Abstract

We present a semianalytical model that quantitatively predicts the scattering of light by a single subwavelength slit in a thick metal screen. In contrast to previous theoretical works related to the transmission properties of the slit, the analysis emphasizes the generation of surface plasmons at the slit apertures. The model relies on a two-stage scattering mechanism, a purely geometric diffraction problem in the immediate vicinity of the slit aperture followed by the launching of a bounded surface-plasmon wave on the flat interfaces surrounding the aperture. By comparison with a full electromagnetic treatment, the model is shown to provide accurate formulas for the plasmonic generation strength coefficients, even for metals with a low conductivity. Limitations are outlined for large slit widths (>λ) or oblique incidence (>30°) when the slit is illuminated by a plane wave.

© 2006 Optical Society of America

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  1. T. Thio, K. M. Pellerin, R. A. Linke, T. W. Ebbesen, and H. J. Lezec, 'Enhanced light transmission through a single subwavelength aperture,' Opt. Lett. 26, 1972-1974 (2001).
    [CrossRef]
  2. F. I. Baida and D. Van Labeke, 'Light transmission by subwavelength annular aperture arrays in metallic films,' Opt. Commun. 209, 17-22 (2002).
    [CrossRef]
  3. K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, 'Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,' Phys. Rev. Lett. 92, 183901 (2004).
    [CrossRef] [PubMed]
  4. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
    [CrossRef] [PubMed]
  5. H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, 'Transmission of light through slit aperture in metallic films,' Opt. Express 12, 6106-6121 (2004).
    [CrossRef] [PubMed]
  12. P. Lalanne, J. C. Rodier, and J. P. Hugonin, 'Surface plasmons of metallic surfaces perforated by nanohole arrays,' J. Opt. A Pure Appl. Opt. 7, 422-426 (2005).
    [CrossRef]
  13. L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
    [CrossRef]
  14. H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
    [CrossRef]
  15. P. Lalanne, J. P. Hugonin, and J. C. Rodier, 'Theory of surface plasmon generation at nanoslit apertures,' Phys. Rev. Lett. 95, 263902 (2005).
    [CrossRef]
  16. Inaccuracies due to numerical truncation in the modal expansions are inevitable in numerics but they are kept at a low level. We estimate that the scattering coefficients are calculated with a relative accuracy better than 1%.
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  18. P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
    [CrossRef]
  19. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, 'Transmission of light through slit apertures in metallic films,' Opt. Express 12, 6106-6121 (2004).
    [CrossRef] [PubMed]
  20. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, 'Use of grating theories in integrated optics,' J. Opt. Soc. Am. A 18, 2865-2875 (2001).
    [CrossRef]
  21. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, 'Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,' J. Opt. Soc. Am. A 12, 1068-1076 (1995).
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  22. Ph. Lalanne and G. M. Morris, 'Highly improved convergence of the coupled-wave method for TM polarization,' J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  23. G. Granet and B. Guizal, 'Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,' J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  24. L. Li, 'Mathematical reflections on the Fourier modal method in grating theory,' in Mathematical Modeling in Optical Science, Frontiers in Applied Mathematics, G.Bao, L.Cowsar, and W.Masters, eds. (Society for Industrial and Applied Mathematics, 2001), pp. 111-139.
    [CrossRef]
  25. J. P. Bérenger, 'A perfectly matched layer for the absorption of electromagnetic waves,' J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  26. J. P. Hugonin and P. Lalanne, 'Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,' J. Opt. Soc. Am. A 22, 1844-1849 (2005).
    [CrossRef]
  27. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).
  28. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
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  30. Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, 'One-mode model and Airy-like formulae for 1D metallic gratings,' J. Opt. A Pure Appl. Opt. 2, 48-51 (2000).
    [CrossRef]
  31. This approximation amounts to considering that the slit is narrow enough so that all modes are evanescent except the fundamental one; see Ref. for a discussion of this approximation with respect to the metallic film thickness h.
  32. Standard techniques, like the adaptive Simpson quadrature method implemented with the function quad in MATLAB (MathWorks, Inc.), can be used.
  33. H. Lochbihler and R. A. Depine, 'Highly conducting wire gratings in the resonance domain,' Appl. Opt. 32, 3459-3465 (1993).
    [CrossRef] [PubMed]

2005 (6)

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

S. H. Chang, S. K. Gray, and G. C. Schatz, 'Surface-plasmon generation and light transmission by isolated nanoholes and nanohole arrays in thin metal films,' Opt. Express 13, 3150-3165 (2005).
[CrossRef] [PubMed]

P. Lalanne, J. C. Rodier, and J. P. Hugonin, 'Surface plasmons of metallic surfaces perforated by nanohole arrays,' J. Opt. A Pure Appl. Opt. 7, 422-426 (2005).
[CrossRef]

P. Lalanne, J. P. Hugonin, and J. C. Rodier, 'Theory of surface plasmon generation at nanoslit apertures,' Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
[CrossRef]

J. P. Hugonin and P. Lalanne, 'Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,' J. Opt. Soc. Am. A 22, 1844-1849 (2005).
[CrossRef]

2004 (4)

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, 'Transmission of light through slit apertures in metallic films,' Opt. Express 12, 6106-6121 (2004).
[CrossRef] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, 'Transmission of light through slit aperture in metallic films,' Opt. Express 12, 6106-6121 (2004).
[CrossRef] [PubMed]

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, 'Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,' Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

2002 (3)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

F. I. Baida and D. Van Labeke, 'Light transmission by subwavelength annular aperture arrays in metallic films,' Opt. Commun. 209, 17-22 (2002).
[CrossRef]

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

2001 (3)

2000 (1)

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, 'One-mode model and Airy-like formulae for 1D metallic gratings,' J. Opt. A Pure Appl. Opt. 2, 48-51 (2000).
[CrossRef]

1996 (2)

1995 (1)

1994 (1)

J. P. Bérenger, 'A perfectly matched layer for the absorption of electromagnetic waves,' J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

1993 (1)

1987 (1)

1954 (1)

C. J. Bouwkamp, 'Diffraction theory,' Rep. Prog. Phys. 17, 35-100 (1954).
[CrossRef]

1944 (1)

H. Bethe, 'Theory of diffraction by small holes,' Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Alkemade, P. F. A.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Astilean, S.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, 'One-mode model and Airy-like formulae for 1D metallic gratings,' J. Opt. A Pure Appl. Opt. 2, 48-51 (2000).
[CrossRef]

Aussenegg, F. R.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Baida, F. I.

F. I. Baida and D. Van Labeke, 'Light transmission by subwavelength annular aperture arrays in metallic films,' Opt. Commun. 209, 17-22 (2002).
[CrossRef]

Bérenger, J. P.

J. P. Bérenger, 'A perfectly matched layer for the absorption of electromagnetic waves,' J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

Bethe, H.

H. Bethe, 'Theory of diffraction by small holes,' Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Blok, H.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Bouwkamp, C. J.

C. J. Bouwkamp, 'Diffraction theory,' Rep. Prog. Phys. 17, 35-100 (1954).
[CrossRef]

Brown, D. E.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Cao, Q.

Chang, S. H.

Chang, S.-H.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Cheng, Y. R.

P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
[CrossRef]

Chou, H. L.

P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
[CrossRef]

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Depine, R. A.

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Ditlbacher, H.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Dubois, G.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Ebbesen, T. W.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, T. W. Ebbesen, and H. J. Lezec, 'Enhanced light transmission through a single subwavelength aperture,' Opt. Lett. 26, 1972-1974 (2001).
[CrossRef]

Eliel, E. R.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Enoch, S.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, 'Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,' Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Fann, W.

P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
[CrossRef]

Felidj, N.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Gaylord, T. K.

Gbur, G.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Granet, G.

Grann, E. B.

Gray, S. K.

S. H. Chang, S. K. Gray, and G. C. Schatz, 'Surface-plasmon generation and light transmission by isolated nanoholes and nanohole arrays in thin metal films,' Opt. Express 13, 3150-3165 (2005).
[CrossRef] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Guizal, B.

Hugonin, J. P.

J. P. Hugonin and P. Lalanne, 'Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,' J. Opt. Soc. Am. A 22, 1844-1849 (2005).
[CrossRef]

P. Lalanne, J. C. Rodier, and J. P. Hugonin, 'Surface plasmons of metallic surfaces perforated by nanohole arrays,' J. Opt. A Pure Appl. Opt. 7, 422-426 (2005).
[CrossRef]

P. Lalanne, J. P. Hugonin, and J. C. Rodier, 'Theory of surface plasmon generation at nanoslit apertures,' Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, 'Use of grating theories in integrated optics,' J. Opt. Soc. Am. A 18, 2865-2875 (2001).
[CrossRef]

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, 'One-mode model and Airy-like formulae for 1D metallic gratings,' J. Opt. A Pure Appl. Opt. 2, 48-51 (2000).
[CrossRef]

Kimball, C. W.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Koerkamp, K. J. K.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, 'Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,' Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Krenn, J. R.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Kuipers, L.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, 'Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,' Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Kuzmin, N.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Lalanne, P.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, 'Theory of surface plasmon generation at nanoslit apertures,' Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

P. Lalanne, J. C. Rodier, and J. P. Hugonin, 'Surface plasmons of metallic surfaces perforated by nanohole arrays,' J. Opt. A Pure Appl. Opt. 7, 422-426 (2005).
[CrossRef]

J. P. Hugonin and P. Lalanne, 'Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,' J. Opt. Soc. Am. A 22, 1844-1849 (2005).
[CrossRef]

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, 'Use of grating theories in integrated optics,' J. Opt. Soc. Am. A 18, 2865-2875 (2001).
[CrossRef]

Lalanne, Ph.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, 'One-mode model and Airy-like formulae for 1D metallic gratings,' J. Opt. A Pure Appl. Opt. 2, 48-51 (2000).
[CrossRef]

Ph. Lalanne and G. M. Morris, 'Highly improved convergence of the coupled-wave method for TM polarization,' J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

Lamprecht, B.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Leitner, A.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Lenstra, D.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Lezec, H. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, T. W. Ebbesen, and H. J. Lezec, 'Enhanced light transmission through a single subwavelength aperture,' Opt. Lett. 26, 1972-1974 (2001).
[CrossRef]

Li, L.

L. Li, 'Mathematical reflections on the Fourier modal method in grating theory,' in Mathematical Modeling in Optical Science, Frontiers in Applied Mathematics, G.Bao, L.Cowsar, and W.Masters, eds. (Society for Industrial and Applied Mathematics, 2001), pp. 111-139.
[CrossRef]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, T. W. Ebbesen, and H. J. Lezec, 'Enhanced light transmission through a single subwavelength aperture,' Opt. Lett. 26, 1972-1974 (2001).
[CrossRef]

Lochbihler, H.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Mansuripur, M.

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Moharam, M. G.

Möller, K. D.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, 'One-mode model and Airy-like formulae for 1D metallic gratings,' J. Opt. A Pure Appl. Opt. 2, 48-51 (2000).
[CrossRef]

Moloney, J. V.

Morris, G. M.

Palamaru, M.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, 'One-mode model and Airy-like formulae for 1D metallic gratings,' J. Opt. A Pure Appl. Opt. 2, 48-51 (2000).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985), Part II.

Pearson, J.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Pellerin, K. M.

Pommet, D. A.

Raether, H.

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

Roberts, A.

Rodier, J. C.

P. Lalanne, J. C. Rodier, and J. P. Hugonin, 'Surface plasmons of metallic surfaces perforated by nanohole arrays,' J. Opt. A Pure Appl. Opt. 7, 422-426 (2005).
[CrossRef]

P. Lalanne, J. P. Hugonin, and J. C. Rodier, 'Theory of surface plasmon generation at nanoslit apertures,' Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

Rydh, A.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Salerno, M.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Schatz, G. C.

S. H. Chang, S. K. Gray, and G. C. Schatz, 'Surface-plasmon generation and light transmission by isolated nanoholes and nanohole arrays in thin metal films,' Opt. Express 13, 3150-3165 (2005).
[CrossRef] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Schider, G.

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

Schouten, H. F.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Segerink, F. B.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, 'Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,' Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Silberstein, E.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

't Hooft, G. W.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Tegenfeldt, J. O.

P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
[CrossRef]

Thio, T.

van Hulst, N. F.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, 'Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,' Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

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F. I. Baida and D. Van Labeke, 'Light transmission by subwavelength annular aperture arrays in metallic films,' Opt. Commun. 209, 17-22 (2002).
[CrossRef]

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C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Visser, T. D.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Vlasko-Vlasov, V. K.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Wannemacher, R.

R. Wannemacher, 'Plasmon-supported transmission of light through nanometric holes in metallic thin films,' Opt. Commun. 195, 107-118 (2001).
[CrossRef]

Wei, C. H.

P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
[CrossRef]

Wei, P. W.

P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
[CrossRef]

Welp, U.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Xie, Y.

Yin, L.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

Zakharian, A. R.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. E. Brown, and C. W. Kimball, 'Surface plasmons at single nanoholes in Au films,' Appl. Phys. Lett. 85, 467-469 (2004).
[CrossRef]

H. Ditlbacher, J. R. Krenn, N. Felidj, B. Lamprecht, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, 'Fluorescence imaging of surface plasmon fields,' Appl. Phys. Lett. 80, 404-406 (2002).
[CrossRef]

J. Comput. Phys. (1)

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[CrossRef]

J. Opt. A Pure Appl. Opt. (2)

P. Lalanne, J. C. Rodier, and J. P. Hugonin, 'Surface plasmons of metallic surfaces perforated by nanohole arrays,' J. Opt. A Pure Appl. Opt. 7, 422-426 (2005).
[CrossRef]

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, 'One-mode model and Airy-like formulae for 1D metallic gratings,' J. Opt. A Pure Appl. Opt. 2, 48-51 (2000).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (3)

R. Wannemacher, 'Plasmon-supported transmission of light through nanometric holes in metallic thin films,' Opt. Commun. 195, 107-118 (2001).
[CrossRef]

F. I. Baida and D. Van Labeke, 'Light transmission by subwavelength annular aperture arrays in metallic films,' Opt. Commun. 209, 17-22 (2002).
[CrossRef]

P. W. Wei, H. L. Chou, Y. R. Cheng, C. H. Wei, W. Fann, and J. O. Tegenfeldt, 'Beaming effect of optical near-field in multiple metallic slits with nanometric linewidth and micrometer pitch,' Opt. Commun. 253, 198-204 (2005).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. (1)

H. Bethe, 'Theory of diffraction by small holes,' Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Phys. Rev. Lett. (3)

P. Lalanne, J. P. Hugonin, and J. C. Rodier, 'Theory of surface plasmon generation at nanoslit apertures,' Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. 't Hooft, D. Lenstra, and E. R. Eliel, 'Plasmon-assisted two-slit transmission: Young's experiment revisited,' Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, 'Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,' Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, 'Diffraction theory,' Rep. Prog. Phys. 17, 35-100 (1954).
[CrossRef]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, 'Beaming light from a subwavelength aperture,' Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Other (8)

Inaccuracies due to numerical truncation in the modal expansions are inevitable in numerics but they are kept at a low level. We estimate that the scattering coefficients are calculated with a relative accuracy better than 1%.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985), Part II.

L. Li, 'Mathematical reflections on the Fourier modal method in grating theory,' in Mathematical Modeling in Optical Science, Frontiers in Applied Mathematics, G.Bao, L.Cowsar, and W.Masters, eds. (Society for Industrial and Applied Mathematics, 2001), pp. 111-139.
[CrossRef]

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

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

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

This approximation amounts to considering that the slit is narrow enough so that all modes are evanescent except the fundamental one; see Ref. for a discussion of this approximation with respect to the metallic film thickness h.

Standard techniques, like the adaptive Simpson quadrature method implemented with the function quad in MATLAB (MathWorks, Inc.), can be used.

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Figures (8)

Fig. 1
Fig. 1

SPP excitations at a metallic interface perforated by a single slit under illumination by (a) the fundamental mode of the slit or (b) a plane wave with an incidence angle θ. The slit width is denoted by w, and n 1 and n 2 refer to the refractive indices inside and below the slit. α + ( x ) , α ( x ) , β + ( x ) , and β ( x ) are the SPP generation coefficients defined for an incident wave with a unit power flow over the slit aperture area.

Fig. 2
Fig. 2

Validation of the rigorous formalism for the calculation of the SPP coupling coefficients. (a)–(d) Near-field patterns generated at a gold interface under illumination by the fundamental slit mode. (a) and (b) H y ; (c) and (d) E z . Red is high magnitude, blue is low, and a saturated scale is used to reinforce the field in the vicinity of the slit. (e) and (f) SPP generation strengths α + ( x ) 2 and α ( x ) 2 obtained from the near-field patterns shown in (a)–(d) by calculating the overlap integrals of Eqs. (6a, 6b). In (f), the circles are numerical data equal to α + ( w 2 ) 2 exp [ 2 Im ( k SP ) ( x w 2 ) ] with α + ( w 2 ) 2 = 0.202 . (a), (c), and (e) are obtained for λ = 1.5 μ m and w = 0.35 μ m ; (b), (d), and (f) are for λ = 0.6 μ m and w = 0.14 μ m . Other parameters are n 1 = n 2 = 1 .

Fig. 3
Fig. 3

Variation of the main physical quantities associated with the scattering problems considered in Fig. 2 as a function of the slit width. α + ( x ) 2 + α ( x ) 2 represents the total SPP excitation, R 0 = r 0 2 is the modal reflectivity, and E FF is the far-field energy radiated in medium 2. The power of the incident slit mode is 1.

Fig. 4
Fig. 4

SPP excitations at a slit perforated in a metal film (thickness h) sandwiched between two uniform media of refractive indices n 2 and n 3 and illuminated by a plane wave with an incidence angle θ. Inside the slit the refractive index is denoted by n 1 . The s 2 , s 2 , s 3 , s 3 represent the SPP generation coefficients at the top and bottom interfaces.

Fig. 5
Fig. 5

Total SPP generation efficiencies e SP = α + ( x ) 2 + α ( x ) 2 for a slit illuminated by its fundamental guided mode as a function of the slit width. Solid curves represent the model predictions. The symbols represent the calculated data obtained with the rigorous formalism. (a) Influence of the metal (gold) permittivity, n 1 = n 2 = 1 . (b) Influence of the substrate refractive index n 2 , n 1 = 1 and λ = 3 μ m .

Fig. 6
Fig. 6

SPP generation efficiencies for a slit illuminated by a plane wave, (a) e SP = β + ( w 2 ) 2 + β ( w 2 ) 2 as a function of the slit width for different wavelengths, λ = 0.6 , 1 , 3 , and 10 μ m , for n 1 = n 2 = 1 and θ = 0 . (b) Influence of the incident angle for λ = 1.5 μ m ; other parameters are n 1 = n 2 = 1 and w λ = 0.3 . The model does not predict any difference between β + ( w 2 ) 2 and β ( w 2 ) 2 for any θ. In (a) and (b), the solid curves represent the model predictions and the symbols represent the calculated data obtained with the rigorous formalism.

Fig. 7
Fig. 7

Total SPP generation efficiency at a groove perforated in a gold substrate as a function of the groove depth h for normally incident light ( θ = 0 ) . Solid curves, model predictions obtained with Eq. (20a) for n 1 = n 2 = 1 and n 3 = ϵ 1 2 . Squares, calculated data using the rigorous formalism and the Fourier modal method. (a) λ = 0.8 μ m , w λ = 0.1 , and n 0 eff = 1.29 + 0.0098 i . (b) λ = 1.5 μ m , w λ = 0.1 , and n 0 eff = 1.16 + 0.0084 i .

Fig. 8
Fig. 8

Phase factors associated with the SPP scattering processes. Solid curve, model predictions obtained with Eqs. (11, 17). Plusses, computational data obtained with the rigorous formalism. From top to bottom, λ = 0.6 , 0.8, and 3 μ m .

Tables (1)

Tables Icon

Table 1 I 1 and I 0 for Gold at λ = 800 nm ( ϵ = 26.27 + 1.85 i )

Equations (25)

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H 1 ( x , z ) = Ψ 0 ( x ) exp ( i k n 0 eff z ) + Σ p r p Ψ p ( x ) exp ( i k n p eff z ) ,
H 2 ( x , z ) = d u t u exp ( i k n 2 u x ) exp ( i k n 2 γ u z ) ,
H y = ( α + ( x ) + α ( x ) ) H SP ( z ) + Σ σ a σ ( x ) H σ ( rad ) ( z ) ,
E z = ( α + ( x ) α ( x ) ) E SP ( z ) + Σ σ a σ ( x ) E σ ( rad ) ( z ) .
H SP ( z ) = ( N SP ) 1 exp ( i γ SP z ) ,
k SP = k ϵ n 2 2 ( ϵ + n 2 2 ) .
d z H y ( x , z ) E SP ( z ) = 2 ( α + ( x ) + α ( x ) ) ,
d z E z ( x , z ) H SP ( z ) = 2 ( α + ( x ) α ( x ) ) .
H 1 ( x , z ) = ( 1 + r 0 ) Ψ 0 ( x , z ) ,
t u = ( N 0 ) 1 2 2 ( n 2 n 1 ) w ( 1 + ( n 2 n 1 ) w I 0 ) sinc ( π w u ) γ u ,
r 0 = ( n 2 n 1 ) w I 0 1 ( n 2 n 1 ) w I 0 + 1 .
2 α + ( w 2 ) = 2 α ( w 2 ) = 0 d z H y ( x , w 2 ) E SP ( z ) ,
α = α + ( w 2 ) = α ( w 2 ) = i ( 4 π n 2 2 n 1 ϵ ( ϵ n 2 2 ) w ) 1 2 I 1 1 + ( n 2 n 1 ) w I 0 ,
I 1 = d u sinc ( π w u ) exp ( i π w u ) γ u ( γ u + n 2 2 ( ϵ + n 2 2 ) ) .
α 2 = α + ( w 2 ) 2 = α ( w 2 ) 2 = 4 w n 2 2 π n 1 ϵ 1 2 ϵ + n 2 2 I 1 1 + ( n 2 n 1 ) w I 0 2 .
H 1 ( x , z ) = t 0 Ψ 0 ( x , z ) ,
H 2 ( x , z ) = ( N P ) 1 2 exp [ i k n 2 sin ( θ ) x i k n 2 cos ( θ ) z ] + d u r u exp ( i k n 2 u x ) exp ( i k n 2 γ u z ) ,
t 0 = ( N 0 N P ) 1 2 2 sinc ( π w sin ( θ ) ) ( n 2 n 1 ) w I 0 + 1
r u = δ ( θ ) t 0 ( N 0 N P ) 1 2 w n 2 n 1 sinc ( π w u ) ( 1 u 2 ) 1 2
β = β + ( w 2 ) = β ( w 2 ) = ( N 0 N p ) 1 2 sin c [ π w sin ( θ ) ] α + ( w 2 ) ,
β 2 = β + ( w 2 ) 2 = β ( w 2 ) 2 = 4 w n 2 3 π cos ( θ ) n 1 2 ϵ 1 2 ϵ + n 2 2 I 1 sinc ( π w sin ( θ ) ) 1 + ( n 2 n 1 ) w I 0 2 .
a = t 21 + r 12 b exp ( i k n 0 eff h ) ,
b = r 13 a exp ( i k n 0 eff h ) ,
s 2 = s 2 = β + t 21 r 13 α exp ( 2 i k n 0 eff h ) 1 r 12 r 13 exp ( 2 i k n 0 eff h ) ,
s 3 = s 3 = t 21 α exp ( i k n 0 eff h ) 1 r 12 r 13 exp ( 2 i k n 0 eff h ) .

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