We measure, simulate, and analyze the optical transmission through arrays of Ag nanorod pairs and U-shaped nanostructures as a function of polarization and angle of incidence. The bianisotropic nature of the metamaterials is exhibited in data and in simulations, and we argue that the electric field rather than the magnetic field excites the low frequency “magnetic” mode. We also observe spatial dispersion in the form of frequency shifts as a function of incident angle which we attribute to coupling effects between neighboring structures. A simple model based upon coupled electromagnetic dipoles is found to provide a qualitative description for the main features observed in the spectra.
© 2010 OSA
The pursuit of negative index materials (NIMs), a special class of metamaterials with simultaneous negative permittivity and permeability, that operate in the visible and infrared (VIS/IR) has largely been driven by the prospect of “perfect lensing” . NIMs were first demonstrated at GHz frequencies  where, compared to the VIS/IR, it easier to build resonant structures that are both complex and strongly subwavelength. In the VIS/IR much work has focused on the resonances of U-shapes and rod pairs towards the goal of achieving negative index [3–6], while some have argued that magnetic responses in such structures are difficult if not impossible at these frequencies given current fabrication techniques [7,8]. With some exceptions , previous studies have concentrated on the behavior of individual resonators and less so on their mutual interaction. In principle, these resonators can couple to resonators in adjacent cells  leading to “spatial dispersion” which is defined as the wave-vector dependence of the permittivity tensor ε(ω,k)  though it could apply to other material responses as well. These effects become more important as the cell size becomes comparable with the wavelength. Additionally, the U-shape and rod-pair resonators are asymmetric with respect to rotations about the z-axis which cause an anisotropic response and a mixing of the electric and magnetic response known as bianisotropy [7,11]; in general the constitutive relations for bianisotropic materials are and where the permittivity ε, permeability μ, and magnetoelectric coupling constant κ are tensors [12,13]. Historically investigators have assigned 3D permittivity and permeability to these kinds of arrays despite strongly subwavelength thicknesses [3–6]; strictly speaking, these types of arrays are 2D “metafilms” for which only surface susceptibilities are unambiguously defined . If one seeks to build 3D lenses by layering these resonator arrays, spatial dispersion and bianisotropy must be accounted for and effective material parameters must be “retrieved” [7,15] correctly from individual resonator cells. In order to gain a greater understanding of these effects, we systematically study the transmission through our resonator arrays as a function of polarization, resonator orientation and incident angle of illumination. Our choice of the unit cell’s plasmonic structures intentionally includes the intrinsically bianisotropic U-shape structure and the more symmetric rod-pair structure. In contrast to previous studies considering U-shapes only [3,4,9,16,17], comparing magnetic responses of these two structures sheds light on the two different types of bianisotropy observed in metamaterials: one caused by the spatial dispersion (as in the case of the rods) while the other dominated by the inherent bianisotropy owed to the lack of the spatial inversion symmetry.
Arrays of the resonators were fabricated by standard lithographic techniques. First 40 μm x 40 μm windows on ITO/glass substrates were fabricated by photolithography. The windows were designed to eliminate light contribution to the transmission spectra from areas outside the patterned area. Rod pairs or U-shapes were patterned by e-beam lithography. 75 nm of Ag was then deposited and followed by a lift-off procedure. The resulting square arrays have a lattice periodicity of 600 nm. The final dimensions of the U-shapes are approximately 320 nm in outer length and width and 90 nm in arm thickness. The rod pairs have the same dimensions as the U-shapes except that they lack the connecting crossbar. The transmission measurements were performed with a DA3.02 Bomem FTIR system in which a collimated beam of diameter 8 cm is focused onto the sample by a lens of focal length 30 cm. These measurements were made at normal and oblique incidence up to 80° with an estimated effective beam divergence of ± 4°. A more complete description of the fabrication procedure and measurement technique may be found in ref .
Figure 1 shows six different sets of transmission curves for 3 different combinations of polarizations and azimuthal orientations of the rod pairs and the U-shapes. For ease of reference, each set for the rod pairs is presented side-by-side with those of the U-shapes corresponding to the same polarization and orientation. We will refer to the components of the electric and magnetic fields of the incident light projected onto the plane of the arrays as and , respectively. We adopt a similar notation for the component of the light propagation vector within this plane, . Figure 1(a) shows the response for s-polarized light with parallel to the long axis of the rods. The spectrum is dominated by a deep, broad, resonant dip in the transmission. A narrower, less intense dip appears at lower frequency by an incident angle of 45° and grows in strength with increasing incident angle. This dip will be characterized as a magnetic resonance and will be one of the main objects of this study. Both resonances exhibit a slight red shift with increasing angle. In Figs. 1(c) and 1(e) with perpendicular to the long-axis of the rods, virtually no resonant behavior is observed. As frequency is increased, a mild transmission decrease does “turn-on” gradually and then flatten out in the spectrum of Fig. 1(c); this we interpret not as a resonance but rather as incident intensity passing into a diffracted beam at a particular onset frequency which we describe in detail in ref . This lack of strong response is attributed to the small polarizability of the plasmonic antennas along their narrow dimension. By contrast, all three sets of transmission curves for the U-shapes, Fig. 1(b), 1(d), 1(f), show a low frequency dip around 150 THz and at least one at higher frequency. The richest set of curves, Fig. 1(b), exhibits at least three resonant dips. Qualitatively, strong optical response for both polarizations is attributed to the composition of the U-shaped structure which contains plasmonic antennas elongated in both in-plane directions.
We simulated our experimental transmission curves using Ansoft HFSS, the results of which we present here. As we described elsewhere , model structures in the simulations were drawn to match SEM images of the actual structures, and the permittivities of the silver, glass, and ITO were inputted with Drude - Lorentzian model fits to reference data. In our simulations we use periodic boundaries and Floquet ports to allow for modeling of transmission at various angles of incidence.
Figure 2(a) shows a set of simulated transmission curves for the rod-pair arrays. The calculated spectra show good agreement with those measured for these structures, as seen in Fig. 1(a). In particular the calculated spectra reproduce the narrow and broad resonances labeled R1 and R2, respectively. The simulations also reveal that the R2 line shape is slightly modified by diffraction. By way of illustration, Fig. 2(a) also shows the calculated intensity of the beam diffracted into the glass substrate for θ = 20° incidence with a “turn-on” of about 280 THz. At larger angles (not shown) the diffracted intensity is smaller in magnitude and grows more gradually from a lower but more poorly defined turn-on frequency. Figure 2(b) shows calculated transmission spectra an array of U-shaped structures for comparison to the measurements presented in Fig. 1(b). Here the agreement to the data is reasonable if we note that the sharp features in the simulations appear to have been “washed out” by experimental limitations such as divergence in the incident beam, inhomogeneity in the U-shapes and array geometries, and incomplete rejection of background light signals,. In agreement with the data, the single dip seen at normal incidence at 280 THz evolves into two dips most noticeable for θ = 20°, and we see a dip emerge around 150 THz at larger angles. The last we label U1, and the former two as U2 and U3. The dramatic distortions of the U2 and U3 line shapes at larger angles we attribute to a strong interaction between the plasmonic resonances and the onset of a diffracted beam into the substrate or into the air at the Wood’s anomaly. We note that simulated spectra (not shown) of the orientations corresponding to Figs. 1(d) and 1(f) show the same sort of cusps and general flattening of the high frequency dip near 300 THz at oblique incidence; it seems reasonable that this effect along with the aforementioned experimental limitations is responsible for the rapid weakening of this dip with increasing angle which would otherwise be unexpected.
In order to help isolate plasmonic resonant modes from diffraction effects, we performed additional simulations. In Fig. 3(a) otherwise identical rod pairs with a reduced sample height of 20 nm (less than the skin depth of silver) were simulated; this red-shifts R2 significantly away from diffraction effects, reduces the frequency spacing between R1 and R2, and reduces the strength of R1. The red shift can be understood in terms of a reduction in the effective index of the surface plasmon responsible for the antenna resonance due to the relatively deeper field penetration into the interior of the nanorods . In Fig. 3(b) we present simulations of U structures with a smaller cell size and thickness similar to those reported by Enkrich, et al. . These curves show the three plasmonic dips more distinctly and in good agreement with their data. Simulated current distributions of the rod-pair modes R1 and R2 are shown in Fig. 4 while those of the U-shape modes U1, U2, and U3 are shown in Fig. 5 ; also shown are idealized illustrations of the predominant direction of current flow for each mode .
4. Bianisotropy of the low frequency modes
We now discuss the nature of the low frequency modes R1 and U1 which appear near 190 THz in the experimental spectra for the rod pairs and near 150 THz for the U-shapes, respectively. These are of interest because of the possibility that they might be magnetically excited, potentially resulting in μ < 0 for 3D structures based on these resonators. We first note that the R1 anti-symmetric mode arises from the coupling of rods within a cell. In noting the smaller mode separation and weaker coupling of R1 to the applied field in the thinner rods of Fig. 3(a) compared to those of Fig. 2(a), we conclude that the coupling mechanism between intracellular rods is capacitive. We now compare the two cases of s-polarization for the rod pairs as illustrated in Figs. 1(a) and 1(c). For either of these two cases, in principle, the anti-symmetric currents of R1 can be excited by the magnetic component of the incident radiation field. However, comparing Figs. 1(a) and 1(c) we see that the dip near 190 THz is only present if is along, but not perpendicular to the long axis of the rods. This indicates that pure magnetic excitation either does not occur or is too weak to be detectable. Figure 1(a) shows dips in the transmission spectra at frequencies corresponding to the R1 mode only at oblique incidence. This mode is not electric-dipole active for normal incidence; rather the lowest order moment is quadrupolar. However, for oblique incidence this mode can be excited electrically if is along the long axis of the rods due to the phase difference in the instantaneous field between individual bars. As noted in ref , this mode is electrically excited much less efficiently for the case of Fig. 1(c) in which is along the short axis of the rods in agreement with the absence of a corresponding dip in Fig. 1(c). These observations imply that the R1 anti-symmetric mode is essentially bianisotropic or magneto-electric in nature, i.e. an electric field induces a circulating current that gives rise to a magnetic field. If rod-pair arrays were to be extended to 3D, our conclusion is of potential importance as it implies that μ≅ 1 along the long axis; the “magnetic” response is contained wholly in the bianisotropic tensorial terms relating B and E. Additionally we note that the low frequency mode is essentially electric quadrupolar  leading to problems in defining an effective dielectric function because classical models used to define and derive effective material parameters typically assume that the higher-order multipole moments can be neglected .
Although U1 might be considered the analog of the anti-symmetric mode R1 observed in the rod pairs, comparing the measured transmission spectra of the two shapes allows us to see some differences arising from the presence of the cross bar. We note that the U-shape is an intrinsically bianisotropic structure owing to its lack of inversion symmetry. Specifically, finite incidence angle is not needed to excite the out-of-plane magnetic moment. For normal incidence, it has been shown that the U1 resonance can be considered as an LC resonance with the effective E-field inducing a voltage across the capacitance formed by the arms [3,7,18]; the cross bar completes the circuit. While the strength of the low frequency U1 mode seen in Figs. 1(b) and 2(b) shows an increase with incident angle similar to R1 in Figs. 1(a) and 2(a), it is broader, and the small red shift with angle observed for the rods is even less. U1 is also observed with parallel to the cross bar and across the capacitor gap in Fig. 1(d), whereas R1 is not observed in Fig. 1(c) where there is no cross bar to complete the circuit. Comparing the strength of the dip in Figs. 1(b) and 1(d), the response of U1 to the electric field is much stronger than any response it might have to the magnetic field as expected. In Fig. 1(d) there is a slight strengthening of the U1 with increasing incident angle, which might be expected from a weak response to the magnetic field fluxing through the current loop. However, U1’s increasing strength with angle in both 1(b) and 1(d) can in principle be accounted for by an increasing difference in phase of the instantaneous electric field at the far ends of the structure. In the former case, the E-field tends to drive opposite currents in the side arms, and in the latter it tends to drive a displacement current across the capacitive gap of the arms in the opposite direction of a real current driven along the cross bar. For the geometry of Fig. 1(f), the strength of the low frequency mode diminishes with incident angle which can be understood as the weakening in electric coupling due to the decrease in with increasing angle. The fact that the strength of this dip even at increasing angle is greater or comparable to the corresponding dip in Fig. 1(b) would further argue that in general the electric coupling is much stronger than any possible magnetic coupling. While there is no one geometrical condition which can distinguish fully between magnetically and electrically excited resonances, we propose that based on our observations, the low frequency mode observed for arrays of U-shaped nanostructures is better explained as an electrically excited resonance rather than a magnetic one. The bianisotropy discussed above for the rod-pair resonators applies to the response of the U-shapes as well and may affect retrieval procedures of effective material parameters accordingly.
5. Higher frequency modes
While the higher frequency modes are interesting in their own right, they are observed as deep and broad resonances onto which the low frequency modes are superimposed, so that proper parameter retrieval will require proper treatment of these modes. In particular, we consider the higher frequency modes observed in our transmission spectra, labeled R2, U2 and U3 in Figs. 2, 3, 4 and 5. While R2 is clearly the symmetric complement of R1, the addition of the cross bar to the bars to make a U-shape modifies the spectra significantly. First, we note that there is a high frequency mode around 300 THz excited for the s-polarization with parallel to the cross bar in Fig. 1(d) that is absent from Fig. 1(c) for the rod pairs. With parallel to the rod pairs at oblique incidence, “adding” the cross bar to form a U-shape causes the single high frequency resonance R2 of Figs. 1(a), 2(a), and 3(a) to split into the doublet modes U2 and U3 of Figs. 1(b), 2(b), and 3(b). If we were to model the three arms of the U-shapes as coupled oscillators, three resonances would be expected, and they are indeed observed. As U1 might be considered the analog of R1, U2 could be considered the analog of the symmetric mode for the rod pairs. Here the analogy is perhaps better; there is essentially no current flow along the cross bar in U2 (as there is in U1) and obviously no flow between rods in R2 (or R1). The additional mode, U3, is in someway also analogous to R1 in that, like U1, it has opposite current flows in its arms and can only be excited at oblique incidence when the E-field is along its arms. Of course U1 and R1 are lower in frequency than U2 and R2, respectively; U3 is higher. Its emergence is seen most clearly when going from θ = 0° to 20° in Fig. 3(b) near 410 THz. It then red shifts slightly and strengthens as the angle increases. We identify this mode essentially as a quadrupole mode, and we note the distinction between U3 and the low frequency mode U1. U2 and U3 are non-degenerate, but U3 has no active dipole moment at normal incidence for the orientation and polarization depicted in Fig. 1(b); it is excited only at oblique incidence due to the phase difference of the exciting E-field on the two side arms. We also note the symmetric resonances R2 and U2 are strongly over-absorbed so that their widths are dominated by their radiative lifetimes. The effect of ohmic losses on the mode decay times is more easily seen on the weaker anti-symmetric, low frequency resonances.
6. Spatial dispersion and coupling of resonators
Another issue in these optical resonators and a potential problem to proper effective parameter retrieval involves spatial dispersion. Ideally, an effective material parameter depends only on frequency and not on the wave vector of the incident light, i.e. ε(ω) as opposed to ε(ω,k ||) where ε in general is a tensor. In these structures the lattice periods are not sufficiently small compared with the optical wavelengths to allow k || ≈0 to be valid in the optical response. Spatial dispersion manifests itself as a shift in resonant frequency as the angle of incidence and thus k || increase. In particular, the plasmonic resonances in these metallic arrays disperse with changing k || corresponding to their photonic band structure. Here we will employ a simple dipole-dipole coupling as a zeroth order tool to qualitatively understand how our resonant structures couple to their neighbors and how this coupling results in the observed resonant shifts with incident angle. We note that a similar coupling has been described theoretically in infinitely long, 1D plasmonic strips . Though non-quantitative, our treatment is very similar in approach to that of Sersic, Frimmer, and Koenderink for U-shapes , except that in their case changing phase differences arise from variations in the cell size and not by changing incident angle. In our approach, we will start with the cases that are easiest to understand. We note that for our samples with a = 600 nm, ka varies from about 2 to 4 for the observed resonances, so it is perhaps not surprising that the effects of spatial dispersion are observed.
We start with R2 since this is essentially just a simple electric dipole. At k || = 0, corresponding to normally incident light, the dipole pairs in all cells oscillate in phase. As is well known, closest neighbors along the axis of the dipole produce a red shift as the spacing between them decreases, while closest neighbors perpendicular to the dipole axis produce a blue shift with decreasing spacing. We consider R2’s behavior in the orientation depicted in Figs. 1(a), 2(a), and 3(a). As k || increases with increasing angle of incidence, the phase difference between neighboring cells perpendicular to the dipole axis increases and the associated k || = 0 blue shift is reduced. The phase of closest neighbors along the dipole axis is unchanged and the overall result is a red shift; this indeed is observed in the data in Fig. 1(a) and the simulations in Figs. 2(a) and 3(a). One thing that tends to suppress the shift comes from the dispersion of ITO’s permittivity which increases with increasing frequency. An observed red shift due to a change in electric dipolar coupling implies an increase in capacitance but also a smaller value of ε(ω), and so the observed shift is smaller than it would have been for a dispersion-less substrate . This will apply in general to shifts due to changes in electric coupling. Having effectively an identical current pattern to R2, U2 red shifts in the same way, as best seen in the simulations in Fig. 3(b) and in going from θ = 0° to θ = 20° in Figs. 1(b) and 2(b). We again note that these spatially dispersive modes provide a significant background upon which the low frequency magnetic modes exist. We also note that at non-normal incidence R2 appears to have some anti-symmetric character, as shown in Figs. 4(e) and 4(f) . Since R1 is the anti-symmetric mode, we now discuss the influence of the anti-symmetry on the resonance shift.
R1 is more complicated since it has both an electric quadrupolar moment and a magnetic dipole moment [9,22]. At k || = 0, the R1 mode with its opposing currents in its rod pairs couples to its nearest neighbors oscillating in phase. The opposing currents give rise to a magnetic dipole in the z-direction perpendicular to the plane of the array. Coupling between these magnetic dipoles produces a blue shift. We now consider R1’s behavior in the orientation depicted in Figs. 1(a), 2(a), and 3(a). As k || increases with increasing angle of incidence, the phase difference between neighbors perpendicular to the long axis of the bars increases and the k || = 0 magnetic dipole blue shift is reduced. The overall result of the magnetic coupling is a red shift with increasing k ||.
The electric coupling affecting R1 is more complicated, but we can approximate it by considering first the dipole-dipole interaction of one rod of a given pair to the nearest neighbor in the adjacent cell. In direct contrast to R2, nearest neighbors perpendicular to the rod’s long axis oscillate 180° out of phase at normal incidence and produce a red shift. When k || increases, as is the case in Figs. 1(a), 2(a), and 3(a), the phase difference between neighbors perpendicular to the bar’s axis increases from 180° and the associated normal-incidence (k || = 0) red shift is reduced. The overall result of the electric coupling should be a blue shift with increasing angle of incidence. However, the next nearest neighboring bars will have the opposite phase as the nearest bars, and so we would expect the electric coupling to produce only a weak shift, certainly weaker for R1 than it is for R2. Additionally, at lower frequencies the ITO becomes even more dispersive, further suppressing the electric shift. Therefore the prevalence of the magnetically induced red shift is reasonably expected.
Similar to R1, U1 should be red shifted with increasing angle due to magnetic coupling. Both the data in Fig. 1(b), 1(d), 1(f) and the simulations in Figs. 2(b) and 3(b) do indeed indicate a very weak red shift. Unlike R1, U1 tends to have its currents highly localized in the interior of the arms , resulting in an inductive loop of smaller area than that of R1; this apparently results in a weaker intercellular coupling and red shift. Also unlike R1, U1 has highly localized electric fields in the interior of the arms of the U-shape , resulting in a short electric dipole, significantly shorter than that of R2, to couple to neighboring dipoles. However, this coupling should produce a blue shift for the low frequency resonance in the geometries depicted in Figs. 1(b) and 1(f) and a red shift for the geometry depicted in Fig. 1(d). Since weak red shifts are observed in all three, we conclude that, although weak, the magnetic coupling must dominate the electric coupling.
Also similar to R1 is U3 which apparently demonstrates a magnetically induced red shift as seen in Fig. 3(b). Even here, the line shape of the resonance may be affected by the sudden onset of the grating mode.
We have investigated the dependence of the resonant behavior of arrays of Ag rod pairs and U-shaped nanostructures on polarization, azimuthal orientation and angle of incidence. From these results we have concluded that the low frequency mode is primarily an electric excitation rather than a magnetic excitation. This implies that μ≅ 1 for 3D materials that might be composed of layers of such resonator arrays. We have also found that both rods and U-shapes can exhibit a bianisotropic response at oblique incidence. The rod pairs have spatial inversion symmetry but spatial dispersion effects at oblique incidence produce a bianisotropic response; in addition to this kind of bianisotropy, the U-shapes demonstrate a bianisotropic response due to the broken inversion symmetry that does not require spatial dispersion. We have shown there is sufficient coupling between neighbors for both U-shapes and rod pairs which leads to spatial dispersion in the form of a k|| dependence of the resonant features associated with the plasmonic band structure at oblique incidence. Therefore we conclude that bianisotropy and spatial dispersion effects can occur together or can be distinct features of the optical response of plasmonic arrays. Many of the features of these optical properties of these arrays can be qualitatively explained in terms of electric and magnetic dipole coupling and can show the relative strengths of these couplings.
G. Shvets, S. H. Mousavi and A. Khanikaev acknowledge funding by Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative grants FA9550-06-1-0279 and FA9550-08-1-0394.
References and links
4. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95(20), 203901 (2005). [CrossRef] [PubMed]
5. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]
6. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007). [CrossRef]
7. Y. A. Urzhumov and G. Shvets, “Optical magnetism and negative refraction in plasmonic metamaterials,” Solid State Commun. 146(5-6), 208–220 (2008). [CrossRef]
8. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. U.S.A. 106(6), 1693–1698 (2009). [CrossRef] [PubMed]
9. I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett. 103(21), 213902 (2009). [CrossRef]
10. J. J. Hopfield and D. G. Thomas, “Theoretical and Experimental Effects of Spatial Dispersion on the Optical Properties of Crystals,” Phys. Rev. 132(2), 563–572 (1963). [CrossRef]
11. D. R. Smith, J. Gollub, J. J. Mock, W. J. Padilla, and D. Schurig, “Calculation and measurement of bianisotropy in a split ring resonator material,” J. Appl. Phys. 100(2), 024507 (2006). [CrossRef]
12. J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990).
13. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).
14. C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: The two dimensional equivalent of metamaterials,” Metamaterials (Amst.) 3(2), 100–112 (2009). [CrossRef]
15. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002). [CrossRef]
18. T. D. Corrigan, P. W. Kolb, A. B. Sushkov, H. D. Drew, D. C. Schmadel, and R. J. Phaneuf, “Optical plasmonic resonances in split-ring resonator structures: an improved LC model,” Opt. Express 16(24), 19850–19864 (2008). [CrossRef] [PubMed]
19. T. D. Corrigan, P. W. Kolb, A. B. Sushkov, H. D. Drew, R. J. Phaneuf, G. Shvets, “Wood’s anomaly in arrays of highly anisotropic plasmonic antennas,” to be published. Here conservation of crystal momentum is used to derive the equation for the onset of diffraction where a is the period of the array, fa = c/a, the incident angle θ is taken to be within the plane defined by k and the substrate normal, n is the index of refraction of the medium (1.00 for air and 1.46 for glass), and m and j are integers specifying the component of momentum transfer along the reciprocal lattice unit vectors along and perpendicular to , respectively.
21. The simulation current plots (Figs. 3 and 4) show the currents in response to an exciting wave of frequency ω. If ω is chosen at or near one of the resonances the corresponding resonance dominates the response. However, the other nearby resonances also contributes to the observed field patterns. In addition the response has in-phase and out-of-phase components that depend both on the dissipation and ω. These factors cause small distortions in the current distribution from the ideal patterns expected from symmetry. The significant deviation seen in Fig. 3 (e, f) can be understood as a mode of coupled oscillators where the individual oscillators are driven out of phase; at the resonant frequency their motion will in general not be in phase.
22. D. J. Cho, F. Wang, X. Zhang, and Y. R. Shen, “Contribution of the electric quadropole resonance in optical metamaterials,” Phys. Rev. B 78, 12101 (2008). [CrossRef]
23. M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75(11), 115104 (2007). [CrossRef]
24. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004). [CrossRef]