Fig. 1. The schematic illustrates how GMRs in silicon waveguides can be used to boost the radiation intensity of SPR through BICs. By modulating the width of a double-period grating on the silicon waveguide to break its symmetry, multiple quasi-BICs are generated within a broad frequency range. These modes are then employed to increase SPR’s radiation intensity. Compared to conventional approaches, the quasi-BIC-enhanced SPR achieves radiation intensity improvements by several orders of magnitude.

Fig. 2. The eigenmodes of the silicon waveguide and their dispersion characteristics after implementing a periodic grating are examined. (a) The field distribution of the silicon waveguide’s intrinsic mode. (b) The dispersion distribution of the silicon waveguide’s intrinsic mode. (c) Excitation characteristics of the TM0 mode following the introduction of a width-modulated grating.

Fig. 3. Using a plane wave at an incident angle φ=0° to validate the effectiveness of constructing BICs based on GMRs in the silicon waveguide. (a), (b) Show the relationship of the scattering spectrum detected at points A and B with the change of the asymmetry parameter α. (c), (d) Display the spectral distribution at the vertical slices in (a) and (b). (e), (f) Display the changes in the Q factor of the GMR at points A and B with the asymmetry parameter α, with the right inset showing the linear relationship between the Q factor at point A and 1/α2, while point B does not exhibit this relationship. (g) The spectra containing higher-order-mode resonance information at α=0.2 and α=0, where the disappeared peaks form BIC.

Fig. 4. Validating the broadband characteristics of BIC. (a)–(c) Respectively show the dispersion relations of the uniform silicon waveguide, the grating silicon waveguide with α≠0 and α=0. (d)–(f) Respectively show the dispersion in the normalized frequency range of 0.12 to 0.28 and the normalized beam range of 0 to 0.5 for (a), (b), and (c). (i), (k) Respectively show the dispersion calculated by simulation when α=0.5 and 0. (g) Shows the relationship between the scattering spectrum and parameter α at points K130° and K230° in diagram (j) at an incident angle φ=30°; (h) shows the relationship between the scattering spectrum and parameter α at points K160° and K260° in diagram (l) at an incident angle φ=60°.

Fig. 5. Dispersion of GMRs and electron beam. (a) Dispersion curve of the silicon waveguide with a symmetry-breaking grating (α≈0.01, p=L) and a 40-keV electron beam, with their intersection points in the illustrated coordinate area as A1,A2,A3, and B1. (b)–(e) Respectively show the enlarged dispersion relationships near these intersection points to determine the operating frequencies.

Fig. 6. Validating that quasi-BICs based on GMRs can enhance the radiation intensity of SPR. (a)–(d) Respectively show the relationship of the detected radiation intensity at the dispersion intersections A1,A2,A3, and B1 with the change of the asymmetry parameter α (first column); the radiation spectrum distribution at the vertical slices (second column); the radiation spectra at α=0.01 and α=1, and the radiation field distribution and directivity at the interaction frequency point at α=0.01 (third column); the relationship between the maximum radiation intensity of |Ez|2 and 1/α2 (fourth column); the relationship between the Q factor calculated from the radiation spectrum and 1/α2 (fifth column); the relationship between the maximum radiation intensity of |Ez|2 and the Q factor calculated from the radiation spectrum (sixth column).

Fig. 7. Validating that BIC can enhance SPR over a wide bandwidth. (a) Dispersion relationship of the grating-loaded silicon waveguide and electron beam at α=0.01. (b) Radiation spectrum as the electron beam voltage U changes, where U is remapped to the wave number space. (c) Relationship between the radiation spectrum and radiation angle as the scanning voltage U changes.