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Abstract
We propose a cascaded asymmetric resonant compound grating (ARCG) for high-performance dual-band refractive index sensing. The physical mechanism of the sensor is investigated using a combination of temporal coupled-mode theory (TCMT) and ARCG eigenfrequency information, which is verified by rigorous coupled-wave analysis (RCWA). The reflection spectra can be tailored by changing the key structural parameters. And by altering the grating strip spacing, a dual-band quasi-bound state in the continuum can be achieved. The simulation results show that the highest sensitivity of the dual-band sensor is 480.1 nm/RIU, and its figure of merit is 4.01 × 105. The proposed ARCG has potential application prospects for high-performance integrated sensors.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical sensors have been widely used in the chemical analysis [1], biological detection [2,3], and environmental detection [4,5] due to their high sensitivity and label-free sensing. Several physical mechanisms have been adopted to enhance the sensing performance, including Fano resonance [6,7], surface plasma resonance [8,9], Fabry–Perot resonance [10,11], and guide-mode resonance (GMR) [12–14]. Among them, the GMR-based sensors have attracted attention owing to their real-time performance (without fluorescent tags) and narrow linewidth. GMR is a valuable mechanism in resonant waveguide gratings for high-performance optical sensing. A sharp resonance peak can be obtained by controlling the grating modulation of the GMR-based structure [15]. In 2006, Magnusson et al. explained GMR biosensor modeling and its mechanism to demonstrate its viability [16]. A high-figure-of-merit (FOM) refractive index sensor based on GMR was proposed by Guilian Lan et al. in 2018 [17]. Sensitivity (S) is crucial in GMR grating-based sensors. Physically, it is highly desirable to improve the optical interactions between the leaky mode and the analyte in GMR grating-based sensors.
In recent years, several methods have been reported to improve the sensitivity of GMR sensors. Wang et al. regulated the electric field distribution of the grating by changing the grating cavity length to achieve a sensitivity of 748 nm/RIU [18], 5.2 times that of the basic structure [19]. Zhou et al. used a metal-layer-assisted system to increase the sensitivity of their GMR sensor to 550.0 nm/RIU [20]. However, these studies improved the S of the sensors with a limited Q-factor. Enhancing the Q-factors effectively and trapping the light field in the sensing region is desirable. As the FOM of the sensor was significantly enhanced, its ability to measure slight resonance wavelength shifts with high accuracy was improved, and the detection limit was reduced [21]. Our previous study proposed a refractive index sensor based on double-layer resonant meta-grating; its sensitivity reached 930 nm/RIU [22]. However, optical trapping is limited in the grating layer, and the fabrication process is complex. To further improve the Q-factor, bound states in the continuum (BICs) have also been widely used in optical devices [23,24].
In general, BICs are separated into two categories: symmetry-protected BICs [25,26] and accidental BICs [27]. The former is a discrete mode at the gamma (Γ) point, and the bound states and continuum belong to separate symmetry classes; thus, coupling cannot occur due to symmetry incompatibility. In the latter case, leakage waves with multiple destructive interferences arise at the non-gamma point, altering the parameter system. The Q-factor of the BIC can approach infinity, and the linewidth vanishes in a perfect lossless condition [28]. However, in actuality, the true BIC cannot be used for reflection or transmission index sensing. When both the Q-factor and linewidth are finite, a quasi-BIC is observed [29]. For optical sensors, a high Q-factor resonance improves the capacity of the grating to bind to the light field, concentrating the light field in the high-refractive index area. The single-mode sensing has received much attention [30,31], whereas the dual-band resonance structure has received little attention [32–34]. Dual-band or even multi-band resonance with a high Q-factor can be applied in high-performance optics devices using bound states in the continuum, such as sensors and absorbers [35–37].
In this study, we have proposed a cascaded asymmetric resonant grating (ARCG) to improve sensitivity while obtaining a high Q-factor in the sensor. The device was constructed using the rigorous coupled-wave analysis (RCWA) [38,39], concentrating on the impact of grating structure parameters on sensor performance. The spectrum line shape can be calculated using a physical model combining the temporal coupled-mode theory (TCMT) [40] and the eigenfrequency information of the grating structure. Interestingly, we find that the quasi-bound states in the continuum (quasi-BICs) can be made by changing the distance between the compound gratings. The simulation results show that the S and FOM of the dual-band quasi-BICs are 479.76 nm/RIU and 4.01 × 105, and 480.2 nm/RIU and 4.02 × 105, respectively.
2. Structure and principle
Figure 1 illustrates the structure of the ARCG-based sensor, which is composed of a liquid chamber and two sets of asymmetric structures. The width of the grating is the only difference between the two asymmetrical constructions. The waveguide layer (nsi = 3.47), grating layer (nsi = 3.47), and substrates (nsio2 = 1.45) all have the same thickness. For the sensor, the period of the grating is $\varLambda $, the distance between the centers of each period of the two grating strips is L, the distance between two gratings in the upper layer is g1, and the distance between two gratings in the lower layer is g2. The overall thickness h of the grating layer and waveguide layer is maintained at 200 nm; dg and ds are the grating layer and waveguide layer thicknesses, respectively. The upper and bottom gratings are separated by a distance of d, and their relative widths are W1 and W2. When the refractive index of a liquid chamber varies, the resonance peak of the reflection spectrum shifts [41]. Furthermore, we investigate the transmission spectra of the proposed structure and eigenfrequency analysis using Comsol Multiphysics software.
The S, FOM, and Q-factor are often used to evaluate the performance of a sensor and could be represented as [42,43]:
here, Δλres represents the resonant wavelength shift, Δnc represents the change in the refractive index of the analyte liquid, and Δλ represents the full width at half maximum (FWHM) of the resonance.When the finite element method (FEM) is used to model a single period, Floquet periodic boundary conditions are used for the lateral boundary, and scattering boundary conditions are used for the upper and lower boundaries [44]. The complex eigenfrequency ω = ω0 - iγ is obtained by the FEM [45]. The Q-factor is expressed as:
here, ${\omega _0}$ is the central resonant frequency, and $\gamma $ is the radiative loss of the leaky mode [46]. A high Q-factor necessitates a small FWHM. The FOM can effectively evaluate the S and Q-factor of the sensor.3. Results and discussion
As an example, we assumed the ARCG parameters: $\varLambda $= 750 nm, L = 370 nm, ds = 85 nm, dg = 115 nm, g1 = L−W1, g2 = L−W2, d = 1630 nm, β = 0 °, and the refractive index of the analyte liquid nc = 1.331. W1 was tentatively set as 135 nm, and W2 was varied. Here, the dual-band resonance is easily generated in the ARCG due to the different widths of W1 and W2. The light source was vertically incident (β = 0 °) and TM-polarized. Using the RCWA, we examined the spectra of ARCGs with different W2 values with W1 = 135 nm, as shown in Fig. 2(a). Comparing it with a symmetric resonant compound grating (W1 = W2 = 135 nm), dual-band resonances were observed with an increase in W2. The proposed dual-band ARCG-based sensor can be used in antibody–antigen binding and refractive index sensing. To investigate the influence of the grating center spacing L on sensor performance, we set W2 at 140 nm, with all other parameters remaining the same, as shown in Fig. 2(b). As L changed, we observed that mode 1 and mode 2 exhibited the same change trend. At L = 375 nm (white circle and red circle), the linewidths of resonance mode 1 and resonance mode 2 disappeared. The distribution of the magnetic field is shown in Fig. 2(c), (d), and (e) for L = 335 nm, 375 nm, and 415 nm, respectively. While L approaches 375 nm, it has been discovered that the energy of each mode gradually converges from the dispersed state, which is the result of the evolution of GMR to BIC. Adjusting the parameter L will cause the distance between the upper and lower gratings to shift synchronously while the overall structure will maintain its vertical symmetry. At L = 375 nm causing accidental BIC, straightforward alteration of L can achieve double-band high-Q resonance. Therefore, it will be applicated to decrease the temperature cross-sensitivity in the proposed sensor due to the double-band resonance.
Fig. 2. (a) ARCG reflection responses for TM polarization with different W2, W2 = 135 nm, W2 = 140 nm, W2 = 145 nm; (b) L of RCWA scanning with TM polarization. The magnetic field distribution |Hy| at different L: (c) L = 335 nm, (d) L = 375 nm, (e) L = 415 nm.
The L was maintained at 370 nm, the positions of the upper compound gratings and the lower left grating were unaltered, and the effect of the lower grating spacing g2 on mode 2 is investigated. With W1 = 135 nm and W2 = 140 nm (W1 and W2 were kept similar to facilitate the observation of the specific simulation conditions), the other conditions were unaltered. At this point, the upper grating spacing g1 remained unchanged (g1 = 235 nm), and the lower grating spacing g2 was altered (g2 = 230 nm). The dimensionless parameter α was introduced (α = (g1 - g2) /g2). The change in α was scanned; it was found that when α = 0 (g2 = 235 nm), it corresponds to point A in Fig. 3(a). When g2 changes, mode 1 remains unchanged, and mode 2 vanishes at a specified point. We calculated the reflection spectrum as the asymmetry parameter changed from -0.12 to 0.12, as shown in Fig. 3(b). On the right is the electric field pattern corresponding to different α. As α approached 0 from -0.12 and 0.12, mode 1 retained an extremely narrow linewidth, whereas mode 2 had a significantly reduced linewidth. The green dashed line represents the peak expected for mode 2 at different α values. This is the result of the diminished evanescent diffraction field and the leaky guided-mode coupling of mode 2. When the spacing of the upper or lower gratings is altered, it can be concluded that mode 1 and mode 2 do not interfere with each other. At α = 0, mode 2 vanishes; this implicit leakage mode is labeled as a misleading BIC as the reflection spectrum does not indicate the abrupt resonance signal shift. The maximum magnetic field energy in the grating occurs at α = 0, as shown by the magnetic field distribution on the right. Figure 3(b) demonstrates that a change in α will impact the resonant peak of mode 2, while the resonant peak of mode 1 remains constant. Here, the direct coupling between the upper compound gratings and the lower compound gratings is negligible because the resonant cavity is rather long. When the spacing of the lower layer compound gratings changes, we observe that it is essentially similar to the change of mode 2 in Fig. 2(b), which is only an BIC generated by engineering the translational symmetry of a unit cell [47]. In Fig. 3(c), the Q-factor of mode 1 remains stable, whereas the Q-factor of mode 2 undergoes a dramatic change. For mode 2, the Q-factor approaches infinity when α = 0 and decreases when it is greater than or less than 0. Thus, it is possible for extremely narrow, high-Q resonance modes to develop in the area close to α = 0; these modes are known as quasi-BICs. To further verify the existence of BICs, the eigenfrequencies were calculated using the FEM, as shown in Table 1. We observe that the real part of the eigenfrequency for mode 2 first decreases and then increases with an increase in α, which is the same as the changing trend for mode 2 in Fig. 3(b). When α = 0 (g2 = 235 nm), the imaginary portion of mode 2 completely disappears, indicating that there is no loss at this time, a typical feature of a BIC. By altering the distance g2 between the two gratings, it is possible to change the GMR into a quasi-BIC and obtain a high Q-factor. The physical representation of Q in the resonator is the ratio of the original energy stored to the energy lost in each oscillation cycle per radian [23]. Consequently, whether α is higher or lower than 0 in the ARCG structure, the energy storage of the optical resonator decreases.
Fig. 3. (a) Dimensionless parameter α in RCWA scanning with TM polarization; (b) Spectrograms of different α with TM polarization and point magnetic field distribution |Hy|; (c) Scan of α with respect to Q-factor by FEM with TM polarization.
Table 1. TM eigenfrequency information with parameter α
The performance of the sensor with TE and TM polarization was compared through multi-parameter variation. Equation (1) and Eq. (2) calculate the S, FOM, and Q-factor of the ARCG structure. We keep W1 = 135 nm, W2 = 140 nm, and other parameters unchanged to investigate the influence of parameters dg and ds on sensor performance. Figure 4(a) shows the S and Q-factors versus ds for TE polarization. As ds increases, S decreases and the Q-factor increases. Figure 4(b) shows the same trend for S and FOM versus ds for TE polarization as in Fig. 4(a). Figure 4(c) shows the S and Q-factors versus ds for TM polarization. With an increase in dg, S first increases and then decreases, and the Q-factor remains stable after decreasing to a certain value. Figure 4(d) shows the same trend for S and FOM versus ds for TE polarization as in Fig. 4(c). With TE or TM polarization, the Q-factor and FOM of the sensor are much larger than the performance parameters of conventional sensors. Most importantly, TM polarization still has high Q-factors while improving the S, solving the mutual restriction between S and the Q-factor. Figure 4(a) and (c) show that the sensitivity of TM polarization with the same dg is roughly two to four times that of TE polarization. To further evaluate the sensor performance with TE and TM polarization, we used the finite element method to predict the intensity distribution of the electric field and magnetic field of the ARCG sensor at ds = 80 nm, as shown in Fig. 4(e) and (f). It was found that most of the electric field in the TE polarization was confined to the waveguide layer, and there was little electric field at the leak site, resulting in slight contact with the analyte liquid solution. In contrast, TM polarization was different as most of the energy leaked out of the grating and waveguide layers, significantly strengthening contact with the analyte liquid and increasing S. Additionally, we considered the high-contrast grating (HCG) when the ds disappears (ds = 0 nm). S is higher with TE polarization, although the Q-factor and FOM are lower. In contrast, S is low and the Q-factor and FOM are high for TM polarization.
Fig. 4. (a, b) Calculated S, Q-factor, and FOM with ds for TE polarization; (c, d) Calculated S, Q-factor, and FOM with ds for TM polarization; (e) The electric field distribution |Ey| at ds = 80 nm with TE polarization; (f) The magnetic field distribution |Hy| at ds = 80 nm with TM polarization.
We combination TCMT with the FEM, resulting in a novel theoretical model that lets us build an ARCG sensor more intuitively. The reflection spectrum of ARCG can be calculated using [48]:
We set ds = 80 nm and dg = 120 nm, with all other parameters remaining unchanged. L was maintained at 370 nm while modifying the dimensionless parameter α; Eq. (4) was used to calculate the TM polarization reflection spectrum, as shown in Fig. 5. A comparison between the reflectance spectra of the theoretical model and RCWA simulations shows that the two spectral shapes are extremely compatible. The calculation results show the feasibility of using eigenfrequency analysis to predict the shape change of the spectrum. This displays the modulation impact of the grating structure on the light field in more detail. The influence of incident angle offset on the sensor was studied when TM was polarized and other parameters were unchanged. As shown in Fig. 6, both peaks with β from 0° to 2° have a redshift, and the drift of the resonance peaks increases with the angle, with no significant change in linewidth. The resonance peak does not disappear with changes in the incident angle; thus, the structure has a certain fault tolerance for the deviation of the incident angle.
Fig. 5. Reflection spectra of ARCG with different α: (a) α = 0.12; (b) α = 0.08; (c) α = 0.04; (d) α = 0.01.
The distance between the centers of the upper and lower grating strips L varies, as shown in Fig. 7(a). With nc = 1.331 and all other parameters held constant, Fig. 7(b) shows the relationship between the ARCG sensor reflection spectrum and the offset parameter K. It is clear that a change in K has no impact on the sensor reflection spectrum. As FP-like resonance is the major coupling mechanism of the structure, the influence of the misalignment of the center spacing of the two gratings is negligible. Not requiring alignment makes building the structure much more feasible.
Fig. 7. (a) Transverse migration of upper and lower grating; (b) Relationship between spectra of ARCG sensor and parameter K.
Investigation of ARCG structural parameters suggests the conditions of $\varLambda $= 750 nm, dg = 120 nm, ds = 80 nm, W1 = 135 nm, W2 = 140 nm, d = 1630 nm, β = 0 °, and L = 370 nm (g1 = 235 nm, and g2 = 230 nm), with TM polarization. The eigenfrequencies were computed using the finite element method as the analyte refractive index changed from 1.331 to 1.339, as shown in Table 2. By analyzing the eigenfrequencies, it can be shown that when nc increases, both peaks are redshifted, yet the line width does not vary much. Figure 8(a) shows the RCWA calculated reflection diagram of the nc change, indicating that the expected trend of eigenfrequencies is the same as for the change in nc. The refractive index of the analyte liquid influences the resonance wavelength of the GMR grating structure. Thus, a variation in the refractive index of the analysis liquid results in a shift in the wavelength of the resonant peak, enabling refractive index sensing. Figure 8(b) illustrates additional study of the fitted curve of the resonance peak location as the refractive index of the analyte liquid varies, indicating a linear connection between the resonance wavelength and the refractive index. From the analysis of these results, the performance parameters of the ARCG sensor were compared with those of GMR sensors in the references, as shown in Table 3. The feasibility of using an asymmetric structure to improve sensor performance was observed.
Fig. 8. (a) ARCG reflection spectra with different nc for TM polarization; (b) Peak wavelength of ARCG resonance has a linear relationship with nc.
Table 2. TM eigenfrequency corresponding to different refractive indices
Table 3. Summary of sensitivity (S) and FOM values of sensor
4. Conclusion
An ARCG-based optical sensor was built, and the impacts of different structural parameters on sensor performance were investigated. When the upper and lower gratings are asymmetrical, the dual-band quasi-BIC can be obtained by adjusting the grating spacing. We calculated the reflection spectra for different grating spacings and the corresponding Q-factors for the two modes using the RCWA and FEM, respectively. This dual-band sensor with BIC based on ARCG is expected to perform well in optical sensing.
Funding
Scientific Research Project of the Department of Education of Hebei Province (ZD2021019); Natural Science Foundation of Hebei Province (B2021402006, F2019402063); National Natural Science Foundation of China (21976049, 61905060).
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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