Abstract

This paper presents theory and finite-difference time-domain (FDTD) calculations for a single and arrays of sub-wavelength cylindrical holes in metallic films presenting large transmission. These calculations are in excellent agreement with experimental measurements. This effect has to be understood in terms of the properties exhibited by the dielectric constant of metals which cannot be treated as ideal metals for the purpose of transmission and diffraction of light. We discuss the cases of well-differentiated metals silver and tungsten. It is found that the effect of surface plasmons or other surface wave excitations due to a periodical set of holes or other roughness at the surface is marginal. The effect can enhance but also can depress the transmission of the arrays as shown by theory and experiments. The peak structure observed in experiments is a consequence of the interference of the wavefronts transmitted by each hole and is determined by the surface array period independently of the material. Without large transmission through a single hole there is no large transmission through the array. We found that in the case of Ag which at the discussed frequencies is a metal there are cylindrical plasmons at the wall of the hole, as reported by Economu et al 30 years ago, that enhanced the transmission. But it turns out, as will be explained, that for the case of W which behaves as a dielectric, there is also a large transmission when compared with that of an ideal metal waveguide at large wavelengths. To deal with this problem one has to use the measured dielectric function of the metals. We discuss thoroughly all these cases and compare with the data. We notice that to discuss these data, for a single hole’s transmission, in terms of the Bethe approximation of ideal metals is misleading. Therefore, the extraordinary enhancement of the transmission for the holes arrays versus the single hole does not exist.

© 2006 Optical Society of America

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  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 (London) 391, 667-669 (1998).
    [CrossRef]
  2. H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through sub-wavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
    [CrossRef] [PubMed]
  3. J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
    [CrossRef]
  4. H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
    [CrossRef]
  5. H. Shin, P. B. Catrysse, and S. Fan, "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
    [CrossRef]
  6. C. A. Pfeiffer, E. N. Economou and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface: Surface plasmons," Phys. Rev. B 10, 3038-3051 (1974).
    [CrossRef]
  7. S. S. Martinos and E. N. Economou, "Excitation of surface plasmons in cylinders by electrons," Phys, Rev. B,  24, 6908-6914 (1981).
    [CrossRef]
  8. P. B. Johnson and R. W. Christy, "Optical constant of the noble metals," Phys. Rev. B 6, 4370 (1972).
    [CrossRef]
  9. J. H. Weaver, C. Krafka, D. W. Lynch, and E. E. Koch, "Optical properties of metals, Physik Daten/Physics Data" No. 18-1 (Fach-Informations-Zentrum, Energie Physick Mathematik GmbH, Karlsruhe, 1981).
  10. A. Taflove, Advances in Computational Electrodynamics, The Finite-Difference Time-Domain Method, (Artech House 1998).
  11. H. Raether, Springer Tracts in Modern Physics, Vol. 111: Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Spinger-Verlag Berlin Heidelberg, 1988).
  12. N. García, "Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polariton resonances," Opt. Commun. 45, 307(1983).
    [CrossRef]
  13. N. García, G. Diaz, J. H. Saenz, and C. Ocal, "Intensities and field enhancement of light scattered from periodic gratings: study of Ag, Au and Cu surface," Surf. Sci. 143, 342 (1984).
    [CrossRef]
  14. John David Jackson, Classical Electrodynamics 3rd edition, (Wiley, 1999).
  15. M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
    [CrossRef]

2006

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

2005

H. Shin, P. B. Catrysse, and S. Fan, "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

2004

M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
[CrossRef]

H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through sub-wavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

1998

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

1984

N. García, G. Diaz, J. H. Saenz, and C. Ocal, "Intensities and field enhancement of light scattered from periodic gratings: study of Ag, Au and Cu surface," Surf. Sci. 143, 342 (1984).
[CrossRef]

1983

N. García, "Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polariton resonances," Opt. Commun. 45, 307(1983).
[CrossRef]

1981

S. S. Martinos and E. N. Economou, "Excitation of surface plasmons in cylinders by electrons," Phys, Rev. B,  24, 6908-6914 (1981).
[CrossRef]

1974

C. A. Pfeiffer, E. N. Economou and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface: Surface plasmons," Phys. Rev. B 10, 3038-3051 (1974).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, "Optical constant of the noble metals," Phys. Rev. B 6, 4370 (1972).
[CrossRef]

1944

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

Bai, M.

M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
[CrossRef]

Bethe, H. A.

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

Bravo-Abad, J.

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

Catrysse, P. B.

H. Shin, P. B. Catrysse, and S. Fan, "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constant of the noble metals," Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Degiron, A.

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

Diaz, G.

N. García, G. Diaz, J. H. Saenz, and C. Ocal, "Intensities and field enhancement of light scattered from periodic gratings: study of Ag, Au and Cu surface," Surf. Sci. 143, 342 (1984).
[CrossRef]

Ebbesen, T. W.

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

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

Economou, E. N.

S. S. Martinos and E. N. Economou, "Excitation of surface plasmons in cylinders by electrons," Phys, Rev. B,  24, 6908-6914 (1981).
[CrossRef]

C. A. Pfeiffer, E. N. Economou and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface: Surface plasmons," Phys. Rev. B 10, 3038-3051 (1974).
[CrossRef]

Fan, S.

H. Shin, P. B. Catrysse, and S. Fan, "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

García, N.

M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
[CrossRef]

N. García, G. Diaz, J. H. Saenz, and C. Ocal, "Intensities and field enhancement of light scattered from periodic gratings: study of Ag, Au and Cu surface," Surf. Sci. 143, 342 (1984).
[CrossRef]

N. García, "Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polariton resonances," Opt. Commun. 45, 307(1983).
[CrossRef]

García-Vidal, F. J.

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

Genet, C.

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

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 (London) 391, 667-669 (1998).
[CrossRef]

Guerrero, C.

M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
[CrossRef]

Ioanid, S.

M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constant of the noble metals," Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Lezec, H. J.

H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through sub-wavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

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

Martín-Moreno, L.

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

Martinos, S. S.

S. S. Martinos and E. N. Economou, "Excitation of surface plasmons in cylinders by electrons," Phys, Rev. B,  24, 6908-6914 (1981).
[CrossRef]

Ngai, K. L.

C. A. Pfeiffer, E. N. Economou and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface: Surface plasmons," Phys. Rev. B 10, 3038-3051 (1974).
[CrossRef]

Ocal, C.

N. García, G. Diaz, J. H. Saenz, and C. Ocal, "Intensities and field enhancement of light scattered from periodic gratings: study of Ag, Au and Cu surface," Surf. Sci. 143, 342 (1984).
[CrossRef]

Paz, E.

M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
[CrossRef]

Pfeiffer, C. A.

C. A. Pfeiffer, E. N. Economou and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface: Surface plasmons," Phys. Rev. B 10, 3038-3051 (1974).
[CrossRef]

Przybilla, F.

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

Saenz, J. H.

N. García, G. Diaz, J. H. Saenz, and C. Ocal, "Intensities and field enhancement of light scattered from periodic gratings: study of Ag, Au and Cu surface," Surf. Sci. 143, 342 (1984).
[CrossRef]

Sanz, M.

M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
[CrossRef]

Shin, H.

H. Shin, P. B. Catrysse, and S. Fan, "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

Thio, T.

H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through sub-wavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

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

Nat. Phys.

J. Bravo-Abad, A. Degiron, F. Przybilla, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and T. W. Ebbesen, "How Light emerges from an illuminated array of sub-wavelength holes," Nat. Phys.,  2, 120-123 (2006).
[CrossRef]

Nature (London)

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

Opt. Commun.

N. García, "Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polariton resonances," Opt. Commun. 45, 307(1983).
[CrossRef]

Opt. Express

Phys, Rev. B

S. S. Martinos and E. N. Economou, "Excitation of surface plasmons in cylinders by electrons," Phys, Rev. B,  24, 6908-6914 (1981).
[CrossRef]

Phys. Rev.

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

Phys. Rev. B

H. Shin, P. B. Catrysse, and S. Fan, "Effect of the plasmonic dispersion relation on the transmission properties of subwavelength cylindrical holes," Phys. Rev. B 72, 085436 (2005).
[CrossRef]

C. A. Pfeiffer, E. N. Economou and K. L. Ngai, "Surface polaritons in a circularly cylindrical interface: Surface plasmons," Phys. Rev. B 10, 3038-3051 (1974).
[CrossRef]

P. B. Johnson and R. W. Christy, "Optical constant of the noble metals," Phys. Rev. B 6, 4370 (1972).
[CrossRef]

M. Bai, C. Guerrero, S. Ioanid, E. Paz, M. Sanz and N. García, "Measuring the speed of a surface plasmons," Phys. Rev. B 69, 115416-115421 (2004)
[CrossRef]

Surf. Sci.

N. García, G. Diaz, J. H. Saenz, and C. Ocal, "Intensities and field enhancement of light scattered from periodic gratings: study of Ag, Au and Cu surface," Surf. Sci. 143, 342 (1984).
[CrossRef]

Other

John David Jackson, Classical Electrodynamics 3rd edition, (Wiley, 1999).

J. H. Weaver, C. Krafka, D. W. Lynch, and E. E. Koch, "Optical properties of metals, Physik Daten/Physics Data" No. 18-1 (Fach-Informations-Zentrum, Energie Physick Mathematik GmbH, Karlsruhe, 1981).

A. Taflove, Advances in Computational Electrodynamics, The Finite-Difference Time-Domain Method, (Artech House 1998).

H. Raether, Springer Tracts in Modern Physics, Vol. 111: Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Spinger-Verlag Berlin Heidelberg, 1988).

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Figures (9)

Fig. 1.
Fig. 1.

Schematics of cylindrical hole structures in metal film from optical transmission calculations, (a) single hole with diameter d and thickness t (b) arrays of holes with periodic P.

Fig. 2.
Fig. 2.

(a). The experimental Ag permittivity dispersion data (dots) [8] and the fitting (lines) by Drude dispersion mode in Eq. (1). ε1 is the real part of the permittivity, ε2 is the imaginary part of the permittivity. (b) Transmission coefficient of single holes in different kind of film. The diameter of the hole is d=270nm. For ideal metal, two cases with t=340nm and t=750nm are compared. For the Ag case, t=340nm. And the thin line is calculated from ideal long waveguide theory.

Fig. 3.
Fig. 3.

(a). Calculated dispersion relations λ sp by Eq. (2) for the SPPs waves in the surface of Ag cylinder with a given cylindricality α, determined by d=270nm and Ag bulk plasmons λ p =325nm. Index n represents the possible SPPs mode number in the cylindrical surface. The cross points between the straight photon line and the SPPs dispersion lines indicate the possible SPPs wave modes that can be excited. For large index n, the dispersion lines tends to the same line. (b) Transmission coefficient of single holes in Ag film with fixed diameter (d=270nm) with different thickness t=340nm, 525nm and 735nm.

Fig. 4.
Fig. 4.

Transmission coefficient of single holes in Ag film (t=340nm). (a) Transmission of single hole with d=250nm (solid line), d=300nm (dash dot line) by experiment [2], with d=250nm oe-14-21-10028-i001, d=270nm oe-14-21-10028-i002 by FDTD calculation and by ideal metal waveguide theory (thick line). The enhancement factor (dash line) is presented by dividing transmission from FDTD calculation by that from waveguide theory. The arrows indicate the positions for different cylindrical plasmons excitation modes. Notice the similar oscillations appear in the experiment data with a little shift. (b) Transmission of holes with d=200nm by experiment (solid line) [2], by FDTD calculation oe-14-21-10028-i003, by the ideal metal waveguide theory (thick line). The enhancement factor (dash line) is presented by dividing transmission from FDTD calculation by that from waveguide theory.

Fig. 5.
Fig. 5.

Transmission coefficient of periodic arrays (P=600nm) of holes in Ag film (a) Transmission of holes (d=270nm, t=225nm) by experiments (solid line) and by theoretical calculations in Fig. 1 of Ref. [3] (dash line). Line oe-14-21-10028-i004 is our FDTD calculation results in excellent agreement with experiments. (b) Transmission of holes (d=250nm, t=340nm) by the experiments (solid line) and the corresponding enhancement factor (dash line) versus single hole transmission (Fig. 2(a), 2(b) in Ref. [2]). Line oe-14-21-10028-i005 is by our FDTD calculation, together with its corresponding enhancement factor versus single hole transmission (dot line). (c) Transmission of holes (d=250nm, t=340nm) by the experiments (dash line) (Fig. 2(c) in Ref. [2]) and by our FDTD calculations (solid line).

Fig. 6.
Fig. 6.

Transmission coefficient of periodic arrays of holes in Ag film (t=340nm) (a) Transmission of holes with fixed d=270nm but for different periodic P=750nm, 870nm and 1050nm. The comparison shows the peak positions is strictly corresponding to the P values. (b) Transmission of holes with fixed P=1200nm but for different diameter holes with d=270nm, d=300nm and 360nm. The comparison shows the peak positions remain fixed because of the same P, the peak intensity is influenced by the hole diameter.

Fig. 7.
Fig. 7.

The experimental W permittivity dispersion data (points) [9] and the fitting (lines) by dispersion relation Eq. (3). ε i is the real part of the permittivity, ε r is the imaginary part of the permittivity.

Fig. 8.
Fig. 8.

Transmission coefficient of single holes in different kind of film. The transmission of single hole in W with d=300nm and t=400nm is compared with that of the same hole in ideal metal.

Fig. 9.
Fig. 9.

Transmission coefficient of periodic arrays of holes in W film with d=200nm, t=340nm and P=600nm. Experimental data (solid line) and calculation oe-14-21-10028-i006 are compared, showing well agreement.

Equations (4)

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ε ( ω ) / ε f = 1 ω p 2 ω 2 + 2 i ω δ
ω sp = ω p [ 1 + 1 2 Q 2 + ( 1 + 1 4 Q 4 ) 1 2 ] 1 2
Q = [ K 2 + ( n α ) 2 ] 1 2 K = k k p
ε ( ω ) = ε r σ i ω ε 0

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