Abstract

We study nonlinear modes in a subwavelength slot waveguide created by a nonlinear dielectric slab sandwiched between two metals. We present the dispersion diagrams of the families of nonlinear guided modes and reveal that the symmetric mode undergoes the symmetry-breaking bifurcation and becomes primarily localized near one of the interfaces. We also find that the antisymmetric mode may split into two brunches giving birth to two families of nonlinear antisymmetric modes.

© 2008 Optical Society of America

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Corrections

Arthur R. Davoyan, Ilya V. Shadrivov, and Yuri S. Kivshar, "Nonlinear plasmonic slot waveguides: erratum," Opt. Express 17, 4833-4833 (2009)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-17-6-4833

References

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  1. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators", Nature 440, 508-511 (2006).
    [CrossRef] [PubMed]
  2. S. A. MaierPlasmonics: Fundamentals and Applications (Springer-Verlag, Berlin, 2007).
  3. E. N. Economou, "Surface plasmons in thin films", Phys. Rev. 182, 539-554 (1969).
    [CrossRef]
  4. J. J. Burke, G. I Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films", Phys. Rev. B 33, 5186-5201 (1986).
    [CrossRef]
  5. B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant", Phys. Rev. B 44, 13556-13572 (1991).
    [CrossRef]
  6. V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, "Nonlinear surface polaritons" Sov. Phys. JETP Lett. 32, 512-515 (1980).
  7. G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, "Nonlinear electromagnetic waves guided by a single interface", J. Appl. Phys. 58, 2453-2459 (1985).
    [CrossRef]
  8. D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface", Opt. Lett. 12, 187-189 (1987).
    [CrossRef] [PubMed]
  9. A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright "Exact theory of nonlinear p-polarized optical waves", Phys. Rev. A 35, 1159-1164 (1987).
    [CrossRef] [PubMed]
  10. K. M. Leung, "p-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions", Phys. Rev. B 32, 5093-5101 (1985).
    [CrossRef]
  11. M. Y. Yu, "Surface polaritons in nonlinear media", Phys. Rev. A 28, 1855-1856 (1983).
    [CrossRef]
  12. J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, "Nonlinear surface polaritons guided by metal films", J. Appl. Phys. 58, 2460-2466 (1985).
    [CrossRef]
  13. We mark S. I. Bozhevolnyi (Denmark) and A. V. Kildishev (USA) for independent verifications of this result.
  14. M. Asobe, T. Kanamori, and K. Kubodera, "Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches", IEEE J. Quantum Electronics 29, 2325-2333 (1993).
    [CrossRef]
  15. P. B. Johnson and R. W. Christy, "Optical constants of nobel metals", Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]

2006 (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators", Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

1993 (1)

M. Asobe, T. Kanamori, and K. Kubodera, "Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches", IEEE J. Quantum Electronics 29, 2325-2333 (1993).
[CrossRef]

1991 (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant", Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

1987 (2)

1986 (1)

J. J. Burke, G. I Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films", Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

1985 (3)

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, "Nonlinear electromagnetic waves guided by a single interface", J. Appl. Phys. 58, 2453-2459 (1985).
[CrossRef]

K. M. Leung, "p-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions", Phys. Rev. B 32, 5093-5101 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, "Nonlinear surface polaritons guided by metal films", J. Appl. Phys. 58, 2460-2466 (1985).
[CrossRef]

1983 (1)

M. Y. Yu, "Surface polaritons in nonlinear media", Phys. Rev. A 28, 1855-1856 (1983).
[CrossRef]

1980 (1)

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, "Nonlinear surface polaritons" Sov. Phys. JETP Lett. 32, 512-515 (1980).

1972 (1)

P. B. Johnson and R. W. Christy, "Optical constants of nobel metals", Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1969 (1)

E. N. Economou, "Surface plasmons in thin films", Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, "Nonlinear surface polaritons" Sov. Phys. JETP Lett. 32, 512-515 (1980).

Ariyasu, J.

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, "Nonlinear surface polaritons guided by metal films", J. Appl. Phys. 58, 2460-2466 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, "Nonlinear electromagnetic waves guided by a single interface", J. Appl. Phys. 58, 2453-2459 (1985).
[CrossRef]

Asobe, M.

M. Asobe, T. Kanamori, and K. Kubodera, "Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches", IEEE J. Quantum Electronics 29, 2325-2333 (1993).
[CrossRef]

Babichenko, V. S.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, "Nonlinear surface polaritons" Sov. Phys. JETP Lett. 32, 512-515 (1980).

Boardman, A. D.

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators", Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Burke, J. J.

J. J. Burke, G. I Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films", Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

Chernyak, V. Ya.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, "Nonlinear surface polaritons" Sov. Phys. JETP Lett. 32, 512-515 (1980).

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of nobel metals", Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators", Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators", Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Economou, E. N.

E. N. Economou, "Surface plasmons in thin films", Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constants of nobel metals", Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Kanamori, T.

M. Asobe, T. Kanamori, and K. Kubodera, "Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches", IEEE J. Quantum Electronics 29, 2325-2333 (1993).
[CrossRef]

Kubodera, K.

M. Asobe, T. Kanamori, and K. Kubodera, "Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches", IEEE J. Quantum Electronics 29, 2325-2333 (1993).
[CrossRef]

Laluet, J.-Y.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators", Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Leung, K. M.

K. M. Leung, "p-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions", Phys. Rev. B 32, 5093-5101 (1985).
[CrossRef]

Maradudin, A. A.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright "Exact theory of nonlinear p-polarized optical waves", Phys. Rev. A 35, 1159-1164 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, "Nonlinear electromagnetic waves guided by a single interface", J. Appl. Phys. 58, 2453-2459 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, "Nonlinear surface polaritons guided by metal films", J. Appl. Phys. 58, 2460-2466 (1985).
[CrossRef]

Mihalache, D.

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant", Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Prade, B.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant", Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Seaton, C. T.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface", Opt. Lett. 12, 187-189 (1987).
[CrossRef] [PubMed]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, "Nonlinear surface polaritons guided by metal films", J. Appl. Phys. 58, 2460-2466 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, "Nonlinear electromagnetic waves guided by a single interface", J. Appl. Phys. 58, 2453-2459 (1985).
[CrossRef]

Stegeman, G. I

J. J. Burke, G. I Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films", Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

Stegeman, G. I.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, "Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface", Opt. Lett. 12, 187-189 (1987).
[CrossRef] [PubMed]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright "Exact theory of nonlinear p-polarized optical waves", Phys. Rev. A 35, 1159-1164 (1987).
[CrossRef] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, "Nonlinear electromagnetic waves guided by a single interface", J. Appl. Phys. 58, 2453-2459 (1985).
[CrossRef]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, "Nonlinear surface polaritons guided by metal films", J. Appl. Phys. 58, 2460-2466 (1985).
[CrossRef]

Tamir, T.

J. J. Burke, G. I Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films", Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

Twardowski, T.

Vinet, J. Y.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant", Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators", Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Wallis, R. F.

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, "Nonlinear surface polaritons guided by metal films", J. Appl. Phys. 58, 2460-2466 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, "Nonlinear electromagnetic waves guided by a single interface", J. Appl. Phys. 58, 2453-2459 (1985).
[CrossRef]

Wright, E. M.

Yu, M. Y.

M. Y. Yu, "Surface polaritons in nonlinear media", Phys. Rev. A 28, 1855-1856 (1983).
[CrossRef]

Zanoni, R.

IEEE J. Quantum Electronics (1)

M. Asobe, T. Kanamori, and K. Kubodera, "Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches", IEEE J. Quantum Electronics 29, 2325-2333 (1993).
[CrossRef]

J. Appl. Phys. (2)

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, "Nonlinear surface polaritons guided by metal films", J. Appl. Phys. 58, 2460-2466 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, "Nonlinear electromagnetic waves guided by a single interface", J. Appl. Phys. 58, 2453-2459 (1985).
[CrossRef]

Nature (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators", Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. (1)

E. N. Economou, "Surface plasmons in thin films", Phys. Rev. 182, 539-554 (1969).
[CrossRef]

Phys. Rev. A (2)

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright "Exact theory of nonlinear p-polarized optical waves", Phys. Rev. A 35, 1159-1164 (1987).
[CrossRef] [PubMed]

M. Y. Yu, "Surface polaritons in nonlinear media", Phys. Rev. A 28, 1855-1856 (1983).
[CrossRef]

Phys. Rev. B (4)

P. B. Johnson and R. W. Christy, "Optical constants of nobel metals", Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

K. M. Leung, "p-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions", Phys. Rev. B 32, 5093-5101 (1985).
[CrossRef]

J. J. Burke, G. I Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films", Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

B. Prade, J. Y. Vinet, and A. Mysyrowicz, "Guided optical waves in planar heterostructures with negative dielectric constant", Phys. Rev. B 44, 13556-13572 (1991).
[CrossRef]

Sov. Phys. JETP Lett. (1)

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, "Nonlinear surface polaritons" Sov. Phys. JETP Lett. 32, 512-515 (1980).

Other (2)

S. A. MaierPlasmonics: Fundamentals and Applications (Springer-Verlag, Berlin, 2007).

We mark S. I. Bozhevolnyi (Denmark) and A. V. Kildishev (USA) for independent verifications of this result.

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Figures (5)

Fig. 1.
Fig. 1.

Dispersion of linear guided modes shown as the dependence of the normalized guide index vs. normalized slot width, for different values of ρ=|εm |/εd : ρ=1.3 (top) and ρ=1.05 (bottom), for symmetric (solid) and two antisymmetric modes (dashed and dotted). Inset shows a schematic of the structure, shaded regions correspond to metal.

Fig. 2.
Fig. 2.

Dispersion of nonlinear guided modes shown as the total power flow vs. guide index for λ=480nm, εm =-8.25, εd =7.84, ρ=1.05,α=1.4×10-18(m 2/V 2), and d=25nm.

Fig. 3.
Fig. 3.

(a–d) Characteristic profiles of nonlinear plasmonic modes shown as the magnetic field distribution for different branches of the dispersion curves marked by points in Fig. 4.

Fig. 4.
Fig. 4.

Dispersion of nonlinear guided modes shown as the total power flow vs. effective guide index for d=35nm. In the linear limit of vanishing power, the structure supports one symmetric and two antisymmetric plasmonic modes.

Fig. 5.
Fig. 5.

Dispersion of nonlinear guided modes shown as the total power flow vs. guide index for d=50nm. In the linear limit of vanishing power, the structure supports one symmetric and one antisymmetric plasmonic modes.

Equations (5)

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

d H y dx = ε E z , β H y = ε E x , d E z dx + β E x = H y ,
ε = { ε d , 0 < x < d , ε m , x < 0 , x > d ,
tanh ( λ d d ) [ ε m 2 λ d 2 + ε d 2 λ m 2 ] + 2 ε m ε d λ m λ d = 0
ε n ln = ε d + α ( E x 2 + E z 2 )
S = ( [ E × H ] · z ) dx .

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