Abstract

New results are reported on investigation of dispersion curves for surface plasmon polaritons (SPPs) at an inhomogenously doped semiconductor/dielectric interface whereby the dielectric is represented by the same undoped semiconductor. The doped semiconductor is described by its frequency-dependent permittivity that varies with the depth. It is shown that a transition layer (TL) with a linear change in carrier concentration supports one branch dispersion curve regardless of the TL thickness. The obtained dispersion curves reach a maximum at a finite frequency depending on the TL thickness, and subsequently asymptotically approach the zero frequency in the shortwave limit. Therefore two surface plasmon modes are supported at a given frequency: a long-wave mode with a positive group velocity and a short-wave mode with a negative group velocity. A condition of a zero group velocity can be satisfied by tuning the TL layer. It is shown that the conventional dispersion relation for SPPs at a TL with a zero thickness is an asymptotic solution, and the convergence of real dispersion curves is point-wise instead of an expected uniform convergence.

© 2015 Optical Society of America

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References

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  1. A. D. Boardman, Electromagnetic Surface Modes (Wiley, 1982).
  2. V. M. Agranovich, “Effects of the transition layer and spatial dispersion in the spectra of surface polaritons,” in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, Modern Problems in Condensed Matter Sciences, V. M. Agranovich and D. L. Mills, eds. (1, 1982), pp.187–236.
  3. S. A. Maier, Plasmonics: Fundamentals and Application, (Springer Science and Business Media LLC, 2007).
  4. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer, 1986).
  5. P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1, 484–588 (2009).
  6. M. Cada and J. Pistora, “Optical Plasmons in Semiconductors,” in Proceedings of International Symposium on Optical and Microwave Technologies, (ISMOT, 2011), pp. 23–29.
  7. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
    [Crossref]
  8. J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and aplications,” J. Phys. D Appl. Phys. 45(11), 113001 (2012).
    [Crossref]
  9. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004).
    [Crossref]
  10. E. A. Vinogradov and T. A. Leskova, “Polaritons in thin films on metal surfaces,” Phys. Rep. 194(5-6), 273–280 (1990).
    [Crossref]
  11. V. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “The surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15(2), 293–295 (1975).
    [Crossref]
  12. T. Lopez-Rios, F. Abeles, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. 39(6), 645–650 (1978).
    [Crossref]
  13. N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, “Metamaterials: electromagnetic enhancement at zero-index transition,” Opt. Lett. 33(20), 2350–2352 (2008).
    [Crossref] [PubMed]
  14. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [Crossref] [PubMed]
  15. J. Yang, M. Huang, C. Yang, Z. Xiao, and J. Peng, “Metamaterial electromagnetic concentrators with arbitrary geometries,” Opt. Express 17(22), 19656–19661 (2009).
    [Crossref] [PubMed]
  16. P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009).
    [Crossref]
  17. S. Foteinopoulou and J. P. Vigneron, “Extended slow-light field enhancement in positive-index/negative-index heterostructures,” Phys. Rev. B 88(19), 195144 (2013).
    [Crossref]
  18. S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface plasmon polariton dispersion curve,” Phys. Rev. B 10(8), 3342–3355 (1974).
    [Crossref]
  19. B. A. Kruger and J. K. S. Poon, “Optical guided waves at graded metal-dielectric interfaces,” Opt. Lett. 36(11), 2155–2157 (2011).
    [Crossref] [PubMed]
  20. C. C. Kao and E. M. Conwell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14(6), 2464–2479 (1976).
    [Crossref]
  21. A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Plasmonic-dielectric systems for high-order dispersionless slow or stopped subwavelength light,” Phys. Rev. Lett. 103(4), 043906 (2009).
    [Crossref] [PubMed]
  22. I. M. Mandel, I. Bendoym, Y. U. Jung, A. B. Golovin, and T. D. Crouse, “Dispersion engineering of surface plasmons,” Opt. Express 21(26), 31883–31893 (2013).
    [Crossref] [PubMed]
  23. M. Fox, Optical Properties of Solids, (Oxford University Press, 2001, 2010, 2012).
  24. M. Cada, D. Blazek, J. Pistora, K. Postava, and P. Siroky, “Theoretical and experimental study of plasmonic effects in heavily dopes gallium arsenide and indium phosphide,” Opt. Mater. Express 5(2), 340–352 (2015).
    [Crossref]
  25. J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003).
    [Crossref] [PubMed]
  26. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
    [Crossref]

2015 (1)

2013 (2)

I. M. Mandel, I. Bendoym, Y. U. Jung, A. B. Golovin, and T. D. Crouse, “Dispersion engineering of surface plasmons,” Opt. Express 21(26), 31883–31893 (2013).
[Crossref] [PubMed]

S. Foteinopoulou and J. P. Vigneron, “Extended slow-light field enhancement in positive-index/negative-index heterostructures,” Phys. Rev. B 88(19), 195144 (2013).
[Crossref]

2012 (1)

J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and aplications,” J. Phys. D Appl. Phys. 45(11), 113001 (2012).
[Crossref]

2011 (1)

2009 (4)

A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Plasmonic-dielectric systems for high-order dispersionless slow or stopped subwavelength light,” Phys. Rev. Lett. 103(4), 043906 (2009).
[Crossref] [PubMed]

J. Yang, M. Huang, C. Yang, Z. Xiao, and J. Peng, “Metamaterial electromagnetic concentrators with arbitrary geometries,” Opt. Express 17(22), 19656–19661 (2009).
[Crossref] [PubMed]

P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009).
[Crossref]

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1, 484–588 (2009).

2008 (2)

2006 (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

2004 (1)

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004).
[Crossref]

2003 (1)

J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003).
[Crossref] [PubMed]

1990 (1)

E. A. Vinogradov and T. A. Leskova, “Polaritons in thin films on metal surfaces,” Phys. Rep. 194(5-6), 273–280 (1990).
[Crossref]

1978 (1)

T. Lopez-Rios, F. Abeles, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. 39(6), 645–650 (1978).
[Crossref]

1976 (1)

C. C. Kao and E. M. Conwell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14(6), 2464–2479 (1976).
[Crossref]

1975 (1)

V. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “The surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15(2), 293–295 (1975).
[Crossref]

1974 (1)

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface plasmon polariton dispersion curve,” Phys. Rev. B 10(8), 3342–3355 (1974).
[Crossref]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[Crossref]

Abeles, F.

T. Lopez-Rios, F. Abeles, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. 39(6), 645–650 (1978).
[Crossref]

Agranovich, V. M.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004).
[Crossref]

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004).
[Crossref]

Bendoym, I.

Berini, P.

Blazek, D.

Bozhevolnyi, S. I.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Cada, M.

Conwell, E. M.

C. C. Kao and E. M. Conwell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14(6), 2464–2479 (1976).
[Crossref]

Crouse, T. D.

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Cunningham, S. L.

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface plasmon polariton dispersion curve,” Phys. Rev. B 10(8), 3342–3355 (1974).
[Crossref]

Ebbesen, T. W.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Foteinopoulou, S.

S. Foteinopoulou and J. P. Vigneron, “Extended slow-light field enhancement in positive-index/negative-index heterostructures,” Phys. Rev. B 88(19), 195144 (2013).
[Crossref]

French, O. E.

P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009).
[Crossref]

Gabitov, I. R.

Genet, C.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Golovin, A. B.

Hopcraft, K. I.

P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009).
[Crossref]

Huang, M.

Ingrey, P. C.

P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009).
[Crossref]

Jakeman, E.

P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009).
[Crossref]

Joannopoulos, J. D.

A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Plasmonic-dielectric systems for high-order dispersionless slow or stopped subwavelength light,” Phys. Rev. Lett. 103(4), 043906 (2009).
[Crossref] [PubMed]

Jung, Y. U.

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Kao, C. C.

C. C. Kao and E. M. Conwell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14(6), 2464–2479 (1976).
[Crossref]

Karalis, A.

A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Plasmonic-dielectric systems for high-order dispersionless slow or stopped subwavelength light,” Phys. Rev. Lett. 103(4), 043906 (2009).
[Crossref] [PubMed]

Knight, J. C.

J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003).
[Crossref] [PubMed]

Kruger, B. A.

Leskova, T. A.

E. A. Vinogradov and T. A. Leskova, “Polaritons in thin films on metal surfaces,” Phys. Rep. 194(5-6), 273–280 (1990).
[Crossref]

Litchinitser, N. M.

Lopez-Rios, T.

T. Lopez-Rios, F. Abeles, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. 39(6), 645–650 (1978).
[Crossref]

Maimistov, A. I.

Mandel, I. M.

Maradudin, A. A.

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface plasmon polariton dispersion curve,” Phys. Rev. B 10(8), 3342–3355 (1974).
[Crossref]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Nazin, V. G.

V. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “The surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15(2), 293–295 (1975).
[Crossref]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Peng, J.

Pistora, J.

Poon, J. K. S.

Postava, K.

Sagdeev, R. Z.

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Shalaev, V. M.

Shen, Y. R.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004).
[Crossref]

Siroky, P.

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Soljacic, M.

A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Plasmonic-dielectric systems for high-order dispersionless slow or stopped subwavelength light,” Phys. Rev. Lett. 103(4), 043906 (2009).
[Crossref] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[Crossref]

Vigneron, J. P.

S. Foteinopoulou and J. P. Vigneron, “Extended slow-light field enhancement in positive-index/negative-index heterostructures,” Phys. Rev. B 88(19), 195144 (2013).
[Crossref]

Vinogradov, E. A.

E. A. Vinogradov and T. A. Leskova, “Polaritons in thin films on metal surfaces,” Phys. Rep. 194(5-6), 273–280 (1990).
[Crossref]

Vuye, G.

T. Lopez-Rios, F. Abeles, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. 39(6), 645–650 (1978).
[Crossref]

Wallis, R. F.

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface plasmon polariton dispersion curve,” Phys. Rev. B 10(8), 3342–3355 (1974).
[Crossref]

Xiao, Z.

Xu, W.

J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and aplications,” J. Phys. D Appl. Phys. 45(11), 113001 (2012).
[Crossref]

Yakovlev, V. A.

V. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “The surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15(2), 293–295 (1975).
[Crossref]

Yang, C.

Yang, J.

Zakhidov, A. A.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004).
[Crossref]

Zhang, J.

J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and aplications,” J. Phys. D Appl. Phys. 45(11), 113001 (2012).
[Crossref]

Zhang, L.

J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and aplications,” J. Phys. D Appl. Phys. 45(11), 113001 (2012).
[Crossref]

Zhizhin, G. N.

V. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “The surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15(2), 293–295 (1975).
[Crossref]

Adv. Opt. Photon. (1)

J. Lumin. (1)

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Optical bulk and surface waves with negative refraction,” J. Lumin. 110(4), 167–173 (2004).
[Crossref]

J. Phys. (1)

T. Lopez-Rios, F. Abeles, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. 39(6), 645–650 (1978).
[Crossref]

J. Phys. D Appl. Phys. (1)

J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and aplications,” J. Phys. D Appl. Phys. 45(11), 113001 (2012).
[Crossref]

Nature (1)

J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003).
[Crossref] [PubMed]

Opt. Commun. (2)

V. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “The surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15(2), 293–295 (1975).
[Crossref]

P. C. Ingrey, K. I. Hopcraft, E. Jakeman, and O. E. French, “Between right and left handed media,” Opt. Commun. 282(5), 1020–1027 (2009).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Opt. Mater. Express (1)

Phys. Rep. (1)

E. A. Vinogradov and T. A. Leskova, “Polaritons in thin films on metal surfaces,” Phys. Rep. 194(5-6), 273–280 (1990).
[Crossref]

Phys. Rev. B (3)

S. Foteinopoulou and J. P. Vigneron, “Extended slow-light field enhancement in positive-index/negative-index heterostructures,” Phys. Rev. B 88(19), 195144 (2013).
[Crossref]

S. L. Cunningham, A. A. Maradudin, and R. F. Wallis, “Effect of a charge layer on the surface plasmon polariton dispersion curve,” Phys. Rev. B 10(8), 3342–3355 (1974).
[Crossref]

C. C. Kao and E. M. Conwell, “Surface plasmon dispersion of semiconductors with depletion or accumulation layers,” Phys. Rev. B 14(6), 2464–2479 (1976).
[Crossref]

Phys. Rev. Lett. (1)

A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Plasmonic-dielectric systems for high-order dispersionless slow or stopped subwavelength light,” Phys. Rev. Lett. 103(4), 043906 (2009).
[Crossref] [PubMed]

Phys. Today (1)

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[Crossref]

Science (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[Crossref]

Other (6)

M. Fox, Optical Properties of Solids, (Oxford University Press, 2001, 2010, 2012).

M. Cada and J. Pistora, “Optical Plasmons in Semiconductors,” in Proceedings of International Symposium on Optical and Microwave Technologies, (ISMOT, 2011), pp. 23–29.

A. D. Boardman, Electromagnetic Surface Modes (Wiley, 1982).

V. M. Agranovich, “Effects of the transition layer and spatial dispersion in the spectra of surface polaritons,” in Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, Modern Problems in Condensed Matter Sciences, V. M. Agranovich and D. L. Mills, eds. (1, 1982), pp.187–236.

S. A. Maier, Plasmonics: Fundamentals and Application, (Springer Science and Business Media LLC, 2007).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer, 1986).

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Figures (8)

Fig. 1
Fig. 1 Studied surface structure; doping concentration and permittivity are functions of x-coordinate; SPPs propagate along z-direction.

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Fig. 2
Fig. 2 Dispersion curves of SP propagating on silicon structure with TL thicknesses x N =0−10μm.

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Fig. 3
Fig. 3 TL thickness versus normalized frequency of surface plasmon with zero group velocity.

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Fig. 4
Fig. 4 Magnetic field distribution; long-wavelength SP ( β=0.5  β p ) ;weakly confined to TL with x N = β p −1 ;positive group velocity.

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Fig. 5
Fig. 5 Magnetic field distribution; SP with zero group velocity ( β=0.4  β p ) ; confined to TL with x N = β p −1 .

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Fig. 6
Fig. 6 Magnetic field distribution; short-wavelength SP ( β=16  β p ) ;strongly confined to TL with x N = β p −1 ;negative group velocity.

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Fig. 7
Fig. 7 Magnetic field distribution; long-wavelength SP ( β=0.5  β p ) ;confined to TL with x N =10 β p −1 ;positive group velocity.

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Fig. 8
Fig. 8 Group velocity versus propagation constant for TL thicknesses x N =0−10μm.

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Tables (1)

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Table 1 Required conditions for spatially stationary plasmon (material: silicon)

View Table

Equations (18)

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ε S ( ω )= ε ∞ ( 1− ω p 2 ω 2 )
ε( ω,x )=  { ε S ( ω )   for   x≤− x N     ε ∞ ( 1+ ω p 2 ω 2 x x N )   for   − x N <x<0 ε ∞    for   x>0  
                D x = β ω H y         D z = 1 jω d H y dx d 2 H y d x 2 = 1 ε dε dx d H y dx +( β 2 − μ 0   ε 0  ε  ω 2 ) H y
d dx ( 1 ε( x ) d H y dx )=( β 2 ε( x ) − μ 0 ε 0 ω 2 ) H y =( β 2 ε( x ) − ω 2 c 2 ) H y
g( x )=jω E x = 1 ε( x ) d H y dx
d dv ( 1 ε( v ) d H y dv )= dg( v ) dv =( n eff  2 ε( v ) −1 ) H y
H y ( v )={ h( − v N )  e i K v1  ( v+ v N )      for      v<− v N        h( v )                          for           − v N <v<0   h( 0 )  e −i K v2 v      for             0<v             
( 1 H y d H y dv ) v=− v N = ε S  g( − v N ) H y ( − v N ) = n eff  2 − ε S
( 1 H y d H y dv ) v=0 = ε ∞  g( 0 ) H y ( 0 ) =− n eff  2 − ε ∞
A= ε ∞ − ε S v N = ε ∞ ω p 2 ω 2 1 v N
g( ν )= h 0   n eff 2 A ln| ν− ν 0 |+O( 1 )   ( ν→ ν 0 )
h( ν )= h 0 + h 0   n eff 2 2 ( ν− ν 0 ) 2 ln| ν− ν 0 |+O( ( ν− ν 0 ) 2 )   ( ν→ ν 0 )
ln| ν− ν 0 |→ln ( ν− ν 0 ) 2 + ( ε i /A ) 2
v e = 〈 S z 〉 〈 W 〉 = 2  ∫ −∞ ∞ E x H y * dν ∫ −∞ ∞ [ ( ∂ε ∂ω )( E x E x * + E z E z * )+μ H y H y * ]  dν
S z ( ν )= E x H y * 2 = β 2 ω ε H y 2 = β  h 0 2 2 ω A { 1 ν− ν 0 + n eff 2 ( ν− ν 0 )ln| ν− ν 0 |+O[ ( ν− ν 0 ) ]( ν→ ν 0 ) }
β p = ω p c ε ∞ 2
β 0 = ω c ε S ε ∞ ε S + ε ∞    →    β ˜ 0 = ω ˜ 2 ω ˜ 2 −2 2 ω ˜ 2 −1
x N mat = ε ∞ ε ∞ mat x N

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