Recently, optical bound states in the continuum (BICs) [1–3] have played a central role in topological polarization manipulation [4–6], chiral light–matter interaction [7–9], chiral emission [10–12], and ultrahigh-quality factor (Q factor) optical resonators [13–17]. However, since the topological properties of BICs are mainly determined by the geometrical symmetry of the photonic structures, it is notoriously difficult to manipulate their properties without modifying the photonic structure. For instance, by breaking the in-plane inversion (C2) symmetry [18] or the C6 symmetry [19,20] of the photonic crystal (PhC) slab, pairs of circularly polarized states (C points) with half topological charge can spawn from the eliminated BICs manifesting as a momentum-space vortex polarization singularity (V point). As for a PhC slab with inversion symmetry but broken out-of-plane mirror (σz) symmetry [21,22], the annihilation of two C points of downward radiation could generate exotic unidirectional guided resonances that radiate only in the upward direction. More interestingly, by breaking the C2 and σz symmetries simultaneously, arbitrarily polarized BICs in bilayer-twisted PhC slabs [23] and chiral BICs in dielectric metasurfaces [24–27] can be realized. Nonetheless, once these photonic structures are fabricated and settled, it is rather difficult to dynamically tune the topological properties of BICs without modifying the geometrical structures, thus seriously limiting their tunability and practical applications.

On the other hand, magneto-optical (MO) modulation of light provides a new strategy to dynamically manipulate the topological properties of BICs without modifying the photonic structure [28–30]. For example, both the light intensity [31] and topological polarization singularities in momentum space [32] can be modulated by the external magnetic field in a one-dimensional MO grating with Faraday configuration. Besides, extreme nonreciprocity and near-unity magnetic circular dichroism can be achieved in a subwavelength magnetic metasurface supporting BICs [33]. These findings significantly promote the applications of the MO effect to modulate the topological properties of BICs actively. However, the magnetically tunable BICs with arbitrary polarization covering the whole Poincaré sphere and intrinsic chirality with near-unity circular dichroism and ultrahigh-Q factor remain elusive so far.

In this work, based on two-dimensional (2D) MO PhC slabs placed in external magnetic fields with time-reversal symmetry (TRS) breaking, we theoretically propose a new paradigm for realizing magnetically tunable BICs with arbitrary polarization, off-Γ chiral emission, and at-Γ intrinsic chirality. By continuously modulating the strength of the external magnetic field, we find that the far-field polarization around BICs can be dynamically tuned to achieve full coverage of arbitrary polarization on the entire Poincaré sphere. Moreover, under a proper external magnetic field, we observe off-Γ chiral emission of circularly polarized light under right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) incidences. Interestingly, by further breaking the C2 symmetry, a pair of C points can spawn from the eliminated BICs. Significantly, by tuning the external magnetic field strength, we can gradually move one C point to the Γ point to achieve the intrinsic chiral BICs, which displays the chiral characteristics on both sides of the PhC slab with near-unity circular dichroism exceeding 0.99 and an ultrahigh-Q factor up to 46,000 while preserving the σz symmetry. Our findings open new avenues for dynamically tuning the topological properties of BICs without modifying the photonic structures and may lead to promising applications in polarization manipulation [34–36], chiral light source and detector [37–41], and magnetic tunable photonic devices [42–44].

To realize magnetically tunable BICs with arbitrary polarization and intrinsic chirality, we start with a free-standing 2D MO PhC slab consisting of a square array of circular air holes exhibiting C4v symmetry, as shown in Fig. 1(a). The inset shows the unit cell of the MO PhC slab with lattice constant a=1.4μm, thickness h=1.2μm, and air hole diameter d=1μm. Under a perpendicular external magnetic field (along the z direction) to break the TRS, the optical response of the MO PhC slab can be expressed by the following permittivity tensor [32]: ε→=(εiδ0−iδε000ε),(1)where ε represents the dielectric constant which is approximately 4 within the frequency regime of interest, and δ describes the magnetization-induced gyration of the material which is proportional to the external magnetic field strength. The governing equation of the MO PhC slab is given as [45] ∇×(∇×E)−ω2μ0ε→·E=0,(2)where ω is the angular frequency and μ0 is the magnetic permeability. By substituting the permittivity in Eq. (1) into Eq. (2), two characteristic solutions can be achieved (see more details in Appendix A): Ez=0,Ex=kx2−ω2μ0εkxky−iω2μ0δEy,β12=ω2μ0(ε2−δ2)ε,(3)Ez≠0,Ex=Ey=0,β22=ω2μ0ε.(4)The characteristic solutions in Eqs. (3) and (4) correspond to the transverse electric (TE) and transverse magnetic (TM) modes, respectively. The third term in Eq. (3) represents the propagation constant as a function of ω and δ, while the third one in Eq. (4) is independent of δ. It can be observed that the two in-plane components of the electric field (Ex and Ey) in the solution of TE modes are related by a function of δ, implying that only the polarization of the TE mode can respond to the external magnetic field. Thus, in this work we consider only the TE mode to investigate the magnetically tunable BICs.

The band structure of the MO PhC slab with zero external magnetic field (δ=0) is shown in Fig. 1(b), in which the lower band of interest is highlighted in red. The inset shows the field profile of the TE-like eigenstate (|Hz|) which exhibits a typical electric quadrupole (EQ) mode. Owing to the C2 and σz symmetries of the MO PhC slab, symmetry-protected BICs can be observed at the Γ point [1]. To confirm the existence of the symmetry-protected BICs, the far-field polarization indicated by the azimuthal angle (ψ) and ellipticity angle (χ) is extracted from the Stokes parameters [19] and shown in the lower panel of Fig. 1(c) (see more details in Appendix B). The symmetry-protected BICs at the Γ point are surrounded by far-field linear polarization indicated by short blue lines. The topological charge (q) of the symmetry-protected BICs can be calculated from the azimuthal angle of the far-field polarization [4]: q=12π∮Ldk‖·∇k‖ψ(k‖),(5)where ψ(k‖) represents the azimuthal angle between the long axis of polarization and the x axis, and L is a closed path (counterclockwise direction) enclosing the target point in the momentum space. Therefore, the topological charge of BICs can be obtained by winding the linear polarization around the Γ point (q=−1). We then calculate the Q factor of the eigenstates around the Γ point. As shown in Fig. 1(d), an ultrahigh-Q factor can be observed at the Γ point. Both the nonzero topological charge and ultrahigh-Q factor confirm the existence of the symmetry-protected BICs at the Γ point. Interestingly, when we continue to increase the external magnetic field strengths, we find that the far-field polarization around the symmetry-protected BICs can evolve gradually from linear (lower panel) to elliptical (middle panel) and eventually to circular polarization (upper panel), as shown in Fig. 1(c), indicating that magnetically tunable BICs with arbitrary polarization covering the entire Poincaré sphere can be achieved by tuning the external magnetic field strength.

We then investigate the influence of external magnetic fields on the working frequency and Q factor of the magnetically tunable BICs, which is crucial for practical applications. Figures 2(a) and 2(b) show the simulated band structures and Q factors of a square MO PhC slab with δ varying from 0 to 0.3, respectively. As the external magnetic field strength increases, the band structures experience a slight blue shift (<0.2%), while the high-Q factors remain almost unchanged near the Γ point. The frequency stability originates from the fact of δ2≪ε2 in Eq. (3); thus as δ increases, the frequency increment is tiny. In contrast, the stability of the high-Q factor at the Γ point is attributed to the symmetry-protected eigenstate (|Hz|). The eigenstate with δ=0.3 is shown as an inset of Fig. 2(b). When the TRS is broken by the external magnetic field, the MO PhC slab still preserves the C2 and σz symmetries that can guarantee the existence of the Γ-point BIC. Thus the eigenstate for δ=0.3 is an EQ, which is insensitive to the external magnetic field along the z direction. Moreover, this can be further explained by multipole expansions [46–49] illustrated in Figs. 2(c) and 2(d); the EQ moment dominates the scattered power even after introducing an external magnetic field (δ=0.3), confirming the stability of eigenstates. The dominant role of the EQ moment is further confirmed by the in-plane electric field distributions at the Γ point, as depicted in Fig. 2(e). The left panel corresponds to δ=0, while the right panel corresponds to δ=0.3. Both electric field distributions clearly illustrate the spatial features of the EQ moment. Although the EQ moment remains dominant in the multipole expansions, its components are influenced by the external magnetic field strengths. The evolution of each component of the EQ moment at (kx,ky)=(0.03π/a,0) is shown in Fig. 2(f), where all components are normalized by the EQ moment. It can be seen that as δ increases from 0 to 0.6, the diagonal components (EQxx,EQyy, and EQzz) increase gradually. These results demonstrate that the components of the EQ moment can be effectively tuned by the external magnetic field, thereby enabling the tunability of far-field polarization around the BICs (details of other points in the momentum space can be seen in Appendix C).

To quantitatively characterize the tunability of the external magnetic field to the symmetry-protected BICs, we investigate the far-field polarization around the BICs under different values of δ (external magnetic field strength) in Figs. 3(a)–3(d). As δ gradually increases from 0 to 0.3, the far-field polarization around BICs changes from linear to elliptical and finally to circular polarization. Figures 3(e)–3(h) show the far-field ellipticity under the corresponding external magnetic fields. As the external magnetic field strength increases, the ellipticity of the far-field polarization gradually evolves from 0 to 1, agreeing well with the far-field polarization shown in Figs. 3(a)–3(d) and further verifying the magnetically tunable BICs with arbitrary polarization covering the entire Poincaré sphere.

Next, we calculate the Stokes parameters around an iso-frequency contour with a radius of 0.03π/a [red dashed circle in Figs. 3(a)–3(d)] and map them onto the Poincaré sphere as red rings, as shown in Fig. 3(i). As δ increases, the red circles and rings become darker for clarity. For δ=0, the red ring is located at the equator of the Poincaré sphere, indicating linear polarization states. When δ increases from 0 to 0.3, the red ring gradually shrinks into a point close to the North Pole corresponding to RCP states. The red rings for δ=0.1 and δ=0.2 are located between the equator and the North Pole, corresponding to elliptically polarized states. The latitude of the red ring gradually varies from 0° to 90° on the Poincaré sphere with increasing magnetic field (the lower hemisphere can be covered by inverting the external magnetic field; see more details in Appendix D), indicating that the far-field polarization of the magnetically tunable BICs can fully cover the entire Poincaré sphere.

The arbitrary polarization tunability of the MO PhC slab can be explained by the MO effect. For instance, an incident wave with linear polarization can be converted to circular polarization emission and vice versa. The ellipticity of the far-field polarization can be derived [50,51]: ρ=tan(12arcsin(2Im(η)1+|η|2)),(6)where η=⟨Ey˜⟩/⟨Ex˜⟩, where ⟨Ex˜⟩ and ⟨Ey˜⟩ are the projection components of s(p)-polarization on the x-y plane.

The bracket indicates the time average of electric fields. According to Eq. (6), the evolution of ellipticity at the point of kx=0.03π/a and ky=0 [triangles in Figs. 3(a)–3(d)] is shown in Fig. 3(j). As δ increases from 0 to 0.3, the ellipticity ρ approximately evolves linearly from 0 to 1 with good agreement between the analytical (gray stars) and simulated (red line) results (see more details for the calculation of the far-field polarization in Appendix B). The excellent consistency between these two approaches indicates that the magnetically tunable arbitrary polarization indeed originates from the MO effect. However, as δ increases beyond 0.3, the far-field polarization around the BICs becomes elliptically polarized again, while the ellipticity ρ decreases as δ increases. The evolution of the far-field polarization and ellipticity at other points in the momentum space can be seen in Appendix D. Note that the existence of magnetically tunable BICs with arbitrary polarization is a universal phenomenon and can also be found in hexagonal MO PhCs consisting of dielectric cylinders (see more details in Appendix E).

In addition to the magnetically tunable arbitrary far-field polarization, the MO effect also can induce TRS breaking and result in intriguing chirality phenomena such as chiral emission. Figure 4(a) shows the circular polarization around the Γ point in the momentum space with δ=0.3. The reflection spectra (color map) of the MO PhC slab illuminated by RCP and LCP waves are shown in Figs. 4(b) and 4(c), respectively. Here the in-plane wavevector kx ranges from −0.05π/a to 0.05π/a and ky=0, corresponding to the red dashed line in Fig. 4(a). For both cases, it can be seen that the reflections are near zero at the Γ point because the eigenstates at the Γ point correspond to symmetry-protected BICs with an infinite Q factor and therefore are decoupled from the incident waves. When deviating from the Γ point, obvious reflection can be observed for the RCP incident wave in the frequency range from 152.5 to 153.5 THz, while the reflection is extremely low for the LCP incidence. The distinct reflection responses for incident waves with opposite circular polarizations demonstrate the unique off-Γ chiral emission of THz wave [red dashed circle in Fig. 4(d)] for the RCP and LCP incidence, respectively, which exhibit chiral emission in the whole iso-frequency contour. Therefore, the MO PhC slab can exhibit chiral emission over a wide range of incident angles rather than only at a specific incident angle, providing a richer degree of freedom for chiral emission. Note that the transmission spectra exhibit similar chiral emission phenomena (see more details in Appendix F).

To fully investigate the intriguing phenomena caused by the MO effect on the BICs, we further break the C2 symmetry of the MO PhC slab by replacing the circular air holes in Fig. 1(a), in which the isosceles triangular air holes have side length L2=1μm and bottom length L1=0.92μm, while keeping the other geometric parameters unchanged, as shown in Fig. 5(a). When no external magnetic field is applied, the far-field polarization around the Γ point is shown in Fig. 5(b), in which the V point (BICs at the Γ point) with an integer topological charge (q=−1) splits into a pair of C points with half-integer topological charge (q=−1/2), governed by the conservation of total topological charge. The chirality of the right-handed and left-handed elliptical polarization states is indicated by red and blue colors, respectively. Due to the σy symmetry of the configuration, these two C points are distributed symmetrically for the ky axis.

Once a nonzero external magnetic field is applied to the MO PhC slab along the z direction, the two C points will move along the direction of the positive kx axis. (See Appendix G for the complete evolution process.) When the external magnetic field δ equals 0.24, the C point with RCP (red dot) moves to the Γ point, indicating the emergence of the intrinsic chiral BICs, as shown in Fig. 5(c). Notably, the intrinsic chiral BICs at the Γ point with near-unity circular dichroism (CD) exceed 0.99 (see more details in Appendix H). Figure 5(d) shows the simulated azimuthal angles of the far-field polarization for the major axis, in which we can see that the total winding angle of the polarization is π along the counterclockwise loop enclosing the Γ point, indicating the topological charge of the polarization singularity at the Γ point is −1/2.

Note that previously reported intrinsic chiral BICs [27] break the C2 and σz symmetries simultaneously, while our design breaks the C2 and TRS symmetries simultaneously. Generally, when both the C2 and σz symmetries are broken, chiral emission can only be found on one side of the photonic structure, and the Q factor of the intrinsic chiral BICs is limited due to the lower structural symmetry. In stark contrast, here the intrinsic chiral BICs exhibit chiral characteristics on both sides of the PhC slab with an ultrahigh-Q factor up to 46,000 at the Γ point owing to the preserved σz symmetry, as shown in Fig. 5(e), significantly enhancing the light–matter interactions and providing new perspectives on the realization of chiral laser [26]. Figures 5(f) and 5(g) show the simulated reflection spectra for the LCP and RCP incidence with δ=0.24, respectively, from which we can see that there is a vanishing point (red dashed circle) in the reflection spectra at the Γ point for LCP incidence, whereas the RCP incident wave can excite the eigenstate at the Γ point, indicating the presence of a C point with right-handed chirality at the Γ point. Moreover, there exists an off-Γ vanishing point (blue dashed circle) in the reflection spectrum for the RCP incidence, corresponding to a C point with left-handed chirality at the off-Γ point, as shown in Fig. 5(g). The results in Figs. 5(f) and 5(g) show good agreement with the polarization distribution in Fig. 5(c), further confirming the existence of the intrinsic chiral BICs.

We then investigate the tunability of the chirality of the at-Γ intrinsic chiral BICs by the external magnetic field. As shown in Fig. 6(a), when δ=0, the far-field polarization around the Γ point is the same as that in Fig. 5(b). As shown in Fig. 6(b), when a downward external magnetic field with δ=−0.24 is applied, the C point with LCP incidence (blue dot) moves to the Γ point, indicating the emergence of an intrinsic chiral BIC. Additionally, Figs. 6(c) and 6(d) show the reflection spectra of the MO PhC slab with δ=−0.24 for RCP and LCP incidences, respectively. The vanishing points of the reflection spectra corresponding to a pair of C points are indicated by the blue and red dashed circles, showing chirality consistent with Fig. 6(b). Thus, it is demonstrated that the dynamically tunable chirality of the intrinsic chiral BICs can be realized by reversing the direction of the external magnetic field.

In conclusion, we have theoretically proposed magnetically tunable BICs with arbitrary polarization and intrinsic chirality in an MO PhC slab for the first time, to the best of our knowledge. By tuning the external magnetic field strength applied to the MO PhC slab, the far-field polarization around the BICs in momentum space can gradually evolve from linear to circular, effectively covering the entire Poincaré sphere. In addition, off-Γ chiral emission can be achieved in a broad frequency range and with multiple incident angles for RCP and LCP incidence. More interestingly, by breaking the C2 and TRS symmetries simultaneously, an intrinsic at-Γ chiral BIC exhibits chiral characteristics on both sides of the PhC slab with near-unity circular dichroism and a high-Q factor can be realized. Furthermore, the chirality of the at-Γ intrinsic chiral BIC can be inverted by flipping the external magnetic field bias. Our work paves the way for exploring the magnetically tunable topological photonics of BICs with TRS breaking.

Acknowledgment. Z. G. acknowledges funding from the National Natural Science Foundation of China, Guangdong Basic and Applied Basic Research Foundation, Shenzhen Science and Technology Innovation Commission, and High Level of Special Funds. Y. M acknowledges support from the National Natural Science Foundation of China, and Guangdong Basic and Applied Basic Research Foundation.