Abstract

In this paper, we experimentally demonstrate the excitation of spoof surface plasmon polaritons (SPPs) on a wire-medium metamaterial slab in the microwave region. The spoof SPPs are excited on the opposite side of the slab from the source, which is desirable for applications such as sensing devices. Using the prism coupling method, we verify the excitation of spoof SPPs by measuring the reflection spectrum and near-field enhancement. The excitation of spoof SPPs is also verified by using the grating coupling method, where we demonstrate transmission enhancement through the metamaterial slab by placing diffraction gratings on both sides of the slab. Numerical investigation shows that the enhanced transmission can be attributed to the dispersion relations of the spoof SPPs and the periodicity of the diffraction grating. These properties can be used to realize extraordinary transmission and directional beaming.

© 2012 OSA

1. Introduction

Surface plasmon polaritons (SPPs) have been the focus of much recent research because of the introduction of the concept of a perfect lens by Pendry [1] and the reporting of extraordinary optical transmission (EOT) by Ebbesen [2]. Both phenomena are attributable to the SPPs’ bound surface states. The bound waves can preserve the diffracted waves that have large momentum, leading to super-resolution [1] and EOT in which re-radiation of the bounded waves enhances the emission from holes in a metal plate [2].

In a bulk metal, SPPs can be excited at only certain optical frequencies where the permittivity of the metal is negative. However, SPPs can be mimicked with structured materials, called spoof SPPs, also introduced by Pendry [3]. This approach opens the possibility of applications of SPP-like phenomena in other frequency regimes (i.e., microwave and terahertz regimes). Therefore, determining how to construct such structured materials (also known as metamaterials) in these regimes is an important research topic in itself.

Early attempts to realize spoof SPPs employed a corrugated perfect electric conductor (PEC) surface and a perforated PEC surface [3]. Reference [4] provided experimental verification of spoof SPPs in the microwave region. The same group subsequently reported experimental results in Ref. [5] that the bound state of an electromagnetic band gap (EBG) structure [6] can also support surface plasmon-like waves. The EBG structure, which is also referred to as a mushroom structure, is well studied in its equivalent circuit representation, and the structure’s surface plasmon-like behavior is also predicted from circuit representations. The operating bandwidth can be improved by employing the equivalent circuit approach [7]. However, these structures are fabricated on a ground plane to ensure tight confinement on the surface; their applications are restricted to thin waveguides and cannot be used for sensing or for lenses in which excitation of the SPPs on the back surface of metallic slabs is utilized.

A straightforward way of realizing the SPPs at arbitrary frequencies is mimicking the electrical properties of metals at optical frequencies. A structure mimicking these properties has been reported; this structure consists of a metallic wire mesh [8]. However, this approach has two disadvantages [9]. First, the electrical response varies with respect to the direction of the electromagnetic wave. This allows a large leakage of the surface-bound waves. Second, the structured material has no flat cutting plane, requiring extra treatment of the surface. However, the spatial dispersion can be reduced by adding auxiliary elements to the wire-medium [10]. This reduces the expression of the effective permittivity of the metamaterial from a tensor to a scalar. This does not imply that the dispersion relation of the surface is described by the scalar-valued permittivity because the surface structure is not considered in the derivation process of the permittivity. In Ref. [11], we found a structure whose surface dispersion almost coincides with the dispersion relation calculated from a scalar-valued permittivity and an assumption that the interface is flat. This allows us to exploit novel applications involving materials that have an intrinsic negative-valued permittivity, at an arbitrary frequency. Using this structure, we have numerically demonstrated the excitation of spoof SPPs on the back surface of the metamaterial slab [11] in the Kretschmann configuration [12].

In this paper, we experimentally demonstrate the excitation of spoof SPPs on the metamaterial structure. Because of their large momentum, the excitation of the spoof SPPs requires a momentum matching procedure. We first employ the prism coupling method, where the excitation of the spoof SPPs is confirmed by a dip in the reflection and field intensity near the surface. Comparison between the experimental result and numerical calculation indicates that the observed field enhancement is related to the excitation of the spoof SPPs on the back surface of the metamaterial. We also conduct an experiment using the grating coupling method, where placing the diffraction gratings on both sides of the metamaterial slab is observed to result in enhanced transmission through the slab. This phenomenon is explained by the dispersion relations of the bound and radiative waves.

2. Principle

The metamaterial structure we have previously reported [11] is shown in the inset of Fig. 1. The basic structure of the metamaterial consists of a three-dimensionally connected metallic wire lattice. The length of the lattice period is 10 mm, the radius of the wire is rw = 0.3 mm, and the radius of the sphere is rs = 1.73 mm. The dispersion relation of the structure is also shown in Fig. 1. In the calculation of the dispersion diagram, we employed a three-dimensional finite difference frequency domain (FDFD) method [13]. By applying Bloch boundary conditions, the computational domain is restricted to the unit cell of the periodic structure. The resulting matrix of the equation system was solved with the aid of a computational library for large-scale sparse eigenvalue problems [14]. The eigenvalues provide the dispersion relation ω(k), where k represents the Bloch wavevector. As we can see in Fig. 1, the two degenerate modes due to the interaction of waves on both sides of the slab will be excited, which is also shown in the inset. It can be also seen that the parallel momentum of the spoof SPPs exceeds any wavenumber that radiative waves can have in the air at the same frequency. This requires momentum-matching methods for coupling the external plane wave to the surface state.

 

Fig. 1 Dispersion relation of the spoof SPPs for a half-plane and finite slab of the meta-material. The slab has two unit cells along the thickness direction, and the cutting plane of the surface is chosen as shown in the inset. The geometrical parameters are defined with respect to the unit length a as the radius of the sphere rs = 0.173a and the radius of the wire rw = 0.03a. In the experiment, the unit length was fixed at a = 10 mm. The shaded region represents the inside of the light cone. The spatial distributions of the electric field Ex corresponding to the lowest two modes are shown in the inset.

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A well-known method for compensating the momentum mismatch between the incident wave and the SPPs is attenuated total reflection (ATR) using a dielectric prism. The parallel component of the wavevector in the prism is given by

Re(kspp)=nk0sinθinc,
where k0 is the wavenumber in free space and n is the refractive index of the dielectric prism. Therefore, if n is sufficiently large, the left-hand side of the equation matches the dispersion relation of the SPPs. The parallel component of the wavevector is also controlled by diffraction gratings, as given by
Re(kspp)=kx+m2πΛ,
where kx = k0 sinθinc, m is a diffraction order, and Λ is the grating period. The diffraction grating is placed near the surface of the metamaterial. This grating layer can be considered as an additional surface structure of the metamaterial. We computed the dispersion relation of the surface with such a grating layer in Section 4.

3. Prism coupling

Figure 2 depicts the measurement setup using the ATR prism coupling method. The prism is a 77-mm-thick 45°–45°–90° triangle made by stacking FR4 dielectric substrates with relative permittivity of 4.9. The metamaterial consists of three-dimensionally connected brass wires with auxiliary spheres made of brass. The structure has dimensions of 11 unit cells in height, 70 unit cells in length, and 2 unit cells in thickness. The TM-polarized electromagnetic wave is incident on one face of the prism with different angles of incidence. The reflection from the prism is used to normalize subsequent measurements of the metamaterial. In the experimental setup, a metal-backed microwave absorber with an aperture is placed in front of the prism to suppress interference. The electric field intensity is measured using a probe mounted on an X–Y position controller. The source and the receiver horn or the probe are connected to a vector network analyzer (VNA). The VNA and X–Y position controller are controlled by an in-house program for recording electric field values.

 

Fig. 2 Schematic representation of a prism coupling experiment setup. Numerical simulation of the experiment by using the finite difference time domain (FDTD) method is also conducted with the same parameters as this setup, except for the height dimension; in the computation, we employed a periodic boundary condition along the height direction.

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Figure 3 shows experimentally determined dispersion relations. Black dots represent the positions of reflection minima. The thickness of the metamaterial slab is 23.5 mm, which is 0.43λ relative to the wavelength of a typical frequency of 5.5 GHz. The distance between the face of the prism and the metamaterial is d = 7.0 mm. There is a qualitative agreement between the measured data and the calculated relations. In the numerical calculation, the presence of the prism is not taken into account. The inset of Fig. 3 shows measured reflection spectra for an angle of incidence of 45° and different d values. It is clear that the frequencies of the reflection minima vary with the distance d. As expected from the dispersion relation, the reflection shows two distinct dips except for the case of d = 3.5 mm, in which case, one single dip with the largest observed depth is found. The presence of two distinct dips is related to the excitation of the spoof SPPs involving both sides of the metamaterial, while the one large dip is related to the excitation of the spoof SPPs only on the air side of the metamaterial. At the frequency corresponding to the dip, we can expect that high field intensity on the metamaterial face will be observed. The electric field intensity at the metamaterial surface measured by scanning using a monopole probe, is shown in Fig. 4(a), where the distance between the prism face and metamaterial surface is fixed at 3.5 mm. The monopole probe was placed perpendicular to the surface and moved along a line from the edge of the left side of the structure depicted in Fig. 2 to 300 mm, at a distance of approximately 3 mm from the surface. We can observe that the highest intensity is found around 5.5 GHz, where the large dip in the reflection is found. The simulated electric field intensity corresponding to this case is shown in Fig. 4(b). The simulation is performed by using the finite difference time domain (FDTD) method [15] and using a periodic boundary condition along the height direction. The frequency components of the field distributions are extracted by applying discrete Fourier transformation at each time step. The calculated spatial field distributions at two frequencies are shown in Fig. 4(c), demonstrating a localization on the air side of the metamaterial. We evaluated the extent of the field enhancement due to the localized field on the metamaterial surface with respect to the intensity on the prism face. The result is shown in Fig. 5. The enhancement of the field is confirmed in both the simulation and the experiment. Therefore, we can conclude that the highest field enhancement observed in the experiment indicates the excitation of the spoof SPPs localized on the back surface of the metamaterial.

 

Fig. 3 Experimentally determined dispersion relations for internal angles between 41.6° and 48.4°. The distance between the face of the prism and the metamaterial is d = 7.0 mm. Black dots represent the positions of reflection minima. The inset shows measured reflection spectra for an angle of incidence of 45°. The distance d is varied as follows: d = 3.5, 5.0, 7.0 and 8.5 mm.

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Fig. 4 Electric field intensity spectra measured by scanning using a monopole probe. The probe was moved along the lateral line depicted in Fig. 2. (a) shows the experimental result and (b) shows the simulated result. In both cases, the distance between the prism and the metamaterial is d = 3.5 mm, the angle of incidence is 45°. (c) shows the calculated spatial distributions of the electric field for 5.37 and 4.40 GHz.

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Fig. 5 Electric field intensity recorded with a monopole probe at the frequency of 5.50 GHz. The calculated results at the frequency of 5.37 GHz are also shown. The field intensity for the metamaterial face is scaled with respect to the field intensity for the prism face.

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4. Grating coupling and plasmon-induced transmission

As we have seen in the previous section, the metamaterial structure can support the excitation of the spoof SPPs on the side opposite the source. This can also be achieved by employing the grating coupling method. From reciprocity, we can expect that the spoof SPPs can be radiated by diffraction gratings. Therefore, if we place the diffraction grating on both sides of the metamaterial slab, enhanced transmission due to the radiation of the SPPs can be observed. This configuration clarifies the characteristics of our structure: i) excitation of the SPPs occurs on the back surface of the metamaterial and ii) the surface state can be manipulated with diffraction gratings. In the following, we discuss the excitation of the spoof SPPs by considering this grating–metamaterial–grating configuration. We first calculated the dispersion relation of the metamaterial slab with diffraction gratings on both sides, as shown in Fig. 6. In contrast to the bound modes shown in Fig. 1, in which non-radiative modes are considered, here the radiative region is considered. The dispersion relations in the radiative region can be coupled with the plane wave. Therefore, the dispersion relations can be characterized by the reflection or transmission from the considered structure. Figure 6(c) shows the transmission spectrum of the metamaterial with diffraction gratings on both sides for the TM-polarization. The width and period of the gratings are 1 and 3 in unit lengths of the metamaterial, respectively. We can observe sharp transmission peaks around ωa/2πc = 0.55 for normal incidence.

 

Fig. 6 (a) Schematic representation of the metamaterial and the grating. The period of the grating is equal to three periods of the unit length of the metamaterial. This introduces modes folded at kxa/π = 1/3 into the dispersion relation. (b) The dispersion relation for the metamaterial slab, in which folding lines kxa/π = n/3 are also depicted. This dispersion relation is normalized with respect to the unit length of the metamaterial. In the experiment described later, the unit length of the metamaterial is fixed at a = 23 mm, and the distance between the grating and the metamaterial face is d = 3.5 mm. (c) The calculated transmission spectra as a function of frequency and wave number parallel to the surface. This dispersion relation is normalized with respect to the period of the grating.

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The grating layer introduces an additional periodicity into the metamaterial structure. Let a be the unit length of a periodic structure along the x direction, and let kx be a wavenumber along the x direction, dispersion relations are folded at kx = 0 and kxa = π. Therefore, the first and second modes appearing from the bound region can be related to the even and odd guided modes folded at kxa = π, where a is the unit length of the diffraction grating. The third and fourth modes are related to the first and second modes folded at kx = 0. These modes cross at around kxa/2π = 0.1. Therefore, with respect to the normal incidence, for which kx = 0, the first and second lowest modes are even modes and the third and fourth modes are odd modes. It should be noted that a band gap is introduced at kx = 0, as can be seen in Fig. 6(c). To confirm the above analysis, we compute the spatial distributions of the electric field at the frequencies corresponding to the peaks in Fig. 6(c). Figure 7(a) shows the spatial distribution of the electric field corresponding to the four modes at kx = 0. As we expected, the first and second modes involve an even distribution and the third and fourth modes involve an odd distribution. For comparison, Fig. 7(b) shows the field distributions when the gratings are placed only on the source side.

 

Fig. 7 (a) The electric Ex field distributions at the frequencies corresponding to four distinct peaks in Fig. 6(c). (b) The electric |Ex| field distributions at the same frequencies, when the grating is placed only on the source side. A, B, C, and D are the 1st, 2nd, 3rd, and 4th lowest modes in Fig. 6(c), respectively.

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To verify this phenomenon experimentally, we conducted an experiment using a parallel-plate waveguide (PPW). The height of the PPW corresponds to three periods of the metamaterial, which also corresponds to the period of the gratings. Therefore, the cutting plane of the PPW along the height direction and the thickness of the metamaterial is identical to the plane shown in Fig. 7. The PPW section that embedded thirteen periods of the metamaterial in the width direction is made as a separable section. An identical PPW section without the metamaterial is also made for normalizing the transmission spectrum. In the experiment, the lattice constant of the metamaterial is a = 23 mm. The gratings are made of a 23-mm-wide aluminum sheet put on a 3.5-mm-thick foam board. The foam boards are placed tight against the metamaterial face. Figure 8 shows the transmission for the metamaterial and the grating–metamaterial–grating configuration installed in the PPW. A large transmission is found around 2.45 GHz in the measured result and around 2.35 GHz in the simulated result. An enhanced transmission of 12 dB, achieved by inserting the gratings on both sides of the slab, is observed in the experiment. In the simulated result, four distinct peaks and two band gaps can be seen, while no such distinctions are found in the measured result. As we have seen, these distinctive peaks involve highly symmetrical spatial distributions. The experimental observation of such details requires more careful fabrication of the metamaterial. It is worth noting that the spatial symmetry can be exploited; the even symmetric modes can be reproduced by placing the PEC plane in the middle of the metamaterial. This can provide another means of experimentally verifying the numerical results and more opportunities for practical applications like frequency dependent beam steering.

 

Fig. 8 Measured transmittance for the metamaterial and grating–metamaterial–grating configuration installed in the parallel plate waveguide (PPW). The height of the PPW corresponds to three periods of the metamaterial in the x direction. Therefore, the cutting plane of the PPW along the xz plane is identical to the plane shown in Fig. 7.

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5. Conclusions

In this paper, we have experimentally demonstrated the excitation of spoof SPPs on a metamaterial slab. A metamaterial slab consisting of a metallic wire-medium with auxiliary metallic spheres was fabricated. By employing the ATR prism coupling method, the excitation of spoof SPPs on the back surface of the metamaterial slab was observed. We have also shown transmission enhancement through the metamaterial slab due to the excitation of spoof SPPs by placing the diffraction gratings on both sides of the slab. Our results show two essential properties of the SPPs for practical applications: the field localization on the back surface and the coupling ability between the SPPs and diffracted waves. Therefore, we believe that our results have potential for plasmonic sensing, waveguiding, and configurable beam forming.

References and links

1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef]   [PubMed]  

2. 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, 667–669 (1998). [CrossRef]  

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

4. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308, 670–672 (2005). [CrossRef]   [PubMed]  

5. M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett. 102, 073901 (2009). [CrossRef]   [PubMed]  

6. D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999). [CrossRef]  

7. M. Navarro-Cía, M. Beruete, S. Agrafiotis, F. Falcone, M. Sorolla, and S. A. Maier, “Broadband spoof plasmons and subwavelength electromagnetic energy confinement on ultrathin metafilms,” Opt. Express 17, 18184–18195 (2009). [CrossRef]   [PubMed]  

8. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely-low-frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef]   [PubMed]  

9. M. A. Shapiro, G. Shvets, J. R. Sirigiri, and R. J. Temkin, “Spatial dispersion in metamaterials with negative dielectric permittivity and its effect on surface waves,” Opt. Lett. 31, 2051–2053 (2006). [CrossRef]   [PubMed]  

10. A. Demetriadou and J. B. Pendry, “Taming spatial dispersion in wire metamaterial,” J. Phys.: Condens. Matter 20, 295222 (2008). [CrossRef]  

11. Y. Kushiyama, T. Uno, and T. Arima, “Novel negative permittivity structure and its application to excitation of surface plasmon in microwave frequency range,” IEICE Trans. Commun. E93-B, 2629–2635 (2010). [CrossRef]  

12. R. Raether, Surface Plasmons (Springer–Verlag, Berlin, 1988).

13. K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech. 40, 540–546 (1992). [CrossRef]  

14. V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Software 31, 351–362 (2005). [CrossRef]  

15. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Third Edition (Artech House Publishers, 2000).

References

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391, 667–669 (1998).
    [CrossRef]
  3. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004).
    [CrossRef] [PubMed]
  4. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
    [CrossRef] [PubMed]
  5. M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett.102, 073901 (2009).
    [CrossRef] [PubMed]
  6. D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47, 2059–2074 (1999).
    [CrossRef]
  7. M. Navarro-Cía, M. Beruete, S. Agrafiotis, F. Falcone, M. Sorolla, and S. A. Maier, “Broadband spoof plasmons and subwavelength electromagnetic energy confinement on ultrathin metafilms,” Opt. Express17, 18184–18195 (2009).
    [CrossRef] [PubMed]
  8. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely-low-frequency plasmons in metallic mesostructures,” Phys. Rev. Lett.76, 4773–4776 (1996).
    [CrossRef] [PubMed]
  9. M. A. Shapiro, G. Shvets, J. R. Sirigiri, and R. J. Temkin, “Spatial dispersion in metamaterials with negative dielectric permittivity and its effect on surface waves,” Opt. Lett.31, 2051–2053 (2006).
    [CrossRef] [PubMed]
  10. A. Demetriadou and J. B. Pendry, “Taming spatial dispersion in wire metamaterial,” J. Phys.: Condens. Matter20, 295222 (2008).
    [CrossRef]
  11. Y. Kushiyama, T. Uno, and T. Arima, “Novel negative permittivity structure and its application to excitation of surface plasmon in microwave frequency range,” IEICE Trans. Commun.E93-B, 2629–2635 (2010).
    [CrossRef]
  12. R. Raether, Surface Plasmons (Springer–Verlag, Berlin, 1988).
  13. K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
    [CrossRef]
  14. V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Software31, 351–362 (2005).
    [CrossRef]
  15. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Third Edition (Artech House Publishers, 2000).

2010 (1)

Y. Kushiyama, T. Uno, and T. Arima, “Novel negative permittivity structure and its application to excitation of surface plasmon in microwave frequency range,” IEICE Trans. Commun.E93-B, 2629–2635 (2010).
[CrossRef]

2009 (2)

2008 (1)

A. Demetriadou and J. B. Pendry, “Taming spatial dispersion in wire metamaterial,” J. Phys.: Condens. Matter20, 295222 (2008).
[CrossRef]

2006 (1)

2005 (2)

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Software31, 351–362 (2005).
[CrossRef]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

2004 (1)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
[CrossRef] [PubMed]

1999 (1)

D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47, 2059–2074 (1999).
[CrossRef]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391, 667–669 (1998).
[CrossRef]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely-low-frequency plasmons in metallic mesostructures,” Phys. Rev. Lett.76, 4773–4776 (1996).
[CrossRef] [PubMed]

1992 (1)

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

Agrafiotis, S.

Alexopolous, N.

D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47, 2059–2074 (1999).
[CrossRef]

Arima, T.

Y. Kushiyama, T. Uno, and T. Arima, “Novel negative permittivity structure and its application to excitation of surface plasmon in microwave frequency range,” IEICE Trans. Commun.E93-B, 2629–2635 (2010).
[CrossRef]

Beilenhoff, K.

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

Beruete, M.

Broas, R.

D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47, 2059–2074 (1999).
[CrossRef]

Demetriadou, A.

A. Demetriadou and J. B. Pendry, “Taming spatial dispersion in wire metamaterial,” J. Phys.: Condens. Matter20, 295222 (2008).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391, 667–669 (1998).
[CrossRef]

Evans, B. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

Falcone, F.

Garcia-Vidal, F. J.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391, 667–669 (1998).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Third Edition (Artech House Publishers, 2000).

Hartnagel, H.

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

Heinrich, W.

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

Hernandez, V.

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Software31, 351–362 (2005).
[CrossRef]

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett.102, 073901 (2009).
[CrossRef] [PubMed]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely-low-frequency plasmons in metallic mesostructures,” Phys. Rev. Lett.76, 4773–4776 (1996).
[CrossRef] [PubMed]

Kushiyama, Y.

Y. Kushiyama, T. Uno, and T. Arima, “Novel negative permittivity structure and its application to excitation of surface plasmon in microwave frequency range,” IEICE Trans. Commun.E93-B, 2629–2635 (2010).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391, 667–669 (1998).
[CrossRef]

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett.102, 073901 (2009).
[CrossRef] [PubMed]

Maier, S. A.

Martín-Moreno, L.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004).
[CrossRef] [PubMed]

Navarro-Cía, M.

Pendry, J. B.

A. Demetriadou and J. B. Pendry, “Taming spatial dispersion in wire metamaterial,” J. Phys.: Condens. Matter20, 295222 (2008).
[CrossRef]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely-low-frequency plasmons in metallic mesostructures,” Phys. Rev. Lett.76, 4773–4776 (1996).
[CrossRef] [PubMed]

Raether, R.

R. Raether, Surface Plasmons (Springer–Verlag, Berlin, 1988).

Roman, J. E.

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Software31, 351–362 (2005).
[CrossRef]

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett.102, 073901 (2009).
[CrossRef] [PubMed]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

Shapiro, M. A.

Shvets, G.

Sievenpiper, D.

D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47, 2059–2074 (1999).
[CrossRef]

Sirigiri, J. R.

Sorolla, M.

Stewart, W. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely-low-frequency plasmons in metallic mesostructures,” Phys. Rev. Lett.76, 4773–4776 (1996).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Third Edition (Artech House Publishers, 2000).

Temkin, R. J.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391, 667–669 (1998).
[CrossRef]

Uno, T.

Y. Kushiyama, T. Uno, and T. Arima, “Novel negative permittivity structure and its application to excitation of surface plasmon in microwave frequency range,” IEICE Trans. Commun.E93-B, 2629–2635 (2010).
[CrossRef]

Vidal, V.

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Software31, 351–362 (2005).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391, 667–669 (1998).
[CrossRef]

Yablonovitch, E.

D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47, 2059–2074 (1999).
[CrossRef]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely-low-frequency plasmons in metallic mesostructures,” Phys. Rev. Lett.76, 4773–4776 (1996).
[CrossRef] [PubMed]

Zhang, L.

D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47, 2059–2074 (1999).
[CrossRef]

ACM Trans. Math. Software (1)

V. Hernandez, J. E. Roman, and V. Vidal, “SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems,” ACM Trans. Math. Software31, 351–362 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech.47, 2059–2074 (1999).
[CrossRef]

IEICE Trans. Commun. (1)

Y. Kushiyama, T. Uno, and T. Arima, “Novel negative permittivity structure and its application to excitation of surface plasmon in microwave frequency range,” IEICE Trans. Commun.E93-B, 2629–2635 (2010).
[CrossRef]

J. Phys.: Condens. Matter (1)

A. Demetriadou and J. B. Pendry, “Taming spatial dispersion in wire metamaterial,” J. Phys.: Condens. Matter20, 295222 (2008).
[CrossRef]

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391, 667–669 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (3)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely-low-frequency plasmons in metallic mesostructures,” Phys. Rev. Lett.76, 4773–4776 (1996).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
[CrossRef] [PubMed]

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, “Microwave surface-plasmon-like modes on thin metamaterials,” Phys. Rev. Lett.102, 073901 (2009).
[CrossRef] [PubMed]

Science (2)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004).
[CrossRef] [PubMed]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

Other (2)

R. Raether, Surface Plasmons (Springer–Verlag, Berlin, 1988).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Third Edition (Artech House Publishers, 2000).

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Figures (8)

Fig. 1
Fig. 1

Dispersion relation of the spoof SPPs for a half-plane and finite slab of the meta-material. The slab has two unit cells along the thickness direction, and the cutting plane of the surface is chosen as shown in the inset. The geometrical parameters are defined with respect to the unit length a as the radius of the sphere rs = 0.173a and the radius of the wire rw = 0.03a. In the experiment, the unit length was fixed at a = 10 mm. The shaded region represents the inside of the light cone. The spatial distributions of the electric field Ex corresponding to the lowest two modes are shown in the inset.

Fig. 2
Fig. 2

Schematic representation of a prism coupling experiment setup. Numerical simulation of the experiment by using the finite difference time domain (FDTD) method is also conducted with the same parameters as this setup, except for the height dimension; in the computation, we employed a periodic boundary condition along the height direction.

Fig. 3
Fig. 3

Experimentally determined dispersion relations for internal angles between 41.6° and 48.4°. The distance between the face of the prism and the metamaterial is d = 7.0 mm. Black dots represent the positions of reflection minima. The inset shows measured reflection spectra for an angle of incidence of 45°. The distance d is varied as follows: d = 3.5, 5.0, 7.0 and 8.5 mm.

Fig. 4
Fig. 4

Electric field intensity spectra measured by scanning using a monopole probe. The probe was moved along the lateral line depicted in Fig. 2. (a) shows the experimental result and (b) shows the simulated result. In both cases, the distance between the prism and the metamaterial is d = 3.5 mm, the angle of incidence is 45°. (c) shows the calculated spatial distributions of the electric field for 5.37 and 4.40 GHz.

Fig. 5
Fig. 5

Electric field intensity recorded with a monopole probe at the frequency of 5.50 GHz. The calculated results at the frequency of 5.37 GHz are also shown. The field intensity for the metamaterial face is scaled with respect to the field intensity for the prism face.

Fig. 6
Fig. 6

(a) Schematic representation of the metamaterial and the grating. The period of the grating is equal to three periods of the unit length of the metamaterial. This introduces modes folded at kxa/π = 1/3 into the dispersion relation. (b) The dispersion relation for the metamaterial slab, in which folding lines kxa/π = n/3 are also depicted. This dispersion relation is normalized with respect to the unit length of the metamaterial. In the experiment described later, the unit length of the metamaterial is fixed at a = 23 mm, and the distance between the grating and the metamaterial face is d = 3.5 mm. (c) The calculated transmission spectra as a function of frequency and wave number parallel to the surface. This dispersion relation is normalized with respect to the period of the grating.

Fig. 7
Fig. 7

(a) The electric Ex field distributions at the frequencies corresponding to four distinct peaks in Fig. 6(c). (b) The electric |Ex| field distributions at the same frequencies, when the grating is placed only on the source side. A, B, C, and D are the 1st, 2nd, 3rd, and 4th lowest modes in Fig. 6(c), respectively.

Fig. 8
Fig. 8

Measured transmittance for the metamaterial and grating–metamaterial–grating configuration installed in the parallel plate waveguide (PPW). The height of the PPW corresponds to three periods of the metamaterial in the x direction. Therefore, the cutting plane of the PPW along the xz plane is identical to the plane shown in Fig. 7.

Equations (2)

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Re ( k spp ) = n k 0 sin θ inc ,
Re ( k spp ) = k x + m 2 π Λ ,

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