Ultrahigh Q-Guided Resonance Sensor Empowered by Near Merging Bound States in the Continuum

Abstract

Bound states in the continuum (BICs) have attracted a lot of interest in the field of nanophotonics, and provide an important physical mechanism to realize high quality (Q) factor resonance. However, in practice, manufacturing error will greatly affect the Q factor. In this paper, we propose an asymmetric metasurface supporting near merging BIC under normal incidence. Such near merging BIC can achieve a higher Q factor (>10⁷) than common structures (Q ~ 10⁵) with the same degree of asymmetry in the structure. Moreover, the near merging BICs also show higher surface sensitivity than other resonant modes. Our work provides a promising approach for the realization of a high-performance biosensing platform.

1. Introduction

A high quality (Q) factor device can produce strong light–matter interaction, which is important for a variety of applications [1], such as nonlinear phenomena [2,3,4], sensors [5,6,7,8], and lasers [9,10,11]. Many structures have been proposed to achieve this goal, such as whispering-gallery mode (WGM), photonic crystal cavities, and so on.

Recently, bound states in the continuum (BICs) have become the most promising scheme to realize high-Q resonators [12,13,14]. Compared with the general bound state (below the light-line), although BIC exists in the radiation spectrum (above the light-line), it still maintains perfect localization, which means, in theory, it has infinite lifetime and infinite Q factor [9].

Generally, there are two types of BIC: symmetric-protected BICs, and accidental BICs [15]. The symmetric-protected BICs cannot couple the bound mode to the radiation channel, as the symmetry of the bound mode is opposite to that of the continuum [16,17,18]. Accidental BICs refer to completely suppressing all radiation channels by adjusting multiple system parameters [19,20]. In reality, an ideal BIC will transform to a quasi-BIC due to the fabrication imperfection and material loss, thus producing finite but high Q resonance. Pioneering works show that quasi-BICs can be supported by periodic nanostructures, such as resonant grating [21,22], photonics crystal slabs [9,12], and metasurfaces [11,23]. In these periodic nanostructures, a symmetric-protected quasi-BIC is more often manipulated than an accidental quasi-BIC, because the symmetric-protected one can be easily realized through oblique incidence or breaking structural symmetry [24,25,26,27,28]. Furthermore, the Q factor of symmetric-protected quasi-BIC can be controlled by the degree of asymmetry or oblique angle [29], which are efficient methods to manipulate high Q resonance.

The practical Q factor (Q_exp) can be described as 1/Q_exp = 1/Q_design + 1/Q_imperfect, where Q_design is the Q factor of theoretical design and Q_imperfect is owing to the loss caused by manufacturing error [30]. Moreover, for a high Q resonator, Q_exp is determined by Q_imperfect rather than Q_design [30]. Hence, a robust structure against fabrication imperfections is necessary for engineering high Q resonator. Recently, Jin et al. proposed a merging BIC with 10 times higher Q factor than common BIC devices. Such a merging BIC is less susceptible to the fabrication losses. However, their work is realized in momentum space and requires precise angle adjustment [31].

In this paper, we propose an asymmetric metasurface composed of nanocube with an asymmetric square hole, supporting triple resonant modes under normal incidence. We systematically investigate the Q factor evolution versus the degree of asymmetry for the structure for different modes. Further, near merging BICs is found by manipulating the structure period. Such a merging BIC can maintain higher Q resonance than that for the structure not in the merging BIC region. Besides, we found such merging BIC shows relative higher surface sensitivity than other modes, which produces a higher figure of merit for sensing performance. Our findings help to understand the accidental BIC and provide valuable information for the design of BIC sensors.

2. Design and Simulation

Figure 1a shows the schematic of asymmetric all-dielectric metasurface structure. It is composed of a square grating with a square hole deviated from the center placed on the cube lattice. The grating layer and waveguide layer are made of silicon nitride, and the substrate is made of quartz. The refractive indices of silicon nitride and quartz are 2 and 1.48, respectively. The thickness of the waveguide layer is h_w = 200 nm and the thickness of grating layer is h_g = 100 nm. The detailed geometric parameters of a single grating are shown in Figure 1b, the top view of a single grating lattice. As shown in the figure, the side lengths of the two squares constituting the grating are w_o = 360 nm and w_i = 120 nm, respectively. The refractive index of the hole in the grating is the same as that of the cover. By moving the inner square outward from the center of the structure by distance d, we create a symmetry break in the x-axis direction. If the asymmetric coefficient is α = d/w_o, then α ∈ (0, 0.667) when the inner square does not move out of the range of the grating structure. The period of the whole structure is p = 600 nm.

Rigorous coupled-wave analysis (RCWA) was used in the simulations to calculate the Q factor from the spectral responses and finite element method (FEM) was used to calculate the electric field distributions of the structures. Q was calculated by resonant wavelength divided by FWHM [32]. The ranges of peaks were partitioned by difference. If the difference was larger than the standard deviation of it, it was supposed to be within the range. In the RCWA method, periodic boundary conditions are applied in the x and y directions, while PMLs are applied in the z direction. The grid sizes and slice grids in the x, y, and z directions are all 1 nm, and the domain range in the z direction is 1300 nm, with 500 nm under and 500 nm above the structure. Alternatively, in the FEM method, periodic boundary conditions are applied in the x and y directions, while periodic ports are applied in the z direction. The mesh is physics-controlled and of normal size. Light strikes the surface at a normal incidence, polarized in the direction of the x-axis, which is the direction of the hole away from the center of the square.

3. Results and Discussion

The theoretical simulation results of the proposed structure are presented here. It should be noted that a resonance with infinite Q factor cannot be displayed. Therefore, in order to avoid this, it is usually only possible to gradually approach the state of BIC by adjusting the parameters of the system [32]. Here we achieve this by adjusting the asymmetry parameter α. Figure 2 shows how the reflection spectrum changes as the asymmetry parameter α changes. There are three TM resonances that are excited by p light in Figure 2a—TM-A, TM-B, and TM-C in order from long wave to short wave. Due to the high Q factor of the TM-C resonance, it is difficult to distinguish in Figure 2a, so Figure 2b,c and Figure 3 show the partial enlarged image and the spectral lines near the BIC point in area B and area C, respectively.

The TM-A resonance with a wavelength near 966 nm is a common guided-mode resonance (GMR) reflection peak, which is formed by the constructive interference of the first-order diffraction of the grating and the reflected light, and can the resonance intensity and linewidth can be adjusted through geometrical design and selection of materials [33]. The TM-B resonance and the TM-C resonance both have BICs under different structure asymmetry parameters; here, we focus on these two resonances. When the asymmetry parameter α approaches 0, the reflection peaks of the two resonances, TM-B and TM-C, are obviously narrowed. When the asymmetry coefficient is close to 0.52, only the reflection peak of the TM-C resonance is narrowed. These areas are denoted by A, B, and C, respectively.

Figure 3a shows the spectral lines near region B, where the abscissa and ordinate are the same for each figure. As α gradually approaches 0, the peak of this resonance disappears, meaning that the resonance can no longer couple to the free space. The spectral lines near region C are shown in Figure 3b. Since the peak width is too narrow relative to the peak displacement, there are four breaks in the figure and the length of each segment is 0.03 nm. It can be seen from the figure that the resonance of the mode disappears when α is close to 0.521, indicating the existence of the BIC.

By observing the Q factor of each reflection peak in Figure 4, the Q factor in the regions A, B, and C all approach infinity; that is, approaching the BIC. The BICs in area A and area B are symmetrical-protective BICs (SP BIC), while those in area C are accidental BICs. SP BICs can be described as follows, with the electric field of a resonance written as:

(1)${\mathit{E}}_{\mathit{k}}\left(\mathit{\rho },z\right)={\mathrm{e}}^{\mathit{i}\mathit{k}\mathit{\rho }}{\mathit{u}}_{\mathit{k}}\left(\mathit{\rho },z\right)$

using Block theorem, where $\mathit{k}=k_x \widehat{x}+k_y \widehat{y}$ is the wave vector, $\mathit{\rho }=x\widehat{x}+y\widehat{y}$ is the coordinate, and ${\mathit{u}}_{\mathit{k}}$ is a period function in $\mathbf{\rho }$. In order to form an SP BIC, $\mathit{c}\left(\mathit{k}\right)$, the zeroth Fourier coefficient of ${\mathit{u}}_{\mathit{k}}$ should be zero, which is given by:

(2)$c_x\left(\mathit{k}\right)=\widehat{x}·⟨u_k⟩,c_y\left(\mathit{k}\right)=\widehat{y}·⟨u_k⟩.$

If the unit cell of metasurface is symmetric with respect to the z→−z transformation and the eigenstates are odd functions, (2) equals to zero and forms an SP BIC [31]. The accidental BIC with broken symmetry was most often formed due to destructive interface between two resonances, because the wavelengths of the two resonances tended to be the same in the process of parameter change [34]. At that value, the two resonances radiated energy outward through the same radiation channel and have destructive interference, and thus forming a Friedrich–Wintgen BIC. Such a BIC can be described by Hamiltonian governed by coupled-mode theory:

(3)$H\left(\alpha \right)=\left[\begin{array}{cc}{\omega }_1& \kappa \\ \kappa & {\omega }_2\end{array}\right]-i\left[\begin{array}{cc}{\gamma }_1& \sqrt{{\gamma }_1{\gamma }_2}{e}^{i\phi }\\ \sqrt{{\gamma }_1{\gamma }_2}{e}^{i\varphi }& {\gamma }_2\end{array}\right]$

where the ${\omega }_i$ and ${\gamma }_i$ are the resonance frequency and damping rate, respectively, $\kappa$ is the coupling strength, and $\phi$ is the phase shift between two modes. When the following condition is satisfied

(4)$\left({\omega }_1-{\omega }_2\right)\kappa =\sqrt{{\gamma }_1{\gamma }_2}\left({\gamma }_1-{\gamma }_2\right),$

such coupling represents one of the eigenmodes has become purely real without an imaginary part, indicating a mode without any loss [34].

In this article, an accidental BIC formed by a single resonance is shown. While adjusting the system parameters, such as period p, when the two modes share one radiation channel, the wavelengths of the resonance peaks of the two modes change similarly, and the wavelengths of the two resonances coupled to the radiation channel also change similarly. In that case, the BIC position in the asymmetry parameter space where destructive interference happens does not move observably while the system parameters p change, which means that the position of the BIC point in the asymmetry parameter space cannot be adjusted by changing the system parameters. However, if it is a BIC formed by a single resonance, the position of the BIC point on the asymmetric coefficient can be adjusted by changing the system parameters, as shown in Figure 5.

When the period p is around 600 nm, the accidental BIC in region C is shown in Figure 5a, and the asymmetry parameter α at which the highest Q factor or the BIC exist is around 0.5. By adjusting the period p to the vicinity of 500 nm, the peak position of the Q factor of the resonance in region C is moved to the vicinity of region B, and the Q factor distribution at such situation is shown in Figure 5b. In the process of changing p from 510 nm to 490 nm, the annihilation of the BIC is demonstrated [31,35]. When the period is away from 500 nm, such as 490 nm, 495 nm, and 510 nm, the Q factor decreases from 10⁸ to around 10⁶. However, when BICs are near merging, 500 nm or 502 nm for example, the Q factor remains greater than 10⁷. By adjusting the system parameters to move the accident BIC and make those two BICs close to the merged state, the sensor can maintain a high Q factor within a relatively large range of the asymmetry parameter, such as from 0 to 0.1. The minimum Q factor is increased from 10⁵ to 10⁷ within the range between the two BICs, which means that the sensor has a strong resistance to manufacturing errors in the structure.

When the asymmetry parameter is zero, the normalized electric field distribution of the GMR and the BIC at region A and B is shown in Figure 6a–c. In contrast, the asymmetry parameter in (d) is 0.52, and it shows the BIC in region C. Among them, Figure 6a is the ordinary GMR mode, and Figure 6b,c show the symmetric-protected BIC mode under the condition of α = 0. Figure 6d presents an accidental BIC with the same resonance as (c), both of which are TM-C. The field enhancement of the BIC mode is significantly higher than that of the GMR mode, reaching the order of 10⁷, which enhances the interaction between the light and the environment. In Figure 6a, the part where the electric field coincides with the environment is mainly distributed on the surface of the grating and that of the waveguide layer on the side of the grating; in (b), it is mainly distributed at the hole in the center of the grating. Since (c) and (d) are the same resonance, the distribution of the electric field is similar, and they are mainly distributed on the surface of the waveguide layer. Figure 7 shows the Ex components of the electric field in the x–z plane for the four cases respectively. Since the resonance are all TM polarization, the Ey component in the x–z plane is zero. Among them, Figure 7a presents the GMR mode, and its Ex component is symmetrical about the plane passing through the z-axis and can radiate outward. Figure 7b,c show an antisymmetric state with respect to the plane passing through the z-axis, and cannot radiate energy outward due to destructive interference, which is a symmetrical-protective BIC mode. Comparing Figure 7c,d, it can be found that the overall electric field distribution of the TM-C mode in Figure 2 looks similar under the condition of different asymmetry coefficients α. However, from the perspective of Ex, due to the absence of the structure symmetry, the electric field in (d) is in an asymmetric state, but its Q factor still has the characteristics of BIC, indicating it is an accidental BIC [12].

Next, the differences in surface sensitivity of different resonances between the dielectric and metal waveguide layer and grating layer are compared. Figure 8 displays the surface sensitivity calculation, containing the structure covered by a molecular layer. Assuming that the molecular to be detected is immunoglobulin G (IgG), then the thickness and refractive index of the layer are 10 nm and 1.33 to 1.5, from reference to a previous paper [36,37] and experiment. Figure 8a displays the structure composed of silicon nitride while Figure 8b displays that of Ag. Figure 9 concerns the dielectric waveguide and grating layer. Figure 9a shows the shifts of the peak positions of the three modes of TM-A, TM-B, and TM-C when the bulk refractive index is 1.33 and the surface refractive index changed from 1.33 to 1.5, with the Q factor of the resonance at 1.33 marked in the figure. The surface sensitivities are 0.279 nm/nm, 0.273 nm/nm, and 0.265 nm/nm, defined as wavelength shift divided by the thickness of the molecular layer. Alternatively, it can be defined as wavelength shift divided by the change of refractive index; then, the surface sensitivities are 16.4 nm/RIU, 16.1 nm/RIU, and 15.6 nm/RIU, respectively. Such surface sensitivities are of the same order of magnitude as other silicon-based sensors, 20 nm/RIU [37] for example. As the Q factor of the reflection peak decreases, the amount of wavelength shift represents a gradual increase in sensitivity.

The wavelength shift of the silicon nitride sensor when the asymmetry parameter α is changed is shown in Figure 9b. The relationship between the sensitivity of the three modes remains stable, that is, the sensitivity of TM-C is greater than that of TM-B, and the sensitivity of TM-B is greater than that of TM-A. Moreover, with the gradual increase of the asymmetry parameter, the gap is gradually increasing, indicating that in this sensor, the relationship between the sensitivity and size is concluded to be stable.

In Figure 10, when both the waveguide and the grating layer are silver, the sensitivities gradually decrease with the decrease in the reflection peak Q factor, and are 0.229 nm/nm and 1.1 nm/nm, or 13.5 nm/RIU and 64.7 nm/RIU. While the sensitivity of the mode near 860 nm is larger, it is not a BIC mode so the Q factor is obviously smaller. Generally, the sensitivity of plasmonic array-based structures is larger than that in dielectric structures. However, in this structure, sensitivities of the BIC mode in the dielectric are larger than that in metal. Moreover, in a sensor composed of a dielectric, the higher the Q factor of resonances with close wavelengths, the higher the sensitivity. In the case where both the waveguide layer and the grating layer are composed of metal, the higher the Q factor of resonances with close wavelengths, the lower the sensitivity.

4. Conclusions

In summary, an all-dielectric metasurface is designed, simulated, and optimized in this paper. By adjusting the system parameters such as period, the accidental BIC generated by asymmetry can approach the symmetrical-protective BIC when the structure is symmetric in the space of asymmetry parameter; therefore, no precise angle adjustment is required. When the system parameters are appropriate, the two BICs can be brought close to the merged state, thereby suppressing out-of-plane scattering caused by asymmetry, and improving the anti-interference ability of the structure against manufacturing errors. In the example of an asymmetric coefficient of 0.1, the Q factor of the ordinary structure, which is about 10⁵, is improved to 10⁷. However, it is not easy to find a mode with two BICs, because there is no current method to find the structure that such modes exist in without mass calculations. The electric field distributions of these resonances are then calculated and analyzed. At the end of this paper, the trends in the variation of sensitivity and Q factor between different resonances of this structure are compared in the case of dielectric and metal. These design methods provide valuable information for the design of BIC sensors.

Author Contributions

Z.L. proposed the idea and conceptualization, and conducted the simulations; Y.Z. and X.W. performed scientific discussions and supervised the work; Z.G., X.Z., M.L. and Y.L. helped with revision and organization of the paper. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data available in a publicly accessible repository.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures

Figure 1. (a) The schematic of asymmetric metasurface structure and (b) the top view of gratings.

Figure 2. (a) The reflection spectrum variation with the asymmetry parameter α and (b,c) partial enlarged view when the period is 600 nm. The color represents reflection, from 0 to 1. Regions A, B, and C include a BIC, respectively.

Figure 3. The reflection spectrum variation with the asymmetry parameter α near (a) region B and (b) region C.

Figure 4. The variation in Q factor of three modes with the asymmetry parameter α when period is 600 nm. TM B has one BIC in region A, and TM C has two BICs in region B and region C.

Figure 5. The variation in Q factor of TM C modes with the asymmetry parameter α under different periods. (a) When the period is near 600 nm, the peak wavelength is around 960 nm, and BIC is around α = 0.5. (b) When the period is near 500 nm, the peak wavelength is around 820 nm, and BIC is around α = 0.07.

Figure 6. (a–c) Normalized electric field distribution of the GMR and the BIC in region A and B when the asymmetry parameter is 0. (d) Normalized electric field distribution in region C when the asymmetry parameter is 0.52, and the square of H2O moves to the side of gratings thereby. The color represents the intensity of electric field.

Figure 7. (a–c) Ex components distribution of the GMR and the BIC in region A and B when the asymmetry parameter is zero. (d) Ex components distribution in region C when the asymmetry parameter is 0.52. The color represents the intensity of Ex.

Figure 8. The demonstration of surface sensitivity calculation. Reflectivity of the molecular layer is changed in calculation. (a) The grating and waveguide layers are made of SiN. (b) The grating and waveguide layers are made of Ag.

Figure 9. (a) The shifts in the peak positions and the Q factor when the surface refractive index changed from 1.33 to 1.5. (b) The wavelength shifts of the silicon nitride sensor variation with the asymmetry parameter α.

Figure 10. The shifts in the peak positions and the Q factor when both the waveguide and the grating layer are metal and the surface refractive index is changed from 1.33 to 1.5.

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