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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 9 — May. 6, 2013
  • pp: 10739–10745
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Highly efficient beam steering with a transparent metasurface

Zeyong Wei, Yang Cao, Xiaopeng Su, Zhijie Gong, Yang Long, and Hongqiang Li  »View Author Affiliations


Optics Express, Vol. 21, Issue 9, pp. 10739-10745 (2013)
http://dx.doi.org/10.1364/OE.21.010739


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Abstract

We propose an ultra-thin planar metasurface with phase discontinuities for highly efficient beam steering. The effect benefits from the broadband transparency and flexible phase modulation of stacked metal/dielectric multi-layers that is perforated with coaxial annular apertures. Proof-of-principle experiments verify that an efficiency of 65% and a deflection angle of 18 o at 10GHz are achieved for the transmitted beam, which are also in good agreement with the finite-difference-method-in-time-domain (FDTD) simulations. The scheme shall be general for the design of beam-steering transmitters in all frequencies.

© 2013 OSA

1. Introduction

According to Fermat’s principle, wavefront of a light beam can be modified by controlling the phase of light wave [1

1. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Pergamon Press, Oxford, Angleterre, 1980).

]. Conventional optical components rely on sophisticated design of permittivity distribution to gradually modulate the phase of light waves for the control of propagation path in bulk material. A propagation length comparable to operational wavelength is required for a specific phase shift. In contrast, metamaterial (MM) [2

2. N. Engheta and R. W. Ziolkowski, Metamaterials: physics and engineering explorations (Wiley-IEEE Press, New York, 2006).

, 3

3. W. Cai and V. Shalaev, Optical metamaterials: fundamentals and applications (Springer, New York, 2009).

] manipulates light waves in subwavelength scale, and exhibits exotic optical properties such as negative refraction [4

4. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005). [CrossRef] [PubMed]

6

6. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef] [PubMed]

], invisibility cloak [7

7. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

9

9. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

], polarization manipulation [10

10. A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. 97(17), 177401 (2006). [CrossRef] [PubMed]

13

13. Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011). [CrossRef]

], as well as extraordinary optical transmission (EOT) [14

14. 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]

19

19. Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: The role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006). [CrossRef] [PubMed]

], etc. Such extreme optical responses arise from local resonances of subwavelength-sized units. An ultra-thin planar MM, i. e., a metasurface, enables abrupt phase discontinuity for scattering waves [20

20. N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]

], opening a new era of light manipulation. An elegant example is the “V-shaped” antenna array, which ensures identical amplitudes of scattering waves at each unit, and provides a constant phase shift between two neighbor units. Beaming shaping, beam steering and phase modulation within an optically thin depth are favorable for various applications [20

20. N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]

32

32. T. Matsui, H. T. Miyazaki, A. Miura, T. Nomura, H. Fujikawa, K. Sato, N. Ikeda, D. Tsuya, M. Ochiai, Y. Sugimoto, M. Ozaki, M. Hangyo, and K. Asakawa, “Transmission phase control by stacked metal-dielectric hole array with two-dimensional geometric design,” Opt. Express 20(14), 16092–16103 (2012). [CrossRef] [PubMed]

].

It has been reported that a thin metal film perforated with an array of subwavelength holes exhibits extraordinary optical transmission (EOT) property due to the excitation of surface plasmon polaritons or localized resonances [14

14. 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]

, 19

19. Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: The role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006). [CrossRef] [PubMed]

]. And it has been found recently that the EOT passband can be very broad for a metal/dielectric multi-layers system that is perforated with an array of coaxial annular apertures (CAAs). In addition, within the EOT passband, the phase of transmitted waves varies smoothly from 0 to 2π due to interlayer coupling [33

33. Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband transparency achieved with the stacked metallic multi-layers perforated with coaxial annular apertures,” Opt. Express 19(22), 21425–21431 (2011). [CrossRef] [PubMed]

]. Both features are advantageous for highly efficient beam steering. As the broadband EOT stems from the evanescent resonant coupling among adjacent perforated metallic layers through local resonance channels provided by CAAs, it is local resonance that primarily determines the EOT property [19

19. Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: The role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006). [CrossRef] [PubMed]

], while the disorder of CAA arrays or the alignment of CAAs among different layers has little influence on it. More control calculations (not shown) indicate that the broadband EOT is robust against the structural disorder and imperfect alignment, which is very advantageous for practical application.

In this paper, we show that, by varying the geometric parameters of CAAs, a multi-layered metasurface can function as a beam steering transmitter. We also perform proof-of-principle experiments in microwave regime to verify the beam steering effect and high efficiency. Measured spatial distributions of electromagnetic local fields, in excellent agreement with finite-difference-time-domain (FDTD) simulations, clearly indicate that varied phase discontinuities on the metasurface dramatically modify the wavefront of transmitted beam. An efficiency of 65% and a deflection angle of 18° are verified by measuring the far-field radiation pattern. Thanks to high transmission and flexible phase modulation in the broad EOT band, the proposed metasurface is capable of refracting the incident beam in any polarization at a predetermined deflection angle.

2. Model description

Figure 1(a)
Fig. 1 (a) Schematic of the metasurface, and the incident and transitted waves. (b) Photograph of a part of the sample.
presents the schematic of the gradient metasurface for beam steering. The metasurface is comprised of three metallic layers and two intermediate dielectric spacer layers. Each metallic layer, with a thickness of 0.035mm, is perforated with an array of CAAs. And metals in our model system are assumed to be perfect electric conductors (PECs) at the microwave frequencies. Each dielectric layer has a thickness of 1.575mm and a permittivity of 2.65. The gradient configuration is realized by varying the inner radius of CAAs along one direction of the array. As illustrated in the top view of the sample [see Fig. 1(b)], the CAAs, perforated through each metal film, are arranged in a square lattice with 9 columns (along x direction) and 12 rows (along y direction). The lattice period is p=12mm, such that the sample has a width of Lx=108mm and a length of Ly=144mm. The outer radius of each CAA is fixed at R=5.8mm. The inner radius rn (n=1,2,...9) of CAAs of the nine columns varies gradually from left to right. The incident plane waves are propagating along +z direction with electric field along x axis and magnetic field along y axis. By tuning the inner radius of CAAs, phase gradient is assigned to the scattering waves along x axis on the metasurface, so that the transmitted waves are refracted toward a prescribed direction.

The spectra of transmission intensity and phase difference present enough information for design the transmitter by selecting appropriate values of radii rn. Table 1

Table 1. Inner radii rn of our designed model

table-icon
View This Table
presents the radii of CAAs in our model. The rule is to ensure a π/4 phase increment at two adjacent CAAs along x axis to cover 0 to 2π for phase variation over the metasurface, while maintaining the amplitude of the transmitted wave above 0.8 [see Fig. 3
Fig. 3 Analytically calculated transmission amplitude (black square dots) and phase difference (red circular dots) at 10GHz by assuming that inner radius of the periodic CAA arrays has the value of rn (see Table 1).
]. Following the generalized Snell’s law for reflection and refraction [20

20. N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]

]
ntsinθtnisinθi=λ2πdΦdx,
(1)
where Φ is the phase discontinuities (i. e., phase difference) at a local point brought by the metasurface, nt (ni) is refractive index of the refracted (incident) medium, while θt (θi) is the refracted (incident) angle, and λ is wavelength. A constant gradient of phase discontinuity dΦ/dx along x axis is expected to deflect the transmitted light beam away from normal (+z) direction. As the sample is placed in free space (i. e., nt=1 and ni=1), for normal incidence (i. e., sinθi=0), the deflection angle is determined by the gradient of phase discontinuity,
θt=arcsin(λ2πdΦdx)=arcsin(λ2π2πLxp),
(2)
giving rise to θt=17.5o in our case.

3. Simulations and measurements of highly efficient beam steering

The theoretical predictions stated above are verified by FDTD simulations. In the calculations, only one period is modeled along y axis for the sake of periodic boundary condition. The metasurface is finite along x axis with the length of Lx=108mm. Perfectly matched layers (PMLs) are adopted along x and z directions to absorb scattering waves at boundaries. A one-way Gaussian beam at 10GHz is positioned 15mm away from the metasurface in xy plane. The beam waist is 50mm. Figures 4(a)
Fig. 4 Simulated magnetic field distributions (Hy) for a Gaussian beam propagated (a) in free space and (b) through the metasurface. Measured magnetic field distributions (Hy) for a horn antenna (c) without and (d) with the sample.
and 4(b) present the calculated magnetic field distributions Hy(x,z) of transmitted waves without and with the metasurface. It is shown that the Gaussian beam is propagating through the metasurface with high efficiency at an angle of about 18o with respect to z axis. Measurements are taken in a microwave chamber. The sample, with a lateral size of 108mm×144mm, is fabricated by printed-circuit-board (PCB) fabrication technology. A standard horn antenna operating at 8.2GHz-12.4GHz with gain coefficient of 24.8dB serves as the Gaussian beam source. A small ring antenna as a probe, lying in xz plane for measuring magnetic field Hy, is fixed on a two-dimensional movable platform. The platform is electrically driven by a computer with a maximum scanning range of 1m×1m and a finest resolution of 0.1mm along both x and z directions. The horn antenna and the ring antenna are connected to a vector network analyzer Agilent8722ES for the measurements. We see that experimental results illustrated in Figs. 4(c) and 4(d), are in excellent agreement with numerical simulations shown in Figs. 4(a) and 4(b), verifying the deflection angle of 17.5° in theoretical prediction.

To quantitatively determine the deflection angle and the transmission efficiency, we also measure the radiation patterns in far field with a horn receiver. The sample is bound to the horn emitter which is placed on a rotary table with a finest angular resolution of 0.1°. We see from Fig. 5(a)
Fig. 5 (a) Measured and (b) simulated far-field radiation patterns for a horn antenna with (blue line) and without (red line) the sample.
that the deflection angle is 18° and the transmittivity is above 65% by comparing with the reference radiation pattern of the horn antenna in free space. It is in excellent agreement with FDTD simulations shown in Fig. 5(b). Both the near-field and far-field measurements confirm that wavefront and high directivity of incident Gaussian beam is preserved. The highly efficient beam steering achieved with an ultra-thin planar scheme shall have great potentials in wavefront engineering and light manipulation for future integrated metaphotonic devices.

4. Conclusion

Acknowledgments

This work was supported by NSFC (Grant No. 10974144, 11174221, 11204218), CNKBRSF (Grant No. 2011CB922001), 863 Major Program (Grant No. 2012AA03A706), the Fundamental Research Funds for the Central Universities, SHEDF (Grant No. 06SG24) and China Postdoctoral Science Foundation (Grant No. 2011M500810, 2012T50433).

References and links

1.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Pergamon Press, Oxford, Angleterre, 1980).

2.

N. Engheta and R. W. Ziolkowski, Metamaterials: physics and engineering explorations (Wiley-IEEE Press, New York, 2006).

3.

W. Cai and V. Shalaev, Optical metamaterials: fundamentals and applications (Springer, New York, 2009).

4.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005). [CrossRef] [PubMed]

5.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]

6.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef] [PubMed]

7.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

8.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

9.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

10.

A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. 97(17), 177401 (2006). [CrossRef] [PubMed]

11.

C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric Transmission of Linearly Polarized Light at Optical Metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010). [CrossRef] [PubMed]

12.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat Commun 3, 870 (2012). [CrossRef] [PubMed]

13.

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011). [CrossRef]

14.

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]

15.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

16.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

17.

S. Carretero-Palacios, F. J. Garcia-Vidal, L. Martin-Moreno, and S. G. Rodrigo, “Effect of film thickness and dielectric environment on optical transmission through subwavelength holes,” Phys. Rev. B 85(3), 035417 (2012). [CrossRef]

18.

Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures 8(2), 94–101 (2010). [CrossRef]

19.

Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: The role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006). [CrossRef] [PubMed]

20.

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]

21.

F. Aieta, P. Genevet, N. F. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (2012). [CrossRef] [PubMed]

22.

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29.

M. Kang, T. H. Feng, H. T. Wang, and J. S. Li, “Wave front engineering from an array of thin aperture antennas,” Opt. Express 20(14), 15882–15890 (2012). [CrossRef] [PubMed]

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32.

T. Matsui, H. T. Miyazaki, A. Miura, T. Nomura, H. Fujikawa, K. Sato, N. Ikeda, D. Tsuya, M. Ochiai, Y. Sugimoto, M. Ozaki, M. Hangyo, and K. Asakawa, “Transmission phase control by stacked metal-dielectric hole array with two-dimensional geometric design,” Opt. Express 20(14), 16092–16103 (2012). [CrossRef] [PubMed]

33.

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband transparency achieved with the stacked metallic multi-layers perforated with coaxial annular apertures,” Opt. Express 19(22), 21425–21431 (2011). [CrossRef] [PubMed]

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OCIS Codes
(160.3918) Materials : Metamaterials
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Metamaterials

History
Original Manuscript: February 25, 2013
Revised Manuscript: March 28, 2013
Manuscript Accepted: March 28, 2013
Published: April 25, 2013

Citation
Zeyong Wei, Yang Cao, Xiaopeng Su, Zhijie Gong, Yang Long, and Hongqiang Li, "Highly efficient beam steering with a transparent metasurface," Opt. Express 21, 10739-10745 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10739


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References

  1. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Pergamon Press, Oxford, Angleterre, 1980).
  2. N. Engheta and R. W. Ziolkowski, Metamaterials: physics and engineering explorations (Wiley-IEEE Press, New York, 2006).
  3. W. Cai and V. Shalaev, Optical metamaterials: fundamentals and applications (Springer, New York, 2009).
  4. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett.95(20), 203901 (2005). [CrossRef] [PubMed]
  5. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008). [CrossRef] [PubMed]
  6. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004). [CrossRef] [PubMed]
  7. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  8. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
  9. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object,” Phys. Rev. Lett.102(25), 253902 (2009). [CrossRef] [PubMed]
  10. A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett.97(17), 177401 (2006). [CrossRef] [PubMed]
  11. C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric Transmission of Linearly Polarized Light at Optical Metamaterials,” Phys. Rev. Lett.104(25), 253902 (2010). [CrossRef] [PubMed]
  12. Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat Commun3, 870 (2012). [CrossRef] [PubMed]
  13. Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett.99(22), 221907 (2011). [CrossRef]
  14. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
  15. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003). [CrossRef] [PubMed]
  16. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys.82(1), 729–787 (2010). [CrossRef]
  17. S. Carretero-Palacios, F. J. Garcia-Vidal, L. Martin-Moreno, and S. G. Rodrigo, “Effect of film thickness and dielectric environment on optical transmission through subwavelength holes,” Phys. Rev. B85(3), 035417 (2012). [CrossRef]
  18. Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures8(2), 94–101 (2010). [CrossRef]
  19. Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: The role of localized waveguide resonances,” Phys. Rev. Lett.96(23), 233901 (2006). [CrossRef] [PubMed]
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