b ) exp ( 2 ln r a ) ) = v 0 diag ( 1 , 1 , r s 4 / r a 4 ) .
v x ( 1 ) = + 1 , v y ( 1 ) = + 1 , v z ( 1 ) = + r s 4 / r c 4 , r r c v x ( 2 ) = 1 , v y ( 2 ) = 1 , v z ( 2 ) = r s 4 / r 4 , r c < r r s v x ( 3 ) = + 1 , v y ( 3 ) = + 1 , v z ( 3 ) = + 1 , r s < r .
J x x = J x r J r r J r x = R ( θ ) diag ( 1 , r , 1 ) diag ( g , 1 , 1 ) diag ( 1 , 1 r , 1 ) R 1 ( θ ) = R ( θ ) diag ( g , g r , 1 ) R ( θ )
R ( θ ) = [ cos θ sin θ 0 sin θ cos θ 0 0 0 1 ] .
T x x 1 = ( J x x T J x x / det ( J x x ) ) 1 = R ( θ ) diag ( g g r , g r g , g g r ) R ( θ ) .
1 r r ( r ε θ ( 1 ) E z r ) + 1 r 2 θ ( ε r ( 1 ) E z θ ) i r r ( r ξ θ ( 1 ) H z r ) i r 2 θ ( ξ r ( 1 ) H z θ ) = ω 2 ε z E z i ω 2 ξ z H z , 1 r r ( r ξ θ ( 1 ) E z r ) + 1 r 2 θ ( ξ r ( 1 ) E z θ ) + i r r ( r μ θ ( 1 ) H z r ) + i r 2 θ ( μ r ( 1 ) H z θ ) = i ω 2 ξ z E z ω 2 μ z H z ,
diag ( v r , v θ , v z ) = v 0 diag ( g g r , g r g , g g r ) .
ε 0 [ r ( g g E z r ) + g g 2 E z θ 2 + ω 2 a g g E z ] i ξ 0 [ r ( g g H z r ) + g g 2 H z θ 2 ω 2 a g g H z ] = 0 , i ξ 0 [ r ( g g E z r ) + g g 2 E z θ 2 ω 2 a g g E z ] + μ 0 [ r ( g g H z r ) + g g 2 H z θ 2 + ω 2 a g g H z ] = 0 ,
E z ( p ) ( r , θ ) = m [ a m ± ( p ) J m ( k ± g ( r ) ) + b m ± ( p ) H m ( 1 ) ( k ± g ( r ) ) ] e i m θ .
r = g ( r ) = { r r c 2 / r s 2 , r r c r s 2 / r , r c < r r s r , r > r s

Metrics