1. Introduction

Metamaterials, the artificial micro- or nano-structures at the subwavelength scale, have attracted enormous interest in recent decades [1,2]. By specifically designing the geometry of individual elements, the electric and magnetic responses of metamaterials can be artificially controlled [3]. Due to the excellent capability in manipulating electromagnetic waves, metamaterials manifest potential applications in imaging [4], sensing [5] and communications [6,7]. Metasurfaces are thin two-dimensional metamaterial layers. Taking the advantages of miniature size and low cost compared to bulky metamaterials, metasurfaces are good candidates for developing antennas, sensors, active components, filters, and integrated technologies [8,9].

Terahertz (THz) radiation is an emerging research field in the last few decades. Because of the wide frequency range and low photon energy, the application of terahertz technologies in communication, imaging and biology has attracted extensive attention [9–11]. At terahertz frequencies, metasurface-based functional devices which can control amplitude [12], wavefront [13], polarization [14,15] and chirality [16] have been widely investigated. One of the most common structures of the metasurface is split ring resonator (SRR) array. Individual SRRs typically support odd/even eigenmodes when the incident electric field is parallel/perpendicular to the gap of the SRRs [17], for example, the fundamental eigenmode (n = 1) and third-order eigenmode (n = 3) are LC and quadrupolar resonances, respectively, and the lowest even eigenmode (n = 2) is dipolar resonance. Many works elaborately designed the metasurfaces concentrated on the resonant response of the eigenmodes [18,19], coupling between different sub-structures in individual elements [20,21], or the Fano resonance induced by the symmetry-breaking unit cells [22,23], and usually neglected the effect of the lattice in the periodic array.

Lattice mode, originated from Wood anomaly, exists in periodic structures [24–26]. Though Wood anomaly is well-known in the optical grating, the lattice mode related works in THz metamaterials are conspicuous recently. Bitzer et al. [27] showed the lattice modes mediated radiative coupling in metamaterial arrays. Schaafsma et al. [28] demonstrated that periodic lattice can enhance the transparency of detuned resonators. Xu et al. [29] proposed a high quality factor metasurface when the lattice mode of the resonator matched to the eigen resonance. Most recently, lattice related novel phenomena have been reported successively, such as the lattice-induced transparency (LIT) [30] and lattice induced plasmon hybridization in metamaterials [31], and the lattice-enhanced Fano resonances from bound states in the continuum metasurfaces [24]. However, the mechanism of lattice mode induced phenomena is insufficient and deserved to be further researched.

In this paper, we access the origin of the lattice mode mediated eigenmode resonance, which is attributed to the phase mismatch of the metasurfaces resulting from the phase difference in neighboring structures. We design three groups of THz metasurfaces consisting of metallic U-shaped SRR arrays deposited on a dielectric substrate, and obtain the transmission spectra via simulation and experiment. The distinguishable modulation of the different eigenmodes by lattice mode in U-shaped metasurfaces is observed, and a remarkable lattice induced suppression of the high order eigenmode resonance is demonstrated. Keeping the lattice mode unchanged, by introducing the phase difference between the neighboring structures, we clarify that the peculiar phenomenon of suppression is owing to the phase mismatch between the adjacent elements of the metasurfaces employing Fano model. The phase mismatch induced suppression of eigenmode resonance can be applied to develop THz multifunctional devices via arbitrarily introducing the phase perturbation in metasurfaces.

2. Sample description and methods

The schematic of the sample G1 is shown in Fig. 1(a), and it consists of a dielectric substrate and a planar array of U-shaped metallic structures. Fig. 1(b) depicts the structures of two groups of the fabricated metasurfaces (samples G1 and G3). L (90 µm) and w (15 µm) indicate the size and width of the U-shaped structure, respectively. In group G1, the period of the unit cell P is varied from 125 µm to 205 µm. In groups G2 and G3, the period is fixed as 145 µm. The metal is 0.2 µm-thickness aluminum, which is deposited on 1 mm-thickness quartz. We investigate the transmission spectra of the metasurfaces with experimental and simulated methods.

 

Fig. 1. (a) Schematic of the U-shaped metasurfaces under oblique incidence. Microscopic images of the fabricated metasurfaces: sample G1 with (b) P = 145 µm and (c) P =165 µm; sample G3 with (c) α = 30° and (e) α = 60°.

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A standard THz time-domain spectroscopy (THz-TDS) system is used to measure the transmission data of the fabricated metasurfaces under dry air environment. The x-polarized terahertz waves are incident on the metasurfaces in the direction of z with angle $\theta$, and the x-polarized transmitted waves are detected. A bare quartz substrate without metasurface is served as a reference. The transmission spectra of the metasurfaces can be extracted from Fourier transforms of the measured time-domain electric fields, which are defined as $T(\omega )= {{{E_{sample}}(\omega )} / {E{{(\omega )}_{ref}}}}$, where ${E_{sample}}(\omega )$ and $E{(\omega )_{ref}}$ are the frequency-domain electric fields of the sample and reference, respectively.

The simulation results are calculated using the frequency domain solver of CST Microwave Studio with unit cell boundary conditions and air environment. The permittivity of quartz is set as 3.82 [20], and the conductivity of aluminum is modeled as 3.56×10⁷ S/m [32].

3. Results and discussion

3.1 Lattice-induced suppression of high order eigenmode

The frequency of lattice mode can be expressed as the following formula [33–36]

From Eq. (1), we can see that the frequency of lattice mode depends on period and incident angle. We first explore the effect of the period on the resonances in metasurfaces with sample G1. The transmission spectra of the U-shaped metasurfaces in normal incidence are shown in Fig. 2(a). Different colors correspond to different transmission spectra of the U-shaped metasurfaces with varying period P. The simulated results (solid lines) agree well with the experimental data (open dots). Apparently, in the minimum period case, there are two resonance peaks produced in the U-shaped metasurfaces, and the low- and high-frequency resonance are the fundamental (LC resonance) and high order mode (quadrupolar resonance) of the eigenmodes [17,27], respectively, which are denoted with A and B. The frequencies of the lowest lattice mode of the U-shaped metasurfaces ${f_{LA}} = f_{LA}^{[1,0]}$ calculated from Eq. (1) are denoted with arrows. As we can see, ${f_{LA}}$ undergoes a redshift caused by the increasing P. With the increment of the period, a spectacle phenomenon is observed: when ${f_{LA}}$ moves to the lower side of the frequency of high order eigenmode, the resonance B is suppressed. For example, when P increases to 185 µm, ${f_{LA}}$ moves to 0.83 THz, which is less than the frequency of resonance B around 1.05 THz, and the amplitude of resonance B considerably weakens compared to the case of P = 125 µm. Especially, when P = 165 µm, the lattice induced sharp resonance is observed. While, since the frequency of the lattice mode is far away from the fundamental eigenmode, the amplitude of the resonance A whose frequency is 0.44 THz is non-significantly affected. Associated with the transmission map obtained from the simulated data as shown in Fig. 2(b), a remarkable suppression of the resonance B caused by the [1, 0] lattice mode can be observed when the frequency of the lattice mode moves to the lower frequency side of the eigenmode. The resonance curve of the high order eigenmode is slightly bent when as the [1, 0] lattice mode approaches, and the two modes overlap around P = 165 µm. And in the frequency region beyond the [1, 0] lattice mode, the eigen resonance seems to be forbidden. Meanwhile, due to the distant separation in the frequency domain, resonance A is barely affected by the lattice modes, and the contribution of the [1,1] lattice mode is also feeble. These results indicate that the varied P of metasurfaces can accurately tune the frequency of the lattice mode, and therefore distinguishably manipulate the different eigenmodes.

 

Fig. 2. (a) Experimental (open dots) and simulated (solid lines) transmission spectra of the U-shaped metasurfaces with different periods. The arrows denote the calculated frequencies of the [1, 0] lattice mode according to Eq. (1). (b) Transmission map obtained from the simulated spectra and lattice modes are denoted with the black curves. (c) Q-factor value of the resonances A and B extracted from the experimental data and the linewidth of the resonance B extracted from the Fano-fitted data. (d) Experimental (open dots) and fitted (solid lines) transmission spectra of the U-shaped metasurfaces with different periods. For clarification, the transmission spectra in (a, d) are vertically shifted.

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Apparently, the reduction of the resonance of high order eigenmode in a large period case should not be simply attributed to the decrease of the packing density, because the amplitude of the fundamental eigenmode only experiences slight change. Consequently, we can treat the phenomenon as lattice-induced suppression of high order eigenmode, which originates from the phase mismatch of adjacent structures. Though many works have investigated the properties of lattice mode in metamaterials [36,37], the case that the period is beyond the diffracted edge is scarce. When the period is in the sub-wavelength scale, if the scattered fields of each structure satisfy in phase with the incident waves, the enhanced coupling of neighboring structures will reinforce the resonance [37], and usually results in a high Q factor resonance. Q factor represents the capability that the resonance sustains energy, which can be calculated as $Q = {{{f_0}} / {\Delta f}}$, where ${f_0}$ and $\Delta f$ are the frequency and the linewidth of the resonance, respectively. The high Q factor value means low loss of the resonance. We extracted the Q factor values of the eigenmodes from the experimental data as shown in Fig. 2(c). As the period increases, the Q factor value of the resonance B undergoes slight enhancement and sharply drops (from 18 to 3) when the period moves across 165 µm (the diffracted edge); while the Q factor value of the resonance A experiences slowly increment (from 5 to 8). These results indicate that the radiation loss of the eigenmodes is mediated by the lattice mode.

To evidence the inference, we conduct the quantitative analysis using Fano model [38] since the transmission spectra of resonance B manifest an asymmetrical Fano profile. The lattice mode and eigenmode act as discrete state and continuum state, respectively. Note that the resonance exhibits a dip in the transmission spectra, so the Fano model is modified as [39,40]

Here A is the scale factor, q is the asymmetry parameter, $\varepsilon = {{2(f - {f_0})} / {\Delta f}}$. Hence, we fit the experimental data of resonance B with Eq. (2), i.e., we select the frequency range denoted with the shadow in Fig. 2(a), and the fitted results are shown in Fig. 2(d). As we can see, the fitted results agree well with the experimental data. We then exact the parameter of linewidth $\Delta f$ from the Fano model, which is plotted in Fig. 2(c) with the blue dots. Apparently, the linewidth of the resonance B undergoes sharply increases, which means that the loss increases when the period moves across 165 µm (the diffracted edge). Combined with the Q factor, the evolution of linewidth strongly supports the inference we mentioned: when the frequency of lattice mode ${f_{LA}}$ moves closer to that of the eigenmodes from the high frequency side, the resonance dip of the high order eigenmode becomes narrow, i.e., the radiation loss of the resonance decreases. However, if the period enlarges than the resonance wavelength, due to the severe phase retardation between the distant adjacent unit cells, the scattered fields of each structure are no longer in phase with the incident waves, therefore, the radiation loss of the high order eigenmode resonance increases significantly, which manifests the broad linewidth in the spectra. Meanwhile, when the period is beyond the diffracted edge, the metasurfaces cannot be effective to a homogenous material anymore, and more incident waves can transmit through the metasurfaces directly. The phase mismatch of adjacent structures and adding transmitted waves lead to the high amplitude of the high order eigenmode in the transmission spectrum.

Notice that in Eq. (1), the frequency of lattice mode ${f_{LA}}$ can be tuned by incident angle θ as well. Hence, keeping the same packing density of the metasurfaces to eliminate the adding transmitted waves, we then verify the lattice-induced suppression beyond the diffracted edge by tuning the lattice mode via changing the incident angle. The simulated results (solid lines) and the experimental data (open dots) of sample G1 (here P is fixed as 145 µm) with varied incident angle θ from 0° to 60° are shown in Fig. 3(a), which are in good agreement. We also map the transmission diagram under different incident angles as shown in Fig. 3(b), in which the lattice modes are denoted with the black curves. As expected, ${f_{LA}}$ redshifts as the incident angle θ increases, and when θ increases to 5°, ${f_{LA}}$ moves to 1 THz coinciding with the high order eigenmode. When ${f_{LA}}$ moves across the frequency of resonance B around 1.02 THz, the amplitude of resonance B obviously weakens compared to the case of θ = 0°. Further, when θ exceeds 30°, the resonance B is suppressed significantly, while the resonance A is barely influenced. These results are consistent with the Q factor variation extracted from the simulated data as shown in Fig. 3(c): the Q factor of resonance B rapidly drops below 8 from 40 and Q factor of resonance A is barely unchanged. Thus, we can conclude that the lattice-induced suppression is also attributed to the phase mismatch of adjacent unit cells caused by additional geometric phase under oblique incidence [41], which results in the increasing loss of the eigenmode beyond the diffracted edge determined by the lattice mode. Hence, we experimentally validate the modulation characteristics on eigenmodes induced by phase mismatch via tuning the lattice mode.

 

Fig. 3. (a) Experimental (open dots) and simulated (solid lines) transmission spectra of the U-shaped metasurfaces with different incident angles. For clarification, the transmission spectra are vertically shifted. The arrows denote the calculated frequencies of the [1, 0] lattice mode according to Eq. (1). (b) Transmission map obtained from the simulated spectra and lattice modes are denoted with the black curves. (c) Q-factor value of the resonances A and B extracted from the simulated data.

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3.2 Phase mismatch induced suppression of high order eigenmode

Since lattice-induced suppression originates from phase mismatch of the array elements, we should observe the similar phenomenon when we vary the phase retardation between the adjacent unit cells, even though the frequency of the lattice mode is invariable.

To verify this explanation, we attempt to introduce the phase difference of neighboring structures by simply adding the thickness of the substrate, namely, the optical path length. We fix the period as 145 µm and simulate the transmission spectra of sample G2 (see Fig. 4(a)) with varying thickness t of adjacent unit cells ranging from 10 µm to 50 µm under normal incidence, as shown in Fig. 4(b). In this situation, the frequency of the lattice mode remains at 1.06 THz, and the period does not access the diffracted edge. Apparently, similar to the results of sample G1 above, the phase mismatch induced suppression of the high order eigenmode resonance is observed as well. With the increasing thickness t, the phase difference of adjacent structures increases, and the high order eigenmode resonance B around 1.02 THz (denoted with the shadow) is gradually weakened. When the t enhances to 50 µm (phase difference is about 0.35π), the suppression of high order eigenmode emerges, while the amplitude of fundamental eigenmode is barely modified due to its insensitivity to the small phase difference (about 0.14π at t = 50 µm). These results can also be supported by the quantitative analysis of Q factor. We extract the Q factor values of the eigenmodes as shown in Fig. 4(c). The thickened t means the increasing phase mismatch, consequently, the Q factorof resonance B monotonically decreases (from 35 to 7) due to the enhanced loss caused by phase mismatch, while the Q factor of resonance A keeps below 10 as usual.

 

Fig. 4. (a) Schematic of the unit cell of sample G2 and (b) the simulated transmission spectra with different thicknesses. For clarification, the transmission spectra are vertically shifted. (c) Q-factor value of the resonances A and B extracted from the simulated data.

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Considering that the sample G2 is hardly fabricated, to experimentally validate the phase mismatch induced suppression, we introduce phase difference in the planar metasurface using sample G3 whose unit cell is shown in Fig. 5(a). We remain the right part and rotate the orientation $\alpha$ of the left part from 0° to 60°. The transmission spectra with varying orientations $\alpha$ are shown in Fig. 5(b), and the experiment results (open dots) agree well with simulated results (solid lines). As expected, the phase mismatch induced amplitude modulation on resonance B (denoted with the shadow) can be observed when the orientation angle increases, and the suppression occurs when $\alpha$ exceeds 45°, while the resonance A is barely affected. Note that in sample G3, the two U-shaped structures form a new supercell which will double the period to 2P. Hence, the [1, 0] lattice mode for the supercell will be aroused in ${f_{LA}}$/2 = 0.52 THz according to the Eq. (1). We map the transmission diagram obtained from the simulated spectra as shown in Fig. 5(c). It is found that the [1, 0] lattice mode for the supercell (denoted with dashed line [1, 0]_2P) apparently emerges when $\alpha$ increases to 30°, which results in the asymmetrical profile around 0.52 THz as shown in the transmission spectra in Fig. 5(b). Consequently, the loss of the resonance B located on the high frequency side will rise. These results coincide with the Q factor variation as shown in Fig. 5(d). Similar to the case of varying thickness of the neighboring substrates, along with the increasing orientation angle $\alpha$, the Q factor of resonance B monotonically decreases (from 38 to 5) due to the enhanced phase mismatch, while the Q factor of resonance A maintains around 8. Besides, we simulate the E-field distributions of resonances A and B shown in Fig. 5(e) and Fig. 5(f, g), respectively. Because of the symmetry breaking, the charge oscillations in the left and right parts in the unit cell will be excited by the external field non-simultaneously, which results in the phase delay on resonance point between the adjacent structures, and further increase the loss of the high order eigenmode. When $\alpha$= 45°, we find that for the resonance B, the phase difference of the adjacent structures is π/4, i.e., the strongest excited E-field of the left part (denoted with the red circle in Fig. 5(f)) is ahead π/4 of that of the right part (denoted with the red circle in Fig. 5(g)). While the resonance A is insensitive of the phase delay because the adjacent structures are excited in-phase (Fig. 5(e)). Hence, we verify that the eigenmode of high order mode can be modulated by phase mismatch via arbitrarily introducing the phase perturbation within the neighboring elements.

 

Fig. 5. (a) Schematic of the unit cell of sample G3 and (b) the experimental (open dots) and simulated (solid lines) transmission spectra with different orientation angles. For clarification, the transmission spectra are vertically shifted. The shadows denote the high order eigenmode resonance. (c) Transmission map obtained from the simulated spectra and lattice modes are denoted with the black dashed lines. (d) Q-factor value of the resonances A and B extracted from the simulated data. E-field distributions of (e) resonance A and (f, g) resonance B of sample G3 with $\alpha$= 45°.

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4. Conclusion

We observe the lattice induced suppression of the eigenmode resonance when the resonance wavelength is beyond the diffracted edge by tuning the frequency of lattice mode by changing the period or incident angle with sample G1. Further, with the quantitative analysis of Fano model, we clarify that the peculiar phenomenon of suppression is originated from the phase mismatch between the adjacent elements of the metasurfaces via introducing the phase difference in neighboring structures with samples G2 and G3, which results in the increasing radiation loss of the eigenmode. These results give insights into the fact how the transmission amplitude of eigenmode resonance is mediated by phase mismatch via arbitrarily introducing the phase perturbation in metasurfaces, which can be flexibly applied to develop THz multifunctional devices, such as adjustable filters and amplitude modulators.