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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 14 — Jul. 2, 2012
  • pp: 16092–16103
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Transmission phase control by stacked metal-dielectric hole array with two-dimensional geometric design

Takayuki Matsui, Hideki T. Miyazaki, Atsushi Miura, Tsuyoshi Nomura, Hisayoshi Fujikawa, Kazuo Sato, Naoki Ikeda, Daiju Tsuya, Masayuki Ochiai, Yoshimasa Sugimoto, Masanori Ozaki, Masanori Hangyo, and Kiyoshi Asakawa  »View Author Affiliations


Optics Express, Vol. 20, Issue 14, pp. 16092-16103 (2012)
http://dx.doi.org/10.1364/OE.20.016092


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Abstract

Transmission phase control is experimentally demonstrated using stacked metal-dielectric hole arrays with a two-dimensional geometric design. The transmission phase varies drastically with small frequency shifts due to structural resonances. Laterally propagating surface plasmon polaritons excited by the periodic hole array roughly determine the resonance frequency, whereas localized resonances in each hole determine the dispersion. The transmission phase at various frequencies is directly evaluated using interferometric microscopy, and the formation of an inclined wavefront is demonstrated using a beam steering element in which the hole shapes gradually change in-plane from square to circular.

© 2012 OSA

1. Introduction

Since Ebbesen et al. published their pioneering work [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]

] describing extraordinary optical transmission (EOT) through metallic films perforated by nanohole arrays, this unique transmission phenomenon has been intensively studied over the past decade [2

2. 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]

4

4. D. Inoue, A. Miura, T. Nomura, H. Fujikawa, K. Sato, N. Ikeda, D. Tsuya, Y. Sugimoto, and Y. Koide, “Polarization independent visible color filter comprising an aluminum film with surface-plasmon enhanced transmission through a subwavelength array of holes,” Appl. Phys. Lett. 98, 093113 (2011). [CrossRef]

]. The mechanism of EOT was first explained in terms of laterally propagating surface plasmon polaritons (SPPs) [5

5. S. A. Maier, Plasmonics: Fundamentals and Applications (Splinger-Verlag, 2007).

], which are resonantly excited by the periodic nanohole array along the metal-dielectric interface [6

6. 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]

]. While lateral SPPs play an important role in EOT, the contribution of resonant tunneling of light through each hole has also been considered [7

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

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]

]. The EOT mechanism is now recognized as a combination of both laterally propagating SPPs (and cylindrical waves for thermal infrared frequencies) excited by the periodic hole array and localized hole resonance [10

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

].

While EOT is a phenomenon involving transmission through a single two-dimensional hole array (2DHA) film, structures comprising stacked metal-dielectric films perforated by nanohole arrays, known as fishnet structures, have also been studied and have been reported to exhibit a negative refractive index [11

11. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

13

13. 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 (London) 455, 376–380 (2008). [CrossRef]

]. The origin of such a negative index has been mainly attributed to both the magnetic response of the double-plate pairs and the electric response of the metal wires, which give rise to a negative permeability and a negative permittivity, respectively. In this interpretation, the structure is treated as an effective medium [14

14. S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4929 (2005). [CrossRef] [PubMed]

, 15

15. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006). [CrossRef] [PubMed]

], with an effective refractive index that is determined by a parameter retrieval method [16

16. X. Chen, T. M. Grzegorczyk, B-I. Wu, J. Pacheco Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004). [CrossRef]

, 17

17. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005). [CrossRef]

]. However, since optical fishnet structures have dimensions comparable to the wavelength of the incident light, instead of using an effective-medium approach, the origin of the negative index was discussed explicitly in terms of gap SPPs that were excited between the metal films [18

18. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. 101, 103902 (2008). [CrossRef] [PubMed]

21

21. M. Iwanaga, “In-plane plasmonic modes of negative group velocity in perforated waveguides,” Opt. Lett. 36, 2504–2506 (2011). [CrossRef] [PubMed]

]. The approach in Ref. [18

18. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. 101, 103902 (2008). [CrossRef] [PubMed]

,19

19. R. Ortuño, C. García-Meca, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays,” Phys. Rev. B 79, 075425 (2009). [CrossRef]

], which focused on the interaction between the gap SPPs and the periodic nanohole array, presented a unified view of transmission through a single and stacked metal hole array (SHA).

In addition to the excitation of lateral SPPs by the periodicity, the effects of the hole shape have also been discussed for both 2DHAs and SHAs. In the 2DHA case, rectangular holes were found to exhibit higher normalized transmittance than circular ones due to localized resonances that occur in each hole [22

22. K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary tTransmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett. 92, 183901 (2004). [CrossRef]

]. In the SHA case also, rectangular holes were found to be more effective in balancing a higher transmittance and a negative index [14

14. S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4929 (2005). [CrossRef] [PubMed]

] than circular holes [23

23. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express 15, 11536–11541 (2007). [CrossRef] [PubMed]

]. In both cases, the transmission properties for circular holes are dominated by lateral SPPs excited by the periodic nanohole array, whereas for rectangular holes, both SPPs and localized resonances in each hole contribute to the transmission.

Here, the relationship is discussed between the transmission phase and the geometric design of a SHA, i.e., the periodicity and the hole shape. Since structural resonances inherently involve phase shifts, both the phase and the amplitude of the transmitted wave undergo large changes. Hence, both 2DHA and SHA have potential as thin phase control elements. In this paper, the applicability of a SHA to a transmission phase control element is experimentally elucidated. The transmission phase is controlled by the shape of the holes. A beam steering element is then proposed that realizes an inclined wavefront by the use of gradually changing hole shapes.

2. SPP excitation by periodic holes and its effect on transmission phase

In this section, the resonance frequencies are first extracted from the dispersion diagram of laterally propagating SPPs and the grating momentum of the nanohole array. Numerical calculations of the transmission amplitude and phase for a SHA around the resonance frequencies are then discussed.

The SHA unit cell structure used for the numerical calculations is depicted in the inset of Fig. 1(b). An alternating stack of six fused silica (SiO2) and five aluminum (Al) layers on a SiO2 substrate is perforated by nanoholes periodically arranged in a square lattice. The thicknesses of the SiO2 and Al layers are 80 nm and 20 nm, respectively, and the periodicity a of the unit cell is 1000 nm. The hole shape is either circular with a diameter of a/2, or square with sides of a/2.

Fig. 1 (a) Dispersion relation for a single interface SPP and a gap SPP. The green line represents the reciprocal lattice vector Gx. (b) Numerically calculated transmittance through the SHA. The geometry of the SHA is depicted in the inset. The model in the inset is for the case of the square-hole SHA. The sidewall slope of the holes is assumed to be 90 degrees to the substrate plane in the calculation. (c) Numerically calculated transmission phase relative to that for a vacuum layer with the same thickness as the SHA. Gray horizontal solid lines in (a) to (c) represent the intersections between laterally propagating SPPs and the grating momentum, ωsingle and ωgap. The rightmost axis shows the normalized frequency a/λ for a = 1000 nm. The arrow in (b) denotes the transmission peak around ωgap for the square-hole SHA.

Since the SHA structure includes ten metal-dielectric interfaces, the dispersion diagram for the laterally propagating SPPs should exhibit a number of modes. However, at the low frequencies discussed here, they can be well represented by just two types of SPP modes [19

19. R. Ortuño, C. García-Meca, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays,” Phys. Rev. B 79, 075425 (2009). [CrossRef]

]: a single-interface SPP [5

5. S. A. Maier, Plasmonics: Fundamentals and Applications (Splinger-Verlag, 2007).

] and a gap SPP [24

24. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]

26

26. Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007). [CrossRef]

]. These dispersion characteristics are analytically calculated and plotted in Fig. 1(a) at wavelengths of around 1.5 μm. The black solid line is the dispersion curve for a single Al/SiO2 interface SPP and the black broken line is for an Al/SiO2/Al gap SPP.

Coupling between a SPP and incident light is achieved by adding a grating momentum derived from the periodic hole array pattern, and expressed as:
ksp=kinc//+iGx+jGy,
(1)
where k⃗sp is the wave vector of the SPP, k k⃗inc// is the in-plane incident wave vector, i and j are integers, and G⃗x and G⃗y are the reciprocal lattice vectors of the periodic holes. For a square lattice, |Gx| and |Gy| become 2π/a and are represented by a green solid line in Fig. 1(a). For the case of x-polarized normal incidence, the diffraction wave corresponding to (i, j) = (1,0) couples with the SPPs at the intersections of the SPP dispersion curves and the green line, i.e., at around 0.85 eV for a single interface SPP and around 0.75 eV for a gap SPP. The transmission amplitude and phase around the frequencies corresponding to these resonances will now be discussed by comparing the results of numerical calculations. The numerical simulations were performed using a commercial finite element software (Comsol Multiphysics with RF module). The frequency dependence of the permittivity of Al is assumed to follow the Lorentz-Drude model [27

27. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998). [CrossRef]

]. The refractive index of the fused silica is set at 1.45. Periodic boundary conditions are defined for the SHA unit cell structure in the x and y directions.

From Fig. 1 (b), bimodal transmittance peaks are observed around resonance frequencies associated with energies of 0.85 eV and 0.75 eV, which correspond to the structural resonance frequencies (horizontal gray solid lines). Hereafter, frequency is expressed by a normalized value a/λ, where λ is the wavelength in vacuum; therefore, the above resonance frequencies correspond to 0.68 and 0.60, respectively. In this paper, these two resonance frequencies are referred to as ωsingle and ωgap, respectively. The higher frequency transmittance band at ωsingle corresponds to the EOT region and the lower frequency one at ωgap to the negative index region [19

19. R. Ortuño, C. García-Meca, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays,” Phys. Rev. B 79, 075425 (2009). [CrossRef]

].

The transmission phase through the SHA is represented in Fig. 1(c). In this article, the phase is expressed as the phase shift relative to that in a vacuum layer with the same thickness as the target structure, so as to correspond straightforwardly to the measured quantity in Sec. 5. Since a time dependence of exp(−iωt) is assumed here, a negative phase means that the structure apparently behaves as a lower index medium than vacuum.

According to this definition, the difference between the phases of the transmitted waves with and without the SHA structure is plotted in Fig. 1(c). The ambiguity of 2π in this phase was removed on the basis of the phase profile of the electromagnetic fields in the z direction. In Fig. 1(c), around the resonance frequencies, the transmission phase drastically changes as expected. Since this occurs in high transmittance regions, the SHA can be used as a phase control element.

3. Nanofabrication of SHA structures and transmission measurements

SHA structures were fabricated using multilayer deposition, electron beam lithography, and layer-by-layer dry etching. A stacked layer structure was first produced on a SiO2 substrate using magnetron sputtering. The structure comprised alternating layers of six SiO2 films (80 nm thick) and five Al films (20 nm thick). The total thickness of the stacked structure was 580 nm. A positive electron beam resist (ZEON, ZEP-520A) was spun onto the film stack and four SHA patterns were defined using an electron beam writer (Elionix, ELS-7000). These patterns had a periodicity a of 1000 nm or 900 nm, and either circular holes with diameters of a/2 or square holes with sides of a/2. The reason for preparing two different periodicities was to allow important frequency regions to be covered using a wavelength-tunable laser with a limited frequency range (λ = 1470 – 1540 nm); the frequency regions of a/ λ = 0.65 – 0.68 and a/λ = 0.58 – 0.61 can be examined for a = 1000 nm and a = 900 nm, respectively. Therefore, both ωsingle and ωgap are covered by the single laser. Finally, holes were produced by dry etching using a resist pattern, in a layer-by-layer manner. Reactive ion etching and inductively coupled plasma etching were used for etching the SiO2 and Al layers, respectively. Figure 2 shows top and cross-sectional view scanning electron microscopy (SEM) images of the fabricated SHAs. From the cross-sectional image, the sidewalls are seen to be non-vertical, having angles of about 80 degrees to the substrate plane. This is due to imperfect anisotropy in the dry etching process for the SiO2 layers.

Fig. 2 Scanning electron micrographs of fabricated SHAs with a = 1000 nm. (a) Top view of the square-hole SHA, (b) top view of the circular-hole hole SHA, and (c) cross-sectional view of the square-hole SHA. The layer on top of the SHA in (c) is a residue from the dry etching process, which is removed before the optical measurements. All scale bars are 1 μm.

The transmission amplitude and phase were measured using the optical microscope system illustrated in Fig. 3. Transmission spectra (squared amplitude) were obtained with a microscopic spectrometer (λ = 900 – 1700 nm) using an incoherent light source.

Fig. 3 Experimental setup based on a near-infrared microscope (Olympus BX-51IR). LD, external-cavity wavelength-tunable diode laser (New Focus Velocity TLB-6326, λ = 1470–1545 nm); W, achromatic half-wave plate; ND, neutral density filter; BS, pellicle beam splitter; S, sample; OL, objective lens (Olympus LMPlanIR, 20X, NA=0.40, NA: numerical aperture); L, achromatic doublet lens (f = 40.0 mm, f: focal length); IL, imaging lens (f = 180.0 mm); C, near-infrared camera (Hamamatsu Photonics, C10633-13, λ = 900 – 1700 nm); X, halogen lamp; PL, polarizer; AL, analyzer; F, visible cut filter; OF, optical fiber (NA = 0.2, diameter: 400 μm); SM, spectrometer (Lambda Vision, TFCAM-7000F/NIR, λ = 900 – 1700 nm).

In order to measure the dependence of the transmission phase on frequency, interferometric microscopy was used. This microscope system consists of a Mach-Zehnder interferometer integrated with an optical microscope and an external-cavity wavelength-tunable diode laser (λ = 1470 – 1545 nm). The beam polarization is first rotated using an achromatic half-wave plate, and the beam is then divided into a sample path and a reference path by a pellicle beam splitter. The sample beam passes through the sample and an objective lens. The reference beam passes through a neutral density filter and an achromatic doublet lens. The wavefront curvature of the reference beam is precisely matched with the sample beam by moving the achromatic doublet lens in order to obtain a flat fringe pattern suitable for phase measurement [28

28. M. B. Sinclair, M. P. de Boer, and A. D. Corwin, “Long-working-distance incoherent-light interference microscope,” Appl. Opt. 44, 7714–7721 (2005). [CrossRef] [PubMed]

]. After the sample and reference paths are recombined by a pellicle beam splitter, the combined beam passes through an imaging lens, and finally forms interferometric images that are viewed by a near-infrared camera.

4. Transmission amplitude for SHA

Figure 4 shows calculated and measured transmission spectra for square- and circular-hole SHAs. The horizontal axis represents transmittance, and the vertical axis is the normalized frequency a/λ. In the numerical calculations, a sidewall slope of 80 degrees is assumed.

Fig. 4 Calculated (calc.) and measured (meas.) transmission spectra for square- and circular-hole SHAs. Symbols and solid lines represent calculated and measured results, respectively. Blue represents the results for square-hole SHAs, and orange for circular-hole SHAs. ωsingle and ωgap are shown by gray horizontal lines. The arrow denotes the transmission peak around ωgap for the square-hole SHA.

Numerical calculations were carried out for a = 1000 nm and a = 900 nm. Since the dependence of the transmittance on a/λ was similar for both a values, only the results for a = 1000 nm are represented in Fig. 4. Note that the structures for a = 1000 nm and a = 900 nm are not perfectly equivalent to each other for the following two reasons: First, the thickness of Al and SiO2 layers remained the same for SHAs with different a values. Second, the permittivity of Al is dependent on λ rather than a/λ. Nonetheless, transmission through the SHAs is dominated by a/λ.

By comparing Fig. 4 with Fig. 1(b), the effects of a sloped sidewall can be discussed. It can be seen that for the square-hole SHA, the transmission peak around ωgap, marked by the arrows in the figures, is strongly diminished by the sloped sidewall. In contrast, for the circular-hole SHA, the transmission around ωgap is somewhat enhanced. This difference is presumably due to the fact that transmission through square-hole SHAs is more strongly affected by hole resonance [22

22. K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary tTransmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett. 92, 183901 (2004). [CrossRef]

, 23

23. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express 15, 11536–11541 (2007). [CrossRef] [PubMed]

]. A sidewall slope would prevent such hole resonance because the square holes in each layer would have different dimensions. In addition, around ωsingle, the transmittance for the circular-hole SHA showed a four-fold increase in the case of a sloped sidewall.

As seen in Fig. 4, there is reasonably good agreement between the measured results for a = 1000 nm and a = 900 nm, as was expected. Hence, the fact that the transmission spectra are dominated by a/λ has also been experimentally confirmed. While the calculated and measured results exhibit relatively similar behavior around ωsingle, a considerable discrepancy exists around ωgap. This is because around ωsingle, the spectral shape is determined by the Rayleigh minimum at a/λ = 0.70 [6

6. 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]

]. However, around ωgap, the transmittance spectrum is affected by the gap SPP, which is strongly dependent on the precise film thickness, permittivity of the metal, and sidewall slope.

5. Transmission phase for SHA

In order to determine the transmission phase for the SHA using interferometric microscopy, a standard region with a known dispersion relation and thickness should be prepared close to the SHA. Therefore, a bare substrate region that partially overlaps the SHA was additionally fabricated using focused ion beam (FIB) milling, and the phase of the light transmitted through the circular- and square-hole SHAs relative to that through the substrate region was evaluated over a range of frequencies. Figure 5(a)–(c) depicts the additional FIB process used.

Fig. 5 Additional fabrication process and result of interferometric observation. (a) – (c) FIB milling process. (a) Left side, stacked Al/SiO2 layers; Right side, SHA. (b) FIB milling of overlap region. (c) Left side, stacked Al/SiO2 layers; Center, standard region (S1) and overetched region (S2); Right side, SHA region (S3). (d) SEM image of fabricated structure. Scale bar is 5 μm. (e) Transmission interferometric image of square-hole SHA with a = 1000 nm at λ = 1530 nm. Scale bar is 150 μm. Another etched area with the same depth as the S1 region is also seen to the right of the SHA region (S3).

The area, which overlaps both the SHA and a surrounding region of the stacked Al/SiO2 layers, is processed by FIB milling until the substrate in the stacked layer region becomes fully exposed. Since the etching rate of the SHA is higher than that of the stacked Al/SiO2 layers, three different regions are obtained (Fig. 5(c), (d)): S1, standard region; S2, overetched region; S3, SHA region. The depth of the S1 region is estimated to be approximately 580 nm, which is consistent with the thickness of the SHA structure, and the S2 region is 700 nm deeper than the S1 region. Figure 5(e) shows a typical interferometric image, with bright and dark fringes superimposed on the normal microscopic image. The separation between adjacent bright fringes represents a phase difference of 2π. The fringe displacement at the border of adjacent regions indicates that a phase difference exists between them.

The transmission phase in the Sj region relative to that in the Si region (i, j = 1,2,3) can be expressed as:
Δϕji=2πΔmji+2πqji,
(2)
where Δmji is the ratio of the fringe displacement between Sj and Si to the distance between adjacent fringes, and qji is an integer.

Here, the phase difference between S2 and S1 should be described as Δϕ12, since a 700-nm-thick SiO2 layer is placed in the S1 region instead of a vacuum layer with the same thickness in the S2 region. Since the optical path length difference is known, q12 is unambiguously determined to be 0. From the sign of Δm12, an upward fringe displacement was confirmed to represent a lower apparent refractive index; in other words, faster light propagation.

The transmission phase of the SHA is measured as the phase difference Δϕ31 between S3 and S1. Since reasonable agreement was obtained between the calculated and experimental results for the transmission amplitude, particularly around ωsingle, q31 was also determined to be 0 on the basis of the numerically obtained field distribution inside the hole of the SHA.

Figure 6 shows the calculated and measured transmission phase through the SHA. In the numerical calculations, a sidewall slope of 80 degrees is again assumed. The horizontal axis represents the transmission phase relative to a vacuum layer with the same thickness as the SHA (i.e., the phase difference of the S3 region with respect to the S1 region), and the vertical axis is the normalized frequency a/λ. As a reference, the gray dashed lines represent the transmission phase through a structure with the refractive indices n indicated in the figure and with the same thickness as the SHA. For example, the gray dashed line labeled n = 1.45 represents the transmission phase through a 580-nm-thick SiO2 layer.

Fig. 6 (a) Calculated (calc.) and measured (meas.) transmission phase for square- and circular-hole SHAs relative to a vacuum layer with the same thickness as the SHAs. All notations, symbols and lines are the same as in Fig. 4. (b)–(e) Transmission interferometric images of SHA. Corresponding data points are indicated in (a). Display ranges of the interferometric images are the same as in Fig. 5(e). (b) circular-hole SHA with a = 1000 nm at λ = 1475 nm. (c) square-hole SHA with a = 1000 nm at λ = 1540 nm. (d) circular-hole SHA with a = 1000 nm at λ = 1540 nm. (e) circular-hole SHA with a = 900 nm at λ = 1530 nm. Gray horizontal solid lines represent ωsingle and ωgap. Gray dashed lines show the transmission phase through media with various apparent refractive indices n (a = 1000 nm and d = 580 nm).

The numerical calculations for a = 1000 nm and a = 900 nm also revealed that the transmission phase is mainly determined by the normalized frequency a/λ, as was also confirmed for the amplitude. Thus, only the calculation results for a = 1000 nm are shown in Fig. 6. The effects of the sidewall slope are also similar to those for the transmission amplitude. The phase around ωsingle exhibits a weak dependence on the sidewall angle for both square- and circular-hole SHAs. However, around ωgap, although the phase for the circular-hole SHA is less sensitive to the sidewall slope, that for the square-hole SHA is strongly influenced by it.

There is reasonably good agreement between the calculated and measured results in the ωsingle region (a/λ = 0.65 – 0.68). While the the square- and circular-hole SHAs exhibit similar phase values at a/λ = 0.68, the phase for the circular-hole SHA changes more quickly and the phase difference between the two types of SHA increases monotonically as the frequency decreases to a/λ = 0.65. Hence, the dispersion characteristics of SHAs can be controlled by changing the hole shape. However, in the low frequency region around ωgap, a large discrepancy exists between the calculated and experimental results for both hole shapes due to the sensitive nature of the gap SPP. As a result, no clear dependence of the phase on the hole shape could be experimentally confirmed around ωgap.

In addition to the ability to control the transmission phase using the hole shape, one of the most important findings in Fig. 6 is the extraordinary phase values. All the phase values obtained both by calculation and experiment were negative throughout the entire frequency range investigated. This implies that both the square- and circular-hole SHAs apparently function as media with a refractive index lower than that of free space. Even transmission phase equivalent to a negative index medium, below the gray dashed line with n = 0.0 in Fig. 6, has been experimentally observed below a/λ = 0.62. For SHA structures around ωgap, a negative refractive index has been extensively demonstrated [11

11. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

13

13. 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 (London) 455, 376–380 (2008). [CrossRef]

], which is consistent with the results obtained in the present study.

However, in this paper, no further discussion will be carried out concerning the effective index or left-handedness. This is firstly because no rigorous values for the effective refractive index can be determined since the amplitude and phase of the reflection cannot be evaluated using the present setup [16

16. X. Chen, T. M. Grzegorczyk, B-I. Wu, J. Pacheco Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004). [CrossRef]

, 17

17. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005). [CrossRef]

]. In addition, the large discrepancy between the measured and calculated results around ωgap indicates that the quality of the present samples is insufficient for a detailed analysis.

Rather than considering the effective index, Beruete et al. directly investigated the phase evolution (dispersion) of transmitted waves through SHAs in the millimeter waveband [29

29. M. Beruete, M. Sorolla, and I. Campillo, “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express 14, 5445–5455 (2006). [CrossRef] [PubMed]

, 30

30. R. Marqués, L. Jelinek, F. Mesa, and F. Medina, “Analytical theory of wave propagation through stacked fishnet metamaterials,” Opt. Express 17, 11582–11593 (2009). [CrossRef] [PubMed]

]. The significance of the present study is that it extends their work to the optical frequency range, and the hole shape is employed as a new parameter for engineering the dispersion.

6. Demonstration of beam steering

In this section, a beam steering element is proposed based on the transmission phase controllability of a SHA around ωsingle. The element consists of a SHA with a hole shape that gradually changes in-plane from circular to square. The beam steering functionality is based on the formation of an inclined wavefront, as will be demonstrated through the interferometric measurements. The region around ωsingle was chosen for two reasons. First, any sidewall slope present in the SHA holes has less effect in this region. Second, based on the results shown in Fig. 6, square and circular-hole SHAs exhibit similar phase values at about ωsingle, and the phase difference between them increases as the normalized frequency decreases from a/λ = 0.68 to 0.65. Hence, a beam steering element designed to operate around ωsingle has the potential to allow a range of outgoing beam angles from normal to the surface to oblique.

Since these SHAs have the same periodicity but different hole shapes, the wavefront of a beam can be deformed if the beam passes through a SHA whose in-plane hole shape gradually changes from circular to square. The proposed structure comprises three regions, as shown in Fig. 7(a). The left region has circular holes and the right region has square holes. Between these two regions is a transition region in which the circular shape is changed to square by gradually decreasing the radius at the four corners. Figure 7(c) shows the phase difference for the beam steering element at various frequencies. The phase differences are determined from the displacements of the fringes relative to that in the circular hole region. Note that the phase difference gradually changes in the transition region as expected, which are also observed from an interferometric image (Fig. 7 (b)). This means that an inclined wavefront can be formed merely by using a planar structure, which suggests the possibility of wavefront shape control based only on the in-plane geometric design, i.e., the hole shape. The maximum phase difference observed for a frequency range from a/λ = 0.685 to 0.667 is 0.125 π, consistent with the results shown in Fig. 6.

Fig. 7 Experimental demonstration of beam steering element. (a) SEM image of the beam steering element. Scale bar is 10 μm. Insets are magnified images of circular and square holes. The double-headed arrow indicates the polarization direction of the incident light. (b) Interferometric image of the beam steering element. Scale bar is 150 μm. (c) Phase distribution plots based on analysis of interferometric images for various frequencies. The horizontal axis shows the position from the center of the transition region.

7. Conclusion

The applicability of stacked metal-dielectric hole arrays to a transmission phase control element was experimentally verified. Laterally propagating SPPs excited by the periodic nanohole array play an important role in light transmission through a SHA. The transmission amplitude and phase were found to drastically change around the resonance frequencies, which were determined mainly by the periodicity of the SHA and the hole shape. A beam steering element was also developed that achieved an inclined wavefront by using a SHA whose in-plane hole shape gradually changes from circular to square.

Although in the present study, the wavefront inclination angle was controlled by sweeping the frequency of the incident light, beam steering at a fixed frequency can be realized by introducing a tunable refractive index material into the holes, such as a liquid crystal whose refractive-index anisotropy can be tuned by an electric field, which would allow the localized resonance in each hole to be changed [31

31. H. Yoshida, T. Matsui, A. Miura, N. Ikeda, M. Ochiai, Y. Sugimoto, H. Fujikawa, and M. Ozaki, “Uniform liquid crystal alignment on metallic nanohole arrays by vapor-phase deposition of silane coupling agent,” Opt. Mater. Express 2, 893–899 (2012). [CrossRef]

].

Acknowledgment

The authors would like to thank Y. Akimoto for technical support. This work was supported in part by the New Energy and Industrial Technology Development Organization (NEDO) and the Nanotechnology Innovation Station of National Institute for Materials Science (NIMS), Japan.

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.

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]

3.

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

4.

D. Inoue, A. Miura, T. Nomura, H. Fujikawa, K. Sato, N. Ikeda, D. Tsuya, Y. Sugimoto, and Y. Koide, “Polarization independent visible color filter comprising an aluminum film with surface-plasmon enhanced transmission through a subwavelength array of holes,” Appl. Phys. Lett. 98, 093113 (2011). [CrossRef]

5.

S. A. Maier, Plasmonics: Fundamentals and Applications (Splinger-Verlag, 2007).

6.

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]

7.

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

8.

P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nanohole arrays,” J. Opt. A: Pure Appl. Opt. 7, 422–426 (2005). [CrossRef]

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.

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

11.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

12.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892–894 (2006). [CrossRef] [PubMed]

13.

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 (London) 455, 376–380 (2008). [CrossRef]

14.

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4929 (2005). [CrossRef] [PubMed]

15.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006). [CrossRef] [PubMed]

16.

X. Chen, T. M. Grzegorczyk, B-I. Wu, J. Pacheco Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004). [CrossRef]

17.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005). [CrossRef]

18.

A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. 101, 103902 (2008). [CrossRef] [PubMed]

19.

R. Ortuño, C. García-Meca, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays,” Phys. Rev. B 79, 075425 (2009). [CrossRef]

20.

J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett. 107, 043903 (2011). [CrossRef] [PubMed]

21.

M. Iwanaga, “In-plane plasmonic modes of negative group velocity in perforated waveguides,” Opt. Lett. 36, 2504–2506 (2011). [CrossRef] [PubMed]

22.

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary tTransmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett. 92, 183901 (2004). [CrossRef]

23.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express 15, 11536–11541 (2007). [CrossRef] [PubMed]

24.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]

25.

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96, 097401 (2006). [CrossRef] [PubMed]

26.

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007). [CrossRef]

27.

A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998). [CrossRef]

28.

M. B. Sinclair, M. P. de Boer, and A. D. Corwin, “Long-working-distance incoherent-light interference microscope,” Appl. Opt. 44, 7714–7721 (2005). [CrossRef] [PubMed]

29.

M. Beruete, M. Sorolla, and I. Campillo, “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express 14, 5445–5455 (2006). [CrossRef] [PubMed]

30.

R. Marqués, L. Jelinek, F. Mesa, and F. Medina, “Analytical theory of wave propagation through stacked fishnet metamaterials,” Opt. Express 17, 11582–11593 (2009). [CrossRef] [PubMed]

31.

H. Yoshida, T. Matsui, A. Miura, N. Ikeda, M. Ochiai, Y. Sugimoto, H. Fujikawa, and M. Ozaki, “Uniform liquid crystal alignment on metallic nanohole arrays by vapor-phase deposition of silane coupling agent,” Opt. Mater. Express 2, 893–899 (2012). [CrossRef]

32.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tatienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011). [CrossRef] [PubMed]

33.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012). [CrossRef]

OCIS Codes
(180.3170) Microscopy : Interference microscopy
(160.3918) Materials : Metamaterials
(220.4241) Optical design and fabrication : Nanostructure fabrication
(250.5403) Optoelectronics : Plasmonics
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 16, 2012
Revised Manuscript: June 25, 2012
Manuscript Accepted: June 26, 2012
Published: June 29, 2012

Virtual Issues
Vol. 7, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Takayuki Matsui, Hideki T. Miyazaki, Atsushi Miura, Tsuyoshi Nomura, Hisayoshi Fujikawa, Kazuo Sato, Naoki Ikeda, Daiju Tsuya, Masayuki Ochiai, Yoshimasa Sugimoto, Masanori Ozaki, Masanori Hangyo, and Kiyoshi Asakawa, "Transmission phase control by stacked metal-dielectric hole array with two-dimensional geometric design," Opt. Express 20, 16092-16103 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-16092


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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. 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]
  3. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature (London)445, 39–46 (2007). [CrossRef]
  4. D. Inoue, A. Miura, T. Nomura, H. Fujikawa, K. Sato, N. Ikeda, D. Tsuya, Y. Sugimoto, and Y. Koide, “Polarization independent visible color filter comprising an aluminum film with surface-plasmon enhanced transmission through a subwavelength array of holes,” Appl. Phys. Lett.98, 093113 (2011). [CrossRef]
  5. S. A. Maier, Plasmonics: Fundamentals and Applications (Splinger-Verlag, 2007).
  6. 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. B58, 6779–6782 (1998). [CrossRef]
  7. F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Morenno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett.95, 103901 (2005). [CrossRef] [PubMed]
  8. P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nanohole arrays,” J. Opt. A: Pure Appl. Opt.7, 422–426 (2005). [CrossRef]
  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. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature (London)452, 728–731 (2008). [CrossRef]
  11. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett.95, 137404 (2005). [CrossRef] [PubMed]
  12. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science312, 892–894 (2006). [CrossRef] [PubMed]
  13. 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 (London)455, 376–380 (2008). [CrossRef]
  14. S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express13, 4922–4929 (2005). [CrossRef] [PubMed]
  15. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett.31, 1800–1802 (2006). [CrossRef] [PubMed]
  16. X. Chen, T. M. Grzegorczyk, B-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E70, 016608 (2004). [CrossRef]
  17. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005). [CrossRef]
  18. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett.101, 103902 (2008). [CrossRef] [PubMed]
  19. R. Ortuño, C. García-Meca, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays,” Phys. Rev. B79, 075425 (2009). [CrossRef]
  20. J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107, 043903 (2011). [CrossRef] [PubMed]
  21. M. Iwanaga, “In-plane plasmonic modes of negative group velocity in perforated waveguides,” Opt. Lett.36, 2504–2506 (2011). [CrossRef] [PubMed]
  22. K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary tTransmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett.92, 183901 (2004). [CrossRef]
  23. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Design-related losses of double-fishnet negative-index photonic metamaterials,” Opt. Express15, 11536–11541 (2007). [CrossRef] [PubMed]
  24. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73, 035407 (2006). [CrossRef]
  25. H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett.96, 097401 (2006). [CrossRef] [PubMed]
  26. Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B75, 035411 (2007). [CrossRef]
  27. A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt.37, 5271–5283 (1998). [CrossRef]
  28. M. B. Sinclair, M. P. de Boer, and A. D. Corwin, “Long-working-distance incoherent-light interference microscope,” Appl. Opt.44, 7714–7721 (2005). [CrossRef] [PubMed]
  29. M. Beruete, M. Sorolla, and I. Campillo, “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express14, 5445–5455 (2006). [CrossRef] [PubMed]
  30. R. Marqués, L. Jelinek, F. Mesa, and F. Medina, “Analytical theory of wave propagation through stacked fishnet metamaterials,” Opt. Express17, 11582–11593 (2009). [CrossRef] [PubMed]
  31. H. Yoshida, T. Matsui, A. Miura, N. Ikeda, M. Ochiai, Y. Sugimoto, H. Fujikawa, and M. Ozaki, “Uniform liquid crystal alignment on metallic nanohole arrays by vapor-phase deposition of silane coupling agent,” Opt. Mater. Express2, 893–899 (2012). [CrossRef]
  32. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tatienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334, 333–337 (2011). [CrossRef] [PubMed]
  33. X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science335, 427 (2012). [CrossRef]

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