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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 10 — May. 19, 2003
  • pp: 1131–1136
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Light transmission by subwavelength square coaxial aperture arrays in metallic films

A. Moreau, G. Granet, F. I. Baida, and D. Van Labeke  »View Author Affiliations


Optics Express, Vol. 11, Issue 10, pp. 1131-1136 (2003)
http://dx.doi.org/10.1364/OE.11.001131


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Abstract

Using the Fourier modal method, we study the enhanced transmission exhibited by arrays of square coaxial apertures in a metallic film. The calculated transmission spectrum is in good agreement with FDTD calculations. We show that the enhanced transmission can be explained when we consider a few guided modes of a coaxial waveguide.

© 2003 Optical Society of America

1. Introduction

2. Statement of the problem

Let us consider a metallic film deposed on a glass substrate with an engraved periodic structure of square coaxial apertures (see Fig. 1).

Fig. 1. Coaxial square aperture in a metallic film.

Fig. 2. Transmission of a square coaxial aperture calculated with the FDTD (blue curve) and the Fourier modal method (red curve).

It can be seen that both methods give resonances at the same place even though a small difference is observed in their intensity.

3. Discussion

3.1. Analysis of the mode

Our goal is to analyze the enhanced transmission by using the guided modes of the coaxial apertures. Since we consider a metallic medium, an aperture is not coupled with its neighbors. A mode for the entire structure thus corresponds exactly to a mode of a sole aperture, and thus no distinction is made in this discussion between them. Indeed, the eigenvalues and the fields inside the apertures corresponding to an eigenmode do not change when the distance between holes varies. As a consequence, the eigenvalues γq give an immediate access to the effective index for each guided mode.

Because of the metal, all the propagating constants are complex but some of them can be considered as guided modes with low losses. For the considered structure we have found that there were three such modes, two of them being degenerated as a result of the square symmetry. The numerically obtained dispersion relations are plotted in Figs. 3 and 4.

Fig. 3. Dispersion curves of the first mode. Blue curve, real part; red curve, imaginary part. The presence of dips is probably due to the right angle corners.
Fig. 4. Dispersion curves of the second mode. Blue curve, real part; red curve, imaginary part.

Figures 5 and 6 show a map of the modulus of the transverse electric field of the first and the second modes.

Fig. 5. Modulus of the transverse electric field of the first guided mode.
Fig. 6. Modulus of the transverse electric field of the second guided mode.

The mode whose effective index has the largest real part and the lowest imaginary part corresponds to the TEM mode of the same coaxial structure with perfect conducting walls. This mode is characterized by an electric field normal to the walls and has no cut-off. In the present case, it is not strictly speaking a TEM mode since its effective index is greater than one. However, when the width of the aperture becomes larger, the coupling between the opposite sides of the coaxial waveguide diminishes resulting in a lower effective index. The two other guided modes have a cut-off ~ λ=845 nm.

3.2. Analysis of the coupling of the modes to free radiation

Fig. 7. Twenty-first modal amplitude coefficients inside the coaxial on the upper face. The red bar corresponds to an attenuated guided wave.

Fig. 8. Twenty-first modal amplitude coefficients inside the coaxial on the lower face. The red bar corresponds to an attenuated guided wave.
Fig. 9. The ten first complex propagating constants associated to the ten first modes that are exited on the upper face inside the coaxial waveguide. The red one corresponds to an attenuated guide wave; its value is γ=1.39—0.006i.

4. Conclusion

We have numerically studied the spectral response of subwavelength coaxial apertures.We have calculated the propagating constants of the modes supported by a square coaxial waveguide. Some of them correspond to attenuated guided modes. However, the excitation of such modes is possible only when the incident wave matches the mode profile. Owing to the electric properties of metals at optical wavelengths, the dispersion relations of the modes of the transmission channel are very specific and very different from those of the same channel with perfectly conducting walls. This preliminary study paves the way for future investigations in order to engineer the modes and their excitation for applications.

References and links

1.

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

2.

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 (1999). [CrossRef]

3.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 114 (2001). [CrossRef]

4.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002). [CrossRef] [PubMed]

5.

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265 (2000). [CrossRef]

6.

M. M. J. Treacy, “Dynamical diffraction in metallic optical gratings,” Appl. Phys. Lett. 75, 606 (1999). [CrossRef]

7.

M. M. J. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through mettallic gratings,” Phys. Rev. B 66, 195105 (2002). [CrossRef]

8.

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A:Pure Appl. Opt. 4, S154 (2002). [CrossRef]

9.

L. Li, “New formulation of the Fourier modal method for corssed surface-relief gratings,” J. Opt. Soc. Am. A 11, 2758–2767 (1997). [CrossRef]

10.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000). [CrossRef]

11.

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

12.

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A: Pure Appl. Opt. 4, S145 (2002). [CrossRef]

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(230.7370) Optical devices : Waveguides
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Research Papers

History
Original Manuscript: March 19, 2003
Revised Manuscript: April 29, 2003
Published: May 19, 2003

Citation
A. Moreau, G. Granet, F. Baida, and D. Van Labeke, "Light transmission by subwavelength square coaxial aperture arrays in metallic films," Opt. Express 11, 1131-1136 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-10-1131


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References

  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, �??Extraordinary optical transmission through sub-wavelength hole arrays,�?? Nature 391, 667 (1998). [CrossRef]
  2. 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 (1999). [CrossRef]
  3. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T.W. Ebbesen, �??Theory of extraordinary optical transmission through subwavelength hole arrays,�?? Phys. Rev. Lett. 86, 114 (2001). [CrossRef]
  4. Q. Cao and P. Lalanne, �??Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,�?? Phys. Rev. Lett. 88, 057403 (2002). [CrossRef] [PubMed]
  5. S. Astilean, P. Lalanne, and M. Palamaru, �??Light transmission through metallic channels much smaller than the wavelength,�?? Opt. Commun. 175, 265 (2000). [CrossRef]
  6. M. M. J. Treacy, �??Dynamical diffraction in metallic optical gratings,�?? Appl. Phys. Lett. 75, 606 (1999). [CrossRef]
  7. M. M. J. Treacy, �??Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,�?? Phys. Rev. B 66, 195105 (2002). [CrossRef]
  8. S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, �??Horizontal and vertical surface resonances in transmission metallic gratings,�?? J. Opt. A:Pure Appl. Opt. 4, S154 (2002). [CrossRef]
  9. L. Li, �??New formulation of the Fourier modal method for corssed surface-relief gratings,�?? J. Opt. Soc. Am. A 11, 2758-2767 (1997). [CrossRef]
  10. E. Popov, M. Nevière, S. Enoch, and R. Reinisch, �??Theory of light transmission through subwavelength periodic hole arrays,�?? Phys. Rev. B 62, 16100 (2000). [CrossRef]
  11. F. I. Baida and D. Van Labeke, �??Light transmission by subwavelength annular aperture arrays in metallic films,�?? Opt. Commun. 209, 17-22 (2002). [CrossRef]
  12. G. Granet and J. P. Plumey, �??Parametric formulation of the Fourier modal method for crossed surface-relief gratings,�?? J. Opt. A: Pure Appl. Opt. 4, S145 (2002). [CrossRef]

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