Abstract

We present the first (to our knowledge) exact dispersion relation for the transverse-magnetic surface plasmon polariton (SPP) modes of a plasmonic slot waveguide, which is formed by a nonlinear Kerr medium sandwiched between two metallic slabs. The obtained relation is then simplified to the case of small field intensities, while retaining nonlinear terms, to derive approximate dispersion equations for the symmetric and antisymmetric SPP modes.

© 2011 Optical Society of America

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References

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  1. D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photonics 4, 83 (2010).
    [CrossRef]
  2. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, Opt. Express 19, 206 (2011).
    [CrossRef] [PubMed]
  3. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, Opt. Express 18, 6191 (2010).
    [CrossRef] [PubMed]
  4. A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, Opt. Express 16, 21209 (2008).
    [CrossRef] [PubMed]
  5. A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
    [CrossRef] [PubMed]
  6. A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, Opt. Express 17, 21732 (2009).
    [CrossRef] [PubMed]
  7. E. Feigenbaum and M. Orenstein, Opt. Lett. 32, 674(2007).
    [CrossRef] [PubMed]
  8. Y. Li and X. Zhang, Opt. Commun. 281, 5009 (2008).
    [CrossRef]
  9. Y. Su, in Communications and Photonics Conference and Exhibition (2010).
  10. A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, in Quantum Electronics and Laser Science Conference, p. QThH5 (2010).
  11. R. I. Joseph and D. N. Christodoulides, Opt. Lett. 12, 826 (1987).
    [CrossRef] [PubMed]

2011 (1)

2010 (3)

D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photonics 4, 83 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef] [PubMed]

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, Opt. Express 18, 6191 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (2)

2007 (1)

1987 (1)

Agrawal, G. P.

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photonics 4, 83 (2010).
[CrossRef]

Christodoulides, D. N.

Davoyan, A. R.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, Opt. Express 17, 21732 (2009).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, Opt. Express 16, 21209 (2008).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, in Quantum Electronics and Laser Science Conference, p. QThH5 (2010).

Feigenbaum, E.

Gramotnev, D. K.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photonics 4, 83 (2010).
[CrossRef]

Hattori, H. T.

Joseph, R. I.

Kivshar, Yu. S.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, Opt. Express 17, 21732 (2009).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, Opt. Express 16, 21209 (2008).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, in Quantum Electronics and Laser Science Conference, p. QThH5 (2010).

Li, Y.

Y. Li and X. Zhang, Opt. Commun. 281, 5009 (2008).
[CrossRef]

Orenstein, M.

Pannipitiya, A.

Premaratne, M.

Rukhlenko, I. D.

Shadrivov, I. V.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, Opt. Express 17, 21732 (2009).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, Opt. Express 16, 21209 (2008).
[CrossRef] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, in Quantum Electronics and Laser Science Conference, p. QThH5 (2010).

Su, Y.

Y. Su, in Communications and Photonics Conference and Exhibition (2010).

Zhang, X.

Y. Li and X. Zhang, Opt. Commun. 281, 5009 (2008).
[CrossRef]

Zharov, A. A.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photonics 4, 83 (2010).
[CrossRef]

Opt. Commun. (1)

Y. Li and X. Zhang, Opt. Commun. 281, 5009 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Yu. S. Kivshar, Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef] [PubMed]

Other (2)

Y. Su, in Communications and Photonics Conference and Exhibition (2010).

A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, in Quantum Electronics and Laser Science Conference, p. QThH5 (2010).

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Figures (2)

Fig. 1
Fig. 1

Representative electric field profiles for the (a) symmetric and (b) antisymmetric SPP modes of a plasmonic slot waveguide, which is created by a nonlinear- dielectric layer of permittivity ε 1 and thickness d placed between two metallic slabs of permittivity ε 2 .

Fig. 2
Fig. 2

Dispersion relations for the symmetric SPP mode calculated with Eq. (7) (blue curve), Eq. (12) (green curve), and Eq. (13) (red curve); dashed is the light line in the nonlinear medium. Insets show profiles of forward- and backward-traveling modes. For simulation parameters refer to text.

Equations (25)

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E ( x , z ) = [ E x ( x ) e x + i E z ( x ) e z ] exp ( i β z ) ,
d E z d x = k j 2 β E x , d ( ε j E x ) d x = β ε j E z ,
E x 2 = A exp ( k 2 x ) , E z 2 = ( k 2 / β ) A exp ( k 2 x ) ,
ε 1 + E x 1 + = ε 2 A exp ( k 2 d / 2 ) , E z 1 + = ( k 2 / β ) A exp ( k 2 d / 2 ) ,
E 1 + = E z 1 + 1 + ( ε 2 / ε 1 + ) 2 ( β / k 2 ) 2 .
d E z 1 d x d 2 E z 1 d x 2 = k 1 2 E x 1 d E x 1 d x ε 1 k 2 E z 1 d E z 1 d x ,
ε 1 [ ε 1 2 ( 2 χ ε 1 ) α E x 1 2 ] = C ,
E z 1 ( E x 1 ) = ( ε 1 ( E x 1 ) ε 1 L α E x 1 2 ) 1 / 2 ,
ε 1 ( E x 1 ) = 2 α E x 1 2 + ( 2 α E x 1 2 ) 2 + C ( 1 + 2 χ α E x 1 2 ) 1 + 2 χ α E x 1 2 .
C ( E 0 ) = ε 10 [ ε 10 2 ( 2 χ ε 10 ) α E 0 2 ] ,
C ( E 0 ) = ε 10 2 .
β d 2 = E x 1 ( 0 ) E x 1 + ( E 0 ) d E x E z ( E x ) ε 1 ( E x ) + 2 α E x 2 ( 2 α E x 2 ) 2 + C ( 1 + 2 χ α E x 2 ) .
E x 1 + ( E 0 ) = ε 1 + ε 1 L α [ 1 + ψ ( ε 1 + ) 2 ] ,
ε 1 + ( 2 ( χ ε 1 + 2 ) ( ε 1 + ε 1 L ) ψ ( ε 1 + ) 2 + 1 + ε 1 + ) = C ( E 0 ) ,
cosh ( k 1 d / 2 ) = E x 1 + / E 0 ,
sinh ( k 1 d / 2 ) = ( k 1 / β ) ( E x 1 + / E 0 ) .
ε 1 + ε 1 L + ( 1 χ ε 1 L ) ( 1 + ψ ε 1 L 2 ) 1 χ ε 1 L ψ ε 1 L 2 α E 0 2 ,
ε 1 + ε 1 L + 1 + ψ ε 1 L 2 χ ε 1 L 1 + ψ ε 1 L 2 α E 0 2 ,
( E 0 E x 1 + ) 2 1 ψ ε 1 L 2 1 χ ε 1 L = 1 ( ε 1 L ε 2 k 2 k 1 ) 2 ,
( E 0 E x 1 + ) 2 ψ ε 1 L 2 1 + χ ε 1 L = ( ε 1 L ε 2 k 2 β ) 2 k 1 2 β 2 ,
tanh ( k 1 d 2 ) = ( ε 1 L | ε 2 | k 2 k 1 ) ± 1 ,
cosh ( k 1 ( d / 2 ) α E 0 2 Φ 1 α E 0 2 Λ ) = q 2 q 2 1 + α E 0 2 Ψ ,
Φ = ( 2 χ ε 1 L ) ( 6 5 χ ε 1 L ) 8 ε 1 L ( 1 χ ε 1 L ) ( q q 2 1 + 1 2 ln q + 1 q 1 ) , Ψ = ψ ε 1 L 2 2 χ ( 1 χ ε 1 L ) + ψ ε 1 L [ 4 χ ε 1 L ( 6 χ ε 1 L ) ] 2 ( 1 χ ε 1 L ψ ε 1 L 2 ) 3 , Λ = [ 4 χ ε 1 L ( 6 χ ε 1 L ) ] / [ 4 ε 1 L ( 1 χ ε 1 L ) ] .
sinh ( k 1 d 2 + α E 0 2 β 2 k 1 2 Φ ) = q 2 1 q 2 + α E 0 2 k 1 2 β 2 Ψ ,
Φ = ( χ / 2 ) f 1 [ ( 2 / ε 1 L ) χ ] f 2 + [ ( 2 / ε 1 L ) χ ] ( 2 3 χ ε 1 L ) ( β / k 1 ) 2 f 3 / 8 , Ψ = 4 ε 1 L [ χ ( 6 χ ε 1 L ) + 2 ψ ε 1 L ( 4 χ ε 1 L ) ] 2 ε 1 L ( ψ ε 1 L 2 1 + χ ε 1 L ) 3 , f 1 = q tan 1 q , f 2 = tan 1 q q / ( 1 q 2 ) , f 3 = q ( 3 2 q 2 ) / ( 1 q 2 ) 3 tan 1 q .

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