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Using terahertz near-field imaging we experimentally investigate the interaction between split-ring resonators (SRRs) in metamaterial arrays. Depending on the inter-SRR spacing two regimes can be distinguished for which strong coupling between SRRs occurs. For dense arrays SRRs couple via their electric and magnetic near-fields. In this case distinct deformations of the SRRs’ characteristic near-field patterns are observed as a signature of their strong interaction. For larger separations with a periodicity matching the resonance wavelength, the SRRs become diffractively coupled via their radiated fields. In this regime hybridization between plasmonic and lattice modes can be clearly identified in the experimentally obtained near-field maps.
©2011 Optical Society of America
1. Introduction
Electromagnetic metamaterials have recently attracted great interest due to their tailorable electric and magnetic properties. They typically consist of two- or three-dimensional arrangements of subwavelength-sized metallic structures, whose response to an incident electromagnetic field defines the properties of the resulting metamaterial. Many different structures have been introduced with split-ring resonators (SRRs) arguably being the most prominent metamaterial subunit [1]. In order to fully understand the response of a metamaterial, near-field studies have proven to be extremely useful [2–5]. With typical structure sizes on the subwavelength scale, however, near-field experiments with the required spatial resolution are highly challenging, in particular at optical frequencies. Experiments with longer wavelength radiation, on the other hand, can overcome this complication. For example, near-field microscopy at THz frequencies has been demonstrated to be a particularly powerful tool to investigate plasmonic interaction between metallic microstructures and electromagnetic waves [6–12]. This approach allows mapping electric and magnetic near-fields in a polarization-, time- and frequency-resolved manner. Its recent application to metamaterial structures has provided valuable information on local fields and current distributions, enabling identification of the fundamental eigenmodes of particular resonator designs and arrangements [13–16].
Most experimental near-field studies so far have focussed on investigating the response of isolated resonator elements only, neglecting their mutual interaction. Owing to their typical arrangement in dense arrays or stacks, however, coupling between the elements plays an important role and the effective response of a metamaterial can not be simply attributed to the properties of the single constituents only. For example, spectral splitting is observed as a consequence of strong coupling between vertically stacked SRRs [17–19] or laterally arranged SRR pairs with broken symmetry [20, 21]. Even in the widely used regular SRR-based metamaterial arrays the separation between the elements has significant influence on frequencies and linewidths of the fundamental resonances [22, 23].
In principle two coupling regimes can be distinguished which could tentatively be classified into (i) near-field and (ii) far-field coupling between the metamaterial unit cells. For small lattice periodicities interaction between the evanescent electric and magnetic near-fields of the SRRs occurs (e.g. dipole-dipole coupling) [22, 24]. This typically leads to enhanced coupling to the incident field and consequently to superradiant decay of the collective mode. As a consequence, for densely packed metamaterial arrays spectral features in far-field transmission may show significant broadening [22, 25]. On the other hand, at rather large separations the plasmonic particles may couple via their radiated far-fields [26]. For example, if the periodicity matches a multiple of the wavelength (e.g. g = λ) a diffractive order radiates into the lattice plane mediating strong coupling between the elements. This effect, also referred to as diffractive coupling, can lead to extremely sharp resonances. It has been investigated in detail for plasmonic particle and nanoantenna arrays [27–31] and was recently found to also occur in split-ring metamaterial arrays [23, 32]. We have characterized the effect of diffractive coupling on the far-field transmission spectra of SRR-based metamaterials in detail in a previous work [23]. Here we significantly extend this study by taking both coupling regimes into account and presenting corresponding electromagnetic near-field maps for the first time, using THz near-field imaging. Our measured field maps of arrays with varying lattice periodicities reveal the underlying microscopic signatures of the interaction both, in the near- and in the far-field coupling regime.
2. Sample fabrication and experiment
Various arrays of U-shaped SRRs have been fabricated by conventional photolithography and subsequent metal etching. The samples consisted of 10 μm thick copper structures on a 120 μm PTFE (teflon) substrate. Periodicities have been varied in steps from 380 μm to 1200 μm in x-and y-directions separately. A microscope image of a section of a typical SRR array is shown in Fig. 1(a).
Fig. 1 (a) Microscope image of a section of a SRR array. The sidelength of an individual SRR structure is l = 300 μm and the width w = 30 μm. Lattice periodicities were varied in steps from 380–1200 μm in x- and y-directions separately. (b) THz transmission spectrum of an array with gx = gy = 500 μm with the electric field of the incident wave polarized along the x-axis. Vertical lines indicate expected spectral positions of plasmonic resonances and lattice modes according to Eqs. (1) and (2).
Transmission spectra of the samples in the range from 50 GHz to 1 THz have been obtained by THz time-domain spectroscopy (THz-TDS) [33]. Near-field maps of the SRR samples were measured by scanning-type THz near-field microscopy [11, 14]. In this setup, the sample is illuminated from one side by broadband THz pulses generated from a photoconductive THz emitter. Another photoconductive antenna acting as near-field probe is raster-scanned across the sample. The probe is oriented to be sensitive to one linear polarization of the electric field in the plane of the sample. From two consecutive measurements of the electric field components Ex and Ey the in-plane electric vector field Exy can be reconstructed. By applying Faraday’s law, the out-of-plane magnetic field component Bz is determined from the measured in-plane electric vectors. For a measurement the sample was positioned in close proximity to the near-field probe (behind a ∼40 μm thick dielectric protective layer, n=1.5) and either the detector, or the sample, was raster-scanned in x- and y-direction in order to map the spatial field distributions. The spatial resolution is on the order of 30 μm which corresponds to λ/20 at 0.5 THz.
3. Plasmonic eigenmodes of SRRs and lattice modes in the array
The spectral signatures of a particular metamaterial array are shown in Fig. 1(b). In this case the THz far-field transmission spectrum was measured for a split-ring square array with gx = gy = 500 μm with the electric field polarization of the incident wave along the x-axis. For this field polarization odd plasmonic eigenmodes of the rings are excited. Their characteristic resonance frequencies can be estimated to a reasonable approximation by
where L = 3l is the length of the unfolded SRR, corresponding to the resonator length, and c * the speed of light divided by an effective refractive index of 1.2, which is between that of the PTFE substrate and air. As indicated by the dashed vertical lines in Fig. 1(b) the two dominant spectral features at lowest frequencies can be assigned to the n=1 and n=3 modes. The significant shift of the n=3 resonance from its estimated spectral position is mainly due to the onset of diffractive coupling between the SRRs [23]. A characteristic drop in transmission above 450 GHz is observed, which shifts with changing lattice periodicity as discussed in section 4.1, indicating its origin as diffractive lattice excitation [34]. In a rectangular array with periodicities gx and gy lattice modes are excited at frequencies which obey where (i, j) is a pair of integers counting the diffractive order.The central frequencies of the (1,0) and the (0,1) lattice mode, which are identical in a square array, are indicated by the coinciding green and blue lines in Fig. 1(b).In order to spatially map the resonant near-fields of the eigenmodes we have initially performed THz near-field imaging on a single SRR structure. The measured near-field maps of the SRR’s resonances for a horizontally polarized incident electric field are shown in Fig. 2(a), with the arrows indicating the in-plane electric field vectors, and the colors the out-of-plane magnetic field component. Note, that the fields are plotted for one particular phase within the oscillation cycle of the mode. To show both fields at their maxima on the same map the magnetic fields are presented with a phase shift of π/2. As expected, the lowest-order eigenmode at 162 GHz (n=1 mode) corresponds to a circularly oscillating current inducing a magnetic dipole in the ring (LC-resonance or magnetic resonance). The next higher order mode (n=3) observed at around 400 GHz corresponds to the formation of an electric quadrupole with the electric field vectors pointing toward and away from the corners of the structure [14]. The corresponding current flows and charge distributions are sketched in the inset. We find that in all near-field measurements resonance frequencies are slightly shifted relative to the features observed in the far-field transmission spectra due to the presence of the near-field detector. The higher order resonance (n=3) is close to the upper limit of the frequency bandwidth of our near-field imaging setup and therefore contains significant noise.
Fig. 2 (a) THz near-field scan of a single SRR at its plasmonic resonances (n=1, n=3) showing the in-plane electric (arrows) and out-of-plane magnetic (color code) near-fields. (b) Simulated near-fields in a plane 60 μm behind the SRR. The insets show the currents and charge distributions associated with each resonance.
To validate that our measurements faithfully reproduce the near-field distributions close to the SRR, we have performed a complementary numerical simulation of the fields around a single SRR using the finite element method (FEM) based program package COMSOL Multiphysics. The result of the simulation is shown in Fig. 2(b) evaluated for a distance of 60 μm behind a copper SRR, which is positioned on top of a 120 μm thick substrate with a refractive index of n=1.5. We find good agreement between our measurements and the simulation, in particular for the magnetic field patterns. Examination of the field maps from experiment and simulation show that the resonant fields extend beyond the dimensions of a single SRR. These extended field contributions act on neighboring resonators invoking coupling between the unit cells in the array as will be discussed in the following section.
4. THz near-field study of electromagnetic SRR coupling
The local electromagnetic field acting on an individual SRR within an array corresponds to the superposition of the incident field and the field generated by all other structures in the lattice. The latter can be further separated into two components: (i) the non-radiative evanescently decaying near-field of directly neighboring SRRs, and (ii) the scattered field radiated from the SRRs into the plane of the array (radiated far-field). Depending on the inter-SRR spacing either near- or far-field coupling becomes the dominant mechanism. In order to investigate the effect of separation on the constituent coupling we have investigated arrays with varying lattice periodicities. By selecting arrays with either small, or large separations we are able to study coupling between the SRRs both, in the near-field as well as in the far-field regime.
4.1. Electromagnetic near-field coupling
First, we investigate the behaviour of the resonances in the array as the SRR periodicity decreases below the extend of their near-fields. Figure 3(a) shows the measured transmission through SRR square arrays as a function of frequency and of their periodicity (gx = gy = 380–620 μm in 20 μm steps). The vertical dashed lines mark the expected frequencies of the eigen-modes n=1 and n=3 of the SRRs calculated according to Eq. (1). The solid green and blue curves indicate the periodicity dependent lowest order lattice modes along x- and y-directions, (1,0) and (0,1), as determined from Eq. (2), which have identical frequencies in a square array. All theoretically predicted features are identified in our measurement. While the resonance frequencies of the plasmonic eigenmodes exhibit only slight dependence on the lattice period, their widths show significant broadening with decreasing separation as signature of electromagnetic coupling between the SRRs.
Fig. 3 (a) Measured transmission spectra of SRR square arrays as a function of periodicity (gx = gy). Resonance frequencies of the plasmonic eigenmodes n=1 and n=3 according to Eq. (1) are indicated by vertical dashed lines and the lowest order lattice modes by solid curves. Circles (A1–A3,B1–B3) mark periodicities and frequencies for which THz near-field maps are shown in Fig. 4. (b) Quality factor versus lattice periodicity gx = gy (solid dots) and gy (open squares) for the two lowest plasmonic eigenmodes (n=1 blue, n=3 red).
The effect of the decreasing periodicity of the SRRs on their local fields is demonstrated in Fig. 4. Electromagnetic near-field maps of the plasmonic resonances have been determined for selected SRR arrays, indicated in Fig. 3(a) by the circles. Note that for these measurements the sample was scanned in x- and y-directions through the focused THz excitation in front of the stationary near-field probe. This mode of operation mimics uniform illumination over a large sample area resulting in relatively homogeneous field maps.
Fig. 4 Measured in-plane electric (vectors) and out-of-plane magnetic near-fields (colors) of SRR arrays (field of view: 1.4 × 1.4 mm) with different lattice periodicities at the plasmonic eigenresonances n=1 (top row) and n=3 (bottom row). A whole oscillation cycle can be seen in Media 1. The near-field maps correlate to the corresponding frequencies and lattice periods, A1–A3 and B1–B3, indicated in Fig. 3(a).
For the LC-resonance (n=1) at the largest periodicity (gx = gy = 550 μm) the near-fields localized around a SRR in the array are nearly identical to the field patterns around a single isolated resonator (see Fig. 2(a)), with a circularly oscillating current forming a magnetic dipole inside the ring. Whereas the magnetic field distribution is basically unchanged for different periodicities, we observe a transition from initially circular electric fields to linearly polarized fields, aligned along the incident field polarization (x-axis) as the spacing is decreased. This is the result of longitudinal coupling between the electric dipoles forming across the SRR gaps [24]. The macroscopic dipole associated with the resulting collective mode enhances coupling to the incident field leading to superradiant damping and the observed spectral broadening [22, 35]. In the case of the n=3 mode, where the magnetic field has a larger spatial extent (Fig. 2(b)), a spatial compression of the overlapping magnetic near-field around the SRRs is observed with decreasing lattice periodicities. However, the structure of the quadrupolar field patterns is preserved both, for the magnetic as well as for the electric near-fields. In this case the collective mode formed in dense arrays consists of individual microscopic quadrupoles centered on each SRR, leaving the coupling to the excitation field essentially unchanged.
This difference in coupling of the modes to the external field is expressed in their periodicity dependent Q-factors shown in Fig. 3(b), calculated according to Q = Δν/ν 0, where Δν is the spectral width and ν 0 the central frequency of the resonance. In the plot the Q-factors with changing periodicity gx = gy are compared with a measurement where only gy was varied and gx = 380 μm was left constant. In all cases the Q-factor rises with increasing periodicity. However, for the LC-resonance (n=1) this increase is significantly smaller, when the SRRs are only separated in y-direction (blue squares) due to the still strong coupling along the x-axis, in comparison to separation in both directions (blue dots). This is in contrast to the n=3 mode where the Q-factor increases equally when the lattice periodicity is changed either in both (red dots) or only in one direction (red squares). This observation indicates that near-field coupling is not contributing significantly to the broadening of this plasmonic resonance. In fact in this case the increasing Q-factor for larger separations can be attributed entirely to enhanced diffractive coupling, as will be shown in the next section.
4.2. Diffractive far-field coupling
For large separations between the SRRs (gx or gy ≥ λ) near-field coupling between neighboring elements becomes increasingly negligible. For particular periodicities, however, radiative coupling between SRRs can be invoked [23]. In the following study only the periodicity along the y-axis was successively changed in our samples while gx was kept constant. This allows us to separate the lattice modes in x- and y-directions due to their different spectral positions. Figure 5(a) shows transmission spectra of SRR arrays with constant gx = 380 μm as gy is varied from 380 μm to 1200 μm. Again, the frequencies of the lowest order odd eigenmodes as well as the lattice modes calculated according to Eqs. (1) and (2) respectively, are indicated in the figure. Most of the expected features are well-reproduced in the measurement. A particularly interesting behavior is observed where a lattice mode coincides spectrally with a SRR eigenmode, e.g. for (0,1) coinciding with n=3 and faintly visible for (0,2) with n=5. In this case we observe an anticrossing behavior between the eigenmodes of the SRRs and the lattice excitations, as signature of strong coupling between the two modes. In addition, significant spectral line-narrowing of the low-frequency coupled mode is observed in this diffractive coupling regime leading to an extremely high Q-factor as shown in Fig. 5(b) [23].
Fig. 5 (a) Measured transmission spectra of SRR arrays as a function of y-periodicity for constant periodicity in x-direction (gx = 380 μm). Resonance frequencies of plasmonic modes n=1, 3 and 5, calculated according to Eq. (1), are indicated by vertical dashed lines. Lattice modes, calculated according to Eq. (2), are shown as solid green and blue curves. Circles mark periodicities and frequencies for which THz near-field maps are shown in Fig. 6. (b) Quality factors of the plasmonic eigenmodes.
Such strong mode coupling is associated with the formation of hybrid modes [36]. In order to experimentally verify hybridization of electromagnetic modes we measured the resonant near-fields of two SRR arrays with different periodicities, indicated by the circles in Fig. 5(a). In the first case (A) plasmonic and lattice modes are spectrally well-separated (weak coupling regime), whereas sample B was chosen to be close to the avoided crossing (strong coupling regime). Figure 6 shows near-field measurements of both arrays at their resonances (A1–A3 and B1–B3). In contrast to the near-field scans in the previous section, here the detector was scanned in x- and y-directions behind the stationary array sample and relative to the stationary THz excitation beam. The THz beam was defocused to a spot size of ∼ 4 mm to ensure efficient excitation of lattice modes through illuminating a sufficient number of SRRs (at least 30 at the largest spacing). Again, for the lowest frequency mode (n=1) we find the formation of a magnetic dipole inside the SRRs for both lattice periodicities (A1 and B1). At 390 GHz for the smaller lattice periodicity (A2), the n=3 resonance (see also Fig. 2) can be clearly identified, representing an electric quadrupole with corresponding four-fold magnetic field pattern, centered on each SRR. The next resonance observed at 465 GHz (A3) corresponds to the (0,1)-lattice mode, exhibiting an oscillating standing wave pattern along the y-direction. The corresponding mode pattern only depends on the y-periodicity showing almost no dependence on the resonator size and structure. On the other hand, resonances B2 and B3 can not be unambiguously assigned to either a plasmonic or a lattice excitation, but they show significant mixing of both modes. This is also expressed in the 2D Fourier transforms of the magnetic field patterns shown in the insets, which reflect the different modal patterns’ spatial periodicity. The color in these plots scales with the amplitude of the spatial Fourier transform, representing the momentum distribution of each mode in reciprocal space. While (A2) follows the x- and y-periodicity of the SRRs, expressed by the four symmetric maxima in x- and y-directions, (A3) is mainly periodic along the y-axis, i.e. along (0,1)-direction. Hence, (A2) and (A3) represent uncoupled modes as expected in this weak coupling regime. The 2D Fourier transform plots of B1 and B2, however, correspond in both cases to the superposition of the pure plasmonic (four-fold symmetry) and the pure lattice mode (two maxima in ky-direction), demonstrating mode hybridization in this diffractive coupling regime.
Fig. 6 Measured THz near-field maps at the resonance frequencies of SRR arrays (field of view: 1.4 × 1.4 mm) with two selected lattice periodicities, gy = 487μm (A) and gy = 700 μm (B); gx = 380 μm in both cases (Media 2). Insets show the corresponding spatial 2D Fourier-transforms of the magnetic field maps.
5. Conclusion
In conclusion, we have used THz far-field spectroscopy and THz near-field microscopy to experimentally investigate coupling between the constituents in split-ring resonator based meta-material arrays. Depending on the inter-SRR spacing two coupling regimes can be distinguished. For densely packed arrays overlapping near-fields of neighboring SRRs result in a characteristic deformation of the local fields for their plasmonic resonances, as observed in our near-field maps. We show that for the fundamental LC-resonance the microscopic dipole moments of the individual SRRs form a macroscopic dipole in the collective coupled mode, while for the higher order quadrupole mode the microscopic quadrupolar field patterns are preserved. For larger periodicities long-range coupling between the SRRs is mediated by diffractive lattice modes. In this regime, as a signature of strong interaction between plasmonic and lattice excitations we observe mode mixing associated with the formation of plasmon-lattice hybrid modes. Our approach based on correlating periodicity-dependent far-field transmission spectra with near-field maps provides microscopic insight into coupling between the meta-atoms in metamaterial arrays.
Acknowledgment
The authors acknowledge Hanspeter Helm for helpful discussion. We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG), grant No. WA 2641/5 and the Open Access Publication fund of the DFG.
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