Bound states in the continuum (BICs), as perfect confinement of light in an open system, have been extensively studied both experimentally and theoretically [1–3], spanning from quantum mechanics to photonics [4,5]. Such non-radiating eigenmodes are peculiar states that remain localized with lifetimes that diverge to infinity, although a leaky channel may be present. Various types of BICs that originate from diverse physical mechanisms have been proposed in numerous geometrical photonics systems, such as symmetry-protected BICs, Friedrich–Wintgen (FW) BICs, and accidental single-resonance BICs . These BICs facilitate exciting applications including enhanced optical nonlinearity , high-efficiency light guiding, and optical sensing  and have been implemented in zero-index plasmonic metamaterials [9,10], photonic slabs , high-index Mie resonators , and others ; they usually have a topological nature  and significantly enhanced light–matter interactions [15–17]. However, current investigations on BICs are restricted in reciprocal systems, and their mechanisms and behaviors in nonreciprocal systems are still elusive, as well as their applications to nonreciprocal devices. Nonreciprocal devices that exhibit different responses of the same transmission channel when their sources and detectors are interchanged have become fundamental building blocks in photonic systems [18–20], and drive widespread applications with asymmetric wave manipulation  assisted by magnetic bias, nonlinearities, or external gains and losses [22,23]. Such applications usually require nonreciprocal eigenmodes with high quality factors (-factors) and large contrast ratios simultaneously , especially in the field of unidirectional or angle-selective sensing and emission . Therefore, nonreciprocal BICs appear to be significantly important and need to be more explored.
However, nonreciprocal BICs are very challenging to realize in generic systems. One reason is that the system supporting stable BICs is invariant under (rotation of about the axis followed by time reversal) and (up–down mirror symmetry of the plane) operators (corresponding to the coordinates in Fig. 1) to achieve zero radiation loss . Generally, the nonreciprocal system is more convenient to achieve by introducing an external magnetic field compared to some metamaterials with complex structural design, and time-reversal symmetry is broken in these magnetic systems, which means , where is the permittivity tensor of materials. So such BICs can hardly be supported. The other is that spectral nonreciprocity is imperative for unidirectional BICs, which have to break inversion symmetry of the band structure in space, i.e., . Such implementation needs to break the rotation-time-reversal symmetry (rotation of about the axis followed by time reversal) [27,28], but usually degrades BICs and inevitably reduces their -factors due to the destruction of geometric symmetry. Consequently, the light emission efficiency and isolation contrast in the system are relatively low, which limits its application in photonic circuits [29,30]. For asymmetric light manipulation operating at different frequencies, schemes in tuning unidirectional BICs are indispensable in designing the optical system. Although some literatures have reported dynamic BICs via photodoping or electric bias [31,32], the significant material losses and imperfection of the structure, such as increased roughness or resultant deformation, largely limit the achievement of tunable BICs. Hence, dynamically tunable BICs  with a distinguished response such as nonreciprocity remain largely unexplored.
Figure 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
In this paper, we propose a feasible paradigm toward unidirectional and dynamic BICs by introducing magnetism asymmetry from magnetic Weyl semimetal (MWS) slabs as shown in Fig. 1(b). By incorporating novel MWS into the system, large nonreciprocity but lower losses can be achieved without external magnetic bias, compared to the conventional counterpart of magnetoplasmonic materials [34,35]. The major findings of this work are that properly pairing MWS in an antiparallel-magnetism configuration could establish unidirectional quasi-BICs and a symmetry-protected quasi-BIC at -point without breaking either or symmetry, and dynamically tunable quasi-BICs are attainable through chemical doping. Our results promise the extreme manipulation of light in emerging quantum and low-dimensional nanomaterials [36–38], taking advantage of superior properties in MWS.
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2. OPTICAL RESPONSE OF MAGNETIC WEYL SEMIMETAL
As a class of emerging low-dimensional quantum materials, MWS simultaneously supports intrinsic optical nonreciprocity and magneto-optic (MO) plasmonics characteristics [39,40]. The large MO response is endowed by its unique topological electronic bands and is easily tunable by the chemical doping and its intrinsic momentum. With net zero chirality, pairs of topologically protected gapless crossing points, known as Weyl nodes, exist at the corners of the Brillouin zone in conical reciprocal space, resulting in monopoles and anti-monopoles with different Berry curvatures [41,42]. These paired Weyl nodes exhibit opposite chirality and are separated by a wave vector in the first Brillouin zone or by an energy offset in the energy spectrum. Direct observations of Weyl nodes, Fermi arcs, and their relation are necessary to confirm the existence of an MSW state. Recently, comprehensive evidence on the MWS state in was reported , verifying that its Fermi-arc connectivity varies with surface termination. Following this work, more experimetal realizations of MWS and its physical properties and effects associated with Weyl nodes were pursued [43,44]. Stimulated by the extensive demand on the application of MWS in nanophotonics, its optical model and electrodynamic response have been explored in previous works [45–48].
A concrete expression of its optical response can be described by the additional axion term in the electromagnetic action and has been applied in studying magnetic surface plasmon polaritons [45,46] and nonreciprocal modes in waveguides . The constitutive relation of MWS can be written as 
Here, denotes the major permittivity function of the Dirac semimetal. The first term denotes the major dielectric response, the second represents the anomalous Hall effect, manifesting itself as an MO effect, and the third describes the chiral magnetic effect, stimulated by an applied magnetic field. Without considering external magnetic bias on MWS in an equilibrium state, the third term can be neglected. Such schemes are experimentally realized in [46,48]. Further, we assume an unperturbed magnetic permeability (i.e., ), and crystal orientation is chosen such that is along direction, i.e., . Thus, the permittivity tensor of MWS can be written as
Moreover, applying the random phase approximation and considering the interband electronic transitions in the Dirac cone, the diagonal term is derived through Kubo formalism as the following form :
3. GENERATION OF UNIDIRECTIONAL QUASI-BIC AND SYMMETRY-PROTECTED QUASI-BIC IN NONRECIPROCAL SYSTEM
Having established the optical responses of MWS, we discuss more details of unidirectional quasi-BICs in our structure composed of a planar dielectric slab covered by MWS, where the direction of magnetism is perpendicular to the incidence plane (Voigt configuration). A leaky mode could exist inside this dielectric core with the complex propagation constant , and the disappearance of radiation shows a feature of quasi-BICs when the dielectric core is covered by plasmonic materials [Fig. 1(a)] . However, unidirectional quasi-BICs appear in the antiparallel-magnetism case, which means the directions of in upper and lower MWS layers are antiparallel to each other [Fig. 1(b)], where the radiation leakage along positive- direction is perfectly forbidden, while that along the negative direction is open. Considering our Voigt configuration under transverse-magnetically polarized oblique incidence with an angle of , no polarization conversion is allowed. The dispersion relation in MWS layer is , where is the component of the wave vector in MWS. denotes effective permittivity. When is close to zero without considering intrinsic loss, tends to become infinite while remains almost unchanged, leading to a singular point in the reflection spectrum. Figures 2(a) and 2(b) show the reflection spectrum in parallel- and antiparallel-magnetism cases, where the vanishing feature of resonances suggests the existence of BICs. The off- BIC arises from the coupling between extreme plasmonic resonance in the coating and Fabry–Perot resonance of the slab . In our cases, considering the reflection at the interface between the MWS and dielectric, the condition of Fabry–Perot resonance is obtained as
Figure 2.Reflection spectra for structure with (a) parallel and (b) antiparallel magnetism when the thickness of MWS is 0.1 μm;
Therefore, the Fabry–Perot resonance is symmetric regardless of propagation directions [Fig. 2(c)]. Such results are obtained from an ideal model in which a dielectric slab is covered by infinite MWS with , which contains a magnetic response outside the cavity. However, in the antiparallel-magnetism case, the extra phase becomes two different forms as and , where the sign represents forward () and backward () propagation, respectively, resulting in asymmetric distribution of the reflection spectrum [Fig. 2(d)]. Such an extra phase influences the performance of unidirectional quasi-BICs because its eigenfrequency is not exactly at Fabry–Perot resonance but close to it. So the resonance cannot be perfectly coupled, thus forming the quasi-BIC, as further explicitly discussed in Appendix A. The white line in Fig. 2(d) represents the frequency when , defined as ENZ frequency, corresponding to plasma frequency. It intersects with Fabry–Perot resonance only in the region, leading to a unidirectional quasi-BIC emerging in this case.
To better illustrate the unidirectional quasi-BIC, we performed an eigenmode analysis in the antiparallel-magnetism case [Fig. 1(b)]. Complex dispersion can be found in the pole of the scattering matrix of the source-free Maxwell equations, which can be derived into the following form:
Here, and are the functions of , and is the component of the wave vector in the central dielectric layer. A concrete expression and detailed derivation can be found in Appendix B. With careful examination by comparing these two equations, one can easily prove that Eq. (6) does not follow spatial-inversion symmetry. In particular, several eigenmodes of interest are plotted in Fig. 3(a), which are responsible for the sharp resonant reflection features in Fig. 2(b), and they are asymmetrically distributed in and regions. Three symmetric modes are labeled as forward mode I (blue), backward mode (green) in the region, and forward mode II (red) in the region, and only of forward mode I is non-zero at the ENZ frequency, which is possible to support quasi-BIC. Compared to Fig. 2(d), the Fabry–Perot resonance is determined by forward mode I and backward mode branches, which are in opposite slopes in the narrow bandwidth a little below the ENZ frequency. Forward mode II dramatically departs from its location in the parallel-magnetism case, remaining only in the high frequency branch, and the other half disappears. Moreover, the symmetry-protected quasi-BIC exists at -point in the antiparallel-magnetism case, determined by antisymmetric modes (yellow) in two wave vector regions, while it vanishes in the parallel-magnetism case. The underlying physics is that the antiparallel-magnetism case preserves and symmetry because the magnetic field, being a pseudovector, gains an additional sign flip under mirror symmetry operation. As the time-reversal symmetry is broken in the system, the permittivity tensor of MWS shown in Eq. (2) under a rotation operator along the axis becomes . Then, combining the operation, we obtain
Figure 3.(a) Angular-frequency dispersions of normalized
Importantly, the imaginary part of an eigenmode, i.e., , vanishes for forward mode I [Fig. 3(b)], indicating the disappearance of energy leakage. The radiative -factor (see the definition in Ref. ) of this mode (blue) is plotted in Fig. 3(d), diverging near the ENZ frequency. Moreover, of other eigenmodes are displayed in Fig. 3(c), where only antisymmetric eigenmodes converge to zero point. That supports the symmetry-protected quasi-BIC at -point, while the other two modes are all leaky. The radiative -factors of symmetry-protected quasi-BICs (yellow) are lower than those of unidirectional quasi-BICs [Fig. 3(d)], and decay more rapidly, indicating that they are sensitive to tiny external perturbations. The unidirectional feature of unidirectional quasi-BIC can be obviously reflected from the radiation loss contrast (red), defined as , where represent the wave vector of eigenmodes in the and regions, respectively. The maximum nonreciprocity is achieved at the ENZ frequency when quasi-BIC propagates in direction without any radiation loss. The field distributions near this BIC are shown in Figs. 3(e) and 3(f), corresponding to the incident angle of . A resonant localized magnetic field in out-of-plane direction exists within the dielectric slab, consistent with the non-radiating nature of the mode, while the electric field is mainly directed along the orthogonal direction, marked by the white arrows. Actually, this strong oscillation of the electric field works as orthogonal electric dipoles to excite volume-plasmon resonance in MWS layers. Only such resonance coupling with Fabry–Perot resonance can achieve the accidental BICs in this transversely homogenous waveguide. Therefore, such a unidirectional quasi-BIC is an FW BIC, which is not limited by the symmetry restrictions mentioned above.
4. TUNABILITY ON UNIDIRECTIONAL BIC
We now discuss the dynamic unidirectional quasi-BICs via tailoring the tunability of MWS based on two avenues available, by the argument of breaking the reciprocal response. First, one could change its intrinsic magnetization value in two MWS layers to break the time-reversal symmetry to generate unidirectional quasi-BICs. As shown in Figs. 4(a)–4(c), we vary the magnitude of its intrinsic magnetism in the lower MWS layer while keeping the upper MWS layer unchanged. A pair of asymmetric quasi-BICs still occur with different wave vectors [Fig. 4(a)], quite as expected due to the parallel magnetism. However, only one quasi-BIC exists in the region when the MO response of the lower MWS disappears [Fig. 4(b)] or becomes enlarged [Fig. 4(c)]. Moreover, such a unidirectional quasi-BIC in forward direction carries a larger momentum and moves to a larger direction in the antiparallel-magnetism case. With too large magnetization, no unidirectional quasi-BIC is found, as the Fabry–Perot resonance is not supported near the ENZ frequency [Fig. 4(d)]. Notably, the symmetry-protected BIC still exists in the antiparallel-magnetism case when and symmetry is preserved [Fig. 4(d)], while it turns into a quasi-symmetry-protected BIC when such symmetry is broken [Fig. 4(c)]. In general, via tailoring the distance between Weyl nodes in momentum space in MWS independently without altering , we could tune quasi-BICs of both forward and backward directions in a designer manner.
Figure 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
Second, via controlling the of MWS, the ENZ frequency could be tuned in a broad range, thus tuning the unidirectional quasi-BIC in our case. The permittivity with different is plotted in Fig. 4(e), and it shows the ENZ frequency appears to be approximately linearly blueshifted with smaller losses when increasing [Fig. 4(f)]. Such a characteristic makes our tunability on quasi-BICs come true. Since the loss is one major limitation in practical implementation in most systems , as a comparison, we calculate the reflection spectra with small intrinsic loss [Figs. 4(i) and 4(j)], calculated by Eq. (3) and without loss [Figs. 4(g) and 4(h)], at and . Indeed, the asymmetric leaky modes change dramatically in both forward and backward directions with varying . The comparison between them shows that the unidirectional quasi-BIC carries larger momentum to satisfy the condition of Fabry–Perot resonance in the region when increases, while the symmetry-protected quasi-BIC still exists, unperturbed by varying the , protected by symmetry as demonstrated above. Therefore, by carefully tuning the and setup of the model, we can obtain the unidirectional quasi-BIC in any appropriate frequency with different radiation angles. If losses are considered, though quasi-BICs are degraded to some extent with the decay of leaky modes and the blurring of spectra contrast, the physical mechanism remains evident as unidirectional quasi-BICs can also be observed at the original location. The effect of intrinsic losses in MWS on BICs is discussed in detail in Appendix F.
In conclusion, we realize unidirectional and dynamically tunable quasi-BICs in paired MWS slabs via tailoring their intrinsic MO responses. They show the extremely different radiation of forward and backward leaky modes—one mode perfectly closed despite being within the radiation channel, and the other open, which can be tuned by changing the geometric pairing and the of MWS. We discuss the symmetry requirement of symmetry-protected BICs in a nonreciprocal system. Our strategies neither rely on the discretized asymmetric lattice [14,52] to break the spatial-inversion symmetry, nor require the external magnetic bias, showcasing a new avenue of extreme wave manipulation with emerging exotic quantum materials.
Acknowledgment. Y. Z. acknowledges NSFC, and C. Q. acknowledges the A*STAR Pharos Program.
APPENDIX A: QUASI-BOUND STATES IN THE CONTINUUM
In this appendix, we discuss the behavior of quasi-BICs in the system with magnetism without considering the loss of MWS. In Fig.
Figure 5.(a) Reflection and transmission spectrum dependence on transverse wave vector
Figure 6.Geometry of MWS sandwiched structure with (a) parallel magnetism and (b) antiparallel magnetism in the coatings.
APPENDIX B: DERIVATION OF SCATTERING MATRIX AND SOLUTION OF EIGENMODE
In this appendix, we present the detailed derivation of the scattering matrix and eigenequations in both parallel-magnetism [Fig.
Here, we assume time harmonic oscillations in the form , and incident light is polarized. The permittivity tensor is the same as Eq. (
As we know, the traditional transfer matrix is the recursion of interface information and propagating factors, which can be written in the following form for a multilayer structure with cells:
is the transfer matrix in units, which connects the field on two sides of one cell. and are the total transmission and reflection coefficients of the structure, respectively, assuming the unit incident intensity. relates field information at the interface between and . The transfer matrix can be simplified to
In MWS layers, and are functions of effective wave vector as the continuity of the tangential field in the boundary condition, which brings in the magnetic related off-diagonal element . For the parallel-magnetism case, the tangential electric field and magnetic field are related in the same form in two MWS layers:
and relate to the coupling relations of the effective wave vectors at the interface of air/MWS and dielectric/MWS, which are described as follows:
We can obtain the concrete form of the scattering matrix and get the eigenmode of the structure from the pole of it, which is the eigenfunction
Therefore, Eq. (
For the antiparallel-magnetism case, the result of the transfer matrix has a similar form except for the expressions of , , , and , which change to the following forms:
Other symbols in the expressions are the same as the ones defined before in the parallel-magnetism case. So the eigenfunction becomes
We simplify this equation in the same way, and the function becomes
From Eq. (
APPENDIX C: MODE PROFILES IN ANTIPARALLEL-MAGNETISM CASE
Figure 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).
APPENDIX D: SYMMETRIC QUASI-BIC IN PARALLEL-MAGNETISM CASE
For comparison of quasi-BICs in the two magnetism cases, we calculate the reflection spectrum [Fig.
Figure 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
APPENDIX E: SYMMETRY REQUIREMENT FOR SYMMETRY-PROTECTED BIC IN NONRECIPROCAL SYSTEM
A symmetry-protected BIC can be supported in a system when it is invariant under and operators, which has been proved in a photonic crystal slab [
The axis is perpendicular to the interface of each layer. The form of can be written as
A similar result can be obtained from :
It shows the permittivity tensor of MWS in a Voigt configuration will not be perturbed by the operation. Next, we apply on each layer in the structure with a similar procedure:
Therefore, we can see that the direction of magnetization in the upper MWS layer flips after the operation, which is exactly the direction of magnetization in the lower MWS layer. We can conclude that only the antiparallel-magnetism configuration is invariant under and operations (a simple derivation can show that the parallel-magnetism configuration is not invariant under these operations). Hence, such conditions can achieve projection of the electric field in the far field onto the plane as purely real [
APPENDIX F: EFFECTS OF INTRINSIC LOSSES ON BICS
In the main text, the BICs and reflection spectra in a three-layer configuration with antiparallel magnetism at are presented, without considering intrinsic losses in MWS. A comparison of unidirectional BICs with and without loss is briefly discussed in larger cases to show the degradation of BIC when loss is involved. However, intrinsic loss of BICs is an important factor to be considered and a major limitation in practical implementation on light confinement [
Figure 9.Imaginary part of diagonal element of permittivity
Figure 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
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