## Anomalies in the disappearance of the extraordinary electromagnetic transmission in subwavelength hole arrays

Optics Express, Vol. 16, Issue 7, pp. 4719-4730 (2008)

http://dx.doi.org/10.1364/OE.16.004719

Acrobat PDF (459 KB)

### Abstract

We studied the evolution of the Extraordinary Electromagnetic Transmission (EET) through subwavelength hole arrays versus hole size. Here, we show that for large holes EET vanishes and is replaced by another unusual transmission. A specific hole size is found where all the characteristics of the EET vanish and where most usual models fail to describe the transmission except full 3D simulations. The transition between these two domains is characterized by the discontinuity of parameters describing the transmission, in particular the resonance frequency. This transition exhibits a first order phase transition like behavior.

© 2008 Optical Society of America

## 1. Introduction

## 2. Experiments and results

*h*very small compared to the wavelength (

*h*≈8

*µ*m in our experiments) and large compared to the skin depth

*δ*of the electromagnetic field (

*δ*≪1

*µ*m). Thus the arrays may be considered to be almost purely two dimensional (

*D*≫

*h*≫

*δ*where

*D*is the size of the holes). On the contrary in the visible, the skin depth is relatively larger, since it scales as √λ. The system is 3D in the visible range (

*D*≈

*h*≫

*δ*), and our results may not be reproducible in the visible range since such conditions are hardly obtained.

*µ*m (figure 1A). Series of twodimensional arrays of square and round subwavelength holes (54 arrays of 16 by 16 holes) have been designed and analyzed. Terahertz spectra are recorded using standard terahertz time-domain spectroscopy [26]. Broadband linearly polarized subpicosecond single cycle pulses of terahertz radiation are generated and coherently detected by illuminating photoconductive antennas with two synchronized femtosecond laser pulses (figure 1B). The sample is positioned on a 10mm circular hole, in the linearly polarized, frequency independent, 4.8mm-waist (1/

*e*in amplitude) Gaussian THz beam. Numerical Fourier transform of the time-domain signals with and without the sample gives access to the transmission spectrum of the subwavelength structures. The dynamics of the EET is recorded over 240 ps, yielding a raw 3GHz frequency precision after numerical Fourier transform, with 10

^{4}signal to noise ratio in 300ms acquisition time. The resonance frequency is obtained by fitting the resonance with a parabolic function taking into account 20 points over the resonance. Therefore the precision is better than the single point precision, and below 1GHz.

1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature **391**, 667–668 (1998). [CrossRef]

4. C. J. Bouwkamp, “Diffraction Theory,” Rep. Prog. Phys. **17**, 35–100 (1954). [CrossRef]

*µ*m. However for larger holes the spectra exhibit unusual features (figure 2, black line). The spectra exhibit large, symmetric resonances with resonance frequencies remaining constant with the hole size, and scaling as a factor 2. These are not characteristics of the EET anymore, even though the transmission energy still exceeds the sum of transmission of all the holes. All resonances not scaling as integers have disappeared. It then appears that the spectral features originate from a complex interplay between the geometrical resonance of the array and the shape and size of the holes. The array periodicity provides approximatively the resonance positions (

*L*=600

*µ*m gives

*ν*=0.5 THz as first resonance). As previously found by [27

27. C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Technol. **21**, 1–6 (1973). [CrossRef]

## 3. Quantitative study

*p*is defined as the ratio between the surface of the metal

*S*and the surface of the holes

_{m}*S*in the lattice,

_{a}*a*,

*p*=

*L*

^{2}/

*a*

^{2}-1 and for round holes of diameter

*d*,

*L*the lattice period. The evolution of frequency

*ν*of the first resonance is given by the normalized frequency shift

*ν*

_{0}is the limit resonance frequency for tiny holes, given by Bloch theory as

*U*as the total electromagnetic energy transmitted through the plate

*A*and

_{t}*A*are the measured amplitudes of the electric fields with and without the sample, respectively. We also define a quality factor by

_{o}*W*is the full width at half maximum of the first resonance. We focused on the observation and evolution of these parameters, which provide fundamental insights on the nature of the generation and propagation of the EET.

*ν*,

*Q*and Δ

*W*versus

*p*are given by figure 3 for square and round holes with a lattice period

*L*=600

*µm*. Each point (filling parameter) is provided by a corresponding free-standing plate. For each hole shape, two domains are clearly distinguishable, limited by a discontinuity of all the parameters, and at the same filling parameter

*p*. However,

_{c}*p*is different for square and round holes:

_{c}*p*=1.03±0.01 for square holes and

_{c}*p*=1.37±0.01 for round holes. Transmission spectra just before and after the transition filling parameter for round holes are shown in figure 4.More precisely, Δ

_{c}*ν*is constant below

*p*and corresponds to the first order of diffraction. Across

_{c}*p*, Δ

_{c}*ν*jumps to lower values, and then continuously decreases toward 0. The jump is small, respectively 5 and 19GHz for square and round holes, but easily observable and is also supported by the slopes of the curves. Evolution of Δ

*W*is similar, except that it not constant before discontinuity, in particular for round holes. The behavior of

*Q*is more complex. First, the evolution in each domain is non monotonic. The discontinuity is clearly visible for square holes, less important for round holes. All the curves with the same hole shape show discontinuity for the same value of

*p*. We emphasize the point that

_{c}*p*is different for square and round holes. It is due to different limit conditions between adjacent interacting holes. As a consequence, the distribution of distance (then the quantity of available metal) is different between square and round holes, much sharper for round holes. And the ability for the electromagnetic wave to couple through surface waves is modified.

_{c}## 4. Modeling

*ab initio*finite element simulations for directly solving Maxwell’s equations.

27. C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Technol. **21**, 1–6 (1973). [CrossRef]

*l*

_{1}=8

*µ*m in our experiments) into account. It is based on the superposition theorem, and decomposes the problem into symmetric and antisymmetric plane wave excitations. Both electric and magnetic fields, on both sides of the holes, are expanded into a set of Floquet modes Φ

*, where*

_{pqr}*p*and

*q*are the spatial modes, and

*r*denotes TE or TM mode [28

28. J.A. Besley, N.N. Akhamediev, and P.D. Miller, “Periodic optical wavequides: exact Floquet theory and spectral properties,” Studies in Applied Mathematics **101**, 343–355 (1998). [CrossRef]

*γ*, and a modal wave admittance ξ

_{pq}*. Inside the holes, the waves are expressed as waveguide modes Ψ*

^{f}_{pqr}*. The amplitude of the electric field is given by*

_{mn}*A*

_{00r}. By matching the boundary conditions on both sides of the holes,

**E**

*is the transverse electric field, and*

_{t}*F*and

_{mnr}*Y*are the modal coefficients of the waveguide modes.

_{mnr}*p*<

*p*, does not catch the discontinuities and Δ

_{c}*ν*diverges rapidly for

*p*≈

*p*.

_{c}29. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124**, 1866–1875 (1961). [CrossRef]

*α*is a numerical constant,

*q*

^{2}is proportional to the ratio of the transition probabilities to the isolated level, and to the portion of the continuum that is not interacting.

*E*is the energy of the isolated level and Hilb(

_{φ}*V*) is the Hilbert transform of

_{E}*V*=〈

_{E}*ψ*|

_{E}*V*|

*ϕ*〉, where

*V*is the interaction potential between the isolated level and a level of the continuum with energy

*E*. Contrary to the approximation of the Fano model previously used in EET [30

30. C. Genet, M.P. van Exter, and J.P. Woerdman, “Fano-Type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. commun. **225**, 331–336 (2003). [CrossRef]

*V*with respect to

_{E}*E*[29

29. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124**, 1866–1875 (1961). [CrossRef]

*V*which provides a Gaussian wave function for the fundamental level of |

*ϕ*〉, and leads to

*A*and

*B*are parameters of the extended model [31]. The calculations are the solid red lines on figure 5A. This model gives good fits for high

*p*values, but begins to differ from experimental curves near

*p*. This model does not catch the discontinuities.

_{c}32. K. J. Klein 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 apertures,” Phys. Rev. Lett. **92**, 183901 (2004). [CrossRef] [PubMed]

33. L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

*q*mode wave vector

^{th}*γ*along

_{q}*z*axis (eigenvalues) and the field distribution Φ

_{q}(

*x*,

*y*) (eigenvectors). The field components around the hole area are written as

*r*=

*x*or

*y*,

*u*is the amplitude of the upward decaying modal fields, and

_{q}*d*is the amplitude of the downward decaying modal fields. This model also partially neglects near field interactions and is based on modal decomposition. The precision of the FMM and its ability to describe experimental results depend in principle on the limit of the development. In describing our experimental data, we show simulation with a 30

_{q}^{th}order development. We also checked the convergence of the algorithm for higher order calculation. We found less than 1% variation in field transmission. Surprisingly, this model is also unable to describe the discontinuity between the two kinds of transmission. It is possible that part of the near field interactions are not entirely described, as well as the complete decomposition of the field on the surface of the metal. The calculations are the dashed red lines on figure 5B. This model gives better result for square holes than for round holes.

35. J.-Y. Laluet, E. Devaux, C. Genet, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Optimization of surface plasmons launching from subwavelength hole arrays: modelling and experiments,” Opt. Express **15**, 3488–3495 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3488. [CrossRef] [PubMed]

36. E. Devaux, T.W. Ebbesen, J.-C Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett. **83**, 4936–4939 (2003). [CrossRef]

**E**

*of wave vector*

_{in}**k**

*is decomposed near the holes into a two dimensional surface wave that is scattered away from the hole as a spherical waves*

_{in}*ε*the dielectric constant of the metal. The total surface plasmon field is evaluated as the sum of all fields emitted from each holes,

_{m}**e**

*is a unitary polarization vector,*

_{SP}**δ**

*is the*

_{n}*n*source from the center of the array. This model is based on the assumption that surface plasmons are excited by the plane wave, or that the arrays generate surface plasmon like interactions [37

^{th}37. E. Ozbay, “Plasmonic : merging photonics and electronics at nanoscale dimensions,” Science **311**, 189–193 (2006). [CrossRef] [PubMed]

*p*>

*p*. Results are still good near

_{c}*p*. But it does not catch the discontinuities and fails to describe experimental results for

_{c}*p*<

*p*.

_{c}*ab initio*Finite Element Method (FEM) analysis [20, 38] of the electric field propagating through the array. This method allows the calculation of the transmitted THz electromagnetic field and takes into account the nearfield effects on the array. To reduce the size of the simulation box, we used a unitary cell with one hole in its center, with adequate boundary conditions. The precision of the simulations are controlled by progressively reducing the adaptive mesh size, in particular in the subwavelength holes. Typical mesh dimensions are λ/700 in the holes and λ/50 outside, yielding to a total precision on the transmitted electric field of 0.1%. The relative permittivity of nickel is

*ε*=-9.7×10

^{3}+1.1×10

^{5}

*i*, and relative permeability is 100 (in Gaussian units) [39

39. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, J. R.W. Alexander, and C. A Ward, “Optical-properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. **22**, 1099–1119 (1983). [CrossRef] [PubMed]

*ν*and Δ

*W*with small shifts, which are the consequence of multiple reflections of the electromagnetic waves inside the simulation box. The evolution of

*Q*is also well described by the simulation.

## 5. Discussion

_{10}mode is λ

*=2.61*

_{c}*d*and is equal to 575

*µ*m at

*p*=1.37 (

_{c}*d*=220

*µ*m). For symmetry reasons, the corresponding mode for square apertures is TE

_{11}. This is justified by the invariance of the shape of the transmission spectra with respect to the incident polarization angle we experimentally observed, as well as in [9

9. 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]

*=√2*

_{c}*a*=595

*µ*m with

*p*=1.03 (

_{c}*a*=421

*µ*m). Both cutoff wavelengths for square and round holes are very close to the lattice period of 600

*µ*m. Therefore, the transition seems to arise when the quantity of metal goes below the quantity given by the corresponding cut-off wavelength. At this point, the number of degrees of freedom of the surface waves diminishes. The propagation at the surface of the metal being prohibited, the system evolves from 2D to 1D characteristics, as hinted by the shapes of the spectra before and after

*p*(see fig. 4): 2D as the resonance frequencies scale as √2 for

_{c}*p*>

*p*and 1D as they scale as integers for

_{c}*p*<

*p*.

_{c}*ψ*(

**x**) behind the aperture and from Kirchhoff’s approximation [41]. Assuming small holes compared to the wavelength [3

3. H. A. Bethe, “Theory of diffraction by small apertures,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

*ψ*

_{0}is the field just before the hole, and

**R**=

**x**-

**x**′. Even though this approximation gives mostly very good results in Dirichlet (or Neumann) boundary conditions, it is known [41] that it does not hold in mixed boundary conditions, where both

*ψ*and ∇′

*ψ*(

**x**′) are fixed. Here, the plate thickness is negligible compared to the wavelength, resulting in a discontinuity of

*ψ*in the hole and then in mixed boundary conditions. To obtain the exact boundary conditions, one then must apply Maxwell’s equations. Otherwise, subsequent calculation of the diffracted fields may contain errors. Similar demonstration can be done with vectorial theory. However, one must emphasize that Kirchhoff’s approximation gives good results for small holes, showing that the boundary conditions are essentially given by Dirichlet conditions. Near the transition, the wave propagation at the surface of the metal is strongly attenuated, and do not anymore only contribute to the boundary conditions. The fields inside the holes also play a role and the boundary conditions are then mixed. Therefore only full 3D resolution of Maxwell’s equations can provide a satisfactory result.

*p*<

*p*, none of the models used here is able to describe the evolution of the three parameters nor do they reproduce the discontinuity of the parameters. Finally only finite element programming is able to describe the discontinuities in the evolution of Δ

_{c}*ν*,

*Q*and Δ

*W*.

3. H. A. Bethe, “Theory of diffraction by small apertures,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

4. C. J. Bouwkamp, “Diffraction Theory,” Rep. Prog. Phys. **17**, 35–100 (1954). [CrossRef]

*p*<

*p*using the sum of one dimensional array of individual Bethe transmissions

_{c}**r**

_{j}the position of the center of hole

*j*,

**r**′

_{j}the coordinates inside hole

*j*, and

**H**

*and*

_{o}**E**

*the magnetic and the electric field, respectively. If we add an effective spatial period corresponding to the first resonant frequency experimentally detected, we obtain spectra fitting reasonably well the experimental ones (figure 6).*

_{o}*ν*to the order parameter of the system consisting in the electromagnetic field coupled to the subwavelength lattice, and the quality factor

*Q*to the heat capacity, one obtains that the system displays a discontinuity of both its order parameter and heat capacity at the same value

*p*. This is literally the definition of a first order phase transition [43, 44]. This parallel could prove to be helpful in conceiving a more general description of the complex interaction between electromagnetic waves and arrays of subwavelength holes, thanks to the important theoretical background of phase transition theories [45].

_{c}## 6. Conclusion

## References and links

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

2. | A. Nahata, R. A. Linke, T. Ishi, and K. Ohashi, “Enhanced nonlinear optical conversion from a periodically nanostructured metal film,” Opt. Lett. |

3. | H. A. Bethe, “Theory of diffraction by small apertures,” Phys. Rev. |

4. | C. J. Bouwkamp, “Diffraction Theory,” Rep. Prog. Phys. |

5. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

6. | T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. |

7. | 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. |

8. | S. C. Hohng, Y. C. Yoon, D. S. Kim, V. Malyarchuk, R. Müller, Ch. Lienau, J. W. Park, K. H. Yoo, J. Kim, H. Y. Ryu, and Q. H. Park, “Light emission from the shadows: surface plasmon nano-optics at near and far fields,” Appl. Phys. Lett. |

9. | 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. |

10. | A. Degiron, H. J. Lezec, W. L. Barnes, and T.W. Ebbesen, “Effects of hole depth on enhanced light transmission through subwavelength hole arrays,” Appl. Phys. Lett. |

11. | E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Plasmon-assisted transmission of entangled photons,” Nature |

12. | E. Devaux, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett. |

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

14. | Y.-H. Ye and J.-Y. Zhang, “Middle-infrared transmission enhancement through periodically perforated metal films,” Appl. Phys. Lett. |

15. | J. Gomez Rivas, C. Schotsch, P. Haring Bolivar, and H. Kurz, “Enhanced transmission of THz radiation through subwavelength apertures,” Phys. Rev. B |

16. | D. Qu and D. Grischkowsky, “Observation of a New Type of THz Resonance of Surface Plasmons Propagating on Metal-Film Hole Arrays,” Phys. Rev. Lett. |

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

18. | J. Gomez Rivas, C. Janke, P. H. Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Express |

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

20. | J.-B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B |

21. | D. Qu, D. Grischkowsky, and W. Zhang, “Terahertz transmission properties of thin, subwavelength metallic hole arrays,” Opt. Lett. |

22. | T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, and T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B |

23. | A. K. Azad, Y. Zhao, and W. Zhang, “Transmission properties of terahertz pulses through an ultrathin subwavelength silicon hole array,” Appl. Phys. Lett. |

24. | F. Przybilla, C. Genet, and T.W. Ebbesen, “Enhanced transmission through Penrose subwavelength hole arrays,” Appl. Phys. Lett. |

25. | T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature |

26. | D. Grischkowsky, S. R. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B |

27. | C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Technol. |

28. | J.A. Besley, N.N. Akhamediev, and P.D. Miller, “Periodic optical wavequides: exact Floquet theory and spectral properties,” Studies in Applied Mathematics |

29. | U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. |

30. | C. Genet, M.P. van Exter, and J.P. Woerdman, “Fano-Type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. commun. |

31. | J.-B. Masson, A. Podzorov, and G. Gallot, “Generalized parabolic Fano model of extraordinary electromagnetic transmission in subwavelength hole arrays,” submitted. |

32. | K. J. Klein 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 apertures,” Phys. Rev. Lett. |

33. | L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

34. | C.M. Soukoulis, “Photonic Crystals and Light Localization in the 21st Century,” NATO Science Series Vol. 563 (Kluwer, Dordrecht, 2001). |

35. | J.-Y. Laluet, E. Devaux, C. Genet, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Optimization of surface plasmons launching from subwavelength hole arrays: modelling and experiments,” Opt. Express |

36. | E. Devaux, T.W. Ebbesen, J.-C Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett. |

37. | E. Ozbay, “Plasmonic : merging photonics and electronics at nanoscale dimensions,” Science |

38. | Comsol Multiphysics, Comsol Inc., Burlington, MA. |

39. | M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, J. R.W. Alexander, and C. A Ward, “Optical-properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. |

40. | Magnetic Properties of Metals, Landolt-Bornstein, Group III: condensed matter, Springer-Verlag, Berlin (1986). |

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44. | H.E. Stanley, Introduction to phase transitions and critical phenomena, Oxford science publications (1971). |

45. | J.-B. Masson and G. Gallot, “Experimental evidence of percolation phase transition in surface plasmons generation,” ArXiv:cond-mat/0611280v1 (2006). |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(260.3090) Physical optics : Infrared, far

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: November 16, 2007

Revised Manuscript: December 12, 2007

Manuscript Accepted: December 13, 2007

Published: March 21, 2008

**Citation**

Jean-Baptiste Masson, Alexander Podzorov, and Guilhem Gallot, "Anomalies in the disappearance of the extraordinary electromagnetic transmission in subwavelength hole arrays," Opt. Express **16**, 4719-4730 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4719

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### References

- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 391, 667-668 (1998). [CrossRef]
- A. Nahata, R. A. Linke, T. Ishi, and K. Ohashi, "Enhanced nonlinear optical conversion from a periodically nanostructured metal film, " Opt. Lett. 28, 423-425 (2003). [CrossRef] [PubMed]
- H. A. Bethe, "Theory of diffraction by small apertures," Phys. Rev. 66, 163-182 (1944). [CrossRef]
- C. J. Bouwkamp, "Diffraction Theory," Rep. Prog. Phys. 17, 35-100 (1954). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, "Control of optical transmission through metals perforated with subwavelength hole arrays," Opt. Lett. 24, 256-258 (1999). [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]
- S. C. Hohng, Y. C. Yoon, D. S. Kim, V. Malyarchuk, R. Muller, Ch. Lienau, J. W. Park, K. H. Yoo, J. Kim, H. Y. Ryu, and Q. H. Park, "Light emission from the shadows: surface plasmon nano-optics at near and far fields," Appl. Phys. Lett. 81, 3239-3241 (2002). [CrossRef]
- 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]
- A. Degiron, H. J. Lezec,W. L. Barnes, and T.W. Ebbesen, "Effects of hole depth on enhanced light transmission through subwavelength hole arrays," Appl. Phys. Lett. 81, 4327-4330 (2002). [CrossRef]
- E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Plasmon-assisted transmission of entangled photons," Nature 418, 304-306 (2002). [CrossRef] [PubMed]
- E. Devaux, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, "Launching and decoupling surface plasmons via micro-gratings," Appl. Phys. Lett. 83, 4936-4939 (2003). [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, 057403 (2002). [CrossRef] [PubMed]
- Y.-H. Ye and J.-Y. Zhang, "Middle-infrared transmission enhancement through periodically perforated metal films," Appl. Phys. Lett. 84, 2977-2980 (2004). [CrossRef]
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