## Optical transmission through circular hole arrays in optically thick metal films

Optics Express, Vol. 12, Issue 16, pp. 3619-3628 (2004)

http://dx.doi.org/10.1364/OPEX.12.003619

Acrobat PDF (340 KB)

### Abstract

In this paper we extend our theoretical treatment of the extraordinary optical transmission through hole arrays to the case of circular holes and beyond the subwavelength limit. Universal curves for the optical transmission in different regimes of the geometrical parameters defining the array are presented. Finally, we further develop the statement by showing that extraordinary transmission phenomena should be expected for any system where transmission is through two localized modes, weakly coupled between them and coupled to a continuum.

© 2004 Optical Society of America

## 1. Introduction

*a*/

*λ*)

^{4}[1

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

*a*≪

*λ*(

*a*being the hole radius and

*λ*the electromagnetic wavelength); considering the finite screen thickness further reduces the transmission [2] as, in the extreme subwavelength regime, all EM modes inside the hole are evanescent. This is why, six years ago, it come as a surprise the experimental finding [3

3. 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]

*d*<

*λ*<

*λ*

_{c}, where

*d*is the array lattice parameter and

*λ*

_{c}is the cutoff wavelength for EM modes inside the hole. In this regime, the high-pass filtering is due to the hole EM cutoff, while the low-pass one is due to the energy redistribution that occurs when the first diffraction mode becomes propagating. A different mechanism must be responsible for the enhanced optical transmission (EOT) found by Ebbesen et al., which occurs for

*λ*

_{c}<

*d*<

*λ*. Already in [3

3. 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]

4. 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**, 1114–1117 (2001). [CrossRef] [PubMed]

## 2. Theoretical formalism

5. F.J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B **66**, 155412 (2002). [CrossRef]

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

7. J.B. Pendry and A. MacKinnon “Calculation of photon dispersion relations,” Phys. Rev. Lett.69, 2772 (1992);P.M. Bell et al., Comp. Phys. Commun.85, 306 (1995). [CrossRef] [PubMed]

4. 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**, 1114–1117 (2001). [CrossRef] [PubMed]

## 3. Transmittance spectra

*L*=750nm, metal thickness

*h*=320nm and hole diameter

*d*=280

*nm*. These are typical experimental parameters in studies of EOT (the experimental transmission spectra,

*T*(

*λ*), for a finite array of 21×21 holes can be found in Fig. 1 of [4

4. 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**, 1114–1117 (2001). [CrossRef] [PubMed]

*T*(

*λ*) (black curve). For each wavelength the effective hole diameter is taken as the nominal value plus twice the skin depth. Notice that, within the considered model, there are surface plasmons in the flat dielectric-metal interfaces (approximated by the SIBC) but there are no surface plasmons running along the surface of the hole. The cutoff wavelength in this case is

*λ*

_{c}=620 nm, so the figure shows an EOT transmission peak at

*λ*≈780 nm. The peak position is in reasonable agreement with the experimental data [4

**86**, 1114–1117 (2001). [CrossRef] [PubMed]

*ε*

_{M}=-∞ in the flat metal-dielectric interfaces. This last calculation shows that the effect of considering a realistic dielectric constant in the flat metal surface is to slightly red-shift the transmission. The calculation also shows that holes in a perfect conductor can also present EOT [4

**86**, 1114–1117 (2001). [CrossRef] [PubMed]

*T*(

*λ*) for the conditions of the red curve, but considering additionally that, in all the spectral range, the hole diameter equals its nominal value. The hole cutoff occurs now at

*λ*

_{c}=477nm, but still a EOT peak appears. This time the peak position (much closer the condition

*λ*=

*d*, when the last diffraction order becomes non-propagating) and the area under the peak is very different from the experimental value.

*T*(

*λ*) for arrays of circular holes (black curve here as in Fig. 1, repeated for visual convenience) and square holes. In the blue curve the nominal square side is 248nm, so that the squares have the same area as the circular holes (which also gives a similar propagation constant for the fundamental EM eigenmode inside both objects) whereas in the red curve the squares have a side equal to the diameter of the circular holes, 280nm. In the square hole calculations an effective square side is also considered, adding to its nominal value twice the skin depth.

12. J. Gomez-Rivas, C. Schotsch, P. Haring Bolivar, and H. Kurz, “Enhanced transmission of Thz radiation through subwavelength holes,” Phys. Rev. B **68**, 201306 (2003). [CrossRef]

13. F. Miyamaru and M. Hangyo, “Finite size effects of trannsmission property for metal hole arrays in subterahertz region,” Appl. Phys. Lett. **84**, 2742 (2004). [CrossRef]

14. H. Cao and A. Nahata, “Resonantly enhanced transmission of terahertz radiation through a periodic array of subwavelength apertures,” Opt. Express **12**, 1004–1010 (2004). [CrossRef] [PubMed]

*λ*/

*d*,

*a*/

*d*, and universal

*T*(

*λ*) curves can be presented. Even in the optical regime, these curves serve as a guide to the different transmission regimes, if the skin depth is taken into account in the definition of an effective hole radius. Figure 3 renders

*T*(

*λ*) within the perfect conductor approximation for a square array of circular holes, for

*r*/

*d*=0.1,0.2,0.3 and 0.4 (panels 3(a), 3(b), 3(c) and 3(d), respectively) and for different values of

*a*/

*h*within each panel.

## 4. EOT and surface modes

**86**, 1114–1117 (2001). [CrossRef] [PubMed]

*TE*

_{11}) eigenmode needs be considered given that it dominates the transmittance in the subwavelength regime. For normal incidence and

*λ*>

*d*,

*T*(

*λ*)=|

*t*

_{0}|

^{2}, where the zero-order transmission amplitude

*t*

_{0}has the form:

*τ*

^{12},

*τ*

^{23}are the transmission amplitudes for crossing the I–II and II–III interfaces, respectively.

*ϕ*

_{P}=exp(

*ik*

_{z}

*h*),

*k*is the EM wavevector in vacuum, and

*ρ*

^{L},

*ρ*

^{R}are the amplitudes for the

*TE*

_{11}mode to be reflected back into the hole at the II-I, II–III interfaces, respectively. In the system we are considering, where dielectric constants in reflection and transmission regions are equal,

*ρ*

^{L}=

*ρ*

^{R}≡

*ρ*.

**86**, 1114–1117 (2001). [CrossRef] [PubMed]

*t*

_{0}(actually, the spectral position is determined by |

*ρ*

^{L}

*ρ*

^{R}

*ρ*|<1, but for evanescent modes only forces Im(

*ρ*)>0, without posing any restriction on |

*ρ*|. Figure 4 shows Re(

*ρ*) and Im(

*ρ*) as a function of wavelength, for an array of circular holes with

*d*=750nm,

*h*=320nm and three different nominal hole radius. The calculation is done within the SIBC approximation, considering enough diffraction orders for achieving convergency. An even more simplified model was presented in Ref. [4

**86**, 1114–1117 (2001). [CrossRef] [PubMed]

*ρ*clearly resembles the the form expected (from Kramers-Kroning relations) for a causal function in the vicinity of a localized mode. The finite width in the Im(

*ρ*) peak indicates that this localized mode is a resonance, i.e. it is coupled to a continuum into which EM energy is lost from the mode. This energy can be lost due both to absorption and radiative losses. In order to show the relative importance of this two mechanisms, we represent in the inset of Fig. 4 (green curve) the calculation for

*r*=140nm within the SIBC but considering a hypothetical lossless silver. The comparison with the full calculation (inset of Fig. 4, red curve) shows that the resonant peak width, and therefore the typical time that the radiation stays at the surface, does not change much and is, within this range of geometrical parameters, limited by radiation losses. Up to this point, it could be thought that the surface modes are the bona fide surface plasmon polaritons of the flat silver surface which, due to the folding of their dispersion curve induced by the hole lattice can now couple to radiative modes. However, surface modes appear even in hole arrays within the perfect conductor approximation (inset of Fig.4, purple curve), although flat perfect conductor interfaces do not present surface plasmons polaritons. In Ref. [4

**86**, 1114–1117 (2001). [CrossRef] [PubMed]

*perfect*conductors. In order to further clarify the situation in this paper we have tried to avoid this (in our opinion minor) point, sticking to the terminology “surface EM modes”. Recently, we have shown that the origin of these modes in corrugated perfect conductors come from the ability of these systems to spoof the response of a real electron plasma [18], which may have interesting applications for controlling light propagation in metal surfaces through geometry.

**86**, 1114–1117 (2001). [CrossRef] [PubMed]

*h*/

*a*.

*V*=30 (in units of

*ĥ*=1, particle mass,

*m*=1),

*L*

_{1}=

*L*

_{2}=5 and

*W*

_{1}=

*W*

_{3}=1. Figure 5. shows the transmission probability as a function of the incident kinetic energy of the particle, for three different widths of the central barrier,

*W*

_{2}. The transmission spectra for this system present the well known resonant tunnelling behavior (notice that

*E*<

*V*) through localized states in the wells formed between each two consecutive barriers. This system also shows the same transition, from a two transmission peaks to a one peak situation, as the transparency of the barrier separating both localized states in decreased. This 1D QM analog also serves to illustrate the properties of the reflection coefficient. Consider now that

*W*

_{2}→∞. For a particle coming from

*x*=-∞, with

*E*>

*V*, the

*total*reflection coefficient,

*r*, must satisfy |

*r*|=1, due to current conservation. But, for

*E*<

*V*, the reflection amplitude for a wave at

*x*=

*x*=

*ρ*, is defined between evanescent modes and, as stated previously, current conservation only forces Im

*ρ*>0. Inset to Figure 5. shows that this is indeed the case, with the peak in Im

*ρ*marking the position of “leaky” (due to radiation) localized modes in the system.

## 5. Conclusions

## References and links

1. | H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. |

2. | A. Roberts, “Electromagnetic Theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A |

3. | T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

4. | 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. |

5. | F.J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B |

6. | E. Popov, M. Nevire, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B |

7. | J.B. Pendry and A. MacKinnon “Calculation of photon dispersion relations,” Phys. Rev. Lett.69, 2772 (1992);P.M. Bell et al., Comp. Phys. Commun.85, 306 (1995). [CrossRef] [PubMed] |

8. | In the Fourier expansion formalism we are not aware of any published data on how convergency as a function of wavenumber cutoff is reached. |

9. | L.D. Landau, E.M. Lifshitz, and L.P. Pitaievskii: |

10. | P.M. Morse and H. Feshbach: |

11. | We take the dielectric constant for silver from: |

12. | J. Gomez-Rivas, C. Schotsch, P. Haring Bolivar, and H. Kurz, “Enhanced transmission of Thz radiation through subwavelength holes,” Phys. Rev. B |

13. | F. Miyamaru and M. Hangyo, “Finite size effects of trannsmission property for metal hole arrays in subterahertz region,” Appl. Phys. Lett. |

14. | H. Cao and A. Nahata, “Resonantly enhanced transmission of terahertz radiation through a periodic array of subwavelength apertures,” Opt. Express |

15. | M. Beruete, M Sorrola, I. Campillo, J.S. Dolado, L. Martin-Moreno, J. Bravo-Abad, and F.J. Garcia-Vidal, “Enhanced millimeter wave transmission through subwavelength hole arrays,” (Opt. Lett., in press). |

16. | Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. |

17. | W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. |

18. | J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science Express, 10.1126, 8 July 2004. |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Focus Issue: Extraordinary light transmission through sub-wavelength structured surfaces

**History**

Original Manuscript: June 14, 2004

Revised Manuscript: July 29, 2004

Published: August 9, 2004

**Citation**

L. Moreno and F. García-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express **12**, 3619-3628 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3619

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### References

- H. A. Bethe, ???Theory of diffraction by small holes,??? Phys. Rev. 66, 163 (1944). [CrossRef]
- A. Roberts, ???Electromagnetic Theory of diffraction by a circular aperture in a thick, perfectly conducting screen,??? J. Opt. Soc. Am. A 4, 1970 (1987).
- 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]
- 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, 1114-1117 (2001). [CrossRef] [PubMed]
- F.J. Garcia-Vidal and L. Martin-Moreno, ???Transmission and focusing of light in one-dimensional periodically nanostructured metals,??? Phys. Rev. B 66, 155412 (2002). [CrossRef]
- E. Popov, M. Nevire, S. Enoch, and R. Reinisch, ???Theory of light transmission through subwavelength periodic hole arrays,??? Phys. Rev. B 62, 16100 (2000). [CrossRef]
- J.B. Pendry and A. MacKinnon, ???Calculation of photon dispersion relations,??? Phys. Rev. Lett. 69, 2772 (1992); P.M. Bell et al., Comp. Phys. Commun. 85, 306 (1995). [CrossRef] [PubMed]
- In the Fourier expansion formalism we are not aware of any published data on how convergency as a function of wavenumber cutoff is reached.
- L.D.Landau, E.M. Lifshitz and L.P. Pitaievskii: Electrodynamics of Continuous Media, (Pergamon Press, Oxford, 1984).
- P.M. Morse and H. Feshbach:Methods of Theoretical Physics, (McGraw-Hill, New York, 1953).
- We take the dielectric constant for silver from: Handbook of Optical Constants of Solids , edited by E.D. Palik (Academic, Orlando, 1985).
- J. Gomez-Rivas, C. Schotsch, P. Haring Bolivar, and H. Kurz, ???Enhanced transmission of Thz radiation through subwavelength holes,??? Phys. Rev. B 68, 201306 (2003). [CrossRef]
- F. Miyamaru and M. Hangyo, ???Finite size effects of trannsmission property for metal hole arrays in subterahertz region,??? Appl. Phys. Lett. 84, 2742 (2004). [CrossRef]
- H. Cao and A. Nahata, ???Resonantly enhanced transmission of terahertz radiation through a periodic array of subwavelength apertures,??? Opt. Express 12, 1004-1010 (2004). [CrossRef] [PubMed]
- M. Beruete, M, Sorrola, I. Campillo, J.S. Dolado, L. Martin-Moreno, J. Bravo-Abad, and F.J. Garcia-Vidal, ???Enhanced millimeter wave transmission through subwavelength hole arrays,??? (Opt. Lett., in press).
- 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]
- W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, ???Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,??? Phys. Rev. Lett. 92, 107401 (2004). [CrossRef] [PubMed]
- J.B. Pendry, L. Martin-Moreno and F.J. Garcia-Vidal, ???Mimicking surface plasmons with structured surfaces,??? Science Express, 10.1126, 8 July 2004.

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