Two Individual Super-Bound State Modes within Band Gap with Ultra-High Q Factor for Potential Sensing Applications in the Terahertz Wave Band

Bound states in the continuum (BICs) garnered significant research interest in the field of sensors due to their exceptionally high-quality factors. However, the wide-band continuum in BICs are noise to the bound states, and it is difficult to control and filter. Therefore, we constructed a top-bottom symmetric cavity containing three high permittivity rectangular columns. The cavity supports a symmetry-protected (SP) superbound state (SBS) mode and an accidental (AC) SBS mode within the bandgap. With a period size of 5 × 15, the bandgap effectively filters out the continuum, allowing only the bound states to exist. This configuration enabled us to achieve a high signal-to-noise ratio and a wide free-spectral-range. The AC SBS and the SP SBS can be converted into quasi-SBS by adjusting different parameters. Consequently, the cavity can function as a single-band sensor or a dual-band sensor. The achieved bulk sensitivity was 38 µm/RIU in terahertz wave band, and a record-high FOM reached 2.8 × 108 RIU−1. The effect of fabrication error on the performance for sensor application was also discussed, showing that the application was feasible. Moreover, for experimental realization, a 3D schematic was presented. These achievements pave the way for compact, high-sensitivity biosensing, multi-wavelength sensing, and other promising applications.


Introduction
Bound states in continuums (BICs) attracted significant attention for open systems without coupling channels from outside radiation, as they possess infinite Q-factors and have become a hot topic of research in interfering systems [1,2], extending from quantum mechanics [3] to photonics [1][2][3][4]. BICs can be divided into several groups based on their intrinsic topological nature, with two of the most conventional kinds being symmetryprotected (SP) BICs, and accidental (AC) BICs [5]. An symmetry-protected (SP) BIC exists in a system that exhibits mirror or rotational symmetry, and this BIC mode belongs to different symmetry classes that completely decouple with the system as long as the symmetry is preserved [1], while an accidental (AC) BIC is a single resonance that evolves into a BIC when enough parameters are tuned. The single resonance itself can be thought of as arising from two (or more) sets of waves, and the radiation of the constituting waves can be tuned to cancel each other [1]. The BIC represents a perfectly localized state with no leakage energy, even when it coexists with a continuous spectrum of radiating waves, Figure 1a shows the schematics and parameter symbols of the proposed structure. Circular cylinders of 5 × 15 were arranged in air to form a two-dimensional PhC and the central three circular columns were replaced with rectangular columns (defect columns). The height of the columns in the z direction was infinite. A TE plane wave with an electric field in the z-direction propagated from the port "in" to the port "out" along the x-direction with wave vector k x . The lattice constant of the square lattice PhC was a and dielectric silicon columns had permittivity ε = 12.25. The three rectangular rutile ceramics pillars at the center, with permittivity ε = 100, are marked in green. Additionally, the width (w) and height (h) of the rectangular defect pillars were the same, and w is the most important parameter for achieving the AC SBS. The center defect column can be moved in y-direction, the amount of movement is ∆y, which is the most significant parameter to tune SP SBS. Two ports marked as the blue solid line indicate the input and output ports of the wave for the cavity. In Figure 1b The remainder of this paper is organized as follows. In Section 2, the property of the AC SBSs and SP SBSs are presented. In Section 3, the sensing property of the AC SBSs and SP SBSs are shown. Finally, Section 4 presents the conclusions of this study. Figure 1a shows the schematics and parameter symbols of the proposed structure. Circular cylinders of 5 × 15 were arranged in air to form a two-dimensional PhC and the central three circular columns were replaced with rectangular columns (defect columns). The height of the columns in the z direction was infinite. A TE plane wave with an electric field in the z-direction propagated from the port "in" to the port "out" along the x-direction with wave vector x k . The lattice constant of the square lattice PhC was a and dielectric silicon columns had permittivity ε = 12.25. The three rectangular rutile ceramics pillars at the center, with permittivity ε = 100, are marked in green. Additionally, the width ( w ) and height ( h ) of the rectangular defect pillars were the same, and w is the most important parameter for achieving the AC SBS. The center defect column can be moved in y-direction, the amount of movement is y Δ , which is the most significant parameter to tune SP SBS. Two ports marked as the blue solid line indicate the input and output ports of the wave for the cavity. In Figure 1b Figure 2a) and a typical w (the blue circle in Figure 2a) were chosen to show the difference between them. The two field patterns had minor differences, while the Poynting vectors showed a significant variation in the center. For the profile of the blue circle point in the left side panel of Figure  2b, the Poynting vectors indicate that the energy radiated to the out space of the cavity. However, for the other panel of Figure 2b, there was no radiation energy from the center field of the BIC point, and the up and down parts had reverse relation, giving no radiating channel to the free space. In momentum space, Q is shown to decay quadratically ( 2 Q k − ∝ ) with respect to the distance k from a single isolated BIC [49]. Here, there is a similar phenomenon: when w tends to SBS w , Q increases quadratically, or fitting constant B. The Q factor here are ultra-high that more than 10 10 . The Q value is calculated using the following formula:

Materials and Methods
where f F is the frequency of the peak in transmission and FWHM (full width of half maximum) is the frequency width at half value of transmission peak. Next, the evolution of transmission, field profiles, and Q factors of the counterpart named SP SBS mode is given in Figure 3. In the previous section, the geometric parameter w was tuned to produce AC SBS, and no structural symmetry was broken until the SP SBS mode was discussed with the parameter y Δ induced. The transmissions in Figure  3a indicate that the amplitudes of the peaks decreased to 0 with the absolute value of y Δ closing to 0. As is known, the infinite Q factor is one of the most specific characteristics of the BIC. Then, the corresponding tendencies of the Q factors and amplitude are detailly shown in Figure 3b. The amplitudes decreased to a small value, while the Q factors increased dramatically with y Δ closing to 0. For the Q factor, it increased quadratically concerning ( / ) a y Δ (   Figure 2a) were chosen to show the difference between them. The two field patterns had minor differences, while the Poynting vectors showed a significant variation in the center. For the profile of the blue circle point in the left side panel of Figure 2b, the Poynting vectors indicate that the energy radiated to the out space of the cavity. However, for the other panel of Figure 2b, there was no radiation energy from the center field of the BIC point, and the up and down parts had reverse relation, giving no radiating channel to the free space. In momentum space, Q is shown to decay quadratically (Q ∝ k −2 ) with respect to the distance k from a single isolated BIC [49]. Here, there is a similar phenomenon: when w tends to w SBS , Q increases quadratically, or Q ∝ a 2 /(w − w SBS ) 2 , as shown in Figure 2c, where Ba 2 /(w − w SBS ) 2 is used to fit the Q-value curve with the fitting constant B. The Q factor here are ultra-high that more than 10 10 . The Q value is calculated using the following formula: where F f is the frequency of the peak in transmission and FWHM (full width of half maximum) is the frequency width at half value of transmission peak. Next, the evolution of transmission, field profiles, and Q factors of the counterpart named SP SBS mode is given in Figure 3. In the previous section, the geometric parameter w was tuned to produce AC SBS, and no structural symmetry was broken until the SP SBS mode was discussed with the parameter ∆y induced. The transmissions in Figure 3a indicate that the amplitudes of the peaks decreased to 0 with the absolute value of ∆y closing to 0. As is known, the infinite Q factor is one of the most specific characteristics of the BIC. Then, the corresponding tendencies of the Q factors and amplitude are detailly shown in Figure 3b. The amplitudes decreased to a small value, while the Q factors increased dramatically with ∆y closing to 0. For the Q factor, it increased quadratically concerning (a/∆y) (Q ∝ (a/∆y) 2 ), as shown in Figure 3b, where B(a/∆y) 2 was used to fit the Q-value curve with the fitting constant B. The Q factor here was also as high as more than 10 10 . To compare the QBIC and BIC points, the field patterns and Poynting vectors at the blue circle point with ∆y = −0.02a and the red circle point with ∆y = 0.001a are illustrated in Figures 3a and 3b, respectively. These patterns and vectors differed from those shown in Figure 2b. The field pattern and Poynting vector in the left side panel of Figure 3c were much weaker in the upper rectangular column than the other two. The topological symmetry was broken to provide a radiation tunnel for the peak is near 1 with ∆y = −0.02a. However, the topological symmetry was reserved to constrain the energy in the cavity in the right panel of Figure 3c, in which the corresponding amplitude was 0.06 with ∆y = 0.001a.

Results
Sensitivity S is a very important parameter in sensor research, which reflects the change of the sensor with detection parameters. The general definition of sensitivity is S = ∆λ/∆n, where ∆λ refers to the change in wavelength under different refractive indices,

Results
Sensitivity S is a very important parameter in sensor research, which reflects the change of the sensor with detection parameters. The general definition of sensitivity is S = ∆λ/∆n, where ∆λ refers to the change in wavelength under different refractive indices, ∆n is the change in refractive index. Another very important parameter in the sensors is the quality factor FOM, which is generally defined as: where S is the sensitivity of the sensor mentioned earlier, Q is the quality factor of the resonator, λ res refers to the resonant frequency of the resonator, and FWHM is the frequency width at half value of transmission peak. From the tuned geometric parameters in the previous sections, the related two parameters w and ∆y can control the AC BIC and SP SBS independently. By adjusting the parameters, we can obtain three kinds of transmission lines in Figure 4. The first one, as in the top panel, shows the result that there was only SP QSBS in the transmission spectra with w = 0.515525a, ∆y = 0.005a. The second one is shown in the middle with both SP QSBS and AC QSBS in the transmission when w = 0.515515a, ∆y = 0.005a. The last one in the bottom panel had only AC QSBS in the transmission spectrum, in which w = 0.515515a, ∆y = 0. The top and the bottom transmission line can be used as ultra-wide FSR high FOM sensors, while the middle panel case can function as multi-mode sensing. For the possibility of fabricating the sensor, we considered that the sensor was applied in the frequency region of THz, and the lattice of the photonic crystal was confirmed as a = 1 mm. Then, the other geometric parameters of h = 0.2 mm and w were established.  The band maps of the even AC SBS mode with different w values are shown in Figure 5a,b. It is evident that this mode resided within the band gap, effectively avoiding interference from background light. The field pattern in Figure 5c is consistent with the even mode pattern in Figure 1d of AC BIC. The transmission in Figure 5d shows a very sharp Lorentz shape line in which the Q factor reached high to 2.83 × 10 9 with h = 0.2 mm, w = 0.51556 mm. For the lone AC BIC mode in the band gap region, no background interference resulted in a relatively broad band suitable for various sensor applications. In Figure 5e, the peak frequencies decrease with the permittivity of the background tuned manually. Meanwhile, the evolution of Q factors and peak frequencies in Figure 5f demonstrate that the Q factors had minor fluctuation and the frequency can be varied linearly with the permittivity. The evaluation parameters FOM and S were obtained by the equation of (2) in Figure 5g. The values of FOM 10 7 and S 38.25 µm/RIU were very high compared to the current research results. The band maps of the even AC SBS mode with different w values are shown in Figure 5a,b. It is evident that this mode resided within the band gap, effectively avoiding interference from background light. The field pattern in Figure 5c is consistent with the even mode pattern in Figure 1d of AC BIC. The transmission in Figure 5d shows a very sharp Lorentz shape line in which the Q factor reached high to 2.83 × 10 9 with h = 0.2 mm, w = 0.51556 mm. For the lone AC BIC mode in the band gap region, no background interference resulted in a relatively broad band suitable for various sensor applications. In Figure 5e, the peak frequencies decrease with the permittivity of the background tuned manually. Meanwhile, the evolution of Q factors and peak frequencies in Figure 5f demonstrate that the Q factors had minor fluctuation and the frequency can be varied linearly with the permittivity. The evaluation parameters FOM and S were obtained by the equation of (2) in Figure 5g. The values of FOM 10 7 and S 38.25 µm/RIU were very high compared to the current research results. To further investigate the impact of varying the parameter w on the properties of AC SBS sensors, we conducted calculations and analyzed the sensor characteristics at w = 0.515515 a . Comparing the amplitudes of transmission in Figure 6b to those in Figure   5d, we observed a significant reduction (0.464), albeit with an accompanying increase in the Q value by an order of magnitude. The evolution of Q factors and peak frequencies with different n values in Figure 6c indicates that the Q factors exhibited minor fluctuations, and the wavelength can be linearly varied. These observations are consistent with those depicted in Figure 5f. The evaluation parameters, FOM and S in Figure 6d, demonstrate that the values of FOM 10 8 were one order of magnitude higher than those of w = 0.51556 a (Figure 5g). However, there was no significant change in sensitivity S. This indicates that, for the same structural mode, altering the parameters did not lead to a considerable change in sensitivity S. On the other hand, FOM was highly influenced by the Q value, and modifying the Q value can effectively alter FOM. To further investigate the impact of varying the parameter w on the properties of AC SBS sensors, we conducted calculations and analyzed the sensor characteristics at w = 0.515515a. Comparing the amplitudes of transmission in Figure 6b to those in Figure 5d, we observed a significant reduction (0.464), albeit with an accompanying increase in the Q value by an order of magnitude. The evolution of Q factors and peak frequencies with different n values in Figure 6c indicates that the Q factors exhibited minor fluctuations, and the wavelength can be linearly varied. These observations are consistent with those depicted in Figure 5f. The evaluation parameters, FOM and S in Figure 6d, demonstrate that the values of FOM 10 8 were one order of magnitude higher than those of w = 0.51556a (Figure 5g). However, there was no significant change in sensitivity S. This indicates that, for the same structural mode, altering the parameters did not lead to a considerable change in sensitivity S. On the other hand, FOM was highly influenced by the Q value, and modifying the Q value can effectively alter FOM.  The other single SBS condition in Figure 4 is the SP SBS, which was protected by topological symmetry. When the geometric parameters h = 0.2 mm and w = 0.515525 mm were fixed, the transmitting tunnel was opened by breaking the symmetry with offset parameter ∆y = 0.005 mm. From Figure 2, we know the peak of transmission disappeared at the AC SBS point (w SBS = 0.515525a). Only a single SP QSBS peak appeared in the transmission spectra due to symmetry broken. The band gaps and the field pattern in Figure 7a,b illustrate that the operating mode was odd in the band gaps, as shown in Figure 1d. The Q factor of the SP QSBS in Figure 7b was 2.1 × 10 10 and the amplitude was 0.64 and both are higher than that of AC QSBS in Figure 6a. The transmissions, the evolution of Q factor, S, and FOM varying with refractive were introduced from Figure 7c-e. The difference from the AC QBIC sensor in Figure 6 was larger values of the peaks of transmission, the operating frequency, Q factors, and FOM, and a smaller value of S = 38. This shows that by adjusting the parameters w and ∆y, we can change the working mode, frequency, FOM, signal amplitude, etc., of the single-mode sensor.
In the more common scenario, when both parameters ∆y = 0 and w = 0.515525 mm, the transmission exhibited two peaks simultaneously at different frequencies. When h = 0.2 mm, w = 0.515515 mm, ∆y = 0.005 mm, both AC QSBS and SP QSBS were in the transmission spectra. The transmission spectra, the evolution of Q factors, S, and FOM varying with the refractive index of a dual-bands sensor are shown in Figure 8. The Q factors of the two modes were larger than 10 10 , inducing an ultra-large FOM that more than 10 8 . The sensitivity of the AC SBS mode and the SP SBS mode was 38 and 38.3 µm/RIU in the terahertz wave band, respectively. It can be utilized for ultra-sensitive multi-parameter measurements. In addition, the dual-band sensor can also be transformed back into a single-band sensor by adjusting the parameters w and ∆y.  Figure 8. The Q factors of the two modes were larger than 10 10 , inducing an ultra-large FOM that more than 10 8 . The sensitivity of the AC SBS mode and the SP SBS mode was 38 and 38.3 µm/RIU in the terahertz wave band, respectively. It can be utilized for ultra-sensitive multi-parameter measurements. In addition, the dual-band sensor can also be transformed back into a single-band sensor by adjusting the parameters w and y Δ .  Compared to traditional 2D PhC activity sensor structures, most of them have larger sizes, typically exceeding 15 × 15 dimensions [50,51]. The Q factor generally does not exceed 10 7 , and the sensitivity remains below 10,000 nm/RIU [50,51]. However, in this study, the 2D PhC cavity structure demonstrated remarkable performance with an ultra-compact size of 5 × 15, achieving a Q value of more than 10 10 and a sensitivity of 383,000 nm/RIU. Furthermore, it exhibited a high signal-to-noise ratio and an exceptionally wide FSR. These achievements open up possibilities for compact, high-sensitivity biosensing, multiwavelength sensing, and other promising applications.

Analysis of the Effect of Fabrication Error and Experimental Scheme
The accuracy of mechanical processing will inevitably affect the sensor parameter indicators. Therefore, it is important to analyze the impact of fabrication errors and determine executable solutions before conducting experiments. In general, the machining accuracy can reach 0.002 to 0.001 mm [52]. We studied the changes in sensor parameters under an error of 0.002 mm. For the SP SBS mode, the accuracy error of ∆y will affect the Q factor, as Q ∝ (a/∆y) 2 . Let Q = B(a/∆y) 2 , then In Figure 3b, when ∆y = 0.005a and 0.009a, Q = 2.1 × 10 10 and 9 × 10 9 , respectively, where a = 1 mm. Therefore, when ∆y = 0.007 mm, even with an error of δ∆y = 0.002 mm, the variation of Q will be δQ Q = −2 δ∆y ∆y = −57%, (4) thus, the value of Q has an evident fall, but it is still high in between 0.903 × 10 10 and 3.87 × 10 9 , which means the sensor can still has high sensitivity and FOM.  Figure 3a, the change of r had no evident effect on both the mode field distribution and the evolution of the transmission spectrum. Figure 9e shows the parameters of the SP SBS mode sensor with changed r, the FOM, and the sensitivity S showed only slight changes compared to that in Figure 7. According to the above, we can see that the SP SBS mode sensor had strong robustness.
Moreover, we investigated the effect of the radius r of the silicon column with an error of 0.002 mm. As shown in Figure 10, the situation was the same with the SP SBS mode sensor except for two things. The first one was that the increase in r induced a slight red shift in the frequency of the AC SBS mode in Figure 10a, not as the blue shift in Figure 9. The second one was that the value of w SBS changed from 0.515525a in Figure 2a to 0.51549a in Figure 10c, which caused a variation in Q value, resulting in the FOM dropping by an order of magnitude. Compared with the SP SBS mode, the AC SBS mode sensor was more affected by changes in r, and if the w value can be adjusted to be closer to w SBS , it can compensate for the loss of the Q value.  Figure 9e shows the parameters of the SP SBS mode sensor with changed r , the FOM, and the sensitivity S showed only slight changes compared to that in Figure 7. According to the above, we can see that the SP SBS mode sensor had strong robustness. Moreover, we investigated the effect of the radius r of the silicon column with an error of 0.002 mm. As shown in Figure 10, the situation was the same with the SP SBS mode sensor except for two things. The first one was that the increase in r induced a slight red shift in the frequency of the AC SBS mode in Figure 10a, not as the blue shift in Figure 9. The second one was that the value of SBS w changed from 0.515525 a in Figure  2a to 0.51549 a in Figure 10c, which caused a variation in Q value, resulting in the FOM dropping by an order of magnitude. Compared with the SP SBS mode, the AC SBS mode sensor was more affected by changes in r , and if the w value can be adjusted to be closer to SBS w , it can compensate for the loss of the Q value. Additionally, we studied the effect of the parameters h and w with an error of 0.002 mm on the AC SBS mode sensor. In Figure 2c, when w = 0.51551 a and 0.5154 a , Q = 1.3 × 10 10 and 1.9 × 10 8 , respectively, where a = 1 mm. The error in w value of 0.00011 mm resulted in a change of two orders of magnitude in Q value, let alone an error of 0.002 mm. The fitted B of the AC SBS mode is 3 in Figure 2c, which is much smaller than that of the SP SBS mode in Figure 3b. So, the AC SBS mode was very sensitive to geometric parameter w . In Figure 11, the value of h increased to 0.202 mm, and the value of SBS w increased to 0.520803 a . The value of w is close to SBS w in Figure 11d, the FOM and S are as high as that in Figure 6. However, the w value in Figure 11e is the same as that in Figure 6, with a 0.002 mm error in h , resulting in the FOM value as low as 5019, reducing it by five orders in magnitude. Therefore, the ability to adjust the w value is very important for the practical application of the AC SBS mode sensor. To construct w adjustable structure, we proposed using long trapezoidal columns Additionally, we studied the effect of the parameters h and w with an error of 0.002 mm on the AC SBS mode sensor. In Figure 2c, when w = 0.51551a and 0.5154a, Q = 1.3 × 10 10 and 1.9 × 10 8 , respectively, where a = 1 mm. The error in w value of 0.00011 mm resulted in a change of two orders of magnitude in Q value, let alone an error of 0.002 mm. The fitted B of the AC SBS mode is 3 in Figure 2c, which is much smaller than that of the SP SBS mode in Figure 3b. So, the AC SBS mode was very sensitive to geometric parameter w. In Figure 11, the value of h increased to 0.202 mm, and the value of w SBS increased to 0.520803a. The value of w is close to w SBS in Figure 11d, the FOM and S are as high as that in Figure 6. However, the w value in Figure 11e is the same as that in Figure 6, with a 0.002 mm error in h, resulting in the FOM value as low as 5019, reducing it by five orders in magnitude. Therefore, the ability to adjust the w value is very important for the practical application of the AC SBS mode sensor.
To construct w adjustable structure, we proposed using long trapezoidal columns instead of rectangular columns in a three-dimensional model with extremely short column lengths in the z-axis direction to achieve adjustment of the w value. We first discussed the construction of 3D models. In this study, the input electromagnetic wave was a TEpolarized wave, with the electric field vector oriented perpendicular to the direction of propagation (in the y-direction). It can be directly input at the wave port. To approach the 2D structure more precisely, very long dielectric columns are usually required, which can be challenging to achieve. Instead, a proposed scheme involves the introduction of two metal plates, creating mirrors for the dielectric columns and electromagnetic waves, thereby achieving an equivalent 2D PhC. To mitigate the loss caused by the metal, silver can be used as the material due to its favorable properties. To prevent oxidation of the silver, an organic film can be deposited on the silver plates. This approach significantly reduces the required length of the dielectric columns in z-direction. The three-dimensional model, depicted in Figure 12a, illustrates this concept with a length (l) of 1 mm. The photonic crystal structure and other parameters in the figure remain the same as that in Figure 1a. A TE electromagnetic wave was excited or input from the input port (left port), guided by the metal plates and the PhC, form resonance in the PhC cavity, and finally went to the output port (right port). Within the resonant cavity, the distribution of the electromagnetic field can be approximated as a TE wave. The band map in Figure 12b displays the odd SP SBS mode (red line) and the even AC SBS mode (blue line) in the 3D cavity. It is similar to the band map of the 2D structure in Figure 1c, except for a minor blue shift in frequency. The frequency shift was understandable because the metal material introduced leads to a change of the effective refractive index of the structure. The electric field patterns in the x-y plane of the two SBS modes in the 3D cavity, shown in Figure 12c,d, are the same as those in the 2D cavity in Figure 1d. The electric field patterns in the x-z plane of the two SBS modes in the 3D cavity, also depicted in Figure 12c,d, illustrate the distribution of the electric field in the z-direction. The transmission spectra of the two SBS modes in the 3D cavity are presented in Figure 12f,h. This demonstrates that it is feasible to reduce the length of the dielectric column by utilizing the metal-plate-assisted construction. Additionally, we studied the effect of the parameters h and w with an error of 0.002 mm on the AC SBS mode sensor. In Figure 2c, when w = 0.51551 a and 0.5154 a , Q = 1.3 × 10 10 and 1.9 × 10 8 , respectively, where a = 1 mm. The error in w value of 0.00011 mm resulted in a change of two orders of magnitude in Q value, let alone an error of 0.002 mm. The fitted B of the AC SBS mode is 3 in Figure 2c, which is much smaller than that of the SP SBS mode in Figure 3b. So, the AC SBS mode was very sensitive to geometric parameter w . In Figure 11, the value of h increased to 0.202 mm, and the value of SBS w increased to 0.520803 a . The value of w is close to SBS w in Figure 11d, the FOM and S are as high as that in Figure 6. However, the w value in Figure 11e is the same as that in Figure 6, with a 0.002 mm error in h , resulting in the FOM value as low as 5019, reducing it by five orders in magnitude. Therefore, the ability to adjust the w value is very important for the practical application of the AC SBS mode sensor. To construct w adjustable structure, we proposed using long trapezoidal columns instead of rectangular columns in a three-dimensional model with extremely short column lengths in the z-axis direction to achieve adjustment of the w value. We first discussed the construction of 3D models. In this study, the input electromagnetic wave was a TE-polarized wave, with the electric field vector oriented perpendicular to the direction of propagation (in the y-direction). It can be directly input at the wave port. To approach the 2D structure more precisely, very long dielectric columns are usually required, which can be challenging to achieve. Instead, a proposed scheme involves the introduction of two metal plates, creating mirrors for the dielectric columns and electromagnetic waves, thereby achieving an equivalent 2D PhC. To mitigate the loss caused by the metal, silver can be used as the material due to its favorable properties. To prevent oxidation of the silver, an organic film can be deposited on the silver plates. This approach significantly  Figure  1a. A TE electromagnetic wave was excited or input from the input port (left port), guided by the metal plates and the PhC, form resonance in the PhC cavity, and finally went to the output port (right port). Within the resonant cavity, the distribution of the electromagnetic field can be approximated as a TE wave. The band map in Figure 12b displays the odd SP SBS mode (red line) and the even AC SBS mode (blue line) in the 3D cavity. It is similar to the band map of the 2D structure in Figure 1c, except for a minor blue shift in frequency. The frequency shift was understandable because the metal material introduced leads to a change of the effective refractive index of the structure. The electric field patterns in the xy plane of the two SBS modes in the 3D cavity, shown in Figure 12c,d, are the same as those in the 2D cavity in Figure 1d. The electric field patterns in the x-z plane of the two SBS modes in the 3D cavity, also depicted in Figure 12c,d, illustrate the distribution of the electric field in the z-direction. The transmission spectra of the two SBS modes in the 3D cavity are presented in Figure 12 f,h. This demonstrates that it is feasible to reduce the length of the dielectric column by utilizing the metal-plate-assisted construction.  Finally, in order to adjust the w value, the rectangular column was designed as a trapezoidal column whose x-z plane was trapezoidal, as shown in Figure 12h. The upper short side of the trapezoid had a length of 0.505 mm, and the lower long side had a length of 0.525 mm, which means the adjustment range of the w value was 0.505-0.525 mm, while h remained fixed. The length of the trapezoidal column in the z-direction was 100 mm, and the part of the trapezoidal column in the resonant cavity could be approximated as a rectangular column. By adjusting the trapezoidal column up and down, the w value can be adjusted to solve the problem caused by insufficient machining accuracy. With this scheme, machining accuracy to 1% is feasible.

Conclusions
In conclusion, this study introduced a top-bottom symmetric cavity consisting of three rectangular columns with high permittivity. It exhibited both an SP SBS mode and an AC SBS mode located within the band gap, with a period size of 5 × 15. By adjusting the parameters w and ∆y, the AC SBS and SP SBS can be converted into QSBS, respectively. This means that the two SBS modes within the band gap can be individually modulated, resulting in a single AC QSBS or SP QSBS in the transmission spectrum. By tuning w and ∆y, it was also possible to obtain both QSBS modes simultaneously in the transmission spectrum. Hence, this cavity can function as a single-band sensor or a dual-band sensor by manipulating the aforementioned parameters. For the single-band sensor configuration, the frequency and figure of merit (FOM) can also be adjusted. The achieved Q factors of the QSBS modes can exceed 10 10 . The obtained bulk sensitivity was 38 µm/RIU with a high signal-to-noise ratio in the terahertz wave band, and a record-breaking figure of merit of 2.8 × 10 8 RIU −1 was achieved. The effect of fabrication error on the performance of the proposed structure was also discussed, showing feasibility for applications. For experimental realization, a 3D schematic was presented. These accomplishments open up avenues for compact, high-sensitivity biosensing, multi-wavelength sensing, and other promising applications.

Data Availability Statement:
The research data will be supplied with reasonable request.