Abstract

The current and field distribution in a 2D metamaterial consisting of resonant elements in a hexagonal arrangement are found assuming magnetic interaction between the elements. The dispersion equation of magnetoinductive (MI) waves is derived with the aid of the direct and reciprocal lattice familiar from solid state theory. A continuous model for the current variation in the elements is introduced leading to the familiar wave equation in the form of a second order differential equation. The current distributions are shown to exhibit a series of spatial resonances for rectangular, circular and hexagonal boundaries. The axial and radial components of the resulting magnetic field are compared with previously obtained experimental results on a Swiss Roll metamaterial with hexagonal boundaries. Experimental and theoretical results are also compared for the near field image of an object in the shape of the letter M followed by a more general discussion of imaging. It is concluded that a theoretical formulation based on the propagation of MI waves can correctly describe the experimental results.

© 2005 Optical Society of America

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References

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  1. R. E. Collin, Field Theory of Guided Waves (Oxford University Press, Oxford, 1991).
  2. E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  3. S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  4. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
    [CrossRef]
  5. J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  6. D. O. S. Melville and R. J. Blaikie,�?? Super-resolution imaging through a planar silver layer,�?? Opt. Express 13, 2127-2134 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2127.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-2127</a>
    [CrossRef] [PubMed]
  7. N. Fang, H. Lee, C. Sun, and X. Zhang, �??Sub-diffraction-limited optical imaging with a silver superlens,�?? Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  8. M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, �??Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging,�?? Science 291, 849-851 (2001).
    [CrossRef] [PubMed]
  9. M. C. K. Wiltshire, J. V. Hajnal, J. B. Pendry, D. J. Edwards, and C. J. Stevens, �??Metamaterial endoscope for magnetic field transfer: near field imaging with magnetic wires,�?? Opt. Express 11, 709-715 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-709.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-709</a>
    [CrossRef] [PubMed]
  10. M. J. Freire and R. Marques, �??Planar magnetoinductive lens for three-dimensional subwavelength imaging,�?? Appl. Phys. Lett. 86, 182505-1-3 (2005).
    [CrossRef]
  11. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, �??Magneto-inductive waveguide,�?? Electron. Lett. 38, 371-373 (2002).
    [CrossRef]
  12. E. Shamonina, V.A. Kalinin, K. H. Ringhofer, and L. Solymar, �??Magnetoinductive waves in one, two, and three dimensions,�?? J. Appl. Phys. 92: 6252-6261 (2002).
    [CrossRef]
  13. M. C. K. Wiltshire, E. Shamonina, I. R. Young, and L. Solymar, �??Dispersion characteristics of magnetoinductive waves: comparison between theory and experiment,�?? Electron. Lett. 39, 215-217 (2003).
    [CrossRef]
  14. M. C. K. Wiltshire, E. Shamonina, I. R. Young, and L. Solymar, �??Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of �??swiss rolls�??,�?? J. Appl. Phys. 95, 4488-4493 (2004).
    [CrossRef]
  15. R. Marques, F. Mesa , J. Martel, and F. Medina,�?? Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments,�?? IEEE Trans. Antennas Prop. 51, 2572-2581 (2003).
    [CrossRef]
  16. J. D. Baena, R. Marques, F. Medina, and J. Martel, �??Artificial magnetic metamaterial design by using spiral resonators,�?? Phys. Rev. B 69, 014402-1-5 (2004).
    [CrossRef]
  17. M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, �??Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring,�?? J. Appl. Phys. 95, 3778-3784 (2004).
    [CrossRef]
  18. F. W. Grover, Inductance Calculations: Working Formulas and Tables (Instrument Society of America, Research Triangle Park, N.C., 1981).
  19. M. C. K. Wiltshire, J. V. Hajnal, J. B. Pendry, and D. J. Edwards, �??RF field transmission through Swiss Rolls �?? an anisotropic magnetic metamaterial,�?? in Proceedings of the 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces, Santiago de Compostela, Spain, 9-11 March, 2004.
  20. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, �??Partial focusing of radiation by a slab of indefinite media,�?? Appl. Phys. Lett. 84, 2244-2246 (2004).
    [CrossRef]
  21. S. Maslovski, S. Tretyakov, and P. Alitalo, �??Near-field enhancement and imaging in double planar polariton-resonant structures,�?? J. Appl. Phys. 96, 1293-1300 (2004).
    [CrossRef]
  22. I.-H. Lin, C. Caloz, and T. Itoh, �??Near-field focusing by a nonuniform leaky-wave interface,�?? Microw. Opt. Techn. Lett. 44, 416-418 (2005).
    [CrossRef]
  23. R. Zengerle, �??Light-propagation in singly and doubly periodic planar wave-guides,�?? J. Mod. Opt. 34, 1589-1617 (1987).
    [CrossRef]
  24. P. A. Belov, C. R. Simovski, and P. Ikonen, �??Canalization of subwavelength images by electromagnetic crystals,�?? Phys. Rev. 71, 193105-1-4 (2005).
    [CrossRef]
  25. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, �??Terahertz magnetic response from artificial materials,�?? Science 303, 1494-1496 (2004).
    [CrossRef] [PubMed]
  26. S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, �??Magnetic response of metamaterials at 100 terahertz,�?? Science 306, 1351-1353 (2004).
    [CrossRef] [PubMed]

Appl. Phys. Lett.

M. J. Freire and R. Marques, �??Planar magnetoinductive lens for three-dimensional subwavelength imaging,�?? Appl. Phys. Lett. 86, 182505-1-3 (2005).
[CrossRef]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, �??Partial focusing of radiation by a slab of indefinite media,�?? Appl. Phys. Lett. 84, 2244-2246 (2004).
[CrossRef]

Electron. Lett.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, �??Magneto-inductive waveguide,�?? Electron. Lett. 38, 371-373 (2002).
[CrossRef]

M. C. K. Wiltshire, E. Shamonina, I. R. Young, and L. Solymar, �??Dispersion characteristics of magnetoinductive waves: comparison between theory and experiment,�?? Electron. Lett. 39, 215-217 (2003).
[CrossRef]

IEEE Trans. Antennas Prop.

R. Marques, F. Mesa , J. Martel, and F. Medina,�?? Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments,�?? IEEE Trans. Antennas Prop. 51, 2572-2581 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. Appl. Phys.

M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, �??Properties of a metamaterial element: analytical solutions and numerical simulations for a singly split double ring,�?? J. Appl. Phys. 95, 3778-3784 (2004).
[CrossRef]

M. C. K. Wiltshire, E. Shamonina, I. R. Young, and L. Solymar, �??Experimental and theoretical study of magneto-inductive waves supported by one-dimensional arrays of �??swiss rolls�??,�?? J. Appl. Phys. 95, 4488-4493 (2004).
[CrossRef]

E. Shamonina, V.A. Kalinin, K. H. Ringhofer, and L. Solymar, �??Magnetoinductive waves in one, two, and three dimensions,�?? J. Appl. Phys. 92: 6252-6261 (2002).
[CrossRef]

S. Maslovski, S. Tretyakov, and P. Alitalo, �??Near-field enhancement and imaging in double planar polariton-resonant structures,�?? J. Appl. Phys. 96, 1293-1300 (2004).
[CrossRef]

J. Mod. Opt.

R. Zengerle, �??Light-propagation in singly and doubly periodic planar wave-guides,�?? J. Mod. Opt. 34, 1589-1617 (1987).
[CrossRef]

Microw. Opt. Techn. Lett.

I.-H. Lin, C. Caloz, and T. Itoh, �??Near-field focusing by a nonuniform leaky-wave interface,�?? Microw. Opt. Techn. Lett. 44, 416-418 (2005).
[CrossRef]

Opt. Express

Phys. Rev.

P. A. Belov, C. R. Simovski, and P. Ikonen, �??Canalization of subwavelength images by electromagnetic crystals,�?? Phys. Rev. 71, 193105-1-4 (2005).
[CrossRef]

Phys. Rev. B

J. D. Baena, R. Marques, F. Medina, and J. Martel, �??Artificial magnetic metamaterial design by using spiral resonators,�?? Phys. Rev. B 69, 014402-1-5 (2004).
[CrossRef]

Phys. Rev. Lett

S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett.

E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Science

N. Fang, H. Lee, C. Sun, and X. Zhang, �??Sub-diffraction-limited optical imaging with a silver superlens,�?? Science 308, 534-537 (2005).
[CrossRef] [PubMed]

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, �??Microstructured magnetic materials for RF flux guides in Magnetic Resonance Imaging,�?? Science 291, 849-851 (2001).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, �??Terahertz magnetic response from artificial materials,�?? Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, �??Magnetic response of metamaterials at 100 terahertz,�?? Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

Other

R. E. Collin, Field Theory of Guided Waves (Oxford University Press, Oxford, 1991).

F. W. Grover, Inductance Calculations: Working Formulas and Tables (Instrument Society of America, Research Triangle Park, N.C., 1981).

M. C. K. Wiltshire, J. V. Hajnal, J. B. Pendry, and D. J. Edwards, �??RF field transmission through Swiss Rolls �?? an anisotropic magnetic metamaterial,�?? in Proceedings of the 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces, Santiago de Compostela, Spain, 9-11 March, 2004.

Supplementary Material (4)

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Figures (7)

Fig. 1.
Fig. 1.

(a) Lattice of resonant elements with hexagonal arrangement. (b) Direct vectors of the hexagonal lattice (d 1,d 2) and vectors of the reciprocal lattice (b 1,b 2).

Fig. 2.
Fig. 2.

2D dispersion equation of MI waves for a hexagonal metamaterial showing the contour lines of frequency as a function of kxd and kyd. The boundaries of the first Brillouin zone are shown by bold lines.

Fig. 3.
Fig. 3.

Current distribution for circular boundary conditions at various frequencies (a–c). (2.05 MB) Movie of current distribution in the frequency range from 1.5 to 0.7 ω 0 (72 frames).

Fig. 4.
Fig. 4.

Current distribution for rectangular boundary conditions at various frequencies (a–d). (2.25 MB). Movie of current distribution in the frequency range from 1.5 to 0.7 ω 0 (77 frames).

Fig. 5.
Fig. 5.

Magnetic field distribution at four resonant frequencies: normal component (a–d) and tangential component (e–h). (2.1 MB, 2.55MB) Corresponding movies in the frequency range from 1.5 to 0.7 ω 0 (86 frames).

Fig. 6.
Fig. 6.

Resonant frequencies of a hexagonal metamaterial. Experimental values are taken from Ref.[19].

Fig. 7.
Fig. 7.

The magnetic field pattern (normal component) at ω= 0.98ω 0 (a) and ω=1.01ω0 (b).

Equations (19)

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Z I n , m + jωM ( I n , m 1 + I n , m + 1 + I n 1 , m + I n + 1 , m + I n 1 , m + 1 + I n + 1 , m 1 ) = 0
r n , m = n d 1 + m d 2
d 1 = i x d , d 2 = ( i x cos 60 ° + i y sin 60 ° ) d
d 1 b 1 = 1 , d 1 b 2 = 0 ,
d 2 b 1 = 0 , d 2 b 2 = 1 .
I n , m = I 0,0 exp ( j k r n , m ) .
k = 2 π ( f 1 b 1 + f 2 b 2 )
ω ω 0 = { 1 + κ [ cos ( 2 π f 1 ) + cos ( 2 π f 2 ) + cos ( 2 π f 1 2 π f 2 ) ] } 1 2
I = Z 1 V .
I ( ν + Δ ν , μ + Δ μ ) = { 1 + Δ ν ν + Δ μ μ +
+ 1 2 [ ( Δ ν ) 2 2 ν 2 + 2 Δ ν Δ μ 2 ν μ + ( Δ μ ) 2 2 μ 2 ] } I ( ν , μ )
2 I ν 2 2 I ν μ + 2 I μ 2 + 1 2 d 2 ( 6 + Z jωM ) I = 0
2 ν 2 2 ν μ + 2 μ 2 = 3 4 ( 2 x 2 + 2 y 2 )
2 I x 2 + 2 I y 2 + k 2 I = 0
k 2 = 4 d 2 [ 1 + 1 3 κ ( 1 ω 2 ω 0 2 ) ]
I = I 0 sin ( k x x ) sin ( k y y )
I = I 0 J 0 ( kr )
kR = ρ i
k x L x = p x π and k y L y = p y π

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