Engineering photonic band structures with desired energy-momentum dispersion is typically achieved by designing periodic structures^[1]. This engineering plays a crucial role in manipulating light and tailoring light–matter interactions within planar photonic structures. Such structures include metasurfaces^[2,3], photonic crystal slabs^[4,5], and gratings^[6], where radiation or scattering can be coupled to free space. Within the realm of energy band engineering, band degeneracy has garnered significant attention, as it often drives topological phenomena^[7]. Numerous band degeneracies, such as Dirac points^[8,9], quadratic degeneracies^[10,11], Weyl points^[12–14], nodal lines^[12], and nodal rings^[15], have been theoretically proposed and experimentally demonstrated in photonic systems. For instance, Dirac points, characterized by linear dispersion, exhibit intriguing physical phenomena in zero-refractive index materials^[9] and hold potential for single-mode laser applications^[16]. In a C4v lattice, a quadratic degeneracy, distinct from a Dirac point due to its quadratic (rather than linear) dispersion around the degeneracy, can arise at high symmetry points in the Brillouin zone, such as the Γ point^[11] or M point^[10]. This quadratic degeneracy is symmetry-protected, meaning it persists regardless of unit cell or material specifics as long as the C4v lattice symmetry remains intact^[10]. Notably, this symmetry-protected quadratic degeneracy in the C4v lattice can transition into Dirac points through additional symmetry breaking^[16]. Beyond the topological transport phenomena associated with band degeneracy, the radiative polarization state also exhibits fascinating polarization singularities at band degeneracies in planar photonic systems. These singularities have been theoretically and experimentally demonstrated^[17–19]. For example, in a C4v photonic crystal slab, symmetry constraints dictate that the degenerate radiative bands at the Γ point share a similar topological structure with the V points but without null intensity. The transition from this quadratic degeneracy at the Γ point, driven by a pair of radiative modes splitting into a pair of Dirac points along a mirror direction, has been demonstrated by breaking the C4v symmetry down to C2v symmetry^[18]. As long as the C4v symmetry is maintained, the lowest two radiative modes remain degenerate, which is a testament to the symmetry protection of this degeneracy. Consequently, symmetry breaking becomes critical for realizing such transitions.

In addition to the polarization singularity that is observed at the band degeneracy point, polarization singularities have also been identified at bound states in the continuums (BICs) across various systems, including but not limited to planar photonic structures^[20–27]. BICs, characterized as localized eigenstates embedded within the radiation continuum, have garnered significant interest for their ability to enhance light–matter interactions in diverse applications, such as lasers^[28–30], sensors^[31,32], and nonlinear photonics^[33,34]. Typically exhibiting singlet behavior, BICs possess inherent polarization singularity properties that have been exploited for generating vector beams^[35,36]. The radiative polarization distribution associated with the polarization singularity in the momentum space can be retrieved by the angle-resolved transmission/reflection spectra^{[18,22,31,35,36]}. Furthermore, the principle of topological charge conservation has been instrumental in understanding the evolution of these singularities, including the annihilation of oppositely charged BICs and the generation of circularly polarized light points (C points) with opposite handedness from a single BIC^[20,37–42]. Research has explored merging BICs on the same band to enhance the quality factors (Q factor) of quasi-BICs^[37,40–42]. Additionally, the generation of C points from a BIC reinforces the conservation of topological charge for both types of polarization singularities, expanding the possibilities for polarization control in the momentum space^[38,43–45]. Despite these separate advancements focused on BICs or band degeneracy points, a thorough investigation into the evolution and interplay of polarization singularities between BICs and band degeneracy points, guided by the law of topological charge conservation, is still lacking. Understanding this relationship is crucial for advancing the field and unlocking new possibilities for manipulating light in the momentum space.

In this Letter, first, we theoretically investigate the emergence of accidental quadratic degeneracy between a pair of transverse electric (TE)-like BICs within a C4v symmetry lattice by precisely tuning the geometry parameters. By transitioning from quadratic to linear degeneracy, we demonstrate the simultaneous generation of two distinct polarization singularities: off-Γ accidental BICs and Dirac-type band degeneracy points. Importantly, our approach diverges from previous methods that relied on symmetry breaking to achieve additional polarization singularities. Instead, we achieve this generation by continuously adjusting a single geometric parameter while preserving the overall structural symmetry. Our findings reveal that Dirac points, characterized by half-integer topological charges associated with radiative polarization, can be continuously tuned along each Γ-M direction, and off-Γ BICs, possessing integer topological charges for radiative polarization, can be synchronously moved along each Γ-X direction. This simultaneous generation adheres to the principle of topological charge conservation within the photonic band^[44,46].

The significance of this work lies in the finding that the emergence of Dirac points is driven by the accidental degeneracy of a pair of TE-like BICs and can be manipulated by precisely tuning geometric parameters without requiring additional symmetry breaking. This stands in stark contrast to previous studies, where symmetry breaking was essential^[18]. The other significance of this work is that we find two types of polarization singularities, including off-Γ accidental BICs and Dirac-type band degeneracy, can be synthesized in the same system. Physically, this conjoint generation can be understood by the conservation of topological charge, which highlights their inherent robustness and stability. To the best of our knowledge, our work presents an unprecedented framework for synthesizing and controlling complex polarization states by strategically combining polarization singularities associated with BICs and band degeneracies in planar photonic devices. The potential application may be found in optical sensors, vector light field lasers, and polarization control of light. The principles outlined here extend beyond photonic crystal slabs and hold potential applications across diverse wave systems^[47–49].

As the first step, to verify the quadratic degeneracy from a pair of symmetry-protected BICs, we consider a freestanding photonic crystal silicon slab formed by a square lattice geometry, as schematically shown in Fig. 1(a). The periodicity of the square lattice is d=450μm, the thickness of the slab is h/d=1/3, and we assume that the permittivity of the silicon we assume is 11.9. Four same rectangular holes with length a/d=0.4 and width w/d=8/45 are arranged in each unit cell. In addition to these constant parameters, the varied distance between the two rectangular holes along each axis direction is px(y)/d=a/d+w/d+gtx(y)/d, and we set gtx=gty=gt. Each rectangular hole can be moved with the value of wx(y) along the direction of the long side of each rectangular hole. We can find that the varied distance gt does not affect the existing symmetry of the lattice, while wx(y) significantly affects the symmetry of this photonic crystal silicon slab without changing the average refractive index of the slab. Here, we consider the simple case wx=wy=0, as schematically shown in Fig. 1(a), which possesses a C4v symmetry. Symmetry-protected BICs at the Γ point can be found in this kind of 2D lattice possessing the C4v symmetry, such as photonic crystal slabs^[20,50–52] and metasurfaces^[53]. Based on the group theory, the symmetry-protected BICs at Γ point are the eigenmodes belonging to A1, A2, B1, and B2 representations of the C4v group, which usually occur at isolated points without degeneracy^[1].

The position of these BICs is usually sensitive to the geometry of the slab structure, and it is possible to achieve accidental degeneracy of two (or even more) BICs. The photonic band structures are numerically calculated by the guided-mode expansion (GME) method^[54]. When we continuously tune gt, the frequencies of the two involved BICs at the Γ point are numerically plotted in Fig. 1(b). We can find that there is a band degeneracy point accompanied by a band inversion around gt/d≃0.158, where the magnetic field distributions at the Γ point clearly show that the two involved BICs are TE-like BICs belonging to the A1 and B1 representations, as shown in Figs. 1(c) and 1(d). When the varied distance is above gt/d≃0.158, such as gt/d≃0.178, the two symmetry-protected TE-like BICs at the Γ point split and no band degeneracy occurs in the investigated momentum region, as displayed in Fig. 1(e). As we tune the varied distance between the two rectangular holes to gt/d≃0.158, as shown in Fig. 1(f), we can clearly generate a degenerate pair of the two symmetry-protected TE-like BICs at the Γ point. While the varied distance is below gt/d≃0.158, such as gt/d≃0.133, the two symmetry-protected TE-like BICs with infinite Q factors occur at different frequency positions at the Γ point, but a new degeneracy occurs at a position along the Γ-M direction, which is a type-II Dirac point as shown in Fig. 1(g).

More importantly, although the degeneracy of the two BICs at the Γ point is accidental and is obtained by finely tuning the geometric parameters, the dispersion in the vicinity of this degeneracy should be quadratic due to the fact that the lattice is guaranteed by the existing symmetry of C2 in the C4v group^[11], and this quadratic degeneracy can be continuously split into four Dirac points with linear dispersions along the Γ-M direction by only tuning the varied distance gt. As shown in Fig. 2(a), the eigenfrequency difference driven by the two BICs defined by log|(f+−f−)/(f++f−)| along the Γ-M direction clearly shows that the transition from the Dirac points to the quadratic degeneracies is continuous, where f+,− is the eigenfrequency of the leaky resonance. We should emphasize that this transition from a quadratic degeneracy into Dirac points does not require any symmetry breaking because the controlling parameter gt does not affect the symmetry of the lattice. To investigate the details of the dispersion behavior around the degeneracy, the system around the Γ point can be approximately described by the Hamiltonian in the form of H=[f0+a(kx2+ky2)]I+b(kx2−ky2)σz+ckxkyσx,(1)and the corresponding eigenfrequencies are^[11]f±=f0−a(kx2+ky2)±b2(kx2−ky2)2+c2kx2ky2.(2)

The fitted polynomial function along the Γ-X direction is f=0.427–1.9kx2 for the B1 type BIC and f=0.427+0.15kx2 for the A1 type BIC, where the eigenfrequency is in units of c/d, and the comment of the wave vector kx,y is in units of 2π/d. The fitted polynomial function along the Γ-M direction is f=0.427–1.4(kx2+ky2) for the B1 type BIC and f=0.427–0.45(kx2+ky2) for the A1 type BIC. The polynomial fitting with order 2 finds good agreement with the obtained numerical results around the Γ point, as shown in Fig. 2(b), which consolidates this quadratic degeneracy. As we tune the controlling parameter gt around the degeneracy at the Γ point, the frequency positions of the two symmetry-protected TE-like BICs can be controlled and approximately fitted along the Γ-M direction by f+=0.427−0.290(gt/d−0.158)−1.4(kx2+ky2) for the B1 type BIC and f−=0.427−0.133(gt/d−0.158)−0.45(kx2+ky2) for the A1 type BIC. Due to different band curvatures along the Γ-M direction and different frequency evolutions at the Γ point for the two involved bands, we find that the degeneracy can be continuously obtained by f−=f+ with kxc2=kyc2=0.157(0.158−gt/d)/1.9 in the two-dimensional parameter space, including the momentum along the Γ-M direction and gt, which finds good agreement with the numerical prediction, as shown by the dashed line in Fig. 2(a). In the neighborhood of (kxc,kyc), the expanded expressions of f+ and f− are f+=0.436−0.059gt/d−1.610(0.158−gt/d)/2(kx2+ky2−kxc2+kyc2) and f−=0.436−0.059gt/d−0.517(0.158−gt/d)/2(kx2+ky2−kxc2+kyc2). We can see the leading term is linear, which implies that the degeneracy along the Γ-M direction is linear. For a fixed parameter possessing linear degeneracy along the Γ-M direction, such as gt/d≃0.133, we find liner fitting finds good agreement with the obtained numerical results with f+≈0.428−0.18(kx2+ky2−0.0452+0.0452) and f−≈0.428−0.06(kx2+ky2−0.0452+0.0452), as shown in Fig. 2(c).

In addition to the exotic dispersion transition behaviors of degeneracies driven by the two TE-like symmetry-protected BICs, we also find that a BIC can be additionally generated for the higher band along the Γ-X direction accompanied by the appearance of the linear degenerated Dirac point, as illustrated in Fig. 2. The Q factors for the two involved bands, as displayed in Figs. 2(d) and 2(e), clearly indicate that the symmetry-protected BICs at Γ point always exist as we change gt, while an additional BIC, i.e., the off-Γ BIC can be continuously found for the higher band accompanied by the generation of the Dirac point. For example, the Q factors for the two bands with the fixed gt/d≃0.133 are shown in Fig. 2(f), the additional divergent Q factor can be found around kx≈0.125(2π/d) beside the Γ point for the higher band, which implies that the two BICs can be found for the higher band along the Γ-X direction, while the Q factor for the lower band gradually decreases as kx increases beside the divergent Q factor at the Γ point. It is worth noting that the generation of the off-Γ BIC can be usually obtained by splitting the at-Γ BIC^[44] or finely tuning structural parameters^[21], while the generation of the off-Γ BIC discussed here is different and continuous along the Γ-X direction accompanied by the appearance of the linear degenerated Dirac point along the Γ-M direction. Thus, our finding provides a systematic approach to generating the off-Γ BIC, and it is necessary to understand the conjoint generation of the off-Γ BIC and the Dirac point.

For a better understanding, we analyze the distribution of radiative polarization on each band. As we know, the degeneracy also implies some interesting polarization properties in addition to the unique topological radiative polarization characteristic of the BIC in the momentum space. For example, a quadratic degeneracy leads to the involved polarization singularities with an integer charge, and a Dirac point of linear degeneracy leads to the involved polarization singularities with a half-integer charge^[18]. When the degeneracy does not occur, such as in the case of gt/d≃0.178, two TE-like BICs occur at different frequencies at the Γ point. The radiative Stokes parameter, −S1/S0=(|dTE|2−|dTM|2)/(|dTE|2+|dTM|2),(3)can be utilized to indicate the polarization topological characteristic in Fig. 3, where dTE,TM indicates the radiation coefficient of the mode in the TE(TM) polarization channel. The radiative Stokes parameter −S1/S0 around the B1 type BIC clearly shows different behavior along the two symmetric directions, as shown in Fig. 3(a), which implies the leaky mode originated from the B1 type BIC can only couple to the TE-incidence light along the Γ-X direction and to the TM-incidence light along Γ-M direction. For the leaky resonant mode on the higher band around the A1 type BIC, the radiative Stokes parameter −S1/S0 clearly shows isotropic behavior in the vicinity of Γ point, as shown in Fig. 3(b), which definitely shows this leaky resonant mode originated from it could only couple to the TE incidence light. To further indicate the topological character of the polarization singularity of the explored BIC, we adopt the phase of ψ=0.5arg[S1/S0+iS2/S0],(4)indicating the orientation angle of the polarization ellipse and the corresponding vector (cos[ψ],−sin[ψ]), where the normalized radiative Stokes parameter S2/S0 is defined by^[44]S2/S0=2Re[dTM*dTE]/(|dTE|2+|dTM|2). The polarization vector distributions, as displayed in Figs. 3(c) and 3(d), clearly indicate polarization singularity at Γ. The topological charge defined by the winding number of the closed loop C in the k∥ space around the polarization singularity^[44]q=12π∮Cdk∥·∇ψ,(5)clearly indicate that the topological charge of the B1 BIC is −1 for the two different closed loops, as shown in Figs. 4(a) and 4(b), and the topological charge of the A1 type BIC is 1 for the two different closed loops, as shown in Figs. 4(c) and 4(d). The closed loops we choose are circles with different radii kr=0.02(2π/d) and 0.18(2π/d), which are schematically shown in the illustration of Fig. 4. The same topological charge for different closed loops on each band implies that the polarization evolution is continuous, and there is no additional polarization singularity in the region between the two closed loops.

When Dirac points occur along each Γ-M direction, such as the case of gt/d≃0.133 where A1 type and B1 type BICs occur at different frequencies at the Γ point, the Dirac points, which are leaky resonant modes, also illustrate some topological characteristics for the radiative polarization. Here, the appearance of a Dirac point is synthesized by a pair of symmetry-protected TE-like BICs, which still hold their topological singularity. The two kinds of polarization singularities coexisting in the involved two BICs would enrich the radiative polarization properties with more feasibility when including Dirac points and BICs. Because both Dirac points and BICs are topological quantities, their simultaneous presentation brings an interesting conservation rule on the topological charge.

As shown in Fig. 5, the radiative Stokes parameter −S1/S0 clearly shows different behaviors around and away from the Γ point. Close to the Γ point, the radiative Stokes parameter −S1/S0 clearly shows isotropic behavior, while illustrating different coupling behavior along the two symmetric directions as the momentum is far away from the Γ point, as shown in Fig. 5(a). The transition point along the Γ-M direction is around (0.045,0.045)2π/d, which coincides with the position of the Dirac point. The vector profile of ψ clearly shows that the polarization singularity still holds at Γ point due to the appearance of the A1 type BIC. The polarization vector direction shows a 90° rotation around the Dirac point and the polarization distribution is similar to the B1 type BIC when the momentum is far away from the Γ point, as shown in Fig. 5(c). The different polarization distributions imply different topological characteristics. For the higher band, the radiative Stokes parameter −S1/S0 clearly shows different coupling behaviors along the two symmetric directions in the vicinity of the Γ point, while illustrating isotropic behaviors along the two symmetric directions when far away from the Γ point, as shown in Fig. 5(b). The transition point along the Γ-M direction is also around (0.045,0.045)2π/d, which coincides with the Dirac point. In addition to the polarization singularity at the Γ point due to the B1 type BIC, the 90° rotation around the Dirac point and the polarization distribution is similar to the A1 type BIC when the momentum is far away from the Γ point. More importantly, the additional polarization singularity can be found due to the generated off-Γ BIC around (0.125,0)2π/d, as displayed in Fig. 5(d), consolidated by the interaction between −S1/S0=−1 with dTE=0 and −S1/S0=1 with dTM=0 in Fig. 5(b). To further indicate the topological characteristic of the polarization distribution, we choose different loops, including these polarization singularities, to obtain each topological charge on each band, as schematically shown in the illustration of Fig. 6.

The closed loops we choose are circles with different radii kr=0.02(2π/d) around the symmetry-protected BIC at the Γ point, the off-Γ BIC, and the Dirac point and 0.18(2π/d) around Γ point. For the lower band, the topological charge around the symmetry-protected BIC at the Γ point is 1, as shown in Fig. 6(a). The topological charge around the Dirac point is −1/2, as displayed in Fig. 6(b), and the topological charge around (0.125,0)2π/d is 0, as illustrated in Fig. 6(c), which implies no polarization singularity at this position on the lower band. The topological charge for the outer ring, including both the symmetry-protected BIC and the Dirac points is −1, as shown in Fig. 6(d), which is equal to the sum of all topological charges in the inner region of the outer ring, qouterring=∑iqi=qΓBIC+4qDirac, with detailed form −1=1+4×(−1/2).

The identity can be explained and understood by the conservation of the topological charge. For the higher band, the topological charge around the symmetry-protected BIC at the Γ point is −1, as shown in Fig. 6(e); the topological charge around the Dirac point is −1/2, as displayed in Fig. 6(f); and the topological charge around the off-Γ BIC is 1, as illustrated in Fig. 6(g), which implies a polarization singularity at this position. The total topological charge for the outer ring, including the symmetry-protected BIC, the Dirac points, and the off-Γ BICs is 1, as shown in Fig. 6(h), which is also equal to the sum of all topological charges in the inner region of the outer ring, qouterring=∑iqi=qΓBIC+4qDirac+4qoff−ΓBIC, with detailed expression 1=−1+4×(−1/2)+4×1. The identity can be also verified by the topological charge conservation law, which clearly demonstrates the generality of the topological charge conservation even including different kinds of polarization singularities. In short, analysis of the polarization distribution for different parameters confirms that the deterministic generation of both the off-Γ BICs and the Dirac points is governed by, and can be explained through, the conservation of topological charge. This insight offers a promising new approach for the controlled synthesis of a wide range of polarization singularities.

In conclusion, we have numerically demonstrated the controlled tuning of dispersion in a silicon photonic crystal slab. By adjusting the geometry of a C4v square lattice, we achieved a transition from quadratic to linear degeneracy involving a pair of TE-like BICs. Importantly, this transition was achieved without breaking the system’s symmetry. This continuous dispersion transition resulted in the deterministic generation of two additional types of polarization singularities, off-Γ BICs, and Dirac-type band degeneracy points, coexisting with the initial symmetry-protected BICs at the Γ points. These findings provide clear validation for the principle of topological charge conservation. Our work establishes that synthesizing degeneracies involving BICs can significantly enrich the radiative polarization functionalities of these states. This opens up new avenues for BIC-based applications across diverse fields. Furthermore, the framework presented here to synthesize and control complex polarization states extends beyond photonic crystal slabs and holds significant promise for other wave systems.