Abstract

The canonical equations of the optical cloaking proposed by Shurig, Pendry and Smith has been proved to be equivalent to the geodesic in a 3-dimensional curved space. Carrying out the argument we extend to the 4-dimensional Riemannian space where the extra time item appears as the potential term in the canonical equations. The physical meaning of the results is interpreted.

© 2009 Optical Society of America

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  2. D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
    [CrossRef] [PubMed]
  3. U. Leonhardt, "Optical Conformal Mapping," Science 312(5781), 1777-1780 (2006).
    [CrossRef]
  4. S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
    [CrossRef]
  5. Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
    [CrossRef]
  6. Z. Ruan, M. Yan, C. Neff, and M. Qiu, "Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007).
    [CrossRef]
  7. H. Chen, B.-I. Wu, B. Zhang, and J. Kong, "ElectromagneticWave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007).
  8. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
    [CrossRef] [PubMed]
  9. L. D.  Landau and E. M. Lifshitz, The Classical Theory of Fields (Oxford: Butterworth-Heinemann, 1995).
  10. J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396 (1960).
    [CrossRef]
  11. Y. Kravtsov and Y. I. Orlov, Geometrical optics of inhomogeneous media (Springer-Verlag, Berlin, 1990).
  12. G. T. del Castillo and C. P. Sánchez, "Uniformly accelerated observers in special relativity," Revista Mexicana De Físic 52, 70-73 (2006).

2008 (1)

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

2007 (2)

Z. Ruan, M. Yan, C. Neff, and M. Qiu, "Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef]

H. Chen, B.-I. Wu, B. Zhang, and J. Kong, "ElectromagneticWave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007).

2006 (6)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
[CrossRef] [PubMed]

U. Leonhardt, "Optical Conformal Mapping," Science 312(5781), 1777-1780 (2006).
[CrossRef]

S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

G. T. del Castillo and C. P. Sánchez, "Uniformly accelerated observers in special relativity," Revista Mexicana De Físic 52, 70-73 (2006).

1960 (1)

J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396 (1960).
[CrossRef]

Chen, H.

H. Chen, B.-I. Wu, B. Zhang, and J. Kong, "ElectromagneticWave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007).

Cummer, S.

S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

del Castillo, G. T.

G. T. del Castillo and C. P. Sánchez, "Uniformly accelerated observers in special relativity," Revista Mexicana De Físic 52, 70-73 (2006).

Jiang, X.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Kong, J.

H. Chen, B.-I. Wu, B. Zhang, and J. Kong, "ElectromagneticWave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007).

Leonhardt, U.

U. Leonhardt, "Optical Conformal Mapping," Science 312(5781), 1777-1780 (2006).
[CrossRef]

Liang, Z.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Neff, C.

Z. Ruan, M. Yan, C. Neff, and M. Qiu, "Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef]

Pendry, J.

S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Plebanski, J.

J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396 (1960).
[CrossRef]

Popa, B.-I.

S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Qiu, M.

Z. Ruan, M. Yan, C. Neff, and M. Qiu, "Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef]

Ruan, Z.

Z. Ruan, M. Yan, C. Neff, and M. Qiu, "Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef]

Sánchez, C. P.

G. T. del Castillo and C. P. Sánchez, "Uniformly accelerated observers in special relativity," Revista Mexicana De Físic 52, 70-73 (2006).

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
[CrossRef] [PubMed]

S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

Smith, D.

S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Sun, X.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Wu, B.-I.

H. Chen, B.-I. Wu, B. Zhang, and J. Kong, "ElectromagneticWave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007).

Yan, M.

Z. Ruan, M. Yan, C. Neff, and M. Qiu, "Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef]

Yao, P.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Zhang, B.

H. Chen, B.-I. Wu, B. Zhang, and J. Kong, "ElectromagneticWave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007).

Appl. Phys. Lett. (1)

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Opt. Express (1)

Phys. Rev. (1)

J. Plebanski, "Electromagnetic Waves in Gravitational Fields," Phys. Rev. 118, 1396 (1960).
[CrossRef]

Phys. Rev. E (1)

S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

Z. Ruan, M. Yan, C. Neff, and M. Qiu, "Ideal Cylindrical Cloak: Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef]

H. Chen, B.-I. Wu, B. Zhang, and J. Kong, "ElectromagneticWave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007).

Revista Mexicana De Físic (1)

G. T. del Castillo and C. P. Sánchez, "Uniformly accelerated observers in special relativity," Revista Mexicana De Físic 52, 70-73 (2006).

Science (3)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, "Optical Conformal Mapping," Science 312(5781), 1777-1780 (2006).
[CrossRef]

Other (2)

L. D.  Landau and E. M. Lifshitz, The Classical Theory of Fields (Oxford: Butterworth-Heinemann, 1995).

Y. Kravtsov and Y. I. Orlov, Geometrical optics of inhomogeneous media (Springer-Verlag, Berlin, 1990).

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Figures (3)

Fig. 1.
Fig. 1.

The direct calculation result of the geodesic by Eq. (12), which gives the same spherical cloaking as in Ref. [2].

Fig. 2.
Fig. 2.

Adding σ(x) cloak: the adding σ(x) term bends the light-rays and can make sure the light does not go into the cloaked region r<a.

Fig. 3.
Fig. 3.

Cloak with a time transformation: The acceleration we choose to draw the picture is very large ��=0.11×(3×108)2m/s2, when �� decreases to one tenth, the light-rays observed in the acceleration reference will be analogous to the light-rays in the original inertia reference.

Equations (63)

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

εij=μij=nij=γγij,
γij=gij +g0ig0jg00,
γij=gij.
nijkikjdet(nij)=0 .
H=f(x)[nijkikjdet(nij)],
dxi=Hki,
dki=Hxi,
kμkμ=0 .
H=f(x)gμvkμkv,
γijkikj(ωc)2=0 ,
εij=μij=nij(x)=ωcγ(x)γij(x).
H=f(x)ωcγ[γijkikj(ωc)2],
dki+Γjlikjkl=0 ,
=fA(x).
ki=dxi=dxi·=2ωcf(x)fA(x)γγliki.
H=fA(x)2[γijkikj(ωc)2].
r=bαbr+α.
ds2=(bbα)2dr2+(bbα)2(rα)22+(bbα)2(rα)2sin2θ2.
εrr=bbα(rα)2sinθ,εθθ=bbαsinθ,εϕϕ=b(bα)·1sinθ.
εrrε0rr=bbα(rαr)2,εθθε0θθ=bbα,εϕϕε0ϕϕ=bbα,
(k1k2)·n=0 ,
H(k2)=0 ,
1γi(γBi)=0 ,
Bit+1γεijkjEk=0 ,
1γxi(γDi)=ρ ,
Dit+1γεijkjHk=Ji;
λFμv+vFλμ+μF=0 ,
1gα(gGαβ)=jβ.
Gαβ=gαμgβμFμv.
Di=ε0γijg00Ej,Bi=μ0γijg00Hj.
Di=εijEj,Bi=μijHj,
εij=ε0γijg00,μij=μ0γijg00.
γijγij=σ (x)γij,
εij=μij=nij=ωcγγij=ωcγσ(x)γij.
H=fA(x)2[σ(x)γijkikj(ωc)2].
dki+Γstikskt=0,Γsti=Γsti+(lnσ(x)xjγijγtsδtilnσ(x)xs).
σ(x)=g(x)=g00·γ.
σ(x)=g00
fA(x)=1(x)(m=h̅ωc2),
H=K(k)+V(x)=γijkikj2m12σ(x)(ch̅)2m.
v˙=12σ(x)2(ch̅)2σ(x)xkikj2m2(γij(x)x),
{xi,kj} =δji,
{xi,kj}=γij
{H,xi}=Hkn=γijkjm,
{H,ki}=Hxi=kskt2m(γstxi)m2σ(x)2(ch̅)2σ(x)xi.
H=f (x)[γijkikjg00(k0)2],··k0=ω0c.
dki+Γjlikjkl+Γ00i(k0)2=dki+(Γjli+γjlγimlng00xm)kjkl=0 ,
Ψ(x)=1g00.
εijkEk(x)xj+μij(x)Hjeiωtt=0 .
εrjk[E'k(x)xj]+[1det(Λii)g00ΛijΛjjμij(x)]Hj(x)eiωg00tt=0 ,
μij(x)=1det(Λii)g00ΛiiΛjjμij(x)
ω=ω g00 .
εij=μij=nij=γg00γij
ds2=(1+𝒜zc2)2dt2dx2dy2dz2.
H=12[γijkikj(1+𝒜zc2)2(ωc)2].
ki=dxi,
ki=γilkl.
fA(x)=2ωcγf(x),
H=fA(x)2(γijkikjg00k0k0).
dkidλ+(Γstikskt+12γijg00,jk0k0)=0
dkidλ+(Γstikskt+12γijg00xj1g00γstkskt)=0
dkidλ+(Γsti+lng00xjγijγst)kskt=0 .
dkjdλ+(Γsti+lng00xjγijγstδtilng00xs)kskt=0 ,

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