Abstract

We examine the propagation of surface-plasmon polaritons on textured surfaces. Specifically, we look at how a grating surface produces a band gap in the propagation of this mode and how the band gap depends on the propagation direction of the mode. This naturally leads to a discussion of a surface profile suitable for blocking surface mode propagation in all directions. By using a diffractive-optics-based theoretical modeling approach we examine the requirements for such a surface. We then confirm these expectations experimentally by producing a surface exhibiting a band gap for surface-plasmon polariton modes in all directions in the energy range 1.91–2.00 eV.

© 1997 Optical Society of America

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References

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [Crossref] [PubMed]
  2. D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor micro cavity,” J. Appl. Phys. 70, 3443–3445 (1991).
    [Crossref]
  3. R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
    [Crossref]
  4. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).
  5. W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
    [Crossref]
  6. W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
    [Crossref]
  7. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [Crossref]
  8. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [Crossref]
  9. S. C. Kitson, W. L. Barnes, J. R. Sambles, “A full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2760–2763 (1996).
    [Crossref]
  10. E. L. Wood, J. R. Sambles, N. P. K. Cotter, S. C. Kitson, “Diffraction grating characterisation using multiple wavelength excitation of surface plasmon polaritons,” J. Mod. Opt. 41, 1343–1349 (1995).
    [Crossref]
  11. S. C. Kitson, W. L. Barnes, G. W. Bradberry, J. R. Sambles, “Surface profile dependence of surface plasmon band gaps on metallic gratings,” J. Appl. Phys. 79, 7383–7385 (1996).
    [Crossref]
  12. S. C. Kitson, W. L. Barnes, J. R. Sambles, “Surface plasmon energy gaps and photoluminescence,” Phys. Rev. B 52, 11441–11445 (1995).
    [Crossref]
  13. I. Pockrand, A. Brilliante, D. Möbius, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 499–504 (1980).
    [Crossref]
  14. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [Crossref]
  15. S. C. Kitson, W. L. Barnes, J. R. Sambles, “The fabrication of submicron hexagonal arrays using multiple-exposure optical interferometry,” IEEE Photonics Technol. Lett. 8, 1662–1664 (1996).
    [Crossref]
  16. See, for example, P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995).
  17. K. H. Drexhage, in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), pp. 165–232.
  18. J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
    [Crossref]

1996 (4)

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “A full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2760–2763 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, G. W. Bradberry, J. R. Sambles, “Surface profile dependence of surface plasmon band gaps on metallic gratings,” J. Appl. Phys. 79, 7383–7385 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “The fabrication of submicron hexagonal arrays using multiple-exposure optical interferometry,” IEEE Photonics Technol. Lett. 8, 1662–1664 (1996).
[Crossref]

1995 (4)

S. C. Kitson, W. L. Barnes, J. R. Sambles, “Surface plasmon energy gaps and photoluminescence,” Phys. Rev. B 52, 11441–11445 (1995).
[Crossref]

E. L. Wood, J. R. Sambles, N. P. K. Cotter, S. C. Kitson, “Diffraction grating characterisation using multiple wavelength excitation of surface plasmon polaritons,” J. Mod. Opt. 41, 1343–1349 (1995).
[Crossref]

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[Crossref]

1994 (1)

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

1991 (1)

D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor micro cavity,” J. Appl. Phys. 70, 3443–3445 (1991).
[Crossref]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

1982 (1)

1980 (1)

I. Pockrand, A. Brilliante, D. Möbius, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 499–504 (1980).
[Crossref]

1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1968 (1)

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[Crossref]

Arakawa, E. T.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[Crossref]

Barnes, W. L.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “A full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2760–2763 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, G. W. Bradberry, J. R. Sambles, “Surface profile dependence of surface plasmon band gaps on metallic gratings,” J. Appl. Phys. 79, 7383–7385 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “The fabrication of submicron hexagonal arrays using multiple-exposure optical interferometry,” IEEE Photonics Technol. Lett. 8, 1662–1664 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “Surface plasmon energy gaps and photoluminescence,” Phys. Rev. B 52, 11441–11445 (1995).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[Crossref]

Birks, T. A.

See, for example, P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995).

Bloemer, M. J.

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

Bowden, C. M.

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

Bradberry, G. W.

S. C. Kitson, W. L. Barnes, G. W. Bradberry, J. R. Sambles, “Surface profile dependence of surface plasmon band gaps on metallic gratings,” J. Appl. Phys. 79, 7383–7385 (1996).
[Crossref]

Brilliante, A.

I. Pockrand, A. Brilliante, D. Möbius, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 499–504 (1980).
[Crossref]

Chandezon, J.

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Cornet, G.

Cotter, N. P. K.

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[Crossref]

E. L. Wood, J. R. Sambles, N. P. K. Cotter, S. C. Kitson, “Diffraction grating characterisation using multiple wavelength excitation of surface plasmon polaritons,” J. Mod. Opt. 41, 1343–1349 (1995).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[Crossref]

Cowan, J. J.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[Crossref]

Deppe, D. G.

D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor micro cavity,” J. Appl. Phys. 70, 3443–3445 (1991).
[Crossref]

Dowling, J. P.

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

Drexhage, K. H.

K. H. Drexhage, in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), pp. 165–232.

Dupuis, M. T.

Hamm, R. N.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[Crossref]

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Kitson, S. C.

S. C. Kitson, W. L. Barnes, J. R. Sambles, “The fabrication of submicron hexagonal arrays using multiple-exposure optical interferometry,” IEEE Photonics Technol. Lett. 8, 1662–1664 (1996).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “A full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2760–2763 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, G. W. Bradberry, J. R. Sambles, “Surface profile dependence of surface plasmon band gaps on metallic gratings,” J. Appl. Phys. 79, 7383–7385 (1996).
[Crossref]

E. L. Wood, J. R. Sambles, N. P. K. Cotter, S. C. Kitson, “Diffraction grating characterisation using multiple wavelength excitation of surface plasmon polaritons,” J. Mod. Opt. 41, 1343–1349 (1995).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “Surface plasmon energy gaps and photoluminescence,” Phys. Rev. B 52, 11441–11445 (1995).
[Crossref]

Lei, C.

D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor micro cavity,” J. Appl. Phys. 70, 3443–3445 (1991).
[Crossref]

Lloyd-Lucas, F. D.

See, for example, P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995).

Maystre, D.

Möbius, D.

I. Pockrand, A. Brilliante, D. Möbius, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 499–504 (1980).
[Crossref]

Nash, D. J.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[Crossref]

Pockrand, I.

I. Pockrand, A. Brilliante, D. Möbius, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 499–504 (1980).
[Crossref]

Preist, T. W.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[Crossref]

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[Crossref]

Raether, H.

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

Ritchie, R. H.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[Crossref]

Sambles, J. R.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “A full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2760–2763 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, G. W. Bradberry, J. R. Sambles, “Surface profile dependence of surface plasmon band gaps on metallic gratings,” J. Appl. Phys. 79, 7383–7385 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “The fabrication of submicron hexagonal arrays using multiple-exposure optical interferometry,” IEEE Photonics Technol. Lett. 8, 1662–1664 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “Surface plasmon energy gaps and photoluminescence,” Phys. Rev. B 52, 11441–11445 (1995).
[Crossref]

E. L. Wood, J. R. Sambles, N. P. K. Cotter, S. C. Kitson, “Diffraction grating characterisation using multiple wavelength excitation of surface plasmon polaritons,” J. Mod. Opt. 41, 1343–1349 (1995).
[Crossref]

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “A scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[Crossref]

Scalora, M.

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

St. J. Russell, P.

See, for example, P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995).

Wood, E. L.

E. L. Wood, J. R. Sambles, N. P. K. Cotter, S. C. Kitson, “Diffraction grating characterisation using multiple wavelength excitation of surface plasmon polaritons,” J. Mod. Opt. 41, 1343–1349 (1995).
[Crossref]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

Chem. Phys. Lett. (1)

I. Pockrand, A. Brilliante, D. Möbius, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 499–504 (1980).
[Crossref]

IEEE Photonics Technol. Lett. (1)

S. C. Kitson, W. L. Barnes, J. R. Sambles, “The fabrication of submicron hexagonal arrays using multiple-exposure optical interferometry,” IEEE Photonics Technol. Lett. 8, 1662–1664 (1996).
[Crossref]

J. Appl. Phys. (3)

J. P. Dowling, M. Scalora, M. J. Bloemer, C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[Crossref]

S. C. Kitson, W. L. Barnes, G. W. Bradberry, J. R. Sambles, “Surface profile dependence of surface plasmon band gaps on metallic gratings,” J. Appl. Phys. 79, 7383–7385 (1996).
[Crossref]

D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor micro cavity,” J. Appl. Phys. 70, 3443–3445 (1991).
[Crossref]

J. Mod. Opt. (1)

E. L. Wood, J. R. Sambles, N. P. K. Cotter, S. C. Kitson, “Diffraction grating characterisation using multiple wavelength excitation of surface plasmon polaritons,” J. Mod. Opt. 41, 1343–1349 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Phys. Rev. B (4)

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227–6244 (1996).
[Crossref]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “Surface plasmon energy gaps and photoluminescence,” Phys. Rev. B 52, 11441–11445 (1995).
[Crossref]

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Phys. Rev. Lett. (3)

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[Crossref]

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

S. C. Kitson, W. L. Barnes, J. R. Sambles, “A full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2760–2763 (1996).
[Crossref]

Other (3)

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

See, for example, P. St. J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995).

K. H. Drexhage, in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), pp. 165–232.

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Figures (11)

Fig. 1
Fig. 1

SPP dispersion curve. The dashed curve is the dispersion curve for a flat surface; the dotted curve, for a corrugated surface. A frequency gap is opened up in the corrugated case. The gap occurs at ±G, with the Bragg vector of the corrugation being 2G. Also shown are the light lines. These are the dispersion curves for photons traveling at grazing incidence to the interface between the metal and the dielectric, i.e., those having the largest possible value of kx. The surface mode always has more momentum than has a photon of the same frequency; the two cannot therefore couple directly.

Fig. 2
Fig. 2

SPP with wave vector kspp propagating at an angle ψ with respect to the Bragg vector of the grating scatters from the 2G component of the grating, resulting in a mode with a wave vector of kspp.

Fig. 3
Fig. 3

Schematic of the upper (ω+) and the lower (ω-) band edges as a function of propagation angle, based on Eq. (4). Band gaps are produced by two corrugations, oriented at an angle of ψ with respect to each other. For the gap to extend between the two corrugation directions there must be some frequency range for which no modes are allowed. Such a situation is shown here: There are no allowed modes in the shaded region. Note: The dotted portions of the band edges in the overlap region, ψ/2, are a guide for the eye; they do not represent allowed modes. The details of the intersections between the two sets of modes are still unclear.9

Fig. 4
Fig. 4

Different methods for coupling to the SPP band-gap region of the dispersion diagram. (a) Dispersion curve for SPP’s on a doubly corrugated surface, that is, with two grating components: one with Bragg vector 2G, the other with Bragg vector G. The dispersion curve scattered by the 2G component (dotted curve) has gaps outside the light lines and so cannot be coupled to photons. To examine the gap experimentally, the second grating component, G, is used. The dispersion curve scattered by the G component (dashed curve) exhibits a gap between the light lines, thus allowing experimental investigation. (b) Dispersion curve for SPP’s examined by prism coupling. Surface modes are excited by light incident from the prism side, and coupling takes place by means of the evanescent field that occurs on total internal reflection (see inset at bottom left). Owing to the relatively high index of the prism, the light line for photons in the prism is increased to higher momentum. It is now possible to couple to modes in the region of the gap that arises from a corrugated surface with just one grating component, G.

Fig. 5
Fig. 5

k-space representation of the grating surface. The solid circle has a radius of k0 and represents the maximum momentum in the plane of the grating available to a photon propagating above the grating surface. The SPP’s may scatter from the grating, increasing or decreasing their effective wave vector in integer multiples of G. To indicate this, the circles representing the SPP’s (dotted circles) are drawn at multiples of ±G along the direction of the grating Bragg vector. A mode satisfying the Bragg condition [Eq. (3)] can scatter from the surface and can couple resonantly to a photon emitted in the plane normal to the Bragg vector of the grating (ϕ=0). Therefore light emitted in this plane always corresponds to modes that have Bragg reflected from the surface corrugation.

Fig. 6
Fig. 6

Intensity of emission normal to the sample as a function of emission wavelength. This sample exhibits an energy gap at 651 nm.

Fig. 7
Fig. 7

Energy of the modes on either side of the energy gap as a function of the direction of propagation, ψ. For this sample we can see that a band gap exists for propagation between the directions -ψc and +ψc.

Fig. 8
Fig. 8

Dependence of the energy gap on the propagation angle ψ. The center of the energy gap is shown in (a). The circles represent the experimental data, the solid line is the calculated dependence assuming a linear SPP dispersion, and the dotted line is the calculated dependence obtained with a quadratic expression for the mode dispersion (see text). The dependence of the gap width on the propagation angle is shown in (b).

Fig. 9
Fig. 9

Scanning electron micrograph of the hexagonal array of dots. The dots were fabricated in photoresist on a glass substrate, and the surface was coated with silver to support the propagation of SPP’s. The schematic above the micrograph shows the dimensions of the array determined by measurement of the diffraction of 457.9-nm light from the surface. The angles are accurate to ±0.2°; the lengths, to ±0.2 nm.

Fig. 10
Fig. 10

Sample set of reflectivity data recorded as a function of the photon energy and kx, the component of the photon momentum in the plane of the silver–air interface. These data are for a propagation angle of ψ=100° (see inset at top left). The light regions represent high reflectivity; the dark regions, low reflectivity. The black triangular region in the bottom right-hand corner is an artifact of the measurement technique. The inset shows how the propagation angle ψ is defined with respect to one of the principal Bragg vectors. Experimentally this angle is determined by diffraction of a 457.9-nm laser beam.

Fig. 11
Fig. 11

Energies of the upper and the lower branches of the SPP energy gap plotted as a function of the propagation direction ψ. The angles are accurate to ±0.2°; the energies, to ±0.01 eV. There is a full gap between 1.91 and 2.00 eV.

Equations (8)

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Δωω0=2Gd2-12,
ω2¯ω02=12 ω+c2+ω-c2=ω0c2[1-(Gd2)2].
kspp cos ψ=G,
kspp(ψ)=Gcos ψ,
ω+(ψ=0)>ω-(ψ=ψ/2);
s(x)=d1 sin(Gx)+d2 sin(2Gx+ϕ2),
tan ψ=k0 sin θG.
nk0 sin θ=kspp,

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