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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 26 — Dec. 21, 2009
  • pp: 23992–24001
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Light trapping cavity enhanced light transmission through a single sub-wavelength aperture in a metal film

Juuso Olkkonen  »View Author Affiliations


Optics Express, Vol. 17, Issue 26, pp. 23992-24001 (2009)
http://dx.doi.org/10.1364/OE.17.023992


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Abstract

We demonstrate that optical transmission of a normally incident, monochromatic plane wave through a single sub-wavelength aperture in an opaque metal film can be substantially enhanced by a thin, semitransparent metal film placed parallel to the opaque metal film in front of the aperture. When the semi-transparent and the opaque metal film are separated by a proper distance, a light trapping cavity is formed and the sub-wavelength aperture exhibits a transmission maximum. An enhancement factor of ~40 is demonstrated for a cylindrical 100 nm diameter hole in a silver film.

© 2009 OSA

1. Introduction

Optical light transmission through sub-wavelength slits and apertures in metal films has gained great interest since the discovery of extraordinary light transmission through arrays of sub-wavelength holes by Ebbesen et al. [1

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(6668), 667–669 (1998). [CrossRef]

]. Later, it was observed that enhanced transmission can also occur in single apertures [2

2. D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen, “Beyond the Bethe Limit: Tunable Enhanced Light Transmission Through a Single Sub-Wavelength Aperture,” Adv. Mater. 11(10), 860–862 (1999). [CrossRef]

4

4. T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: Physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]

] and slits [5

5. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901–1 (2003). [CrossRef] [PubMed]

,6

6. D. A. Thomas and H. P. Hughes, “Enhanced optical transmission through a subwavelength 1D aperture,” Solid State Commun. 129(8), 519–524 (2004). [CrossRef]

] that are surrounded by periodic surface corrugations. García-Vidal et al. [5

5. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901–1 (2003). [CrossRef] [PubMed]

] associated the transmission enhancement in corrugated structures to groove cavity mode excitation, in-phase groove reemission, and slit waveguide mode. Also surface plasmon polaritons (SPPs) excited by the surface corrugations are seen to have a significant role in the transmission process [2

2. D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen, “Beyond the Bethe Limit: Tunable Enhanced Light Transmission Through a Single Sub-Wavelength Aperture,” Adv. Mater. 11(10), 860–862 (1999). [CrossRef]

4

4. T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: Physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]

,7

7. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B Condens. Matter 58(11), 6779–6782 (1998). [CrossRef]

]. Lezec et al. [8

8. H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12(16), 3629–3651 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3629. [CrossRef] [PubMed]

] have introduced the so called diffracted evanescent wave model, which explains the enhanced transmission as being due to interference between the normally incident plane wave and diffracted surface waves generated by the periodic surface corrugations. F. Wu et al. [9

9. F. Wu, D. Han, X. Li, X. Liu, and J. Zi, “Enhanced transmission mediated by guided resonances in metallic gratings coated with dielectric layers,” Opt. Express 16(9), 6619–6624 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6619. [CrossRef] [PubMed]

] have studied a metallic grating structure that is coated symmetrically with a dielectric layer on both sides. Besides cavity resonances in slits and SPPs, they have found that guided resonances in the dielectric layers provide another mechanism for enhanced transmission.

In this paper, we introduce a completely new way to enhance transmission of a normally incident, monochromatic plane wave through a sub-wavelength aperture in a metal film. It is based on the Fabry-Pérot resonance between two parallel metal films from which the first is semi-transparent and the second opaque. By properly selecting the thickness of the semitransparent film and the gap between the films, light gets trapped to the space between the metal films and the electric field is substantially increased in the gap region. That is, the structure works as a light trapping cavity. We demonstrate that a sub-wavelength aperture formed into the opaque metal film of the light trapping structure exhibits much higher transmittance than the same aperture in a bare opaque metal film. This new transmission enhancement mechanism works also with the transverse electric (TE) polarized light, in contrary to the surface wave -based mechanisms.

The paper is organized as follows. Section 2 introduces the numerical modeling methods used in the study. The optimal structure for the light trapping cavity is found in Section 3. In Section 4, we study light transmission through an ordinary sub-wavelength slit in a metal film. The effect of the light trapping cavity on light transmission through a slit is studied in Section 5 and through a cylindrical aperture in Section 6. The summary and conclusions are presented in Section 7.

2. Numerical models

The thickness of the semitransparent metal film and the distance from the opaque metal film in the absence of the aperture structure was optimized using the analytical multi-layer calculation method [10

10. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2000).

]. Light transmission through the slit structure was modeled by the two-dimensional finite difference time domain (FDTD) method [11

11. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]

,12

12. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Second Edition, Artech House, INC., 2000).

]. The Yee cell size was 2.5 nm both in the transverse and the longitudinal direction. The computation domain was terminated by the perfectly matched layer (PML) absorbing boundary condition [13

13. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996). [CrossRef]

]. The simulations results presented for the cylindrical hole were obtained with the body of revolution finite difference time domain (BOR-FDTD) method [12

12. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Second Edition, Artech House, INC., 2000).

]. The Yee cell size both in the z and in the radial direction was 2.5 nm.

The optical response of silver was included into the FDTD models via the Lorentz model that was implemented using the auxiliary differential equation [14

14. M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guid. Wave Lett. 7(5), 121–123 (1997). [CrossRef]

]. Parameters of the Lorentz model were optimized so that refractive index of silver matches with the experimental data presented by Johnson et al. [15

15. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev., B, Solid State 6(12), 4370–4379 (1972). [CrossRef]

]. The used Lorentz model for silver produced refractive index of 0.054 + 4.41i at the free space wavelength of 650 nm.

3. Light trapping multilayer cavity

Using a semitransparent and an opaque metal film, which are parallel to each other and separated by a proper distance, it is possible to form a multilayer structure that absorbs nearly all incident monochromatic light at a particular angle of incidence. Such a structure is illustrated in Fig. 2(d)
Fig. 2 (a) Reflectance of a normally incident plane wave (Ex, Ez ≡ 0, Hy) having wavelength of 650 nm from the multilayer structure shown in the inset (d) as a function of t and g. The uppermost silver film is semitransparent (t = 0-100 nm) while the second is opaque (h = 200 nm), i.e., the transmittance through the entire structure is zero for all values of g and t. n 0 = 1.0. (b) Cross profiles of the reflectance map shown in (a) for t = 20, 30, 40, and 50 nm. (c) Electric field amplitude in the multilayer structure shown in (d) (g = 277.5 nm, h = 200 nm) without (t = 0 nm) and with the semitransparent metal film (t = 40 nm).
, in which t, g, and h denote the thickness of the semitransparent silver film, the gap between the silver films, and the thickness of the opaque silver film, respectively. Figure 2(a) shows the reflectance of a normally incident plane having free space wavelength (λ 0) of 650 nm from the free standing structure (n 0 = 1.0) as a function of t and g for a fixed value h = 200 nm. (At visible frequencies, a silver film is practically opaque when its thickness is greater than 100 nm.) It is seen, especially from Fig. 2(b), that nearly all light is trapped into the structure when t = 40 nm and g = 277.5 nm. When t = 40 nm and h = 200 nm, the light trapping occurs always when g = 277.5 nm + mλ 0/2, where m is a positive integer. This periodicity is clearly observable in Fig. 2(a). Due to the opaque silver film, light transmission through the structure is always zero for all values of g and t.

Figure 2(c) shows the electric field amplitude in the light trapping structure in two cases: (a) t = 40 nm, (b) t = 0 nm, i.e., in the presence and absence of the 40 nm thick semitransparent silver film. In the first case, light is trapped into the structure and the electric field amplitude is substantially enhanced in the gap between the metal films, whereas in the second case, practically all the incident light is reflected from the opaque metal film. We note that in both cases the shape of the electric field amplitude is exactly the same in the gap region and inside the opaque metal film. At resonance, the electric field amplitude is enhanced in these regions by a factor of 6.5 but not altered in shape.

As illustrated in Fig. 2, the reflectance of the light trapping structure is very sensitive to the gap thickness (g) between the metal films. Due to this fact, it might be thought that the presented structure is not very practical. The situation, however, can be eased, for example, by assuming that the angle of incidence of the illuminating plane wave is an adjustable parameter. Figure 3
Fig. 3 (a) Reflectance of an obliquely incident, TM polarized plane wave having free space wavelength of 650 nm from the multilayer structure (t = 40 nm, h = 200 nm, n 0 = 1) shown in the inset (c) as a function of the incident angle (θ) and the thickness of the air gap (g). (b) Cross profiles of the reflectance map shown in (a) for g = 277.5, 300, and 400 nm.
shows the reflectance of the resonance structure (t = 40 nm, h = 200 nm) as a function of the gap and the angle of incidence. It is seen that when the gap is equal or larger than 277.5 nm, we can find an incident angle that satisfies the resonance condition.

4. Light transmission through a slit in a silver film

Before describing light transmission through a sub-wavelength slit in a light trapping multilayer cavity, we shall discuss the basic transmission properties of metallic sub-wavelength slits. We will utilize these results when interpreting results in the forthcoming sections.

When light transmission through a slit in a metal film is considered, the slit can be thought as a scatterer that resides in a stratified background structure. The scattered field due to the slit can be obtained by subtracting the field, which would be present if the metal film contained no slit, from the total field of the slit structure. Note that in the region below the metal film, the scattered field is the same as the total field since no incident field penetrate through the opaque metal film. The scattered field emanates from the slit to the forward ( + z) and backward (-z) directions. Traditionally, the light transmission through the slit is characterized by the normalized transmission efficiency (η+) that is the ratio of the transmitted power in the geometrical exit of the slit and the incident power at the slit entrance. In this paper, we also examine a parameter that is referred to as the normalized back-scattering efficiency (η-). This parameter is defined as the ratio of the scattered and incident power at the geometrical entrance of the slit. Figure 4(a)
Fig. 4 (a) Normalized transmission (η + ) and back-scattering (η-) efficiency as a function of the silver film thickness (h) for a 100 nm wide (w) slit under normally incident TM polarized plane wave illumination (ns = 1.0, n0 = 1.0, λ0 = 650 nm). (b) The same as (a) but now the slit is illuminated by a TE polarized plane wave and the slit is filled by a dielectric medium having refractive index (ns) of 3.8. (c) The dependency of the phase difference between the illuminating plane wave and the scattered field on the slit length in the middle of the slit entrance (point p shown in the inset). In the TM case, the phase difference is calculated from the magnetic field and from the electric field in the TE case.
shows the normalized transmission (η+) and back-scattering (η-) efficiency as a function of the silver film thickness (h) for a 100 nm wide slit under TM polarized plane wave illumination. It is observable from Fig. 4(a) that the normalized transmission efficiency exhibits a clear Fabry-Pérot-like behavior, i.e., η + varies periodically with the slit length (h). At resonance (e.g. h = 165 nm), the η +- and η --curve cross, i.e., the slit radiates equally much power into + z- and –z-directions.

When the slit width (w) is smaller than λ 0/2, only the TM polarized light propagates through the slit. However, the transmittance of the TE polarized light can be substantially enhanced by filling the slit by a high index dielectric medium. The high index filling has two functions in the transmission process: (1) it enables the formation of a guided TE01 mode having the propagation constant with a relatively small imaginary part; (2) it forms a Fabry-Pérot-like resonance structure due to reflecting entrance and exit interfaces of the slit. Figure 4(b) shows the η+ and η- as a function of the silver film thickness for a 100 nm wide slit under TE polarized illumination. The slit is filled with a medium having refractive index of 3.8. It is seen that efficient light transmission through the 100 nm wide slit is achieved. We note that under TM polarized illumination, the high index filling was not observed to increase the transmission efficiency of a 100 nm wide slit in a silver film.

Figure 4(c) shows how the phase difference between the backward scattered light and illuminating plane wave depends on the slit length in the middle of the slit entrance (point p in the inset of Fig. 4(c)). Since the TM polarized light contains only one magnetic field component (Hy), and the TE polarized light only one electric field component (Ey), these components were used to determinate the phase difference. The phase difference in the TM case is between 145 and 203 degrees, and between −47 and 87 degrees in the TE case.

5. Light trapping cavity enhanced light transmission through a sub-wavelength slit

Figure 5(b) shows the normalized transmission efficiency as a function of the slit length (h) when the gap between the 40 nm thick semitransparent silver film and the opaque film is fixed to 277.5 nm. By comparing these results to the results of Figs. 4(a) and 4(b), it is observable that transmission peaks are now shifted to the slit lengths that are in close proximity of minimums of the normalized back-scattering efficiency (η-) curves shown in Figs. 4(a) and (b). The back-coupling effect depends also on the phase of the back-coupled light, and thus the transmission peaks are not observed at the minimums of η- -curves. The η +TE curve in Fig. 5(b) exhibits a transmission maximum when h = 210 nm. In the absence of the semitransparent metal film, the normalized transmission efficiency of the slit (Fig. 4(a)) is about 0.36. Assuming that there is no back-coupling effect, the multilayer resonance enhances the transmission by 42.25, which means that the transmission efficiency of the multilayer resonance slit could be about 15, which is in agreement with the results of Fig. 5(b).

6. Light trapping cavity enhanced light transmission through a cylindrical aperture

Light transmission through a cylindrical aperture reduces significantly when the aperture diameter gets smaller than λ 0/2. We have earlier shown [16

16. J. Olkkonen, K. Kataja, and D. Howe, “Light transmission through a high index dielectric-filled sub-wavelength hole in a metal film,” Opt. Express 13(18), 6980–6989 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6980. [CrossRef] [PubMed]

] that light transmission through sub-wavelength apertures can be greatly enhanced by filling the aperture by a high index dielectric medium. This effect is also illustrated in Fig. 6(a)
Fig. 6 (a) Transmission efficiency of a normally incident plane wave (λ 0 = 650 nm) through a cylindrical hole in a free standing silver film as a function the film thickness (h). The hole radius (r) is 50 nm and it is filled with a dielectric medium having the refractive index of n a. The red line is magnified by a factor of 500. (b) Transmission efficiency of a cylindrical hole (r = 50 nm, n a = 2.8, h = 180 nm) with a light trapping cavity as a function of the distance between the semitransparent metal film (t = 40 nm) and the opaque metal film (g = 277.5 nm + mλ 0/2, m = 1, 2,…, 8). (c) Transmission efficiency as a function of the opaque metal film thickness with the fixed gap thickness of 277.5 nm. Other parameters are the same as in (b).
that shows the normalized transmission efficiency for a cylindrical aperture, 100 nm in diameter, in a silver film with and without the high refractive index (2.8) dielectric filling.

Figure 6(c) shows the normalized transmission efficiency as a function of the thickness of the opaque film (h) with the fixed gap thickness of 277.5 nm. The maximum transmission efficiency is obtained with h = 170 nm, i.e., the optimum thickness is shifted by ~10 nm from that of the aperture in a bare metal film.

7. Conclusions

In this paper, we have presented a new technique to enhance transmission of a normally incident, monochromatic plane wave through a sub-wavelength aperture in a metal film. The technique is based on the light trapping cavity formed in front of the aperture. The presented idea was verified by FDTD simulations for both sub-wavelength slits and cylindrical apertures. The transmission enhancement factor of ~40 was demonstrated for cylindrical hole filled with a high index dielectric medium. The presented idea may be applicable to periodic slit and hole arrays if the spacing between the holes is large enough to support formation of the light trapping cavity. It may also find applications in biosensors since the transmission efficiency is highly sensitive to refractive index changes occurring in the slit and in the gap region between the metal films.

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 391(6668), 667–669 (1998). [CrossRef]

2.

D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen, “Beyond the Bethe Limit: Tunable Enhanced Light Transmission Through a Single Sub-Wavelength Aperture,” Adv. Mater. 11(10), 860–862 (1999). [CrossRef]

3.

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26(24), 1972–1974 (2001). [CrossRef] [PubMed]

4.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: Physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]

5.

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901–1 (2003). [CrossRef] [PubMed]

6.

D. A. Thomas and H. P. Hughes, “Enhanced optical transmission through a subwavelength 1D aperture,” Solid State Commun. 129(8), 519–524 (2004). [CrossRef]

7.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B Condens. Matter 58(11), 6779–6782 (1998). [CrossRef]

8.

H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12(16), 3629–3651 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3629. [CrossRef] [PubMed]

9.

F. Wu, D. Han, X. Li, X. Liu, and J. Zi, “Enhanced transmission mediated by guided resonances in metallic gratings coated with dielectric layers,” Opt. Express 16(9), 6619–6624 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6619. [CrossRef] [PubMed]

10.

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2000).

11.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]

12.

A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Second Edition, Artech House, INC., 2000).

13.

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996). [CrossRef]

14.

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guid. Wave Lett. 7(5), 121–123 (1997). [CrossRef]

15.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev., B, Solid State 6(12), 4370–4379 (1972). [CrossRef]

16.

J. Olkkonen, K. Kataja, and D. Howe, “Light transmission through a high index dielectric-filled sub-wavelength hole in a metal film,” Opt. Express 13(18), 6980–6989 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6980. [CrossRef] [PubMed]

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(240.0240) Optics at surfaces : Optics at surfaces

ToC Category:
Diffraction and Gratings

History
Original Manuscript: October 9, 2009
Revised Manuscript: November 25, 2009
Manuscript Accepted: December 2, 2009
Published: December 16, 2009

Citation
Juuso Olkkonen, "Light trapping cavity enhanced light transmission through a single sub-wavelength aperture in a metal film," Opt. Express 17, 23992-24001 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23992


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References

  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(6668), 667–669 (1998). [CrossRef]
  2. D. E. Grupp, H. J. Lezec, T. Thio, and T. W. Ebbesen, “Beyond the Bethe Limit: Tunable Enhanced Light Transmission Through a Single Sub-Wavelength Aperture,” Adv. Mater. 11(10), 860–862 (1999). [CrossRef]
  3. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26(24), 1972–1974 (2001). [CrossRef] [PubMed]
  4. T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: Physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]
  5. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90(21), 213901–1 (2003). [CrossRef] [PubMed]
  6. D. A. Thomas and H. P. Hughes, “Enhanced optical transmission through a subwavelength 1D aperture,” Solid State Commun. 129(8), 519–524 (2004). [CrossRef]
  7. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B Condens. Matter 58(11), 6779–6782 (1998). [CrossRef]
  8. H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12(16), 3629–3651 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3629 . [CrossRef] [PubMed]
  9. F. Wu, D. Han, X. Li, X. Liu, and J. Zi, “Enhanced transmission mediated by guided resonances in metallic gratings coated with dielectric layers,” Opt. Express 16(9), 6619–6624 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6619 . [CrossRef] [PubMed]
  10. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2000).
  11. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]
  12. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Second Edition, Artech House, INC., 2000).
  13. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996). [CrossRef]
  14. M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guid. Wave Lett. 7(5), 121–123 (1997). [CrossRef]
  15. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev., B, Solid State 6(12), 4370–4379 (1972). [CrossRef]
  16. J. Olkkonen, K. Kataja, and D. Howe, “Light transmission through a high index dielectric-filled sub-wavelength hole in a metal film,” Opt. Express 13(18), 6980–6989 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-18-6980 . [CrossRef] [PubMed]

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