Abstract

Although surface polariton modes supported by finite-width interfaces can guide electromagnetic energy in three dimensions, we demonstrate for the first time to our knowledge that such modes can be modeled by the solutions of two-dimensional dielectric slab waveguides. An approximate model is derived by a ray-optics interpretation that is consistent with previous investigations of the Fresnel relations for surface polariton reflection. This model is compared with modal solutions for metal stripe waveguides obtained by full vectorial magnetic-field finite-difference methods. The field-symmetric modes of such waveguides are shown to be in agreement with the normalized dispersion relationship for analogous TE modes of dielectric slab waveguides. Lateral confinement is investigated by comparison of power-density profiles, and implications for the diffraction limit of guided polariton modes are discussed.

© 2005 Optical Society of America

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  1. J. Takahara and T. Kobayashi, Opt. Photonics News 15(10), 54 (Oct. 2004).
    [CrossRef]
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
    [CrossRef] [PubMed]
  3. N. Ocelic and R. Hillenbrand, Nature Mater. 3, 606 (2004).
    [CrossRef]
  4. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, Opt. Lett. 22, 475 (1997).
    [CrossRef] [PubMed]
  5. P. Berini, Opt. Lett. 24, 1011 (1999).
    [CrossRef]
  6. J. C. Weeber, Y. Lacroute, and A. Dereux, Phys. Rev. B 68, 115401 (2003).
    [CrossRef]
  7. R. Zia, M. D. Selker, and M. L. Brongersma, Phys. Rev. B 71, 165431 (2005).
    [CrossRef]
  8. For a review, see A. A. Maradudin, R. F. Wallis, and G. I. Stegeman, Prog. Surf. Sci. 33, 171 (1990).
    [CrossRef]
  9. G. I. Stegeman, N. E. Glass, A. A. Maradudin, T. P. Shen, and R. F. Wallis, Opt. Lett. 8, 626 (1983).
    [CrossRef] [PubMed]
  10. S. J. Al-Bader, IEEE J. Quantum Electron. 40, 325 (2004).
    [CrossRef]
  11. P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, J. Lightwave Technol. 12, 487 (1994).
    [CrossRef]
  12. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, J. Opt. Soc. Am. A 21, 2442 (2004).
    [CrossRef]
  13. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, J. Lightwave Technol. 17, 929 (1999).
    [CrossRef]
  14. Interestingly, our model well approximates the field-antisymmetric sab1 modes of P. Berini, Phys. Rev. B 61, 10484 (2000). Berini uses the method of lines, which, similarly to our model, involves analytical treatment in the vertical direction.
    [CrossRef]
  15. For example, see H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Chap. 6.
  16. H. A. Jamin and S. J. Al-Bader, IEEE Photonics Technol. Lett. 7, 321 (1995).
    [CrossRef]

2005

R. Zia, M. D. Selker, and M. L. Brongersma, Phys. Rev. B 71, 165431 (2005).
[CrossRef]

2004

J. Takahara and T. Kobayashi, Opt. Photonics News 15(10), 54 (Oct. 2004).
[CrossRef]

N. Ocelic and R. Hillenbrand, Nature Mater. 3, 606 (2004).
[CrossRef]

S. J. Al-Bader, IEEE J. Quantum Electron. 40, 325 (2004).
[CrossRef]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, J. Opt. Soc. Am. A 21, 2442 (2004).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

J. C. Weeber, Y. Lacroute, and A. Dereux, Phys. Rev. B 68, 115401 (2003).
[CrossRef]

2000

Interestingly, our model well approximates the field-antisymmetric sab1 modes of P. Berini, Phys. Rev. B 61, 10484 (2000). Berini uses the method of lines, which, similarly to our model, involves analytical treatment in the vertical direction.
[CrossRef]

1999

1997

1995

H. A. Jamin and S. J. Al-Bader, IEEE Photonics Technol. Lett. 7, 321 (1995).
[CrossRef]

1994

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, J. Lightwave Technol. 12, 487 (1994).
[CrossRef]

1990

For a review, see A. A. Maradudin, R. F. Wallis, and G. I. Stegeman, Prog. Surf. Sci. 33, 171 (1990).
[CrossRef]

1983

Al-Bader, S. J.

S. J. Al-Bader, IEEE J. Quantum Electron. 40, 325 (2004).
[CrossRef]

H. A. Jamin and S. J. Al-Bader, IEEE Photonics Technol. Lett. 7, 321 (1995).
[CrossRef]

Anemogiannis, E.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Berini, P.

Interestingly, our model well approximates the field-antisymmetric sab1 modes of P. Berini, Phys. Rev. B 61, 10484 (2000). Berini uses the method of lines, which, similarly to our model, involves analytical treatment in the vertical direction.
[CrossRef]

P. Berini, Opt. Lett. 24, 1011 (1999).
[CrossRef]

Brongersma, M. L.

R. Zia, M. D. Selker, and M. L. Brongersma, Phys. Rev. B 71, 165431 (2005).
[CrossRef]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, J. Opt. Soc. Am. A 21, 2442 (2004).
[CrossRef]

Catrysse, P. B.

Dereux, A.

J. C. Weeber, Y. Lacroute, and A. Dereux, Phys. Rev. B 68, 115401 (2003).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Gaylord, T. K.

Glass, N. E.

Glytsis, E. N.

Haus, H. A.

For example, see H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Chap. 6.

Hillenbrand, R.

N. Ocelic and R. Hillenbrand, Nature Mater. 3, 606 (2004).
[CrossRef]

Jamin, H. A.

H. A. Jamin and S. J. Al-Bader, IEEE Photonics Technol. Lett. 7, 321 (1995).
[CrossRef]

Kobayashi, T.

Lacroute, Y.

J. C. Weeber, Y. Lacroute, and A. Dereux, Phys. Rev. B 68, 115401 (2003).
[CrossRef]

Lusse, P.

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, J. Lightwave Technol. 12, 487 (1994).
[CrossRef]

Maradudin, A. A.

For a review, see A. A. Maradudin, R. F. Wallis, and G. I. Stegeman, Prog. Surf. Sci. 33, 171 (1990).
[CrossRef]

G. I. Stegeman, N. E. Glass, A. A. Maradudin, T. P. Shen, and R. F. Wallis, Opt. Lett. 8, 626 (1983).
[CrossRef] [PubMed]

Morimoto, A.

Ocelic, N.

N. Ocelic and R. Hillenbrand, Nature Mater. 3, 606 (2004).
[CrossRef]

Schule, J.

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, J. Lightwave Technol. 12, 487 (1994).
[CrossRef]

Selker, M. D.

R. Zia, M. D. Selker, and M. L. Brongersma, Phys. Rev. B 71, 165431 (2005).
[CrossRef]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, J. Opt. Soc. Am. A 21, 2442 (2004).
[CrossRef]

Shen, T. P.

Stegeman, G. I.

For a review, see A. A. Maradudin, R. F. Wallis, and G. I. Stegeman, Prog. Surf. Sci. 33, 171 (1990).
[CrossRef]

G. I. Stegeman, N. E. Glass, A. A. Maradudin, T. P. Shen, and R. F. Wallis, Opt. Lett. 8, 626 (1983).
[CrossRef] [PubMed]

Stuwe, P.

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, J. Lightwave Technol. 12, 487 (1994).
[CrossRef]

Takahara, J.

Taki, H.

Unger, H. G.

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, J. Lightwave Technol. 12, 487 (1994).
[CrossRef]

Wallis, R. F.

For a review, see A. A. Maradudin, R. F. Wallis, and G. I. Stegeman, Prog. Surf. Sci. 33, 171 (1990).
[CrossRef]

G. I. Stegeman, N. E. Glass, A. A. Maradudin, T. P. Shen, and R. F. Wallis, Opt. Lett. 8, 626 (1983).
[CrossRef] [PubMed]

Weeber, J. C.

J. C. Weeber, Y. Lacroute, and A. Dereux, Phys. Rev. B 68, 115401 (2003).
[CrossRef]

Yamagishi, S.

Zia, R.

R. Zia, M. D. Selker, and M. L. Brongersma, Phys. Rev. B 71, 165431 (2005).
[CrossRef]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, J. Opt. Soc. Am. A 21, 2442 (2004).
[CrossRef]

IEEE J. Quantum Electron.

S. J. Al-Bader, IEEE J. Quantum Electron. 40, 325 (2004).
[CrossRef]

IEEE Photonics Technol. Lett.

H. A. Jamin and S. J. Al-Bader, IEEE Photonics Technol. Lett. 7, 321 (1995).
[CrossRef]

J. Lightwave Technol.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, J. Lightwave Technol. 17, 929 (1999).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, J. Lightwave Technol. 12, 487 (1994).
[CrossRef]

J. Opt. Soc. Am. A

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Nature Mater.

N. Ocelic and R. Hillenbrand, Nature Mater. 3, 606 (2004).
[CrossRef]

Opt. Lett.

Opt. Photonics News

J. Takahara and T. Kobayashi, Opt. Photonics News 15(10), 54 (Oct. 2004).
[CrossRef]

Phys. Rev. B

J. C. Weeber, Y. Lacroute, and A. Dereux, Phys. Rev. B 68, 115401 (2003).
[CrossRef]

R. Zia, M. D. Selker, and M. L. Brongersma, Phys. Rev. B 71, 165431 (2005).
[CrossRef]

Interestingly, our model well approximates the field-antisymmetric sab1 modes of P. Berini, Phys. Rev. B 61, 10484 (2000). Berini uses the method of lines, which, similarly to our model, involves analytical treatment in the vertical direction.
[CrossRef]

Prog. Surf. Sci.

For a review, see A. A. Maradudin, R. F. Wallis, and G. I. Stegeman, Prog. Surf. Sci. 33, 171 (1990).
[CrossRef]

Other

For example, see H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Chap. 6.

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Figures (3)

Fig. 1
Fig. 1

Dielectric waveguide treatment of surface polaritons along finite interfaces: (a) TIR of a surface polariton wave, (b) ray-optics interpretation of a surface polariton mode, (c) top view of the ray-optics interpretation, (d) equivalent two-dimensional dielectric slab waveguide.

Fig. 2
Fig. 2

Surface plasmons supported by coupled interfaces of a finite-width silver stripe of varying thickness ( t ) embedded in a silicon matrix. Markers denote approximate solutions obtained from the dielectric waveguide model for M 00 (circles), M 01 (triangles), and M 10 (squares) modes. Solid curves denote solutions previously presented by J. Al-Bader.[10] Relevant parameters are W = 1 μ m , λ = 1.55 μ m , ϵ Ag = 125.735 + i 3.233 , ϵ Si = 12.25 .

Fig. 3
Fig. 3

Normalized dispersion curves for surface plasmon modes of silver stripe waveguide (as in Fig. 2). Filled and open markers denote solutions obtained by FVH-FDM for the M 00 and M 10 modes, respectively. For three stripe thicknesses [ t = 25 nm (triangles), 50 nm (circles), and 100 nm (squares)], width W varied from 0.5 to 4 μ m . Solid and dashed curves represent the TE 0 and TE 1 modes of a dielectric slab waveguide, respectively.[15] Inset, comparison of the lateral power densities for a surface plasmon waveguide (solid curve, W = 1 μ m ; t = 50 nm ) and an approximate dielectric waveguide (dashed curve with gray shading, n eff = 3.628 + i 0.002267 ) normalized to unit power.

Equations (4)

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

n eff = k sp k 0 ,
V k 0 W ( n eff 2 n d 2 ) 1 2 ,
b [ ( β + i α ) 2 k 0 2 ] n d 2 n eff 2 n d 2 ,
Δ x λ 0 2 n eff .

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