Abstract

We study one- and two-dimensional transmission of electromagnetic waves through a finite slab of a dielectric material with negative refraction. In the case when the dielectric slab possesses an intensity-dependent nonlinear response, we observe the nonlinearity-induced wave transmission through an opaque slab accompanied by the generation of spatiotemporal solitons. We solve this problem numerically, by employing the finite-difference time-domain simulations, for the parameters of microstructured materials with the negative refractive index in the microwave region, but our results can be useful for a design of nonlinear metamaterials with the left-handed properties in other frequency range.

© 2005 Optical Society of America

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References

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  1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
    [CrossRef] [PubMed]
  2. J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  3. P. Markos and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E 65, 036622–8 (2002).
    [CrossRef]
  4. P. Markos and C. M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401–4 (2002).
    [CrossRef]
  5. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [CrossRef] [PubMed]
  6. M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
    [CrossRef]
  7. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–4 (2003).
    [CrossRef] [PubMed]
  8. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Usp. Fiz. Nauk 92, 517–526 (1967) (in Russian) [English translation: Phys. Usp. 10, 509 (1968)].
    [CrossRef]
  9. 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]
  10. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, “Plasmon modes and negative refraction in metal nanowire composites,” Opt. Express 11, 735–745 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-735
    [CrossRef] [PubMed]
  11. A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401–4 (2003).
    [CrossRef] [PubMed]
  12. S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101(R) (2004).
  13. M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601–4 (2003).
    [CrossRef]
  14. M. W. Feise, I. V. Shadrivov, and Yu. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. 85, 1451–1453 (2004).
    [CrossRef]
  15. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112–165117 (2004).
    [CrossRef]
  16. I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Yu. S. Kivshar, “Defect modes and transmission properties of left-handed bandgap structures,” Phys. Rev. E 70, 046615–6 (2004).
    [CrossRef]
  17. M. Born and E.Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh ed. (Cambridge University Press, UK, 2002).
    [PubMed]
  18. A. A. Zharov and A. K. Kotov, “Nonlinear matching of electromagnetic waves with a plane plasma slab,” Fiz. Plazmy 10, 615–618 (1984).
  19. A. V. Kochetov and A. M. Feigin, “Bleaching of dense plasma by an intense TM wave,” Fiz. Plazmy 14, 716–726 (1988).
  20. For an overview of optical solitons, see Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

Appl. Phys. Lett. (2)

M. Bayindir, K. Aydin, E. Ozbay, P. Markos, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Yu. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. 85, 1451–1453 (2004).
[CrossRef]

Fiz. Plazmy (2)

A. A. Zharov and A. K. Kotov, “Nonlinear matching of electromagnetic waves with a plane plasma slab,” Fiz. Plazmy 10, 615–618 (1984).

A. V. Kochetov and A. M. Feigin, “Bleaching of dense plasma by an intense TM wave,” Fiz. Plazmy 14, 716–726 (1988).

IEEE Trans. Microw. Theory Tech. (1)

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

Opt. Express (1)

Phys. Rev. B (3)

P. Markos and C. M. Soukoulis, “Transmission studies of left-handed materials,” Phys. Rev. B 65, 033401–4 (2002).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112–165117 (2004).
[CrossRef]

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101(R) (2004).

Phys. Rev. E (3)

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601–4 (2003).
[CrossRef]

I. V. Shadrivov, N. A. Zharova, A. A. Zharov, and Yu. S. Kivshar, “Defect modes and transmission properties of left-handed bandgap structures,” Phys. Rev. E 70, 046615–6 (2004).
[CrossRef]

P. Markos and C. M. Soukoulis, “Numerical studies of left-handed materials and arrays of split ring resonators,” Phys. Rev. E 65, 036622–8 (2002).
[CrossRef]

Phys. Rev. Lett. (4)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401–4 (2003).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401–4 (2003).
[CrossRef] [PubMed]

Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Usp. Fiz. Nauk 92, 517–526 (1967) (in Russian) [English translation: Phys. Usp. 10, 509 (1968)].
[CrossRef]

Science (1)

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]

Other (2)

M. Born and E.Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh ed. (Cambridge University Press, UK, 2002).
[PubMed]

For an overview of optical solitons, see Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

Supplementary Material (2)

» Media 1: GIF (1250 KB)     
» Media 2: GIF (1828 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Reflection (solid red) and transmission (dashed green) coefficients for a slab of nonlinear metamaterial vs. the normalized incident field intensity in a stationary regime, for the case of (a) defocusing nonlinearity (α=-1), and (b) focusing nonlinearity (α=+1). Insets show real (solid blue) and imaginary (dashed red) parts of the magnetic permeability inside the slab of a composite material.

Fig. 2.
Fig. 2.

(a) Temporal evolution of the reflected (solid) and incident (dashed) wave intensity. (b,c) Spatial distribution of the magnetic and electric fields, respectively, at the end of the simulation domain; the metamaterial region is shaded. Left: low-amplitude (linear) regime, right: weakly nonlinear regime.

Fig. 3.
Fig. 3.

(a) Temporal evolution of the reflected (solid) and incident (dashed) wave intensity in the strongly nonlinear regime (i.e., for the overcritical amplitude of the incident wave). (b,c) Spatial distribution of the magnetic and electric fields, respectively, at the end of simulation domain; the metamaterial is shaded. Left: generation of an oscillating localized state at the surface, right: soliton generation in the overcritical regime. [Media 1]

Fig. 4.
Fig. 4.

Magnetic field distribution (in logarithmic scale) for the beam scattering by a metamaterial slab in low-intensity regime (top). Bottom – plot shows that the metamaterial is opaque (yellow) for the beam incident at 45 degrees from the left. Red color indicates the high field areas outside the slab. Coordinates are normalized on the free-space wavelength.

Fig. 5.
Fig. 5.

Magnetic field distribution (in logarithmic scale) for the beam scattering by a metamaterial slab in high-intensity regime (top). Bottom – plot shows the transparent left-handed domain (black) formed in initially opaque metamaterial (yellow) by the beam incident at 45 degrees. Red color indicates the high field areas outside the slab. Coordinates are normalized on the free-space wavelength. [Media 2]

Equations (9)

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

× E = 1 c B t ,
× B = 1 c E t + 4 π c j + 4 π × M ,
σ L w S d j d t + j = σ S d cell 2 E ,
M = n m 2 c π a 2 I R B B ,
L d I R d t = π a 2 c d H d t U R I R ,
C ( U ) d U d t = I R ,
C ( U ) = r 2 ε 4 d g ( 1 + α U 2 U c 2 ) ,
H = B 8 π 3 M .
μ eff ( H ) = 1 + F ω 2 ω 0 N L 2 ( H ) ω 2 ( 1 + F 3 ) + i Γ ω ,

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