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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17876–17882
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Optical transmission anomalies in a double-layered metallic slit array

Koichi Akiyama, Keisuke Takano, Yuji Abe, Yasunori Tokuda, and Masanori Hangyo  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17876-17882 (2010)
http://dx.doi.org/10.1364/OE.18.017876


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Abstract

We theoretically predicted the existence of an anomalous optical transmittance dip, which must be observed for a metamaterial structure with two metallic slabs with cut-through slit arrays of a constant period d under a normal incident condition. By changing the relative lateral displacement l between the two slabs, the dip frequency varies across that of a so-called Rayleigh-Wood’s (RW) anomaly frequency. The mechanism of this anomaly is quite different from that of the RW anomaly and interpreted in terms of the interference between the propagating and evanescent waves. For the present double-layered system, furthermore, it is suggested that the RW anomaly vanishes for l = 0 and d/2. In experiments in the terahertz region, we observe that the fundamental features agree with these theoretical predictions.

© 2010 OSA

1. Introduction

Light scattering from sub-wavelength structures has been the subject of long standing interest. Especially periodic structures such as a grating support many wave phenomena depending on the periodic unit cell structure and periodicity. One of the most basic electromagnetic phenomena is a Rayleigh-Wood’s (RW) anomaly [1

1. R. W. Wood, “XLII. On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

3

3. L. Rayleigh, “III. Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

], in which rapid variations in the angle and frequency dependence of the scattering coefficients are manifested. In such anomalies, incident plane wave is diffracted in the direction parallel to the grating surface with exciting resonant surface waves, which plays an important role in the recently discovered and much debated phenomenon, extraordinary optical transmission (EOT) in periodically perforated metal films [4

4. 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]

6

6. M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. 67(8), 085415 (2003). [CrossRef]

]. To develop the potential application of light manipulation on a sub-wavelength scale by whole understanding of the EOT, there has been much effort to elucidate the underlying physics for the interplay between the incident (propagating) plane wave and the structurally induced surface (evanescent) waves [7

7. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]

9

9. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium: Unusual resonators: Plasmonics metamaterials, and random media,” Rev. Mod. Phys. 80(4), 1201–1213 (2008). [CrossRef]

]. In addition to the EOT, the transmission enhancement originated from the shape resonance [10

10. F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95(10), 103901 (2005). [CrossRef] [PubMed]

12

12. J. W. Lee, M. A. Seo, D. H. Kang, K. S. Khim, S. C. Jeoung, and D. S. Kim, “Terahertz electromagnetic wave transmission through random arrays of single rectangular holes and slits in thin metallic sheets,” Phys. Rev. Lett. 99(13), 137401 (2007). [CrossRef] [PubMed]

] and the cavity resonance of the wave guide modes [5

5. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999). [CrossRef]

] have been reported, which are induced by the interference between the surface wave and the modes in the local resonators [9

9. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium: Unusual resonators: Plasmonics metamaterials, and random media,” Rev. Mod. Phys. 80(4), 1201–1213 (2008). [CrossRef]

].

In this paper, we investigated novel optical transmission anomalies induced by the interference between the propagating and evanescent waves in a narrow air gap region sandwiched by metallic slit arrays. Theoretical consideration under a normal incident condition predicted the existence of an optical transmission dip, whose frequency varies across the RW anomaly frequency [1

1. R. W. Wood, “XLII. On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

,2

2. R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48(12), 928–936 (1935). [CrossRef]

,3

3. L. Rayleigh, “III. Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

,13

13. A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4(10), 1275–1297 (1965). [CrossRef]

] with changing the relative lateral position between the two metallic slabs. For the present double-layered configuration, moreover, it is suggested that the RW anomaly vanishes when the displacement between the two metallic slabs is 0 or a half of the slit pitch. In the experiment for the transmission spectra in the terahertz region [14

14. F. Miyamaru and M. Hangyo, “Anomalous terahertz transmission through double-layer metal hole arrays by coupling of surface plasmon polaritons,” Phys. Rev. B 71(16), 165408 (2005). [CrossRef]

], we observed the fundamental features of these predicted phenomena.

2. Transmission anomalies in a double-layered metallic slit array

We investigate the transmission properties for electromagnetic waves incident normally on a metamaterial structure with a narrow air gap sandwiched by two metallic slabs with the slit arrays of the same period, as schematically illustrated in Fig. 1(a)
Fig. 1 (a) Schematic illustration of the two metallic slabs with the cut-through slit arrays. The structures of the upper and lower slabs are identical with the same geometrical parameters of the slit width a, the slit periodicity d, and the height h. The lateral displacement and the gap length between the slabs are represented by l and s, respectively. (b) A cross-sectional view of the configuration for simulations, where h = for elimination of the Fabry-Pérot interferences by the metallic slit layers as dielectric materials.
. Here, a, d, and h are the slit width, the slit period, and the slab thickness, respectively, whereas the lateral displacement and the vertical air gap between the upper and lower slabs are represented by l and s, respectively. Here, under the normal incident configuration, a transverse magnetic (TM) mode, whose magnetic field is parallel to the x-axis, can propagates in both the slits and the air gap while a transverse electric (TE) mode decays and cannot penetrate below the cut-off frequency ( = c/a, where c is the light velocity in vacuum).

It was revealed that a metallic slab with the cut-through slit array may act as a dielectric material whose effective refractive index is determined by the width and periodicity of the slit array [15

15. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94(19), 197401 (2005). [CrossRef] [PubMed]

]. To extract phenomena inside the narrow air gap region by eliminating the Fabry-Pérot interference in the metallic slabs, we consider a case of h = ∞ as a cross-sectional view shown in Fig. 1(b). Here, im and rm are the amplitudes of the electric fields of the incident and reflected lights in the upper m-th slit, respectively, whereas tn is that of the transmitted light in the lower n-th slit. Under the normal incident condition, im is normalized to be unity (im = 1). Then the wave interactions between the upper (lower) m(n)-th and m’(n’)-th slits and between the upper m-th and lower n-th slits, represented by the dyadic Green’s functions Gmm'(nn')intra and Gmninter, respectively, are governed by the following equations [16

16. J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004). [CrossRef] [PubMed]

,17

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

],
m'(δmm'Gmm'intra)Am'+n'Gmn'interBn'=1,
(1)
m'Gnm'interAm'+n'(δnn'Gnn'intra)Bn'=0,
(2)
where δmm’ is the Kronecker’s delta, Am = 1 - rm and Bn = tn. Furthermore, the dyadic Green’s functions can be expressed by the site representation as follows,
Gmm'intra=p1Zpwp+wp1wpwp1<m|p><p|m'>,
(3)
Gmn'inter=p1Zp1wpwp1<m|p><p|n'>,
(4)
where |p> and |m> represent the p-th diffracted wave components of the incident light in the air gap region (a propagating wave component is given for p = 0) and the waveguide mode at the m-th slit, respectively. wp(=eiαps) and Zp(=k/αp) specify a phase propagation factor and an effective impedance of the slit, respectively. αp(=k2(2πp/d)2) is a wave vector along the z direction and k is the vacuum wave number. Then, we can describe the complex transmittance amplitude t ( = Bn) by
t=Ginter(1Gintra)2(Ginter)2,
(5)
where Gintra=mGm0intra, Ginter=nG0ninter.

Figure 2
Fig. 2 (a) Transmission spectra calculated for the various lateral displacement l values, and (b) electric field distributions for the Rayleigh-Wood’s anomaly with l = d/4 and (c) for the new anomaly with l = 0. The slit width a and the gap length s are fixed at d/6 and d/4, respectively.
indicates the transmittance as a function of the wave frequencyν calculated by use of Eq. (5) for various l values. Here, it should be noticed that the dip at ν = c/d, which should be referred as the RW anomaly [1

1. R. W. Wood, “XLII. On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

,2

2. R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48(12), 928–936 (1935). [CrossRef]

,3

3. L. Rayleigh, “III. Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

] and observed usually for a grating interface connecting to an open space [13

13. A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4(10), 1275–1297 (1965). [CrossRef]

], disappears for l = 0 and d/2. Moreover, we find another dip that moves with changing l (new anomaly). The distributions of the electric fields are quite different for the RW and new anomalies. For the RW anomaly, the propagating wave along z direction completely disappears and the remaining evanescent wave forms a standing wave along y direction with a period of d with its nodes are located at the upper entrance of the slits (Fug.2(b)). Then the transmission dip is induced. On the other hand the electric field distributions shows spatial decay profiles from the upper entrance slits along both direction for the new anomaly, leading to another transmission dip [Fig. 2(c)]. These results strictly show that the mechanism of the transmission anomaly is quite different between the two cases.

In Fig. 3
Fig. 3 Transmission dip frequencies as a function of the lateral displacement l for the various gap lengths s. All lines cross the RW anomaly frequency of c/d at l = d/4.
, we summarized these moving dip frequencies as a function of l for various s values. In the present case of s < d/2 [18

18. The reverse cases that the dip frequency is higher than the RW anomaly one for l < d/4 and lower for l > d/4 may occur for s > d/2.

], the dip frequencies are lower than c/d (the RW anomaly frequency) for l < d/4 and approach to c/d with l → d/4 for all the s values.

Here, we explore the physical mechanism of the moving anomaly. To extract the anomaly condition for the transmittance t = 0, we calculated the amplitudes and the phases of G inter [a numerator of Eq. (5)] for the propagating and evanescent waves. The propagating wave corresponds to p = 0 whereas the sum of the contributions of G inter with p≠0 gives the amplitude for the evanescent waves. Figure 4
Fig. 4 Amplitudes and phases of G inter for the propagating (p = 0) and evanescent (p≠ 0) waves as a function of the frequency for the various l values with s = d/4.
shows the amplitudes and the phases as a function of the frequency for various l values for s = d/4. It should be noticed that there are two crossing points for the amplitudes of the propagating and evanescent waves for each s values, one is observed for ν < c/d and another for ν > c/d. As the contribution of the propagating and evanescent waves is canceled out owing to their opposite sign at the amplitude crossing point for ν < c/d, the lower frequency crossing point may give the anomaly frequencies of t = 0. On the other hand, for another crossing point for ν > c/d, the propagating and evanescent waves have the same sign and thus t ≠0. In addition we can understand from the figure that the frequency of the crossing point approaches c/d as the displacement l approaches d/4. These characteristics are the same for all the s values. Thus we conclude that the destructive interference between the propagating and evanescent waves explains the features of Figs. 2 and 3. Here we can also understand from the figure and Eq. (5) that the transmission anomaly (t→0) occurs at the RW frequency because Ginter →∞ for the evanescent waves.

The interference between the propagating and evanescent waves plays a crucial role in optical processes in the metallic slit array. When we consider a single interface of an air-slab with the slit array system, a quasi surface (evanescent) wave is formed from the diffracted incident wave by the surface corrugation. The resonant evanescent wave is enhanced around the Rayleigh wavelength λ R (corresponding to d in our case) and interferes with the undiffracted propagating waves, giving rise to both the transmission minima and maxima around λ R, known as the Fano interference profiles [6

6. M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. 67(8), 085415 (2003). [CrossRef]

,19

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

]. The situation becomes quite different by introducing cascaded structures with the dual interface of the air-slab with the slit array. Our obtained result indicates that in such structures, even non-resonant evanescent waves far from λ R can interfere with the propagating wave, leading to a transmission minima. Similar anomalous transmission was also observed by numerical simulations where a simple explanation that such anomalous behavior is induced by destructive interference of surface plasmon-polaritons emitted by the neighboring slit apertures was given [20

20. C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78(7), 075406 (2008). [CrossRef]

].

In addition to the moving anomaly, we consider the physical mechanism of the unusual behavior of the RW anomaly in the present system, i.e., the calculated result that the RW anomaly vanishes for l = 0 and d/2. As the spatial decay constant of the field intensity of the evanescent wave is very small along the gap direction at this frequency, G inter is less dependent on the gap distance s. Thus G inter becomes equal to G intra under the symmetrical conditions of l = 0 and d/2, leading to the transmittance t = - G intra/(1 - 2G intra) in Eq. (5). Hence the transmittance t keeps finite value even when G intra → ∞ at the RW frequency, i.e. the fact that the RW anomaly vanishes for l = 0 and d/2. In this case, the electric fields form a standing wave as the same for l ≠ 0, d/2, in which the anti-nodes locate at both the upper entrance and the lower exit slits, leading to finite transmission.

3. Experimental results in the terahertz frequency region

Lastly we present preliminary experimental results. For measurements of the transmission spectra, we prepared the double-layered cut-through slit arrays made of brass, where a = 160 μm, d = 600 μm, and h = 1300 μm [21

21. M. Hangyo, K. Akiyama, K. Takano, F. Miyamaru, K. Shibuya, Y. Abe, Y. Tokuda, and H. Miyazaki, “Artificial dielectric metamaterials made of metals and their terahertz-wave propagation properties,” Proc. SPIE 7935, 53 (2009).

], while s is set at 80 μm. Here, the RW anomaly frequency ( = c/d) should be 0.5 THz corresponding to c/d. The measurements were performed in the terahertz frequency region of 0.05 ~0.65 THz by the terahertz time-domain spectroscopy (THz-TDS) [14

14. F. Miyamaru and M. Hangyo, “Anomalous terahertz transmission through double-layer metal hole arrays by coupling of surface plasmon polaritons,” Phys. Rev. B 71(16), 165408 (2005). [CrossRef]

]. In Fig. 5(a)
Fig. 5 (a) Transmission spectra measured by the terahertz time domain spectroscopy for the double-layered cut-through slit arrays of a = 160 μm, d = 600 μm, and h = 1300 μm. The gap length s is fixed at 80 μm whereas the lateral displacement l is from 0 to 300 μm ( = d/2). The black circles are the transmission spectra of the single cut-through slits. (b) Transmission spectra in which the Fabry-Pérot interference and reflection effect at the boundary between air and the double-layered cut-through slits are removed by the procedure described in the text.
, we show transmission spectra for l from 0 to 300 μm ( = d/2). The transmission spectra show the periodic oscillations and sharp dips. These spectra are composed of the moving anomaly discussed above, Fabry-Pérot interference in the cut-through slit arrays, and reflection effects at the upper and lower boundaries between air and the double-layered cut-through arrays.

The Fabry-Pérot interference can be eliminated with the time-domain signal processing in the THz-TDS. The multi-pulses due to the multi-reflection in the slit structures can be eliminated by using the time-domain data until 15.5 ps after the main pulses, which makes the frequency resolution 0.065 THz (0.13c/d). This procedure also eliminates the interference between the upper and lower surfaces of the double-layered cut-through slit arrays and the air-gap region. Then, the reflection effects of the boundary between the air and double-layered cut-through arrays can be removed by normalizing the transmission spectra with that of the single cut-through slit array. Finally, the effects of the moving anomaly are separated from the transmission spectra in Fig. 5(a) by these procedures as shown in Fig. 5(b).

4. Conclusion

In summary, theoretical investigations predicted the existence of a new anomaly below the RW anomaly frequency and disappearance of the RW anomaly for two metallic slabs with the cut-through slit arrays, and the experiments in the terahertz regions strongly support the theoretical predictions. The mechanism of the observed anomaly is quite different from that of the RW anomaly and interpreted in terms of the interference between the propagating and evanescent waves. At present, we are planning more detailed experiments, whose results will be presented elsewhere.

Acknowledgment

This work was supported partly by the Japan Society for the Promotion of Science (JSPS). Grant-in-Aid for Scientific Research A (No. 20246022).

References and links

1.

R. W. Wood, “XLII. On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

2.

R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48(12), 928–936 (1935). [CrossRef]

3.

L. Rayleigh, “III. Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

4.

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]

5.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999). [CrossRef]

6.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. 67(8), 085415 (2003). [CrossRef]

7.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]

8.

F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007). [CrossRef]

9.

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium: Unusual resonators: Plasmonics metamaterials, and random media,” Rev. Mod. Phys. 80(4), 1201–1213 (2008). [CrossRef]

10.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95(10), 103901 (2005). [CrossRef] [PubMed]

11.

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]

12.

J. W. Lee, M. A. Seo, D. H. Kang, K. S. Khim, S. C. Jeoung, and D. S. Kim, “Terahertz electromagnetic wave transmission through random arrays of single rectangular holes and slits in thin metallic sheets,” Phys. Rev. Lett. 99(13), 137401 (2007). [CrossRef] [PubMed]

13.

A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4(10), 1275–1297 (1965). [CrossRef]

14.

F. Miyamaru and M. Hangyo, “Anomalous terahertz transmission through double-layer metal hole arrays by coupling of surface plasmon polaritons,” Phys. Rev. B 71(16), 165408 (2005). [CrossRef]

15.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94(19), 197401 (2005). [CrossRef] [PubMed]

16.

J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004). [CrossRef] [PubMed]

17.

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

18.

The reverse cases that the dip frequency is higher than the RW anomaly one for l < d/4 and lower for l > d/4 may occur for s > d/2.

19.

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

20.

C. Cheng, J. Chen, D. Shi, Q. Wu, F. Ren, J. Xu, Y. Fan, J. Ding, and H. Wang, “Physical mechanism of extraordinary electromagnetic transmission in dual-metallic grating structures,” Phys. Rev. B 78(7), 075406 (2008). [CrossRef]

21.

M. Hangyo, K. Akiyama, K. Takano, F. Miyamaru, K. Shibuya, Y. Abe, Y. Tokuda, and H. Miyazaki, “Artificial dielectric metamaterials made of metals and their terahertz-wave propagation properties,” Proc. SPIE 7935, 53 (2009).

22.

K. Akiyama, K. Shibuya, K. Takano, Y. Abe, Y. Tokuda, and M. Hangyo, “Variable effective refractive index of a gap layer between two cut-through metal slit array metamaterial slabs,” in Proceedings of the 3rd International Congress on Advanced Electromagnetic Materials in Microwaves and Optics, (London, UK, 2009), pp. 436–437.

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(240.6690) Optics at surfaces : Surface waves
(250.5403) Optoelectronics : Plasmonics
(300.6495) Spectroscopy : Spectroscopy, teraherz

ToC Category:
Diffraction and Gratings

History
Original Manuscript: June 7, 2010
Revised Manuscript: July 12, 2010
Manuscript Accepted: July 18, 2010
Published: August 4, 2010

Citation
Koichi Akiyama, Keisuke Takano, Yuji Abe, Yasunori Tokuda, and Masanori Hangyo, "Optical transmission anomalies in a double-layered metallic slit array," Opt. Express 18, 17876-17882 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17876


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References

  1. R. W. Wood, “XLII. On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
  2. R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48(12), 928–936 (1935). [CrossRef]
  3. L. Rayleigh, “III. Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
  4. 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]
  5. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999). [CrossRef]
  6. M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. 67(8), 085415 (2003). [CrossRef]
  7. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]
  8. F. J. García de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007). [CrossRef]
  9. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium: Unusual resonators: Plasmonics metamaterials, and random media,” Rev. Mod. Phys. 80(4), 1201–1213 (2008). [CrossRef]
  10. F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95(10), 103901 (2005). [CrossRef] [PubMed]
  11. 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]
  12. J. W. Lee, M. A. Seo, D. H. Kang, K. S. Khim, S. C. Jeoung, and D. S. Kim, “Terahertz electromagnetic wave transmission through random arrays of single rectangular holes and slits in thin metallic sheets,” Phys. Rev. Lett. 99(13), 137401 (2007). [CrossRef] [PubMed]
  13. A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4(10), 1275–1297 (1965). [CrossRef]
  14. F. Miyamaru and M. Hangyo, “Anomalous terahertz transmission through double-layer metal hole arrays by coupling of surface plasmon polaritons,” Phys. Rev. B 71(16), 165408 (2005). [CrossRef]
  15. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94(19), 197401 (2005). [CrossRef] [PubMed]
  16. J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant transmission of light through finite chains of subwavelength holes in a metallic film,” Phys. Rev. Lett. 93(22), 227401 (2004). [CrossRef] [PubMed]
  17. F. J. Garcia-Vidal, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]
  18. The reverse cases that the dip frequency is higher than the RW anomaly one for l < d/4 and lower for l > d/4 may occur for s > d/2.
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