In 1953, E. M. Purcell and his student S. J. Smith discovered a unique form of Cherenkov radiation (CR) capable of radiating into free space [1,2]. This phenomenon, later named Smith–Purcell radiation (SPR), occurs when free electrons interact with periodic structures, without the usual constraints of voltage or speed thresholds required for radiation from charged particles. The working principle of SPR is based on the Brillouin zone folding effect [3]. After being diffracted by a periodic structure, the wave vector of the high-order space harmonics of the space-charge wave excited by free electrons will be superimposed with 2mπ/d (where m is the order of the space harmonics and d is the period length), effectively bridging the gap between the slow-wave wave vector and the free-space wave vector, thereby removing the speed threshold for charged particles in CR [4,5]. This free-space radiation is established through phase matching between the space harmonics of charged particles and the free-space plane wave, known as the Smith–Purcell effect [6]: λ=L|m|(1β−cosθ),(1)where λ is the wavelength of the radiation and θ is the angle between the direction of radiation propagation and the direction of electron motion. SPR can produce electromagnetic radiation in frequency regions typically difficult for conventional radiation sources to cover, such as terahertz, far-infrared, ultraviolet, and X-ray bands, indicating significant potential for applications in non-destructive testing [7], free-electron lasers [8,9], and tunable light sources [10–13]. Despite its broad application prospects, SPR, as a special type of CR, relies on the diffraction of periodic structures to extract energy from the electron beam, usually with an efficiency of about 0.1% [4]. This creates challenges related to low beam-wave coupling impedance and low radiation intensity, limiting the integration and performance of SPR-based terahertz electron vacuum devices. To address these challenges, scientists have been exploring ways to enhance SPR radiation intensity by introducing special electromagnetic structures or modes. For instance, in 2005, Korbly et al. from MIT observed coherent super-Smith–Purcell radiation (Super-SPR) in the terahertz band for the first time, discovering that the radiation intensity of SPR under higher-order harmonics could be significantly enhanced by interactions with bunched micro electron beams, resulting in high-intensity coherent terahertz radiation [14]. In 2014, Shenggang Liu and his team proposed a mechanism to realize tunable coherent terahertz SPR using graphene surface plasmons (GSPs) [15]. This mechanism significantly enhanced the radiation intensity near the excitation frequency through the low-loss characteristics of GSP and the localized field enhancement effect.

Bound states in the continuum (BICs) represent localized wave phenomena commonly found in systems involving photons and phonons [16,17]. While BICs are part of a continuum of extended states, they do not couple with these extended states, resulting in a fully localized state without energy leakage [18]. As a result, BICs are theoretically characterized by infinitely high Q-values and extremely narrow resonance widths, making them an excellent choice for enhancing light–matter interactions [19]. Since von Neumann and Wigner first introduced the concept of BIC in quantum mechanics in 1929 [20], the development of BIC has driven research into various high-Q-value applications, including lasers [21], nonlinear frequency conversion [22], and sensors [23]. BICs can also boost free-electron radiation. In 2018, Song et al. proposed BIC-enhanced SPR, constructing BICs with parameter tunability through one-dimensional or two-dimensional dielectric gratings [24]. This approach uses quasi-BICs with extremely high Q-values and frequencies close to BIC to interact with free electrons, thereby enhancing the diffraction intensity of free-electron radiation in traditional interaction systems. This offers a novel approach to developing high-efficiency on-chip terahertz radiation. Additionally, Yang et al. at MIT constructed symmetry-protected BICs by breaking the symmetry at the Γ-point in a Si/SiO2 two-dimensional grating [25]. On this basis, they enhanced the radiation intensity of SPR and theoretically demonstrated the general upper limit of photon emission and energy loss in free-electron interaction processes. However, these methods often work only at specific frequencies, which can limit their application over a broad spectral range, reducing the tuning capability of light sources and the energy detection range of charged particle detectors. To address the tunability challenge, we proposed a new approach in 2023, where we created BICs based on spoof surface plasmons (SSPs) on metal plasmonic grating surfaces, covering a broad frequency spectrum [26]. This led to a significant enhancement in the radiation intensity of free electrons over a wide bandwidth. However, as the operating frequency increases above 220 GHz, the loss in sub-wavelength metal SSP systems rises considerably, thus limiting the high-frequency performance of these devices [27].

Recently, researchers realized momentum-mismatch-driven BICs in compound grating waveguides [23,28]. This category of BICs originates from the momentum mismatch, enabling high Q factors over a broad frequency spectrum [23,28–31]. For instance, momentum-mismatch-driven BICs in compound grating waveguide structures exhibit angularly robust Q factors and ellipsometric phase singularities, which enable ultrasensitive refractive index sensing and phase-based optical devices [28]. Empowered by momentum-mismatch-driven quasi-BICs, the Goos-Hänchen shift can be enhanced to 104 times the wavelength [23]. Similarly, dual-momentum-mismatch-driven quasi-BICs in lithium niobate grating waveguides significantly enhance sum-frequency generation efficiency, providing a tunable and robust platform for nonlinear light sources and quantum photonics [32]. Broadband quasi-guided modes have been realized through perturbation-induced guided-mode resonances, enabling high-Q resonances with tunable polarization properties and distinguishing between BICs and guided modes [30]. Moreover, ultrahigh-Q resonances in silicon compound lattices have been achieved using a double-band-folding strategy, advancing applications in terahertz devices, nonlinear optics, and quantum computing [29]. Additionally, Brillouin-zone folding in mid-infrared grating structures has produced coherent thermal emissions with narrowband and spatial coherence, highlighting their potential for practical applications [31].

To address the high losses in metallic circuit structures at terahertz frequencies and the limitations of traditional methods that generate BICs only within narrow frequency bands, this study employs silicon grating waveguides to construct broadband momentum-mismatch-driven quasi-BICs [23,28]. These broadband momentum-mismatch-driven quasi-BICs leverage enhanced light confinement near BICs to significantly boost the radiation intensity of SPR. By loading gratings on the surface of silicon waveguides, conventional guided modes, which are typically confined within the dielectric, are transformed into guided-mode resonances (GMRs) that couple with free-space plane waves. Further, modulating the groove width between dual-period gratings introduces symmetry-breaking conditions that isolate these GMRs from free-space traveling waves, converting them into quasi-BICs with high radiation quality factors (Q). This approach enables the creation of multiple quasi-BICs across a broad frequency range, allowing simultaneous enhancement of free-electron radiation at multiple frequencies. The resulting SPR intensity is approximately six orders of magnitude higher than conventional SPR, demonstrating the potential of high-Q quasi-BICs for radiation enhancement. Additionally, the system supports beam scanning from 0° to nearly 90° by adjusting the operating voltage, providing precise directional control of radiation. In summary, this study presents a robust scheme for enhancing SPR using broadband multiple quasi-BICs, laying the foundation for compact, tunable, and efficient silicon-based terahertz free-electron radiation sources [33].

First, based on the schematic in Fig. 1, we briefly introduce a scheme and its features for constructing multiple momentum-mismatch-driven quasi-BICs and enhancing radiation intensity of SPR through guided-mode resonances (GMRs). The basic interaction structure consists of a silicon (relative permittivity εr=11.9) waveguide loaded with a double-period grating. By loading a grating onto the surface of the silicon waveguide, traditional guided modes that are localized within the dielectric are converted into GMRs that can couple with free-space plane waves. Further, by modulating the slot width between the double-period gratings, the symmetry-protected structure will prevent energy leakage from these GMRs, cutting off their coupling pathways with free-space traveling waves, thus constructing multiple BICs based on these GMRs. When free electrons are phase-matched with the quasi-BIC near the BIC frequency, significant SPR enhancement occurs over a broad spectral range.

To explore the mechanism behind these broadband GMRs-induced BICs, a detailed description of the dispersion characteristics of width-modulated gratings is required. As shown in the left inset of Fig. 1 the silicon waveguide has a thickness of d=0.1mm. The propagation and modulation periods are d=0.1mm and L=2d=0.2mm, respectively. The widths of both gratings are a=0.04mm, and the depths of the slots are h=d/2=0.05mm. Additionally, the slot widths between the two gratings are defined as {b1=(1+α)d,b2=(1−α)d.(2)

The process of modulating the grating width is characterized by an asymmetric parameter with a range of α∈[0,1]. The condition α=0 occurs only when the widths of the two grooves are equal (b1=b2); otherwise, when b1≠b2, α is not equal to zero. To further discuss how BICs are formed in this system, it is necessary to calculate and analyze the dispersions of the silicon waveguide, as shown in Fig. 2(a). In this configuration, regions 1 and 3 are filled with air, while region 2 consists of the silicon. Since the propagation constants for guided modes in both air and the medium should be the same, the wave vector must meet the following condition: {kz2−τ2=k02=ω2μ0ε0,kz2+T2=kd2=ω2μdεd,(3)where kz is the propagation constant, τ is the transverse attenuation constant in air, T is the transverse wave number in the waveguide, k0 and kd are the wave numbers of plane waves in free space and the waveguide, respectively, ω indicates the operating angular frequency, μ0 and μd represent the permeabilities in vacuum and the waveguide, and ε0 and εd are the permittivities in vacuum and the waveguide. Assuming the relative permeability to be one, the dispersion relation for the TM modes in a dielectric waveguide is given as follows [4]: ε0Tdtan(Td2−mπ2)=εdτd,(4)where m is the order of the guided mode. By combining Eqs. (3) and (4) and using the graphical method to find their intersection, the dispersion relation diagram shown in Fig. 2(b) can be obtained, revealing the dispersion curves for the lowest five modes. The solid dots indicate the dispersion relation derived from CST simulations, validating the consistency between theoretical analysis and simulation results. The calculated frequency and wave number components can be applied to the field expressions to obtain the electric field distribution of these guided modes, as depicted in the field distributions in Fig. 2(a), where the wave number kz used in the calculation equals 2π/d, as indicated by the gray solid line in Fig. 2(b).

Next, we will discuss the formation process of BIC based on the TM0 mode. Initially, for a uniform silicon waveguide, the dispersion relationship of the TM0 mode and a plane wave incident at angle φ in free space is juxtaposed in the first column of Fig. 2(c). At this juncture, the TM0 mode’s dispersion curve resides in the slow-wave region, whereas that of the incident wave is in the fast-wave region, preventing their coupling due to wave number mismatch. However, as shown in the second column of Fig. 2(c), upon the addition of a grating with period p=L=2d to the waveguide surface, the incident wave, via grating diffraction, generates multiple space harmonics, translating the dispersion curve of the incident wave by 2π/L periodically in the ω−kz space. In this scenario, some higher-order space harmonics intersect with the TM0 mode’s dispersion curve at points A,B,C, and D corresponding to −1st, −2nd, −3rd, and −4th order harmonics. These intersections indicate wave number matching between the diffractions of the incident wave and the TM0 mode at specific frequencies, thus enabling the excitation of the TM0 mode within the waveguide. Following this, by progressively reducing the asymmetry parameter α to zero and adjusting the slot widths in the bi-periodic grating to be equal, the grating’s minimum period adjusts to p=d, as illustrated in the third column of Fig. 2(c). Here, the dispersion curve of the incident wave shifts periodically by 2π/d in the ω−kz space, and since L=2d, the new period of shift is twice the original. This modification eliminates the appearance of odd-order space harmonic components like −1st and −3rd, transitioning previously excitable frequencies such as fA and fC at points A and C into non-excitable states, hence forming what are termed as dark modes. This alteration leads to the formation of a series of BICs at the intersection frequencies of these odd-order space harmonics with the TM0 mode dispersion. In contrast, even-order space harmonics like −2nd and −4th orders remain, and previously excitable frequencies like fB and fD at points B and D continue to be excitable; these modes do not become BICs.

To delve deeper and validate the processes forming these BICs, a plane wave incident at angle φ=0° is employed to excite the eigenmode of the grating-loaded dielectric waveguide and the scattering spectrum is then probed in the far-field region. As the asymmetry parameter α is reduced from one to zero, Figs. 3(a) and 3(b) display the variations in the scattering spectra near points A and B, respectively. Simulation results demonstrate that near point A, as α decreases, the resonance peak evolves from weak to strong and from broad to narrow; this transformation is not observed at point B. Additionally, Fig. 3(c) further details the specific alterations in the frequency and width of the resonance peak at point A as the α value changes (1, 0.8, 0.6, 0.4, 0.2, 0), illustrating the resonance peak’s frequency increases, intensity heightens, and width contracts, elucidating the BIC formation process. To enhance readability, the scattering spectra for different α values are represented using sequence numbers on the vertical axis, where the numbers 1 to 6 correspond to α=0, 0.2, 0.4, 0.6, 0.8, 1, respectively. For each parameter, the relative minimum value of the scattering efficiency is normalized to zero (aligned with the lower bound of its respective sequence number), and the relative maximum value is normalized to one (aligned with the upper bound of the next sequence number). As α approaches zero, the resonance peak’s width nearly vanishes and the intensity markedly augments, signifying the emergence of quasi-BICs. Upon setting α to zero, a BIC theoretically forms with an infinite Q factor, and the resonance peak width vanishes, leading to the disappearance of the resonance peak from the spectrum entirely. In contrast, the resonance peak at point B does not display these characteristics, as illustrated in Fig. 3(d). Additionally, Figs. 3(e) and 3(f) visually depict how the Q factor of the resonant state changes with α. For point A, as α nears zero, the Q factor surges, displaying a relationship inversely proportional to the square of α, i.e., Q∝α−2. This relationship aligns with the theoretical descriptions for symmetry-protected BICs, thereby validating this as a symmetry-protected BIC. Conversely, at point B, the Q factor of the resonance peak does not exhibit a rapid increase as α approaches zero, nor does it show an inverse quadratic relationship. Thus, point B does not develop into a BIC, aligning with earlier results from the analysis of the dispersion of diffracted harmonics: as α nears zero, intersection points like A and C of odd-order space harmonics with the TM0 mode dispersion will form a series of BICs; in contrast, points like B and D of even-order space harmonics with TM0 mode dispersion will not develop into BICs.

The analysis presented above, while based on the waveguide’s lowest-order mode TM0, is also applicable to higher-order eigenmodes, which exhibit similar characteristics. To explore this, the spectral range of the simulation was extended to encompass the frequencies where wave vector matching occurs between the incident wave and higher-order eigenmodes (such as TM1 and TM2). The simulation results and the field distributions around these frequencies are depicted in Fig. 3(g). Initially, with the structural parameter at α=0.2, the spectral intensity distribution, shown with a yellow curve, reveals a series of resonance peaks within the specified frequency range. These resonance peaks are correlated with specific eigenmodes and space harmonic orders, with their corresponding electric field distributions organized in ascending mode order, as presented in the first row of Fig. 3(g). From the relationship between field distributions and frequency magnitudes, these peaks correspond to the intersections of the −1st, −2nd, and −3rd order space harmonics with TM0, and −1st order space harmonic with TM1,TM2, and TM3, respectively. Setting the structural parameter α to zero, intersections of odd-order space harmonics with the eigenmodes theoretically form BICs with infinite Q factors. This results in the vanishing of the resonance peak widths and consequently, the disappearance of these peaks from the spectrum. Specifically, under α=0 conditions, resonance peaks at intersections like −1st and −3rd order space harmonics with TM0, and −1st order space harmonic with TM1,TM2, and TM3 will vanish. Conversely, peaks at intersections of −2nd order space harmonics with TM0 and TM1 remain observable, as evidenced by comparing spectral results depicted with orange and red solid lines in Fig. 3(g).

By altering the parameter α, the formation process of BIC was demonstrated for an incident angle of φ=0°. However, with a constant incident angle, GMRs can be excited only at specific frequencies corresponding directly to the wave number kz=k0sinφ. Therefore, the previous discussions validated the formation of BICs only at isolated frequency points or wave vectors. To illustrate the broadband nature of BICs, initially, the dispersion relations of TM0 and TM1 modes in the unloaded silicon waveguide are depicted in Fig. 4(a). These modes reside in the slow-wave region, and their dispersion curves gradually converge towards the light cone of the dielectric waveguide with increasing frequency. Following this, based on Floquet’s theorem, the dispersion curves in Fig. 4(a) are both translated and folded in the ω−kz space at periods of 2π/d and 2π/L, as illustrated in Figs. 4(b) and 4(c). This procedure approximates the dispersion relationships within the grating-loaded dielectric waveguide for periods p=d and p=L (with actual dispersion curves ideally computed via commercial simulation software, which should closely resemble these approximations). The modified dispersion curves in the ω−kz space, now highly complex, are simplified for analysis by presenting the magnified dispersion relations within normalized frequency ranges of 0.12 to 0.31 and wave vector ranges of 0 to 0.5, as shown in Figs. 4(d) to 4(f). In contrast to the dispersion of an unloaded silicon waveguide as shown in Fig. 4(d), the grating-loaded configurations introduce new space harmonics, denoted by red (odd-order harmonics) and blue (even-order harmonics) curves shown in Figs. 4(e) and 4(f). These curves, along with the original deep blue curves, form the guided-mode dispersion under asymmetric grating conditions. Importantly, segments of the red and blue curves fall within the fast-wave region, where the guided modes, not entirely confined to the grating surface, can radiate into free space as their wave numbers match those of the extended modes (i.e., plane waves). Thus, these guided modes inside the light cone are resonant states within the continuum of free-space extended modes, thereby defined as GMRs. From the comparison between Figs. 4(e) and 4(f), it is evident that with the structural symmetry protection when α transitions from a non-zero value to zero, the dispersion curves of the odd-order space harmonics are represented in red shift from resonant states to BICs. Despite these resonances residing within the spectrum of extended states in free space, they exhibit perfect localization and are non-radiative. This establishes the GMR-based BIC, protected by structural symmetry, characterized by theoretically infinite Q-values and the absence of detectable resonance spectral widths. Additionally, this BIC, derived from GMRs localizing the fast-wave region, persists across a wide spectral range, evidenced by the red dispersion curves situated within the light cone. Conversely, even-order harmonic GMRs, depicted by the blue curves, remain in a resonant state even as the asymmetry parameter shifts from α≠0 to α=0, preventing their transition into BICs.

Subsequent simulations validate the broadband presence of BICs. Initially, Fig. 4(i) shows dispersion curves at α=0.5, generated using CST, where red and blue lines represent odd and even space harmonic GMRs, respectively, and the deep blue line indicates the guided-mode dispersion. As established, satisfying phase matching conditions facilitates the excitation of GMRs within the fast-wave area. Variations in the incident angle φ alter the longitudinal wave number kinc=k0sinφ, modifying phase matching conditions accordingly. Adjusting the incident angle allows the excitation of GMRs at diverse frequencies and wave vectors. As φ incrementally approaches 90°, the wave vector of the incident wave spans the entire light cone, triggering a series of GMRs therein, with the detected spectral distribution illustrated in Fig. 4(j). The spectral peaks align with the theoretical dispersion curves within the light cone, confirming the presence of GMRs over an extensive bandwidth. As α reduces from 0.5 to 0, the odd-order harmonic GMRs transition first to high-Q quasi-BICs and subsequently to BICs with absent resonance widths. The dispersion curve of the GMRs at α=0 is shown in Fig. 4(k), where the red dashed line represents the dispersion distribution of symmetry-protected BICs. Given that BICs cannot couple with any free-space modes, the detected spectrum should reveal no resonance peaks. Following this, the scattering analysis performed at α=0 replicates the conditions at α=0.5, with the detected scattering spectrum displayed in Fig. 4(l). As analyzed, the locations previously exhibiting resonances (red dashed line) now show no peaks, as the odd-order harmonic GMRs have transitioned to BICs that remain undetectable using conventional methods. This demonstrates that BICs, situated within a broad spectral range in the light cone and initiated by odd-order harmonic GMRs, gradually emerge as α decreases to zero, confirming their extensive spectral existence. When the incident angles are φ=30° and 60°, Figs. 4(g) and 4(h) respectively show the changes in the scattering spectrum near the points K130°, K230°, K160°, and K260° as the asymmetry parameter α is reduced from one to zero. The findings reveal a marked intensification and sharpening of resonance peaks near these points, confirming the transformation of the odd-order harmonic GMRs within the light cone into multi-BICs that are undetectable by traditional means.

In practical applications, the performance of high-Q resonances in the THz band, including the GMR-based BICs proposed in this study, is often limited by fabrication imperfections such as processing accuracy and sidewall roughness. These imperfections introduce additional losses that can significantly suppress the radiation Q factor. For GMR-based BICs, maintaining the structural symmetry is critical to sustaining the high Q factor, as deviations during fabrication can shift the resonance frequency and hinder the formation of quasi-BICs or BICs. Similarly, sidewall roughness can scatter light and increase non-radiative losses, further degrading the guided-mode resonances. To overcome these challenges, advanced fabrication techniques such as e-beam lithography, focused ion beam milling, and improved etching processes can enhance accuracy and reduce surface imperfections. Additionally, the use of low-loss materials in the THz regime is crucial for minimizing absorption losses, while post-fabrication optimizations, such as thermal annealing or surface smoothing, can improve the modal field quality and reduce scattering effects. Addressing these limitations is essential for translating the theoretical potential of GMR-based BICs into practical devices, enabling their application in THz sensing, communication, and spectroscopy with enhanced Q factors and performance reliability.

To substantiate the assertion that quasi-BICs can enhance the radiation intensity of SPR, a system excited by a single electron was modeled using the frequency-domain solver in COMSOL. When the asymmetry parameter α=0.01, the system’s dispersion curve is shown in Fig. 5(a), where the red and blue solid lines respectively represent the dispersion of odd and even space harmonic GMRs. A 40-keV electron beam moving along the +z direction facilitated the excitation of these resonances, and its dispersion relation is illustrated by a purple solid line to clarify the wave vector matching relationship. Intersection points labeled A1,A2,A3, and B1 denote where wave vector matching occurs, signifying interaction points between guided modes and free electrons. As free electrons skim the grating, GMRs at wave numbers kA1,kA2,kA3, and kB1 are excited simultaneously, and these intersection points lie within the light cone, enabling high-Q odd space harmonics to diffract into free space and yield enhanced SPR. Figures 5(b)–5(e) zoom in at these points to more distinctly reveal the matching frequencies.

To further elaborate on how quasi-BIC enhances the radiation intensity of SPR, Fig. 6(a1) portrays the far-field detected radiation spectrum near the frequency fA1 as α decreases from one to zero, displaying the evolution of the radiation peak from weak to strong. Furthermore, Fig. 6(a2) presents the radiation spectra at different α values (1, 0.8, 0.6, 0.4, 0.2, and 0), visually demonstrating how the frequency and width of the radiation peaks change. To assess the enhancement effect, Fig. 6(a3) displays the spectral intensity of ordinary and enhanced radiation at α=1 and 0.01. At α=1, only SPR coherent with conventional GMRs is generated, and because of the low radiation Q-value of conventional GMRs, the peak radiation intensity is only 1.87×10−45. However, as α gradually approaches zero, conventional GMRs transform into high-Q quasi-BICs. When α=0.01, the peak radiation intensity generated by this quasi-BIC is 4.44×10−39, an increase of six orders of magnitude compared to the situation at α=1, clearly showing the potential of high-Q-value quasi-BICs in enhancing SPR radiation intensity. Additionally, the inset on the right of Fig. 6(a3) displays the Ez distribution and far-field directivity at the peak radiation frequency at α=0.01, with the radiation angle aligning at 86.4° to the +z direction, conforming to the Smith–Purcell relation [Eq. (1)]. Results in Figs. 6(a4), 6(a5), and 6(a6) further confirm that during the reduction of α to zero, the trends of enhanced SPR’s Q-value and radiation intensity IR exhibit a distinct inverse quadratic relation with α, i.e., Q∝IR∝α−2. This indicates that as α approaches zero, the radiation Q-value of GMRs escalates, thereby augmenting the radiation intensity. These findings align with the BIC formation results depicted in Fig. 3, substantiating that high-Q-value quasi-BICs can substantially enhance SPR’s radiation intensity. Points A2 and A3, similar to A1, are situated on the dispersion curves of odd-order space GMRs within the light cone. As depicted in Figs. 6(b) and 6(c), their related simulations of radiation spectra and field distributions corroborate that BICs based on GMRs-induced quasi-BICs at A2 and A3 effectively enhance SPR’s radiation intensity. In contrast, point B1, positioned on the curve for even-order resonances, does not evolve into a BIC as α decreases from one to zero; thus, the radiation intensity of SPR near this point is not enhanced, as evidenced by the simulations shown in Fig. 6(d).

As evidenced in Fig. 4, the quasi-BICs presented in this study exhibit a presence across a broad bandwidth. This bandwidth aligns with the frequency range of odd-order space harmonic GMRs within the light cone as α is progressively reduced to zero. Consequently, quasi-BICs, existing across such a wide spectral range, are anticipated to significantly enhance the radiation intensity of SPR across a comparable range. Previously, the electron beam’s operating voltage was fixed at 40 kV, limiting quasi-BIC excitation to specific frequencies tied to particular wave numbers. This constraint meant that past analyses showcased the system’s potential for radiation enhancement only from a singular frequency or wave vector perspective. To delve deeper into the wide-band radiation enhancement capabilities afforded by BICs, it is crucial to examine how variations in operating voltage U influence both the radiation spectrum and its directional characteristics. At α=0.01, GMR dispersion curves are depicted within a normalized frequency range of 0.15 to 0.21 in Fig. 7(a), where red lines denote even-order harmonic GMRs. Additionally, as the operating voltage U is adjusted from 53.5 kV down to 9.5 kV, the electron beam’s dispersion curve, illustrated with a purple dashed line, comprehensively spans the quasi-BIC’s existence region. Consequently, within this range of operating voltages, quasi-BICs at all frequencies can be excited, yielding an SPR that is both enhanced and coherent with quasi-BIC and features broadband tunability. Figure 7(b) showcases the resulting enhanced SPR radiation spectrum. For clarity, theoretical dispersion curves are reillustrated as white solid lines within this diagram. These results confirm that the radiation peaks conform to theoretical dispersion curves inside the light cone, validating the successful excitation of enhanced SPR over an extensive frequency spectrum. Additionally, Fig. 7(c) delineates how the radiation spectrum adjusts with changes in the radiation angle θ. Through modulation of the operating voltage, beam scanning from 0° up to near 90° is feasible, illustrating the capability to manipulate radiation direction via voltage adjustments. This flexibility underscores the transformative potential of quasi-BICs in broadening the applicative scope of SPR technology.

In practical applications, the performance of high-Q optical resonances in the THz frequency range is often limited by fabrication imperfections, such as processing accuracy and sidewall roughness, which introduce additional losses and significantly suppress the radiation Q factor. Deviations from the designed geometric parameters during fabrication can shift the resonance frequency and reduce the Q factor, as the precise structure is critical for maintaining the symmetry required for BIC formation. Similarly, sidewall roughness can scatter light and increase non-radiative losses, further degrading the guided-mode resonances. To address these challenges, advanced fabrication techniques such as e-beam lithography, focused ion beam milling, and improved etching processes can be employed to enhance accuracy and reduce roughness. Additionally, using low-loss materials in the THz frequency range and applying post-fabrication optimizations like thermal annealing or surface smoothing can further improve optical performance. Overcoming these fabrication limitations is essential to translate theoretical advancements into practical devices, enabling the realization of high-Q resonances in experimental settings for applications in sensing, communication, and spectroscopy.

To address losses in metallic circuits at terahertz frequencies and to overcome the limitation of traditional methods that only enable BICs within narrow frequency bands, this research introduces a method employing the Brillouin zone folding effect. This approach constructs BICs across a wide spectrum, substantially enhancing the radiation intensity of free electrons over a broad frequency range. Initially, the study involves applying a grating to the surface of a silicon waveguide and adjusting the slot width in a dual-period silicon waveguide system to disrupt its structural symmetry. This manipulation facilitates the construction of wide-bandwidth BICs using GMRs, which, in turn, enhance the free-electron radiation intensity across a broad spectrum. Analysis of the dispersion relations in dual-period modulated grating systems under various symmetric structural parameters reveals that the Brillouin zone folding effect transforms the confined guided modes in the silicon waveguide into GMRs. These resonances are capable of coupling with free-space plane waves. Further reduction in structural asymmetry allows the in-light-cone GMRs to evolve into a series of wide-bandwidth symmetry-protected BICs. These modes boast theoretically infinite Q-values and exceedingly narrow resonance bandwidths, significantly amplifying the intensity of free-electron radiation. The simultaneous matching of free electrons with multiple BIC wave vectors enhances radiation at multiple frequency points. The high radiation Q-value of the quasi-BIC elevates the system’s radiation intensity by approximately six orders of magnitude compared to conventional SPR, showcasing the potential of high-Q quasi-BICs to boost SPR radiation intensity. Overall, this study offers novel insights into using wide-band BICs to enhance SPR and opens new avenues for the development of next-generation integrated, compact terahertz free-electron radiation sources.

Acknowledgment. The authors acknowledge discussions with Fan-Hong Li.