Abstract

The propagation of a surface plasmon polariton along a finite grating is studied numerically. It is shown that there is a forbidden band of frequencies. The transmittivity of the system exhibits oscillations that are attributed to finite size effects. We also investigate the properties of acceptor defects, which produce localized states. The excitation of these modes yields a peak of the transmittivity within the gap and an enhancement of the near field. Finally, we study the transmittivity and the near field for random gratings that are periodic on average. We observe that the gap width decreases. In addition, we show that the reflectivity of the finite grating increases when the length of the finite grating increases for a frequency out of the gap but close to the band edge. This behavior indicates the presence of Anderson localization.

© 1996 Optical Society of America

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References

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  1. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283–295 (1993).
    [CrossRef]
  2. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).
  3. A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 405–510.
  4. F. Pincemin, A. Sentenac, and J.-J. Greffet, “Near field scattered by a dielectric rod below a metal surface,” J. Opt. Soc. Am. A 11, 1117–1127 (1994).
    [CrossRef]
  5. F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B 50, 15,261–15, 275 (1994).
    [CrossRef]
  6. K. Arya, B. Z. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1561 (1985).
    [CrossRef] [PubMed]
  7. A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  8. S. John, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 1–97.
    [CrossRef]
  9. M. Saillard, “Numerical evidence of Anderson localization for electromagnetic surface waves,” Opt. Commun. 96, 1–7 (1993).
    [CrossRef]
  10. J. Sanchez-Gil, “Anderson localization of surface electromagnetic waves on rough, perfectly conducting surfaces that are periodic on average,” Opt. Commun. (to be published).
  11. D. Sornette, L. Macon, and J. Coste, “Transfer matrix theory of leaky guided waves,” J. Phys. (Paris) 49, 1683–1689 (1988).
    [CrossRef]
  12. J. Sanchez-Gil and A. A. Maradudin, Department of Physics, University of California Irvine, Calif. 92717 (personal communication).
  13. D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
    [CrossRef]
  14. J. W. Haus, “A brief review of theoretical results for photonic band structures,” J. Mod. Opt. 41, 195–207 (1994).
    [CrossRef]
  15. P. R. Villeneuve and M. Piché, “Photonic bandgaps: what is the best numerical representation of periodic structures?” J. Mod. Opt. 41, 241–256 (1994).
    [CrossRef]
  16. E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif.1985).
  17. R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
    [CrossRef]
  18. A. Sentenac, F. Pincemin, and J.-J. Greffet, “Finite size effects on the scattering by a photonic band-gap structure,” submitted to J. Opt. Soc. Am. B.

1994 (5)

F. Pincemin, A. Sentenac, and J.-J. Greffet, “Near field scattered by a dielectric rod below a metal surface,” J. Opt. Soc. Am. A 11, 1117–1127 (1994).
[CrossRef]

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B 50, 15,261–15, 275 (1994).
[CrossRef]

J. W. Haus, “A brief review of theoretical results for photonic band structures,” J. Mod. Opt. 41, 195–207 (1994).
[CrossRef]

P. R. Villeneuve and M. Piché, “Photonic bandgaps: what is the best numerical representation of periodic structures?” J. Mod. Opt. 41, 241–256 (1994).
[CrossRef]

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

1993 (3)

1988 (1)

D. Sornette, L. Macon, and J. Coste, “Transfer matrix theory of leaky guided waves,” J. Phys. (Paris) 49, 1683–1689 (1988).
[CrossRef]

1985 (2)

K. Arya, B. Z. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1561 (1985).
[CrossRef] [PubMed]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Arya, K.

K. Arya, B. Z. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1561 (1985).
[CrossRef] [PubMed]

Aspect, A.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

Birman, J. L.

K. Arya, B. Z. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1561 (1985).
[CrossRef] [PubMed]

Boardman, A. D.

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B 50, 15,261–15, 275 (1994).
[CrossRef]

Celli, V.

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Coste, J.

D. Sornette, L. Macon, and J. Coste, “Transfer matrix theory of leaky guided waves,” J. Phys. (Paris) 49, 1683–1689 (1988).
[CrossRef]

Dalichaouch, R.

Greffet, J.-J.

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B 50, 15,261–15, 275 (1994).
[CrossRef]

F. Pincemin, A. Sentenac, and J.-J. Greffet, “Near field scattered by a dielectric rod below a metal surface,” J. Opt. Soc. Am. A 11, 1117–1127 (1994).
[CrossRef]

A. Sentenac, F. Pincemin, and J.-J. Greffet, “Finite size effects on the scattering by a photonic band-gap structure,” submitted to J. Opt. Soc. Am. B.

Haus, J. W.

J. W. Haus, “A brief review of theoretical results for photonic band structures,” J. Mod. Opt. 41, 195–207 (1994).
[CrossRef]

John, S.

S. John, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 1–97.
[CrossRef]

Kaiser, R.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

Kroll, N.

Leipold, D.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

Lévy, Y.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

Macon, L.

D. Sornette, L. Macon, and J. Coste, “Transfer matrix theory of leaky guided waves,” J. Phys. (Paris) 49, 1683–1689 (1988).
[CrossRef]

Maradudin, A. A.

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B 50, 15,261–15, 275 (1994).
[CrossRef]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

J. Sanchez-Gil and A. A. Maradudin, Department of Physics, University of California Irvine, Calif. 92717 (personal communication).

A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 405–510.

McCall, S. L.

McGurn, A. R.

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Mlynek, J.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif.1985).

Piché, M.

P. R. Villeneuve and M. Piché, “Photonic bandgaps: what is the best numerical representation of periodic structures?” J. Mod. Opt. 41, 241–256 (1994).
[CrossRef]

Pincemin, F.

F. Pincemin, A. Sentenac, and J.-J. Greffet, “Near field scattered by a dielectric rod below a metal surface,” J. Opt. Soc. Am. A 11, 1117–1127 (1994).
[CrossRef]

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B 50, 15,261–15, 275 (1994).
[CrossRef]

A. Sentenac, F. Pincemin, and J.-J. Greffet, “Finite size effects on the scattering by a photonic band-gap structure,” submitted to J. Opt. Soc. Am. B.

Platzman, P. M.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).

Saillard, M.

M. Saillard, “Numerical evidence of Anderson localization for electromagnetic surface waves,” Opt. Commun. 96, 1–7 (1993).
[CrossRef]

Sanchez-Gil, J.

J. Sanchez-Gil, “Anderson localization of surface electromagnetic waves on rough, perfectly conducting surfaces that are periodic on average,” Opt. Commun. (to be published).

J. Sanchez-Gil and A. A. Maradudin, Department of Physics, University of California Irvine, Calif. 92717 (personal communication).

Schultz, S.

Seifert, W.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

Sentenac, A.

F. Pincemin, A. Sentenac, and J.-J. Greffet, “Near field scattered by a dielectric rod below a metal surface,” J. Opt. Soc. Am. A 11, 1117–1127 (1994).
[CrossRef]

A. Sentenac, F. Pincemin, and J.-J. Greffet, “Finite size effects on the scattering by a photonic band-gap structure,” submitted to J. Opt. Soc. Am. B.

Smith, D. R.

Sornette, D.

D. Sornette, L. Macon, and J. Coste, “Transfer matrix theory of leaky guided waves,” J. Phys. (Paris) 49, 1683–1689 (1988).
[CrossRef]

Su, B. Z.

K. Arya, B. Z. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1561 (1985).
[CrossRef] [PubMed]

Vansteenkiste, N.

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

Villeneuve, P. R.

P. R. Villeneuve and M. Piché, “Photonic bandgaps: what is the best numerical representation of periodic structures?” J. Mod. Opt. 41, 241–256 (1994).
[CrossRef]

Yablonovitch, E.

J. Mod. Opt. (2)

J. W. Haus, “A brief review of theoretical results for photonic band structures,” J. Mod. Opt. 41, 195–207 (1994).
[CrossRef]

P. R. Villeneuve and M. Piché, “Photonic bandgaps: what is the best numerical representation of periodic structures?” J. Mod. Opt. 41, 241–256 (1994).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

J. Phys. (Paris) (1)

D. Sornette, L. Macon, and J. Coste, “Transfer matrix theory of leaky guided waves,” J. Phys. (Paris) 49, 1683–1689 (1988).
[CrossRef]

Opt. Commun. (2)

R. Kaiser, Y. Lévy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leipold, and J. Mlynek, “Resonant enhancement of evansecent waves with a thin dielectric waveguide,” Opt. Commun. 104, 234–240 (1994).
[CrossRef]

M. Saillard, “Numerical evidence of Anderson localization for electromagnetic surface waves,” Opt. Commun. 96, 1–7 (1993).
[CrossRef]

Phys. Rev. B (2)

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B 50, 15,261–15, 275 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

K. Arya, B. Z. Su, and J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1561 (1985).
[CrossRef] [PubMed]

Other (7)

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).

A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 405–510.

S. John, in Scattering and Localization of Classical Waves in Random Media, P. Sheng, ed. (World Scientific, Singapore, 1990), pp. 1–97.
[CrossRef]

J. Sanchez-Gil, “Anderson localization of surface electromagnetic waves on rough, perfectly conducting surfaces that are periodic on average,” Opt. Commun. (to be published).

A. Sentenac, F. Pincemin, and J.-J. Greffet, “Finite size effects on the scattering by a photonic band-gap structure,” submitted to J. Opt. Soc. Am. B.

J. Sanchez-Gil and A. A. Maradudin, Department of Physics, University of California Irvine, Calif. 92717 (personal communication).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif.1985).

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Figures (19)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

(a) Transmission, (b) reflection, and (c) scattering coefficients versus frequency for SPP impinging upon a finite grating of 30 periods. Two period lengths have been studied, d = 1.5γp and d = λp, with 2π/λp = ωp/c. The other parameters are a = 4λp/10, b = 3λp/10, 2 = 2.25, 3 = 1 − ωp2/ω2, 1 = 1, and Δx = Δz = λ/p/10.

Fig. 3
Fig. 3

Transmission coefficient versus the number (Nb.) of periods. The parameters are d = 1.5λp, a = 4λp/10, b = 3λp/10, 2 = 2.25, 3 = 1 − ωp2/ω2, 1 = 1, and Δx = Δz = λ/p/10.

Fig. 4
Fig. 4

Square of the modulus of the electric field above the finite grating versus x and at constant z = 3.5λp/10. The edges of the grating are marked by arrows. Two frequencies have been studied: (a) one in a band ω/ωp = 0.27 and (b) one in the gap ω/ωp = 0.3. The grating is 30 periods long. The other parameters are those for Fig. 3.

Fig. 5
Fig. 5

Same as Fig. 4 but for two other frequencies, (a) ω/ωp = 0.29 and (b) ω/ωp = 0.283.

Fig. 6
Fig. 6

Dispersion curve for a SPP propagating along the grating with the following parameters: d = 1.5λp, a = 4λp/10, b = 3λp/10, 2 = 2.25, 3 = 1 − ωp2/ω2, and 1 = 1.

Fig. 7
Fig. 7

Transmission coefficient versus frequency. The system consists of two finite gratings of 15 periods separated by a long zone 8 periods long without ridges (sold curve) or 4 periods long without ridges (dashed curve). The parameters for the grating are the same as for Fig. 6.

Fig. 8
Fig. 8

Transmission coefficient versus frequency. The system consists of two finite gratings of 25 periods separated by a long zone 8 periods long without ridges. The parameters for the grating are the same as for Fig. 6.

Fig. 9
Fig. 9

Square of the modulus of the electric field above the system studied in Fig. 8 versus x and at constant z = 3.5λp/10. The edges of the vacancy region (zone without ridges) are marked by arrows. Two frequencies in the gap have been studied: (a) ω/ωp = 0.305 and (b) ω/ωp = 0.2992. The other parameters are the same as for Fig. 3.

Fig. 10
Fig. 10

Scattering coefficient versus frequency. The system consists of two finite gratings of 25 periods separated by a long zone 8 periods long without ridges (solid curve) or a 50-period-long grating (dashed curve). The other parameters are the same as for Fig. 6, and Δx = Δz = λ/p/10.

Fig. 11
Fig. 11

Transmission coefficient versus frequency. The system consists of two finite gratings of 25 periods separated by a long zone 8 periods long with a random defect (see text). The parameters for the grating are the same as for Fig. 6.

Fig. 12
Fig. 12

Square of the modulus of the electric field above the system studied in Fig. 11 versus x and at constant z = 3.5λp/10. Two frequencies in the gap have been studied: ω/ωp = 0.297 (solid curve) and ω/ωp = 0.305 (dashed curve). The other parameters are the same as for Fig. 3.

Fig. 13
Fig. 13

Transmission coefficient versus frequency. The system consists of a 50-period random grating that is periodic on average. Two deviations Δ have been studied: Δ = 1 (long-and short-dashed curve) and Δ = 0.25 (solid curve). Only one realization has been done (no average). The other parameters for grating are the same as for Figs. 3.

Fig. 14
Fig. 14

Square of the modulus of the electric field above the system studied in Fig. 13 with Δ = 1 versus x and at constant z = 3.5λp/10. The other parameters are ω/ωp = 0.312 and those for Fig. 3.

Fig. 15
Fig. 15

Average of the logarithm of the transmission coefficient versus the number (Nb.) of periods. The parameters are Δ = 1, ω/ωp = 0.3 and those for Fig. 3. The numbers of realizations for the average are 60 for a 20-period grating, 40 for a 30-period grating, 30 for a 40-period grating, 25 for a 50-period grating, 20 for a 60-period grating, and 15 for a 70-period grating.

Fig. 16
Fig. 16

Localization length versus frequency. The parameters are Δ = 1 and those for Fig. 3. The arrow indicates the presumed edge of the gap of the band structure of the underlying periodic grating.

Fig. 17
Fig. 17

Reflection and scattering coefficients versus the number (Nb.) of periods for (b) a periodic grating and (a) a random grating with Δ = 1 for a frequency near the band edge ω/ωp = 0.29. The common parameters are those for Fig. 3. The numbers of realization for the average are 60 for a 20-period grating, 50 for a 25-period grating, 40 for a 30-period grating, 35 for a 35-period grating, 30 for a 40- or 45-period grating, 25 for a 50- or 55-period grating, 20 for a 60- or 65-period grating and 15 for a 70-period grating.

Fig. 18
Fig. 18

Three-layer system corresponding to the case in which a = ∞.

Fig. 19
Fig. 19

Schematic of the different waves interfering in the grating.

Equations (10)

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E ( x , z ) = E f ( x , z ) + k 0 2 d x d z G ( x x , z , z ) × [ ( x , z ) f ( z ) ] E ( x , z ) .
R = | E x , sc ( x , z ) E x , f ( x , z ) | 2 .
T = | E x , sc ( x , z ) E x , f ( x , z ) | 2 ,
S = P sc P inc
ω ω p = λ p ( 1 + c κ sp / ω ) d
E ( x ) = A { exp [ i ( π / d δ ) x ] ± exp [ i ( π / d + δ ) x ] } , E ( x ) = A exp [ i ( π / d δ ) x ] { 1 ± exp [ i 2 π x / d ] } ,
E ( x ) = 2 A exp [ i δ x ] cos ( π x / d ) or E ( x ) = 2 i A exp [ i δ x ] sin ( π x / d ) .
2 κ sp L vr + arg ( r 1 ) + arg ( r 2 ) = 2 p π ,
2 κ ( 2 ) sp F L dz + 2 κ sp ( 1 F ) L dz + arg ( r 1 ) + arg ( r 2 ) = 2 p π ,
κ ( 2 ) sp a + κ sp ( d a ) = π + n 2 π .

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