1. Introduction

In 1929, von Neumann and Wigner found special bound solutions with discrete eigenvalues which are embedded in the continuum energy spectrum by solving single-particle Schrödinger equation [1]. This kind of special bound states can be called bound states in the continuum (BICs). In addition to quantum systems, the concept of BICs has been extended to acoustic systems and optical systems [2]. In optical systems, BICs with infinite quality (Q) factors exist in ideal infinite structures [2]. Real finite structures can have strong resonances with finite ultra-high Q factors close enough to BICs, which can partially couple to the input light [3–9]. This kind of strong resonant states can be called quasi-BICs. In nano-optics, strong resonances with ultra-high Q factors have attracted researchers’ great interests since they can be utilized in narrow-band filtering [10] and sensing [11]. Very recently, various all-dielectric metasurfaces have been proposed to realize symmetry-protected BICs in the near-infrared and visible bands [12–15]. As the mirror symmetry of the metasurfaces is broken, BICs turn into quasi-BICs with ultra-high Q factors [12–15]. Besides, BICs in the microwave (MW) and terahertz (THz) bands have attracted researchers’ great interests. In 2019, Yan et al. utilized an all-metallic metasurface comprising rectangular-hole dimers to realize symmetry-protected BICs in the MW bands, where a strong trapped-mode resonance arose from the destructive interference between two anti-phased dipole modes of the asymmetric dimer holes [16]. As a special frequency band, THz band ranging from 0.1 to 10 THz possesses lots of applications such as lasing and sensing due to THz waves’ non-ionization and low photon energy (4.1 meV at 1 THz) [17]. Achieving BICs will be of great interest for THz frequencies [18–28]. To this end, different subwavelength structures such as metamaterials, photonic crystals, and resonators have been proposed, accelerating the basic components toward practical applications [18–28]. Particularly, researchers proposed metallic metasurfaces with dielectric substrates to realize BICs in the THz band [26–28]. By suitable in-plane C2 symmetry breaking, BICs can be produced in metallic metasurfaces. For example, quasi-BICs emerge in metasurfaces formed by arrays of detuned resonant dipolar dimers [22]. This resonance evolves continuously from a Fano resonance into a symmetry-protected BIC as the dipole detuning vanishes. In addition, a quasi-BIC resonance in metasurfaces consisted of split-ring resonators has been applied for THz sensing, which enables sensing of a 7 nm thick analyte [26]. In 2020, Zhao et al. proposed a metallic resonator array with a polyimide substrate to realize a quasi-BIC with a Q factor of 27 in the THz band [27]. Without breaking the structural symmetry, the BICs evolve to an observable quasi-BIC high-Q Fano resonance by tuning the structural parameters of the metasurface. In 2021, Niu et al. proposed another metallic resonator array with a cyclic olefin copolymer substrate to realize a quasi-BIC with a Q factor of ∼20 in the THz band [28]. Compared to the above two metallic structures with dielectric substrates [26–28], free-standing metallic structures that do not require any holder or substrate show higher flexibility [29]. Furthermore, free-standing metal structures no need to consider additional loss from other materials. A free-standing structure of THz metal structures with surface plasmon polaritons (SPPs) has been investigated frequently in past decades [30–35]. Owing to the SPPs, electromagnetic waves are efficiently trapped at the metal edges through the interaction with the free electrons of the metallic surfaces [30]. Typical structures including metal hole arrays [30–33], metal grooves arrays [34], and metal slits [35] have been reported recently. However, oblique waves are applied to excite the sharp resonance in metal hole arrays (MHAs), restricting it toward practical applications [32–33]. Although the Ohmic loss of metal contributes to the total losses, a resonator based on all-metal structures with high-Q BIC and strong SPPs is highly desirable in THz applications, particularly in sensing [36–38].

In this work, a free-standing terahertz metallic resonator based on double-slit arrays (DSAs) is demonstrated for achieving a sharp resonance. Such sharp resonance with a Q-factor of about 60.9 can be termed as quasi-BIC, which could be further improved by reducing the structural asymmetry factor. The influence of the imperfection in experimental samples such as the round edge and the trapezoid shape on the transmission properties of DSAs is analyzed. Because of Ohmic losses, the copper DSAs show a lower Q-factor than that of ideal perfect electrical conductor (PEC) DSAs. The transformation from BIC to quasi-BIC is analyzed in detail by calculating the electric field and vector distributions. In contrast to the symmetry-protected BIC with weak surface current, the surface current is strongly excited and the orientation as well as magnitude are rearranged in the gap of metal slits when the structural symmetry is broken, indicates the bound state couples to the incident radiation thus giving rise to quasi-BICs. We also investigated the effect of metal thickness on the quasi-BICs for DSAs. Results exhibit that thinner resonators can achieve stronger quasi-BICs. The achievement of high Q-factors in the THz regions proves that such metal structures especially interesting for practical THz applications in electromagnetic wave filtering and biomolecular sensing.

2. Fabrication and simulation of DSAs

The configuration of DSAs is schematically shown in Fig. 1(a). Different from these metasurfaces with dielectric substrates [39–40], the proposed structure is free-standing, i.e., without using a dielectric substrate. The DSAs consist of one layer of metal slit arrays in which the period (Λ) is 320 µm (both X and Y directions) and the width (W) of slits is 80 µm [Fig. 1(b)]. The thickness of metal slit arrays is t=50 µm. The geometric parameters of the unit cell are as follows: s=60 μm, L=200 μm. L2 is a variable parameter, in the range of 170–220 μm and 180–210 μm for simulations and experiments, respectively. Different from Ref. 16 introduced two small square holes as the asymmetric element, the structural symmetry of DSAs can be broken by changing the value of L2. The top-view photo of an experimental sample is shown in Fig. 1(c), which is fabricated on a 50-µm thick copper foils by femtosecond laser processing. The corners in the experimental samples are rounded, which are right angles in the simulated models. Apart from the rounded corners, the laser-fabricated slits look quite rectangular. This all-metal structure can be bent, presenting high flexible properties. In addition, copper has a better cost–stability balance than other common metals, which makes the copper DSAs cheaply accessible and reusable due to the cleanability under acetone or alcohol after use for material sense.

 

Fig. 1. (a) The configuration of DSAs. (b) The unit cell of DSAs. (c) The photography of experimental samples in the X-Y direction. (d) The cross-section of perfect and fabricated slits in the Z-X direction. (e) The measured and simulated structural parameters of DSAs.

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Due to the Gaussian energy distribution of laser pulses, the laser-ablated hole has an inverted triangle profile [41]. Hence, a rectangle processing pattern on the foil surface shows an inverted trapezoid profile. As shown in Fig. 1(d), in the vertical direction (the Z-X direction), the cross-section of the laser-fabricated slit shows an inverted trapezoid profile rather than a modeled rectangle profile. Therefore, the top width (W) of fabricated slits is larger than the bottom width (W2), while W is equal to W2 in the perfect model of the simulations. Figure 1(e) shows the structural parameters for four experimental samples. In finite-difference time-domain method (FDTD) simulation, the cells along the X- and Y-axes are infinitely and periodically extended. The perfectly matched layers are occupied along the Z-axis. In FDTD, a plane wave is used, where the electric fields of the input transverse magnetic (TM) and transverse electric (TE) waves are parallel to the X- and Y-axes, respectively. The mesh size in X-, Y- and Z-directions is set as 10 µm, 10 µm, and 5 µm, respectively. In the simulation, the material of DSAs is assumed to be copper with a conductivity of 5.96×10⁷ S/m.

The schematic of the experimental setup of the laser processing system is illustrated in Fig. 2(a). Femtosecond-laser pulses from a Yb: KGW laser with a central wavelength of 1030 nm (pulse duration: 700 fs, PHAROS PH1-10, Light Conversion, Lithuania) were irradiated and focused upon a 50-µm thick copper foil by an F-Theta lens (focus length: 200 mm, SILL 297358, Sill Optics GmbH, Germany) at normal incidence. The laser beam (repetition rate: 100 kHz, pulse energy: 7.5 μJ) was guided by a galvanometer scanner head (SUPERSCAN IV-30, Raylase GmbH) to directly write rectangle patterns on the copper foil at a constant speed of 5,000 mm/s. The rectangle patterns were written for 1000 times to penetrate the foil and obtain a slit, which is corresponding an effective pulse number of 700 [41–42]. Through careful adjustments of the scanned pattern, periodically arranged slits are successfully fabricated. Figure 2(b) shows the three-dimensional graphics by a confocal laser scanning microscope (CLSM, LEXT ILS4100, Olympus, Japan). The black rings around the slits are debris deposited during laser ablation, whose height is less than 8 μm. Because the irradiated spot is a circle with a diameter of ∼35 μm, the fabricated slit shows a rounded rectangle shape.

 

Fig. 2. Schematic of the femtosecond laser processing system. 30 × 30 slit pairs (1800 slits) were periodically fabricated in a 10 × 10 mm² square on a 15 × 15 mm² copper foil with a thickness of 50 µm. (b) The three-dimensional graphics of the laser-fabricated DSAs.

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3. Results and discussion

The experimental results are measured using a terahertz time-domain spectroscopy (THz-TDS). With the femtosecond laser (780 nm), and a computer together with the home-made software, this spectrometer can equip an ultrafast optical lab for extending into the research field of THz spectroscopy with a high cut-off frequency (3 THz). A parallel THz wave, which was radiated from a photoconductive antenna (PCA) emitter pumped by the 780 nm laser pulse, was incident on the DSAs [29]. Noted that the measurement resolution is 9.6 GHz and the THz beam size is about 4 mm. Figure 3 shows the measured and simulated transmittance spectra of TM modes for four samples (samples 0, 1, 2, and 3), where the simulated structural parameters are the same as that of measurement in Fig. 1(e). To reproduce the details (shape and dimensions) of the laser-fabricated DASs, the simulation models are plotted from their microscopy images. Thus, the simulated model has the round edge and the trapezoid shape, which is shown in the Fig. 3. The structural parameter L2 of DSAs is 200 μm, which means that the structure is symmetrical (sample 0). As shown in Fig. 3(a), the lowest resonant peak occurs at 0.90 THz, corresponding to a wavelength of about 0.33 mm. Compared to the MHAs with the same period, the DSAs realize a resonance with a lower frequency because of stronger coupling [31]. The green dot line is the experimental spectrum for TE modes, which shows very low transmittance in the working frequency range and hence will not discuss in the following sections.

 

Fig. 3. (a) The experimental and simulated transmission spectra of TM and TE modes for sample 0. (b) The experimental and simulated transmission spectra of TM modes for sample 1. (c) The experimental and simulated transmission spectra of TM modes for sample 2. (d) The experimental and simulated transmission spectra of TM modes for sample 3. The inset of simulated models is plotted based on its microscopy images.

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The DSAs consist of double slits, the structural symmetry can be broken by increasing or reducing the length of one slit. As shown in Fig. 3(b), for sample 1 with the ideal parameter L2 = 180 µm, a sharp resonant dip occurs at 0.792 THz. This resonance can be fitted by a Fano line shape, which is given by [43]

To better understand the effect of the slit with the round edge and the trapezoid shape on the transmission properties of THz waves, we have simulated the transmittance spectra of three models of DSAs having the round edge and the trapezoid shape slits. Here, sample 1 is selected as an example to investigate. Figure 4(a) shows the schematic of three models of DSAs. The slit in Model A has the round edge and the trapezoid shape, which is the real model plotted from its microscopy images. Model B is the model with the 90-degree edge and the trapezoid shape, which is set up based on the measured structural parameters, where the structural parameters are depicted in Fig. 1(e). Model C is the perfect model with the 90-degree edge and the rectangular shape, which is the designed model. The corresponding simulated transmittance spectra are demonstrated in Fig. 4(b). Clearly, the transmittance spectrum of model A is almost perfectly matched the experimental results, which shows a sharp dip at 0.792 THz. For model B with the 90-degree edge and the trapezoid shape, its transmittance spectrum has a dip at 0.767 THz. Compared to that of Model A, the resonant dip of model B shows a redshift of 25 GHz. Results proved that the imperfection of the round edge affects the resonant dip, resulting in the redshift of dip frequency. The perfect model C is our designed model, which shows a resonant dip at 0.685 THz (Blue line). In contrast to that of model A, the redshift is equal to 107 GHz. Based on these results, we can find that the trapezoid shape affects the resonant dip greatly because of the reduced width (W2) of the bottom pattern.

 

Fig. 4. (a) The schematic of three models of DSAs. The slit in Model A has a round edge and a trapezoid shape, which is the real model from the experimental samples. Model B is the model with the 90-degree edge and the trapezoid shape, which is set up based on the measured structural parameters. Model C is the perfect model with the 90-degree edge and the rectangular shape, which is the designed model. (b) Simulated transmittance spectra for three DSAs with different models.

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For symmetry-protected BIC, it has an infinite Q-factor and can be transformed into quasi-BIC by breaking the symmetry of the unit cell, leading to a sharp resonance response [2]. To deeply understand BIC in DSAs, we simulated the transmission spectra for DSAs with various L2, ranging from 180 to 220 μm with a step of 10 μm [ Fig. 5(a)]. The simulated structural parameters are as follows: Λ=320 µm, s=60 μm, L=200 μm. To simplify the simulation model, here, the perfect slit is set in the simulation, where W = W2 = 80 µm. Clearly, for symmetric DSAs with L2 = 200 μm, no transmission dip can be observed and the resonance linewidth vanishes, where the symmetry-protected BIC state can be supported. Once the symmetry of DSAs is broken, in other words, the length of L2 deviates from 200 μm, a quasi-BIC with the sharp feature is induced, which is shown in Fig. 5(a). Figure 5(b) shows the transmission map of DSAs for various L2. As L2 decreases, from 220 to 200 μm, corresponding to removing a smaller length from the slit, the resonance becomes sharper and finally vanishes. That vanished resonance at around 0.650 THz when L2 = 200 μm means that no leaky energy from the bound state to the free space, demonstrates the existence of a BIC state. Along with the broken symmetry of DSAs, obvious asymmetric Fano resonances are produced, demonstrating the emergence of a quasi-BIC. While further decreasing L2, the resonance grows wider again.

 

Fig. 5. (a) Simulated spectra of the DSAs with varying structural parameters L2. (b) Simulated transmittance spectra map of copper DSAs for various L2. (c) Electric field magnitude distributions for 0.667 THz when L2 is 190 μm. (d) Electric field magnitude distributions for 0.650 THz when L2 is 200 μm. (e) Electric field magnitude distributions for 0.639 THz when L2 is 210 μm. (f) Electric field vector distributions for 0.667 THz when L2 is 190 μm. (g) Electric field vector distributions for 0.650 THz when L2 is 200 μm. (h) Electric field vector distributions for 0.639 THz when L2 is 210 μm. The color bar value in (c-h) is the normalization of the local electric field to the incident electric field.

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We also simulated the electric field magnitude and vector distributions of DSAs with different lengths of L2, which are plotted in Figs. 5(c)–5(h), where the color bar value is the normalization of the local electric field to the incident electric field. As shown in Figs. 5(c)–5(d), the strong fields locate in the slits of L2 = 190 μm-DSAs in comparison to that of symmetric DSAs with L2 = 200 μm. It means that the scattering fields of symmetric DSAs are strongly suppressed, which hardly coupling to the free space radiation and is completely confined as a bound state without leakage channel [2,23]. Specially, the symmetry-broken DSAs with L2 = 190 μm achieves a strong localization SPP field. Similarly, the strong SPP fields of 0.639 THz can also be found in DSAs with L2 = 210 μm. Interestingly, the strong fields move from the right side to the left side when L2 changed from 190 to 210 μm. Results indicated that the electric field is strongly concentrated in shorter slits because the smaller aperture shows stronger interference between the EM waves and SPPs on metal surface [45]. The asymmetric field distribution can be modulated by altering the length of slits. Figures 5(f)–5(h) show the corresponding electric field vector distributions for L2 = 190, 200, and 210 μm at featured frequencies. At 0.667 THz, when L2 is 190 μm, the electric dipoles (blue arrow) locate inside the slits and its alignments are out-phase (or asymmetry). The in-phase alignment between the electric dipoles in the horizontal direction can be found at the symmetry-protected BIC of about 0.650 THz, which is different from the quasi-BIC at 0.667 THz. It means that the dipole symmetry is broken in terms of the orientation, which allows coupling of the mode to incident TE waves for quasi-BICs [22–23,46]. Results exhibit that strong coupling between the horizontal dipoles are occurred and thus giving rise to the quasi-BIC because of the symmetry-broken effect [5]. A similar dipole alignment can also be found in Fig. 5(h) when L2 is 210 μm. At 0.639 THz, the electric dipoles align out-phase but the direction is contrary to that of 0.667 THz due to the different phases.

The resonance characteristics strongly depend on the asymmetry parameter, defined as δ=|(L-L2)|/L=|(ΔL)|/L. For ΔL larger than zero, with the asymmetry parameter increasing and the mirror symmetry broken, the transmission spectra show distinct quasi-BIC features and the resonant frequency redshifts with the linewidth increasing. It means the BIC state is unstable against the perturbation with breaking mirror symmetry, which induces leakage of BIC and leads to quasi-BICs [2,22–23]. The Q-factors with different asymmetry parameters are also investigated, where the Q-factors are extracted from the simulated transmission spectra of the ideal metal structures (PEC DSAs) and the loss metal structures (copper DSAs). The Q-factors for PEC and copper DSAs as a function of ΔL are shown in Fig. 6(a). As ΔL gradually approaches zero, the Q-factor increases rapidly. For the copper DSAs, owing to the Ohmic loss, the corresponding Q-factor is lower than that of PEC DSAs with zero Ohmic loss. The Ohmic loss is inversely proportional to the conductivity. Thus, for a metal resonator with certain structural parameters, higher conductivity implies higher Q factors [47]. When ΔL is equal to zero, the resonant width vanishes and the Q-factor becomes infinity. Such resonance corresponds to a symmetry-protected BIC, while degenerates to quasi-BICs manifesting as finite Q-factor when ΔL slightly deviating from zero. The dependence of the Q-factors on the asymmetry parameter is also demonstrated. The relation of the Q-factor for the PEC and copper DSAs with 1/δ² is plotted in Fig. 6(b). The results clearly show the linear dependence for PEC DSAs but not copper DSAs because of the Ohmic loss, which is a typical feature of the BICs. The result is consistent with previous reports [5–6]. Therefore, breaks the structural symmetry of DSAs is an efficient way to achieve quasi-BICs, where the Q-factor can be manipulated by the asymmetry parameter.

 

Fig. 6. (a) Dependence of Q-factor on ΔL. (b) Linear relationship between Q-factor and 1/δ². The black line is a linear fitting line.

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We also investigated the effect of metal thickness on the quasi-BICs for DSAs. Figure 7(a) shows the simulated transmittance spectra of DSAs with various metal thicknesses, where L2 is kept as 180 μm. Obviously, with the metal thickness increasing, the quasi-BIC blue shifts with the linewidth increases. The relation among the metal thickness, resonant dip frequency, and Q-factor of quasi-BICs is summarized in Fig. 7(b). The frequency of resonant dips increases as the metal thickness increases from 10 μm to 100 μm. In addition, the Q-factor is significantly decreased when the metal thickness increases. For instance, the Q-factor reduces from 92.9 to 50 when the metal thickness up from 10 μm to 100 μm. It means that the thinner resonators can achieve stronger quasi-BICs, where the field coupling between top and bottom metal surfaces is enhanced [30–31]. It is noted that such high Q-factors are not easy to realize experimentally because of the limited fabricated conditions. The 10-μm copper sheet is easy to be bent when the high-power laser beam incidents on the sheet. We have simulated transmittance spectra for DSAs with a lower asymmetric parameter, i.e. L2 = 190 µm. As shown in Fig. 7(c), the quasi-BIC shifts towards higher frequencies as the metal thickness increases. We summarized the relation among the metal thickness, resonant dip frequency, and Q-factor of quasi-BICs in Fig. 7(d). In contrast to 180 µm-DSAs, 190 µm-DSAs shows higher Q-factors. For example, when t=10 µm, L2 = 190 µm-DSAs achieves a high Q-factor of 230. Therefore, decreases the asymmetric parameter of DSAs, the Q-factors can be significantly improved. Similarly, as the metal thickness increases, the Q-factor decreases. But the decreasing trend is smooth when the Q-factor is higher than 200. It means that when the asymmetric parameter is very low, the little difference in metal thickness shows a small effect on the Q-factors. Figures 7(e)–7(g) show the electric field magnitude distributions for DSAs (L2 = 180 µm) with the metal thickness of 10 μm, 40 μm, and 100 μm, where the cut plane is the center of DSAs. Obviously, the electric field becomes weaker as the metal thickness increased, which indicates that thick metal makes quasi-BIC hardly coupling to the free-space radiation. Hence, for DSAs with a certain asymmetric parameter, by reducing the metal thickness, the Q-factor of quasi-BICs can also be improved.

 

Fig. 7. (a) Simulated transmittance spectra of DSAs (L2 = 180 µm) with various metal thicknesses. (b) The relation among the metal thickness, resonant dip frequency, and Q-factor of quasi-BICs (L2 = 180 µm). (c) Simulated transmittance spectra of DSAs (L2 = 190 µm) with various metal thicknesses. (d) The relation among the metal thickness, resonant dip frequency, and Q-factor of quasi-BICs (L2 = 190 µm). (e) Electric field magnitude distributions for 0.641 THz when t is 10 μm. (f) Electric field magnitude distributions for 0.677 THz when t is 40 μm. (g) Electric field magnitude distributions for 0.712 THz when t is 100 μm. The color bar value in (e-g) is the normalization of the local electric field to the incident electric field.

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4. Conclusion

Free-standing metallic resonators based on DSAs are experimentally and numerically investigated at THz frequencies. Investigation results demonstrate that a quasi-BIC with a high Q-factor can be induced via breaking the structural symmetry of DSAs, where the Q-factor can be manipulated by the asymmetry parameter. The transformation from BIC to quasi-BIC is analyzed in detail by calculating the electric field and vector distributions. The out-phase alignment between the electric dipoles in the horizontal direction can be found at the quasi-BIC, which is different from the symmetry-protected BIC with in-phase alignment. This means that strong coupling between the horizontal dipoles are occurred and thus giving rise to the quasi-BIC because of the symmetry-broken effect. The influence of the imperfection in experimental samples such as the round edge and the trapezoid shape on the transmission properties of DSAs is analyzed. Results indicated that the trapezoid shape affects the resonant dip greatly because of the reduced width of the bottom pattern. We investigated the effect of Ohmic losses on the quasi-BIC, compared to the PEC DSAs with zero Ohmic loss, the copper DSAs show lower Q-factor. We also investigated the effect of metal thickness on the quasi-BICs for DSAs. Results exhibit that thinner resonators can achieve stronger quasi-BICs. Based on these results, we can find that high Q-factor quasi-BIC can be realized by modulating the asymmetry parameter, materials loss, and metal thickness. In contrast to the sensor based on metamaterials with high cost, this metal structure is cheap and thus very suitable for low-cost sensing applications. Furthermore, this structure is made of full metal, which can be used for reusable sensors because it is easy to clean by acetone and alcohol after used for material sense. Therefore, the achievement of high Q-factors in the THz regions proves that such metal structures especially interesting for practical THz applications in electromagnetic wave filtering and biomolecular sensing.