Fig. 1. (a) Geometry of the plasmonic coating sanwiched structure. (b) Antiparallel-magnetism configuration with MWS coating. Unidirectional quasi-BICs and their corresponding radiation leakage in left (blue) and right (red) propagation are presented. BICs are achieved in both propagation directions in (a), while only the right propagating BIC (red) is in (b). (c) Angular-frequency dispersions of MWS in different amounts of magnetization. Red, yellow, green, blue, and purple lines correspond to different εa when 2b equals q, 3q, 5q, 10q, and 16q, respectively, where q=1×108  m−1. The inset is the dependence of εa on 2b at ENZ frequency (vertical dotted line). (d) Angular-frequency dispersions of permittivity and effective permittivity in narrow frequency range.

Fig. 2. Reflection spectra for structure with (a) parallel and (b) antiparallel magnetism when the thickness of MWS is 0.1 μm; εc=5 and dc=1  μm. Reflection spectra for structure with infinite thickness of MWS layers at two sides with (c) parallel and (d) antiparallel magnetism when εd=1 and εa=0.2. The white dashed line denotes ENZ frequency.

Fig. 3. (a) Angular-frequency dispersions of normalized Re(β) for some eigenmodes related to the quasi-BIC. The antisymmetric mode has antisymmetric field distribution, shown in Appendix C. Angular-frequency dispersions of normalized Im(β) of (b) forward mode I and (c) other eigenmodes. (d) Radiative Q-factor of unidirectional quasi-BIC (blue) and symmetry-protected quasi-BIC (yellow), and radiation loss contrast between two modes in two opposite directions. (e) Time-snapshot of the magnetic field profile at the frequency close to ENZ frequency. (f) Time-snapshot of magnetic field intensity and electric field.

Fig. 4. (a)–(d) Reflection spectra with different intrinsic magnetization values in two MWS layers. The insets are the corresponding schematic models. (e) Angular-frequency dispersions of the bulk MWS in different EF. Solid and dashed lines correspond to Re(εd) and Im(εd), respectively. (f) Dependence of ωp on EF. (g), (h) Reflection spectra with and without losses for EF=0.36  eV and EF=0.4  eV, corresponding to ENZ frequency of ωp=4.961×1014  rad/s and ωp=6.892×1014  rad/s, marked by horizontal dashed lines. In this case, the model is the same as the antiparallel-magnetism configuration in Fig. 2(b) except for dc=0.75  μm, with magnetization value being 2b=q in upper layer and 2b=−q in lower layer.

Fig. 5. (a) Reflection and transmission spectrum dependence on transverse wave vector kx at frequency ω=4.2167×1014  rad/s. (b) Reflection and transmission spectrum dependence on the frequency of incident light ω at certain incident angle sin θ=0.778. Solid and dashed lines represent the spectrum in linear coordinates and log scale, respectively.

Fig. 6. Geometry of MWS sandwiched structure with (a) parallel magnetism and (b) antiparallel magnetism in the coatings.

Fig. 7. Profile of each eigenmode in antiparallel-magnetism case. (a) Mode distribution of the real part of propagation constant, illustrated in the main text. (b)–(d) Magnetic field profile of the corresponding mode pointed out by the arrow in (a).

Fig. 8. (a) Amplitude of reflection coefficients dependence on angular frequency and transverse wave vector in parallel-magnetism configuration with the thickness of MWS layers being 0.1 μm. The permittivity and thickness of the dielectric slab are εc=5.5 and dc=1  μm, respectively. (b) Angular-frequency dispersion of the real part of the wavenumber for some eigenmodes related to the quasi-BIC in (a). (c) Angular-frequency dispersion of the imaginary part of the wavenumber of the mode in blue in (b) at +kx and −kx regions. (d) Dependence of the Q-factor of the mode in blue on the angular frequency of incident light.

Fig. 9. Imaginary part of diagonal element of permittivity εd versus Fermi level. The inset is an enlarged plot of losses in a higher Fermi level range.

Fig. 10. Reflection spectra of anti-magnetism case with different losses of (a) 0.001, (b) 0.005, and (c) 0.008, and the geometric parameter is the same as that in the main text. (d) Reflection spectrum at the transverse wave vector of kx=0.81k0, where k0 is a wave vector in air. This resonance caused by a leaky mode is near the frequency of BICs.