Extraordinary transmission through dielectric screens with 1D sub-wavelength metallic inclusions

doi:10.1364/OE.19.013612

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Extraordinary transmission through dielectric screens with 1D sub-wavelength metallic inclusions

V. Delgado, R. Marqués, and L. Jelinek »View Author Affiliations

Optics Express, Vol. 19, Issue 14, pp. 13612-13617 (2011)
http://dx.doi.org/10.1364/OE.19.013612

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– Abstract
1. Introduction
2. Theory
3. Results
4. Conclusion
– References and links

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Abstract

Extraordinary optical transmission (EOT) through dielectric screens periodically loaded with sub-wavelength 1D discontinuities, such as apertures or metallic insets is analyzed. The results of the analysis and computational electromagnetic simulations show that the transmission is higher for for metallic inclusions than for empty slits. This effect confirms that EOT is a quite general property of weakly transparent periodic diffraction screens and opens the door to optically induced EOT in photo-conductive semiconductor screens.

1. Introduction

The first reports on extraordinary optical transmission (EOT) through metallic screens periodically perforated by sub-wavelength holes or slits [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

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]

] stimulated many subsequent experimental and theoretical studies of this effect. The reader is referred to the excellent reviews by C. Genet et al. [3

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

], F. J. García de Abajo [4

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007). [CrossRef]

] or F.J. García Vidal et al. [5

F. J. García Vidal, L. Martín-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]

] in order to have a complete overview of the topic. Soon after the seminal work of Ebbesen was published, it was reported that EOT also appears in periodically corrugated thin metal films without apertures [6

U. Schroter and D. Heitmann, “Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration,” Phys. Rev. B 60(7), 4992–4999 (1999). [CrossRef]

, 7

I. Avrutsky, Y. Zhao, and V. Kochergin , “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. 25(9), 595–597 (2000). [CrossRef]

], and in non-metallic dielectric screens with apertures [8

M. Sarrazin and J. P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E 68, 016603 (2003). [CrossRef]

, 9

H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Lett. 12(16), 3629–3651 (2004).

]. Therefore, it seems that nor the metallic nature of the screen nor the presence of apertures are essential for the presence of EOT. On the contrary, it seems that EOT will appear, for the appropriate light polarization, when: (i) There is some periodic modulation of the electrical properties of the screen and (ii) there is some weak coupling mechanism between both surfaces of the screen.

Following this ansatz, we will study EOT through high permittivity thin dielectric screens, which will provide the required weak coupling between both sides of the screen. Our analysis and numerical simulations will show that dielectric screens periodically loaded either with apertures or with metallic inclusions may present EOT in the vicinity of Wood’s anomaly. Surprisingly, transmittance will be higher for metallic inclusions than for apertures.

2. Theory

In order to simplify the analysis we will consider EOT of p-polarized light through an array of 1D empty slits or parallel metallic inclusions in a dielectric screen. One-dimensional configurations have been previously analyzed in order to simplify the computational problem, see for example reference [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, 2845–2848 (1999). [CrossRef]

], and subsequently they have taken on a life of its own. The unit cell of this structure is shown in Fig. 1. In our analysis we will take advantage of the equivalence between the studied problem and the equivalent problem of a symmetric discontinuity in a parallel plate waveguide, as it is illustrated in Fig. 1. First of all, the electric and magnetic fields at both sides of the screen are expanded in series of TM waveguide modes:

E_{y}^{-} = 1 + R + \sum_{n = 1}^{N} R_{n} cos (2 n π y / a)

(1)

E_{y}^{+} = T + \sum_{n = 1}^{N} T_{n} cos (2 n π y / a)

(2)

H_{x}^{-} = Y_{0} (- 1 + R) + \sum_{n = 1}^{N} Y_{n} R_{n} cos (2 n π y / a)

(3)

H_{x}^{+} = - Y_{0} T - \sum_{n = 1}^{N} Y_{n} T_{n} cos (2 n π y / a),

(4)

where T and R are the transmission and reflection coefficients, T_n and R_n are the amplitudes of the different modes excited in the screen,

Y_{0} = \sqrt{ɛ_{0} / μ_{0}}

is the admittance of free space and Y_n are the modal admittances of the empty waveguide. It is now assumed that the empty slits or metallic inclusions are small (b ≪ a), so that the modal components of the electric and magnetic fields at both sides of the screen can be linked through the matrix surface impedance of the dielectric slab:

[\begin{matrix} E_{y, n}^{+} & + & E_{y, n}^{-} \\ E_{y, n}^{+} & - & E_{y, n}^{-} \end{matrix}] \approx [\begin{matrix} Z_{n}^{(1)} & 0 \\ 0 & Z_{n}^{(2)} \end{matrix}] [\begin{matrix} H_{x, n}^{+} & - & H_{x, n}^{-} \\ H_{x, n}^{+} & + & H_{x, n}^{-} \end{matrix}],

(5)

where

E_{y, n}^{\pm}

and

H_{x, n}^{\pm}

are the modal components of

E_{y}^{\pm}

and

H_{x}^{\pm}

, i.e. the different terms in the expansion (Eqs. (1)–(4)), and

Z_{n}^{(1)} = \frac{[1 + cos (k_{z, n} t)]}{i sin (k_{z, n} t) Y_{d, n}} and Z_{n}^{(2)} = \frac{i sin (k_{z, n} t)}{[1 + cos (k_{z, n} t)] Y_{d, n}},

(6)

where k_z,n and Y _d,n are the wave-vector and the modal admittances of the TM modes of the dielectric filled waveguide, respectively. From Eq. (5) we can define the quantities

A = E_{y, n}^{+} + E_{y, n}^{-} - Z_{n}^{(1)} (H_{x, n}^{+} - H_{x, n}^{-})

and

B = E_{y, n}^{+} - E_{y, n}^{-} - Z_{n}^{(2)} (H_{x, n}^{+} + H_{x, n}^{-})

which must vanish in the intervals a/2 ≥ |y| ≥ b/2. Therefore, for b ≪ a

\int_{- a / 2}^{a / 2} A cos (\frac{2 m π y}{a}) dx = \int_{- b / 2}^{b / 2} A cos (\frac{2 m π y}{a}) dx \approx \int_{- b / 2}^{b / 2} Adx = \int_{- a / 2}^{a / 2} Adx,

(7)

with a similar relation for B, where m is an integer in the interval 1 ≤ m ≲ a/b. After substitution of Eqs. (1)–(4) into the above relations, we find

(1 + Z_{n}^{(1)} Y_{n}) (T_{n} + R_{n}) = 2 (1 + R + T) - 2 Z_{0}^{(1)} Y_{0} (1 - R - T); n ≲ a / b

(8)

(1 + Z_{n}^{(2)} Y_{n}) (T_{n} - R_{n}) = 2 (- 1 - R + T) - 2 Z_{0}^{(2)} Y_{0} (- 1 + R - T); n ≲ a / b

(9)

which determine the auxiliary coefficients T_n and R_n as functions of the transmission and reflections coefficients T and R. These coefficients can now be determined if the surface impedances associated with the inclusions

Z_{s}^{(1)}

and

Z_{s}^{(2)}

are known (for instance, for perfect conductors it is

Z_{s}^{(1)} = Z_{s}^{(2)} = 0

). In order to determine T and R we simply integrate the quantities

C = E_{y, n}^{+} + E_{y, n}^{-} - Z_{s}^{(1)} (H_{x, n}^{+} - H_{x, n}^{-})

and

D = E_{y, n}^{+} - E_{y, n}^{-} - Z_{s}^{(2)} (H_{x, n}^{+} + H_{x, n}^{-})

along the aperture (i.e. in the interval –b/2 ≤ y ≤ b/2) and obtain

1 - Z_{s}^{(1)} Y_{0} + (1 + Z_{s}^{(1)} Y_{0}) (T + R) + \sum_{n = 1}^{N} (1 + Z_{s}^{(1)} Y_{n}) (T_{n} + R_{n}) sinc (\frac{bn π}{a}) = 0

(10)

- 1 + Z_{s}^{(2)} Y_{0} + (1 + Z_{s}^{(2)} Y_{0}) (T - R) + \sum_{n = 1}^{N} (1 + Z_{s}^{(2)} Y_{n}) (T_{n} + R_{n}) sinc (\frac{bn π}{a}) = 0,

(11)

where sinc(x) ≡ sin(x)/x. It is worth to mention here that this analysis is also valid for empty slits, which can be characterized by the surface impedances corresponding to the fundamental quasi-TEM mode in the slit, as well as for realistic metallic inclusions, which can be characterized by the appropriate surface impedances [10

V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express 18, 6506–6515 (2010). [CrossRef] [PubMed]

Fig. 1 Front and side views of the unit cell of the analyzed structures. Due to the polarization of the incident wave and due to periodicity, upper and lower planes of the unit cell are virtual perfect conducting plates (PEC).Therefore, the structures are equivalent to a symmetrical metallo-dielectric discontinuity in a parallel-plate waveguide. (a) and (b): Front and side views of the unit cell of a dielectric screen with apertures. (c): Side view of the unit cell of a dielectric screen with metallic insets.

3. Results

EOT through low-loss high permittivity dielectric screens will now be considered. Transmittance through these screens is low (except at Fabry-Perot resonances) due to the high reflectance associated to the jump in the dielectric constant at the interfaces. However, if the screen is periodically perforated by an array of periodic parallel slits (Figs. 1(a) and 1(b)), an extraordinary increase of transmittance appears near Wood’s anomaly, just as in EOT through metallic screens. This effect is shown in Fig. 2 for a periodic array of slits perforated in a slab of zirconium-tin-titanate with ε = 92.7(1 + 0.005i) [11

P. H. Bolivar, J. G. Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51(4), 1062–1066 (2003). [CrossRef]

]. The results provided by the model reported in the previous Section are compared with numerical simulations carried out using the electromagnetic solver CST Microwave Studio. Electromagnetic simulations considered a parallel plate host waveguide with a length equal to four periods, and with 3 modes (the fundamental TEM mode and two evanescent modes) at the input and output ports. Lateral perfect magnetic walls, with an adaptive tetrahedral mesh were used for numerical computations. Computation time was about 1 min for each frequency point in CST, whereas computation time using the analytical model was negligible (less than 1 micro-second). As it can be seen in Fig. 2 there is a reasonable qualitative and quantitative agreement between the results of our analytical model and the numerical simulations.

Fig. 2 Transmittance through a zirconium-tin-titanate (ε = 92.7(1 + 0.005i)) screen of thickness t = 0.12 mm, periodically perforated by an array of parallel slits of periodicity a = 3 mm and width b = a/6, which can be either empty or filled by a PEC. Solid lines are our results computed from Eqs. (8)–(11). Dashed lines are results from CST. Upper scale shows the ratio f/f _w, where f _w is the frequency corresponding to Wood’s anomaly f _w = c/a.

In the THz range of frequencies most metals behave as PEC. Thus, transmittances through the same diffraction screen, but with the slits filled with a PEC, was analyzed. The results of this analysis, also shown in Fig. 2, illustrate how EOT can be induced by metallic inclusions in low-loss, high permittivity, thin dielectric screens operating in the THz frequency range and below. Surprisingly, transmittance is higher with metallic inclusions than with apertures.

Similar results are shown at infrared frequencies in Fig. 3 for screens of amorphous silicon (a-Si), periodically loaded with silver inclussions. In this analysis, the metallic inclusions were modeled using standard Drude theory with the constants already used in [13

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Photonic band gaps in three-dimensional metallic lattices,” Phys. Rev. B 62, 15299–15302 (2000). [CrossRef]

] for silver. The results of Fig. 3 show that the increase of transmittance in a-Si screens loaded with metallic insets is much higher than the corresponding increase of transmittance due to the presence of open slits in the screen, which is actually very weak.

Fig. 3 Transmittance through an a-Si screen periodically perforated by an array of parallel slits which can be either empty or filled with silver. Solid lines are our results computed from Eqs. (8)–(11). In (a), the periodicity of the array is a = 3 μm, the thickness of the slab is t = 200 nm and the permittivity of a-Si is ε = 11.8(1+0.007i) [12

D. T. Pierce and W. E. Spicers, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1971). [CrossRef]

]. In (b) a = 1.55 μm, t = 100 nm and the permittivity of a-Si is ε = 12.4(1 + 0.016i). In both cases the width of the slits/inclusions is b = a/4. Dashed lines are results from CST. Upper scale shows the ratio f / f _w, where f _w is the frequency corresponding to Wood’s anomaly f _w = c/a.

It is well known that EOT can be associated to the excitation of weakly coupled leaky surface waves at both sides of the screen. In standard EOT, these surface waves are coupled through the apertures perforated in the screen. In low-loss dielectric screens, this coupling is ensured by the screen itself, and the role of the metallic inclusions is merely to allow for the excitation of the surface waves. In order to support this interpretation, we show in Fig. 4 the electric field distribution at both sides of the screen in the configuration of Fig. 3 at the frequency of maximum transmission. A clear standing wave pattern corresponding to the simultaneous excitation of two surface waves traveling at opposite directions along each side of the screen can be appreciated in both cases. These field distributions are very similar to those found in simulations of metallic screens at optical frequencies, thus confirming that the excitation of coupled surface waves with wavelengths close to the periodicity is the physical mechanism behind EOT in dielectric screens with conductive insets.

Fig. 4 Normalized electric field distribution (absolute value) at both sides of the screen for the configuration analyzed in Fig. 2 with empty slits (upper figure) and slits filled by PEC (lower figure). Calculations were made using CST, and correspond to the frequency of maximum transmission in both cases. Green color corresponds to low field values and red color to maximum field value.

4. Conclusion

An analytical model for EOT through high permittivity 1D dielectric screens periodically loaded with open slits or metallic inclusions has been provided and validated with full wave electromagnetic simulations. Analytical and numerical results show that EOT through high permittivity dielectric screens loaded with a periodic distribution of metallic inclusions is higher than for the corresponding screens with empty slits.

Regions with a high density of free electrons can be induced in most semiconductors by using lasers of the appropriate wavelength. Thus, we feel that the reported effect opens the door to the design of photo-induced EOT in semiconductor screens at THz and infrared frequencies.

Acknowledgments

This work has been supported by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds (project No. CSD2008-00066), by the Czech Grant Agency (project No. 102/09/0314), and by the Czech Technical University in Prague (project No. SGS10/271/OHK3/3T/13).

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 (London) 391, 667–669 (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, 2845–2848 (1999). [CrossRef]
3.	C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 445, 39–46 (2007). [CrossRef]
4.	F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007). [CrossRef]
5.	F. J. García Vidal, L. Martín-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]
6.	U. Schroter and D. Heitmann, “Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration,” Phys. Rev. B 60(7), 4992–4999 (1999). [CrossRef]
7.	I. Avrutsky, Y. Zhao, and V. Kochergin , “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. 25(9), 595–597 (2000). [CrossRef]
8.	M. Sarrazin and J. P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E 68, 016603 (2003). [CrossRef]
9.	H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Lett. 12(16), 3629–3651 (2004).
10.	V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express 18, 6506–6515 (2010). [CrossRef] [PubMed]
11.	P. H. Bolivar, J. G. Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51(4), 1062–1066 (2003). [CrossRef]
12.	D. T. Pierce and W. E. Spicers, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1971). [CrossRef]
13.	I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Photonic band gaps in three-dimensional metallic lattices,” Phys. Rev. B 62, 15299–15302 (2000). [CrossRef]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(050.1960) Diffraction and gratings : Diffraction theory

ToC Category:
Diffraction and Gratings

History
Original Manuscript: March 22, 2011
Revised Manuscript: June 2, 2011
Manuscript Accepted: June 3, 2011
Published: June 29, 2011

Citation
V. Delgado, R. Marqués, and L. Jelinek, "Extraordinary transmission through dielectric screens with 1D sub-wavelength metallic inclusions," Opt. Express 19, 13612-13617 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13612

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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 (London) 391, 667–669 (1998). [CrossRef]
J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]
C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 445, 39–46 (2007). [CrossRef]
F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007). [CrossRef]
F. J. García Vidal, L. Martín-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]
U. Schroter and D. Heitmann, “Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration,” Phys. Rev. B 60(7), 4992–4999 (1999). [CrossRef]
I. Avrutsky, Y. Zhao, and V. Kochergin , “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. 25(9), 595–597 (2000). [CrossRef]
M. Sarrazin and J. P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E 68, 016603 (2003). [CrossRef]
H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Lett. 12(16), 3629–3651 (2004).
V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express 18, 6506–6515 (2010). [CrossRef] [PubMed]
P. H. Bolivar, J. G. Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51(4), 1062–1066 (2003). [CrossRef]
D. T. Pierce and W. E. Spicers, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1971). [CrossRef]
I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Photonic band gaps in three-dimensional metallic lattices,” Phys. Rev. B 62, 15299–15302 (2000). [CrossRef]

Avrutsky, I.
Biswas, R.
Bolivar, P. H.
de Maagt, P.
Delgado, V.
Ebbesen, T. W.
Ederra, I.
El-Kady, I.
García de Abajo, F. J.
García Vidal, F. J.
Garcia-Vidal, F. J.
Genet, C.
Ghaemi, H. F.
Gonzalo, R.
Heitmann, D.
Ho, K. M.
Holker, M.
Jelinek, L.
Kochergin, V.
Kuipers, L.
Lezec, H. J.
Marqués, R.
Martín-Moreno, L.
Pendry, J. B.
Pierce, D. T.
Porto, J. A.
Reynolds, A. L.
Rivas, J. G.
Sarrazin, M.
Schroter, U.
Sigalas, M. M.
Soukoulis, C. M.
Spicers, W. E.
Thio, T.
Vigneron, J. P.
Wolff, P. A.
Zhao, Y.

IEEE Trans. Microwave Theory Tech.

P. H. Bolivar, J. G. Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51(4), 1062–1066 (2003). [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]
C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London) 445, 39–46 (2007). [CrossRef]

Opt. Express

V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express 18, 6506–6515 (2010). [CrossRef] [PubMed]

Opt. Lett.

I. Avrutsky, Y. Zhao, and V. Kochergin , “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. 25(9), 595–597 (2000). [CrossRef]
H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Lett. 12(16), 3629–3651 (2004).

Phys. Rev. B

U. Schroter and D. Heitmann, “Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration,” Phys. Rev. B 60(7), 4992–4999 (1999). [CrossRef]
D. T. Pierce and W. E. Spicers, “Electronic structure of amorphous Si from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1971). [CrossRef]
I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Photonic band gaps in three-dimensional metallic lattices,” Phys. Rev. B 62, 15299–15302 (2000). [CrossRef]

Phys. Rev. E

M. Sarrazin and J. P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E 68, 016603 (2003). [CrossRef]

Phys. Rev. Lett.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]

Rev. Mod. Phys.

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007). [CrossRef]
F. J. García Vidal, L. Martín-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010). [CrossRef]

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