Abstract

Transformational optics allows for unprecedented control of light with cylindrical cloaks, concentrators, rotators and superscatterers. These are made of different heterogeneous anisotropic media. Can one cloak an s-polarized field and concentrate (or rotate) a p-polarized field with the same metamaterial (or vice versa)? We show the answer is positive provided the geometric transforms underpinning these functionalities take the same values on the outer boundary of what we call a bicephalous metamaterial. In this way, one can also make a metallic cylinder appear invisible for one light polarization, and larger for the other.

© 2014 Optical Society of America

1. Introduction

Eight years ago, it was suggested that an object surrounded by a coating consisting of a heterogeneous anisotropic [1] or even isotropic heterogeneous [2] metamaterial might become invisible for electromagnetic waves. The theoretical idea based on geometric transformations in the Maxwell’s equations has been since then confirmed by some experiments. Plasmonic routes to invisibility were also proposed [3, 4], which rely on a specific knowledge of the shape and material properties of the object being concealed, and possibly include negatively refracting metamaterials [5]. These studies sparked an interest in transformational optics, giving rise to miscellenaous optical devices such as concentrators [6], rotators [7] and superscatterers [8].

However until now transformational optics has been used to design a metamaterial responsible for a single optical function (e.g. cloak, concentrator, rotator, superscatterer) expected not to be dependent on the illumination conditions (e.g. wavelength, incidence and polarization). In this paper we adress the opportunity to modify this statement, that is, to design a unique cloak able to deliver a specific and predetermined effect for each polarization of the incident light.

2. Merging two transformation matrices in one metamaterial

Let us map a co-ordinate system {u, v, w} to the co-ordinate system {x, y, z}, where x(u, v, w), y(u, v, w) and z(u, v, w). The Jacobian of the transformation is J=(x,y,z)(u,v,w) and the transformation matrix T=JTJdet(J) is a representation of the metric tensor. This tells us that in the transformed coordinates the materials are inhomogeneous (their characteristics are no longer piecewise constant but merely depend on u, v, w co-ordinates) and anisotropic (tensorial nature) [9]:

ε̳=ε0εrT1andμ̳=μ0μrT1,
where ε0μ0 = 1/c2 with c the speed of light in vacuum and εr and μr are the relative permittivity and permeability. We shall impose no restriction on the coefficients of T, which can notably have different signs. We note also that there is no change in the impedance of the media since (μ̳ε̳1)1/2=μ0μrε0εrI, where I is the 3 × 3 identity matrix.

We would like in this paper to study the electromagnetic scattering of a metamaterial from an incident plane wave. Importantly, we restrict our analysis to media which are invariant along the z-direction. Thanks to this cylindrical geometry, the problem splits into p and s polarizations:

×(μ̳1×El)ω2ε̳El=0,
(ε̳1×Hl)ω2μ̳Hl=0,
where El = Ez(x, y)ez (p-polarization), Hl = Hz(x, y)ez (s-polarization), ε̳ and μ̳ are defined by (1). To model the outgoing waves, we use perfectly matched layers. The finite element method is implemented in the commercial software package COMSOL MULTIPHYSICS.

It is illuminating to recast these two equations with the longitudinal fields as unknowns. For this we recall a result useful in the context of duality correspondences in checkerboards[10]:

Property: Let M be a spatially varying real symmetric matrix defined as follows

M(x,y)=(m11(x,y)m(x,y)0m(x,y)m22(x,y)000m33(x,y))=(MT(x,y)00m33(x,y)).

Then for R(θ) denoting a rotation matrix through an angle θ, one has:

×(M×(u(x,y)ez))=(R(π/2)MTR(π/2)u(x,y))ez.

Let us now assume we perform two different kinds of geometric transforms for the s and p polarizations. They are characterized by transformation matrices T and T in (1). Using the property with M=μ01μr1T in Eq. (2) and M=ε01εr1T in Eq. (3) we obtain

(μr1R(π/2)TTR(π/2))Ez)+μ0ε0εr(T33)1ω2Ez=0,
(εr1R(π/2)TTR(π/2))Hz)+μ0ε0μr(T33)1ω2Hz=0.

By inspection of (6) and (7), we note that a metamaterial defined by tensors

ε̳1=ε01εr1T1=(T11T120T21T22000T33),μ̳1=μ01μr1T2=(T11T120T21T22000T33),
has the remarkable property of performing simultaneously the two geometric transforms described by matrices T and T. This peculiarity only occurs in cylindrical geometries thanks to the p and s discoupling of light. Importantly, the metamaterial so designed will be impedanced matched with the surrounding medium provided that (μ̳ε̳1)=μ0μrε0εrT21T1=μ0μrε0εrI at the metamaterial’s outer boundary. This will be the case if the geometric transformations associated with T and T coincide at this boundary. There is obviously an infinite class of geometric transforms that fullfil this criterion, but we will illustrate this proposal with four transforms underpinning the design of concentrator, rotator, cloak and superscatterer.

3. Metamaterial concentrating p-polarized field and rotating s-polarized field

The above construction is fairly general. To illustrate this fact, we shall design a bicephal meta-material concentrating the field in one polarization and rotating the field in the other polarization, which has inner and outer boundaries described as follows:

{R1(θ)=0.4R(1+0.2sin(3θ));R2(θ)=0.6R(1+0.2sin(3θ))R=0.4;R3(θ)=R(1+0.2sin(3θ)+cos(4θ)))

More precisely, we consider the following geometric transform for the concentrator:

r=αr+β,0r<+,θ=θ,0<θ2π,z=z,<z<
with
{α=R1(θ)R2(θ)β=0(0rR1(θ))α=R3(θ)R1(θ)R3(θ)R2(θ)β=R3(θ)R1(θ)R2(θ)R3(θ)R2(θ)(R1(θ)rR3(θ))
and α = 1, β = 0 for r > R3(θ).

The transformation matrix T of the resulting concentrator is defined through its inverse:

T1=((T1)11(T1)120(T1)21(T1)22000(T1)33)=R(θ)((rβ)2+c222α2(rβ)rc22αrβ0c22αrβrrβ000rβα2r)R(θ)T
where
  • 0 ≤ rR1(θ) : c22=rθ=rR1(θ)2(R2(θ)R1(θ)θR1(θ)R2(θ)θ),
  • R1(θ) ≤ rR3(θ) : c22=1(R2(θ)R1(θ))2{R1(θ)θ(R3(θ)r)(R3(θ)R2(θ))+R2(θ)θ(R1(θ)R3(θ))(R3(θ)r)R3(θ)θ(R2(θ)R1(θ))(R1(θ)r)}.

We now consider the geometric transform for a rotator:

r=r,0r<+,θ=αr+β,0<θ2π,z=z,<z<
with
α=θoR1(θ)R2(θ),β=θ+R2(θ)θoR2(θ)R1(θ),R1(θ)rR2(θ),
and α = 0, β = θ elsewhere.

The transformation matrix T of the resulting rotator is defined through its inverse:

T1=((T1)11(T1)120(T1)21(T1)220001)=R(θ)(c22αr0αr1+α2r2c22000c22)R(θ)T
where c22=θθ=1θor(R2(θ)R1(θ))2(R1(θ)θR1(θ)θ)θo(R2(θ)R1(θ))2(R2(θ)R1(θ)θR1(θ)R2(θ)θ).

Permittivity and permeability tensors as in (8) where the entries are given by (12) and (15) define a unique bicephalous metamaterial which rotates the electromagnetic field in p-polarization and concentrates it in s-polarization. The numerical computation for the bifunctional metamaterial which is a rotator in p-polarization and a concentrator in s-polarization is given for the telecommunication wavelength 1.4μm in Fig. 1.

 

Fig. 1 Bicephalous metamaterial rotating Ez field and concentrating Hz field: A plane wave of wavelength 1.4μm incident from the top rotates the longitudinal electric field Ez in p-polarization and concentrates the longitudinal magnetic field Hz in s-polarization. The difference in color scale is due to the large field inside the rotator in (a).

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4. Metamaterial cloaking in p polarization and super-scattering in s polarization

Let us now consider two radially symmetric geometric transforms f and g from (r′, θ, z) to (r, θ, z) such that r = f (r′) and r = g(r′) which leads us to the transformation matrices T = Diag(f(r′)/(f′(r′)r′),r′f′(r′)/f(r′),f(r′)/(r′f′(r′)) and T = Diag(g(r′)/(g′(r′)r′),r′g′(r′)/g(r′),g(r′)/(r′g′(r′)).

Let us write the tensors of permittivity and permeability of (8) in the stretched polar coordinates as ε̳=εrDiag(εrr,εθθ,εzz)=εrDiag(f(r)/(f(r)r),rf(r)/f(r),g(r)/(rg(r)) and μ̳=μrDiag(μrr,μθθ,μzz)=μrDiag(g(r)/(g(r)r),rg(r)/g(r),f(r)/(rf(r)),. The p and s polarized electromagnetic fields satisfy the two equations

1rεzzr(rμθθEzr)+1r2εzzθ(rμrrEzθ)+μ0ε0ω2Ez=0,
1rμzzr(rεθθHzr)+1r2μzzθ(rεrrHzθ)+μ0ε0ω2Hz=0,
in the stretched polar coordinate system, where we assumed the sources are outside the transformed medium. To design the cloak, we consider the geometric transformation which maps the field within the annulus R1r′R2 onto the disc 0 < rR2:
r=f(r)=(rR1)/α,0<r<+,θ=θ,0<θ2π,z=z,<z<
where α = (R2R1)/R2 for 0 < rR2 and α = 1 for r > R2.

To design the superscatterer, we take the following transform:

r=g(r)=rR12/R22,0<r<R22/R1,θ=θ,0<θ2π,z=z,<z<r=g(r)=R22/r,R1rR2,θ=θ,0<θ2π,z=z,<z<
which maps the disc 0<rR22/R1 onto r < R1 and the annulus R22/R1rR1 onto the annulus R1r′R2 and we take the identity elsewhere.

The bicephalous cloak/superscatterer is thus described by:

ε̳=εrDiag(f/(fr),rf/f,g/(rg))=εrDiag(R1+αrαr,αrR1+αr,±1),μ̳=μrDiag(g/(gr),rg/g,f/(rf))=μrDiag(±1,±1,R1+αrαr),
where + is for the medium inside the inner disc and − for the medium inside the shell.

Let us now illustrate numerically such a bicephalous cloak. We show in Fig. 2 some finite element computations for this metamaterial which demonstrate that it works as a superscatterer in Ez polarization and as a cloak in Hz polarization. Moreover, if we now interchange the roles of permittivity and permeability, we design a second bicephalous cloak which works as a superscatterer in Hz polarization and as a cloak in Ez polarization, as shown in Fig. 3.

 

Fig. 2 Superscatterer in Ez and cloak in Hz with inner radius 2.5μm and outer radius 4μm:: A plane wave of wavelength 1.4μm incident from the top enhances the scattering by an infinite conducting obstacle of radius 2.5μm for the longitudinal electric field Ez in p-polarization (b), which scatters like an infinite conducting obstacle of radius 6.4μm (a). This first bicephalous cloak/superscatterer makes an infinite conducting obstacle of radius 2.5μm (c) invisible for the longitudinal magnetic field Hz in s-polarization (d).

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Fig. 3 Superscatterer in Hz and cloak in Ez with inner radius 2.5μm and outer radius 4μm: A plane wave of wavelength 1.4μm incident from the top enhances the scattering by an infinite conducting obstacle of radius 2.5μm for the longitudinal magnetic field Hz in s-polarization (b), which scatters like an infinite conducting obstacle of radius 6.4μm (a). This second bicephalous cloak/superscatterer makes an infinite conducting obstacle of radius 2.5μm (c) invisible for the longitudinal electric field Ez in p-polarization (d).

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5. Conclusion

We have introduced the notion of electromagnetic bicephalous metamaterials, which have one functionality (e.g. rotator or superscatterer) in one polarization and a second one (e.g. rotator or cloak) in the other polarization. We stress that such designs only work for cylindrical meta-materials as they heavily rely upon the property stated in section 2 (decoupling Ez/Hz). Finally, we note that some birefringent dielectric metamaterials were introduced in [11], which also perform two optical functions at the same time, depending on polarization, as experimentally demonstrated in metal-dielectric waveguides in [12]. Although [11] is an inspirational work, we note that it relies upon manipulations of Maxwell’s equations in a curved virtual space in the geometrical optics limit, which is not the case for our bicephalous transformed media.

Acknowledgments

S.G. is thankful for an ERC funding through ANAMORPHISM project.

References and links

1. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef]   [PubMed]  

2. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef]   [PubMed]  

3. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005). [CrossRef]  

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

5. J. B. Pendry and S. Anantha Ramakrishna, “Focussing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345 (2003).

6. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008). [CrossRef]  

7. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]  

8. T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16, 18545–18550 (2008). [CrossRef]   [PubMed]  

9. A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. Phys. J. Appl. Phys. 28, 153–157 (2004). [CrossRef]  

10. F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003). [CrossRef]  

11. A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photon. 5, 357–359 (2011). [CrossRef]  

12. V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013). [CrossRef]  

References

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  1. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [Crossref] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
    [Crossref] [PubMed]
  3. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
    [Crossref]
  4. G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London A 462, 3027–3059 (2006).
    [Crossref]
  5. J. B. Pendry and S. Anantha Ramakrishna, “Focussing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345 (2003).
  6. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
    [Crossref]
  7. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
    [Crossref]
  8. T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16, 18545–18550 (2008).
    [Crossref] [PubMed]
  9. A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. Phys. J. Appl. Phys. 28, 153–157 (2004).
    [Crossref]
  10. F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003).
    [Crossref]
  11. A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photon. 5, 357–359 (2011).
    [Crossref]
  12. V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013).
    [Crossref]

2013 (1)

V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013).
[Crossref]

2011 (1)

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photon. 5, 357–359 (2011).
[Crossref]

2008 (2)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16, 18545–18550 (2008).
[Crossref] [PubMed]

2007 (1)

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[Crossref]

2006 (3)

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

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

2005 (1)

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
[Crossref]

2004 (1)

A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. Phys. J. Appl. Phys. 28, 153–157 (2004).
[Crossref]

2003 (2)

F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003).
[Crossref]

J. B. Pendry and S. Anantha Ramakrishna, “Focussing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345 (2003).

Alu, A.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
[Crossref]

Anantha Ramakrishna, S.

J. B. Pendry and S. Anantha Ramakrishna, “Focussing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345 (2003).

Chan, C. T.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[Crossref]

Chen, H.

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16, 18545–18550 (2008).
[Crossref] [PubMed]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[Crossref]

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

Danner, A. J.

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photon. 5, 357–359 (2011).
[Crossref]

Engheta, N.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
[Crossref]

Ermer, H. K.

V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013).
[Crossref]

Guenneau, S.

A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. Phys. J. Appl. Phys. 28, 153–157 (2004).
[Crossref]

F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003).
[Crossref]

Leonhardt, U.

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photon. 5, 357–359 (2011).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Luo, X.

Ma, H.

Milton, G.

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

Nicolet, A.

A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. Phys. J. Appl. Phys. 28, 153–157 (2004).
[Crossref]

Nicorovici, N. A. P.

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

Pendry, J. B.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

J. B. Pendry and S. Anantha Ramakrishna, “Focussing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345 (2003).

Piazza, A.

V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013).
[Crossref]

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

Schaefer, D.

V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013).
[Crossref]

Schurig, D.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

Shurig, D.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Smith, D. R.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Smolyaninov, I. I.

V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013).
[Crossref]

Smolyaninova, V. N.

V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013).
[Crossref]

Tyc, T.

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photon. 5, 357–359 (2011).
[Crossref]

Yang, T.

Zolla, F.

A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. Phys. J. Appl. Phys. 28, 153–157 (2004).
[Crossref]

F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003).
[Crossref]

Appl. Phys. Lett. (1)

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105 (2007).
[Crossref]

Eur. Phys. J. Appl. Phys. (1)

A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. Phys. J. Appl. Phys. 28, 153–157 (2004).
[Crossref]

J. Phys.: Condens. Matter (1)

J. B. Pendry and S. Anantha Ramakrishna, “Focussing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345 (2003).

Nat. Photon. (1)

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photon. 5, 357–359 (2011).
[Crossref]

Opt. Express (1)

Photon. Nanostruct. Fundam. Appl. (1)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[Crossref]

Phys. Rev. B (1)

V. N. Smolyaninova, H. K. Ermer, A. Piazza, D. Schaefer, and I. I. Smolyaninov, “Experimental demonstration of birefrigent transformation optics devices,” Phys. Rev. B 87, 075406 (2013).
[Crossref]

Phys. Rev. E (2)

F. Zolla and S. Guenneau, “A duality relation for the Maxwell system,” Phys. Rev. E 67, 026610 (2003).
[Crossref]

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005).
[Crossref]

Proc. R. Soc. London A (1)

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

Science (2)

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

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Figures (3)

Fig. 1
Fig. 1 Bicephalous metamaterial rotating Ez field and concentrating Hz field: A plane wave of wavelength 1.4μm incident from the top rotates the longitudinal electric field Ez in p-polarization and concentrates the longitudinal magnetic field Hz in s-polarization. The difference in color scale is due to the large field inside the rotator in (a).
Fig. 2
Fig. 2 Superscatterer in Ez and cloak in Hz with inner radius 2.5μm and outer radius 4μm:: A plane wave of wavelength 1.4μm incident from the top enhances the scattering by an infinite conducting obstacle of radius 2.5μm for the longitudinal electric field Ez in p-polarization (b), which scatters like an infinite conducting obstacle of radius 6.4μm (a). This first bicephalous cloak/superscatterer makes an infinite conducting obstacle of radius 2.5μm (c) invisible for the longitudinal magnetic field Hz in s-polarization (d).
Fig. 3
Fig. 3 Superscatterer in Hz and cloak in Ez with inner radius 2.5μm and outer radius 4μm: A plane wave of wavelength 1.4μm incident from the top enhances the scattering by an infinite conducting obstacle of radius 2.5μm for the longitudinal magnetic field Hz in s-polarization (b), which scatters like an infinite conducting obstacle of radius 6.4μm (a). This second bicephalous cloak/superscatterer makes an infinite conducting obstacle of radius 2.5μm (c) invisible for the longitudinal electric field Ez in p-polarization (d).

Equations (20)

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ε ̳ = ε 0 ε r T 1 and μ ̳ = μ 0 μ r T 1 ,
× ( μ ̳ 1 × E l ) ω 2 ε ̳ E l = 0 ,
( ε ̳ 1 × H l ) ω 2 μ ̳ H l = 0 ,
M ( x , y ) = ( m 11 ( x , y ) m ( x , y ) 0 m ( x , y ) m 22 ( x , y ) 0 0 0 m 33 ( x , y ) ) = ( M T ( x , y ) 0 0 m 33 ( x , y ) ) .
× ( M × ( u ( x , y ) e z ) ) = ( R ( π / 2 ) M T R ( π / 2 ) u ( x , y ) ) e z .
( μ r 1 R ( π / 2 ) T T R ( π / 2 ) ) E z ) + μ 0 ε 0 ε r ( T 33 ) 1 ω 2 E z = 0 ,
( ε r 1 R ( π / 2 ) T T R ( π / 2 ) ) H z ) + μ 0 ε 0 μ r ( T 33 ) 1 ω 2 H z = 0 .
ε ̳ 1 = ε 0 1 ε r 1 T 1 = ( T 11 T 12 0 T 21 T 22 0 0 0 T 33 ) , μ ̳ 1 = μ 0 1 μ r 1 T 2 = ( T 11 T 12 0 T 21 T 22 0 0 0 T 33 ) ,
{ R 1 ( θ ) = 0.4 R ( 1 + 0.2 sin ( 3 θ ) ) ; R 2 ( θ ) = 0.6 R ( 1 + 0.2 sin ( 3 θ ) ) R = 0.4 ; R 3 ( θ ) = R ( 1 + 0.2 sin ( 3 θ ) + cos ( 4 θ ) ) )
r = α r + β , 0 r < + , θ = θ , 0 < θ 2 π , z = z , < z <
{ α = R 1 ( θ ) R 2 ( θ ) β = 0 ( 0 r R 1 ( θ ) ) α = R 3 ( θ ) R 1 ( θ ) R 3 ( θ ) R 2 ( θ ) β = R 3 ( θ ) R 1 ( θ ) R 2 ( θ ) R 3 ( θ ) R 2 ( θ ) ( R 1 ( θ ) r R 3 ( θ ) )
T 1 = ( ( T 1 ) 11 ( T 1 ) 12 0 ( T 1 ) 21 ( T 1 ) 22 0 0 0 ( T 1 ) 33 ) = R ( θ ) ( ( r β ) 2 + c 22 2 α 2 ( r β ) r c 22 α r β 0 c 22 α r β r r β 0 0 0 r β α 2 r ) R ( θ ) T
r = r , 0 r < + , θ = α r + β , 0 < θ 2 π , z = z , < z <
α = θ o R 1 ( θ ) R 2 ( θ ) , β = θ + R 2 ( θ ) θ o R 2 ( θ ) R 1 ( θ ) , R 1 ( θ ) r R 2 ( θ ) ,
T 1 = ( ( T 1 ) 11 ( T 1 ) 12 0 ( T 1 ) 21 ( T 1 ) 22 0 0 0 1 ) = R ( θ ) ( c 22 α r 0 α r 1 + α 2 r 2 c 22 0 0 0 c 22 ) R ( θ ) T
1 r ε z z r ( r μ θ θ E z r ) + 1 r 2 ε z z θ ( r μ r r E z θ ) + μ 0 ε 0 ω 2 E z = 0 ,
1 r μ z z r ( r ε θ θ H z r ) + 1 r 2 μ z z θ ( r ε r r H z θ ) + μ 0 ε 0 ω 2 H z = 0 ,
r = f ( r ) = ( r R 1 ) / α , 0 < r < + , θ = θ , 0 < θ 2 π , z = z , < z <
r = g ( r ) = r R 1 2 / R 2 2 , 0 < r < R 2 2 / R 1 , θ = θ , 0 < θ 2 π , z = z , < z < r = g ( r ) = R 2 2 / r , R 1 r R 2 , θ = θ , 0 < θ 2 π , z = z , < z <
ε ̳ = ε r Diag ( f / ( f r ) , r f / f , g / ( r g ) ) = ε r Diag ( R 1 + α r α r , α r R 1 + α r , ± 1 ) , μ ̳ = μ r Diag ( g / ( g r ) , r g / g , f / ( r f ) ) = μ r Diag ( ± 1 , ± 1 , R 1 + α r α r ) ,

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