Abstract

Highly asymmetrical plasmonic waveguides exhibit guiding even below the expected cut-off dimensions. A new family of discrete complex guided modes of asymmetrical waveguides with losses - which may assist in nano plasmonic guiding, is found by employing the effective index method. These modes having a real effective index lower than the substrate index of refraction can be exhibited also in regular (lossy) photonics.

© 2008 Optical Society of America

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References

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  1. E. Feigenbaum and M. Orenstein, “Modeling of Complementary (Void) Plasmon Waveguiding,” J. Lightwave Technol. 25, 2547–2562 (2007).
    [CrossRef]
  2. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. 31, 213–222 (1941).
    [CrossRef]
  3. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
    [CrossRef]
  4. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802-1–4 (2005).
    [CrossRef] [PubMed]
  5. R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon polariton waves supported by a thin metal film of finite width,” Opt. Lett. 25, 844–846 (2000).
    [CrossRef]
  6. J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
    [CrossRef]
  7. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,”  Nature 440, 508–511 (2006).
  8. F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
    [CrossRef]
  9. S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14, 9467–9476 (2006).
    [CrossRef] [PubMed]
  10. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13, 6645–6650 (2005).
    [CrossRef] [PubMed]
  11. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007).
    [CrossRef]
  12. Y. Satuby and M. Orenstein, “Surface plasmon polariton waveguiding: From multimode stripe to a slot geometry,” Appl. Phys. Lett. 90, 251104 (2007).
    [CrossRef]
  13. D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
    [CrossRef]
  14. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
  15. I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002).
    [CrossRef]
  16. W. O. Schlosser, “Gain-induced modes in planar structures,” Bell Syst. Tech. J. 52, 887–905 (1973).
  17. A. Sommerfeld, “Uber die Fortpflanzung elektromagnetischer Wellen langs eines Drahtes,” Wiedermann Ann. Phys. Chem. 67, 233–290 (1899).
    [CrossRef]
  18. J. Zenneck, “Uber die Fortpflanzung ebener elektromagnetischer Wellen langs einer ebenen Leiterflache,” Ann. Phys. 23, 846–866 (1907).
    [CrossRef]

2007 (4)

E. Feigenbaum and M. Orenstein, “Modeling of Complementary (Void) Plasmon Waveguiding,” J. Lightwave Technol. 25, 2547–2562 (2007).
[CrossRef]

F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
[CrossRef]

G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007).
[CrossRef]

Y. Satuby and M. Orenstein, “Surface plasmon polariton waveguiding: From multimode stripe to a slot geometry,” Appl. Phys. Lett. 90, 251104 (2007).
[CrossRef]

2006 (2)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,”  Nature 440, 508–511 (2006).

S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14, 9467–9476 (2006).
[CrossRef] [PubMed]

2005 (3)

L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13, 6645–6650 (2005).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802-1–4 (2005).
[CrossRef] [PubMed]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

2002 (1)

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002).
[CrossRef]

2001 (1)

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
[CrossRef]

2000 (1)

1973 (1)

W. O. Schlosser, “Gain-induced modes in planar structures,” Bell Syst. Tech. J. 52, 887–905 (1973).

1969 (1)

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

1941 (1)

1907 (1)

J. Zenneck, “Uber die Fortpflanzung ebener elektromagnetischer Wellen langs einer ebenen Leiterflache,” Ann. Phys. 23, 846–866 (1907).
[CrossRef]

1899 (1)

A. Sommerfeld, “Uber die Fortpflanzung elektromagnetischer Wellen langs eines Drahtes,” Wiedermann Ann. Phys. Chem. 67, 233–290 (1899).
[CrossRef]

Berini, P.

Berolo, E.

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,”  Nature 440, 508–511 (2006).

S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14, 9467–9476 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802-1–4 (2005).
[CrossRef] [PubMed]

Charbonneau, R.

Chen, H. S.

F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
[CrossRef]

Dereux, A.

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
[CrossRef]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,”  Nature 440, 508–511 (2006).

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802-1–4 (2005).
[CrossRef] [PubMed]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,”  Nature 440, 508–511 (2006).

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802-1–4 (2005).
[CrossRef] [PubMed]

Economou, E. N.

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

Fan, S.

G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007).
[CrossRef]

Fano, U.

Feigenbaum, E.

Fukui, M.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Goudonnet, J. P.

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
[CrossRef]

Gramotnev, D. K.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Han, Z.

Haraguchi, M.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

He, S.

Huang, H.

F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
[CrossRef]

Kong, F. M.

F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
[CrossRef]

Kong, J. A.

F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
[CrossRef]

Krenn, J. R.

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
[CrossRef]

Lacroute, Y.

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
[CrossRef]

Lamprecht, B.

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
[CrossRef]

Li, K.

F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
[CrossRef]

Lisicka-Shrzek, E.

Liu, L.

Maradudin, A. A.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002).
[CrossRef]

Matsuzaki, Y.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Novikov, I. V.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002).
[CrossRef]

Ogawa, T.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Okamoto, T.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Orenstein, M.

Y. Satuby and M. Orenstein, “Surface plasmon polariton waveguiding: From multimode stripe to a slot geometry,” Appl. Phys. Lett. 90, 251104 (2007).
[CrossRef]

E. Feigenbaum and M. Orenstein, “Modeling of Complementary (Void) Plasmon Waveguiding,” J. Lightwave Technol. 25, 2547–2562 (2007).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

Pile, D. F. P.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Satuby, Y.

Y. Satuby and M. Orenstein, “Surface plasmon polariton waveguiding: From multimode stripe to a slot geometry,” Appl. Phys. Lett. 90, 251104 (2007).
[CrossRef]

Schlosser, W. O.

W. O. Schlosser, “Gain-induced modes in planar structures,” Bell Syst. Tech. J. 52, 887–905 (1973).

Sommerfeld, A.

A. Sommerfeld, “Uber die Fortpflanzung elektromagnetischer Wellen langs eines Drahtes,” Wiedermann Ann. Phys. Chem. 67, 233–290 (1899).
[CrossRef]

Vernon, K. C.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Veronis, G.

G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007).
[CrossRef]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,”  Nature 440, 508–511 (2006).

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802-1–4 (2005).
[CrossRef] [PubMed]

Weeber, J.-C.

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
[CrossRef]

Wu, B. I.

F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
[CrossRef]

Yamaguchi, K.

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Zenneck, J.

J. Zenneck, “Uber die Fortpflanzung ebener elektromagnetischer Wellen langs einer ebenen Leiterflache,” Ann. Phys. 23, 846–866 (1907).
[CrossRef]

Ann. Phys. (1)

J. Zenneck, “Uber die Fortpflanzung ebener elektromagnetischer Wellen langs einer ebenen Leiterflache,” Ann. Phys. 23, 846–866 (1907).
[CrossRef]

Appl. Phys. Lett. (2)

Y. Satuby and M. Orenstein, “Surface plasmon polariton waveguiding: From multimode stripe to a slot geometry,” Appl. Phys. Lett. 90, 251104 (2007).
[CrossRef]

D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87, 261114 (2005).
[CrossRef]

Bell Syst. Tech. J. (1)

W. O. Schlosser, “Gain-induced modes in planar structures,” Bell Syst. Tech. J. 52, 887–905 (1973).

IEEE J. Lightwave Tech. (1)

G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” IEEE J. Lightwave Tech. 25, 2511–2521 (2007).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. (1)

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

Phys. Rev. B (2)

J.-C. Weeber, J. R. Krenn, A. Dereux, B. Lamprecht, Y. Lacroute, and J. P. Goudonnet, “Near-field observation of surface plasmon polariton propagation on thin metal stripes,” Phys. Rev. B 64, 045411 (2001).
[CrossRef]

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66, 035403 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802-1–4 (2005).
[CrossRef] [PubMed]

PIER (1)

F. M. Kong, K. Li, B. I. Wu, H. Huang, H. S. Chen, and J. A. Kong, “Propagation properties of the SPP modes in nanoscale narrow metallic gap, channel, and hole geometries,” PIER 76, 449–466 (2007).
[CrossRef]

Wiedermann Ann. Phys. Chem. (1)

A. Sommerfeld, “Uber die Fortpflanzung elektromagnetischer Wellen langs eines Drahtes,” Wiedermann Ann. Phys. Chem. 67, 233–290 (1899).
[CrossRef]

Other (2)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,”  Nature 440, 508–511 (2006).

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Figures (5)

Fig. 1.
Fig. 1.

(a). Slot waveguide; (b) the equivalent gap waveguide of the intermediate layer – as the first stage of the EIM; (c) effective slab waveguide to be solved as the second stage of the EIM.

Fig. 2.
Fig. 2.

Schematic solution of the asymmetric plasmonic slab waveguide Air/Au/SiO2; Vertical double-dashed lines emphasize the cut-off point of the discrete complex modes.

Fig. 3.
Fig. 3.

Effective index of the basic mode of an asymmetrical slot (Substrate/Au/Air) as a function of slot height for different substrate materials with slot width 20nm. Vertical dashed lines show the cut-off values: SiO2 (A), Si3N4 (B), Si (C), Au (D).

Fig. 4.
Fig. 4.

(a). Dependence of the asymmetrical slab solution on refractive index of the substrate material for different slab heights. Vertical line emphasizes the transition point between two types of modes. (b) Regular mode corresponding to the left yellow circle in (a) at slab height 2µm. (c) Novel complex mode corresponding to the right yellow circle in (a) at slab height 2µm; Small inset schematically indicates the direction of the Poynting vector S in the substrate.

Fig. 5.
Fig. 5.

Cut-off (a) and vertical confinement factor (b) as a function of substrate index for various core layer losses of a 1µm high slab waveguide Air/(n=1.76-iLoss)/Substrate. Vertical line indicates the transition point between two types of modes

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