## Extended Fano model of Extraordinary Electromagnetic Transmission through subwavelength hole arrays in the terahertz domain

Optics Express, Vol. 17, Issue 17, pp. 15280-15291 (2009)

http://dx.doi.org/10.1364/OE.17.015280

Acrobat PDF (259 KB)

### Abstract

We developed an extended Fano model describing the Extraordinary Electromagnetic Transmission (EET) through arrays of subwavelength apertures, based on terahertz transmission measurements of arrays of various hole size and shapes. Considering a frequency-dependent coupling between resonant and non-resonant pathways, this model gives access to a simple analytical description of EET, provides good agreement with experimental data, and offers new parameters describing the influence of the hole size and shape on the transmitted signal.

© 2009 Optical Society of America

## 1. introduction

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 **391**, 667–668 (1998). [CrossRef]

2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

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

4. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**, 728–731 (2008). [CrossRef] [PubMed]

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

5. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science **305**, 847–848 (2004). [CrossRef] [PubMed]

4. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**, 728–731 (2008). [CrossRef] [PubMed]

6. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature **445**, 39–46 (2007). [CrossRef] [PubMed]

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. **86(6)**, 1114–1117 (2001). [CrossRef]

8. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of Surface Plasmon Generation at Nanoslit Apertures,” Phys. Rev. Lett. **95**, 263,902 (2005). [CrossRef]

9. J. Bravo-Abad, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of Extraordinary Transmission of Light through Quasiperiodic Arrays of Subwavelength Holes,” Phys. Rev. Lett. **99**, 203,905 (2007). [CrossRef]

10. G. Gay, O. Alloschery, B. V. de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nature Physics **2**, 262–267 (2006). [CrossRef]

11. K. G. Lee and Q. H. Park, “Coupling of Surface Plasmon Polaritons and Light in Metallic Nanoslits,” Phys. Rev. Lett. **95**, 103,902 (2005). [CrossRef]

13. J. M. Brok and and H. P. Urbach, “Extraordinary transmission through 1, 2 and 3 holes in a perfect conductor, modelled by a mode expansion technique,” Opt. Exp. **14(7)**, 2552–2572 (2006). [CrossRef]

14. A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Exp. **16(13)**, 9601–9613 (2008). [CrossRef]

4. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**, 728–731 (2008). [CrossRef] [PubMed]

## 2. Extended Fano model

25. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66(7–8)**, 163–182 (1944). [CrossRef]

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

*i*〉 and excited state |

*ψ*〉. The latter is the result of the coupling between a non resonant continuum {|

_{E}*E*〉} and a resonant state |

*φ*〉. Without coupling between {|

*E*〉} and |

*φ*〉, the matrix elements of the non perturbed Hamiltonian

*H*

_{0}are

*δ*is the Dirac function. Considering a coupling between {|

*E*〉} and |

*φ*〉, the total Hamiltonian becomes

*V*is the coupling Hamiltonian. |

*ψ*〉 are the eigenstates of

_{E}*H*of eigenvalues

*E*and the new matrix elements are

*i*〉 and either the discrete or continuum states. The transmission efficiency through the periodic arrays of subwavelength holes is then given by the probability of transition from |

*i*〉 to the final state |

*ψ*〉 with coupling, |〈

_{E}*ψ*|

_{E}*T*|

*i*〉|

^{2}, normalized by the transition probability in absence of coupling, |〈

*E*|

*T*|

*i*〉|

^{2}. According to Fano derivation [21

21. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124(6)**, 1866–1875 (1961). [CrossRef]

*q*(

*E*) the Breit-Wigner-Fano coupling coefficient defined as

## 3. Experimental results

27. D. Grischkowsky, S. R. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B **7(10)**, 2006–2015 (1990). [CrossRef]

*µ*m at 1 THz), the corresponding mechanical precision on the hole geometry allows a very accurate design and shape control of the apertures. Therefore, polyhedral geometries can be investigated: triangle, square, pentagon and round holes of various sizes. 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. Numerical Fourier transform of the time-domain signals gives access to the transmission spectrum of the arrays. The samples are free-standing 10-

*µ*m-thick nickel arrays of subwavelength polyhedral holes, fabricated by electroforming. Influence of substrate or plate thickness is then negligible, and the plates are still much thicker than skin depth in the terahertz range. All arrays have a

*L*=600

*µ*m period, and are positioned on a 10 mm circular aperture, in the linearly polarized, frequency independent 4.8 mm-waist (1/e in amplitude) Gaussian THz beam. The precision over the hole size and periodicity is 1

*µ*m. The dynamics of the EET is then recorded during 250 ps, yielding to a 4 GHz frequency precision after numerical Fourier transform, with 10

^{4}signal to noise ratio in a 300 ms acquisition time. A reference scan is taken with empty aperture. The transmission of the array is then calculated by taking the amplitude ratio of the complex spectra of the metal plate and reference scans.

19. W. Zhang, A. K. Azad, J. Han, J. Xu, J. Chen, and X.-C. Zhang, “Direct Observation of a Transition of a Surface Plasmon Resonance from a Photonic Crystal Effect,” Phys. Rev. Lett. **98**, 183,901 (2007). [CrossRef]

*ν*

^{0}

_{i,j}. The observed resonance frequencies

*ν*

_{i,j}are shifted from

*ν*

^{0}

_{i,j}as usually found [20

20. J.-B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B **73**, 121,401(R) (2006). [CrossRef]

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

29. J.-B. Masson, A. Podzorov, and G. Gallot, “Anomalies in the disappearance of the extraordinary electromagnetic transmission in subwavelength hole arrays,” Opt. Exp. **16(7)**, 4719–4730 (2008). [CrossRef]

*D*for which

*S*=

*πD*

^{2}/4 equals the real surface of an individual hole. Contrary to some previous papers, we do not observe anti-resonance at Bloch frequencies, probably due to the ultra thin metal plates used in the experiments.

*q*and

*v*are left free to evolve with respect to

*E*. The first important result is that parameter

*q*remains constant for all shapes and sizes, within experimental uncertainty (

*q*=-6±0.5). On the contrary,

*v*is not constant, and exhibits a strong dependence with

*E*. It then appears that

*q*is no more an important parameter of our model. The peak asymmetry will be much more sensitive now to the simultaneous evolution of

*v*(

*E*) and Γ(

*E*) rather than

*q*. The coupling Hamiltonian is clearly of Gaussian shape, whose height and width depend on the shape and size of the apertures, as shown in figure 3.

*v*(

*E*) can then be written in terms of Gaussian parameters, as

*A*represents the integral of the Gaussian, as

*A*=∫

^{∞}

_{0}

*v*(

*E*)

*dE*, and Δ is the width of the Gaussian. Since

*v*

^{2}(

*E*) has the dimension of an energy (taken in THz for purpose of simplicity here), dimensions of

*A*and Δ are then in THz

^{3/2}and THz, respectively. Evolution of Gaussian parameters

*A*and Δ can be found in figures 4 and 5.

*A*evolves monotonously with respect to

*D*(see figure 4). Moreover, its profile is the same for all the hole shapes. Every curves can be superimposed within uncertainty range if normalized. This parameter can then be decomposed into shape-dependent and size-dependent functions as

*s*refers to round, square, triangle or pentagon shapes. We found that all the curves are homothetic to a unique hyperbolic function. The inset of figure 4 shows

*A*(

_{D}*D*) and the solid curve is a fit with the following hyperbolic function

*D*in

*µ*m. Evolution of the shape-dependent parameter

*A*(

*s*,0) is also given by figure 6A. To compare the different hole shapes, a rugosity parameter Δ

*r*has been introduced as the mean deviation of the hole profiles compared to the mean radius

*r*̄,

*n*=3, 4, 5 or ∞ for triangle, square, pentagon and round shapes, respectively, and

*r*(

*θ*) is the polar coordinate of the hole with respect to its center.

*A*(

*s*,0) is an increasing function of rugosity.

*D*. Therefore, the slope only depends on the hole shape, and one can write

*α*(

*s*) is an increasing linear function of the rugosity (see figure 6B). Each linear 1/Δ curves crosses the X axis at a point

*D*(

*s*) comprised between 420 and 470

*µ*m, corresponding to a state of infinitely broad coupling Hamiltonian.

*x*) where erfi is the imaginary error function defined as erfi(

*x*)=-

*i*erf(

*ix*) [30]. Then

*ν*

_{1,0}of the first resonance, compared to the one of the extended Fano model. Both show that

*ν*

_{1,0}is larger for big apertures, and converges toward

*ν*

^{0}

_{1,0}for tiny apertures. Evolutions of

*ν*

_{1,0}for round and square hole lattice are very different, highlighting the complex relationship between EET and the geometry of the screen [29

29. J.-B. Masson, A. Podzorov, and G. Gallot, “Anomalies in the disappearance of the extraordinary electromagnetic transmission in subwavelength hole arrays,” Opt. Exp. **16(7)**, 4719–4730 (2008). [CrossRef]

## 4. Discussion

*A*and Δ on the shape of the resonance is not as straightforward as in the original Fano model, since the coupling now depends on the frequency. However it is possible to obtain a general behavior of

*A*and Δ. Figure 8 shows several calculated Fano profiles, for various values of

*A*and Δ. It results in first order that

*A*mainly affects the width of the resonance, with Δ constant (Figure 8A), whereas the ratio

*A*/Δ controls the asymmetry and shift (Figure 8B).

## 5. Conclusion

## A. Numerical calculation of the extended Fano model parameters

*v*(

*E*) and

*q*(

*E*) are derived from the inversion calculation of Laplace transform [32

32. D. G. Duffy, “On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications,” ACM Trans. Math Soft. **19(3)**, 333–359 (1993). [CrossRef]

*v*(

*E*) over orthonormal Laguerre functions

*ϕ*(

_{k}*E*) defined from Laguerre polynomials

*L*(

_{k}*E*) by [33]

*b*are the Laguerre polynomial coefficients. The coupling

_{l}*v*(

*E*) is decomposed over these orthonormal functions as

*c*are coefficients straightforwardly obtained from the

_{n}*a*and

_{k}*b*coefficients. At last, the latter integral is calculated using the saddle point method [33] for any value of

_{l}*n*. The summation is truncated at a given value of

*k*. We carefully checked that the values of Γ rapidly converge for increasing values of

*k*, and we assumed here that

*k*=5 (

*i.e.*

*n*=10). Higher order Laguerre polynomial decomposition was checked to have negligible effect on the fitting precision, but it increases the calculation time. As a result, one obtains an expression of the transmission

*T*using eq. 6. Transmission is then a function of

*q*and Laguerre coefficients

*a*. These parameters are calculated using the nonlinear least-square method [34] on the difference between the measured transmission and theoretical expressions (eq.6 and following), depending on parameters

_{k}*a*,

_{k}*q*and

*E*.

*v*(

*E*). Consequently, the value Γ(

*E*) is known for any desired value of

*E*with little additional cost since most computational cost is spent in calculating the coefficients of the Laguerre expansion.

## B. Harmonic oscillator model

*H*be the total Hamiltonian

*H*refers to the continuum {|

_{E}*E*〉},

*H*to the resonant state |

_{φ}*φ*〉 and

*V*is the coupling Hamiltonian between resonant and non-resonant states whose matrix elements are

*φ*〉 be the resonant ground state depending on the state coordinate

*ρ*. Since

*V*does not depend on

*ρ*,

*E*〉∝

*e*, one obtains

^{ikρ}*H*=

_{φ}*aρ*

^{2}.

## References and links

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 |

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

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

4. | H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature |

5. | J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science |

6. | C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature |

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. | P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of Surface Plasmon Generation at Nanoslit Apertures,” Phys. Rev. Lett. |

9. | J. Bravo-Abad, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of Extraordinary Transmission of Light through Quasiperiodic Arrays of Subwavelength Holes,” Phys. Rev. Lett. |

10. | G. Gay, O. Alloschery, B. V. de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nature Physics |

11. | K. G. Lee and Q. H. Park, “Coupling of Surface Plasmon Polaritons and Light in Metallic Nanoslits,” Phys. Rev. Lett. |

12. | 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. |

13. | J. M. Brok and and H. P. Urbach, “Extraordinary transmission through 1, 2 and 3 holes in a perfect conductor, modelled by a mode expansion technique,” Opt. Exp. |

14. | A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Exp. |

15. | A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, “Squeezing MillimeterWaves into Microns,” Phys. Rev. Lett. |

16. | 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. Comm. |

17. | M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role ofWood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B |

18. | S.-H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Exp. |

19. | W. Zhang, A. K. Azad, J. Han, J. Xu, J. Chen, and X.-C. Zhang, “Direct Observation of a Transition of a Surface Plasmon Resonance from a Photonic Crystal Effect,” Phys. Rev. Lett. |

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

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

22. | J. Han, A. K. Azad, M. Gong, X. Lu, and W. Zhang, “Coupling between surface plasmons and nonresonant transmission in subwavelength holes at terahertz frequencies,” Appl. Phys. Lett. |

23. | S. Bandopadhyay, B. Dutta-Roy, and H.S. Mani, “Understanding the Fano Resonance : through Toy Models,” Am. J. Phys. |

24. | C.-M. Ryu and S. Y. Cho, “Phase evolution of the transmission coefficient in an Aharonov-Bohm ring with Fano resonance,” Phys. Rev. B |

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

26. | C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. |

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

28. | C.-C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theo. Tech. |

29. | J.-B. Masson, A. Podzorov, and G. Gallot, “Anomalies in the disappearance of the extraordinary electromagnetic transmission in subwavelength hole arrays,” Opt. Exp. |

30. | B. D. Fried and S. D. Conte, The plasma dispersion function. |

31. | C. Cohen-Tannoudji, B. Diu, and F. Laloe, |

32. | D. G. Duffy, “On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications,” ACM Trans. Math Soft. |

33. | K. F. Riley, M. P. Hobson, and S. J. Bence, |

34. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(260.3090) Physical optics : Infrared, far

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: May 11, 2009

Revised Manuscript: June 30, 2009

Manuscript Accepted: June 30, 2009

Published: August 14, 2009

**Citation**

Jean-Baptiste Masson, Alexander Podzorov, and Guilhem Gallot, "Extended Fano model of Extraordinary Electromagnetic Transmission through subwavelength hole arrays in the terahertz domain," Opt. Express **17**, 15280-15291 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-15280

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### References

- 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, 667-668 (1998). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- E. Ozbay, "Plasmonic: merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006). [CrossRef] [PubMed]
- H. Liu and P. Lalanne, "Microscopic theory of the extraordinary optical transmission," Nature 452, 728-731 (2008). [CrossRef] [PubMed]
- J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with Structured Surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
- C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007). [CrossRef] [PubMed]
- 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(6), 1114-1117 (2001). [CrossRef]
- P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of Surface Plasmon Generation at Nanoslit Apertures," Phys. Rev. Lett. 95, 263,902 (2005). [CrossRef]
- J. Bravo-Abad, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, and L. Martin-Moreno, "Theory of Extraordinary Transmission of Light through Quasiperiodic Arrays of Subwavelength Holes," Phys. Rev. Lett. 99, 203,905 (2007). [CrossRef]
- G. Gay, O. Alloschery, B. V. de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Physics 2, 262-267 (2006). [CrossRef]
- K. G. Lee and Q. H. Park, "Coupling of Surface Plasmon Polaritons and Light in Metallic Nanoslits," Phys. Rev. Lett. 95, 103,902 (2005). Now with Institut Pasteur, CNRS URA 2171, Unit In Silico Genetics, 75724 Paris Cedex 15, France [CrossRef]
- 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. 93(19), 196,804 (2004).
- J. M. Brok and H. P. Urbach, "Extraordinary transmission through 1, 2 and 3 holes in a perfect conductor, modelled by a mode expansion technique," Opt. Exp. 14(7), 2552-2572 (2006). [CrossRef]
- A. Agrawal, Z. V. Vardeny, and A. Nahata, "Engineering the dielectric function of plasmonic lattices," Opt. Exp. 16(13), 9601-9613 (2008). [CrossRef]
- A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and J. R. Brown, "Squeezing MillimeterWaves into Microns," Phys. Rev. Lett. 92(14), 143,904 (2004).
- 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. Comm. 225(4-6), 331-336 (2003). [CrossRef]
- M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, "Role ofWood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes," Phys. Rev. B 67, 085,415 (2003). [CrossRef]
- S.-H. Chang, S. K. Gray, and G. C. Schatz, "Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films," Opt. Exp. 13(8), 3150-3165 (2005). [CrossRef]
- W. Zhang, A. K. Azad, J. Han, J. Xu, J. Chen, and X.-C. Zhang, "Direct Observation of a Transition of a Surface Plasmon Resonance from a Photonic Crystal Effect," Phys. Rev. Lett. 98, 183,901 (2007). [CrossRef]
- J.-B. Masson and G. Gallot, "Coupling between surface plasmons in subwavelength hole arrays," Phys. Rev. B 73, 121,401(R) (2006). [CrossRef]
- U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124(6), 1866-1875 (1961). [CrossRef]
- J. Han, A. K. Azad, M. Gong, X. Lu, and W. Zhang, "Coupling between surface plasmons and nonresonant transmission in subwavelength holes at terahertz frequencies," Appl. Phys. Lett. 91, 071,122 (2007). [CrossRef]
- S. Bandopadhyay, B. Dutta-Roy, and H. S. Mani, "Understanding the Fano Resonance : through Toy Models," Am. J. Phys. 72, 1501 (2004). [CrossRef]
- C.-M. Ryu and S. Y. Cho, "Phase evolution of the transmission coefficient in an Aharonov-Bohm ring with Fano resonance," Phys. Rev. B 58(7), 3572 (1998). [CrossRef]
- H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66(7-8), 163-182 (1944). [CrossRef]
- C. J. Bouwkamp, "Diffraction theory," Rep. Prog. Phys. 17, 35-100 (1954). [CrossRef]
- D. Grischkowsky, S. R. Keiding, M. van Exter, and C. Fattinger, "Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors," J. Opt. Soc. Am. B 7(10), 2006-2015 (1990). [CrossRef]
- C.-C. Chen, "Transmission of microwave through perforated flat plates of finite thickness," IEEE Trans. Microwave Theo. Tech. 21(1), 1-6 (1973). [CrossRef]
- J.-B. Masson, A. Podzorov, and G. Gallot, "Anomalies in the disappearance of the extraordinary electromagnetic transmission in subwavelength hole arrays," Opt. Exp. 16(7), 4719-4730 (2008). [CrossRef]
- B. D. Fried and S. D. Conte, The plasma dispersion function. The Hilbert transform of the Gaussian (Academic Press, New York, 1961).
- C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Wiley and Hermann, Paris, 1977).
- D. G. Duffy, "On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications," ACM Trans. Math Soft. 19(3), 333-359 (1993). [CrossRef]
- K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical methods for physics and engineering (Cambridge University Press, 2006).
- W. H. Press, S. A. Teukolsky,W. T. Vetterling, and B. P. Flannery, Numerical recipes in C (Cambridge University Press, Cambridge, 1992).

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