Abstract

Kerr-type left-handed metamaterial (LHM) slab is proved to have an effect of focusing paraxial Gaussian beams and changing their waist radius, as conventional lens can do. The expressions for the focusing distance and the spot radius at the focal point are derived by the variational approach. We show that the incident Gaussian beams can be compressed or expanded by a single Kerr LHM slab, according to the sign of the Kerr nonlinearity and the divergence of the incident beam. Especially, it is demonstrated the focusing properties are significantly tuned by the slab thickness, the beam power and the divergence of the incident Gaussian beam.

© 2008 Optical Society of America

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  1. J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  2. J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-44 (2004).
    [CrossRef]
  3. N. Garcia and M. Nieto-Vesperinas, "Left-handed materials do not make a perfect lens" Phys. Rev. Lett. 88, 207403 (2002).
    [CrossRef] [PubMed]
  4. X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (2003).
    [CrossRef]
  5. D. Maystre and S. Enoch, "Perfect lenses made with left-handed materials: Alice’s mirror?," J. Opt. Soc. Am. A 21, 122-131 (2004).
    [CrossRef]
  6. E. Schonbrun, T. Yamashita, W. Park, and C. J. Summers, "Negative-index imaging by an index-matched photonic crystal slab," Phys. Rev. B 73, 195117 (2006).
    [CrossRef]
  7. J. J. Chen, T. M. Grzegorczyk, B. Wu, and J. A. Kong, "Imaging properties of finite-size left-handed material slabs," Phys. Rev. E. 74, 046615 (2006).
    [CrossRef]
  8. L. Zhao and T. J. Cui, "Super-resolution imaging of dielectric objects using a slab of left-handed material," Appl. Phys. Lett. 89, 141904 (2006).
    [CrossRef]
  9. M. W. Feise and Y. S. Kivshar, "Sub-wavelength imaging with a left-handed material flat lens," Phys. Lett. A 334, 326 (2005).
    [CrossRef]
  10. J. J. Chen, T. M. Grzegorczyk, B.-I. Wu, and J. A. Kong, "Limitation of FDTD in simulation of a perfect lens imaging system," Opt. Express 13, 108409 (2005).
    [CrossRef] [PubMed]
  11. A. N. Lagarkov and V. N. Kissel, "Near-Perfect Imaging in a Focusing System Based on a Left-Handed-Material Plate," Phys. Rev. Lett. 92, 077401(2004).
    [CrossRef] [PubMed]
  12. V. N. Kissel and A. N. Lagarkov, "Superresolution in left-handed composite structures: From homogenization to a detailed electrodynamic description," Phys. Rev. B 72, 085111 (2005).
    [CrossRef]
  13. H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
    [CrossRef]
  14. R. Ziolkowski, "Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs," Opt. Express 11, 662-681 (2003).
    [CrossRef] [PubMed]
  15. J. Lu and S. He, "Numerical study of a Gaussian beam propagating in media with negative permittivity and permeability by using a bidirectional beam propagation method," Microwave Opt. Technol. Lett. 37, 292-296 (2003).
    [CrossRef]
  16. J. A. Kong, B.-I. Wu, and Y. Zhang, "A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
    [CrossRef]
  17. A. Husakou and J. Herrmann, "Superfocusing of light below the diffraction limit by photonic crystals with negative refraction," Opt. Express 12, 6491-6497 (2004).
  18. P. P. Banerjee, and G. Nehmetallah, "Linear and nonlinear propagation in negative index materials," J. Opt. Soc. Am. B. 23, 2348-2355 (2006).
    [CrossRef]
  19. V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
    [CrossRef]
  20. V. Yannopapas and N. V. Vitanov, "Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength," Phys. Rev. B 74, 193304 (2006).
    [CrossRef]
  21. V. Yannopapas, "Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices," Phys. Stat. Sol.(RRL) 1, 208-210 (2007).
  22. V. Yannopapas, "Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies," Appl. Phys. A 87, 259-264 (2007).
    [CrossRef]
  23. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear Properties of Left-Handed Metamaterials," Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  24. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
    [CrossRef]
  25. P. P. Banerjee and G. Nehmetallah, "Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials," J. Opt. Soc. Am. B 24, A69-A76 (2007).
    [CrossRef]
  26. 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]
  27. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
    [CrossRef] [PubMed]
  28. I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S86-S72 (2005).
    [CrossRef]
  29. I. Kourakis and P. K. Shukla, "Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials," Phys. Rev. E 72, 016626 (2005).
    [CrossRef]
  30. S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
    [CrossRef]
  31. S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, "Role of the anomalous self-steepening effect in modulation instability in negative-index material," Opt. Express 14, 1568-1575 (2006).
    [CrossRef] [PubMed]
  32. S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
    [CrossRef]
  33. D. Anderson, M. Bonneal, and M. Lisak, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
    [CrossRef]

2007 (5)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

V. Yannopapas, "Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices," Phys. Stat. Sol.(RRL) 1, 208-210 (2007).

V. Yannopapas, "Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies," Appl. Phys. A 87, 259-264 (2007).
[CrossRef]

P. P. Banerjee and G. Nehmetallah, "Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials," J. Opt. Soc. Am. B 24, A69-A76 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
[CrossRef]

2006 (8)

S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, "Role of the anomalous self-steepening effect in modulation instability in negative-index material," Opt. Express 14, 1568-1575 (2006).
[CrossRef] [PubMed]

V. Yannopapas and N. V. Vitanov, "Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength," Phys. Rev. B 74, 193304 (2006).
[CrossRef]

E. Schonbrun, T. Yamashita, W. Park, and C. J. Summers, "Negative-index imaging by an index-matched photonic crystal slab," Phys. Rev. B 73, 195117 (2006).
[CrossRef]

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J. A. Kong, "Imaging properties of finite-size left-handed material slabs," Phys. Rev. E. 74, 046615 (2006).
[CrossRef]

L. Zhao and T. J. Cui, "Super-resolution imaging of dielectric objects using a slab of left-handed material," Appl. Phys. Lett. 89, 141904 (2006).
[CrossRef]

H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

P. P. Banerjee, and G. Nehmetallah, "Linear and nonlinear propagation in negative index materials," J. Opt. Soc. Am. B. 23, 2348-2355 (2006).
[CrossRef]

2005 (6)

V. N. Kissel and A. N. Lagarkov, "Superresolution in left-handed composite structures: From homogenization to a detailed electrodynamic description," Phys. Rev. B 72, 085111 (2005).
[CrossRef]

M. W. Feise and Y. S. Kivshar, "Sub-wavelength imaging with a left-handed material flat lens," Phys. Lett. A 334, 326 (2005).
[CrossRef]

J. J. Chen, T. M. Grzegorczyk, B.-I. Wu, and J. A. Kong, "Limitation of FDTD in simulation of a perfect lens imaging system," Opt. Express 13, 108409 (2005).
[CrossRef] [PubMed]

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

I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S86-S72 (2005).
[CrossRef]

I. Kourakis and P. K. Shukla, "Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials," Phys. Rev. E 72, 016626 (2005).
[CrossRef]

2004 (6)

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]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

A. N. Lagarkov and V. N. Kissel, "Near-Perfect Imaging in a Focusing System Based on a Left-Handed-Material Plate," Phys. Rev. Lett. 92, 077401(2004).
[CrossRef] [PubMed]

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-44 (2004).
[CrossRef]

D. Maystre and S. Enoch, "Perfect lenses made with left-handed materials: Alice’s mirror?," J. Opt. Soc. Am. A 21, 122-131 (2004).
[CrossRef]

A. Husakou and J. Herrmann, "Superfocusing of light below the diffraction limit by photonic crystals with negative refraction," Opt. Express 12, 6491-6497 (2004).

2003 (4)

R. Ziolkowski, "Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs," Opt. Express 11, 662-681 (2003).
[CrossRef] [PubMed]

J. Lu and S. He, "Numerical study of a Gaussian beam propagating in media with negative permittivity and permeability by using a bidirectional beam propagation method," Microwave Opt. Technol. Lett. 37, 292-296 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (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 (2)

N. Garcia and M. Nieto-Vesperinas, "Left-handed materials do not make a perfect lens" Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

J. A. Kong, B.-I. Wu, and Y. Zhang, "A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

2000 (1)

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1979 (1)

D. Anderson, M. Bonneal, and M. Lisak, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
[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 Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Anderson, D.

D. Anderson, M. Bonneal, and M. Lisak, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
[CrossRef]

Banerjee, P. P.

P. P. Banerjee and G. Nehmetallah, "Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials," J. Opt. Soc. Am. B 24, A69-A76 (2007).
[CrossRef]

P. P. Banerjee, and G. Nehmetallah, "Linear and nonlinear propagation in negative index materials," J. Opt. Soc. Am. B. 23, 2348-2355 (2006).
[CrossRef]

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 Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Boardman, A. D.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Bonneal, M.

D. Anderson, M. Bonneal, and M. Lisak, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
[CrossRef]

Chen, J. J.

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J. A. Kong, "Imaging properties of finite-size left-handed material slabs," Phys. Rev. E. 74, 046615 (2006).
[CrossRef]

J. J. Chen, T. M. Grzegorczyk, B.-I. Wu, and J. A. Kong, "Limitation of FDTD in simulation of a perfect lens imaging system," Opt. Express 13, 108409 (2005).
[CrossRef] [PubMed]

Cui, T. J.

L. Zhao and T. J. Cui, "Super-resolution imaging of dielectric objects using a slab of left-handed material," Appl. Phys. Lett. 89, 141904 (2006).
[CrossRef]

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 Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Dai, X.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Egan, P.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Enoch, S.

Fan, D.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, "Role of the anomalous self-steepening effect in modulation instability in negative-index material," Opt. Express 14, 1568-1575 (2006).
[CrossRef] [PubMed]

Fan, D. Y.

S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Feise, M. W.

M. W. Feise and Y. S. Kivshar, "Sub-wavelength imaging with a left-handed material flat lens," Phys. Lett. A 334, 326 (2005).
[CrossRef]

Fu, X.

Fu, X. Q.

S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, "Left-handed materials do not make a perfect lens" Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

Grzegorczyk, T. M.

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J. A. Kong, "Imaging properties of finite-size left-handed material slabs," Phys. Rev. E. 74, 046615 (2006).
[CrossRef]

J. J. Chen, T. M. Grzegorczyk, B.-I. Wu, and J. A. Kong, "Limitation of FDTD in simulation of a perfect lens imaging system," Opt. Express 13, 108409 (2005).
[CrossRef] [PubMed]

He, S.

J. Lu and S. He, "Numerical study of a Gaussian beam propagating in media with negative permittivity and permeability by using a bidirectional beam propagation method," Microwave Opt. Technol. Lett. 37, 292-296 (2003).
[CrossRef]

Herrmann, J.

Hu, W.

H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

Hu, Y.

Husakou, A.

Kissel, V. N.

V. N. Kissel and A. N. Lagarkov, "Superresolution in left-handed composite structures: From homogenization to a detailed electrodynamic description," Phys. Rev. B 72, 085111 (2005).
[CrossRef]

A. N. Lagarkov and V. N. Kissel, "Near-Perfect Imaging in a Focusing System Based on a Left-Handed-Material Plate," Phys. Rev. Lett. 92, 077401(2004).
[CrossRef] [PubMed]

Kivshar, Y. S.

M. W. Feise and Y. S. Kivshar, "Sub-wavelength imaging with a left-handed material flat lens," Phys. Lett. A 334, 326 (2005).
[CrossRef]

I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S86-S72 (2005).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[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]

Kong, J. A.

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J. A. Kong, "Imaging properties of finite-size left-handed material slabs," Phys. Rev. E. 74, 046615 (2006).
[CrossRef]

J. J. Chen, T. M. Grzegorczyk, B.-I. Wu, and J. A. Kong, "Limitation of FDTD in simulation of a perfect lens imaging system," Opt. Express 13, 108409 (2005).
[CrossRef] [PubMed]

J. A. Kong, B.-I. Wu, and Y. Zhang, "A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

Kourakis, I.

I. Kourakis and P. K. Shukla, "Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials," Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Lagarkov, A. N.

V. N. Kissel and A. N. Lagarkov, "Superresolution in left-handed composite structures: From homogenization to a detailed electrodynamic description," Phys. Rev. B 72, 085111 (2005).
[CrossRef]

A. N. Lagarkov and V. N. Kissel, "Near-Perfect Imaging in a Focusing System Based on a Left-Handed-Material Plate," Phys. Rev. Lett. 92, 077401(2004).
[CrossRef] [PubMed]

Li, F.

H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

Lisak, M.

D. Anderson, M. Bonneal, and M. Lisak, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
[CrossRef]

Lu, J.

J. Lu and S. He, "Numerical study of a Gaussian beam propagating in media with negative permittivity and permeability by using a bidirectional beam propagation method," Microwave Opt. Technol. Lett. 37, 292-296 (2003).
[CrossRef]

Luo, H. L.

H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (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 Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Maystre, D.

Nehmetallah, G.

P. P. Banerjee and G. Nehmetallah, "Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials," J. Opt. Soc. Am. B 24, A69-A76 (2007).
[CrossRef]

P. P. Banerjee, and G. Nehmetallah, "Linear and nonlinear propagation in negative index materials," J. Opt. Soc. Am. B. 23, 2348-2355 (2006).
[CrossRef]

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, "Left-handed materials do not make a perfect lens" Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

Ong, C. K.

X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (2003).
[CrossRef]

Park, W.

E. Schonbrun, T. Yamashita, W. Park, and C. J. Summers, "Negative-index imaging by an index-matched photonic crystal slab," Phys. Rev. B 73, 195117 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-44 (2004).
[CrossRef]

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

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 Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Rao, X. S.

X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (2003).
[CrossRef]

Ren, Z. Z.

H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[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 Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Schonbrun, E.

E. Schonbrun, T. Yamashita, W. Park, and C. J. Summers, "Negative-index imaging by an index-matched photonic crystal slab," Phys. Rev. B 73, 195117 (2006).
[CrossRef]

Shadrivov, I. V.

I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S86-S72 (2005).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[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]

Shalaev, V. M.

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

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]

Shu, W. X.

H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

Shukla, P. K.

I. Kourakis and P. K. Shukla, "Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials," Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Smith, D. R.

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-44 (2004).
[CrossRef]

Su, W.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, "Role of the anomalous self-steepening effect in modulation instability in negative-index material," Opt. Express 14, 1568-1575 (2006).
[CrossRef] [PubMed]

Su, W. H.

S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Summers, C. J.

E. Schonbrun, T. Yamashita, W. Park, and C. J. Summers, "Negative-index imaging by an index-matched photonic crystal slab," Phys. Rev. B 73, 195117 (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 Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Tang, Z.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Vitanov, N. V.

V. Yannopapas and N. V. Vitanov, "Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength," Phys. Rev. B 74, 193304 (2006).
[CrossRef]

Wang, Y. W.

S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Wen, S.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, "Role of the anomalous self-steepening effect in modulation instability in negative-index material," Opt. Express 14, 1568-1575 (2006).
[CrossRef] [PubMed]

Wen, S. C.

S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Wu, B.

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J. A. Kong, "Imaging properties of finite-size left-handed material slabs," Phys. Rev. E. 74, 046615 (2006).
[CrossRef]

Wu, B.-I.

J. J. Chen, T. M. Grzegorczyk, B.-I. Wu, and J. A. Kong, "Limitation of FDTD in simulation of a perfect lens imaging system," Opt. Express 13, 108409 (2005).
[CrossRef] [PubMed]

J. A. Kong, B.-I. Wu, and Y. Zhang, "A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

Xiang, Y.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, "Role of the anomalous self-steepening effect in modulation instability in negative-index material," Opt. Express 14, 1568-1575 (2006).
[CrossRef] [PubMed]

Xiang, Y. J.

S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Yamashita, T.

E. Schonbrun, T. Yamashita, W. Park, and C. J. Summers, "Negative-index imaging by an index-matched photonic crystal slab," Phys. Rev. B 73, 195117 (2006).
[CrossRef]

Yannopapas, V.

V. Yannopapas, "Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies," Appl. Phys. A 87, 259-264 (2007).
[CrossRef]

V. Yannopapas, "Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices," Phys. Stat. Sol.(RRL) 1, 208-210 (2007).

V. Yannopapas and N. V. Vitanov, "Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength," Phys. Rev. B 74, 193304 (2006).
[CrossRef]

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.

J. A. Kong, B.-I. Wu, and Y. Zhang, "A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

Zhao, L.

L. Zhao and T. J. Cui, "Super-resolution imaging of dielectric objects using a slab of left-handed material," Appl. Phys. Lett. 89, 141904 (2006).
[CrossRef]

Zharov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[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]

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 Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Ziolkowski, R.

Appl. Phys. A (1)

V. Yannopapas, "Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies," Appl. Phys. A 87, 259-264 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

L. Zhao and T. J. Cui, "Super-resolution imaging of dielectric objects using a slab of left-handed material," Appl. Phys. Lett. 89, 141904 (2006).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

I. V. Shadrivov and Y. S. Kivshar, "Spatial solitons in nonlinear left-handed metamaterials," J. Opt. A: Pure Appl. Opt. 7, S86-S72 (2005).
[CrossRef]

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

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

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

P. P. Banerjee, and G. Nehmetallah, "Linear and nonlinear propagation in negative index materials," J. Opt. Soc. Am. B. 23, 2348-2355 (2006).
[CrossRef]

Microwave Opt. Technol. Lett. (2)

J. Lu and S. He, "Numerical study of a Gaussian beam propagating in media with negative permittivity and permeability by using a bidirectional beam propagation method," Microwave Opt. Technol. Lett. 37, 292-296 (2003).
[CrossRef]

J. A. Kong, B.-I. Wu, and Y. Zhang, "A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability," Microwave Opt. Technol. Lett. 33, 136-139 (2002).
[CrossRef]

Nat. Photonics (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Commun. (1)

H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, "Focusing and phase compensation of paraxial beams by a left-handed material slab," Opt. Commun. 266, 327-331 (2006).
[CrossRef]

Opt. Express (4)

Phys. Fluids (1)

D. Anderson, M. Bonneal, and M. Lisak, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
[CrossRef]

Phys. Lett. A (1)

M. W. Feise and Y. S. Kivshar, "Sub-wavelength imaging with a left-handed material flat lens," Phys. Lett. A 334, 326 (2005).
[CrossRef]

Phys. Rev. A (1)

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, "Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials," Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Phys. Rev. B (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]

V. N. Kissel and A. N. Lagarkov, "Superresolution in left-handed composite structures: From homogenization to a detailed electrodynamic description," Phys. Rev. B 72, 085111 (2005).
[CrossRef]

E. Schonbrun, T. Yamashita, W. Park, and C. J. Summers, "Negative-index imaging by an index-matched photonic crystal slab," Phys. Rev. B 73, 195117 (2006).
[CrossRef]

V. Yannopapas and N. V. Vitanov, "Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength," Phys. Rev. B 74, 193304 (2006).
[CrossRef]

Phys. Rev. E (4)

X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (2003).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

I. Kourakis and P. K. Shukla, "Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials," Phys. Rev. E 72, 016626 (2005).
[CrossRef]

S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, "Modulation instability in nonlinear negative-index material," Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Phys. Rev. E. (1)

J. J. Chen, T. M. Grzegorczyk, B. Wu, and J. A. Kong, "Imaging properties of finite-size left-handed material slabs," Phys. Rev. E. 74, 046615 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

N. Garcia and M. Nieto-Vesperinas, "Left-handed materials do not make a perfect lens" Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

A. N. Lagarkov and V. N. Kissel, "Near-Perfect Imaging in a Focusing System Based on a Left-Handed-Material Plate," Phys. Rev. Lett. 92, 077401(2004).
[CrossRef] [PubMed]

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

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

Phys. Stat. Sol. (1)

V. Yannopapas, "Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices," Phys. Stat. Sol.(RRL) 1, 208-210 (2007).

Phys. Today (1)

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-44 (2004).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Propagation model for the focusing of a Gaussian beam by a Kerr LHM slab.

Fig. 2.
Fig. 2.

(a) The focusing distance L f0 and (b) the focal spot radius w 0,3 of the focused Gaussian beam as functions of L LHM and RP for the case χ p,2>0. (Available in color)

Fig. 3.
Fig. 3.

(a) The focusing distance L f0 and (b) the focal spot radius w 0,3 of the focused Gaussian beam as functions of LLHM and RP for the case χ p,2<0 and RP <1/(-µ rl,2). (Available in color)

Fig. 4.
Fig. 4.

(a) The spot radius of the beam waist in D3 and (b) the focusing distance versus the thickness of the Kerr LHM slab. The solid curve and the dashed curve are theoretical predictions; the rectangular-dotted curve and the circular-dotted curve are simulated results.

Fig. 5.
Fig. 5.

The focal spot radius versus |θ o,1|. The thicknesses of Kerr LHM slabs in (a) and (b) are 10 cm and 4 cm, respectively. In both figures, the solid curves are the theoretical results for LLHM =0, where w 0,3=w 0,1, other curves are simulated results.

Fig. 6.
Fig. 6.

The focal spot radius versus |θ o,1| for χ p,2<0. Solid curve, as that in Fig. 5, is the theoretical result for LLHM =0, where w 0,3=w 0,1, other curves are simulated results. |Δθ o,1|=0.1.

Equations (13)

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

A 1 ( x , y , z ) = A 0 , 1 w 0 , 1 w 1 ( z ) exp [ x 2 + y 2 w 1 ( z ) 2 ] exp [ i k 1 x 2 + y 2 2 R 1 ( z ) ]
i 2 k 2 A 2 z + ( 2 x 2 + 2 y 2 ) A 2 + 2 k 2 2 μ r l , 2 n L , 2 C N L , 2 A 2 2 A 2 = 0
w 0 , 2 i = w 0 , 1 , Z w , 2 i = Z 0 , 2 + [ ( Z w , 1 Z 0 , 2 ) n L , 2 n L , 1 ] ,
A 2 i ( x , y , Z 0 , 2 ) = A 0 , 2 i w 0 , 2 i w 2 i ( Z 0 , 2 ) exp [ x 2 + y 2 w 2 i ( Z 0 , 2 ) 2 ] exp [ i k 2 x 2 + y 2 2 R 2 i ( Z 0 , 2 ) ] ,
A 2 = a 2 ( z ) exp [ x 2 + y 2 w 2 2 ( z ) + i b 2 ( z ) ( x 2 + y 2 ) + i φ 2 ( z ) ] .
w 2 ( z ) = C 0 1 + 2 k 2 C 0 C 1 ( z Z 0 , 2 k 2 C 0 2 ) + [ K + k 2 2 C 0 2 C 1 2 ] ( z Z 0 , 2 k 2 C 0 2 ) 2 ,
K = 4 { 1 [ μ r l , 2 sgn ( C N L , 2 ) R p ] } ,
w 2 ( z ) = C 0 1 4 θ o , 1 ( z Z 0 , 2 k 2 C 0 2 ) + [ K + 4 θ o , 1 2 ] ( z Z 0 , 2 k 2 C 0 2 ) 2 .
b 2 ( z ) = K 2 C 0 2 { 1 + K [ ( z Z 0 , 2 ) ( k 2 C 0 2 ) ] 2 } ( z Z 0 , 2 k 2 C 0 2 ) .
w 2 ( Z 0 . 3 ) = C 0 1 + K [ L LHM ( k 2 C 0 2 ) ] 2 , b 2 ( Z 0 . 3 ) = K 2 C 0 2 { 1 + K [ L LHM ( k 2 C 0 2 ) ] 2 } ( L LHM k 2 C 0 2 ) . }
Z w , 3 = Z 0,3 + 1 2 k 3 b 2 ( Z 0,3 ) w 2 ( Z 0,3 ) 4 1 + b 2 ( Z 0,3 ) 2 w 2 ( Z 0,3 ) 4 , w 0,3 = w 2 ( Z 0,3 ) 1 + b 2 ( Z 0,3 ) 2 w 2 ( Z 0,3 ) 4 ,
L f 0 = 1 2 k 3 b 2 ( Z 0,3 ) w 2 ( Z 0,3 ) 4 1 + b 2 ( Z 0,3 ) 2 w 2 ( Z 0,3 ) 4 .
Δ Ω z 0,3 = Δ Ω z 0,2 [ 1 + K 2 ( L LHM k 2 C 0 2 ) 2 ] [ 1 + K ( L LHM k 0 C 0 2 ) 2 ] .

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