Abstract

A three-dimensional cone-shaped concentrator was designed and analyzed through an approach of coordinate transformation theory. The device can provide varying performances for concentrating along the symmetric axis. The physical picture regarding concentrating ability of this structure was revealed and quantitative analyses were performed for the purpose of investigating the dependence of the concentrating properties on the structural parameters. Moreover, reduced material parameters were theoretically derived and the corresponding mismatched impedance at boundaries was analyzed. Finite element method-based numerical simulations results of the device were further presented to verify our theoretical design.

© 2008 Optical Society of America

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  1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equation,” J. Mod. Opt. 43, 773–793 (1996).
    [Crossref]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
    [Crossref] [PubMed]
  4. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” http://www.arxiv.org:physics/0708.0262v1, (2007).
  5. 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]
  6. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett,  90, 241105 (2007).
    [Crossref]
  7. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).
  8. W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
    [Crossref]
  9. S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phy. Rev. E 74, 036621 (2006).
    [Crossref]

2007 (2)

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

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

2006 (4)

S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phy. 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]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and 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]

1996 (1)

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

Cai, W.

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

Chan, C. T.

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

Chen, H.

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

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

Cummer, S. A.

S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phy. Rev. E 74, 036621 (2006).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Kildishev, A.

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and 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]

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

Popa, B. I.

S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phy. Rev. E 74, 036621 (2006).
[Crossref]

Psaltis, D.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” http://www.arxiv.org:physics/0708.0262v1, (2007).

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

Schurig, D.

S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phy. Rev. E 74, 036621 (2006).
[Crossref]

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]

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

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

Smith, D. R.

S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phy. 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]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and 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]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Tsang, M.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” http://www.arxiv.org:physics/0708.0262v1, (2007).

Ward, A. J.

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

Appl. Phys. Lett (1)

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

J. Mod. Opt. (1)

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

Nat. Photonics (1)

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[Crossref]

Opt. Express (1)

Phy. Rev. E (1)

S. A. Cummer, B. I. Popa, D. Schurig, and D. R. Smith, “Full-wave simulations of electromagnetic cloaking structures,” Phy. Rev. E 74, 036621 (2006).
[Crossref]

Science (2)

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

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Other (2)

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” http://www.arxiv.org:physics/0708.0262v1, (2007).

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s Equations,” http://www.arxiv.org:physics/0706.2452v1, (2007).

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Figures (4)

Fig. 1.
Fig. 1.

(Color online) (a). A schematic diagram of our proposed cone-shaped concentrator. The yellow region is the concentrating area. The incident wave is normal to the symmetric axis. (b). A schematic diagram of the space transformation in a cross-section where region I (the concentrating area in yellow color) and region II (blue) are marked respectively.

Fig. 2.
Fig. 2.

(Color online) (a). Distribution of Ez in the calculated region. The x-y plane is at z=0 and the y-z plane is at x=0. (b). Distribution of Ez at x-y plane at site of z=6 cm.

Fig. 3.
Fig. 3.

(Color online) (a). Energy distribution along the z axis at the site of x=0 and y=0. (b). Energy distribution in free space is shown by the red line. The other three lines are obtained through the concentrator along the y axis at x=0 with different positions of z.

Fig. 4.
Fig. 4.

(Color online) Impedance of four conditions at the site of x=z=0. The blue line represents the ideal case. The other three lines are derived by three sets of the reduced parameters obtained in section 2.

Equations (16)

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r = { R 1 R 2 r ( 0 r R 2 ) R 3 R 1 R 3 R 2 r R 2 R 1 R 3 R 2 R 3 ( R 2 r R 3 )
θ = θ
z = z
Q r = Q θ = R 2 R 1 , Q z = 1
Q r = R 3 R 2 R 3 R 1
Q θ = [ 1 + R 3 ( R 2 R 1 ) r ( R 3 R 2 ) ] R 3 R 2 R 3 R 1
Q z = 1
η ij = η Q 1 Q 2 Q 3 ( Q i Q j ) 1 η = ε , μ
ε ij = μ ij = { ( 1 0 0 0 1 0 0 0 ( R 2 R 1 ) 2 ) ( 0 r R 1 ) ( R 2 R 1 R 3 R 2 R 3 r + 1 0 0 0 ( R 2 R 1 R 3 R 2 R 3 r + 1 ) 1 0 0 0 ( R 3 R 2 R 3 R 1 ) 2 × R 2 R 1 R 3 R 2 R 3 r + 1 ) ( R 1 r R 3 )
k = R 2 R 1 R 3 R 2 R 3 r + 1
R 2 R 1 = R 20 z × t R 10 z × t = 1 + R 20 R 10 R 10 z × t
Set 1 : ε z = 1 , μ φ = ( R 3 R 2 R 3 R 1 ) 2 , μ r = ( R 3 R 2 R 3 R 1 ) 2 k 2
Set 2 : ε z = μ φ = R 3 R 2 R 3 R 1 , μ r = k 2 R 3 R 2 R 3 R 1
Set 3 : ε z = R 3 R 2 R 3 R 1 × R 2 R 1 , μ φ = R 3 R 2 R 3 R 1 × R 1 R 2 , μ r = k 2 × R 3 R 2 R 3 R 1 × R 1 R 2
n φ = μ r ε z = R 3 R 2 R 3 R 1 k
n r = μ φ ε z = R 3 R 2 R 3 R 1

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