Abstract

The previously reported simplified cylindrical linear cloak is improved so that the cloak’s outer surface is impedance-matched to free space. The scattering characteristics of the improved linear cloak is compared to the previous counterpart as well as the recently proposed simplified quadratic cloak derived from quadratic coordinate transformation. Significant improvement in invisibility performance is noticed for the improved linear cloak with respect to the previously proposed linear one. The improved linear cloak and the simplified quadratic cloak have comparable invisibility performances, except that the latter however has to fulfill a minimum shell thickness requirement (i.e. outer radius must be larger than twice of inner radius). When a thin cloak shell is desired, the improved linear cloak is much more superior than the other two versions of simplified cloaks.

© 2007 Optical Society of America

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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. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
    [CrossRef] [PubMed]
  3. A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
    [CrossRef]
  4. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113,903 (2007).
    [CrossRef]
  5. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Improvement of cylindrical cloaking with the SHS lining," Opt. Express 15, 12,717-12,734 (2007).
    [CrossRef]
  6. 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]
  7. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (2006).
    [CrossRef]
  8. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
    [CrossRef]
  9. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (2007).
    [CrossRef]
  10. M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233,901 (2007).
    [CrossRef]
  11. U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
    [CrossRef]

2007

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113,903 (2007).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Improvement of cylindrical cloaking with the SHS lining," Opt. Express 15, 12,717-12,734 (2007).
[CrossRef]

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

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (2007).
[CrossRef]

M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233,901 (2007).
[CrossRef]

2006

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (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]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (2006).
[CrossRef]

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, 1777-1780 (2006).
[CrossRef] [PubMed]

2005

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
[CrossRef]

Alù, A.

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
[CrossRef]

Cai, W.

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

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (2007).
[CrossRef]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. 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, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (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]

Engheta, N.

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
[CrossRef]

Greenleaf, A.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Improvement of cylindrical cloaking with the SHS lining," Opt. Express 15, 12,717-12,734 (2007).
[CrossRef]

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. V.

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

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (2007).
[CrossRef]

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Improvement of cylindrical cloaking with the SHS lining," Opt. Express 15, 12,717-12,734 (2007).
[CrossRef]

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Improvement of cylindrical cloaking with the SHS lining," Opt. Express 15, 12,717-12,734 (2007).
[CrossRef]

Leonhardt, U.

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Milton, G. W.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (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]

Neff, C. W.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113,903 (2007).
[CrossRef]

Pendry, J.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (2006).
[CrossRef]

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]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Popa, B.-I.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (2006).
[CrossRef]

Qiu, M.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113,903 (2007).
[CrossRef]

M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233,901 (2007).
[CrossRef]

Ruan, Z.

M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233,901 (2007).
[CrossRef]

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113,903 (2007).
[CrossRef]

Schurig, D.

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]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (2006).
[CrossRef]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. 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, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (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]

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

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]

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Improvement of cylindrical cloaking with the SHS lining," Opt. Express 15, 12,717-12,734 (2007).
[CrossRef]

Yan, M.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113,903 (2007).
[CrossRef]

M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233,901 (2007).
[CrossRef]

Appl. Phys. Lett.

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (2007).
[CrossRef]

Nat. Photonics

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

New J. Phys.

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Opt. Express

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Improvement of cylindrical cloaking with the SHS lining," Opt. Express 15, 12,717-12,734 (2007).
[CrossRef]

Phys. Rev. E

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (2006).
[CrossRef]

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016,623 (2005).
[CrossRef]

Phys. Rev. Lett.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113,903 (2007).
[CrossRef]

M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," Phys. Rev. Lett. 99, 233,901 (2007).
[CrossRef]

Science

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, 1777-1780 (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]

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Figures (6)

Fig. 1.
Fig. 1.

Illustrations of interactions between a plane wave and perfect cylindrical cloaks resulted from coordinate transformations. (a) Original EM space; (b) Ideal linear cloak; (c) Ideal quadratic cloak. The colormap is for Ez field. Green lines are for energy flow (poynting vector). a=1, b=3, λ=1.5 (a.u.). Domain size is at 7.5×7.5. For the quadratic cloak, p=a/b 2. (Read the text for parameter descriptions.) Invariant coordinate lines are imposed to show how the EM fields are compressed.

Fig. 2.
Fig. 2.

(a) Material parameters for an ideal linear cloak (black curves), a simplified linear cloak (green curves), and an improved linear cloak (red curves). (b) Material parameters for an ideal quadratic cloak (black curves), and an simplified quadratic cloak (blue curves). Solid line: µ r ; dashed line: µ θ ; dotted line: ε z . All cloaks have a=1 and b=3 (a.u.).

Fig. 3.
Fig. 3.

The scattering coefficients in each cylindrical order as a function of either b, a or λ. (a1)–(a3) Previously proposed simplified linear cloak; (b1)–(b3) Improved simplified linear cloak; (c1)–(c3) Simplified quadratic cloak. In each row, the left panel shows the effect of b as a=0.1m and λ=0.15m; the middle panel shows the effect of a as b=0.6m and λ=0.15m; the right panel shows the effect of λ as a=0.2m and b=0.6m.

Fig. 4.
Fig. 4.

Scattering coefficients in each cylindrical order as a function of b (a), a (b) or λ (c). For each parameter study, the other two parameters are fixed in the same way as described in Fig. 3 caption.

Fig. 5.
Fig. 5.

The far field scattering patterns by the simplified cloaks with an identical plane wave incidence (amplitude at 1). Green curves are for the previously reported simplified linear cloak. The red curves are for the improved simplified linear cloak. The blue curves are for the simplified quadratic cloak. Dashed black curves are for the bare PEC cylinder. (a) a=0.1m, b=0.15m; (b) a=0.1m, b=0.3m.

Fig. 6.
Fig. 6.

Snapshots of Ez fields around (a) a previously reported simplified linear cloak, (b) an improved linear cloak, and (c) a simplified quadratic cloak. a=0.1m, b=0.3m. Poynting vectors are shown in green lines. The scattered Ez fields of the three cloaks in (a)–(c) are shown respectively in (d)–(f). Domain for (a)–(c): 0.75×0.75m2, and for (d)–(f): 12×12m2.

Tables (2)

Tables Icon

Table 1. Material parameters for linear cloaks

Tables Icon

Table 2. Material parameters for quadratic cloaks

Equations (10)

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

r = b a b r + a .
1 r [ r ( r μ θ E z r ) ] + 1 r 2 θ ( 1 μ r E z θ ) + k 0 2 ε z E z = 0 ,
d dr ( r μ θ d Ψ dr ) + k 0 2 r ε z Ψ m 2 1 r μ r Ψ = 0 .
1 μ θ ε z d dr ( r d Ψ dr ) + k 0 2 r Ψ m 2 1 r μ r ε z Ψ = 0 .
( r a ) 2 d 2 Ψ d r 2 + ( r a ) 2 r d Ψ dr + [ ( r a ) 2 ( b b a ) 2 k 0 2 m 2 ] Ψ = 0 .
r 2 d 2 Ψ d r 2 + r d Ψ dr + r 2 k 0 2 ( b b a ) 2 Ψ = 0 .
r = [ b a b + p ( r b ) ] r + a ,
E z 1 = m 𝓐 m 1 J m ( k 0 r ) exp ( im θ ) ,
E z 2 = m { 𝓐 m 2 Q m + 𝓑 m 2 R m } exp ( im θ ) ,
E z 3 = m { 𝓐 m 2 J m ( k 0 r ) + 𝓑 m 3 H m ( 2 ) ( k 0 r ) } exp ( im θ ) .

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