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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 3 — Feb. 2, 2009
  • pp: 1859–1864
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Tailoring optical transmission via the arrangement of compound subwavelength hole arrays

Jian-Qiang Liu, Meng-Dong He, Xiang Zhai, Ling-Ling Wang, Shuangchun Wen, Li Chen, Zhe Shao, Qing Wan, B. S. Zou, and Jianquan Yao  »View Author Affiliations


Optics Express, Vol. 17, Issue 3, pp. 1859-1864 (2009)
http://dx.doi.org/10.1364/OE.17.001859


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Abstract

The transmission properties of light through metal films with compound periodic subwavelength hole arrays is numerically investigated by using the finite-difference time-domain (FDTD) method. The sharp dips in the transmission bands, together with the suppression of surface plasmon resonance (SPR) (0, 1) peak, are found when two square holes in every unit cell are arranged asymmetrically along the polarization direction of the incident light. However, the shape of transmission spectra is not sensitive to the symmetry if the holes are arranged perpendicular to the propagation direction of surface plasmon polaritons (SPPs). The physics origin of these phenomena is explained qualitatively by the phase resonance of SPPs.

© 2009 Optical Society of America

1. Introduction

An optically thick metal film perforated with a periodic array of subwavelength holes transmits light remarkably efficiently at certain resonant wavelengths[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 (London) 391, 667–669 (1998). [CrossRef]

, 2

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

], which is called extraordinary optical transmission (EOT). Such a phenomenon has evoked great scientific interest and attracted attention for its significant potential applications[3

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

]. To understand it, both of one-dimensional (1D) periodic arrays of slits[4

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

, 5

5. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express 13, 4485–4491 (2005). [CrossRef] [PubMed]

] and two-dimensional (2D) subwavelength periodic hole arrays with different shapes[6

6. L. Mart́in-Moreno and F. J. Garćia-Vidal, “Optical transmission through circular hole arrays in optically thick metal films,” Opt. Express 12, 3619–3628 (2004). [CrossRef]

, 7

7. L. Mart́in-Moreno, F. J. Garćia-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]

, 8

8. K. J. K. 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 holes,” Phys. Rev. Lett. 92, 183901 (2004). [CrossRef] [PubMed]

] were studied intensively. It has been established that two main mechanisms are responsible for the enhanced transmission: surface plasmon resonance (SPR) and localized waveguide resonance[9

9. Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: The role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006). [CrossRef] [PubMed]

, 10

10. C. Rockstuhl and F. Lederer, “Enhanced transmission of periodic, quasperiodic and random nanoaperture arrays,” Appl. Phys. Lett. 91, 151109 (2007). [CrossRef]

, 11

11. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garćia-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007). [CrossRef]

]. The former mechanism relies on the excitation of surface plasmon polaritons (SPPs) on either side of the metal film, and the latter depends on the shape of single hole.

Fig. 1. Schematic illustrations of the compound subwavelength arrays of square hole. (a) A unit cell of compound subwavelength hole arrays, consisting of two square holes width identical width. (b) corresponds to dx = 0. (c) the general case. (d) the case dy = 0. The width of square hole are a = 200 nm, filled with air, and the periodicity is p = 600 nm. The thickness of Ag film is 100 nm.

2. Simulated model and method

In Fig. 1, we schematically show the 2D compound subwavelength hole arrays with different arrangement style, perforated on the Ag film deposited on a substrate assumed as quartz with dielectric constant ϕd = 2.16. Fig. 1(a) illustrates one unit cell of compound subwavelength hole arrays, consisting of two square holes with identical width a = 200 nm. The hole width a and period of compound hole arrays px = py = 600 nm are fixed for the whole paper. In the complex unit cell, the hole B is inserted to the position with both lateral and the longitudinal displacements with respect to the hole A, labeled as dx and dy , respectively. The interspace between two holes in a unit cell along the y direction is denoted as dAB. Changing the dx or dy, different geometrical arrangement can be obtained. Figs. 1(b) and (d) depict the cases of dx = 0 and dy = 0, and the holes are aligned to the y and x direction, respectively. Also, the general case is demonstrated in Fig. 1(c). The three-dimensional Finite Difference Time Domain (3D-FDTD) method[18

18. A. Taflove and S. C. Hagness, Computational Electrodynamics. The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston. 2000).

] was employed to simulate the interaction between metal sliver film and the incident wave. In our simulation, the spatial mesh steps were set Δx = Δy = Δz = 10 nm and the time step was set Δt = Δx/2c (c is the velocity of light). Also, perfectly matched layer (PML) absorbing boundary conditions were applied at either end of the computing space where periodic boundary conditions were used on other boundaries. The structure is illuminated by a p-polarized plane wave at normal incidence (i.e., the incoming electric field points along the y direction). The thickness of sliver film is h = 100 nm and the frequency-dependent permittivities of sliver are referred to Ref. [19

19. E. D. Palik, Handbook of Optical Constants and Solids(Academic, New York, 1985).

].

3. Results and discussion

At normal incidence, the position of transmission peak due to the excitation SPPs is given approximately by the surface plasmon dispersion for a smooth film[20

20. 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 58, 6779–6782 (1998). [CrossRef]

],

λmax=pi2+j2εdεmεd+εm,
(1)

where i, j are integers defining the different diffraction orders, ϕd and ϕm are the dielectric constant of the interface medium and metal, respectively. Fig. 2(a) shows the zero-transmission spectra of the compound hole arrays shown as in Fig. 1(b). For comparison, we also give the transmission spectrum of square hole arrays with only one hole in each cell (dashed line). According to Eq. (1), these transmission peaks can be indexed as Q (0, 1) at 930 nm, Q (1, 1) at 760 nm, and A (0, 1) at 710 nm, A (1, 1) at 590 nm as shown in Figs. 2(a) and (b). Here, Q denotes the SPPs at the quartz-silver interface and A represents the SPPs at air-sliver interface, respectively. With respect to the physics origin of these peaks, it has been demonstrated that the reflected SPPs Bloch waves from the holes produce a multiple beam interference pattern of the symmetry determined by the orientation of the hole rows[21

21. L. Salomon, F. Grillot, A. V. Zayats, and F. de. Fornel, “Near-field distribution of optical transmission of periodic subwavelength holes in a metal film,” Phys. Rev. Lett. 86, 1110–1113 (2001). [CrossRef] [PubMed]

].

Fig. 2. Transmission spectra for different arrangement of compound subwavelength hole arrays. (a)-(b) Transmission spectra correspond to the arrangement style shown as in Figs. 1(b) and (c). The transmission for simple cell arrays with the same width and periodicity as the compound structures is also plotted (dashed line), shown in the inset as “simple”.

We first consider the effect of the distance dAB on the transmission spectrum for compound hole arrays. When dAB =100 nm, the interspace between every two holes in the y direction is equal, which actually becomes rectangular arrays of square hole with py = px/2 = 300 nm. In Fig. 2(a), two distinct peaks are observed in the transmission spectra (line with circles). However, if we change the distance dAB, both of the two transmission bands split and two sharp dips are formed, one at wavelength λ = 620 nm and the other at λ = 870 nm (the arrow denoted). Moreover, as the distance dAB changes from 80 nm to 60 nm, the dip of A (1, 1) peak at λ = 620 nm is further deepened while the other dip at λ = 870 nm is not so sensitive to dAB. Comparing the transmission spectra with square hole arrays consisting of simple unit cell(dashed line), another interesting feature is the suppression of two (0, 1) peaks (A (0, 1) and Q (0, 1)) in the transmission spectra of compound hole arrays. The SPR peak arises from the constructive interference of SPPs, which is determined by the phase-match condition[22

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

]: Re(kSpp)px(y)+arg(τ) = 0 modulo 2π. Here, kspp=2π/λεdεmεd+εm is the SPPs wave vector, and τ is the transmittance coefficient of the SPPs mode. As pointed out in Ref. [21

21. L. Salomon, F. Grillot, A. V. Zayats, and F. de. Fornel, “Near-field distribution of optical transmission of periodic subwavelength holes in a metal film,” Phys. Rev. Lett. 86, 1110–1113 (2001). [CrossRef] [PubMed]

], the SPR (0, 1) peak is mainly induced by the Bragg scattering from the holes of y direction. When the square hole arrays B are inserted along to the y direction, the additional phase difference is introduced and the condition of constructive interference for the SPPs mode is broken, leading to the suppression of two (0,1) peaks. This phenomenon also agrees well with the experimental results in Ref. [16

16. Q. J. Wang, C. P. Huang, J. Q. Li, and Y. Y. Zhu, “Suppression of transmission minma and maxima with structural metal surface,” Appl. Phys. Lett. 89, 221121 (2006). [CrossRef]

], where the suppression of transmission minima and maxima for compound graphite arrays and rectangular arrays were observed. While for the two (1, 1) peaks (A (1, 1) and Q (1, 1)), the asymmetrical compound hole arrays can only be regarded as rectangular arrays of square hole with structural defects, for the SPPs (1,1) modes are mainly scattered by the holes along the diagonal of the alternating arrays. As a result, one or more forbidden channels are formed within the allowed transmission bands, resulting in sudden variations of diffracted efficiency at a certain wavelength. On the other hand, due to the different dispersion relation of SPPs on two interfaces, the phase difference accumulated by the variation of dAB is different for two dips at λ = 620 nm and λ = 870 nm, which gives rise to their different sensitivity to dAB.

Fig. 3. Transmission spectra for various dx, corresponds to the compound subwavelength square hole arrays illustrated in Fig. 1(d).

Since the SPPs modes mainly propagate along the y direction, the condition of SPPs resonance almost would not be affected if the two holes in every unit cell are arranged along the x axis shown as in Fig. 1(d). Thus, the above mentioned phenomena will not exist. This prediction is also verified by our FDTD simulation, as the transmission spectra shown in Fig. 3. We can find that all the transmission peaks for square hole arrays with simple unit cell appear again, except the Q (1, 1) peak and A (0, 1) peak combine together in some cases. Also, the Q (0, 1) peak for compound hole arrays exhibits redshift compared to the transmission for hole arrays with simple unit cell , which is caused by the variation of dispersion of SPPs by the additional holes. We also have calculated other cases for different dx (not shown here), which all indicate that the transverse distance dx is of minor importance to the transmission spectra for the compound hole arrays when illuminated by p-polarized incident light. This can be attribute to the fact that SPPs does not propagate along the x direction, confirming the above mentioned splitting of transmission peak and the formation of dips are caused by phase resonance of SPPs.

To further verify the explanation about the above results, we also give the field distribution at resonant and off-resonant wavelength in Fig. 4. The field distribution at wavelength λ = 610 nm, dAB = 100 nm and λ= 620 nm, dAB = 60 nm are depicted in Figs. 4(a) and (b), respectively. From Fig. 4(a), the hole arrays are distributed symmetrically, the SPPs Bloch wave interferes constructively at the edge of hole, leading to the transmission peak in Fig. 2(a). However, as to the asymmetrical case, the SPPs Bloch wave interferes destructively (Fig. 4(b)), which give rise to a dip in the transmission shown in Fig. 2(a). In order to better understand the suppression of two SPR (0, 1) peaks at metal surface, we also plot the field distribution at wavelength λ = 970 nm in Figs. 4(c) and (d), corresponding to the structure displayed in Figs. 1(d) and (b), respectively. The electric field distribution confirms that the interference behavior of SPPs is modulated by the arrangement of compound hole arrays.

Fig. 4. The calculated time-averaged density distribution of the electric field at dielectric-metal interfaces, the xy cross section. (a) corresponds to the case of dAB = 100 nm, λ= 610 nm at the air-metal interface. (b) corresponds to dAB = 60 nm, λ = 620 nm at the air-metal interface. (c)-(d) the electric field distribution at quartz-metal interface for Q (0, 1) peak at wavelength λ = 970 nm, corresponding to the arrangement of Figs. 1(d) and (b), respectively. The dotted lines mark the position of the holes.

4. Conclusion

In summary, we have shown that the enhanced transmission can be tailored by changing the geometrical arrangement of compound subwavelength hole arrays. When the symmetry of hole arrays along the direction of polarized incident light is broken, the transmission bands show splitting behavior and the depth of the dip depends on the dielectric contacted with metal, as well as the distance between two holes. Additionally, as the symmetry of hole arrays changes, the two SPR (0, 1) peaks are suppressed, if the phase difference of SPPs resulting from the scattering by the holes is appropriated. However, the transmission spectra are not sensitive to the symmetry in the direction of perpendicular to the polarized light. We ascribe these phenomena to the phase resonance of SPPs, produced by the broking of symmetry of hole arrays. The proposed method to tailor enhanced transmission spectra is expected to have applications such as frequency selector or filter.

Acknowledgments

The authors would acknowledge financial supports from the Natural Science Foundation of Hunan Province, China (Grants No. 07JJ3114 and No. 07JJ107), the National Natural Science Foundation of China (Grants No. 90606001, No. 50602014, No. 10674045 and No. 60538010) and “973” National Key Basic Research Program of China (Grants No. 2007CB310403 and No. 2007CB310500).

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.

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

3.

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

4.

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

5.

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express 13, 4485–4491 (2005). [CrossRef] [PubMed]

6.

L. Mart́in-Moreno and F. J. Garćia-Vidal, “Optical transmission through circular hole arrays in optically thick metal films,” Opt. Express 12, 3619–3628 (2004). [CrossRef]

7.

L. Mart́in-Moreno, F. J. Garćia-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]

8.

K. J. K. 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 holes,” Phys. Rev. Lett. 92, 183901 (2004). [CrossRef] [PubMed]

9.

Z. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: The role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006). [CrossRef] [PubMed]

10.

C. Rockstuhl and F. Lederer, “Enhanced transmission of periodic, quasperiodic and random nanoaperture arrays,” Appl. Phys. Lett. 91, 151109 (2007). [CrossRef]

11.

A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garćia-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007). [CrossRef]

12.

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005). [CrossRef] [PubMed]

13.

D. C. Skigin and R. A. Depine, “Narrow gaps for transmission through metallic structured gratings with sub-wavelength slits,” Phys. Rev. E 74, 046606 (2006). [CrossRef]

14.

Y. G. Ma, X. S. Rao, G. F. Zhang, and C. K. Ong, “Microwave transmission modes in compound metallic gratings,” Phys. Rev. B 76, 085413 (2007). [CrossRef]

15.

Q. J. Wang, J. Q. Li, C. P Huang, C. Zhang, and Y. Y. Zhu, “Enhanced optical transmission through metal films with rotation-symmetrical hole arrays,” Appl. Phys. Lett. 87, 091105 (2005). [CrossRef]

16.

Q. J. Wang, C. P. Huang, J. Q. Li, and Y. Y. Zhu, “Suppression of transmission minma and maxima with structural metal surface,” Appl. Phys. Lett. 89, 221121 (2006). [CrossRef]

17.

Y. H Ye, Z. B Wang, and Y. Cao, “Enhanced transmission through metal films perforated with circular and cross-dipole apertures,” Appl. Phys. Lett. 91, 251105 (2007). [CrossRef]

18.

A. Taflove and S. C. Hagness, Computational Electrodynamics. The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston. 2000).

19.

E. D. Palik, Handbook of Optical Constants and Solids(Academic, New York, 1985).

20.

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 58, 6779–6782 (1998). [CrossRef]

21.

L. Salomon, F. Grillot, A. V. Zayats, and F. de. Fornel, “Near-field distribution of optical transmission of periodic subwavelength holes in a metal film,” Phys. Rev. Lett. 86, 1110–1113 (2001). [CrossRef] [PubMed]

22.

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

OCIS Codes
(120.7000) Instrumentation, measurement, and metrology : Transmission
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: November 14, 2008
Revised Manuscript: January 18, 2009
Manuscript Accepted: January 19, 2009
Published: January 29, 2009

Citation
Jian-Qiang Liu, Meng-Dong He, Xiang Zhai, Ling-Ling Wang, Shuangchun Wen, Li Chen, Zhe Shao, Qing Wan, B. S. Zou, and Jianquan Yao, "Tailoring optical transmission via the arrangement of compound subwavelength hole arrays," Opt. Express 17, 1859-1864 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1859


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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 (London) 391, 667-669 (1998). [CrossRef]
  2. F. J. García de Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007). [CrossRef]
  3. C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature (London) 445, 39-46 (2007). [CrossRef]
  4. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonance on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999). [CrossRef]
  5. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through a periodic array of slits in a thick metallic film," Opt. Express 13, 4485-4491 (2005). [CrossRef] [PubMed]
  6. L. Martín-Moreno and F. J. García-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619-3628 (2004). [CrossRef]
  7. L. Martín-Moreno, F. J. García-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]
  8. K. J. K. 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 holes," Phys. Rev. Lett. 92, 183901 (2004). [CrossRef] [PubMed]
  9. Z. Ruan and M. Qiu, "Enhanced transmission through periodic arrays of subwavelength holes: The role of localized waveguide resonances," Phys. Rev. Lett. 96, 233901 (2006). [CrossRef] [PubMed]
  10. C. Rockstuhl and F. Lederer, "Enhanced transmission of periodic, quasperiodic and random nanoaperture arrays," Appl. Phys. Lett. 91, 151109 (2007). [CrossRef]
  11. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcıa-Vidal, "Theory of light transmission through an array of rectangular holes," Phys. Rev. B 76, 195414 (2007). [CrossRef]
  12. D. C. Skigin and R. A. Depine, "Transmission resonances of metallic compound gratings with subwavelength slits," Phys. Rev. Lett. 95, 217402 (2005). [CrossRef] [PubMed]
  13. D. C. Skigin and R. A. Depine, "Narrow gaps for transmission through metallic structured gratings with subwavelength slits," Phys. Rev. E 74, 046606 (2006). [CrossRef]
  14. Y. G. Ma, X. S. Rao, G. F. Zhang, and C. K. Ong, "Microwave transmission modes in compound metallic gratings," Phys. Rev. B 76, 085413 (2007). [CrossRef]
  15. Q. J. Wang, J. Q. Li, C. P Huang, C. Zhang, and Y. Y. Zhu, "Enhanced optical transmission through metal films with rotation-symmetrical hole arrays," Appl. Phys. Lett. 87, 091105 (2005). [CrossRef]
  16. Q. J. Wang, C. P. Huang, J. Q. Li, and Y. Y. Zhu, "Suppression of transmission minma and maxima with structural metal surface," Appl. Phys. Lett. 89, 221121 (2006). [CrossRef]
  17. Y. H Ye, Z. B Wang, and Y. Cao, "Enhanced transmission through metal films perforated with circular and cross-dipole apertures," Appl. Phys. Lett. 91, 251105 (2007). [CrossRef]
  18. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Boston. 2000).
  19. E. D. Palik, Handbook of Optical Constants and Solids (Academic, New York, 1985).
  20. 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 58, 6779-6782 (1998). [CrossRef]
  21. L. Salomon, F. Grillot, A. V. Zayats, and F. de. Fornel, "Near-field distribution of optical transmission of periodic subwavelength holes in a metal film," Phys. Rev. Lett. 86, 1110-1113 (2001). [CrossRef] [PubMed]
  22. H. Liu and P. Lalanne, "Microscopic theory of the extraordinary optical transmission," Nature,  452, 728-731 (2008). [CrossRef] [PubMed]

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