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Abstract
Bound states in the continuum (BIC) represent distinct non-radiative states endowed with infinite lifetime and vanishing resonance linewidth. Introducing asymmetric perturbation to the system can convert true BICs into high quality leaky modes which is useful in many photonic applications. Previously, such perturbation and resonance of interest is only limited to a single factor. However, different perturbations by unit cell gap, geometry and rotation angle result distinctive resonance modes. The combination of two perturbation factors can excite multi-mode resonance contributed from each asymmetric factor which coexist simultaneously; thus, the number of reflectance peaks can be controlled. In addition, we have carefully analyzed the electric field variations under different perturbation factors, followed by a multipolar decomposition of resonances to reveal underlying mechanisms of distinct resonance modes. Through simulations, we find that the introduction of multiple asymmetric perturbations also influences the metasurface sensitivity in refractive index sensing and compare the performance of different resonance modes. These observations provide structural design insights for achieving high quality resonance with multiple modes and ultra-sensitive sensing.
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1. Introduction
High-quality electromagnetic resonance modes are of paramount significance in optics. Recently, bound states in the continuum (BIC) have attracted widespread attention because of its infinite quality factor (Q-factor) and vanishing resonance linewidth. The concept of BIC was initially introduced by Neumann and Wigner within the context of quantum mechanics, referring to a localized confined mode within dynamic systems that coexist alongside continuous radiation spectra [1]. They represent a ubiquitous phenomenon with observations spanning diverse domains including microwaves, acoustics, and optics [2–4]. BICs can be systematically categorized into two major classes: accidental or tunable BIC and symmetry-protected BIC [5]. Accidental BIC arises from the continuous tuning of one or multiple system parameters, resulting in the accidental vanish of radiation wave coupling coefficients. Common accidental BIC includes the Fabry-Pérot BIC and Friedrich-Wintgen BIC. Within a dual-resonator system, the Fabry-Pérot BIC emerges, with each resonator serving as an ideal mirror [6,7], for example Gao et al. [8] introduced a structural design guiding the realization of Fabry-Pérot BICs within lossy systems. Friedrich-Wintgen BIC (FW-BIC) arises through adjusting interactions between two or more resonances, resulting in the complete disruption of interference mechanisms [9]. FW-BIC was initially observed through experimental exploration conducted by Lepetit and Kanté, involving a metal waveguide incorporating two ceramic plates [10]. Symmetry-protected BIC inhabits the center of the Brillouin zone, the Γ point, where the coupling to the free space is forbidden, the vanishing of coupling coefficients stems from the symmetry disruptive interference. Compare to accidental or tunable BIC, symmetry-protected BIC provides more freedom for design and application, translation [11,12], rotation [13], structural destruction [13–15], and modulation of dielectric constants [16] are usually employed to break symmetry. Consequently, BICs can transform into leaky modes which are commonly referred to as quasi-BICs [17], characterized by sharp resonances observed prominently in the spectrum with ultra-high quality factors. Nonetheless, the ultra-high Q resonance only exist when approaching the BIC point in momentum space under ideal theoretical situations. Recently, Brillouin zone folding (BZF) concept has been introduced to engineer BICs with disorder-robust high Q resonance [18].
Metasurfaces are artificial arrangements of subwavelength meta-atoms capable of strong light field manipulation and modulation, noted for many advanced applications within the realm of modern optics [19]. BIC designed in dielectric metasurfaces can support high Q-factor resonances with low optical losses, showcasing considerable potential in diverse application domains, including nonlinear optics [20,21], optical switching [22], laser [23], ultra-sensitive sensing [11], narrowband filters [24,25] etc. Recently, there has been extensive research on quasi-BIC modes resonating in the near-infrared spectrum. For example, Algorri et al. [26] introduced the square slotted silicon metasurface based on quasi-BIC supporting electromagnetically induced transparency (EIT) theoretically, where eigenmode analysis is provided and later provided an experimental investigation of a slotted silicon metasurface supporting such quasi-BIC in the near IR [27]. Dielectric metasurfaces based on rectangular silicon bar design have also been investigated intensively. For instance, symmetric array of all dielectric silicon bars can achieve strong electrical dipole resonance with high Q-factor [28]. Breaking the symmetry of the silicon bar dimension, can greatly enhance the nonlinear response [29] and breaking the symmetry of the rotation angle of silicon bar pairs can employ phase interrogation for highly sensitive refractive index sensing [30]. However, in most designs, the BIC only introduces one type of perturbation to form quasi-BIC, the concurrent introduction of different types of perturbations within a structure remains unexplored. In this work, we take the simplest rectangular silicon bar as the fundamental unit cell, under the scheme of symmetry-protected BICs, by changing the length or rotation of the unit cell under the condition of gap perturbation, we find that different types of symmetry perturbations can excite corresponding quasi-BIC modes simultaneously, thus the number of peaks can be controlled. By analyzing the electric field and performing multipolar decomposition of various quasi-BIC modes, we carefully investigate the mechanisms of quasi-BIC resonance features. Finally, the sensitivity of refractive index sensing is compared among different perturbation conditions. Our research presents a more thorough design methodology for achieving multi-mode resonating quasi-BIC, thus advancing the utilization of BIC for high performance sensing devices.
2. Structure design
We choose a square periodic unit cell with size Px = 450 nm, Py = 900 nm. The length, width, and height of the nanobars are 450 nm, 150 nm and 400 nm respectively. Finite Domain Time Domain (Lumerical, Inc) was used for simulation. The plane wave was incident in the normal direction on the metasurface, the reflectance spectrum has no response in 1100 - 1500 nm wavelength when there is no perturbation. After introducing a perturbation, such as gap perturbation, length perturbation or rotation perturbation as shown in Fig. 1(a) and Fig. 1(b), the symmetry of the structure is broken, hence the symmetry-protected quasi-BIC was motivated. In this paper, we found that different types of symmetry perturbation can exist simultaneously as shown in Fig. 1(c).
Fig. 1. Symmetry-protected bound states in the continuum triggered by different perturbation factors. (a) a schematic diagram of the metasurface and perturbation factors. (b) a schematic diagram of the combined perturbation of length/gap and rotation/gap. (c) a reflection spectrum diagram of the multi-mode resonance from combined perturbation.
3. Results and discussions
First, we introduce a gap perturbation by changing the distance between two adjacent structures in X direction as shown in Fig. 2(a). The asymmetry parameter α was be defined as:
Where Δx is half of the variation of the distance between two adjacent structures and D is the distance between two adjacent structures (initial value =300 nm). Initially, the two nanobar structure is located at the center of each half unit cell, thus α1 = 0. After the nanobar is shifted towards (or away from) each other, each half unit cell is no longer symmetric, instead, the period of the unit cell in the x direction is turned to double (900 nm), which leads to the emergence of quasi-BIC which we term it as mode 1 resonance. We consider this mode 1 resonance is originated from the gap (or periodicity) perturbation. Figure 2(b) shows the reflection spectrum of the structures in 1100 nm - 1500 nm wavelength range with different asymmetry parameter α. When α1 is 0, the period of structure is 450 nm and the resonance is in dark state, thus there is no resonance peak observable in the reflection spectrum. When the asymmetry is introduced and the α1 is not 0, the period turns to 900 nm, the quasi-BIC resonance mode is excited and sharp resonance peak appears in the reflection spectrum. As α1 is increased, the reflection and the bandwidth of the resonance peak become larger, and the blue shift occurs. Q- factor can be calculated as: Where fr is the resonance frequency, and FWHM is the full width at half maximum. Figure 2(c) shows the correlation curve of the Q-factor and asymmetry parameter α1, it is obvious that Q- factor decreases as α1 increases and satisfies $\textrm{Q} \propto {\; }{\mathrm{\alpha }^{ - 2}}$. which is consistent with conclusions of previous studies on BIC [13]. The electric field diagram was shown in Fig. 2(d), the initial distance between two adjacent structures in x direction is 300 nm when the structure is in perfect symmetry. Following the introduction of the perturbation, a substantial enhancement in the electric near-field intensity is observed, with an increased maximum value of the electric near-field at the gap region. The intensified electric field remains localized on the resonator surface. When Δx approaches 0, α1 becomes smaller, and the enhancement of the electric field becomes larger.Fig. 2. Gap perturbation. (a) schematic diagram of the unit cell and definition of asymmetric factor. (b) Reflection spectrum for different α. (c) the Q-factor as a function of α1. (d) When α=0.16, 0.33, 0.50, the electric field enhancement of mode 1. (e) resonance multipolar decomposition diagram.
To assess the dominant contribution mechanisms behind distinct resonances, the multipole decomposition is conducted in the Cartesian coordinate system. Initially, the displacement current density J(r) within the unit cell is integrated to derive diverse multipole moments. These multipole moments subsequently facilitate the calculation of the sum of scattering cross-sections for all moments at a particular point. The displacement current density J(r) is derived from the electric field distribution E(r):
The introduction of gap perturbation changes the unit cell size into double (900 nm) as shown in Fig. 1(b) which make the two adjacent nanobars form a new unit cell and we get one resonance peak. On this basis, we change the geometric symmetry within the cell (such as changing the length of one of the nanobars shown in Fig. 1(b)) to get two resonance peaks, and another symmetry-protected quasi-BIC pattern is motivated. We defined the new asymmetry parameter α as:
Where ΔL is the variation of length of one of the nanobars from each end, and L is the initial value of the length of nanobars (450 nm). For ease of distinction here we name the gap asymmetry parameter as α1 and the new length asymmetry parameter as α2. The two kinds of symmetry-protected quasi-BIC patterns respectively motivated by α1 and α2 can be concurrence. Figure 3(a) shows the reflection spectrum of the structures in 1100 nm - 1500 nm wavelength with different dimension asymmetry parameter α2 when α1 is fixed (Δx = 75 nm).Fig. 3. Introduce gap perturbation and length perturbation simultaneously. (a) reflection spectrum diagram, (b)and the near electric field enhancement of mode 1 and mode 2 under different ΔL when Δx = 75 nm. (c) reflection spectrum diagram, (d) and the near electric field enhancement of mode 1 and mode 2 under different Δx when ΔL = 50 nm. (e) the Q-factor as a function of α2. (f) multipolar decomposition of resonance.
There are two resonance peaks, and their resonance features can be affected by both asymmetric factors. As the increasing of α2 (ΔL increase from 0 nm to112.5 nm), the reflectivity and the bandwidth of mode 2 become larger with a blue shift occurrence while the reflectivity and the bandwidth of mode 1 remain unaffected with observed blue shift as well. Since α1 is relatively large when Δx is 75 nm, the Q-factor of mode 1 is quite low. The complete spectral shifts from ΔL = 0 to 115 nm are presented in Fig. S1, and the true BIC resonance occurs at ΔL = 0 nm. According to Formula 2, we can calculate that the Q-factor of mode 1 decreases when α2 decreases. As shown in Fig. 3(b), when ΔL approaches to 0, α2 becomes smaller, and the electric field enhancement of mode 1 becomes smaller, the intensified electric field remains predominantly concentrated along the peripheries of the two rectangular unit cells. While the electric field enhancement of mode 2 becomes larger which has the same variation tendency with Q-factor, and the near-field augmentation is predominantly focused on the right-side truncated rectangular silicon pillar. Figure 3(c) shows the reflection spectrum of the structures in 1100 nm - 1500 nm wavelength range with different dimension asymmetry parameter α1 when α2 is fixed (ΔL = 50 nm). The mode 1 exhibits the same variation tendency shown in Fig. 2(b) while the reflectivity and bandwidth of mode 2 become larger with a red shift occurrence. When Δx approaches to 0, α1 becomes smaller, and the electric field enhancement of both mode 1 and mode 2 becomes larger, as depicted in Fig. 3(d). Moreover, in comparison to the resonator featuring solely a gap perturbation, the introduction of a length perturbation under identical conditions leads to a more substantial augmentation of the electric field. Figure 3(e) shows the correlation curve of the Q-factor of mode 2 and α2, it is obvious that Q-factor decreases as α increases and satisfies $\textrm{Q} \propto {\; }{\mathrm{\alpha }^{ - 2}}$. Further, we decomposed the resonance mode 2 as shown in Fig. 3(f), the main resonance mode is ED resonance. When two perturbations are introduced, there is no influence on each other's major electromagnetic modes.
To get three quasi-BIC resonance peaks, we demonstrate another kind of symmetry-protected quasi-BIC pattern motivated by rotation perturbation (shown in Fig. 1(a) and 1(b)). Figure 4(a) shows the reflection spectrum of the structures with rotation asymmetry parameter α3 when there is no other perturbation. The new asymmetry parameter α3 is defined as:
As shown in Fig. 4(a), there are two resonance peaks (mode 3 and mode 4) in reflection spectrum in 1100 nm - 1500 nm wavelength range motivated only by rotation perturbation. As α3 increases (θ increase from 1° to 8°), the reflectivity and the bandwidth of both mode 3 and mode 4 become larger with a blue shift occurrence. Through an observation of the intensified near-field phenomena, we find that the near electric field of mode 3 is predominantly localized on the upper and lower exterior surfaces of the resonator which leads to better sensitivity, while for mode 4, the electric field finds confinement within the resonator which can hardly be utilized for surface sensing applications. Upon an increase in α3 (with θ ranging from 1° to 8°), the electric field enhancement of mode 3 and mode 4 both become smaller as shown in Fig. 4(b) corresponding to the decrease of Q-factor shown in Fig. 4(c). Further, we decomposed the resonance mode 3 and mode 4 as shown in Fig. 4(d), the main resonance of mode 3 is MD resonance while the main resonance of mode 4 is MQ resonance.
Fig. 4. Rotation perturbation. (a) structural reflection spectrum diagram in different angles. (b) the near electric field enhancement of mode 3 and mode 4 under different θ. (c) Q-factor as a function of α3 for modes III and IV. (d) multipole decomposition in the Cartesian coordinate system.
Next, we introduce rotation perturbation and gap perturbation at the same time. As shown in Fig. 5(a), there are three resonance peaks in reflection spectrum, and respectively are mode 1 motivated by α1 (the period asymmetry parameter) and mode 3, mode 4 motivated by α3 (the rotation asymmetry parameter). As the increasing of α3 (θ increase from 1° to 8°), the reflectivity and the bandwidth of both mode 3 and mode 4 become larger with a blue shift occurrence which has the same variation tendency without introducing gap perturbation shown in Fig. 4(a) while mode 1 is barely affected. The complete spectral shifts from θ ranging from 0 to 12° are presented in Fig. S1 and the true BIC resonance occurs at θ = 0°. Figure 5(b) and 5(c) are respectively the correlation curve of the Q- factor and α3−2 of mode 3 and mode 4. When the distance between two adjacent structures in x direction approach the initial value (300 nm), the slope of the Q- α3−2 curve of mode 3 remains small while the slope of the Q- α3−2 curve of mode 4 is much larger and more sensitive to the change of α1, in both cases, the highest Q-factors of mode 3 and mode 4 are obtained at the condition of no gap perturbation where α1 = 0. When the gap perturbation of α1 is changed under θ = 2°, the trend of near-field enhancement of mode 1 is consistent with that when there is only gap perturbation, the field enhancement and Q-factor increase with smaller α1 (or Δx). As Δx increases, the electric field enhancement in both mode 3 and mode 4 becomes larger, as shown in Fig. 5(d). We observe that the introduction of rotation perturbation is beneficial to the electric field enhancement of mode 1 compared to only gap perturbation, and large gap perturbation is also beneficial to the electric field enhancement of mode 3 and mode 4. Again, when two perturbations are introduced, there is no influence on each other's major electromagnetic modes.
Fig. 5. Introduce gap perturbation and rotation perturbation simultaneously. (a) reflection spectrum diagram in different angles when Δx = 75 nm. (b) Q-factor as a function of α3−2 for mode 3 in different values of Δx. (c) Q-factor as a function of α3−2 for mode 4 in different values of Δx. (d) the near electric field enhancement of mode 1, mode 3 and mode 4 under different Δx when θ = 2°.
Furthermore, we simulate the sensitivity in refractive index sensing by covering a layer of material with a changed refractive index above the original structure. Figure 6(a)-(c) shows the movement of resonance peak in reflection spectrum with coating of different refractive indices when there is only a gap perturbation as shown in Fig. 2(a). when Δx is 25 nm, the resonance peak has a 55 nm movement when the refractive index of coating changed from 1 to 1.1 which shows a sensitivity of 550 nm per RIU (shown in Fig. 6(a)). When Δx is 50 nm and 75 nm as shown in Fig. 6(b) and 6(c), the sensitivity respectively become 540 nm per RIU and 520 nm per RIU since α1 become larger and the Q-factor become smaller. Next, we introduced the geometric dimension perturbation as shown in Fig. 3(a), the movement of resonance peak of mode 1 and mode 2 in reflection spectrum with coating of different refractive indices are shown in Fig. 6(d)-(f). At this time ΔL is fixed at 50 nm, and when Δx is 25 nm, 50 nm and 75 nm, the resonance peak of mode 1 respectively has a 53 nm, 54 nm and 56 nm movement when the refractive index of coating changed from 1 to 1.1 which shows a sensitivity of 530 nm, 540 nm and 560 nm per RIU, showing a relatively stable and good performance, while the resonance peak of mode 2 shows a smaller sensitivity of 410 nm, 430 nm and 510 nm per RIU respectively.
Fig. 6. Reflection spectrum of structure with different refractive index coatings. (a) under gap perturbation, when Δx = 25 nm, (b) Δx = 50 nm, (c) Δx = 75 nm, (d) under gap perturbation and length perturbation, Δx = 25 nm, ΔL = 50 nm (e) Δx = 50 nm, ΔL = 50 nm (f) Δx = 75 nm, ΔL = 50 nm, the reflection spectrum of structure with different refractive index coatings in the wavelength range of 1100 nm −1500 nm.
We also demonstrate the sensitivity in refractive index sensing with the gap perturbation and rotation perturbation at the same time. Figure 7(a) shows the movement of resonance peak in reflection spectrum with coating of different refractive indices when there is only a rotation perturbation as shown in Fig. 4(a). Since there is only rotation perturbation, only mode 3 and mode 4 exist. θ is fixed at 2°, and the resonance peak of mode 3 and mode 4 respectively has a 70 nm and 11 nm movement when the refractive index of coating changed from 1 to 1.1 which shows a sensitivity of 700 nm, and 110 nm per RIU. The electric field enhancement of mode 3 is mainly concentrated on the upper and lower outer surfaces of the resonator, while the electric field of mode 4 is confined to the resonator, so mode 3 shows better sensitivity. After adding a gap perturbation, the sensitivity of mode 3 and mode 4 decrease slightly as shown in Fig. 7(b) to 7(d). The sensitivity of mode 3 and mode 4 is 690 nm and 100 nm per RIU when Δx is 25 nm. 680 nm and 90 nm per RIU when Δx is 50 nm, and 680 nm and 80 nm per RIU when Δx is 75 nm. The sensitivity of mode 1 also decreases slightly compared with the structures only have a gap perturbation (shown in Fig. 6(a)-(c)), when Δx is 25 nm, 50 nm and 75 nm, the sensitivity of mode 1 is respectively 510 nm, 500 nm and 490 nm per RIU. Though the gap perturbation and rotation perturbation can interfere with each other, the sensitivity of mode 3 and 4 does not change much. Of course, different structural perturbations and broken symmetry can result in different resonance mode and optical sensitivity. For example, if we introduce a slotted silicon column structure as shown in Fig. S5, the sensitivity of mode 5 can be increased by 20% from 310 nm per RIU to 370 nm per RIU under simultaneous gap perturbation. Furthermore, the comparison of four peaks in resonance mode, Q-factor and refractive index sensitivity is shown in Table 1. According to the table, we find that the refractive index sensitivity corresponding to MD resonance is higher than that of ED and MQ resonances.
Fig. 7. (a) Reflection spectrum of structure with different refractive index coatings under rotation perturbation (θ = 2°). (b) Under gap perturbation and rotation perturbation, Δx = 25 nm, θ = 2° (c) Δx = 50 nm, θ = 2° (d) Δx = 75 nm, θ = 2°, the reflection spectrum of structure with different refractive index coatings in the wavelength range of 1100 nm −1500 nm.
Table 1. Comparison of four resonance modes
Considering the performance of high-Q structures usually drops dramatically when considering increased loss, we simulate a 30% higher imaginary part for the permittivity of silicon corresponding to sub-optimal material conditions. The results show that when the imaginary part of the permittivity increases, the Q-factor decreases 20% on average, as shown in Fig. S2. In addition, all our results are based on ideal infinite arrayed structures, however, in practical applications, only finite array is made possible and the array size can impact the BIC resonance. Here we tested a small array size of 5 × 5 and a larger array size of 11 × 11 in Fig. S3, it is proved that array size plays an important role of the BIC resonance performance and only sufficient large array of periodic unit cells can excite BIC resonances effectively. The feasibility of fabrication is considered and discussed in the Supplement 1 Fig. S4. Although many previous BIC studies have demonstrated that a strong correspondence between simulated and experimental results, only that unavoidable fabrication errors could lead to much lower experimental Q-factor compared to ultrahigh Q-factors in ideal simulations [32,33].
4. Conclusion
Although BICs have received substantial attention in recent years, the concurrent introduction of dual perturbations into the unite cell has remained less explored. In this study, we use rectangular silicon nanobar as unit cells, introducing gap perturbation alongside length perturbation or angular perturbation. Our investigation encompasses an analysis of variations in the Q-factor, electric field enhancement, and resonant modes, coupled with a comparison of refractive index sensitivity for different BIC resonance modes. Upon introducing dual perturbations, there is an intensified electric field enhancement, and we can regulate the number and position of resonance peaks. Moreover, the combination of length perturbation and gap perturbation results in elevated structural sensitivity in refractive index sensing. Despite the interference between different kinds of asymmetric perturbation may exist, they can still have high Q-factor and show high performance at refractive index sensing at the same time. This study provides innovative design strategies for future optical devices, encompassing multi-band responsive ultra-sensitive sensors.
Funding
National Natural Science Foundation of China (61974069, 62022043, 62235008, 62288102, 62375139); National Key Research and Development Program of China (2021YFA1202904); Natural Science Foundation of Jiangsu Province Major Project (BK20212012); Project of State Key Laboratory of Organic Electronics and Information Displays (GDX2022010007).
Acknowledgment
L. G. conceived the project and co-wrote the article. L. W. performed the FDTD simulations, analyzed the data, and co-wrote the article, Q. D., T. Z. H. Z and L. W. helped with the data analysis and discussion.
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.
Supplemental document
See Supplement 1 for supporting content.
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