Abstract

We theoretically prove that electromagnetic beams propagating through a nonlinear cubic metamaterial can exhibit a power flow whose direction reverses its sign along the transverse profile. This effect is peculiar of the hitherto unexplored extreme nonlinear regime where the nonlinear response is comparable or even greater than the linear contribution, a condition achievable even at relatively small intensities. We propose a possible metamaterial structure able to support the extreme conditions where the polarization cubic nonlinear contribution does not act as a mere perturbation of the linear part.

© 2010 Optical Society of America

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References

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  1. J. B. Pendry, “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. 85, 3966 (2000).
    [Crossref] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science 312, 1780 (2006).
    [Crossref] [PubMed]
  3. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter”, Opt. Lett. 22, 475 (1997).
    [Crossref] [PubMed]
  4. N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials”, Science 317, 1698 (2007).
    [Crossref] [PubMed]
  5. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials”, Phys. Rev. Lett. 91, 037401 (2003).
    [Crossref] [PubMed]
  6. I. V. Shadrivov and Y. S. Kivshar, “Spatial solitons in nonlinear left-handed metamaterials”, J. Opt. A: Pure Appl. Opt. 7, 68 (2005).
    [Crossref]
  7. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. 99, 153901 (2007).
    [Crossref] [PubMed]
  8. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized Nonlinear Schrdinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials”, Phys. Rev. Lett. 95, 013902 (2005).
    [Crossref] [PubMed]
  9. A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwells equations”, J. Opt. Soc. Am. B 22, 1384 (2005).
    [Crossref]
  10. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1994).
  11. Note that the presented scheme can be improved by considering more than two basic layers constituents. This can simplify the identification of suitable active media (not coinciding with the nonlinear medium) to steer loss compensation.
  12. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1998).
  13. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. 27, 1296 (1991).
    [Crossref]
  14. S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain”, Phys. Rev. B 67, 201101(R) (2003).
  15. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).

2007 (2)

N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials”, Science 317, 1698 (2007).
[Crossref] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. 99, 153901 (2007).
[Crossref] [PubMed]

2006 (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science 312, 1780 (2006).
[Crossref] [PubMed]

2005 (3)

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

A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwells equations”, J. Opt. Soc. Am. B 22, 1384 (2005).
[Crossref]

I. V. Shadrivov and Y. S. Kivshar, “Spatial solitons in nonlinear left-handed metamaterials”, J. Opt. A: Pure Appl. Opt. 7, 68 (2005).
[Crossref]

2003 (2)

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain”, Phys. Rev. B 67, 201101(R) (2003).

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

2000 (1)

J. B. Pendry, “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. 85, 3966 (2000).
[Crossref] [PubMed]

1997 (1)

1991 (1)

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. 27, 1296 (1991).
[Crossref]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).

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

Bartal, G.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. 99, 153901 (2007).
[Crossref] [PubMed]

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

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1994).

Ciattoni, A.

Crosignani, B.

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

Engheta, N.

N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials”, Science 317, 1698 (2007).
[Crossref] [PubMed]

Genov, D. A.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. 99, 153901 (2007).
[Crossref] [PubMed]

Hagan, D. J.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. 27, 1296 (1991).
[Crossref]

Hutchings, D. C.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. 27, 1296 (1991).
[Crossref]

Kivshar, Y. S.

I. V. Shadrivov and Y. S. Kivshar, “Spatial solitons in nonlinear left-handed metamaterials”, J. Opt. A: Pure Appl. Opt. 7, 68 (2005).
[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]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).

Kobayashi, T.

Liu, Y.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. 99, 153901 (2007).
[Crossref] [PubMed]

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

Morimoto, A.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1998).

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science 312, 1780 (2006).
[Crossref] [PubMed]

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain”, Phys. Rev. B 67, 201101(R) (2003).

J. B. Pendry, “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. 85, 3966 (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 Schrdinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials”, Phys. Rev. Lett. 95, 013902 (2005).
[Crossref] [PubMed]

Porto, P. Di

Ramakrishna, S. A.

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain”, Phys. Rev. B 67, 201101(R) (2003).

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

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science 312, 1780 (2006).
[Crossref] [PubMed]

Shadrivov, I. V.

I. V. Shadrivov and Y. S. Kivshar, “Spatial solitons in nonlinear left-handed metamaterials”, J. Opt. A: Pure Appl. Opt. 7, 68 (2005).
[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]

Sheik-Bahae, M.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. 27, 1296 (1991).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science 312, 1780 (2006).
[Crossref] [PubMed]

Stryland, E. W. Van

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. 27, 1296 (1991).
[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 Schrdinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials”, Phys. Rev. Lett. 95, 013902 (2005).
[Crossref] [PubMed]

Takahara, J.

Taki, H.

Yamagishi, S.

Yariv, A.

Zhang, X.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. 99, 153901 (2007).
[Crossref] [PubMed]

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

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. 27, 1296 (1991).
[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, 68 (2005).
[Crossref]

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

Opt. Lett. (1)

Phys. Rev. B (1)

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain”, Phys. Rev. B 67, 201101(R) (2003).

Phys. Rev. Lett. (4)

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. 99, 153901 (2007).
[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 Schrdinger 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]

J. B. Pendry, “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. 85, 3966 (2000).
[Crossref] [PubMed]

Science (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science 312, 1780 (2006).
[Crossref] [PubMed]

N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials”, Science 317, 1698 (2007).
[Crossref] [PubMed]

Other (4)

R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1994).

Note that the presented scheme can be improved by considering more than two basic layers constituents. This can simplify the identification of suitable active media (not coinciding with the nonlinear medium) to steer loss compensation.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1998).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).

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Figures (2)

Fig. 1.
Fig. 1.

Nonlinear guided waves transverse profile of ux (panel (a)) and of uz (panel (b)) at different values of u x in the range of Eq. (5), for γ = 0.5. (c) Profiles of the z- component of the Poynting vector (see Eq. (6)) normalized with S 0 = ( ε 0 ε 3 ) / ( 4 μ 0 μ χ 2 ) corresponding to the fields reported in Fig. 1(a) and 1(b). Each profile is characterized by an off-center positive part (black portion) and a central negative part (red portion). (d) Plot of the field S/S 0 (arrows) in the plane (ξ, ζ) corresponding to the nonlinear guided wave with u x = 0.65 of Fig. 1(a) and 1(b). The color is related to the local value of Sz /S 0. Note the reversing of S along the transverse ξ axis.

Fig. 2.
Fig. 2.

Metamaterial layered structure able to support transverse power flow reversing of TM fields, consisting of alternating slabs of a negative permittivity dielectric (ND) and a nonlinear cubic medium (NL).

Equations (14)

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

D = ε 0 ε E ε 0 χ [ ( E · E * ) E + γ ( E · E ) E * ] ,
B = μ 0 μ H ,
E ( x , z ) = e iβζ ε χ [ u x ( ξ ) e ̂ x + i u z ( ξ ) e ̂ z ] ,
H ( x , z ) = e iβζ ε 0 ε 2 μ 0 μχ [ β u x ( ξ ) d u z ( ξ ) ] e ̂ y
β d u z = [ ( β 2 1 ) + ( 1 + γ ) u x 2 + ( 1 γ ) u z 2 ] u x ,
d 2 u z d ξ 2 β d u x = [ 1 + ( 1 γ ) u x 2 + ( 1 + γ ) u z 2 ] u z
F ( u x , u z ) = ( β 2 1 ) u x 2 u z 2 + 1 2 ( 1 + γ ) ( u x 4 + u z 4 ) + ( 1 γ ) u x 2 u z 2 +
1 β 2 [ ( β 2 1 ) + ( 1 + γ ) u x 2 + ( 1 γ ) u z 2 ] 2 u x 2
1 2 + ( γ + 1 ) 2 γ < u x < 1 2
S = ε 0 ε 3 4 μ 0 μ χ 2 1 β { 1 [ ( u x 2 + u z 2 ) + γ ( u x 2 u z 2 ) ] } u x 2 e ̂ z
S = c 2 β εμ D x E x * e ̂ z
ε = f ε 1 + ( 1 f ) ε 2 ,
χ = ( 1 f ) χ 2 ,
μ 1 = f μ 1 1 + ( 1 f ) μ 2 1

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