Abstract

We discuss a mode expansion technique to rigorously model the diffraction from three-dimensional pits and holes in a perfectly conducting layer with finite thickness. On the basis of our simulations we predict extraordinary transmission through a single hole, caused by the Fabry-Perot effect inside the hole. Furthermore, we study the fundamental building block for extraordinary transmission through hole arrays: two and three holes. Coupled electromagnetic surface waves, the perfect conductor equivalent of a surface plasmon, are found to play a key role in the mutual interaction between two or three holes.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H.A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163 (1944).
    [CrossRef]
  2. J. Meixner and W. Andrejewski, "Strenge Theorie der Beugung ebener elektromagnetischer Wellen and der vollkommen leitenden Kreisscheibe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm," Annalen der Physik 7, 157-168 (1950).
    [CrossRef]
  3. C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. I. Oblate spheroidal vector wave functions," J. Appl. Phys. 24, 1218-1223 (1953).
    [CrossRef]
  4. C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. II. The diffraction problems," J. Appl. Phys. 24, 1224-1231 (1953).
    [CrossRef]
  5. C.J. Bouwkamp, "Diffraction theory," Reports on progress in physics 17, 35-100 (1954).
    [CrossRef]
  6. T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
    [CrossRef]
  7. H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036,608 (2003).
    [CrossRef]
  8. J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, "Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026,601 (2004).
    [CrossRef]
  9. H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
    [CrossRef] [PubMed]
  10. M.G. Moharam and T.K. Gaylord, "Rigorous coupled-wave analysis of metallic surface-relief gratings," J. Opt. Soc. Am. A 3, 1780-1787 (1986).
    [CrossRef]
  11. J.A. Porto, F.J. Garcia-Vidal, and J.B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
    [CrossRef]
  12. Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057,403 (2002).
    [CrossRef] [PubMed]
  13. L. Martin-Moreno and F.J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619-3628 (2004).
    [CrossRef]
  14. S.H. Chang, S.K. Gray, and G.C. Schatz, "Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films," Opt. Express 13, 3150-3165 (2005).
    [CrossRef] [PubMed]
  15. F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103,901 (2005).
    [CrossRef] [PubMed]
  16. F.J.G.I. de Abajo, "Light transmission through a single cylindrical hole in a metallic film," Opt. Express 10, 1475-1484 (2002).
    [PubMed]
  17. A. Roberts, "Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen," J. Opt. Soc. Am. A 4, 1970-1983 (1987).
    [CrossRef]
  18. J.M. Brok and H.P. Urbach, "A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording," J. Mod. Opt. 51, 2059-2077 (2004).
    [CrossRef]
  19. J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
    [CrossRef] [PubMed]
  20. A.P. Hibbins, B.R. Evans, and J.R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670 (2005).
    [CrossRef] [PubMed]
  21. Here and henceforth the square root of a complex number z is defined such that for real z > 0 we have √z > 0 and √z = +i√z, with the branch cut along the negative real axis.
  22. J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
  23. Although υ¯α(x,y) is real, we include the conjugation for consistency of the notation.
  24. J. van Bladel, Singular Electromagnetic Fields and Sources, 1st ed. (Clarendon Press, Oxford, 1991).
  25. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, 1st ed. (Springer-Verlag, Berlin, 1988).
  26. <other>. Although the ± for α4 now does not have anything to do with the propagation direction, for consistency, we stick to this notation. </other>
  27. The reader might wonder why we do not use the sine and cosine form always instead of using the exponential form only for | γz/kp|< ε. However, for large and purely imaginary γz, the functions cos (γzz) and sin (γzz) increase exponentially with |z|, which is not very convenient for numerical implementation.
  28. The interaction integral also has some other properties that can save a lot of computation time. These properties are outside the scope of this paper.
  29. This routine is a multi-dimensional adaptive quadrature over a hyper-rectangle. For a description, see the internet link: http://www.nag.com/nagware/mt/doc/d01fcf.html.

2005 (4)

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103,901 (2005).
[CrossRef] [PubMed]

A.P. Hibbins, B.R. Evans, and J.R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670 (2005).
[CrossRef] [PubMed]

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

2004 (4)

L. Martin-Moreno and F.J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619-3628 (2004).
[CrossRef]

J.M. Brok and H.P. Urbach, "A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording," J. Mod. Opt. 51, 2059-2077 (2004).
[CrossRef]

J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, "Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026,601 (2004).
[CrossRef]

2003 (1)

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036,608 (2003).
[CrossRef]

2002 (2)

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057,403 (2002).
[CrossRef] [PubMed]

F.J.G.I. de Abajo, "Light transmission through a single cylindrical hole in a metallic film," Opt. Express 10, 1475-1484 (2002).
[PubMed]

1999 (1)

J.A. Porto, F.J. Garcia-Vidal, and J.B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

1998 (1)

T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

1987 (1)

1986 (1)

1954 (1)

C.J. Bouwkamp, "Diffraction theory," Reports on progress in physics 17, 35-100 (1954).
[CrossRef]

1953 (2)

C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. I. Oblate spheroidal vector wave functions," J. Appl. Phys. 24, 1218-1223 (1953).
[CrossRef]

C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. II. The diffraction problems," J. Appl. Phys. 24, 1224-1231 (1953).
[CrossRef]

1950 (1)

J. Meixner and W. Andrejewski, "Strenge Theorie der Beugung ebener elektromagnetischer Wellen and der vollkommen leitenden Kreisscheibe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm," Annalen der Physik 7, 157-168 (1950).
[CrossRef]

1944 (1)

H.A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163 (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. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

Andrejewski, W.

J. Meixner and W. Andrejewski, "Strenge Theorie der Beugung ebener elektromagnetischer Wellen and der vollkommen leitenden Kreisscheibe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm," Annalen der Physik 7, 157-168 (1950).
[CrossRef]

Bethe, H.A.

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

Blok, H.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036,608 (2003).
[CrossRef]

Bouwkamp, C.J.

C.J. Bouwkamp, "Diffraction theory," Reports on progress in physics 17, 35-100 (1954).
[CrossRef]

Bravo-Abad, J.

J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, "Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026,601 (2004).
[CrossRef]

Brok, J.M.

J.M. Brok and H.P. Urbach, "A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording," J. Mod. Opt. 51, 2059-2077 (2004).
[CrossRef]

Cao, Q.

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057,403 (2002).
[CrossRef] [PubMed]

Chang, S.H.

de Abajo, F.J.G.I.

Dubois, G.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

Ebbesen, T.W.

T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Eliel, E.R.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

Evans, B.R.

A.P. Hibbins, B.R. Evans, and J.R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670 (2005).
[CrossRef] [PubMed]

Flammer, C.

C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. I. Oblate spheroidal vector wave functions," J. Appl. Phys. 24, 1218-1223 (1953).
[CrossRef]

C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. II. The diffraction problems," J. Appl. Phys. 24, 1224-1231 (1953).
[CrossRef]

Garcia-Vidal, F.J.

F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103,901 (2005).
[CrossRef] [PubMed]

L. Martin-Moreno and F.J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619-3628 (2004).
[CrossRef]

J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, "Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026,601 (2004).
[CrossRef]

J.A. Porto, F.J. Garcia-Vidal, and J.B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

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. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

Ghaemi, H.F.

T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Gray, S.K.

Hibbins, A.P.

A.P. Hibbins, B.R. Evans, and J.R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670 (2005).
[CrossRef] [PubMed]

Hooft, G.W.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

Kuzmin, N.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

Lalanne, P.

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057,403 (2002).
[CrossRef] [PubMed]

Lenstra, D.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036,608 (2003).
[CrossRef]

Lezec, H.J.

T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Martin-Moreno, L.

F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103,901 (2005).
[CrossRef] [PubMed]

L. Martin-Moreno and F.J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619-3628 (2004).
[CrossRef]

J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, "Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026,601 (2004).
[CrossRef]

Meixner, J.

J. Meixner and W. Andrejewski, "Strenge Theorie der Beugung ebener elektromagnetischer Wellen and der vollkommen leitenden Kreisscheibe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm," Annalen der Physik 7, 157-168 (1950).
[CrossRef]

Moharam, M.G.

Moreno, E.

F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103,901 (2005).
[CrossRef] [PubMed]

Pendry, J.B.

J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

J.A. Porto, F.J. Garcia-Vidal, and J.B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Porto, J.A.

F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103,901 (2005).
[CrossRef] [PubMed]

J.A. Porto, F.J. Garcia-Vidal, and J.B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Roberts, A.

Sambles, J.R.

A.P. Hibbins, B.R. Evans, and J.R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670 (2005).
[CrossRef] [PubMed]

Schatz, G.C.

Schouten, H.F.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036,608 (2003).
[CrossRef]

Thio, T.

T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Urbach, H.P.

J.M. Brok and H.P. Urbach, "A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording," J. Mod. Opt. 51, 2059-2077 (2004).
[CrossRef]

Visser, T.D.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036,608 (2003).
[CrossRef]

Wolff, P.A.

T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Annalen der Physik (1)

J. Meixner and W. Andrejewski, "Strenge Theorie der Beugung ebener elektromagnetischer Wellen and der vollkommen leitenden Kreisscheibe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm," Annalen der Physik 7, 157-168 (1950).
[CrossRef]

J. Appl. Phys. (2)

C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. I. Oblate spheroidal vector wave functions," J. Appl. Phys. 24, 1218-1223 (1953).
[CrossRef]

C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. II. The diffraction problems," J. Appl. Phys. 24, 1224-1231 (1953).
[CrossRef]

J. Mod. Opt. (1)

J.M. Brok and H.P. Urbach, "A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording," J. Mod. Opt. 51, 2059-2077 (2004).
[CrossRef]

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

Nature (1)

T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Opt. Express (3)

Phys. Rev. (1)

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

Phys. Rev. E (2)

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036,608 (2003).
[CrossRef]

J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, "Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026,601 (2004).
[CrossRef]

Phys. Rev. Lett. (4)

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005).
[CrossRef] [PubMed]

J.A. Porto, F.J. Garcia-Vidal, and J.B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057,403 (2002).
[CrossRef] [PubMed]

F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103,901 (2005).
[CrossRef] [PubMed]

Reports on progress in physics (1)

C.J. Bouwkamp, "Diffraction theory," Reports on progress in physics 17, 35-100 (1954).
[CrossRef]

Science (2)

J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004).
[CrossRef] [PubMed]

A.P. Hibbins, B.R. Evans, and J.R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670 (2005).
[CrossRef] [PubMed]

Other (9)

Here and henceforth the square root of a complex number z is defined such that for real z > 0 we have √z > 0 and √z = +i√z, with the branch cut along the negative real axis.

J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).

Although υ¯α(x,y) is real, we include the conjugation for consistency of the notation.

J. van Bladel, Singular Electromagnetic Fields and Sources, 1st ed. (Clarendon Press, Oxford, 1991).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, 1st ed. (Springer-Verlag, Berlin, 1988).

<other>. Although the ± for α4 now does not have anything to do with the propagation direction, for consistency, we stick to this notation. </other>

The reader might wonder why we do not use the sine and cosine form always instead of using the exponential form only for | γz/kp|< ε. However, for large and purely imaginary γz, the functions cos (γzz) and sin (γzz) increase exponentially with |z|, which is not very convenient for numerical implementation.

The interaction integral also has some other properties that can save a lot of computation time. These properties are outside the scope of this paper.

This routine is a multi-dimensional adaptive quadrature over a hyper-rectangle. For a description, see the internet link: http://www.nag.com/nagware/mt/doc/d01fcf.html.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Problem under consideration. Multiple rectangular holes in a perfectly conducting layer with finite thickness.

Fig. 2.
Fig. 2.

Relative error FNN ̃, with N = 2600 as a function of the number of waveguide modes (Ñ = 120,440,960,1680). Setup is a single hole with a perpendicular incident, linearly polarized plane wave. As a reference, 1/N is also plotted.

Fig. 3.
Fig. 3.

Line scan of the absolute value of the x-component of the electric field in a plane with constant z, through the center of the hole, for various numbers of waveguide modes. Setup is a single hole,Lx =Ly =D = λ/5, with a perpendicular incident, linearly polarized plane wave.

Fig. 4.
Fig. 4.

Energy flux through single hole as a function of layer thickness, normalized by the energy that is incident on the area of the hole. Incident field is a perpendicular, linearly polarized plane wave. The hole is square: Lx = Ly = L with different values of L for every curve, as listed in the legend. The number at the right of each curve is the number of waveguide modes above cut-off.

Fig. 5.
Fig. 5.

Polar plot of the near field scattering from a single square pit with a depth d = λ/4. For different sizes, the Poynting vector in the radial direction is shown, at a half circle with radius λ, with its center coinciding with the center of the pit, at z = D/2. Black line is for the (y,z)-plane, gray for the (x,z)-plane. Incident field is a perpendicular plane wave, with its electric field linearly polarized along the x-direction. The radial scale is arbitrary, but equal for all three figures.

Fig. 6.
Fig. 6.

Energy flux through a hole, normalized by the energy flux through an identical single hole, as a function of the distance between two holes (or one hole and one pit). Pits and holes are all square, with Lx = Ly = L = λ/4. Incident field is a linearly polarized plane wave, with polarization as stated at the top of the figures. Incidence is always perpendicular, except for the dashed and dotted lines in Fig. 6(b). Here, kxi ≠ 0 means that the plane of incidence is the (x,z)-plane. Fig. 6(d) shows cross-sections of the four used geometries. The bold arrows denote where the energy flux is calculated. The thickness of the layer is λ/2 and the depth of the pits is λ/4.

Fig. 7.
Fig. 7.

Comparison between the scalar optics normalization (solid line) and the single hole normalization (dashed line). The vertical axis shows the energy flux through one of two holes, the horizontal axis the distance between the centers of the two holes. The holes are square, with L = 0.49λ (black) and L = 0.6λ (gray). The thickness of the conducting layer is λ/2.

Fig. 8.
Fig. 8.

Cartesian components of the Poynting vector in a plane of constant z, at a distance of λ/20 below the metal layer. Gray scale is in arbitrary units, the same for all figures. The layer contains two holes with Lx = Ly = λ/4 and the thickness of the layer is λ/2. A parallel polarized plane wave is incident from above, perpendicular incidence. Top figure corresponds to a maximum in energy throughput, lower figure to a minimum. See arrows in Fig. 6(b).

Fig. 9.
Fig. 9.

The division of the integration area into 12 domains. Not on scale.

Equations (67)

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

[ E α ( r ) H α ( r ) ] = [ E α x y H α x y ] e ± z z ,
γ Z = k p 2 γ x 2 γ y 2 ,
γ x = m x π L x p , γ y = m y π L y p ,
e x y z î Z × [ î Z × E x y z ] = ( E x E y 0 ) ,
h x y z î Z × H x y z = ( H y H x 0 ) ,
[ e α x y z h α x y z ] = υ α ̄ x y [ η α ( Z ) ζ α ( Z ) ] ,
υ α ̄ υ α ̄ Ω p ∫∫ Ω p [ υ α ̄ , x x y υ α ̄ , x x y * + υ α ̄ , y x y υ α ̄ , y x y * ] d x d y = 1 ,
υ α ̄ υ α′ ̄ Ω p = 0 , if α ̄ α ̄
S α , z = 1 2 Re ( E α , x H α , y * E α , y H α , x * ) = 1 2 Re ( η α ζ α * ) ( υ α ̄ , x υ α ̄ , x * + υ α ̄ , y υ α ̄ , y * ) ,
[ E pit ( r ) H pit ( r ) ] = α a α [ E α ( r ) H α ( r ) ] ,
[ E ( r ) H ( r ) ] = [ E i ( r ) H i ( r ) ] + [ E r ( r ) H r ( r ) ] + [ E s ( r ) H s ( r ) ] .
[ E s ( r ) H s ( r ) ] = β 1 b β [ E β ( r ) H β ( r ) ] d β 2 ,
k z u = + k u 2 k x 2 k y 2 ,
k z = k 2 k x 2 k y 2 .
[ e β x y z h β x y z ] = υ β x y [ η β ( z ) ζ β ( z ) ] .
υ β υ β 2 ∫∫ 2 [ υ β , x ( x , y ) υ β , x ( x , y ) * + υ β , y ( x , y ) υ β , y ( x , y ) * ] d x d y = δ β 1 β 1 δ ( β 2 β 2 ) .
δ ( β 2 ) = 1 4 π 2 e i ( k x x + k y y ) d x d y , β 2 = k x k y .
h β x y z = k z ωμ 0 e β x y z , β 1 = S ,
h β x y z = ωεε 0 k z e β x y z β 1 = P .
e s x y z = β 1 b β e β x y z 2 ,
A ( f ) k z ωμ 0 f υ β 2 S 2 υ β 2 s 2 + ωεε 0 k z f υ β 2 P 2 υ β 2 P d β 2 ,
h s ( x , y , ± D 2 ) ̄ = A [ e s ( x , y , ± D 2 ) ] .
e pit = e i + e r + e s , x y , z = ± D 2 ,
h pit = h i + h r + h s , x y α 1 Ω α 1 , z = ± D 2 ,
e pit = e s , x y , z = ± D 2 .
h pit = h i + h r + A ( e pit ) ,
α 4 a α ς α ( ± D 2 ) α a α η α ( ± D 2 ) A ( v α ̄ ) v α ̄ Ω p = h i + h r v α ̄ Ω p ,
b β = α a α e α e β .
[ E s ( r ) H s ( r ) ] = β 1 α a α e α e β [ E β ( r ) E β ( r ) ] 2 .
F N N ˜ = ∫∫∫ V p ( ε 0 ε p E N ˜ E N 2 + μ 0 H N ˜ H N 2 ) d x d y d z ∫∫∫ V p ( ε 0 ε p E N ˜ 2 + μ 0 H N ˜ 2 d x d y d z ,
∫∫ Ω p S z d x d y = α ̄ α 4 α ˜ 4 1 2 Re [ a α ̄ α 4 a α ̄ α ˜ 4 * η α α 4 ( z 0 ) ς α α ˜ 4 ( z 0 ) * ] ,
[ e α x y z h α x y z ] = υ α ̄ x y [ η α ( z ) ζ α ( z ) ] .
Γ α = γ x 2 + γ y 2 ,
Λ α = { 2 ( L x p L y p ) 1 2 , 2 ( L x p L y p ) 1 2 , if m x 0 and m y 0 , if m x = 0 or m y = 0 ,
υ α ̄ x ̄ p y ̄ p = { Λ α Γ α x ̄ p y ̄ p [ γ y cos ( γ y x ̄ p ) sin ( γ y y ̄ p ) γ x sin ( γ x x ̄ p ) cos ( γ y y ̄ p ) ] , α 2 = TE , Λ α Γ α ( x ̄ p , y ̄ p ) [ γ x cos ( γ x x ̄ p ) sin ( γ y y ̄ p ) γ y sin ( γ x x ̄ p ) cos ( γ y y ̄ p ) ] , α 2 = TM ,
ϑ α ̄ x ̄ p y ̄ p = { i Λ α Γ α x ̄ p y ̄ p cos ( γ x x ̄ p ) cos ( γ y y ̄ p ) , α 2 = TE , i Λ α Γ α x ̄ p y ̄ p sin ( γ x x ̄ p ) sin ( γ p y ̄ p ) , α 2 = TM ,
[ H ( x ̄ p ) H ( x ̄ p L x p ) ] [ H ( y ̄ p ) H ( y ̄ p L y p ) ] ,
υ α ̄ υ α ̄ Ω p ∫∫ Ω p [ υ α ̄ , x ( x , y ) υ α ̄ , x x y * + υ α ̄ , y ( x , y ) υ α ̄ y ( x , y ) * ] d x d y = 0 , α ̄ α ̄ .
f α ( z ) = { exp [ z ( z z 1 p ) ] , exp [ z ( z z 2 p ) ] , ik p γ z 1 cos ( γ z z ) , i sin ( γ z z ) , γ z k p ε , α 4 = , γ z k p ε , α 4 = + , γ z k p < ε , α 4 = , γ z k p < ε , α 4 = + ,
g α ( z ) = { exp [ z ( z z 1 p ) ] , exp [ z ( z z 2 p ) ] , k p γ z 1 sin ( γ z z ) , cos ( γ z z ) , γ z k p ε , α 4 = , γ z k p ε , α a = + , γ z k p < ε , α 4 = , γ z k p < ε , α 4 = + ,
η α ( z ) = { ωμ 0 g α ( z ) , γ z μ 0 ε p ε 0 f α ( z ) , α 2 = TE , α 2 = TM ,
ζ α ( z ) = { γ z f α ( z ) , k p g α ( z ) , α 2 = TE , α 2 = TM .
[ e α x y z h α x y z ] = υ α ̄ x y [ η α ( z ) ζ α ( z ) ] ,
E α x y z = { 0 , μ 0 ε p ε 0 ϑ α ̄ x y ( z ) , α 2 = TE , α 2 = TM ,
E α , z x y z = { ϑ α ̄ x y ( z ) 0 , α 2 = TE , α 2 = TM .
[ e β x y z h β x y z ] = υ β x y [ η β ( z ) ζ β ( z ) ] .
Γ β = k x 2 + k y 2 ,
Λ β = 1 2 π ,
υ β x y = { Λ β Γ β e i ( k x x + k y y ) ( k y k x ) , β 1 = S , k x 2 + k y 2 > 0 , Λ β ( 0 1 ) , β 1 = S , k x 2 + k y 2 = 0 , Λ β Γ β e i ( k x x + k y y ) ( k x k y ) β 1 = P , k x 2 + k y 2 > 0 , Λ β ( 1 0 ) , β 1 = P , k x 2 + k y 2 > 0 ,
ϑ β x y = Λ β Γ β e i ( k x x + k y y ) ,
f β ( z ) = { exp [ ik z u ( z D 2 ) ] , z > D 2 , exp [ ik z l ( z + D 2 ) ] , z < D 2 ,
η β ( z ) = { ωμ 0 f β ( z ) , β 1 = S , k z μ 0 εε 0 f β ( z ) , β 1 = P ,
ζ β ( z ) = { k z f β ( z ) , β 1 = S , kf β ( z ) , β 1 = P .
E β , z ( x , y , z ) = { 0 , β 1 = S , μ 0 εε 0 ϑ β ( x , y ) f β ( z ) , β 1 = P ,
H β , z x y z = { ϑ β x y f β ( z ) β 1 = S , 0 , β 1 = P .
A ( υ α ̄ ) υ α ̄ Ω p = k z ω μ 0 υ α υ β 2 S Ω p υ α υ β 2 S Ω p * d β 2 + ωε ε 0 k z υ α υ β 2 P Ω p υ α υ β 2 P Ω p * d β 2 .
υ α ̄ υ β Ω p = x 0 p x 0 p + L x p y 0 p y 0 p + L y p υ α ̄ x y υ β x y * d x d y ,
e i ( k x x 0 p + k y y 0 p ) 0 L p x 0 L p y υ α ̄ x ̄ ̄ p y ̄ p υ β x ̄ p y ̄ p * d ̄ x p d ̄ x ̄ p ,
F α ̄ β 1 ( k x , k y ) 0 L x p 0 L y p υ α ̄ ( x ̄ p , y ̄ p ) υ β ̄ ( x ̄ p , y ̄ p ) * d x ̄ p d x ̄ p ,
= Λ α Λ β Γ α Γ β { γ y k y c m x p ( k x ) s m y p ( k y ) + γ x k x s m x p ( k x ) c m y p ( k y ) , α 2 = TE , β 1 = S , γ x k y c m x p ( k x ) s m y p ( k y ) γ y k x s m x p ( k x ) c m y p ( k y ) = 0 , α 2 = TE , β 1 = S , γ y k x c m x p ( k x ) s m y p ( k y ) γ x k y s m x p ( k x ) c m y p ( k y ) , α 2 = TE , β 1 = P , γ x k x c m x p ( k x ) s m y p ( k y ) + γ y k y s m x p ( k x ) c m y p ( k y ) , α 2 = TE , β 1 = P ,
c m j p ( k j ) 0 L j p cos ( γ j z ) e ik j z dz = { ik j γ j 2 k j 2 [ 1 ( 1 ) m j e ik j L j ] , k j ± γ j , 1 2 L j , k j = ± γ j , k j 0 , L j , k j = γ j = 0 ,
s m j p ( k j ) 0 L j p sin ( γ j z ) e ik j z dz = { γ j γ j 2 k j 2 [ 1 ( 1 ) m j e ik j L j ] , k j ± γ j , 1 2 iL j , k j ± γ j , k j 0 , 0 , k j = γ j = 0 .
A ( υ α ̄ ) υ α ̄ Ω p ̄ = k z ωμ 0 e i ( k x Δ x + k y Δ y ) F α ̄ S ( k x , k y ) F α ̄ ' S ( k x , k y ) * d k x d k y + ω εε 0 k z e i ( k x Δ x + k y Δ y ) F α ̄ P ( k x , k y ) F α ̄ P ( k x , k y ) * d k x d k y ,
A ( υ α ̄ ) υ α ̄ Ω p = N N in m f Re ( I N ) + N N out m f i Im ( I N ) ,
for α 1 = α 1 : m f = 4 , N in = { 1 , 11 } , N out = { 2 , 3 , 4 , 5 , 12 } ,
for α 1 α 1 : m f = 2 , N in = { 1 , 6 , 11 } , N out = { 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 , 12 } ,
I N = ∫∫ domain N I α α ̄ ̄ ( k x , k y ) d k x d k y .

Metrics