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Abstract
Conventionally, a symmetry-protected quasi bound state of the continuum (BIC) becomes achievable by breaking the C2 symmetry of meta-atoms. Our work exhibits a novel approach to achieving dual band quasi-BIC by breaking the C2v symmetry into Cs symmetry. Also, we show that a single band quasi-BIC can be achieved by breaking the C2v symmetry into C2 symmetry. Our metasurface of C2v symmetry is composed of double gaps split ring resonator (DSRR), and it degrades to C2 symmetry when the double gaps are displaced in opposite directions. One band quasi-BIC can be observed occurring at around 0.36 and 0.61 THz respectively with the metasurface excited by x- and y-polarized terahertz radiation, respectively. A couple of dark dipole oscillator dominates the quasi-BIC at 0.36 THz, while a quadruple-like oscillator dominates the quasi-BIC at 0.61 THz. The damping ratio and coupling coefficients of the above single quasi-BIC are close to the orthogonal polarization of the incident terahertz wave. However, the metasurface of the DSRR array degrades down to Cs symmetry when the double gaps are displaced in the same directions. A dual band quasi-BIC (0.23 THz and 0.62 THz) is found to be sensitive to the y-polarized terahertz radiation. It is found that the inductive-capacitive (LC) resonance results in quasi-BIC at 0.23 THz, while a quadrupole-like oscillation results in quasi-BIC at 0.62 THz. The quasi-BIC at 0.62 THz has a higher coupling coefficient and lower damping ratio than quasi-BIC at 0.23 THz in a metasurface of Cs symmetry. The realization of the above locally symmetric breaking on the quasi-BIC of terahertz metasurfaces is helpful for the innovation of multi-band terahertz biosensors.
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1. Introduction
Terahertz band (1 THz = 1012 Hz) within the full-electromagnetic spectrum, shows a wide range of application prospects in non-invasive imaging and label-free sensing due to its special properties of strong penetrability of low photon energy in soft matter, such as DNA, protein, polypeptide, tumour, etc. [1–6]. The detection or characterization of extremely subtle analytes is one such case that has now inspired a burgeoning branch of terahertz spectroscopy or imaging techniques. Alternatively, small local disturbances can cause detectable changes in the electromagnetic spectra due to the sharp-resonance with a high quality (Q) factor of planar metamaterials or metasurfaces, which can be used for ultra-sensitive terahertz biosensors [7]. Ideally, the bound states in the continuum (BIC) provide a robust method to achieve infinite Q factor in theory and has recently attracted strong attention from researchers [8–10]. However, it is impossible to observe the infinite Q factor due to the disappearance of the resonance linewidth from the frequency resonance spectrum. Such BIC that exists in a preserved symmetric system is classified as symmetry-protected BIC (SP-BIC). In practice, BIC can be transformed into a quasi-BIC by introducing perturbation to break the symmetry to produce an observable Fano line-shaped leaky resonance, in which both the Q factor and the resonance linewidth become measurable [11–13].
In a system of reflection or rotational symmetry, resonant modes of different symmetry classes completely decouple. For a bound state of one symmetry class, embedded in the continuous spectrum of another symmetry class, their coupling is forbidden if the symmetry is preserved. When the symmetry of the structure is broken (in most cases the C2 group needs to be broken), an energy leaky channel results in radiation to the external continuum, such as crossing structure, rods pairs, asymmetric split-ring resonators (SRRs) [12,14,15]. However, this is usually only accompanied by the observation of a single band quasi-BIC. According to group theory, C2 group is the sub-group of C2v. The degradation of the C2v to C2 or Cs is e;;ssentially one type of localized symmetry. In this case, does a higher-order symmetry breaking affect the number of observed bands in quasi-BIC? Theoretically, multipolar scattering power by introducing symmetric perturbation can achieve dual band quasi-BIC [16,17]. Compared to the single band quasi-BIC, dual band quasi-BIC can tune the spectral lines of two frequency resonance positions at the same time, which is very suitable for designing multi-band sensors.
In this work, we investigate the evolution of quasi-BIC in a system of both reflectional and rotational symmetry (C2v) degrading to only rotational symmetry (C2) or reflectional symmetry (Cs). The proposed metasurfaces are composed of classical sub-wavelength meta-atoms of double-gaps split ring resonators (DSRR). By tuning the position of the double gaps, C2v symmetry degrades to C2 symmetry or Cs symmetry. BIC turns to observable quasi-BIC relying on the symmetry of meta-atoms, as well as the polarization of the incident radiation. The physical mechanism of single band and dual band quasi-BIC with different symmetry under orthogonal polarization states are realized.
2. Experiments
Two types of meta-atoms are illustrated in Fig. 1(a). Initially, a symmetric DSRR of both gaps at the center of the top and bottom arms, which belongs to the C2v group. The C2v group has two sub-groups: C2 rotational symmetry relative to the z axis; while Cs reflectional symmetry relative to the x and y axis. Two gaps divide the metal arm into two branches. When the top and bottom gaps move in opposite directions, the original structure of C2v symmetry degrades to C2 symmetry. The metal arms of the two branches divided by gaps are equal in length (l1 = l2). Conversely, when the top and bottom gaps move in the same direction, the original structure of C2v symmetry degrades to Cs symmetry. The metal arms of the two branches are not equal in length (l1 ≠ l2). The structural parameters of the two symmetrical types of units are as follow: the period of the unit cell is p = 300 µm, the size of DSRR is l = 200 µm, width w = 20 µm, and gap size is g = 20 µm, the distance between the gap center and central axes is d (dmax = 70 µm). A dimensionless parameter is defined as α = 100% × d / dmax to quantitatively measure the degree of asymmetry. The simulations are performed by finite element method-based platform CST Microwave StudioTM. All materials assumed lossy in simulation. The transmittance T(ν) spectrum of the proposed meta-atoms can be calculated as a function of S-parameter T(ν) |S21|2. Experimentally, seven meta-atoms with two types of symmetry at different d, are fabricated, that is d = 0, ±30, ±50, and ±70 µm, respectively. After photolithography and lift-off process, 0.2 µm thick gold DSRR are deposited on a 25 µm thick polyimide thin-film substrate. The specific fabrication process is similar to our previous work [18]. Fig. 1(b) shows the samples of two symmetrical types of our proposed meta-atoms.
Fig. 1. (a) The meta-atom with C2 symmetry and Cs symmetry. The horizontal and vertical dash lines and black dots represent the symmetric axis of x, y, and z, respectively. (b) Microscopic images of the unit cell of the fabricated samples with C2, C2v, and Cs symmetry. (c) A schematic diagram showing terahertz transmittance through the proposed metasurface.
A schematic diagram demonstrates terahertz transmittance measurement shown in Fig. 1(c). The metasurface is in the x-y plane, and the incident terahertz pulse is along the z axis. The transmittance spectra of the metasurfaces were measured by a fibre-coupled terahertz-time domain spectroscopy (THz-TDS) system (TERA K15, Menlosystem GmbH). The temporal window is set at 100 ps, corresponding frequency revolution is 10 GHz. The transmission spectrum, extracted from Fourier transforms of the measured time-domain electric fields, is defined as [19]
where T(ν) is the transmittance as a function of terahertz frequency, Esample(ν) and Eref(ν) are the amplitudes of Fourier transformed electric fields across the sample and reference, respectively.3. Results and discussions
As shown in Fig. 2(a), the transmittance spectra of C2 meta-atom as a function of frequency are plotted for different d excited by x-polarized and y-polarized terahertz waves. Initially, when d = 0 µm (α = 0), the structure is in the bound state due to the symmetry protected state, which only has a wider dipole response of intrinsic mode at 0.62 THz for x-polarization and 0.38 THz for y-polarization without leaky mode. At the introduction of asymmetry, C2v symmetry of the metasurface degrades to C2 symmetry, which breaks the reflectional symmetry along the x and y axis. Therefore, the quasi-BIC phenomenon in which the line width gradually increases can be observed. It is worth noting that the frequency response of C2 meta-atom excited by x-polarized terahertz wave occurs at around 0.36 THz, which is lower than 0.61 THz for y-polarization.
Fig. 2. (a) Transmittance of C2 meta-atom as a function of frequency for different d excited by x-polarized and y-polarized terahertz waves. Black solid line: simulation, red solid line: measurement, blue solid line: coupled model theory. (b) Diagram of the coupled two oscillators model for the C2 meta-atom. γ: damping rate. κ: coupling coefficient. (c), (d) The simulated terahertz transmittance as functions of asymmetry and frequency of C2 meta-atom excited by x-polarized and y-polarized terahertz waves. (e), (f) Extracted Q factors of the quasi-BICs excited by x-polarized and y-polarized terahertz waves. Black dotted line: simulation, Red Cross: measurement, blue solid line: theory (lossless).
To reveal the physical mechanism underlying the frequency response, we employed the two coupled oscillator theory to mimic the behavior of the quasi-BIC in terahertz transmittance spectra. In this coupled oscillator model, the bright mode is the intrinsic resonance considered as the continuum and the dark mode is the leaky mode as shown in Fig. 2(b). The interaction between intrinsic and leaky modes under incident terahertz wave E = E0e-iωt in a coupled system can be described by the following equations: [20]
Table 1. The fitted parameters for C2 meta-atoms.
Obviously, the intrinsic modes and leaky modes of the C2 symmetric meta-atom excited by x-polarized terahertz radiation occur around 0.60 and 0.40 THz, respectively. However, the frequencies of intrinsic modes and leaky modes excited by y-polarized THz radiation are interchanged. With the asymmetric growth of C2 meta-atom, the value of the coupling strength κ, as well as the leaky ratio γ1 gradually increases, which means the increase of ability to build leaky channels to free space. To further investigate the frequency response in detail, finer maps of transmittance are simulated as a function of frequency ν and the asymmetry α, in Figs. 2(c), (d). The maps show a clear process of converting BIC into quasi-BIC, as the asymmetry increases. Further evidence of BIC is the disappearance of the leaky mode, giving a divergence of the Q factor tending to infinity. The Q factors of the resonances can be calculated as Q = ν0/2γ, where ν0 is the center resonance frequency. When d = ±30, the maximum Q factor measured experimentally under the excitation of y-polarized terahertz radiation is 20.7. Our result is comparable to previously reported values for the Q factor of quasi-BIC based on metallic structures [21]. It is worth mentioning that the ohmic loss of gold and polyimide substrate are considered in the simulation, so that the fitted Q factor trend is much real-world scenarios, different from that of perfect electric conductor (PEC), as shown in Figs. 2(e) and (f). It is obvious that the Q factor without considering the loss is significantly higher than the real case. For an ideal BIC, the theoretical Q factor should be close to infinity. The loss mechanism of the limited Q factors originates from the ohmic loss in the metal. The Drude model indicates that the imaginary part of the dielectric constant of gold reaches 106 at terahertz band, which is much higher than that of the PEC [22].
To further understand the quasi-BIC behavior of C2 meta-atoms, the distributions of surface current and magnetic field are monitored when d = ±50 µm at the leaky mode presented in Fig. 3. For x-polarization when d = 50 µm, the induced currents are generated weakly along the metal left and right arms of DSRR, which flow in the same direction but perpendicular to the direction of the incident electric field. When d = −50 µm, the current direction is opposite due to the out-of-phase caused by the gaps moving to the opposite side of the equidistant. For y-polarization, there are four current branches of equal strength along the metal arm of DSRR. The strength of the induced current at y-polarization is stronger than x-polarization. The resonance frequency is also higher, as shown in Fig. 3(a). Due to the introduction of asymmetry, the current distribution seems to be twisted, but the individual current amounts are equal to satisfy C2 symmetry. The corresponding magnetic field distributions depend on the induce current presented in Fig. 3(b). Unlike dipoles that usually appear in bright mode, the direction of the currents that produce magnetic dipoles here are perpendicular to the direction of the incident electric field at x-polarization, indicating the leaky mode ω1 (2π × 0.38 THz) is due to broken symmetry. For y-polarization, the magnetic quadrupole dominates the leaky mode ω1 (2π × 0.62 THz).
Fig. 3. (a) Surface current distribution and (b) corresponding magnetic field distribution of the quasi-BICs when d = ±50 µm for C2 meta-atom.
Figure 4(a) shows the transmittance spectra of Cs meta-atom as a function of frequency for different d excited by x-polarized and y-polarized terahertz waves. The difference from C2 meta-atom is that Cs meta-atom only has reflectional symmetry along the x axis, thus the quasi-BIC cannot be observed, but the redshift of the intrinsic mode, with increasing of the asymmetry under x-polarized excitation. However, dual-band quasi-BICs can be observed at around νL = 0.22 THz and νH = 0.61 THz at y-polarization. To better explain this phenomenon, Eq. (3) needs to be added with a new coupling term, given as:
Fig. 4. (a) Transmittance spectra of Cs meta-atoms as a function of frequency for different d excited by x-polarized and y-polarized terahertz waves. Black solid line: simulation, red solid line: measurement, blue solid line: coupled model theory. (b) Diagram of the coupled three oscillators model for Cs meta-atom. γ: damping rate. κ: coupling coefficient. (c), (d) The simulated terahertz transmittance as functions of asymmetry and frequency of Cs meta-atom excited by x-polarized and y-polarized terahertz waves. (e), (f) Extracted Q factors of the quasi-BICs at the frequencies of νL and νH. Black dotted line: simulation, red cross: measurement, blue solid line: theory (lossless).
Therefore, the three coupled oscillators model is composed of one intrinsic mode and two leaky modes shown in Fig. 4(b). By solving the Eqs. (2), (7), and (8), the linear susceptibility expression of the Cs meta-atom can be written as:
where:The transmittances of the three coupled oscillators model are in good agreement with the measured and simulated results shown in Fig. 4(a). The corresponding fitted parameters are listed in Table 2.
Table 2. The fitted parameters for Cs meta-atoms.
As shown in Table 2, the dual band quasi-BIC phenomenon is attributed to the introduction of asymmetry, which results in the increase of coupling coefficient (κ12, κ23) as well as the leaky ratio (γ1, γ2). Intuitive maps of transmittance are simulated as a function of frequency ν and the asymmetry α, in Figs. 4(c), (d). It is obvious that two narrow resonances (νL, νH) of leaky modes are distributed on both sides of the intrinsic mode at y-polarization, presented in Fig. 4(d). Herein, we fit Q factor trend with loss of materials and compare it with the simulation using PEC, as shown in Figs. 4(e), (f). It can be found that the maximum Q factor measured in the experiment is 20.3.
For Cs meta-atom, the two leaky modes of the y-polarization are monitored simultaneously, shown in Fig. 5. For the mode νL, when d = −50 µm, the surface current circulates anti-clockwise along the metal arm, which indicates an inductive-capacitive (LC) resonance mode corresponding to ω1 (0.25 × 2π THz) in coupled oscillator model. While when d = 50 µm, the surface current circulates clockwise along the metal arm. The current is mainly distributed on the side of the long arm. For the mode νH mode corresponding to ω2 (0.58 × 2π THz), there are four currents distributed on the metal arm. Herein, the current on the metal arm along the direction of the electric field is parallel to the same direction, and the direction of the current on the metal arm perpendicular to the direction of the electric field is antiparallel. The strength of the current along the longer arm is also stronger than the short ones. The corresponding magnetic field distributions are shown in Fig. 5(b). Unlike C2 meta-atoms, the current distribution is not equal due to the symmetry incompatibility (l1 ≠ l2), forming the unequal-sized magnetic dipoles and quadrupoles at resonance frequencies νL and νH, respectively. The magnetic field formed by the current distribution influenced by the geometrical structure determines the way of coupling with the far field.
Fig. 5. (a) Surface current distribution and (b) corresponding magnetic field distribution of the quasi-BICs at y-polarization when d = ±50 µm for Cs meta-atoms.
4. Summary
In summary, we investigated the evolution of quasi-BIC in a system of C2v symmetry degrading down to only C2 symmetry or Cs symmetry when double gaps on opposite arms of DSRR are displaced in the same direction or opposite direction. For C2 meta-atoms, a couple of dark dipole oscillator dominates the quasi-BIC excited by x-polarized terahertz waves at 0.36 THz, while a quadruple-like oscillator dominates the quasi-BIC excited by y-polarized terahertz waves at 0.36 THz and 0.61 THz. While for Cs meta-atoms, a dual-band quasi-BIC (0.23 THz and 0.62 THz) is derived from LC and quadrupole-like resonance, which can only be excited by y-polarized terahertz radiation. Both two and three coupled oscillators models reveal that coupling strength and the leaky ratio with free space are attributed to the asymmetry of the structure. The maximum Q factor measured in the experiment of the two symmetry types both exceeds 20. Our results demonstrate the influence of locally geometric symmetry breaking on the quasi-BIC of terahertz meta-atoms, which provide a feasible solution for the design of multi-band sensors.
Funding
General Research Fund of Shanghai Normal University (309-C-9000-21-309119); Chinese Academy of Sciences (21ZD01).
Acknowledgments
This work was performed in part at the Micro Nano Research Facility at RMIT University in the Victorian Node of the Australian National Fabrication Facility (ANFF). Lei Wang and Zhenyu Zhao contributed equally to this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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