Abstract

We derive the d-dimensional scattering cross section for homogeneous and composite hyper-particles inside a metamaterial. The polarizability of the hyper-particles is expressed in multi-dimensional form and is used in order to examine various scattering characteristics. We introduce scattering bounds that display interesting results when d → ∞ and in particular consider the special limit of hyper-particle cloaking in some detail. We demonstrate cloaking via resonance for homogeneous particles and show that composite hyper-particles can be used in order to obtain electromagnetic cloaking with either negative or all positive constitutive parameters respectively. Our approach not only considers cloaking of particles of integer dimension but also particles with non-integer dimension such as fractals. Theoretical results are compared to full-wave numerical simulations for two interacting hyper-particles in a medium.

© 2010 OSA

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  1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509 (1968).
    [Crossref]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
    [Crossref] [PubMed]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E 84, 4184 (2000).
  4. R. W. Ziolkowski and A. D. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag. 51, 2626–2640 (2003).
    [Crossref]
  5. N. Engheta and R. W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53, 1535–1556 (2005).
    [Crossref]
  6. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
    [Crossref]
  7. A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express,  15(6), 3318–3332 (2007).
    [Crossref] [PubMed]
  8. A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15(12), 7578–7590 (2007).
    [Crossref] [PubMed]
  9. A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
    [Crossref] [PubMed]
  10. A. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatters,” Prog. Electromagn. Res. 66, 191–198 (2006).
    [Crossref]
  11. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (New York: Wiley, 1983).
  12. X. Zhou and G. Hu, “Design for electromagnetic wave transparency with metamaterials,” Phys. Rev. E 74, 026607 (2006).
    [Crossref]
  13. J. S. McGuirk and P. J. Collins, “Controlling the transmitted field into a cylindrical cloak’s hidden region,” Opt. Express 16, 17560–17572 (2008).
    [Crossref] [PubMed]
  14. S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, and S. Hayashi, “Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials,” Opt. Express 16, 9942–9950 (2008).
    [Crossref] [PubMed]
  15. H. R. Stuart and R. W. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
    [Crossref]
  16. J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
    [Crossref]
  17. A. Alexopoulos, “Effective response and scattering cross section of spherical inclusions in a medium,” Phys. Lett. A 373(35), 3190–3196 (2009).
    [Crossref]
  18. A. Alexopoulos, “Effective-medium theory of surfaces and metasurfaces containing two-dimensional binary inclusions,” Phys. Rev. E 81, 046607 (2010).
    [Crossref]
  19. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  20. 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 312, 1780–1782 (2006).
    [PubMed]
  21. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
    [Crossref]
  22. 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]
  23. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirrage effect,” Opt. Lett. 32, 1069–1071 (2007).
    [Crossref] [PubMed]
  24. A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43 (2008).
    [Crossref]
  25. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref] [PubMed]
  26. U. Leonhardt, “Notes on conformal invisibility devices,” N. J. Phys. 8, 118 (2006).
    [Crossref]
  27. U. Leonhardt, “General relativity in electrical engineering,” N. J. Phys. 8, 247 (2006).
    [Crossref]
  28. A. Alù and N. Engheta, “Robustness in design and background variations in metamaterial/plasmonic cloaking,” Radio Sci. 43, RS4S01 (2008).
    [Crossref]
  29. V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express 18(10), 9780 (2010).
    [Crossref] [PubMed]
  30. G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. Lond. A Math Phys. Sci. 462, 3027–3059 (2006).
    [Crossref]
  31. M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
    [Crossref]
  32. W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007).
    [Crossref]
  33. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
    [Crossref]
  34. W. Li, J. Guan, Z. Sun, W. Wang, and Q. Zhang, “A near perfect-invisibility cloak constructed with homogeneous materials,” Opt. Express 17, 23410–23416 (2009).
    [Crossref]
  35. J. Gleick, Chaos: Making a New Science, (New York: Penguin Books, 1988).
  36. J. W. Harris and H. Stocker, “Koch’s Curve” and “Koch’s Snowflake,” 4.11.5-4.11.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 114–115, (1998).

2010 (2)

A. Alexopoulos, “Effective-medium theory of surfaces and metasurfaces containing two-dimensional binary inclusions,” Phys. Rev. E 81, 046607 (2010).
[Crossref]

V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express 18(10), 9780 (2010).
[Crossref] [PubMed]

2009 (2)

W. Li, J. Guan, Z. Sun, W. Wang, and Q. Zhang, “A near perfect-invisibility cloak constructed with homogeneous materials,” Opt. Express 17, 23410–23416 (2009).
[Crossref]

A. Alexopoulos, “Effective response and scattering cross section of spherical inclusions in a medium,” Phys. Lett. A 373(35), 3190–3196 (2009).
[Crossref]

2008 (5)

2007 (6)

A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express,  15(6), 3318–3332 (2007).
[Crossref] [PubMed]

A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15(12), 7578–7590 (2007).
[Crossref] [PubMed]

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[Crossref]

W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (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]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirrage effect,” Opt. Lett. 32, 1069–1071 (2007).
[Crossref] [PubMed]

2006 (11)

A. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatters,” Prog. Electromagn. Res. 66, 191–198 (2006).
[Crossref]

X. Zhou and G. Hu, “Design for electromagnetic wave transparency with metamaterials,” Phys. Rev. E 74, 026607 (2006).
[Crossref]

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

U. Leonhardt, “Notes on conformal invisibility devices,” N. J. Phys. 8, 118 (2006).
[Crossref]

U. Leonhardt, “General relativity in electrical engineering,” N. J. Phys. 8, 247 (2006).
[Crossref]

G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. Lond. A Math Phys. Sci. 462, 3027–3059 (2006).
[Crossref]

H. R. Stuart and R. W. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (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 312, 1780–1782 (2006).
[PubMed]

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

2005 (2)

N. Engheta and R. W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53, 1535–1556 (2005).
[Crossref]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

2003 (1)

R. W. Ziolkowski and A. D. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag. 51, 2626–2640 (2003).
[Crossref]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[Crossref] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E 84, 4184 (2000).

1999 (1)

J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

1968 (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509 (1968).
[Crossref]

Alexopoulos, A.

A. Alexopoulos, “Effective-medium theory of surfaces and metasurfaces containing two-dimensional binary inclusions,” Phys. Rev. E 81, 046607 (2010).
[Crossref]

A. Alexopoulos, “Effective response and scattering cross section of spherical inclusions in a medium,” Phys. Lett. A 373(35), 3190–3196 (2009).
[Crossref]

Alù, A.

A. Alù and N. Engheta, “Robustness in design and background variations in metamaterial/plasmonic cloaking,” Radio Sci. 43, RS4S01 (2008).
[Crossref]

A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
[Crossref] [PubMed]

A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express,  15(6), 3318–3332 (2007).
[Crossref] [PubMed]

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[Crossref]

A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15(12), 7578–7590 (2007).
[Crossref] [PubMed]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (New York: Wiley, 1983).

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, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007).
[Crossref]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (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]

Collins, P. J.

Cummer, S. A.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. 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 312, 1780–1782 (2006).
[PubMed]

Engheta, N.

A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
[Crossref] [PubMed]

A. Alù and N. Engheta, “Robustness in design and background variations in metamaterial/plasmonic cloaking,” Radio Sci. 43, RS4S01 (2008).
[Crossref]

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[Crossref]

A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express,  15(6), 3318–3332 (2007).
[Crossref] [PubMed]

A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15(12), 7578–7590 (2007).
[Crossref] [PubMed]

N. Engheta and R. W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53, 1535–1556 (2005).
[Crossref]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

Fujii, M.

Gleick, J.

J. Gleick, Chaos: Making a New Science, (New York: Penguin Books, 1988).

Grigorenko, A. N.

V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express 18(10), 9780 (2010).
[Crossref] [PubMed]

Guan, J.

Guenneau, S.

Harris, J. W.

J. W. Harris and H. Stocker, “Koch’s Curve” and “Koch’s Snowflake,” 4.11.5-4.11.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 114–115, (1998).

Hayashi, S.

Holden, A. J.

J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Hu, G.

X. Zhou and G. Hu, “Design for electromagnetic wave transparency with metamaterials,” Phys. Rev. E 74, 026607 (2006).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (New York: Wiley, 1983).

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 312, 1780–1782 (2006).
[PubMed]

Kildishev, A. V.

A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43 (2008).
[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, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007).
[Crossref]

Kipple, A. D.

R. W. Ziolkowski and A. D. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag. 51, 2626–2640 (2003).
[Crossref]

Kravets, V. G.

V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express 18(10), 9780 (2010).
[Crossref] [PubMed]

Leonhardt, U.

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

U. Leonhardt, “Notes on conformal invisibility devices,” N. J. Phys. 8, 118 (2006).
[Crossref]

U. Leonhardt, “General relativity in electrical engineering,” N. J. Phys. 8, 247 (2006).
[Crossref]

Li, W.

McGuirk, J. S.

Milton, G. W.

W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007).
[Crossref]

G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. Lond. A Math Phys. Sci. 462, 3027–3059 (2006).
[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 312, 1780–1782 (2006).
[PubMed]

Nasser, N.

D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E 84, 4184 (2000).

Nicolet, A.

Nicorovici, N. A.

G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. Lond. A Math Phys. Sci. 462, 3027–3059 (2006).
[Crossref]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E 84, 4184 (2000).

Pendry, J.

J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Pendry, J. B.

S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, and S. Hayashi, “Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials,” Opt. Express 16, 9942–9950 (2008).
[Crossref] [PubMed]

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirrage 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 312, 1780–1782 (2006).
[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, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. 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, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[Crossref] [PubMed]

Pidwerbetsky, R. W.

H. R. Stuart and R. W. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
[Crossref]

Popa, B. I.

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

Robbins, D. D.

J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Schedin, F.

V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express 18(10), 9780 (2010).
[Crossref] [PubMed]

Schultz, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E 84, 4184 (2000).

Schurig, D.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. 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 312, 1780–1782 (2006).
[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]

Shalaev, V. M.

A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43 (2008).
[Crossref]

W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (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]

Sihvola, A.

A. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatters,” Prog. Electromagn. Res. 66, 191–198 (2006).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[Crossref]

Smith, D. R.

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]

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 312, 1780–1782 (2006).
[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, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[Crossref]

D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E 84, 4184 (2000).

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 312, 1780–1782 (2006).
[PubMed]

Stewart, W. J.

J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Stocker, H.

J. W. Harris and H. Stocker, “Koch’s Curve” and “Koch’s Snowflake,” 4.11.5-4.11.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 114–115, (1998).

Stuart, H. R.

H. R. Stuart and R. W. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
[Crossref]

Sun, Z.

Taylor, S.

V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express 18(10), 9780 (2010).
[Crossref] [PubMed]

Tomita, S.

Veselago, V. G.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509 (1968).
[Crossref]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E 84, 4184 (2000).

Viita, D.

V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express 18(10), 9780 (2010).
[Crossref] [PubMed]

Wang, W.

Wood, B.

Yanagi, H.

Yokoyama, T.

Zhang, Q.

Zhou, X.

X. Zhou and G. Hu, “Design for electromagnetic wave transparency with metamaterials,” Phys. Rev. E 74, 026607 (2006).
[Crossref]

Ziolkowski, R. W.

N. Engheta and R. W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53, 1535–1556 (2005).
[Crossref]

R. W. Ziolkowski and A. D. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag. 51, 2626–2640 (2003).
[Crossref]

Zolla, F.

Appl. Phys. Lett. (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111105 (2007).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

H. R. Stuart and R. W. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
[Crossref]

IEEE Trans. Antennas Propag. (1)

R. W. Ziolkowski and A. D. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag. 51, 2626–2640 (2003).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

N. Engheta and R. W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53, 1535–1556 (2005).
[Crossref]

J. Pendry, A. J. Holden, D. D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

N. J. Phys. (2)

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[Crossref]

U. Leonhardt, “General relativity in electrical engineering,” N. J. Phys. 8, 247 (2006).
[Crossref]

Nat. Photonics (1)

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

Opt Express (1)

V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt Express 18(10), 9780 (2010).
[Crossref] [PubMed]

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A. Alexopoulos, “Effective response and scattering cross section of spherical inclusions in a medium,” Phys. Lett. A 373(35), 3190–3196 (2009).
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A. Alexopoulos, “Effective-medium theory of surfaces and metasurfaces containing two-dimensional binary inclusions,” Phys. Rev. E 81, 046607 (2010).
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D. R. Smith, W. J. Padilla, D. C. Vier, N. Nasser, and S. C. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. E 84, 4184 (2000).

X. Zhou and G. Hu, “Design for electromagnetic wave transparency with metamaterials,” Phys. Rev. E 74, 026607 (2006).
[Crossref]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
[Crossref]

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

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
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J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
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A. Alù and N. Engheta, “Multifrequency optical cloaking with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
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Proc. R. Soc. Lond. A Math Phys. Sci. (1)

G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. Lond. A Math Phys. Sci. 462, 3027–3059 (2006).
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A. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatters,” Prog. Electromagn. Res. 66, 191–198 (2006).
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A. Alù and N. Engheta, “Robustness in design and background variations in metamaterial/plasmonic cloaking,” Radio Sci. 43, RS4S01 (2008).
[Crossref]

Science (3)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (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 312, 1780–1782 (2006).
[PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509 (1968).
[Crossref]

Other (3)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (New York: Wiley, 1983).

J. Gleick, Chaos: Making a New Science, (New York: Penguin Books, 1988).

J. W. Harris and H. Stocker, “Koch’s Curve” and “Koch’s Snowflake,” 4.11.5-4.11.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 114–115, (1998).

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Figures (11)

Fig. 1.
Fig. 1.

A hyper-particle is shown in two dimensions (d = 2) with all constitutive parameters defined. Higher or lower dimensional particles have a similar configuration depending on their dimension. The definitions here hold for integer dimensions while for fractal dimensions we consider a different approach, refer to Section 4.

Fig. 2.
Fig. 2.

Scattering cross section for d = 1,2,3 and 4 non-homogeneous composite layered particles for the parameters chosen. The scattering cross section for particles with dimension d = 1 is given by the lightest curve while particles with increasing dimension d = 2,3 and 4 are represented by the darker curves respectively. Hence a particle with dimension d = 4 (darkest curve) has the smallest scattering cross section compared to other dimensions. In all cases, ε 2 = 1 means that σ approaches the holes limit (σH ) while at ε 2 = 2 it means that σ tends towards the superconducting limit (σS ). As the polarizability of the particles decreases, the scattering cross section becomes singular as shown by the sharp dips (σ → 0).

Fig. 3.
Fig. 3.

Field plots for d = 3 binary interacting particles (spheres) using full wave numerical analysis. The top plot (a) with ε 2 = 5.0 shows scattering by two homogeneous spherical particles. Particles cloaked via resonance (b) with ε 2 = 9 and via negative constitutive parameters for the inner core volume (c) with a 1 = 2.5 cm, ε 2 = 2.0, ε 1 = −1.69231 and β = (0.5)3. In all cases the spherical particles are surrounded by air so that ε 0 = 1 or μ 0 = 1. Common parameters are: a 2 = a = 5 cm and f 0 = 2 GHz.

Fig. 4.
Fig. 4.

Equation (17) is solved for fixed parameters as shown but varying γ 1 in order to determine which integer-dimensional particles, for the parameters given, result in σ = 0. Curve (1) (orange) indicates that for γ 1 = 0.3 the intersection with the horizontal axis at zero gives a value d > 4 which is rejected-(this would involve fractal particles). The value γ 1 = 1.1 represented by (3) (blue) indicates a value of d ≈ 6, again rejected because it is a non-integer dimension. Finally when γ 1 = 1.72 we see that the curve (2) (red) intersects the horizontal axis at d = 5. Hence a d = 5 particle is best suited in this case to cancel all scattering for the parameters given.

Fig. 5.
Fig. 5.

The optimum ratio for the hyper-radii is calculated as a function of the permittivity (or permeability) of the outer hyper-volume that makes cloaking for d = 1,2,3,4 particles possible for all positive constitutive parameters as considered here. Notice that for any arbitrary value for ε 2, cloaking can be achieved by more than one choice of hyper-particle, provided that the ratio a 1/a 2 is adjusted accordingly. This gives us enormous insight as to the cloaking behavior of higher dimensional particles and vice-versa.

Fig. 6.
Fig. 6.

Scattering and cloaking of d = 3 spherical particles embedded in a medium with permittivity (or permeability) of ε 0 = 2.5 (μ 0 = 2.5). Field plots on the left (a) and (c) show normal scattering of homogeneous particles while the field plots on the right (b) and (d) show cloaking via layered particles and using all positive constitutive parameters.

Fig. 7.
Fig. 7.

Scattering and cloaking of d = 2 particles (disks or cylinders). The top field plots (a) and (b) correspond to the case where the particles are surrounded by air so that ε 0 = 1 or μ 0 =1 and the cloaking effect shown top right (b) is via negative constitutive parameters for the inner layer. The bottom field plots (c) and (d) are for the particles surrounded by a medium with ε 0 = 2.5 or μ 0 = 2.5. Cloaking in plot (d) is achieved via all positive constitutive parameters.

Fig. 8.
Fig. 8.

The scattering cross section (RCS) of d = 1 homogeneous and composite layered line segments, ie, d = 1 particles. The full-wave numerical simulations shown here confirm our theoretical predictions.

Fig. 9.
Fig. 9.

The optimum ratio for the radii is shown for a frequency dependent outer layer based on the Drude model that makes d = 1,2,3,4 particles invisible to the field. As the particle dimension increases d → ∞, we find that the curves approach unity, ie, a 1a 2. Notice that the vertical axis implies that a realistic solution exists iff a 1a 2 so a 1/a 2 ∈ [0,1].

Fig. 10.
Fig. 10.

Numerical simulations showing cloaking effect for d = 3 (spherical) particles. In plots (a) and (b) the parameters are the same as those of Fig. 6(d) and we show here that the condition for cloaking is independent of the particle separations, even when the gap between them is zero as in (b) (touching). In (c) and (d) we have scaled the size of the particles by a factor of 1/10 and changed the frequency to f 0 = 20 GHz to verify that the cloaking effect is also independent of such things as incident frequency, particle size or number of particles.

Fig. 11.
Fig. 11.

For an n = 1 iterated Koch fractal with dimension d = 1.13093, normal scattering is shown bottom left for a homogeneous binary pair and the cloaked version on the right for a composite layered Koch snowflake fractal pair predicted by our d-dimensional theory.

Equations (34)

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σ = 8 π 3 3 λ 0 4 γ eff 2
σ = 8 π 3 3 λ 0 4 p tot E 0 2
p tot = E 0 d π d 2 a 2 d Γ ( d 2 + 1 ) γ
γ = γ 1 1 γ 1 + d 1 [ 1 + β ( γ 2 1 ) [ ( d 1 ) γ 1 + 1 ] ( γ 1 1 ) ( γ 2 + d 1 ) ] [ 1 + β ( d 1 ) ( γ 1 1 ) ( γ 2 1 ) ( γ 1 + d 1 ) ( γ 2 + d 1 ) ] 1
σ = 8 d 2 π d + 3 a 2 2 d 3 λ 0 4 Γ 2 ( d 2 + 1 ) γ 2
σ = 8 d 2 π d + 3 a 2 2 d 3 λ 0 4 Γ 2 ( d 2 + 1 ) ( γ 1 1 γ 1 + d 1 ) 2
γ 1 = ε 2 ε 0
γ 2 = ε 1 ε 2 = 1
γ 1 = ε 2 ε 0 0
γ = 1 d 1
1 d 1 γ 1
σ H σ σ S
σ H = 8 d 2 π d + 3 a 2 2 d 3 λ 0 4 ( d 1 ) 2 Γ 2 ( d 2 + 1 )
σ S = 8 d 2 π d + 3 a 2 2 d 3 λ 0 4 Γ 2 ( d 2 + 1 )
n 2 = ε 2 μ 2 = ( λ 0 λ 2 )
λ 2 = λ 0 ε 2
2 a = m λ 2 m = 2 a ε 2 λ 0
Ω F ( d ) ( γ 1 + d 1 ) ( γ 2 + d 1 ) + ( d 1 ) ( γ 2 1 ) ( γ 1 1 ) ( a 1 a 2 ) d = 0
Ω C ( d ) = Ω ( d ) ( γ 1 1 ) ( γ 2 + d 1 ) ( γ 2 1 ) [ ( d 1 ) γ 1 + 1 ] + ( a 1 a 2 ) d = 0
γ 2 = [ 1 ( d 1 ) ( γ 1 1 ) β ( γ 1 ( d 1 ) + 1 ) ] [ 1 + ( γ 1 1 ) β ( γ 1 ( d 1 ) + 1 ) ]
ε 1 = ε 2 [ ε 2 ( d 1 ) + ε 0 ( d 1 ) ( a 2 a 1 ) d ( ε 2 ε 0 ) ε 2 ( d 1 ) + ε 0 + ( a 2 a 1 ) d ( ε 2 ε 0 ) ]
( a 1 a 2 ) d = ( γ 2 + d 1 ) [ ( γ 1 1 ) f mn ( d ) ( γ 1 + d 1 ) ] ( γ 2 1 ) [ ( d 1 ) γ 1 + 1 ] f mn ( d ) ( d 1 ) ( γ 1 1 ) ( γ 2 1 )
f mn ( d ) = ( 1 ) m ( d 1 ) m δ mn
f 00 ( d ) = 1 ( superconducting )
lim d ( 1 d 1 ) = f 01 ( d ) = f 10 ( d ) 0
( a 1 a 2 ) d = ( γ 1 1 ) ( γ 2 + d 1 ) ( γ 2 1 ) [ ( d 1 ) γ 1 + 1 ]
ε 2 ( ω ) = 1 ω p 2 ω 2 + α ω i
μ 2 ( ω ) = 1 ω 2 ω 2 ω m 2 + α ω i
a 1 a 2 = ( ε 0 ε 2 ( ω ) ) ( ε 1 + ( d 1 ) ε 2 ( ω ) ) ( ε 1 ε 2 ( ω ) ) ( ε 0 + ( d 1 ) ε 2 ( ω ) ) d
d n = log ( S n ) log ( N n )
S n = 4 S n 1 n 2 and S 1 = 12
N n = 3 N n 1 n 2 and N 1 = 9
A n = 3 s 2 4 ( 1 + 3 k = 1 n 4 k 1 9 k )
s 1 s 2 = ( γ 1 1 ) ( γ 2 + d 1 ) ( γ 2 1 ) [ ( d 1 ) γ 1 + 1 ]

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