Abstract

We study the coupling between negative diffraction and direct dispersion in a nonlinear ring cavity containing slabs of Kerr nonlinear right-handed and left-handed materials. Within the mean field approximation, we show that a portion of the homogeneous response curve is affected by a three-dimensional modulational instability. We show numerically that the light distribution evolves through a sequence of three-dimensional dissipative structures with different lattice symmetry. These structures are unstable with respect to the upswitching process, leading to a premature transition to the upper branch in the homogeneous hysteresis cycle.

© 2006 Optical Society of America

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  1. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer-Verlag, Berlin, 2002).
  2. K. Staliunas and J. V. Sanchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer-Verlag, Berlin, 2003).
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    [CrossRef]
  5. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).
  6. M. Tlidi, M. Haelterman, and P. Mandel, "Three-dimensional structures in diffractive and dispersive nonlinear ring cavities," Europhys. Lett. 42, 505-509 (1998).
    [CrossRef]
  7. K. Staliunas, "Three-Dimensional Turing Structures and Spatial Solitons in Optical Parametric Oscillators," Phys. Rev. Lett. 81, 81-84 (1998).
    [CrossRef]
  8. M. Tlidi and P. Mandel, "Three-Dimensional Optical Crystals and Localized Structures in Cavity Second Harmonic Generation," Phys. Rev. Lett. 83, 4995-4998 (1999).
    [CrossRef]
  9. S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003).
    [CrossRef]
  10. M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator," Phys. Rev. Lett. 93, 203901 (2004).
    [CrossRef] [PubMed]
  11. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  14. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
    [CrossRef]
  15. J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 39663969 (2000).
    [CrossRef]
  16. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear Properties of Left-Handed Metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  17. N. Lazarides and G. P. Tsironis, "Coupled nonlinear Schr¨odinger field equations for electromagnetic wave propagation in nonlinear left-handed materials, " Phys. Rev. E 71, 036614 (2005).
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    [CrossRef]
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    [CrossRef]
  22. N. Engheta, "An Idea for Thin Subwavelength Cavity Resonators Using Metamaterials With Negative Permittivity and Permeability," IEEE Ant. Wireless Prop. Lett. 1, 10-13 (2002).
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2006 (2)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

M. Marklund, P. K. Shukla, and L. Stenflo, "Ultrashort solitons and kinetic effects in nonlinear metamaterials," Phys. Rev. E 73, 037601 (2006).
[CrossRef]

2005 (2)

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

N. Lazarides and G. P. Tsironis, "Coupled nonlinear Schr¨odinger field equations for electromagnetic wave propagation in nonlinear left-handed materials, " Phys. Rev. E 71, 036614 (2005).
[CrossRef]

2004 (4)

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 (2004).
[CrossRef]

M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator," Phys. Rev. Lett. 93, 203901 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

P. Mandel and M. Tlidi, "Transverse dynamics in cavity nonlinear optics (2000-2003)," J. Opt. B: Quantum Semiclass. Opt. 6, R60-R75 (2004).
[CrossRef]

2003 (2)

S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear Properties of Left-Handed Metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

2002 (1)

N. Engheta, "An Idea for Thin Subwavelength Cavity Resonators Using Metamaterials With Negative Permittivity and Permeability," IEEE Ant. Wireless Prop. Lett. 1, 10-13 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 39663969 (2000).
[CrossRef]

1999 (1)

M. Tlidi and P. Mandel, "Three-Dimensional Optical Crystals and Localized Structures in Cavity Second Harmonic Generation," Phys. Rev. Lett. 83, 4995-4998 (1999).
[CrossRef]

1998 (2)

M. Tlidi, M. Haelterman, and P. Mandel, "Three-dimensional structures in diffractive and dispersive nonlinear ring cavities," Europhys. Lett. 42, 505-509 (1998).
[CrossRef]

K. Staliunas, "Three-Dimensional Turing Structures and Spatial Solitons in Optical Parametric Oscillators," Phys. Rev. Lett. 81, 81-84 (1998).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Agranovich, V. M.

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 (2004).
[CrossRef]

Akozbek, N.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Baughman, R. H.

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 (2004).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Brambilla, M.

M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator," Phys. Rev. Lett. 93, 203901 (2004).
[CrossRef] [PubMed]

Columbo, L.

M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator," Phys. Rev. Lett. 93, 203901 (2004).
[CrossRef] [PubMed]

D’Aguanno, G.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Engheta, N.

N. Engheta, "An Idea for Thin Subwavelength Cavity Resonators Using Metamaterials With Negative Permittivity and Permeability," IEEE Ant. Wireless Prop. Lett. 1, 10-13 (2002).
[CrossRef]

Fedorov, S. V.

S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003).
[CrossRef]

Firsov, A. A.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

Geim, A. K.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

Gleeson, H. F.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

Grigorenko, A. N.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

Haelterman, M.

M. Tlidi, M. Haelterman, and P. Mandel, "Three-dimensional structures in diffractive and dispersive nonlinear ring cavities," Europhys. Lett. 42, 505-509 (1998).
[CrossRef]

Khrushchev, I. Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

Kivshar, Y. S.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear Properties of Left-Handed Metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Lazarides, N.

N. Lazarides and G. P. Tsironis, "Coupled nonlinear Schr¨odinger field equations for electromagnetic wave propagation in nonlinear left-handed materials, " Phys. Rev. E 71, 036614 (2005).
[CrossRef]

Maggipinto, T.

M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator," Phys. Rev. Lett. 93, 203901 (2004).
[CrossRef] [PubMed]

Mandel, P.

P. Mandel and M. Tlidi, "Transverse dynamics in cavity nonlinear optics (2000-2003)," J. Opt. B: Quantum Semiclass. Opt. 6, R60-R75 (2004).
[CrossRef]

M. Tlidi and P. Mandel, "Three-Dimensional Optical Crystals and Localized Structures in Cavity Second Harmonic Generation," Phys. Rev. Lett. 83, 4995-4998 (1999).
[CrossRef]

M. Tlidi, M. Haelterman, and P. Mandel, "Three-dimensional structures in diffractive and dispersive nonlinear ring cavities," Europhys. Lett. 42, 505-509 (1998).
[CrossRef]

Marklund, M.

M. Marklund, P. K. Shukla, and L. Stenflo, "Ultrashort solitons and kinetic effects in nonlinear metamaterials," Phys. Rev. E 73, 037601 (2006).
[CrossRef]

Mattiucci, N.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Patera, G.

M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator," Phys. Rev. Lett. 93, 203901 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 39663969 (2000).
[CrossRef]

Petrovic, J.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

Poliakov, E. Y.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Rosanov, N. N.

S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003).
[CrossRef]

Scalora, M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shadrivov, I. V.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear Properties of Left-Handed Metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Shatsev, A. N.

S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shen, Y. R.

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 (2004).
[CrossRef]

Shukla, P. K.

M. Marklund, P. K. Shukla, and L. Stenflo, "Ultrashort solitons and kinetic effects in nonlinear metamaterials," Phys. Rev. E 73, 037601 (2006).
[CrossRef]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Staliunas, K.

K. Staliunas, "Three-Dimensional Turing Structures and Spatial Solitons in Optical Parametric Oscillators," Phys. Rev. Lett. 81, 81-84 (1998).
[CrossRef]

Stenflo, L.

M. Marklund, P. K. Shukla, and L. Stenflo, "Ultrashort solitons and kinetic effects in nonlinear metamaterials," Phys. Rev. E 73, 037601 (2006).
[CrossRef]

Syrchin, M. S.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Tlidi, M.

P. Mandel and M. Tlidi, "Transverse dynamics in cavity nonlinear optics (2000-2003)," J. Opt. B: Quantum Semiclass. Opt. 6, R60-R75 (2004).
[CrossRef]

M. Tlidi and P. Mandel, "Three-Dimensional Optical Crystals and Localized Structures in Cavity Second Harmonic Generation," Phys. Rev. Lett. 83, 4995-4998 (1999).
[CrossRef]

M. Tlidi, M. Haelterman, and P. Mandel, "Three-dimensional structures in diffractive and dispersive nonlinear ring cavities," Europhys. Lett. 42, 505-509 (1998).
[CrossRef]

Tsironis, G. P.

N. Lazarides and G. P. Tsironis, "Coupled nonlinear Schr¨odinger field equations for electromagnetic wave propagation in nonlinear left-handed materials, " Phys. Rev. E 71, 036614 (2005).
[CrossRef]

Veretenov, N. A.

S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003).
[CrossRef]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vladimirov, A. G.

S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Zakhidov, A. A.

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 (2004).
[CrossRef]

Zhang, Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

Zharov, A. A.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear Properties of Left-Handed Metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Zheltikov, A. M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Europhys. Lett. (1)

M. Tlidi, M. Haelterman, and P. Mandel, "Three-dimensional structures in diffractive and dispersive nonlinear ring cavities," Europhys. Lett. 42, 505-509 (1998).
[CrossRef]

IEEE Ant. Wireless Prop. Lett. (1)

N. Engheta, "An Idea for Thin Subwavelength Cavity Resonators Using Metamaterials With Negative Permittivity and Permeability," IEEE Ant. Wireless Prop. Lett. 1, 10-13 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

P. Mandel and M. Tlidi, "Transverse dynamics in cavity nonlinear optics (2000-2003)," J. Opt. B: Quantum Semiclass. Opt. 6, R60-R75 (2004).
[CrossRef]

Nature (1)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006).
[CrossRef]

Phys. Rev. B (1)

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 (2004).
[CrossRef]

Phys. Rev. E (2)

N. Lazarides and G. P. Tsironis, "Coupled nonlinear Schr¨odinger field equations for electromagnetic wave propagation in nonlinear left-handed materials, " Phys. Rev. E 71, 036614 (2005).
[CrossRef]

M. Marklund, P. K. Shukla, and L. Stenflo, "Ultrashort solitons and kinetic effects in nonlinear metamaterials," Phys. Rev. E 73, 037601 (2006).
[CrossRef]

Phys. Rev. Lett. (6)

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

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Supplementary Material (2)

» Media 1: MPG (8334 KB)     
» Media 2: MPG (2277 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

A driven ring resonator filled with left- (LHM) and right-handed (RHM) materials.

Fig. 2.
Fig. 2.

Bistability curve of the resonator. Modulationally stable regions are denoted with blue solid lines, unstable regions with red dashed lines. Parameters are ��=β=-2 and Δ=10.

Fig. 3.
Fig. 3.

At the bifurcation point, the unstable modes form (a) an ellipsoid or (b) a hyperboloid in the Fourier space (k x,k y,k τ). Parameters are ��=β=-2 and Δ=10.

Fig. 4.
Fig. 4.

(2.2 MB) Movie of the 3D structures observed in the cavity (8.1 MB version). (a)–(d) Representative frames of the movie at times T=100, T=215, T=236 and 245. Parameters are ��=β=-2 and Δ=10.

Equations (4)

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A z = i γ R , L A 2 A + i [ 1 2 k R , L ( 2 x 2 + 2 y 2 ) + β R , L 2 τ 2 ] A ,
A m + 1 ( 0 , x , y , τ ) = ρ A m ( L , x , y , τ ) exp ( i Φ ) + θ A i .
A T = ( 1 + i Δ ) A + 𝓔 + i Γ A 2 A + i [ 𝓓 ( 2 x 2 + 2 y 2 ) + β 2 τ 2 ] A .
𝓓 = 𝓕 π ( l R 2 k R + l R 2 k L ) , β = 𝓕 π ( β R l R + β L l L ) .

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