Abstract

A waveguide mode of a subwavelength rectangular hole in a real metal is analyzed. Due to coupling between surface plasmons on the long edges of the hole, the cut-off wavelength increases as the hole-width is reduced. The cut-off wavelength is found to be much larger than Rayleigh’s criterion for perfect metals — 2.3 times as large for a 15 nm wide hole. The analytical results are verified by finite-difference calculations. The finite difference calculations also show the influence of including material loss.

© 2005 Optical Society of America

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References

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  1. Lord Rayleigh, �??On the Passage of Electric Waves Through Tubes,�?? Philos. Mag. 43, 125-132 (1897).
  2. H. A. Bethe, �??Theory of Diffraction by Small Holes,�?? Phys. Rev. 66, 163-182 (1944).
    [CrossRef]
  3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, �??Extraordinary Optical Transmission through Sub-Wavelength Hole Arrays,�?? Nature 391, 667-669 (1998).
    [CrossRef]
  4. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, �??Beaming Light from a Subwavelength Aperture,�?? Science 297, 820-822 (2002).
    [CrossRef] [PubMed]
  5. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, �??Strong Polarization in the Optical Transmission through Elliptical Nanohole Arrays,�?? Phys. Rev. Lett. 92, 037401 (2004).
    [CrossRef] [PubMed]
  6. 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]
  7. A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, �??Optical Transmission Properties of a Single Subwavelength Aperture in a Real Metal,�?? Opt. Commun. 239, 61-66 (2004).
    [CrossRef]
  8. Johnson and Christy, �??Optical Constants of the Nobel Metals,�?? Phys. Rev. B 12, 4370-4379 (1972).
    [CrossRef]
  9. R. C. Booton Jr., Computational Methods for Electromagnetics and Microwaves, (John Wiley & Sons, New York, 1992).

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, �??Extraordinary Optical Transmission through Sub-Wavelength Hole Arrays,�?? Nature 391, 667-669 (1998).
[CrossRef]

Opt. Commun. (1)

A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, �??Optical Transmission Properties of a Single Subwavelength Aperture in a Real Metal,�?? Opt. Commun. 239, 61-66 (2004).
[CrossRef]

Philos. Mag. (1)

Lord Rayleigh, �??On the Passage of Electric Waves Through Tubes,�?? Philos. Mag. 43, 125-132 (1897).

Phys. Rev. (1)

H. A. Bethe, �??Theory of Diffraction by Small Holes,�?? Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Phys. Rev. B (1)

Johnson and Christy, �??Optical Constants of the Nobel Metals,�?? Phys. Rev. B 12, 4370-4379 (1972).
[CrossRef]

Phys. Rev. Lett. (2)

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, �??Strong Polarization in the Optical Transmission through Elliptical Nanohole Arrays,�?? Phys. Rev. Lett. 92, 037401 (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]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, �??Beaming Light from a Subwavelength Aperture,�?? Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Other (1)

R. C. Booton Jr., Computational Methods for Electromagnetics and Microwaves, (John Wiley & Sons, New York, 1992).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of hollow rectangular waveguide in a metal, with co-ordinates shown. The lowest-order mode is analyzed by considering the TM mode of a slab of width w, to derive an effective dielectric constant, and then solving for the TE mode component in a slab of separation l filled with the effective dielectric. This provides the effective dielectric constant for the lowest-order mode of the rectangular waveguide.

Fig. 2.
Fig. 2.

The top graph shows the effective relative permittivity of the dielectric as calculated from Eq. (4). The bottom graph shows the TE01 mode for rectangular holes in silver and for a perfect electric conductor (PEC), as calculated using the effective index method. +, ×, □, and ○ are the results from numerical simulations.

Fig. 3.
Fig. 3.

E, H fields polarized along the x, y and z directions for the lowest order mode in a 105 nm by 270 nm rectangular waveguide in silver at the wavelength of 750 nm. These mode profiles were calculated numerically by the finite difference technique. Each figure shows 220 nm along the y-direction, and 380 nm along the x-direction. The graphs were scaled in the ratios Ex:Ey:Ez given by 0.48:1.0:0.012, and Hx:Hy:Hz given by 1.0:0.016:13.4.

Fig. 4.
Fig. 4.

Attenuation from material losses and cut-off attenuation for a 105 nm by 270 nm rectangular hole in silver.

Equations (5)

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β = π ( 2 λ ) 2 ( 1 a ) 2
tan ( k o 2 ε d β TE 2 l 2 ) = β 2 k o 2 ε m k o 2 ε d β TE 2
λ cut-off = π l ε d arctan ε m ε d
tanh ( β TM 2 k o 2 ε air w 2 ) = ε air ε m β TM 2 k o 2 ε m β TM 2 k o 2 ε air .
ε d = ( β TM k o ) 2 .

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