1. Introduction

As the front end of the communication system and the design basis of the duplexer, radio frequency (RF) microwave filters play a vital role in modern communication. Due to their low insertion loss [1] and ease of integration [2], substrate integrated waveguides (SIWs) have become the best choice for RF microwave filters and duplexers serving modern communication systems. Therefore, the Q-factor of the SIW resonator is one of the important indicators affecting the performance of modern communication systems. The conventional approach to SIW resonator design involves creating metalized via holes with specific dimensions and spacing on the substrate of a printed circuit board (PCB), forming a closed resonator-like structure [3]. By adjusting the dimension and shape of the resonator, the Q-factor can be improved. Subsequently, more and more researchers have begun to use negative feedback networks to compensate for the energy loss inside the SIW resonator [4,5], and the Q-factor of active SIW resonators can be improved by one to two orders of magnitude. On the other hand, this also increases the complexity and power consumption of the circuit and may introduce additional noise and mismatch [6]. In recent years, there has been a growing academic interest in an SIW structure that employs air or other low-loss dielectrics [7,8]. A hybrid SIW and periodically drilled SIW structure was proposed in Ref. [9], in which periodic air holes were added to synthesize a lower effective permittivity, thus achieving a Q-factor of 815. However, it is difficult to derive theoretically a Q_3dB (defined in Eq. (1)) of more than 1000 by traditional methods. Here we suggest a novel approach based on the concept of bound states in the continuum (BICs).

Bound states in the continuum (BICs), distinct from the conventional bound states whose frequencies are outside the continuum spectrum, lie inside the continuum while remaining perfectly localized without radiation [10,11]. BICs attracted significant attention due to their infinite lifetime and have become a hot topic of research in interfering systems, extending from quantum mechanics [12] to photonics [13]. Nevertheless, any losses occurring in the actual systems result in the bound states transforming into resonant states with a finite lifetime [14]. BICs can be divided into several groups based on their intrinsic physical mechanisms [15], one of which is called symmetry-protected (SP) BIC if its coupling constants vanish due to symmetry [16]. An 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 [10]. The BIC represents a perfectly localized state with no leakage energy, even when it coexists with a continuous spectrum of radiating waves. Ideally, BIC is invisible in its spectra with zero linewidth and an infinite Q-factor [15,17]. When collapsing into quasi-BIC, it can be observed experimentally with a highly enhanced Q-factor. Thus, BICs found prospects in various material systems of photonics crystals [18–20], topological insulators [21,22], and metamaterials [23–25]. However, to the best of our knowledge, there are currently no reports about BICs in SIW. We believe that the introduction of the concept of BIC can greatly improve the Q-factor of the SIW structure.

This work proposes a stacked SIW structure to achieve SP BIC. When the electromagnetic wave oscillates in the traditional SIW resonator made of PCB, the loss of the dielectric layer of PCB hinders the formation of BICs. Therefore, the key to achieving BICs by PCB lies in using air as the dielectric layer in the SIW resonator, thus reducing the loss of the dielectric in this work. By collaboratively employing simulation, experimentation, and theoretical analysis methods, the underlying physics of the modes of SP BIC have been demonstrated. The high Q value of BIC is utilized to make this system have a higher Q-factor. Our results offer a new approach to building high-performance microwave devices.

2. Simulation and experiment

This study proposed a structure of three stacked PCB layers with each layer consisting of a 1 mm thick FR-4 (${\varepsilon _r} = 4.7,\; \; tan\delta = 0.022$) substrate (as illustrated in Fig. 1(a)) and copper on both sides, with a thickness of 0.035 mm and an electric conductivity of $5.8 \times {10^7}$ S/m. Figures 1(b) and 1(c) show schematics of the structure, where the three layers are identified as P₁, P₂, and P₃, respectively. P₁ and P₂ are cleverly used to form a resonator with internal dielectric air. P₂ and P₃ form an SIW with impedance matched with two SMA ports as in-out ports. A vector network analyzer is applied to the two ports to obtain the S parameters. A coupling slot parallel to the x-axis opened in the middle of P₂ to guide the income waves into the resonator. The center shift y of the coupling slot allows the symmetry of the resonator to be broken along the y-axis, but the symmetry is maintained along the x-axis [26]. The values of the simulated and experimental parameters are the same, as shown in Table 1.

 

Fig. 1. (a) Overall structure of the model. (b) The cross-section of the sample and the experimental schematic. (c) The structure of each layer of PCB and its geometric parameters. (d) The fabricated prototype of the SIW and the resonator. The comparison of simulated and experimental (e) S₂₁ spectrums and (f) Q factors with various y values. (g) The simulated and experimental S₂₁ with the circular gap at y = 0.5 mm.

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Table 1. The values of the parameters of the structure

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As the three-dimensional schematic shown in Fig. 1(c), d is the diameter of the metallic via hole, and p is the distance between two adjacent via holes. L₁ and L₃ represent the length and width of the SIW, respectively. L₂ is the length of the top metal layer of P₂. A reasonable choice of its value can attenuate the propagation of electromagnetic waves in the P₂ and thus reduce the interference with the coupling process. L₄ and L₅ represent the length and width of the rectangular resonator, respectively. In addition, the length of the coupling slot is denoted as L_slot, and its width is denoted as W_slot. h_c and h_s represent the heights of the resonator and SIW, respectively. The experimental procedure is to adjust the center shift y of the SIW resonator concerning the symmetric axis of the coupling slot. Figure 1(d) illustrates the fabricated prototype of the SIW and the resonator.

Using the frequency domain solver in CST Microwave Studio, simulations are performed to confirm the presence of the SP BIC in the system. The mesh type is set to “Tetrahedral”, and the boundary conditions in all directions are set to “Open”. The simulated and experimental S parameters are shown in Fig. 1(e). For a distinct observation of the experimental spectrums, they uniformly employ decibels (dB) as the unit of measurement. When y = 0, the structure is symmetrical, and the resonator mode is BIC mode, so there is no peak or dip in the transmission spectra. When y increases, the resonance mode collapses to quasi-BIC, the resonance amplitude increases, and the Q_3dB decreases [27]. The Q₃dB [28,29] is computed as the ratio between the resonance frequency ${\omega _0}$ and the full width at half maximum of $\Delta \omega $ in transmission:

As shown in Fig. 1(e), when y $\le $ 0.2 mm, the resonance bandwidth is relatively narrow, with the resonance amplitude in its S₂₁ falling less than 3dB. This makes it impractical to calculate Q_3dB factor for this range using the half-power bandwidth method. In momentum space, Q-factor is shown to decay quadratically (Q${\propto} $1/k²) with respect to the distance k from a single isolated BIC [30]. There is a similar phenomenon: when y tends to 0, the Q_3dB factor increases quadratically concerning 1/y (Q_3dB${\propto} $1/y²), as shown in Fig. 1(f) where 95/y² is used to fit the Q_3dB curve.

In Fig. 1(e) and 1(f), the experimental and simulation results show similar trends, but there is a disparity. It is because there are small gaps between the metallized vias and the PCB during assembly, resulting in more energy leakage. As illustrated in Fig. 1(g), we added a circular gap of uniform width $\Delta gap$ around each via hole to simulate this discrepancy. The results show that $\Delta gap$ increases the bandwidth of the quasi-BIC both in the simulation and the experiment. Furthermore, a portion of the electromagnetic waves within the resonator adheres to the vicinity of the metalized via holes and escapes into the space, as presented in the inset of the electric field distribution. In the experiment, the distribution of $\Delta gap$ is not uniform, so it is difficult to get a simulation that is completely consistent with the experimental result. This difference can be effectively eliminated by using integrated circuit board fabrication processes during manufacture.

3. Discussion and analysis

Coupled mode theory [31] is usually used to analyze coupled resonances, which also forms a significant theoretical analytical method for comprehending the formation of BICs. To facilitate the analysis, a coupling model is established, as shown in Fig. 2. The incident source ${S_{{1^ + }}}$ enters from port 1, where a portion of the energy enters the SIW resonator with a resonance amplitude a with the coupling coefficient $\kappa $, while the remaining power flows into port 2. Notably, the energy in the resonator can also be directly coupled back to the SIW with the leak coefficient d, flowing evenly to port 1 and port 2. These leakages are respectively denoted as ${S_{a_1^ - }}$ and ${S_{a_2^ - }}$, possessing identical amplitudes but divergent directions. Furthermore, the overall outgoing wave at port 2 is represented as ${S_{{2^ - }}}$.

 

Fig. 2. The coupling model of CMT.

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In accordance with the classical CMT proposed by Fan S. H. [32] in optical resonators, the following dynamic equations can be derived as:

Based on the properties of time-reversal symmetry and energy conservation in Maxwell's equations [33], it can be determined that $\kappa ={-} d = \sqrt {2/{\tau _r}} $. Consequently, the expression for the amplitude a within the resonator is as follows:

The conditions of $\omega = {\omega _0}$, ${\Gamma _o} = 0$, and $|{{S_{21}}} |= 0$ mean that the resonator completely binds the energy without loss. While (${\Gamma _o} \ne 0$, $|{{S_{21}}} |> 0$) indicates some energy loss in absorption and radiation. The non-negligible dielectric loss in the PCB compels us to construct the resonator filled with air, as illustrated in Fig. 1(b), where a portion of the energy is inevitably radiated to the external space when it is transferring from one SMA port to another. These losses bring up $T \ne 1$, and obtaining the exact T without affecting the other coefficients is the primary task. According to the inset of Fig. 3(a), the transmission in the red region at y = 0 mm is approximately a linear function versus $\omega $, which means no energy leaks from the resonator. T in this section stands for the systematic loss and does not affect ${\Gamma _r}$. Therefore, the precise value of T is the guarantee to solve the fitting problems in Eq. (6) with different W_slot as in Fig. 3(a). Here, T is a function proportional to $\omega $, when W_slot is 0.6 mm, 0.8 mm, etc., until 1.4 mm, T is $T = 0.6933\omega - 2.4848,\; \; T = 1.03\omega - 4.1332,\; \; T = 1.27\omega - 5.327,\; \; T = 1.4\omega - 6.0,\; $ and $T = 1.48\omega - 6.435$, respectively.

 

Fig. 3. Simulated transmission spectrums of different ${W_{slot}}$. (a) y = 0 mm (The red curve enclosed by the dashed line can be seen as nearly linear, while other ${W_{slot}}$ resemble this). (b) y = 0.2 mm (solid line) and the corresponding fitting curves (dotted line).

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We investigated the transmission of the Fano resonance with W_slot varying and fitted the curves using revised CMT as shown in Fig. 3(b). It can be concluded that the resonance frequency has a significant red shift with decreasing W_slot. The smaller the slot width, the narrower the resonance bandwidth is. This can be understood by treating W_slot as a perturbation of the resonator [26]. The fitted values of $|{{\Gamma _r}} |$, $|{{\Gamma _o}} |$, and their corresponding Q_f are presented in Table 2 for W_slot from 0.6 mm to 1.4 mm. Specifically, ${\Gamma _o}$ and ${\Gamma _r}$ were originally complex numbers, only the absolute values are displayed for the sake of illustrating the coupling strength. The results indicate that $|{{\Gamma _r}} |$ and $|{{\Gamma _o}} |$ increase with the increment of W_slot. With more energy radiating into the free space, the less confined energy induced a minor Q_f. The increase in absorption loss is related to the leakage through the coupling slot to the dielectric substrate of P₂. This is considered an absorption loss. As the W_slot increases, the radiation loss increases and more energy leaks to the dielectric substrate, increasing the absorption loss. Based on the fitting results above and Fig. 3(b), W_slot is also one of important parameters determined the Q_3dB. The maximum values of Q_3dB, i.e., Q_3dB-max, with varying y are listed in Table 2.

Table 2. Fitted values of ${{ \omega }_0}$, $|{{{ \varGamma }_{ r}}} |$, $|{{{ \varGamma }_{ o}}} |$ and their corresponding Q-factor.

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When y = 0, the whole structure is symmetrical along the central axis, but the mode in the resonator is oddly symmetrical as Fig. 4(c) shown. Different symmetry types result in the BIC mode totally decoupling with the structure by the Q-factor tending to infinity, as the red dot circle in Fig. 4(b) shows. When y ≠ 0, the symmetry of the entire structure is broken, the BIC mode turns into a guided mode or partly transmitted mode, i.e., a quasi-BIC mode. As |y| decreases, the Fano resonance bandwidth reduces till it disappears at y = 0. The Q_3dB of the proposed system reaches up to 2556 near y = 0 as shown in Fig. 4(a).

 

Fig. 4. (a) Trend of Q₃dB factor versus y with different W_slot. Incorporating the Q_3dB factor in Fig. 1(f), the characteristic that the Q_3dB factor tends to infinity with the increasement in the structure symmetry is more clearly shown. (b) The normalized transmission spectrum with different offset y. (The plus or minus sign of y represents the direction in which the resonator is moving in the y-axis.) (c)-(f) The electric field pattern in SIW with different y.

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The underlying physics also can be obtained from the electric field patterns. In Fig. 4(c), y = 0, due to the symmetrical protection, there is a small amount of energy in the resonator (the electric field amplitude is very low compared to Fig. 4(d)–4(f)). When y $\ne $ 0, the symmetry of the structure is broken, and a large amount of waveguide energy enters into the resonator, and the strong coupling effect leads to a great change in the electric field distribution, as shown in Fig. 4(d)–4(f). As y decreases, the amplitude of the electric field increases, indicating that the capacity of the resonator for local energy increases, i.e., the Q value increases. Thus, we get the Q_3dB high to 2556 near y = 0.

4. Conclusion

In conclusion, we illustrate the evolution of a stacked SIW structure from BIC to quasi-BIC when the resonator is displaced along the y-axis direction. According to the analytical model for the coupling within this system utilizing CMT, the correlation between absorption loss, radiation loss, and transmission spectrum is explored to investigate the impact of the coupling slot width on resonance generation; Moreover, we considered the absorption loss and engaged in a discussion about T, which made our theory more in line with real-life situations. Both the structural symmetry and the electric field mode symmetry of this system demonstrate typical properties of SP BIC, i.e., this resonance exhibits a higher Q-factor when symmetry is enhanced. The results of this study provide new ideas and theoretical analyses for the design of BIC devices by SIW technology, which will be helpful for future applications in the field of integrated circuits.