Homogenous Metamaterial Description of Localized Spoof Plasmons

Letter pubs.acs.org/journal/apchd5

Homogenous Metamaterial Description of Localized Spoof Plasmons in Spiral Geometries Zhen Liao,‡,□ Antonio I. Fernández-Domínguez,⊥,□ Jingjing Zhang,†,□ Stefan A. Maier,*,∥ Tie Jun Cui,*,‡,# and Yu Luo*,† †

School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, 639798, Singapore State Key Laboratory of Millimetre Waves, Southeast University, Nanjing 210096, China ⊥ Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain ∥ The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, U.K. # Cooperative Innovation Centre of Terahertz Science, No. 4, Section 2, North Jianshe Road, Chengdu 610054, China ‡

ABSTRACT: It has been recently shown that ultrathin spiral metamaterials can support localized spoof plasmon modes whose resonant wavelength is much larger than the size of the structure. Here, an analytical model is developed to describe the electromagnetic properties of the two-dimensional version of these devices: a perfect conducting wire corrugated by spiral grooves. The emergence of localized spoof plasmons in this geometry is quantitatively investigated. Calculations show that these modes can be engineered through the spiral angle and the number of grooves. The theory also allows us to elucidate the contribution of magnetic and electric localized spoof plasmons to the optical response of these metamaterial devices. Finally, experimental evidence of the existence of these modes in extremely thin textured copper disks is also presented. KEYWORDS: spoof surface plasmons, metamaterials, magnetic resonances, effective medium theory

S

confinement) of spoof SPs and the geometrical parameters of the surface texturing of the supporting structure. Since then, highly confined spoof SPs have been reported at microwave21−25 and terahertz26−28 frequencies in both two-dimensional (2D)21,25,26,28,29 and three-dimensional (3D)22,23,30 configurations. Like flat surfaces, periodically corrugated particles also support spoof plasmon modes, usually termed spoof localized surface plasmon resonances (spoof LSPRs).31 Thus, a corrugated PEC wire is found to exhibit an EM response similar to a Drude metal wire,31 transferring the distinct properties of optical LSPRs to much lower operating frequencies. On the basis of this principle, the realization of a number of spoof LSPR-based devices has already been demonstrated.32−34 Particles sustaining spoof LSPRs, as those proposed in ref 31, have a large scattering cross-section, which enables them to interact very strongly with incident radiation.32 However, their localized modes do not lead to strong EM field localization since the resonant wavelength, which is dictated by the depth of the corrugation decorating the particle surface, is comparable to its overall size. Lately, a strategy has been proposed to improve the mode-confinement characteristics of

urface plasmons (SPs) are collective oscillations of the delocalized electrons that exist at metal−dielectric interfaces. Owing to their ability to confine light in subwavelength dimensions with high efficiency, SPs offer a route to overcome the diffraction limit of classical optics.1,2 This enables a wide range of applications including surfaceenhanced spectroscopy,3−5 biomedical sensing,6−8 solar cell photovoltaics,9−11 and optical antennas.12−16 These applications, however, are generally limited to high electromagnetic (EM) frequencies (the UV, visible, and near-infrared ranges). This is because metals behave akin to perfect electric conductors (PECs) at lower frequency regimes and do not support SP modes. Early works showed that by corrugating metal surfaces, this limitation can be overcome, and they reported the excitation of highly confined SP-like electromagnetic modes at microwave frequencies.17,18 In 2004, Pendry et al. introduced the concept of spoof surface plasmons (spoof SPs) and demonstrated that plasmonic metamaterials constructed by patterning metal surfaces with subwavelength periodic features can mimic, at low frequencies (far IR, terahertz, or microwave regimes), the EM guiding characteristics of optical SPs.19,20 This metamaterial approach greatly facilitated exporting SP-related phenomena to lower frequency regimes of the EM spectrum, as it provided a direct link between the dispersion relation (and hence the spatial © 2016 American Chemical Society

Received: July 12, 2016 Published: September 27, 2016 1768

DOI: 10.1021/acsphotonics.6b00488 ACS Photonics 2016, 3, 1768−1775

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arm and the radial direction is the same along the spiral length; see Figure 1a. These are the so-called logarithmic spirals,38 characterized by a spiral angle θ, and can be parametrized as

spoof LSPR devices. This is based on corrugating the metal surface with long spiral grooves.35 An appropriate choice of the spirals’ length allows the spectral shifting of spoof LSPRs to wavelengths much larger than the particle size, giving rise to strong EM field localization at its surface.35,37 This property also adds additional degrees of freedom for the design of spoof plasmon structures, greatly increasing the tunability of these devices. Importantly, an analytical effective medium treatment of spiral-shaped spoof LSPRs is still lacking. The quantitative description of their EM properties is limited to only numerical methods,36,37 which cannot provide insight into the intricate relationship between geometry and field confinement in spoof plasmon particles, especially for those presenting deep subwavelength dimensions. In this work, we develop an effective medium theoretical model for the analytical description of spiral-shaped spoof LSPR devices and use it to investigate quantitatively their spectral and light-confinement properties. This model allows for a comprehensive understanding of the electric and magnetic plasmonic resonances (both fundamental and higher order modes) supported by the structure. Decreasing the number of spiral arms gives rise to nonlocal effects in the optical response of these structures, which are beyond our effective medium model. Numerical simulations show that these nonlocal effects result in a blue-shift of spiral LSP modes and a decrease in the corresponding density of states. Finally, we present experimental measurements and numerical calculations on thin-film spiral spoof LSPR devices operating in the microwave. This allows us to discuss the differences in the optical response of these geometries compared to infinitely long spiral wires.

x(t ) = r et /cotθ cos t y(t ) = r et /cotθ sin t

(1)

In the limit a < d ≪ λ, a metamaterial approximation, in which the textured PEC is treated as a homogeneous effective medium, can be applied.19 In this homogeneous metamaterial picture, the effective permittivity and permeability in the region between the inner and outer radii acquire a diagonal, spatially independent tensor form. The component of the permittivity normal to the spiral arms (z-direction) is given by εn = ng2d/a, while the parallel components diverge. The permeability components are set so that the EM radiation propagates at the speed of light inside the spiral grooves.20 This procedure yields the following effective permittivity and permeability tensors for TM waves expressed in cylindrical coordinates: ε −1 =

a ⎡1 − cos 2θ −sin 2θ ⎤ ⎢ ⎥, 2ng 2d ⎣−sin 2θ 1 + cos 2θ ⎦

μz−1 =

d a

(2)

where the permittivity tensor is restricted to the xy-plane. Consider an incident plane wave with the magnetic field polarized along the z-direction, propagating along the x-axis with a wavenumber k0. Substituting eq 2 into Maxwell’s equations, we obtained the following partial differential equation for the magnetic field:

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RESULTS AND DISCUSSION Theoretical Model. Our starting point is a 2D wire whose surface is decorated by N spiral-shaped grooves filled with a dielectric material of refractive index ng. The resulting inner and outer radii, which correspond to the bottom and opening of the grooves, are r and R, respectively, as shown in panel (a) of Figure 1. The period and width of the grooves along the wire perimeter are d and a. The spiral geometry is built in such a way that the intersection angle between the tangent to each spiral

1 + cos 2θ ∂ ∂ sin 2θ ∂ 2 ρ Hz + Hz 2ρ ∂ρ ∂ρ ρ ∂ρ∂φ +

1 − cos 2θ ∂ 2 H + ng 2k 0 2Hz = 0 2 2 z 2ρ ∂φ

(3)

By solving eq 3, the analytical form of Hz can be found as ⎛ e φ ⎞in⎡ ⎛ ng k 0ρ ⎞ ⎛ ng k 0ρ ⎞⎤ ⎟⎥ ⎟ + C2Y0⎜ Hz = ⎜ tanθ ⎟ ⎢C1J0 ⎜ ⎝ cos θ ⎠⎥⎦ ⎝ ρ ⎠ ⎢⎣ ⎝ cos θ ⎠

(4)

where J(·) and Y0(·) denote the zeroth-order Bessel and Neumann function, respectively; n is an arbitrary integer denoting the azimuthal mode order, and C1 and C2 are constants that can be determined by applying the appropriate boundary conditions. Equation 4 allows for the calculation of the scattering cross-section of the corrugated wire (detailed derivations can be found in the Methods section): σ sca =

4 k0

+∞

∑

|bn|2

(5)

n =−∞

where bn denotes the scattering coefficient assoicated with the nth-order cylindrical harmonics. It takes the form

Figure 1. Localized spoof surface plasmons in a 2D subwavelength PEC wire corrugated with spiral grooves. (a) Cross-section of the corrugated PEC wire with the inner and outer radii r and R, periodicity d, groove width a, and the spiral angle θ. (b) Analytical calculations of scattering cross-sections for 2D corrugated wires with different spiral angles θ = 0° (cyan), 30° (red), 60° (green), and 78.94° (black). In these calculations, we set r = 1.1 mm, R = 13.8 mm, and a/d = 2/3.

bn = −i n

pa cos θJn (k 0R ) − qng dJ ′n (k 0R ) pa cos θHn(1)(k 0R ) − qng dHn(1) ′(k 0R )

(6)

with the constants p and q given by 1769


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Figure 2. Real (left) and imaginary (right) parts of the resonance frequencies of different spoof plasmon modes (lowest radial order modes) as a function of the spiral angle θ. The geometric parameters are taken from Figure 1.

Figure 3. Localized spoof surface plasmons in a 2D corrugated wire with r = 1.1 mm, R = 13.8 mm, a/d = 2/3, and θ = 78.94°. The first column (a1−c1) shows the scattering peaks at different frequencies and the contribution due to spoof LSPRs of different orders. The red dashed lines, blue dot lines, and green dashed lines plot the SCS contribution due to the magnetic dipolar (n = 0), the electric dipolar (n = 1), and the electric quadrupolar (n = 2) modes. Magnetic field amplitudes corresponding to different spoof plasmon modes are plotted in the right column.

⎛ ng k 0R ⎞ ⎛ ng k 0r ⎞ ⎛ ng k 0r ⎞ ⎛ ng k 0R ⎞ ⎟ ⎟Y1⎜ ⎟ − J1⎜ ⎟Y1⎜ p = J1⎜ ⎝ cos θ ⎠ ⎝ cos θ ⎠ ⎝ cos θ ⎠ ⎝ cos θ ⎠ ⎛ ng k 0r ⎞ ⎛ ng k 0R ⎞ ⎛ ng k 0R ⎞ ⎛ ng k 0r ⎞ ⎟ ⎟Y0⎜ ⎟ − J1⎜ ⎟Y1⎜ q = J0 ⎜ ⎝ cos θ ⎠ ⎝ cos θ ⎠ ⎝ cos θ ⎠ ⎝ cos θ ⎠

resonances are characterized by the azimuthal angular momentum n and fulfill ng q Hn(1)(k 0R ) ⎛ na ⎞ 2a cos θ sin c 2⎜ ⎟ (1) = (1) ⎝ 2R ⎠ H (k R ) − H (k R ) p d n−1 0 n+1 0 (7)

(8)

The divergence condition for bn in eq 6 provides us with the transcendental equation for the complex resonant frequencies of the various spoof LSPRs supported by the geometry. These

Analytical and Numerical Results. Figure 1b depicts the scattering cross-section (SCS) of various spiral wires with the same inner and outer radii (r = 1.1 mm and R = 13.8 mm), a fixed filling factor a/d = 2/3, and different spiral angles. The 1770


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Figure 4. (a) Numerical scattering cross-section for spoof plasmon wires formed of N spirals: N = 2 (black), 5 (red), 8 (green), 10 (blue), and 20 (orange), and the analytical scattering cross-section obtained with the effective medium theory (gray dashed line). The right (left) insets render the magnetic field amplitude at the electric dipole (quadrupole) resonance for different N’s. The color scale codes the magnetic field amplitude from red (positive maximum) to blue (negative maximum). Note that for N ≤ 5 the quadrupole mode is no longer supported by the structure. Hence, only three contour plots of the quadrupole resonance are given here. Panels (b) and (c) show the numerical and analytical scattering cross-section for the same structures at higher frequency scattering bands.

Figure 3a1−c1 depict the SCS for the corrugated wire in Figure 1b. The spectra are evaluated at frequency windows corresponding to the three lowest multipeaked features supported by the geometry. The magnetic field distributions for the spoof plasmon modes originating the various crosssection maxima in each case are shown in the right panels. As reflected by our analytical calculations, each scattering band results from the spectral overlapping of several spoof LSPRs, which share the same radial characteristics (the same number of field nodes along the groove length) but present distinct azimuthal properties. The red dashed, blue dotted, and green dash-dotted lines in Figure 3a1−c1 render the SCS contribution corresponding to the magnetic dipole, the electric dipole, and the electric quadrupole, respectively. The direct comparison among the three panels evidences that, as anticipated above, magnetic modes are weakly excited and hence always overlap spectrally with electric resonances. A strategy that makes magnetic dipolar modes observable consists in truncating infinitely long spiral wires into finite ones.36 This will be discussed in more detail in the Methods section. Our analytical model is based on the homogeneous metamaterial approximation given by eq 2, whose validity requires the period (number) of the corrugations to be sufficiently small (large). Otherwise, the effect of spatial nonlocality40 resulting from the finite size of the surface textures must be considered. Since the fabrication of very small corrugations is experimentally challenging, it is vital to understand how spatial nonlocality affects the optical responses of the spoof LSP particles and how this effect can be minimized. To address these concerns, we performed numerical simulations (COMSOL Multiphysics) on wires corrugated by a different number of spiral grooves. Figure 4 renders the numerical scattering spectra calculated for different numbers of grooves, N, perforated on corrugated

spectra, whose peaks originate from the excitation of spoof LSPRs, were calculated using our effective metamaterial approach. They show that spoof resonances red-shift for increasing θ, in agreement with ref 33. Remarkably, our analytical results also reveal that higher order electric modes (electric quadropole, n = 2, and hexapole, n = 3) dominate the cross-section for small spiral angles (θ < 60°); see Figure 1b. On the contrary, the electric dipolar resonance (n = 1) becomes obversable only for relatively large θ, as shown in Figure 1b. In this panel, the emergence of higher radial order spoof LSPRs39 is also apparent, leading to similar multipeaked scattering profiles at increasing spectral ranges. The occurrence of these different radial order modes illustrates the fact that spoof LSPRs are fundamentally different from conventional localized plasmons at optical frequencies, since the EM fields inside the particle are no longer evanescent but propagating along the grooves.39 Figure 2 explores the dependence on θ of the scattering characteristics of the different electric and magnetic azimuthal multipolar spoof LSPRs (for lowest radial order). It plots the real (left) and imaginary (right) parts of the spoof LSPR frequency for n between 1 and 4 obtained from eq 8. At small θ, the real part of the eigenfrequency for the electric and magnetic dipole spoof LSPRs (which overlap spectrally) is highest. On the contrary, all the modes merge and shift toward zero frequency as θ increases. The large imaginary part of the dipolar spoof plasmon modes explains their negligible contribution to the SCS, as they lead to broad (very low quality factor) features in the cross-section spectra. This explains why only quadrupole and hexapole LSPRs can be identified in Figure 1b. At larger θ, the imaginary frequencies become similar, which causes the emergence of an apparent n = 1 LSPR peak in the lowest scattering band in Figure 1b. 1771


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Figure 5. (a) Schematic of the fabricated ultrathin textured coper disk (thickness t = 0.018 mm) on a flexible dielectric substrate with permittivity ε = 4.9 and thickness t = 0.8 mm. The geometrical parameters for the spiral structure are d = 0.45 mm, a = 0.3 mm, r = 1.1 mm, and R = 13.8 mm. (b) Experimental setup to measure the near-field resonance of the textured metallic disk, in which two monopole antennas are used as the source and the probe.

Figure 6. Measured (red line) and simulated (black line) resonance spectrum for the textured copper disk shown in Figure 5. The insets show the corresponding measured near-field distributions at the resonance peaks: (1) fundamental electric dipole; (2) fundamental magnetic dipole; (3) second-order electric dipole; (4) second-order magnetic dipole; (5) third-order electric quadrupole; (6) third-order electric dipole; (7) third-order magnetic dipole.

confinement capabilities of the plasmonic devices, its reduction implies that the corrugated wires lose their subwavelength light confinement abilities when N decreases. This point is confirmed by the spoof plasmon field maps in the insets of Figure 4a, which show how the magnetic field amplitude increases with N. Our findings suggest that in order to realize spiral spoof LSPRs with significant field confinement abilities, the number of corrugations has to be on the order of 20. Notably, the analytical calculation of the scattering crosssection for the spiral cylinder is also plotted in Figure 4 (gray dashed lines) for comparison. We can clearly observe that the

wires with the same geometric parameters as in Figure 3. Panel (a) corresponds to the lowest radial modes, and panels (b) and (c) show consecutive scattering bands. Four different N’s are considered: 2 (black), 5 (red), 8 (green), 10 (blue), and 20 (orange). Our numerical simulations indicate that by reducing the number of corrugations, spoof LSPRs red-shift, while the associated density of states (DOS) decreases in all scattering bands (the number of spoof plasmon modes supported by the wire drops). This effect is similar to the nonlocal screening of metallic electrons observed for conventional LSPRs at optical frequencies.41,42 Since the DOS is directly related to the field 1772


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from 0.5 to 1.5 GHz, from 2 to 3 GHz, and from 3.5 to 4.5 GHz, respectively. Both the electric and magnetic dipolar resonances are clearly apparent in all three bands. Remarkably, the experimental near-field profiles are in excellent agreement with the analytical field maps in Figure 3, which enables us to identify precisely the nature of the spoof LSPRs behind the measured scattering maxima. In this way, we can conclude that our experiments allow us to observe high-order magnetic modes not reported before. In order to verify the experimental results, we performed CST Microwave Studio simulations to calculate the extinction cross-section (ECS) of the structure; see Figure 5a. The system is excited by a linearly polarized plane wave incident from the top. The numerical ECS plotted as the black solid line in Figure 6a shows excellent agreement with the measured resonance spectrum.

numerical results approach the analytical results as the groove number N increases and already show good agreement as N reaches 20. In the next section, we will use this value for the experimental implementation of spoof LSPRs. Experiment. To further demonstrate the existence of the spoof LSP modes in subwavelength spiral particles, we fabricated an ultrathin disk with a spiral pattern and characterized experimentally its resonant field confinement characteristics through a 2D EM field mapper.32 The ultrathin design has several advantages. First, an infinite spiral wire cannot be fabricated in practice and the ultrathin structure is easier to implement. Second, truncating the 2D spiral cylinder can shift the magnetic modes from the electric ones and increase their excitation strength and quality factor,36,37 thus allowing the magnetic resonances predicted by the theoretical studies to be clearly observed. Figure 5a shows the schematic of our experimental sample, which consists of a copper disk whose perimeter is etched with a spiral pattern, placed on top of a 0.8mm-thick substrate with dielectric constant ε = 4.9. Other geometrical parameters of the structure are given in the figure caption. The fabricated sample and the measurement setup are displayed in Figure 5b. Note that by decreasing the thickness of the spiral disk, the number of high-order magnetic resonances increases. Two monopole antennas connected to an Agilent N5230A network analyzer are used to record the relative amplitude and phase of the local electric fields. The measured resonance spectrum and the near-field distributions for the spoof LSPRs supported by the sample are given in Figure 6. The comparison with Figure 3 shows that the LSPRs supported by the spiral disk shift to lower frequencies. This red-shift can be attributed to the finite thickness of the copper disk and the presence of the dielectric substrate, which lower the energy of the modes.37 The fundamental, second, and third scattering bands now span

■

CONCLUSION To conclude, we have developed a theoretical model to design and investigate subwavelength localized spoof plasmon devices with spiral textures. This model allows quantitative study of fundamental and higher order electric and magnetic plasmonic resonances, which can be engineered by the angle of the spiral and the number of grooves corrugating the perfectly conducting wire surface. Our method can be extended to study a 3D spiral disk of finite thickness, where the optical response under oblique incidence must be considered. The insightful character of this approach provides new paths for optimizing plasmonic devices in long-wavelength regimes.

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METHODS Under plane wave illumination, the magnetic fields in the background medium and spiral wire can be written as

∞ ⎧ ⎪ ρ ≥ R2 ∑ [inJn(k 0ρ) + bnHn(1)(k 0ρ)]einφ , ⎪ ⎪ n =−∞ Hz = ⎨ ⎛ ng k 0 ⎞⎤ ⎪ ∞ ⎡ ⎛ ng k 0 ⎞ ρ⎟ + dnY0⎜ ρ⎟⎥e in(φ − tanθ lnρ), R 2 ≥ ρ ≥ R1 ⎪ ∑ ⎢cnJ0 ⎜ ⎪ ⎢ cos θ cos θ ⎝ ⎠ ⎝ ⎠⎥⎦ ⎩ n =−∞ ⎣

(9)

Applying the boundary conditions (continuity of Hz and Eφ) yields ⎡ ⎛ ng k 0R 2 ⎞ ⎛ ng k 0R 2 ⎞⎤ ⎟⎥ ⎟ + dnY0⎜ i nJn (k 0R 2) + bnHn(1)(k 0R 2) = R 2−intan θ ⎢cnJ0 ⎜ ⎢⎣ ⎝ cos θ ⎠ ⎝ cos θ ⎠⎥⎦ i nJ ′n (k 0R 2) + bnHn(1) ′(k 0R 2) = −

⎛ ng k 0R 2 ⎞⎤ a cos θ −intan θ ⎡ ⎛ ng k 0R 2 ⎞ ⎢cnJ1⎜ ⎟⎥ ⎟ + dnY1⎜ R2 ⎢⎣ ⎝ cos θ ⎠ ng d ⎝ cos θ ⎠⎥⎦

⎛ ng k 0R1 ⎞ ⎛ ng k 0R1 ⎞ ⎟=0 ⎟ + dnY1⎜ cnJ1⎜ ⎝ cos θ ⎠ ⎝ cos θ ⎠

(10)

Solving eq 10, the unknown expansion coefficients can be obtained:

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bn = −i n

(pa /d)Jn (k 0R 2) − (qng /cos θ )J ′n (k 0R 2) (pa /d)Hn(1)(k 0R 2) − (qng /cos θ )Hn(1) ′(k 0R 2)

cn = [i nJn (k 0R 2) + bnHn(1)(k 0R 2)]Y1(ng k 0R1/cos θ )R 2intan θ dn = −cn

J1(ng k 0R1/cos θ ) Y1(ng k 0R1/cos θ )

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where p and q are given by eq 7.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. Author Contributions □

Z. Liao, A. I. Fernán dez-Domı ń guez, and J. Zhang contributed equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J.Z. and Y.L. are supported by Singapore Ministry of Education (MOE) AcRF TIER 1 under Grant No. RG72/15, AcRF TIER 2 under Grant No. MOE2015-T2-1-145, and the Program Grant (11235100003) from NTU-A*STAR Silicon Technologies Centre of Excellence. Z.L. and T.J.C. are supported by National Science Foundation of China (61171024, 61171026, and 61138001), National High Tech Projects (2012AA030402), 111 Project (111-2-05), Scientific Research Foundation of Graduate School of Southeast University (YBJJ1436), and Program for Postgraduates Research Innovation in University of Jiangsu Province (3204004910). A.I.F.-D. acknowledges funding from the EU Seventh Framework Program under Grant Agreement FP7-PEOPLE-2013CIG-630996 and the Spanish MINECO under contract FIS2015-64951-R. S.A.M. acknowledges the EPSRC (EP/L 204926/1), the Royal Society, and the Lee-Lucas Chair in Physics.

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ACS Photonics

Letter

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Homogenous Metamaterial Description of Localized Spoof Plasmons

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