Abstract

Discrete systems of infinitely long polarizable line dipoles are considered in the quasistatic limit, interacting with a two-dimensional cloaking system consisting of a hollow plasmonic cylindrical shell. A numerical procedure is described for accurately calculating electromagnetic fields arising in the quasistatic limit, for the case when the relative permittivity of the cloaking shell has a very small imaginary part. Animations are given which illustrate cloaking of discrete systems, both for the case of induced dipoles and induced quadrupoles on the interacting particles. The simulations clarify the physical mechanism for the cloaking.

© 2007 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. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (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. A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24413-419 (2003).
    [CrossRef] [PubMed]
  5. A. Greenleaf, Y. Kurylev,M. Lassas, and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," http://arxiv.org/abs/math.AP/0611185.
  6. G.W. Milton,M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New Journal of Physics 8, 248-267 (2006).
    [CrossRef]
  7. W. Cai, U. K. Chettiar, A. V. Kildishev, and V.M. Shalaev, "Optical Cloaking with Non-MagneticMetamaterials," http://arxiv.org/pdf/physics/0611242.
  8. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 490, 8479-8482 (1994).
    [CrossRef]
  9. T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005).
    [CrossRef]
  10. A. D. Boardman and K. Marinov, "Non-radiating and radiating configurations driven by left-handed metamaterials," J. Opt. Soc. Am. B 23, 543-552 (2006).
    [CrossRef]
  11. V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, "Planar electromagnetic metamaterial with a fish scale structure," Phys. Rev. E 72, 056613 (2005).
    [CrossRef]
  12. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, "A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance," Proc. R. Soc. Lond. A 461, 3999-4034 (2005).
    [CrossRef]
  13. G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. A 462, 3027-3059 (2006).
    [CrossRef]
  14. G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, "Opaque perfect lenses," Physica B, in press, http://www.arxiv.org/abs/physics/0608225.
    [CrossRef]
  15. M. Kerker, "Invisible bodies," J. Opt. Soc. Am. 65, 376-379 (1975).
    [CrossRef]
  16. A. Alu and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
    [CrossRef]
  17. A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
    [CrossRef]
  18. A. G. Ramm, "Invisible obstacles," Ann. Polon. Math. 90, 145-148 (2007).
    [CrossRef]
  19. D. A. B. Miller, "On perfect cloaking," Opt. Express 14, 12457-12466 (2006).
    [CrossRef] [PubMed]
  20. P. Sheng, "Waves on the horizon," Science 313, 1399-1400 (2006).
    [CrossRef] [PubMed]
  21. P. Weiss, "Out of Sight: Physicists get serious about invisibility shields," Science News 170, 42-44 (2006).
    [CrossRef]
  22. Supporting Online Material, http://www.physics.usyd.edu.au/cudos/research/plasmon.html.
  23. O. P. Bruno and S. Lintner, "Superlens-cloaking of small dielectric bodies in the quasistatic regime," submitted.

2007 (1)

A. G. Ramm, "Invisible obstacles," Ann. Polon. Math. 90, 145-148 (2007).
[CrossRef]

2006 (9)

D. A. B. Miller, "On perfect cloaking," Opt. Express 14, 12457-12466 (2006).
[CrossRef] [PubMed]

P. Sheng, "Waves on the horizon," Science 313, 1399-1400 (2006).
[CrossRef] [PubMed]

P. Weiss, "Out of Sight: Physicists get serious about invisibility shields," Science News 170, 42-44 (2006).
[CrossRef]

G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. A 462, 3027-3059 (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]

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]

G.W. Milton,M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New Journal of Physics 8, 248-267 (2006).
[CrossRef]

A. D. Boardman and K. Marinov, "Non-radiating and radiating configurations driven by left-handed metamaterials," J. Opt. Soc. Am. B 23, 543-552 (2006).
[CrossRef]

2005 (4)

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, "Planar electromagnetic metamaterial with a fish scale structure," Phys. Rev. E 72, 056613 (2005).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, "A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance," Proc. R. Soc. Lond. A 461, 3999-4034 (2005).
[CrossRef]

T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005).
[CrossRef]

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

2003 (2)

A. Alu and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24413-419 (2003).
[CrossRef] [PubMed]

1994 (1)

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 490, 8479-8482 (1994).
[CrossRef]

1975 (1)

Alu, A.

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

A. Alu and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Boardman, A. D.

Briane, M.

G.W. Milton,M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New Journal of Physics 8, 248-267 (2006).
[CrossRef]

Cheng, Q.

T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005).
[CrossRef]

Cui, T. J.

T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005).
[CrossRef]

Cummer, S. A.

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. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

A. Alu and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Fedotov, V. A.

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, "Planar electromagnetic metamaterial with a fish scale structure," Phys. Rev. E 72, 056613 (2005).
[CrossRef]

Greenleaf, A.

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24413-419 (2003).
[CrossRef] [PubMed]

Jiang, Q.

T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005).
[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]

Kerker, M.

Kong, J. A.

T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005).
[CrossRef]

Lassas, M.

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24413-419 (2003).
[CrossRef] [PubMed]

Leonhardt, U.

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

Lu, W. B.

T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005).
[CrossRef]

Marinov, K.

McPhedran, R. C.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, "A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance," Proc. R. Soc. Lond. A 461, 3999-4034 (2005).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 490, 8479-8482 (1994).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, "Opaque perfect lenses," Physica B, in press, http://www.arxiv.org/abs/physics/0608225.
[CrossRef]

Miller, D. A. B.

Milton, G. W.

G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. A 462, 3027-3059 (2006).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, "A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance," Proc. R. Soc. Lond. A 461, 3999-4034 (2005).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 490, 8479-8482 (1994).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, "Opaque perfect lenses," Physica B, in press, http://www.arxiv.org/abs/physics/0608225.
[CrossRef]

Milton, G.W.

G.W. Milton,M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New Journal of Physics 8, 248-267 (2006).
[CrossRef]

Mladyonov, P. L.

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, "Planar electromagnetic metamaterial with a fish scale structure," Phys. Rev. E 72, 056613 (2005).
[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]

Nicorovici, N. A.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 490, 8479-8482 (1994).
[CrossRef]

Nicorovici, N.-A. P.

G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. A 462, 3027-3059 (2006).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, "A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance," Proc. R. Soc. Lond. A 461, 3999-4034 (2005).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, "Opaque perfect lenses," Physica B, in press, http://www.arxiv.org/abs/physics/0608225.
[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]

Podolskiy, V. A.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, "A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance," Proc. R. Soc. Lond. A 461, 3999-4034 (2005).
[CrossRef]

Prosvirnin, S. L.

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, "Planar electromagnetic metamaterial with a fish scale structure," Phys. Rev. E 72, 056613 (2005).
[CrossRef]

Ramm, A. G.

A. G. Ramm, "Invisible obstacles," Ann. Polon. Math. 90, 145-148 (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]

Sheng, P.

P. Sheng, "Waves on the horizon," Science 313, 1399-1400 (2006).
[CrossRef] [PubMed]

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. 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]

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, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24413-419 (2003).
[CrossRef] [PubMed]

Weiss, P.

P. Weiss, "Out of Sight: Physicists get serious about invisibility shields," Science News 170, 42-44 (2006).
[CrossRef]

Willis, J. R.

G.W. Milton,M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New Journal of Physics 8, 248-267 (2006).
[CrossRef]

Zheludev, N. I.

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, "Planar electromagnetic metamaterial with a fish scale structure," Phys. Rev. E 72, 056613 (2005).
[CrossRef]

Ann. Polon. Math. (1)

A. G. Ramm, "Invisible obstacles," Ann. Polon. Math. 90, 145-148 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. Alu and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

New Journal of Physics (1)

G.W. Milton,M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New Journal of Physics 8, 248-267 (2006).
[CrossRef]

Opt. Express (1)

Phy. Rev. B (1)

T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005).
[CrossRef]

Phys. Rev. B (1)

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 490, 8479-8482 (1994).
[CrossRef]

Phys. Rev. E (2)

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, "Planar electromagnetic metamaterial with a fish scale structure," Phys. Rev. E 72, 056613 (2005).
[CrossRef]

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

Physica B (1)

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, "Opaque perfect lenses," Physica B, in press, http://www.arxiv.org/abs/physics/0608225.
[CrossRef]

Physiol. Meas. (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24413-419 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A (1)

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, "A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance," Proc. R. Soc. Lond. A 461, 3999-4034 (2005).
[CrossRef]

Proc. Roy. Soc. A (1)

G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. A 462, 3027-3059 (2006).
[CrossRef]

Science (4)

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]

P. Sheng, "Waves on the horizon," Science 313, 1399-1400 (2006).
[CrossRef] [PubMed]

Science News (1)

P. Weiss, "Out of Sight: Physicists get serious about invisibility shields," Science News 170, 42-44 (2006).
[CrossRef]

Other (4)

Supporting Online Material, http://www.physics.usyd.edu.au/cudos/research/plasmon.html.

O. P. Bruno and S. Lintner, "Superlens-cloaking of small dielectric bodies in the quasistatic regime," submitted.

A. Greenleaf, Y. Kurylev,M. Lassas, and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," http://arxiv.org/abs/math.AP/0611185.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V.M. Shalaev, "Optical Cloaking with Non-MagneticMetamaterials," http://arxiv.org/pdf/physics/0611242.

Supplementary Material (5)

» Media 1: MOV (4569 KB)     
» Media 2: MOV (3159 KB)     
» Media 3: MOV (3607 KB)     
» Media 4: MOV (3597 KB)     
» Media 5: MOV (5050 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

A coated cylinder centered about the origin of coordinates. Polarizable line dipoles are placed at the points P 1,P 2,…,Pn . E appl is an electric field with sources at infinity. The annular domain D, between the shell of the cylinder and the dashed circle, is chosen so that it contains no field sources. P( z ) is an arbitrary point in the plane where the total electric potential is evaluated.

Fig. 2.
Fig. 2.

Potential distribution showing cloaking of a single polarizable dipole (α= 2, E appl = 1, εm = εc = 1, εs = -1+10-12i, rs = 4, rc = 2, r # = 5.66, plot range = [-15 (blue),15 (red)] for the potential). The concentric circles, shown in this and the following figures, bound the core (solid line), the shell (solid line), and the cloaking region (dashed line).

Fig. 3.
Fig. 3.

The equipotential lines (generally vertical) and the electric field lines (generally horizontal) for a polarizable line dipole on the x-axis (α = 2, E appl = 1, εm = εc = 1, εs = -1+10-12i, rs = 4, rc = 2, r # = 5.7, plot range = [-22 (blue), 25 (yellow)]).

Fig. 4.
Fig. 4.

The equipotential lines and the direction of the electric field (arrows) for a polariz-able line dipole on the x-axis (α= 2, E appl = 1, εm = εc = 1, εs = -1+10-12i, rs = 4, rc = 2, r # = 5.7, plot range = [-22 (blue), 25 (red)]).

Fig. 5.
Fig. 5.

Two polarizable dipoles, one cloaked and the other close to the cloaking region (α= 20, E appl = 0.05, εm =εc = 1, εs = -1+10-12i, rs = 4, rc = 2, r # = 5.66, plot range = [-2.5 (blue), 2.5 (red)]).

Fig. 6.
Fig. 6.

Cloaking of a cluster of 7 polarizable dipoles in a hexagonal pattern (α= 20, E appl = 0.05, εm =εc = 1, εs = -1+10-12i, rs = 4, rc = 1.1, r # = 7.63, plot range = [-2.5 (blue), 2.5 (red)]).

Fig. 7.
Fig. 7.

Cloaking of a non-symmetric cluster of 25 polarizable dipoles plus one separate polarizable dipole (α= 20, E appl = 0.05, εm =εc = 1, εs = -1+10-12i, rs = 4, rc = 1.2, r # = 7.3, plot range = [-2 (blue), 2 (red)]).

Fig. 8.
Fig. 8.

A quadrupole close to the cloaking region (α = 20, E appl = 0.05, εm = εc = 1, εs = -1 +10-12i, rs = 4, rc = 2, r # = 5.66, plot range = [-15 (blue), 15 (red)]).

Equations (5)

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V r θ = A 0 e + = 1 ( A e r + B e r ) cos ( ℓθ ) + ( A o r l + B o r ) sin ( ℓθ ) ,
V m ( z ) = V appl ( z ) + n = 1 N V n ( z ) + V ˜ appl ( z ) + n = 1 N V ˜ n ( z ) ,
[ d x ( j ) d y ( j ) ] = α j [ E x ( j ) E y ( j ) ] = α j V ̂ ( j ) z = z j ,
V ̂ ( j ) = V appl ( z ) + n j N V n ( z ) + V ˜ appl ( z ) + n = 1 N V ˜ n ( z ) = V m ( z ) V j ( z ) .
[ k e ( j ) k o ( j ) ] = [ cos θ j sin θ j sin θ j cos θ j ] [ d x ( j ) d y ( j ) ] ,

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