## Abstract

It is demonstrated that higher-order plasmonic modes in the split-ring resonators (SRRs) are strongly enhanced when SRRs are arranged in a densely spaced two-dimensional array. The mode enhancement results from the near-field electrical coupling between the adjacent resonators. The effect is most pronounced in narrow gap SRRs which allows to observe experimentally plasmon modes up to the seventh order. In the array of narrow gap SRRs, the fifth-order resonance demonstrates high Q-factor, high resonance strength and wide tunability which opens up attractive features for practical applications of planar SRR structures.

© 2017 Optical Society of America

## 1. Introduction

Recently developed electromagnetic metamaterials (MM) have attracted a tremendous amount of interest because of unique electromagnetic responses which have never been found in natural materials. Metamaterials are periodic structures of artificial “atoms” with “lattice constants” that are smaller than the wavelength of light. Pendry et al. showed that a combination of metallic split ring resonators (SRRs) (“magnetic atoms”) [1] and metallic wires (“electric atoms”) [2] with negative permeability and permittivity, respectively, can lead to materials exhibiting a negative index of refraction. First demonstration of negative refraction index was reported in microwaves [3]. This has led to intense theoretical, computational and experimental studies and the negative refraction issues have covered broadband scale of electromagnetic radiation starting from microwaves up to optical wavelengths [4].

Special emphasis needs to be given to terahertz (THz) range. Electromagnetic waves at these frequencies have the advantage of allowing the use of quasi-optical measurement methods, while the MM elements remain still relatively large and can be easily fabricated with high precision using standard photolithography. Therefore, THz range is very attractive for experimental demonstration of delicate physical effects in MM [5]. On the other hand, THz MM provide a promising approach towards filling the “THz gap” [6]. A variety of components based on terahertz MM have been proposed. These include tunable filters [7], perfect absorbers [8], ultrathin lenses [9], compact zone plates [10] amplitude and phase modulators [11], biosensors [12].

In most experiments, planar MM are manufactured using SRRs as fundamental building blocks. Many terahertz MM consist of tightly packed arrays of SRRs, mainly to increase the material's overall response. Moreover, mutual coupling can play a significant role and also provides an extra degree of freedom for the manipulation of metamaterial response. Coupling of electric and magnetic fields of the adjacent resonators produces a red shift or blue shift of the resonance, depending on the nature of the near-field coupling [13–15]. By modifying the period and geometrical arrangement of the SRRs, variations of light extinction and resonance Q-factor have been obtained [13–16]. In addition, a long-range radiative coupling at larger separations between SRRs via scattered far-fields were reported [17]. This long-range diffractive coupling occurs when the lattice period of the SRR array matches the wavelength of incident wave, leading to a significant reduction in radiation damping that enhances the Q-factor of resonances [17,18]. The sharp transparency has been observed as a result of coupling of the first-order lattice mode to the structural resonance of SRRs [19]. The lattice modes have been observed in a number of works as a characteristic minima in far-field transmission spectra in U-type SRRs [20], C-type SRRs [16], O-type resonators [21]. It has been also revealed that radiative coupling causes anti-crossing and hybridization between the SRR eigenmodes and the diffractive lattice modes [22,23].

For polarization parallel to the gap-bearing side of SRRs, plasmon modes of the first-order and third-order are usually at the focus of interest [18,22–25]. Modes of a higher-order, although they have the advantage of allowing the use of MM at higher frequencies [26], receive much less attention. The fifth-order mode has been observed mainly in model simulations [20,27]. Experimentally, this mode has been found in U-type SRRs in the mid-infrared [15]. Usually the fifth-order mode is difficult to detect experimentally, because imperfections of sample geometry result in strong broadening and damping of this resonance [20]. It is interesting to note that the transmission spectra reported in [28,29] show that the fifth-order plasmonic mode appears with a high resonance strength in C-type SRRs in THz region. However, the origin of this resonance is not discussed in [28,29].

In this work, we study in detail the collective excitation of SRRs using planar arrays of different lattice periods and SRRs of different designs. We show for the first time that the electric near-field coupling between the SRRs is responsible for enhancement of the higher-order plasmon modes in dense SRR arrays. In particular, we discuss the conditions which allow to achieve high Q-factor, strong THz extinction and wide tunability of the fifth-order plasmon mode, thus making it a promising candidate for various practical applications such as filters, modulators and sensors. Experimental observations are confirmed with a numerical simulations using a custom-made software based on finite-difference time-domain method.

## 2. Sample fabrication, experimental and simulation details

A typical periodically arranged structure along with a sketch of a single unit cells and relevant geometrical parameters is shown in Fig. 1. The studied MM samples were fabricated on a reinforced PTFE substrate laminated with a copper foil (Rogers RT/duroid 5880). Planar arrays of SRRs have been fabricated by conventional photolithography and wet etching techniques. PTFE film of thickness 125 μm, dielectric permittivity 2.2 laminated with copper foil of thickness 9 μm was used. To explore the SRR resonant modes we employed rectangular SRR arrays with the lattice period in the range 600 – 1000 μm. In the samples, geometrical parameters of the split gap were varied, which led to different effective capacitance in the gap. In Fig. 1, samples S1 – S4 are shown in ascending order of the effective capacitance of the gap.

The power transmission was measured using commercial frequency-domain terahertz spectrometer (Toptica TeraScan 780). Far-field transmission spectra were obtained by normalizing the measured transmission to the reference transmission of air. In terahertz regime, dissipative losses are negligible since the metals are almost perfect conductors and the PTFE substrate is practically non-absorbing. In addition, the use of a 125 μm thick PTFE membrane as a substrate shifts the second Fabry–Pérot transmission peak behind 500 GHz, i.e. outside of our region of interest. Therefore, all spectral features can be interpreted in terms of fundamental excitations of SRRs in planar arrays.

Experimental observations are confirmed with numerical simulations using a custom-made program based on finite-difference time-domain method. For the simulation of the SRR array, one cell containing one SRR was modeled with periodic boundary conditions at the lateral edges. The differentiated Gaussian pulse was generated using total-field-scattered-field plane wave source [30]. The incident wave vector was perpendicular to the SRR array surface which was placed at the first third of the modeling space on the side of the excitation plane. The modeling domain was truncated by the uniaxial perfectly matched layers [30] to introduce the absorption of waves without reflection. To calculate the transmission spectra, a method based on the generalized Goertzel algorithm was used [31].

## 3. Plasmonic eigenmodes and the lattice modes in SRR arrays

Before proceeding to the experimental results, we will discuss the anticipated resonant effects in planar SRR arrays. Transmission spectra of samples were studied at normal incidence with electric field polarized in parallel to the gap bearing side of SRRs. In this case, the SRRs do not couple to the incident magnetic field which is parallel to the plane on which SRRs are deposited. However, circular current is induced in the metallic rings due to asymmetry of SRR in the direction of electric field. The oscillating current leads to charge accumulation across the gap resulting in the first-order plasmonic resonance (also called *LC* resonance). The resonant frequency is determined by inductance of the loop and effective capacitance of the gap. A magnetic dipole is coupled to electric excitation as a result of the circular current [27]. For the considered polarization only asymmetric higher-order plasmonic modes in SRRs are excited. As a crude picture we can imagine the plasmonic resonances in the SRRs as charge density waves on a rod of a length equal to the arc length of the SRR ring [27]. At resonance the length (perimeter) of SRR approximately equals to an odd multiple of half-wavelength [23]. Thus, the characteristic resonance frequencies are given by:

*P*is the perimeter of a SRR,

*c*is the speed of light in vacuum and

*n*is the effective refractive index of media surrounding SRR.

_{ef}In periodic arrays, one can observe the Wood anomaly [32], also called lattice mode. The lattice mode appears at the frequency at which the radiation scattered by individual SRRs interferes constructively along the substrate. The resonance frequency of the first-order lattice mode in a rectangular periodic array is determined by [19]:

where*L*is the period of the lattice.

The higher-order lattice modes [22] are of no particular interest for the present study and will not be discussed further.

#### 3.1 “Wide gap” SRR arrays

First, we investigate the behavior of the resonances in the U-type SRR array (sample S1). Mutual coupling between resonators is controlled by varying the period of rectangular lattice. Fig. 2 shows the measured far-field transmission spectra of U-type SRR array. The first-order resonance occurs at approximately 95 GHz which is clearly visible as a characteristic transmission minimum in the far-field spectra. Beside the typical *LC* resonance, SRRs also exhibit higher-order excitation modes. Second pronounced dip in transmission is found at around 230 GHz. It corresponds to the third-order (*n* = 3) mode associated with the formation of the electric quadrupole structure with the electric field lines pointing toward and away from the corners of SRRs [22,27]. The third resonance occurs at a frequency in the range 300 – 350 GHz. Our investigation revealed that this resonance represents the fifth-order (*n* = 5) mode excited in the SRRs. This is confirmed by the fact that at *L* = 600 μm, the first three plasmon modes are excited at frequencies proportional to the series 1: 2.6: 4.0, which is similar to that reported in [18]. Another small transmission minima denoted as LM appears at 335 – 430 GHz as shown in Figs. 2(a)−2(c). This type of resonance corresponds to the frequency at which the first diffraction order in the substrate changes from an evanescent mode to the propagating one [18]. Excitation of the lattice modes in U-type SRR arrays has been explored in detail in [22,23]. At higher frequencies (above the LM frequency) hybrid resonances are observed, characterized by a complex distribution of the amplitudes of the electric field. This is outside the scope of this paper.

The spectral results are summarized in Fig. 2(d). To highlight evolution of the lattice mode, the resonant frequencies are plotted as functions of reciprocal lattice constant. The horizontal dashed lines in Fig. 2(d) indicate the resonant frequencies of plasmonic eigenmodes calculated according to Eq. (1). Calculating the spectral position of the resonances, we take into account the fact that the surface current is concentrated at the inner edge of the SRR [33]. Thus, we take the perimeter *P* = 1300 μm. The effective refractive index was taken *n _{ef}* = 1.2 as reported in [23]. Dots in Fig. 2(d) correspond to resonant frequencies determined from the measured transmission spectra. Figure 2(d) shows that resonant frequencies of the third-order and the fifth-order resonances are significantly below their spectral positions estimated from Eq. (1). It is worth noting that the proposed Eq. (1) represents the eigenmodes of a straight rod. In U-type SRR, the geometric inductance [27] and the effective capacitance are larger than for the straight rod, which results in the red shift of the resonant frequency with respect to that predicted by Eq. (1). In addition, the resonant frequency of the plasmonic modes is affected by the coupling between the SRRs as will be discussed in Section 3. We find that the lattice mode frequency obtained from the transmission spectra demonstrate functional dependence on the lattice period according to Eq. (2). Solid line demonstrates linear fit of the experimental data with the effective refractive index

*n*= 1.12. Similar result (

_{ef}*n*= 1.14) can be extracted from data reported in [23].

_{ef}Whereas resonant frequency of *n* = 1 and *n* = 3 plasmonic modes is basically unchanged for different periods, we observe a decline of the resonant frequency of *n* = 5 mode as the period is increased as it is seen in Fig. 2(d). More detailed explanation of this phenomenon is presented in Section 3. We find that the extinction of THz radiation at the resonant frequency slightly declines with increasing period [16]. In addition, from the transmission spectra shown in Figs. 2(a)–2(c) one can see that resonances *n* = 1 and *n* = 3 undergo significant broadening as the array period is reduced. Similar results have been observed in subdiffraction lattices of U-type SRRs (i. e. in the case when the lattice resonance frequency remains above the plasmonic eigenfrequencies) [15,17]. Broadening of the plasmonic resonances with decreasing lattice period has been attributed to superradiant decay of the electric dipoles [15]. Such a spectral reshaping provides an efficient way to adjust Q-factor of frequency-selective surfaces for various applications.

#### 3.2 “Narrow gap” SRR arrays

Transmission properties of C-type SRR arrays (sample S3) corresponding to different periods of the lattice are presented in Fig. 3. From the comparison of U-type SRR and C-type SRR arrays shown in Figs. 2 and 3, it is evident that a decrease in the width of the split gaps shifts all plasmon resonances to lower frequencies. As Eq. (1) indicates, this is associated with the increase in the perimeter of SRRs. As in U-type arrays, a significant narrowing of the resonances *n* = 1 and *n* = 3 is observed as the lattice period is increased.

As can be seen from Fig. 3, a third resonance corresponding to deep minimum in the transmission spectra is excited in the range 280-330 GHz (Fig. 3). We suppose that this resonance represents the fifth-order plasmonic mode (*n* = 5) which is much stronger in C-type array than its counterpart in U-type array. Both higher-order resonances *n* = 3 and *n* = 5 are spectrally shifting as the lattice period is varied, in contrast to the fundamental mode *n* = 1 which is practically independent of *L*. Wood anomaly in C-type SRR array is observed in the range 340 – 480 GHz as a characteristic drop in transmission in Fig. 3. At the smallest period, as shown in Fig. 3(a), a resonant structure is clearly seen at frequency 421 GHz, i.e. between the fifth-order plasmon mode and the lattice mode. At this frequency, seven zones of electric field concentration are observed in the calculated surface electric field map shown in Fig. 5(g), indicating that this resonance represents genuine seventh-order plasmon mode excited in the SRRs. To our knowledge, this mode has never been observed in SRRs.

The range of the SRR lattice periods that can be investigated experimentally is limited. On the one hand, terahertz beam spot size limits the number of SRRs that can be excited coherently during the transmission measurements. On the other hand, at large lattice periods resonances are closely located in the far-field transmission spectra, which makes their experimental investigation difficult. Therefore, arrays with periods above 1000 µm were studied using model simulations. Calculation results shown in Fig. 3 demonstrate that our numerical model reproduces the experimental data correctly, including plasmonic resonances as well as the first-order lattice mode.

The results of the simulations together with the experimental data are summarized in Fig. 3(d). Calculation results are presented for SRR arrays in which the lattice period changes in 10 steps from 600 to 2000 μm. Horizontal lines indicate spectral positions of plasmonic eigenfrequencies according to Eq. (1), where perimeter of SRRs (1550 μm) is taken along the inner edges of the rings and the effective refractive index is taken *n _{ef}* = 1.2. Like plasmon resonances in U-type arrays, their resonant frequencies are red shifted with respect to those predicted by Eq. (1). Solid line in Fig. 3(d) indicates variation of the first-order lattice mode with the inverse lattice constant at effective refractive index

*n*= 1.12 according to Eq. (2). Whereas resonant frequency of the fundamental plasmonic mode is basically the same for different periods, we observe that frequency of the modes

_{ef}*n*= 3 and

*n*= 5 declines with increased lattice period and approaches the lattice mode line in the limit of large

*L*. According to reports published by other groups [22,23] we believe that this effect is a signature of a strong interaction of the plasmonic modes and the lattice excitations. There is slight difference between the frequency positions of the lattice mode in U-type and C-type arrays, but, in general, the lattice mode is correctly described by Eq. (2) as follows from Figs. 2(d) and 3(d). This claims that the lattice mode is practically independent of the resonator geometry.

As the period of the array is increased, we observe a significant spectral line-narrowing of the plasmonic modes leading to gradually increasing of Q-factor as demonstrated in Fig. 4(a). Q-factors are calculated according to Q = *Δf* /*f _{0}*, where

*Δf*is the spectral width and

*f*is the central frequency of the resonance. A particularly interesting behavior is observed where the lattice excitation approaches spectrally the SRR eigenmodes, e.g. around

_{0}*L*= 1500 μm for

*n*= 3 and around

*L*= 1100 μm for

*n*= 5. During the excitation of the lattice, the diffracted fields are trapped in the SRR array, which leads to a sharp increase in the resonance Q-factor [18] as it can be seen in Fig. 4 (a). We note that at small lattice periods, Q-factor of

*n*= 5 mode significantly exceeds Q-factor of other modes, since the resonance frequency of the

*n*= 5 mode is relatively close to the lattice mode frequency in dense SRR arrays. We also noticed that an increase in

*L*leads to a strong decrease in the resonance strength. Therefore, at large lattice periods,

*n*= 3 and

*n*= 5 resonances are only weakly seen in the far field transmission spectra.

To highlight the collective behavior of the SRRs, we studied the properties of a single SRR, which corresponds to the limit of an infinitely large lattice period. For spectral measurements, a single C-type SRR was fabricated on PTFE film and placed in the center of 3 mm diameter aperture. As a rough illustration, the transmission of one SRR is shown in Fig. 4(b) instead of the extinction cross-section [34]. Transmission of empty aperture was taken as a reference. To screen the THz beam, a 2 mm thick water-saturated cardboard was used. In principle, a metal plate would ideally block the terahertz beam, however, interaction between the conductive currents in the metal and the currents in the SRR leads to false resonances in the transmission spectrum. The water screen transmits several percent in the lowest frequency edge and attenuates radiation (with transmission below 0.1%) at frequencies above 60 GHz. As shown in Fig. 4(b), the first-order resonance at 70 GHz in a single C-type SRR is narrow and rather strong in contrast to higher-order modes which are rather wide and weak. In particular, the fifth-order mode is hardly seen in the spectrum.

From these observations we conclude that plasmonic modes of two types – uncoupled and coupled – can manifest themselves in SRRs. In what follows plasmon modes in a single SRR will be referred to as uncoupled plasmon modes. Meanwhile, plasmonic modes in dense arrays of SRRs will be called coupled plasmon modes. Whereas the fundamental plasmon mode is basically unaffected by the presence of neighboring SRRs, we observe that the character of the higher-order modes is strongly dependent on the lattice constant. The uncoupled higher-order modes exhibit rather broad spectral lines and a rapid decrease of the resonance strength with an increase in the mode number as can be seen from Fig. 4(b), whereas the coupled plasmon modes *n* = 3 and *n* = 5 are practically as strong as the fundamental mode as shown in Fig. 3. In addition, it is worthwhile to note that at large lattice periods (> 1300 μm) the third order plasmon modes of both types can be detected in the simulated spectra (not shown). The third order mode of both types has the same number of current nodes, however, Q-factors and resonant frequencies are different.

## 4. Enhancement of higher-order plasmonic modes in SRR arrays

A remarkable increase in the resonance strength of the coupled *n* = 3 and *n* = 5 modes compared to the resonance strength of these modes in a single SRR we call enhancement. Interaction between the plasmonic modes and the lattice modes can lead to an increase in Q-factor of the resonance and a shift in the frequency of the plasmonic modes [17,18,22,23], but there are no reports that the plasmonic modes are enhanced because of their interaction with the lattice modes. We suppose that the enhancement mechanism can be understood by considering the induced electric fields in the gaps of each SRR, as well as the surface currents in the rings.

First, consider the behavior of plasmon resonances at a lattice period of 600 μm, which corresponds to the interval between SRRs well below the extent of their near fields [23]. Figures 5(a), 5(c), 5(e), and 5(g) illustrate the calculated electric field maps of the SRR’s resonances for *n* = 1, *n* = 3, *n* = 5 and *n* = 7 plasmon modes along with the corresponding current flows and the induced charge diagrams shown in Figs. 5(b), 5(d), 5(f), and 5(h). At *n* = 1 (*LC*) mode, the incident electric field excites circular oscillating current in the SRRs, where oscillating charges in the entire structure are excited due to coupling of the electromagnetic field at the gap [35]. Additionally, the charge accumulation at the ends of the gap, as can be seen in Fig. 5(a), gives rise to a zone of strong electric field parallel to the incident polarization. This mode is also efficiently excited in a single SRR with an infinitely large distance between the SRRs as demonstrated in Fig. 4(b), which indicates that the influence of local electric fields has small effect on the fundamental oscillation mode.

The third-order mode (*n* = 3), observed at a frequency of 208 GHz, is markedly enhanced in the presence of a coupling between neighboring resonators. Detailed studies show that this mode of electric current oscillation corresponds to the formation of an electric quadrupole associated with three spots of greatly enhanced electric field as shown in Fig. 5(c). The corresponding current at *n* = 3 resonance is segmented into three sectors, demonstrated in Figs. 5(d) and 6, oscillating in antiphase. Figure 5(d) also shows that the middle current sectors are located near the zones of a strong electric field localized in the split gaps. Therefore, if the phase matching condition is satisfied, the strong electric-field zones in the gaps contribute to the enhancement of the third-order mode in closely spaced SRRs as a result of the electrical coupling between neighboring resonators.

In general, the electrical coupling and the magnetic coupling are competing interactions in dense SRR arrays [15], and the relative importance of each of them depends on the relative orientation and separation of neighboring SRRs [14]. Detailed studies show that the interaction between two SRRs in the gap-to-gap configuration is dominated by the electric coupling [14]. Similar interaction distance should be taken into account in the case of the interaction between a split gap and the middle current sector in the neighboring resonator. Therefore, we can assume that in dense SRR array the enhancement of a third-order mode is dominated by the electric interaction due to the close proximity of the neighboring resonators.

The fifth-order mode (*n* = 5) corresponds to five zones of enhanced electric field and five sectors of current oscillating in antiphase as shown in Figs. 5(e) and 5(f), respectively. In a single SRR, this mode appears as a dip with a very small strength seen in Fig. 4(b), whereas in coupled SRRs, the electric field at the split gaps enhances the current in the middle sectors in neighboring resonators, which leads to a pronounced fifth-order resonance in the far field transmission spectrum. It can be seen from Fig. 3 that plasmonic mode *n* = 5 in C-type array demonstrates resonance as strong as lower-order modes *n* = 1 and *n* = 3 as a result of the strong electric field enhancement. As follows from Fig. 4(a), the coupled mode *n* = 5 demonstrates a rather high Q-factor (Q = 20 – 25) in densely packed arrays and even higher in less densely packed arrays.

The described enhancement mechanism also holds for the plasmon mode of the seventh order which can be resolved only at *L* = 600 μm. The electric field map for the seventh mode for this period contains seven spots of strong electric field shown in Fig. 5(g). The extinction of THz radiation is rather low in the experiment, as demonstrated in Fig. 3(a), as the excitation of this mode is extremely sensitive to the quality of the fabricated structures [20]. The map of the surface electric field was also calculated at frequency corresponding to the first-order lattice modes. In contrast to the plasmon modes, the lattice modes exhibit an oscillating standing wave pattern of electric field (not shown) along the direction perpendicular to the polarization of the incident wave as reported in [23].

The proposed interaction model is further supported by a numerical calculation of the current distribution in C-SRRs. Figure 6 shows the total current (current density integrated along the cross-section of the current path) along the length of the arc around the ring for the three different lattice periods. At period 600 μm, we see the five strong current peaks associated with *n* = 5 plasmon mode. The position of the current nodes is slightly shifted with the lattice period, which indicates a change in the oscillation frequency. Furthermore, we observe a significant reshaping of the current profile with a change in the period. A noteworthy observation is a reduction of the middle current peak with increasing period. This is the result of decreasing longitudinal coupling between the electric dipoles forming at the split-gaps and the current paths in nearest neighboring resonators in agreement with above considerations.

The evolution of the plasmon mode frequency with inverse lattice period shown in Fig. 3(d) reveals that at large periods the higher-order coupled plasmon modes converge to the lattice mode line calculated according to Eq. (2). This means that in low-density C-type arrays, the plasmon modes are coupled via the long-range lattice modes. In dense SRR arrays, the plasmon modes are coupled by the short-range electric interaction between neighboring SRRs, whereas at large periods the enhancement of electron oscillations in SRRs is associated with the coupling of the plasmon modes with the lattice modes and formation of the hybrid modes [23].

It is interesting to note that the frequency of the higher-order coupled plasmon modes shifts with the lattice period, as shown in Fig. 3(a), in addition, the shift of the fifth-order mode is much stronger compared to the third-order mode. We observe that proximity of the plasmonic mode with the lattice mode results in strong interaction between the modes which shifts the plasmonic mode frequency as the lattice period is changed [22,23]. However, we suppose that the proximity effect of the plasmonic mode with the lattice mode cannot explain the significant shift of the plasmonic modes (in our case, the fifth-order mode) at small lattice periods where frequency separation between plasmonic modes and the lattice modes is large. The aforesaid mode enhancement model can explain this phenomenon as follows: a change in the period changes the relative phase difference between the electric field in the gaps and the current in the neighboring SRRs. The increase in the inter-SRR distance shifts down the frequency of the coupled plasmon modes, since the phase matching condition is satisfied at longer wavelengths. According to this explanation, the reason for the different shifts of the third-order and fifth-order modes is related to the different degrees of enhancement of these modes. Moreover, the different behavior of these modes may be due to the different number and configuration of strong electric field zones, which is not taken into account in our model.

## 5. Control of the coupled plasmon modes in SRR arrays

To sum up, all plasmon modes and lattice modes demonstrate the same behavior both in U-type and C-type arrays as it can be seen from comparison of Figs. 2 and 3. Nevertheless, the fifth-order plasmon mode is rather strong in C-type arrays whereas appears with very small strength in U-type arrays. To obtain a complete picture of the behavior of plasmon resonances, we used two additional samples corresponding to a larger gap capacitance (S4) and a smaller gap capacitance (S2) with respect to the C-type SRR. Figure 7(a) displays experimental far-field transmission of samples S1-S4 for the lattice period of 700 μm. Note that the sample with the widest gap (S1) represents U-type SRRs and sample S3 – C-type SRRs, both discussed above.

We observe that similar behavior is common to all four samples. The main difference is that the decrease in the gap capacitance shifts the plasmon resonances associated with the SRRs to higher frequency. Furthermore, we noticed that a decrease in the resonance strength of the higher-order modes correlates with a decrease in the geometric capacitance of the gap. This effect is most noticeable for plasmon resonance of the fifth-order which becomes hardly noticeable in “wide-gap” U-type SRRs. From this observation we conclude that widening the split gap leads to weakening of the electric coupling between neighboring SRRs. This is a consequence of a weak enhancement of the surface electric field in wide split gaps [36]. In addition, we note that the deepest transmission minimum of the fifth-order plasmonic resonance occurs in C-type SRRs. This result suggests that inter-SRR coupling is determined by the electric field in the split gap rather than by the gap capacitance.

Finally, we want to emphasize that the excitation of higher-order plasmon modes in dense SRR arrays can be advantageous for many practical applications. To reach higher operating frequencies, SRRs can be made to oscillate at one of their higher-order modes. In particular, the fifth-order mode demonstrates a fairly wide tunability with a change in the capacitance of the split gap and the lattice period as shown in Figs. 7(a) and 7(b), respectively, as well as a rather strong attenuation of electromagnetic waves at resonant frequency. In addition, this resonance shows a rather high Q-factor which practically does not dependent on the capacitance of the gap and the lattice period.

## 6. Conclusions

In conclusion, we used simulation as well as THz far-field spectroscopy to study coupling between components in metamaterial arrays of split-ring resonators. Our results provide a microscopic insight into the coupling between the meta-atoms in metamaterial arrays. It is shown that in tightly packed SRR arrays the short-range electrical coupling between the nearest neighboring SRRs maintains oscillations of electric current corresponding to higher-order plasmon modes with extremely high strength. In low-density SRR arrays, plasmon modes of higher-order are mediated by the diffractive lattice modes. In particular, the fifth-order resonance is characterized by a high Q-factor and a wide tunability with a change in the lattice period and geometric parameters of the SRRs. This could be relevant for the design and optimization of frequency-selective surfaces for various applications.

## Funding

Research Council of Lithuania Foundation of Researcher teams (Project MIP-059/2014).

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