Abstract

We designed an electromagnetic refractor and a phase transformer using form-invariant coordinate transformation of Maxwell’s equations. The propagation direction of electromagnetic energy in these devices can be modulated as desired. Unlike the conventional dielectric refractor, electromagnetic fields at our refraction boundary do not conform to the Snell’s law in isotropic materials and the impedance at this boundary is matched which makes the reflection extremely low; and the transformation of the wave front from cylindrical to plane can be realized in the phase transformer with a slab structure. Two dimensional finite-element simulations were performed to confirm the theoretical results.

© 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. 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. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  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]
  5. 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]
  6. F. Zolla, A. Nicolet, and J. B. Pendry, "Electromagnetic fields analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007).
    [CrossRef] [PubMed]
  7. W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photo. 1, 224-227 (2007).
    [CrossRef]
  8. 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).
  9. H. Chen and C. T.  Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007).
    [CrossRef]

2007

F. Zolla, A. Nicolet, and J. B. Pendry, "Electromagnetic fields analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007).
[CrossRef] [PubMed]

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photo. 1, 224-227 (2007).
[CrossRef]

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

2006

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]

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]

1996

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. Photo. 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, Huanyang

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. Photo. 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]

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. Photo. 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]

Nicolet, A.

Pendry, J. B.

F. Zolla, A. Nicolet, and J. B. Pendry, "Electromagnetic fields analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007).
[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]

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]

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]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photo. 1, 224-227 (2007).
[CrossRef]

Smith, D. R.

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]

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]

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]

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]

Zolla, F.

Appl. Phys. Lett.

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

J. Mod. Opt.

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell�??s equation," J. Mod. Opt. 43, 773-793 (1996).
[CrossRef]

Nat. Photo.

W. Cai, U. K. Chettiar, A. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photo. 1, 224-227 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phy. Rev. E

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

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]

Other

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 (6)

Fig. 1.
Fig. 1.

(Color online) Schematic diagram of the space transformation for the refractor. The deflection angle θ is defined as the refraction angle under normal incidence.

Fig. 2.
Fig. 2.

Schematic diagram of the space transformation for the phase transformer. O is the position of the source and supposed to be the origin of the coordinate. Wave front I, II and III are changed to line 1, 2 and 3 respectively.

Fig. 3.
Fig. 3.

The middle part is the refractor; the other parts are set to be free space. (a) Incident angle is -45°. (b) Incident angle is -30°. (c) Incident angle is 0°. (d) Incident angle is 30°.

Fig. 4.
Fig. 4.

(Color online) Refraction angle and incidence angle in both conventional refractors (black line and blue line) and the designed refractor (red line).

Fig. 5.
Fig. 5.

(Color online) (a) Distribution of Ez of the splitting by two metamaterials combination. Region I and IV are air, region II and region III are two different metamaterials. (b) Time averaged energy density of the splitting by two metamaterials combination.

Fig. 6.
Fig. 6.

(Color online) (a) Region I and III are set to be free space, region II is the phase transformer. (b) Distribution of Ez without the phase transformer. (c) Ez at x=4µm with (red curve) and without (the blue curve) the phase transformer.

Equations (12)

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

x = x + tan ( θ ) y ,
y = y ,
z = z
A i i = x i x i
η i j = det ( A i i ) 1 A i i A j j η ij , η = ε , μ
ε i j = ( 1 + tan ( θ ) 2 tan ( θ ) 0 tan ( θ ) 1 0 0 0 1 ) ε
x = x 2 + y 2
y = y
z = z
ε i j = ( x x 2 y 2 y x 2 y 2 0 y x 2 y 2 x x 2 y 2 0 0 0 1 ) ε
θ 2 = arcsin ( n 1 × sin ( θ 1 ) n 2 )
θ 2 = arctan ( tan ( θ 1 ) + tan ( Δ θ ) )

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