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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 28513–28522
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Enhanced and suppressed infrared transmission through germanium subwavelength arrays

Wei Dong, Toru Hirohata, Kazutoshi Nakajima, and Xiaoping Wang  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 28513-28522 (2013)
http://dx.doi.org/10.1364/OE.21.028513


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Abstract

We have studied the zero-order transmission of periodic germanium (Ge) subwavelength arrays in an infrared range by using finite-difference time-domain simulations. A special wavelength-selective peak in a triangular hole array of Ge film is observed with an enhanced transmission accompanied by a drastic suppression nearby, which cannot be found in a one-dimensional Ge subwavelength array and is different from the extraordinary transmission related to surface plasmons in a metal film. The electromagnetic field is found to be concentrated on both surfaces of the Ge film at this peak. The unique transmission is verified through measurements on fabricated samples and is interpreted using the photonic band structure.

© 2013 Optical Society of America

1. Introduction

SPs are essentially electromagnetic (EM) waves localized at a metallic surface through interaction with the free electrons of the metal [8

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

]. This nature indicates that there is always intrinsic absorption (SP loss), such that the structure would hardly be expected to be highly transparent although transmissions that are 1000 times greater than predicted by the standard aperture theory have been observed [1

1. T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

]. At lower frequencies such as in the infrared range, dissipation caused by the imaginary part of the permittivity becomes more important. Such dissipation always leads to a lower enhancement of the transmission and to broader resonances [9

9. T. Thio, H. Ghaemi, H. Lezec, P. Wolff, and T. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16(10), 1743–1748 (1999). [CrossRef]

], which is considered to be a feature that has somewhat limited applications, especially in wavelength-selective sources and detectors. Some researchers tried to use other materials in the infrared range. For example, an SU-8 film [10

10. Y.-H. Ye and J.-Y. Zhang, “Middle-infrared transmission enhancement through periodically perforated metal films,” Appl. Phys. Lett. 84(16), 2977–2979 (2004). [CrossRef]

] was exploited to make a symmetrical structure to achieve high transmission, but it contains many layers and its fabrication requires several different processes.

In the transmission spectra of metallic subwavelength arrays, the enhanced transmissions are always accompanied by drastic suppressions in a nearby wavelength range [11

11. C. Genet, M. P. van Exter, and J. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Commum. 225(4-6), 331–336 (2003). [CrossRef]

, 12

12. 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). [CrossRef] [PubMed]

]. Lezec and Thio discovered similar transmission features in nonmetallic systems that do not support SPs, and they interpreted the mechanism as a composite diffracted evanescent wave (CDEW) [12

12. 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). [CrossRef] [PubMed]

]. Their findings and other theories of subwavelength arrays such as Bloch-wave modes of dynamic diffraction [13

13. M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66(19), 195105 (2002). [CrossRef]

], show some possibilities of achieving wavelength selectivity along with high transmission by using non-metallic material to overcome the SP loss.

2. FDTD simulation

Frequency-dependent dielectric constants obtained from the literature [17

17. E. D. Palik, Handbook of Optical Constants of Solids: Index (Academic, 1998).

] are applied in our modeling. It should be noted that in the investigated infrared region, Ge has a higher refractive index (n ~4.1) than other nonmetallic materials, which is beneficial for light confinement, and almost zero imaginary part, and thus the transmission of the non-perforated film is originally larger than that of the metal film. The typical parameters of an analyzed model consist of a lattice constant a = 1.8 μm, a width d = 1.0 μm and a thickness t = 0.36 μm as shown in Figs. 1(a) and 1(b). Comparisons of the simulation results for the 1D and 2D structures are shown in Figs. 1(c) and 1(d), respectively. Both the transmissions of the periodic structures (red lines) and of single-lattice structures (black lines) are recorded. In Fig. 1(d), which shows the case of a Ge-THA, there is a resonant peak at 3150 cm−1, which represents an enhanced transmission accompanied by a suppression; these features share similar characteristics to those described in [12

12. 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). [CrossRef] [PubMed]

]. The transmission maximum becomes as high as 100%, which is enhanced by a factor of 2 compared with the single-lattice case, while the nearby minimum can be as low as 0 (since it is the first peak observed when we initially examine the spectrum from low to high frequency, we call it the “First Peak”). However, in the case of periodic Ge stripes as shown in Fig. 1(c), only one broad-band resonance (~3150 cm−1) is obtained, and it seems like an absorption peak rather than a transmission peak. Although we present here a specific example for the structure based on periodic Ge stripes, we have studied different parameters and the results show similar phenomena.

3. Fabrication and measurement

Since the optical properties of the quartz substrate used in real samples are more complicated than the simple SiO2 model in the simulation, all the peaks in the fabricated samples suffer blue-shift deviations from the simulation results. However, there are the same patterns of red shifts in the peak positions in the measurement and the simulation as the lattice constant a increases from 1.6 to 2.0 μm. The results indicate the reliability of our simulations in revealing the nature of this subwavelength structure. The differences between the simulations and the measurements can also be caused by other factors such as processing errors, edge roughness, slopes in the holes, additional optical property of the real material and the 5° focus angle in the collinear configuration. In particular, the measured transmission maxima of 83%, which is less than the predicted value of nearly 100% in the simulation, is strongly affected by the 5° angle since the optical properties of this kind of structure are very sensitive to any dispersion [1

1. T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

, 18

18. J. Braun, B. Gompf, G. Kobiela, and M. Dressel, “How holes can obscure the view: suppressed transmission through an ultrathin metal film by a subwavelength hole array,” Phys. Rev. Lett. 103(20), 203901 (2009). [CrossRef] [PubMed]

]. The zero-order transmission of the 1D periodic Ge stripes is also measured. Instead of enhanced transmission, only absorption bands are observed, which also coincides with our simulation.

4. Distinctive features of the First Peak

4.1. EM field

4.2. Spectral positions of resonant peaks

4.3. Field distributions of the cross-section

The cross-section field distributions of the above three structures are investigated in order to determine what kind of discrete states they have. Note that in our simulation, the model is built upside down from Fig. 1(b) so that the source is located at the bottom. The TM mode is mainly considered, and both the Ez and Hy fields are studied. The Ez field patterns have evanescent features mainly concentrated at the corners of the openings, and thus show no big differences in all three cases. However, the Hy field patterns change drastically from case to case. The patterns in the region of one period can be seen in Fig. 4.
Fig. 4 Cross-section field distributions of Hy at λsp in Au-THA (a), at λstripe in periodic Ge stripes (b), at λGe-THA in Ge-THA (c), and at the First Peak in Ge-THA (d). The source is located at the bottom of the structure. The white dashed rectangles indicate the area of Au or Ge.
In the case of Au-THA, since the gold film is opaque in this frequency range, the Hy field decays quickly in the material and can only exist on the surface, as shown in Fig. 4(a). It has been explained that there are SPs tunneling through the holes [2

2. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef] [PubMed]

] so that the Hy field on the back surface (the top side in the figure) is much stronger than that on the irradiated surface (the bottom side) indicating the transmission enhancement. As noted previously, Ge has a high refractive index and no absorption in the infrared range, so that in periodic Ge stripes, the Hy field concentrates almost inside the Ge material forming a localized pattern, as shown in Fig. 4(b). A similar Hy pattern is also found at λGe-THA in the case of Ge-THA, as indicated in Fig. 4(c). However, in Fig. 4(d), the First Peak, generated simultaneously in Ge-THA, surprises us with a Hy pattern that shares the general features with the Au case: although the Hy field spreads partly inside the Ge, it mainly concentrates on both surfaces of the film. In addition, the Hy field on the outgoing surface is also enhanced so that there is an enhanced transmission with a sharp suppression nearby in the spectrum.

5. Basic analysis of the phenomena

Fig. 5 Geometry of optical diffractions by a subwavelength hole in a Ge screen on quartz substrate.
The phenomena of enhanced and suppressed transmissions in Ge-THA can be interpreted using the CDEW theory. As described in [12

12. 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). [CrossRef] [PubMed]

], when light emerges from a subwavelength aperture, it is diffracted partly into a continuum of radiative modes of which the in-plane component of the wave vector is Kx < k0ns (blue circle in Fig. 5), and partly into a continuum of evanescent modes propagating along the surface with real Kx > k0ns. In the structure of Ge-THA on a quartz substrate, Ge has a much larger refractive index than that of the substrate, which makes it a good channel as a waveguide for confining light in-plane. As indicated with the yellow curve in Fig. 5, part of the diffracted light can be coupled into the guided modes inside the Ge film, which has been verified in our latest published paper [21

21. W. Dong, T. Hirohata, K. Nakajima, and X. Wang, “Near-field effect in the infrared range through periodic Germanium subwavelength arrays,” Opt. Express 21(22), 26677–26687 (2013). [CrossRef]

]. According to the waveguide theory, the in-plane component of the wave vector of the guided modes satisfies the condition of Kx > k0ns, and their electric fields on the surface of the Ge film are evanescent. Therefore, these modes act as a continuum of evanescent modes forming the “composite diffracted evanescent wave”, which will have constructive or destructive interference with the light directly incident on the hole. In consequence, the transmission peaks as well as the suppressions occur.

The CDEW theory works well in explaining the origin of the enhanced and suppressed transmissions. However, the unique feature of the First Peak is the sharpness and the large amplitude of the peaks or the valleys, whose determinants is not explicitly indicated in the CDEW theory. In fact, the band at 3600 cm−1 for Ge-THA in Fig. 3(b) also shows a peak along with a valley owing to the guided mode inside the Ge film, whose field can be seen in Fig. 4(c), but the amplitude of the peak is much smaller than that of the First Peak. The lineshape and the sharpness of a resonant band can be better estimated by introducing the Fano model as mentioned in Section 4.2. Details of the model can be found in [11

11. C. Genet, M. P. van Exter, and J. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Commum. 225(4-6), 331–336 (2003). [CrossRef]

]. The model emphasizes the importance of a discrete state in forming a sharp resonance. The field of the discrete state is always bound to the surface, like the one caused by SPs or the one that we find at the First Peak in Ge-THA, as shown in Fig. 4(d). The field pattern in Fig. 4(d) existing in the dielectric periodic structure is reminiscent of an odd discrete mode in photonic crystals [22

22. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

], which can be treated as a Bloch wave.

When a normal incidence reaches the surface of a periodic structure, many eigenstates are obtained by solving Hermitian eigenfunctions converted from Maxwell equations. All of the eigenstates can be cast in Bloch form: a periodic function modulated by a plane wave. The field can propagate through the crystal in a coherent manner, as a Bloch wave [13

13. M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66(19), 195105 (2002). [CrossRef]

, 16

16. Z. 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]

, 22

22. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

].

6. Conclusion

In summary, we have studied the transmission property of Ge subwavelength arrays. A band with nearly 100% transmission and a drastic suppression nearby is observed in Ge-THAs, and we name it the “First Peak”. The near-field EM responses (only Ez was presented) and the calculations of the spectral position show the distinctions from the 1D structure and the normal metallic subwavelength structure, while similar features of Hy field patterns are found in both triangular-hole-array structures made of germanium and gold. The position of the First Peak is well verified by the band edge at the Γ point, but a thorough theoretical analysis would be useful for gaining better insight into the mechanism of creating nearly 100% transmission. The First Peak in Ge-THAs provides a wavelength-selective property together with a high transmission, which can have many applications in optical MEMS such as a single-layer optical filter due to the unique large transmission region surrounded by regions of suppressed transmission, a window for the infrared light sources to improve the emission, and chemical or biological sensors on the subwavelength scale.

Acknowledgments

We thank Shohei Hayashi for FIB fabrication of our samples, and Yoshitaka Kurosaka and Kazuyoshi Hirose for discussions. We are also grateful for some helpful advice given by T. W. Ebbesen regarding FTIR measurement.

References and links

1.

T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

2.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef] [PubMed]

3.

W. Wu, A. Bonakdar, and H. Mohseni, “Plasmonic enhanced quantum well infrared photodetector with high detectivity,” Appl. Phys. Lett. 96(16), 161107 (2010). [CrossRef]

4.

Y. Cui and S. He, “Enhancing extraordinary transmission of light through a metallic nanoslit with a nanocavity antenna,” Opt. Lett. 34(1), 16–18 (2009). [CrossRef] [PubMed]

5.

S. C. Lee, S. Krishna, and S. R. Brueck, “Quantum dot infrared photodetector enhanced by surface plasma wave excitation,” Opt. Express 17(25), 23160–23168 (2009). [CrossRef] [PubMed]

6.

T. Ishi, J. Fujikata, K. Makita, T. Baba, and K. Ohashi, “Si nano-photodiode with a surface plasmon antenna,” Jpn. J. Appl. Phys. 44(12), L364–L366 (2005). [CrossRef]

7.

C.-Y. Chang, H.-Y. Chang, C.-Y. Chen, M.-W. Tsai, Y.-T. Chang, S.-C. Lee, and S.-F. Tang, “Wavelength selective quantum dot infrared photodetector with periodic metal hole arrays,” Appl. Phys. Lett. 91(16), 163107 (2007). [CrossRef]

8.

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

9.

T. Thio, H. Ghaemi, H. Lezec, P. Wolff, and T. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16(10), 1743–1748 (1999). [CrossRef]

10.

Y.-H. Ye and J.-Y. Zhang, “Middle-infrared transmission enhancement through periodically perforated metal films,” Appl. Phys. Lett. 84(16), 2977–2979 (2004). [CrossRef]

11.

C. Genet, M. P. van Exter, and J. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Commum. 225(4-6), 331–336 (2003). [CrossRef]

12.

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). [CrossRef] [PubMed]

13.

M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66(19), 195105 (2002). [CrossRef]

14.

M. Sarrazin and J.-P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(1), 016603 (2003). [CrossRef] [PubMed]

15.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]

16.

Z. 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]

17.

E. D. Palik, Handbook of Optical Constants of Solids: Index (Academic, 1998).

18.

J. Braun, B. Gompf, G. Kobiela, and M. Dressel, “How holes can obscure the view: suppressed transmission through an ultrathin metal film by a subwavelength hole array,” Phys. Rev. Lett. 103(20), 203901 (2009). [CrossRef] [PubMed]

19.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

20.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007). [CrossRef] [PubMed]

21.

W. Dong, T. Hirohata, K. Nakajima, and X. Wang, “Near-field effect in the infrared range through periodic Germanium subwavelength arrays,” Opt. Express 21(22), 26677–26687 (2013). [CrossRef]

22.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

23.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991). [CrossRef] [PubMed]

OCIS Codes
(120.7000) Instrumentation, measurement, and metrology : Transmission
(130.3060) Integrated optics : Infrared
(240.6690) Optics at surfaces : Surface waves
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Plasmonics

History
Original Manuscript: September 18, 2013
Revised Manuscript: November 2, 2013
Manuscript Accepted: November 5, 2013
Published: November 13, 2013

Citation
Wei Dong, Toru Hirohata, Kazutoshi Nakajima, and Xiaoping Wang, "Enhanced and suppressed infrared transmission through germanium subwavelength arrays," Opt. Express 21, 28513-28522 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28513


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References

  1. T. W. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
  2. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature445(7123), 39–46 (2007). [CrossRef] [PubMed]
  3. W. Wu, A. Bonakdar, and H. Mohseni, “Plasmonic enhanced quantum well infrared photodetector with high detectivity,” Appl. Phys. Lett.96(16), 161107 (2010). [CrossRef]
  4. Y. Cui and S. He, “Enhancing extraordinary transmission of light through a metallic nanoslit with a nanocavity antenna,” Opt. Lett.34(1), 16–18 (2009). [CrossRef] [PubMed]
  5. S. C. Lee, S. Krishna, and S. R. Brueck, “Quantum dot infrared photodetector enhanced by surface plasma wave excitation,” Opt. Express17(25), 23160–23168 (2009). [CrossRef] [PubMed]
  6. T. Ishi, J. Fujikata, K. Makita, T. Baba, and K. Ohashi, “Si nano-photodiode with a surface plasmon antenna,” Jpn. J. Appl. Phys.44(12), L364–L366 (2005). [CrossRef]
  7. C.-Y. Chang, H.-Y. Chang, C.-Y. Chen, M.-W. Tsai, Y.-T. Chang, S.-C. Lee, and S.-F. Tang, “Wavelength selective quantum dot infrared photodetector with periodic metal hole arrays,” Appl. Phys. Lett.91(16), 163107 (2007). [CrossRef]
  8. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003). [CrossRef] [PubMed]
  9. T. Thio, H. Ghaemi, H. Lezec, P. Wolff, and T. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B16(10), 1743–1748 (1999). [CrossRef]
  10. Y.-H. Ye and J.-Y. Zhang, “Middle-infrared transmission enhancement through periodically perforated metal films,” Appl. Phys. Lett.84(16), 2977–2979 (2004). [CrossRef]
  11. C. Genet, M. P. van Exter, and J. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Commum.225(4-6), 331–336 (2003). [CrossRef]
  12. H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express12(16), 3629–3651 (2004). [CrossRef] [PubMed]
  13. M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B66(19), 195105 (2002). [CrossRef]
  14. M. Sarrazin and J.-P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.68(1), 016603 (2003). [CrossRef] [PubMed]
  15. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett.88(5), 057403 (2002). [CrossRef] [PubMed]
  16. Z. 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]
  17. E. D. Palik, Handbook of Optical Constants of Solids: Index (Academic, 1998).
  18. J. Braun, B. Gompf, G. Kobiela, and M. Dressel, “How holes can obscure the view: suppressed transmission through an ultrathin metal film by a subwavelength hole array,” Phys. Rev. Lett.103(20), 203901 (2009). [CrossRef] [PubMed]
  19. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124(6), 1866–1878 (1961). [CrossRef]
  20. T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature446(7135), 517–521 (2007). [CrossRef] [PubMed]
  21. W. Dong, T. Hirohata, K. Nakajima, and X. Wang, “Near-field effect in the infrared range through periodic Germanium subwavelength arrays,” Opt. Express21(22), 26677–26687 (2013). [CrossRef]
  22. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).
  23. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter44(16), 8565–8571 (1991). [CrossRef] [PubMed]

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