Abstract

We introduce a class of conformal versions of the previously introduced quasi-conformal carpet cloak, and show how to construct such conformal cloaks for different cloak shapes. Our method provides exact refractive-index profiles in closed mathematical form for the usual carpet cloak as well as for other shapes. By analyzing their asymptotic behavior, we find that the performance of finite-size cloaks becomes much better for metal shapes with zero average value, e.g., for gratings.

© 2010 Optical Society of America

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References

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  1. J. Li, and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
    [CrossRef] [PubMed]
  2. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
    [CrossRef] [PubMed]
  3. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
    [CrossRef]
  4. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
    [CrossRef]
  5. J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17, 12922–12928 (2009).
    [CrossRef] [PubMed]
  6. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
    [CrossRef] [PubMed]
  7. H. F. Ma, and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1, 1–6 (2010).
    [CrossRef]
  8. I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. 102, 213901 (2009).
    [CrossRef] [PubMed]
  9. B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104, 233903 (2010).
    [CrossRef] [PubMed]
  10. N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” N. J. Phys. 10, 115020 (2008).
    [CrossRef]
  11. V. M. Shalaev, “Transforming light,” Science 322, 384–386 (2008).
    [CrossRef] [PubMed]
  12. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
    [CrossRef]
  13. P. Zhang, M. Lobet, and S. He, “Carpet cloaking on a dielectric half-space,” Opt. Express 18, 15158–18163 (2010).
  14. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef] [PubMed]

2010 (5)

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

H. F. Ma, and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1, 1–6 (2010).
[CrossRef]

B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104, 233903 (2010).
[CrossRef] [PubMed]

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

P. Zhang, M. Lobet, and S. He, “Carpet cloaking on a dielectric half-space,” Opt. Express 18, 15158–18163 (2010).

2009 (5)

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. 102, 213901 (2009).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17, 12922–12928 (2009).
[CrossRef] [PubMed]

2008 (3)

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

V. M. Shalaev, “Transforming light,” Science 322, 384–386 (2008).
[CrossRef] [PubMed]

J. Li, and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

2006 (1)

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

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Blair, J.

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Chan, C. T.

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

Chan, T.

B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104, 233903 (2010).
[CrossRef] [PubMed]

Chen, H.

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

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Cui, T. J.

H. F. Ma, and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1, 1–6 (2010).
[CrossRef]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Enoch, S.

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

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

He, S.

P. Zhang, M. Lobet, and S. He, “Carpet cloaking on a dielectric half-space,” Opt. Express 18, 15158–18163 (2010).

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Kildishev, A. V.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. 102, 213901 (2009).
[CrossRef] [PubMed]

Lee, J. H.

Leonhardt, U.

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

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

J. Li, and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Lobet, M.

P. Zhang, M. Lobet, and S. He, “Carpet cloaking on a dielectric half-space,” Opt. Express 18, 15158–18163 (2010).

Ma, H. F.

H. F. Ma, and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1, 1–6 (2010).
[CrossRef]

McPhedran, R. C.

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

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Nicorovici, N.-A. P.

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

Park, W.

Pendry, J. B.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

J. Li, and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Rhee, S. J.

Shalaev, V. M.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. 102, 213901 (2009).
[CrossRef] [PubMed]

V. M. Shalaev, “Transforming light,” Science 322, 384–386 (2008).
[CrossRef] [PubMed]

Sheng, P.

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

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Smolyaninov, I. I.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. 102, 213901 (2009).
[CrossRef] [PubMed]

Smolyaninova, V. N.

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. 102, 213901 (2009).
[CrossRef] [PubMed]

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

Summers, C. J.

Tamma, V. A.

Tayeb, G.

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

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

Wu, B.-I.

B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104, 233903 (2010).
[CrossRef] [PubMed]

Wu, Q.

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Zhang, B.

B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104, 233903 (2010).
[CrossRef] [PubMed]

Zhang, P.

P. Zhang, M. Lobet, and S. He, “Carpet cloaking on a dielectric half-space,” Opt. Express 18, 15158–18163 (2010).

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

N. J. Phys. (1)

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

Nat. Mater. (2)

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

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Nat. Photonics (1)

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Nature Commun. (1)

H. F. Ma, and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1, 1–6 (2010).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (3)

J. Li, and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
[CrossRef] [PubMed]

I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. 102, 213901 (2009).
[CrossRef] [PubMed]

B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104, 233903 (2010).
[CrossRef] [PubMed]

Science (4)

V. M. Shalaev, “Transforming light,” Science 322, 384–386 (2008).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

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

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Left: illustration of the original coordinates (x,y) (gray) and the transformed coordinates (u, v) (red) resulting from the cloak of Eq. (4) for w = 1.66 and h = 0.77 (matching the width and height of the cloak in Ref. [13], see Fig. 4). All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane at y = 0, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of 9 × 4.5 normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.86 to 2.10.

Fig. 2
Fig. 2

Left: illustration of the original coordinates (x,y) (gray) and the transformed coordinates (u, v) (red) resulting from Eq. (9) with c = i/12 and k = 2π. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane indicated in gray, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a height of one normalized unit. Above it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.66 to 2.10.

Fig. 3
Fig. 3

Left: illustration of the original coordinates (x,y) (gray) and the transformed coordinates (u, v) (red) resulting from Eq. (11) with w = 2, h = 0.5, and κ = 0.5. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane indicated in gray, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of 9 × 4.5 normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.78 to 2.14.

Fig. 4
Fig. 4

Left: illustration of the original coordinates (x, y) (gray) and the transformed coordinates (u, v) (red) resulting from the cloak of Ref. [13], i.e., for the conformal map zf(z) = z −1/(z + iδy) with δy = 1.3. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane at y = 0, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of 9 × 4.5 normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.92 to 2.45.

Equations (13)

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f ( z ) = z + 0 c k e i kz d k ,
n ( f ( z ) ) = n 0 ( z ) | d f / d z | .
c k = i hw π e ( k w / 2 ) 2 .
f ( z ) = z + i h e ( z / w ) 2 [ 1 + erf ( i z / w ) ] ,
v ( u ) h e ( u / w ) 2 .
v ( x , 0 ) = 0 [ a k sin ( kx ) + b k cos ( kx ) ] d k
v ( u ) 0 [ a k sin ( ku ) + b k cos ( ku ) ] d k .
n ( u + i v = ρ e i φ ) = 1 hw cos ( 2 φ ) π ρ 2 + 𝒪 ( ρ 4 ) .
f ( z ) = z + c e i kz
n ( ζ = u + i v ) = | 1 + W 0 ( i c k e i k ζ ) | 1 = 1 + [ a sin ( ku ) + b cos ( ku ) ] k e kv + 𝒪 ( e 2 kv ) ,
f ( z ) = z + 0 c k e i ( k + κ ) z d k = z + i h e ( z / w ) 2 [ 1 + erf ( i z / w ) ] e i κ z .
n ( u + i v = ρ e i φ ) = 1 + hw κ sin ( φ κ ρ cos φ ) π × e κ ρ sin φ ρ + 𝒪 ( e κ ρ sin φ ρ 2 ) .
n ( u + i v = ρ e i φ ) = 1 cos ( 2 φ ) ρ 2 + 𝒪 ( ρ 3 ) .

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