The bound state in the continuum (BIC) is a state of wave that coexists with a continuous radiation wave but can be localized, i.e., the energy of the wave is completely confined within the system without any radiation^[1–4]. It can be considered a resonance with zero linewidth or an infinite $Q$-factor. Two-dimensional artificial structures, metasurfaces, and photonic crystal (PhC) slabs provide an excellent platform for the implementation of BICs. Up to now, various BICs based on different mechanisms have been reported, including symmetry-protected (SP) BICs^[5–9], Friedrich–Wintgen (F–W) BICs^[10–13], accidental BICs^[14–17], and guided mode resonance (GMR)-related BICs^[18–21]. Based on the extreme electromagnetic field confinement and ultrahigh $Q$-factors, BICs have been applied in optical nonlinearities^[7,8], low-threshold lasers^[22,23], and high-sensitivity sensing^[24,25]. Moreover, the working frequency of BICs can be adjusted arbitrarily owing to the structural scalability of metasurfaces and PhC slabs. Particularly, the terahertz wave, as an important part of the electromagnetic spectrum, has been widely applied in communication^[26], sensing^[27], and imaging^[28]. In recent years, BICs have been demonstrated in terahertz metasurfaces^[29–33] or PhC slabs^[34], and $Q$-factors exceeding 1000 have been achieved in all-silicon metasurfaces composed of an array of air holes^[35,36]. This lays the foundation for further study of high-$Q$ resonances in the terahertz band.

As true BICs possessing infinite $Q$-factors are unavailable, they are required to be turned into quasi-BICs with limited but high $Q$-factors in realistic applications. SP BICs usually occur at the Γ-point, and their $Q$-factors drop with the increase of the in-plane wavevector^[22]. Therefore, quasi-BICs derived from the SP BICs can be excited under oblique incidence, which is widely adopted in PhC slabs^[37]. SP BICs in PhC slabs exhibit a vortex center in the polarization direction of far-field radiation^[38], which has led to the advancement of ultra-low threshold vortex lasers^[23,39]. In addition to BICs at the Γ-point, PhC slabs also support accidental BICs at the off-Γ point, and their $Q$-factors tend to infinity at some incident angles^[14]. Moreover, merging the SP BIC and accidental BIC in momentum space can suppress the out-of-plane scattering; thus, an extremely high-$Q$ quasi-BIC is achieved^[40]. Although high-$Q$ quasi-BICs can be realized by oblique incidence, they are highly sensitive to the incident angle. Therefore, accurate control of the incident angle is required, which adds to the complexity of application systems. Recent investigations indicate that quasi-BICs in PhC slabs can also be obtained by breaking the structural symmetry, and the manipulation of polarization was studied in some works^[41,42]. However, so far, high-$Q$ quasi-vector BICs reported in PhC slabs are mainly excited through oblique incidence, and the discussion on that induced via structural symmetry breaking remains limited. A complete investigation of quasi-BICs generated by breaking the symmetry of incident light and structure is still rare in THz PhC slabs. Additionally, although accidental BICs can be realized in PhC slabs at oblique incidence, it remains unclear whether accidental BICs could be generated by breaking the structural symmetry at normal incidence.

In this paper, we numerically and experimentally investigate quasi-BICs in a terahertz PhC slab by breaking the excitation field and structural symmetry. The PhC slab supports four SP BICs in the frequency range of 0.6–0.8 THz. The near-field distribution of these BICs at the Γ-point shows different dipole properties, including electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ). Furthermore, the far-field polarization in momentum space reveals their vector vortex and anti-vortex characteristics. All BICs are turned into quasi-BICs under oblique incidence, with two quasi-BICs excited by the $s$-polarized wave while the other two quasi-BICs are induced by the $p$-polarized wave. In addition, breaking the in-plane ${\mathrm{C}}_2$ symmetry of the PhC slab, four quasi-BICs can also be generated under $x$- and $y$-polarized normal incidences. Moreover, for the asymmetric PhC slab, several accidental BICs can be formed at some degrees of asymmetry. Finally, symmetric and asymmetric PhC slabs were fabricated using a combination of photolithography and deep reactive etching. Quasi-BICs in the symmetric PhC slab were measured under oblique incidence using a THz frequency domain spectroscopy system, while quasi-BICs in the asymmetric PhC slab were measured under normal incidence, with the highest $Q$-factor up to 358. High-$Q$ quasi-BICs in asymmetric terahertz PhC slabs have potential applications in terahertz sensors, filters, and modulators.

A terahertz PhC slab with an array of air holes in a silicon plate is illustrated in Fig. 1(a). The lattice constants of the PhC slab in the $x$ and $y$ directions are $a=150\mathrm{\mu m}$, the radius of the air holes is $r=40\mathrm{\mu m}$, and the thickness of the silicon plate is $t=150\mathrm{\mu m}$. First, the finite element method (COMSOL Multiphysics software) was utilized to investigate the eigenmodes and transmission spectra of the PhC slab. In calculations, period boundary conditions were employed in both the $x$ and $y$ directions, along with perfectly matched layers (PMLs) in the $z$-direction. The dielectric constant of silicon was set to 11.67. Figures 1(b) and 1(c) display the calculated transverse magnetic (TM) and transverse electric (TE) eigenmodes of the PhC slab and their $Q$-factors, respectively. We can see that there are eight eigenmodes existing in the frequency range of 0.6–0.8 THz, named TM 1–TM 4 and TE 1–TE 4, as shown in Fig. 1(b). At the Γ-point, the frequencies of the four TM modes are 0.761, 0.672, 0.695, and 0.695 THz, respectively, while the frequencies of the four TE modes are 0.669, 0.727, 0.756, and 0.756 THz, respectively. Among them, TM 3 and TM 4, as well as TE 3 and TE 4, have the same frequency due to mode degeneracy (dashed line), while the $Q$-factors of these modes are finite at the Γ-point and off the Γ-point. These modes are GMRs caused by grating Rayleigh diffraction^[19,37]. However, non-degenerate modes (solid lines of various colors) differ from them. The $Q$-factors of TM 1, TM 2, TE 1, and TE 2 tend to infinity at the Γ-point, and they decrease rapidly as they deviate from the Γ-point, as presented in Fig. 1(c). This occurs due to the spatial symmetry mismatch between the electromagnetic wave and the structure at the Γ-point, which leads to the failure of these modes to couple with the electromagnetic wave and consequently form the SP BICs. The electric and magnetic fields of these modes will be localized inside the PhC slab.

To further understand the characteristics of the four BICs, the electric and magnetic near-field distributions of these BICs in the $x–y$ plane at $k=0$ were calculated and displayed in Figs. 1(d) and 1(e). Specifically, for TM 1, the electric near field is concentrated between the diagonal air holes, creating an opposite pole of the electric field in the $z$-direction. A circular magnetic field is formed around it, leading to an electric dipole in the $z$-direction (${\mathrm{ED}}_z$). For TM 2, the electric near field is mainly distributed between adjacent holes, forming a magnetic field vector along the directions of $\pm 45°$ to the $x$-axis, resulting in a MQ mode as shown in Fig. 1(d). However, the TE 1 mode presented in Fig. 1(e) exhibits opposite characteristics in electromagnetic field distribution to the TM 1 modes, where the magnetic near field of TE 1 is mainly concentrated between the diagonal holes, forming a magnetic dipole in the $z$-direction (${\mathrm{MD}}_z$). Additionally, TE 2 shows the feature of the EQ mode opposite to that of TM 2. Therefore, the four SP BICs possess different physical origins in electromagnetic near fields. In addition to the analysis of electromagnetic near-field distributions, we also discuss the polarization distributions of these four SP BICs in the far fields. The polarization field in momentum space can be defined by projecting the radiative polarization state onto the structural plane and then mapping it onto the Brillouin zone^[43–45]. The $Q$-factor and polarization field distributions of TM 1, TM 2, TE 1, and TE 2 calculated in momentum space are illustrated in Fig. 2. The $Q$-factor of these modes approaches infinity at $k=0$ (Γ-point). In the polarization field, the polarization cannot be defined at these points. In other words, these points will manifest as singularities of far-field polarization or polarization vectors in momentum space, serving as vortex centers that carry conserved and quantized topological charges^[46].

The topological charge ($q$) carried by BICs is defined as^[38]$$q=\frac{1}{2\pi }{\oint }_C\mathrm{d}k·{\nabla }_k\varphi (k),q\in Z,$$(1)where $\varphi (k)$ represents the angle between the polarization vector and the $x$-axis, and $C$ refers to a closed trajectory within the momentum space that travels around the BICs in the counterclockwise direction. The polarization vector must return to its original position after completing the closed loop, meaning the total angle change must be a multiple of $2\pi$, and $q$ must be an integer. In Figs. 2(a) and 2(c), the polarization vectors of TM 1 and TE 1 return to their original positions after completing a counterclockwise loop, resulting in a change in the overall angle of $+2\pi$. Therefore, the topological charge ($q$) is $+1$. In Figs. 2(b) and 2(d), the overall angle of the polarization vectors of TM 2 and TE 2 undergoes a change of $-2\pi$ in their overall angle after completing a counterclockwise loop, resulting in a topological charge ($q$) of $-1$.

As we discussed previously, the PhC slab can support four types of SP BICs with infinite $Q$-factors. Since ideal BICs are unavailable by external excitations, they should be transformed into quasi-BICs with high but limited $Q$-factors in realistic applications. Oblique incidence is widely adopted in PhC slabs to induce quasi-BICs^[37]. Here, we also investigate the influence of oblique incidence on the four types of SP BICs. $s$-polarization and $p$-polarization are considered at oblique incidence, as illustrated in Figs. 3(a) and 3(d), respectively. Figure 3(b) shows the simulated transmission spectra when the PhC slab is illuminated by an $s$-polarized THz wave with an incidence angle of $\theta$. At $\theta =0°$, TE 1 and TE 2 modes are not observed because of their ideal BIC property. As the angle $\theta$ increases, these two modes are turned into quasi-BICs and appear as Fano resonances in the transmissions. The $Q$-factors of TE 1 and TE 2 can be extracted from the transmissions by fitting the Fano formula^[47,48]: $$T(\omega )=T_0+A_0\frac{{[u+2(\omega -{\omega }_0)/\gamma ]}^2}{1+{[2(\omega -{\omega }_0)/\gamma ]}^2},$$(2)where $u$ is the Fano fitting parameter that determines the asymmetry of the resonance curve, ${\omega }_0$ and $\gamma$ represent the resonance angular frequency and bandwidth, respectively, $T_0$ is the transmittance baseline, and $A_0$ is the coupling coefficient. Therefore, $Q={\omega }_0/\gamma$. As shown in Fig. 3(c), the $Q$-factors of TE 1 and TE 2 decrease sharply at first when $\theta$ increases from 0° to 5° and then decrease slowly as $\theta$ further increases. Using the equation $ka/2\pi =(\omega /c)$ sin $\theta$, it can be concluded that the attenuation of the $Q$-factors of these modes follows $Q\propto k^{-2}$ as illustrated in the inset of Fig. 3(c)^[6]. Moreover, two GMRs (TM 3 and TE 4) are always excited under $s$-polarized incidence. As the angle $\theta$ increases, TE 4 experiences a blueshift, while TM 3 is less sensitive to $\theta$. When the PhC slab is illuminated by an oblique $p$-polarized incidence, the other two quasi-BICs (TM 1 and TM 2) and two GMRs (TE 3 and TM 4) are excited and observed in the transmission spectra, as presented in Fig. 3(e). Similarly, as the incidence angle $\phi$ increases, the $Q$-factors of TM 1 and TM 2 decrease quickly, following $Q\propto k^{-2}$, as shown in Fig. 3(f). With the increase of the angle $\phi$, TM 1 undergoes a blueshift, while TM 2 experiences a redshift. It should be noted that the quasi-BICs excited under $p$-polarized incidence are different from those under $s$-polarized incidence, which can be understood based on symmetry analysis. As observed in Fig. 1(d), the electric fields of TM 1 and TM 2 exhibit even characteristics under the mirror reflection operation around the $x$-axis (${\sigma }_y$ operation). Similarly, the magnetic fields of TE 1 and TE 2 are even under ${\sigma }_y$ operation, as seen in Fig. 1(e). Since the electric field of $p$-polarized light and the magnetic field of $s$-polarized light display even features under ${\sigma }_y$ operation, thus TE 1 and TE 2 are induced by $s$-polarized light while TM 1 and TM 2 are excited by $p$-polarized light.

Although high-$Q$ quasi-BICs can be induced through oblique incidence, the incident angle needs to be controlled accurately, which is a large challenge in real applications. In addition, SP BICs can be transformed into quasi-BICs under normal incidence by breaking the in-plane ${\mathrm{C}}_2$ symmetry, which is widely investigated in dielectric metasurfaces with asymmetry particle structures^[6,49,50]. However, there are very few discussions on quasi-BICs induced by breaking the hole symmetry in PhC slabs. Here, we adjust the hole shape to disrupt the in-plane ${\mathrm{C}}_2$ symmetry, as illustrated in Fig. 4(a). In the design, we extended the circular air hole located below the $x$-axis by a length of $L$ in the directions of $\pm x$ (red part) and denoted the cross-sectional area of the red part as $S_{\mathrm{red}}$. The relationship between the $S_{\mathrm{red}}$ and $L$ is given as $$S_{\mathrm{red}}=\{\begin{array}{ll}2Lr-Lr\cos \eta -\eta r^2,& 0\le L<r\\ 2Lr-\pi r^2/2,& r\le L\le a/2\end{array},$$(3)where $L/r=\sin \eta (0\le \eta <\pi /2)$. The asymmetry parameter here is defined as the cross-sectional area of the red part divided by the area of the semicircle, i.e., $\alpha =2S_{\mathrm{red}}/S_{\mathrm{circle}}$, where $S_{\mathrm{circle}}=\pi r^2$. The calculated resonant frequencies and $Q$-factors of TE 1 and TE 2 with respect to the asymmetry parameter $\alpha$ are shown in Figs. 4(b) and 4(c), respectively, when the PhC slab is illuminated by an $x$-polarized THz wave at normal incidence. As $\alpha$ increases, the volume of air holes increases; correspondingly, the effective dielectric constant decreases, leading to blueshift in both TE 1 and TE 2. Meanwhile, the $Q$-factors of these two quasi-BICs decrease significantly and follow a trend of $Q\propto {\alpha }^{-2}$^[6]. However, an “accidental” phenomenon occurs during the increase of $\alpha$, and the $Q$-factor again approaches infinity, i.e., BIC, at $\alpha =0.11$ for TE 1 and $\alpha =0.07$ for TE 2, respectively. Unlike the SP BIC of the TE modes at $\alpha =0$, this isolated BIC is stimulated by destructive interference of several leakage channels, known as an accidental BIC^[31,51,52]. Within the PhC slab, the propagation field can be described as a combination of waves with different propagation constants along the $z$-axis. When these waves reach the interface between the PhC slab and air, they interfere destructively, thus the leakage vanishes completely. Moreover, TM 1 and TM 2 are excited when the PhC slab is illuminated by a $y$-polarized terahertz wave at normal incidence. Their frequencies and $Q$-factors with respect to the asymmetry parameter $\alpha$ are shown in Figs. 4(d) and 4(e). Similarly, they exhibit blueshifts as $\alpha$ increases, and their $Q$-factors approach infinity when $\alpha =0$, but decrease significantly with the increase of $\alpha$. When $\alpha =0.06$, the $Q$-factor of TM 2 approaches infinity again, which is also called accidental BIC, but it does not occur for TM 1. Here, we should mention that these accidental BICs do not have vortex characteristics and are not evolved from those topologically protected accidental BICs in the PhC slab at $\alpha =0$^[14,38]. We take TE 1 as an example to illustrate as follows: The dispersion curves of TE 1 at different thicknesses of a PhC slab when $\alpha =0.11$ was calculated are shown in Fig. 5. When $t=170\mathrm{\mu m}$, there exists a quasi-SP BIC at the Γ-point and multiple quasi-accidental BICs at the off Γ-points, due to small asymmetry disturbance of the structure. As $t$ decreases to 165 µm, these quasi-accidental BICs approach and merge with the quasi-SP BIC at the Γ-point. However, when $t$ further decreases from 165 µm, only the isolated quasi-SP BIC remains, while the merged quasi-accidental BICs usually band transit to a corresponding GMR^[53].

To demonstrate previous simulations, the designed PhC slabs were fabricated on a high-resistivity silicon wafer ($>5000\mathrm{\Omega }·\mathrm{cm}$) using conventional photolithography and deep reactive ion etching (Bosch process)^[35,54,55]. High-resistance silicon is used because of its low absorption and low dispersion at terahertz frequencies. According to the simulations, both a symmetric PhC slab ($L=0\mathrm{\mu m}$) and an asymmetric PhC slab ($L=60\mathrm{\mu m}$ or $\alpha =0.91$) with a size of $15\text{\hspace{0.17em} mm}\times 15\text{\hspace{0.17em} mm}$ were fabricated, respectively, and their scanning electron microscope (SEM) images are shown in Figs. 6(a) and 6(b). A high-spectral-resolution (140 MHz) THz frequency domain photomixing system was utilized to measure the transmission spectra of the PhC slabs, as illustrated in Fig. 6(c), where four off-axis parabolic lenses with 75 mm focal length and 2.54 cm diameter make up the 4f optical system. The measurements were conducted under room temperature and dry air conditions ($\text{humidity}\le 1\%$) to minimize the absorption of water vapor on the THz wave. The integration constant was set to 300 ms to enhance the signal-to-noise ratio (SNR) of the system. The measured and simulated transmission spectra of the symmetric PhC slab ($L=0\mathrm{\mu m}$) under $s$- and $p$-polarized oblique incidences are shown in Figs. 7(a) and 7(c), respectively. When $\theta =0°$, a GMR of TM 3 is clearly measured at a frequency of 0.71 THz, which agrees well with simulations; another GMR of TE 4 with a simulated $Q$-factor over 70,000 is not measured, while two BICs of TE 1 and TE 2 are not observed as expected. When $\theta$ increases to 10°, TE 1 and TE 2 as quasi-BICs are excited and measured at 0.663 and 0.733 THz, with $Q$-factors of 133 and 235, respectively, while the simulated TE 1 and TE 2 lie at 0.664 and 0.738 THz, with $Q$-factors of 384 and 2226, respectively. The measured $Q$-factor of TE 2 at $\theta =10°$ is obtained by Fano fitting the transmission as shown in Fig. 7(b). When $\theta$ increases to 20°, the measured two quasi-BIC resonances show a redshift of 21 and 1 GHz, respectively, and the $Q$-factors decrease to 89 and 229, respectively.

Conversely, the $Q$-factor of TM 3 increases as $\theta$ increases, reaching a simulated value over 10,000 at $\theta =20°$. However, TM 3 at $\theta =20°$ and TE 4 at all three different incident angles were not observed in the experiment, possibly due to the fabrication defects and material’s absorption loss, which could affect the achievement of high-$Q$ resonance. Furthermore, it is mainly limited by the resolution and SNR of the commercially available measurement instruments. For the same reason, TE 1 and TE 2 were not observed at $\theta \le 5°$. The transmission spectra under $p$-polarized incidence are shown in Fig. 7(c), and only the GMR of TM 4 with a specific frequency of $\sim 0.71\mathrm{THz}$ is clearly observed at $\phi =0°$ and $\phi =10°$, but it is not seen at $\phi =20°$. No quasi-BIC resonances of TM 1 and TM 2 were observed in the measurements due to their very large $Q$-factors, as well as the GMRs of TE 3, although they indeed existed in the simulations.

For the asymmetric PhC slab with $\alpha =0.91$ ($L=60\mathrm{\mu m}$) under $x$- polarized incidence, both the measured and simulated transmissions are shown in Fig. 7(d). In the simulated transmissions, two quasi-BICs, TE 1 and TE 2, are located at frequencies of 0.716 and 0.811 THz, respectively, while two GMRs of TM 3 and TE 4 are at 0.733 and 0.848 THz. In the measured transmissions, TE 1, TM 3, and TE 4 can be clearly observed at 0.715, 0.735, and 0.852 THz, respectively, which agrees well with the simulations. The measured $Q$-factor of TE 1 by Fano fitting, as shown in Fig. 7(e), is 358, whereas 792 and 342 for TM 3 and TE 4, respectively. However, quasi-BIC TE 2 is not measured. For the $y$-polarized incidence shown in Fig. 7(f), the simulated two quasi-BIC resonances, TM 1 and TM 2, are located at 0.812 and 0.719 THz, respectively, while the two GMRs, TE 3 and TM 4, are situated at 0.808 and 0.767 THz, respectively. The measured results show that TM 1 and TM 2 are located at 0.814 and 0.720 THz, with $Q$-factors of 370 and 85, respectively. TM 4 is measured at 0.768 THz, but TE 3 is not observed. The measured frequencies and $Q$-factors of these TE and TM modes are displayed in Table 1. Larger $Q$-factors of quasi-BICs could be achieved in PhC slabs with smaller asymmetry parameters and larger device size; however, a limited $Q$ value of about 1000 was measured by commercially available THz time-domain spectroscopy and frequency-domain photomixing systems^[35,36], while a $Q$ value of 3700 was achieved using a higher resolution THz system^[56].

In conclusion, we propose and demonstrate quasi-BICs in a symmetry-breaking terahertz PhC slab. The PhC slab supports four SP BICs in the frequency range of 0.6–0.8 THz, which shows ED, MD, EQ, and MQ properties in the near fields, and vector vortex characteristics in the far fields. Quasi-BIC resonances were excited in both ${\mathrm{C}}_2$ symmetry-breaking PhC slabs at normal incidence and symmetric PhC slabs with oblique incidence. Besides SP BICs, accidental BICs also appear as the asymmetry parameter varies. Finally, both symmetric and asymmetric PhC slabs were fabricated using a combination of photolithography and deep reactive etching. The experimental results obtained from terahertz high-resolution spectroscopy agree well with the simulations. The measured $Q$-factors of a quasi-BIC and a GMR in symmetry-breaking terahertz PhC slabs can reach 358 and 792, respectively. Although the demonstrations were conducted at terahertz, similar phenomena could be realized at other frequencies by adjusting the structural parameters. We believe that the terahertz PhC slab presented in this work has great potential in terahertz sensing and other functional devices.