Topological photonics represents a rapidly evolving research frontier in optical science, extending fundamental principles of topological phenomena initially established in condensed matter systems to photonics.1^–3 In recent years, three-dimensional (3D) photonic topological phases have attracted widespread interest due to their inherent resilience to disorders or imperfections, paving the way for groundbreaking advancements in device applications.4^–6 In general, 3D photonic topological phases are broadly categorized into two types, gapped and gapless cases. The 3D gapped photonic topological phases are characterized by a complete 3D energy gap between the conduction and valence bands, such as 3D topological insulators7^–9 and axion insulators.10^,11 On the other hand, the 3D gapless photonic topological phases, where the band gap is absent, are characterized by the existence of multitype symmetry-protected 3D band degeneracies, such as Dirac points,12^–14 triple degenerate points,15^–17 and Weyl points.18^–20 These 3D band degeneracies play an important role in topological band theory, serving as the origin for multiple photonic gapped topological insulators and gapless topological semimetals. A prominent example is the photonic Weyl point, which serves as the physical origin of Fermi arc surface states and chiral zero modes. In particular, in Weyl semimetals, each Weyl point is associated with a quantized topological charge, functioning as either a source or sink of Berry curvature.21 Due to the inherent topological constraints, Weyl points with opposite Chern numbers are always generated or annihilated in pairs, ensuring the conservation of topological charge within the system.22^–25

In photonic systems, the propagation characteristics of light within bulk media can be effectively represented by the 3D equifrequency surface, which serves as an optical analog to the Fermi surface in electronic band structures of crystalline solids.26 This correspondence provides a fundamental framework for understanding wave dynamics and dispersion properties in complex photonic materials.27 Within the phase space of electromagnetic waves at a fixed frequency, the property of 3D equifrequency surfaces reflects distinct topological characteristics.27^–29 Typically, these equifrequency surfaces remain fundamentally different and cannot undergo a continuous transformation into one another unless a topological transition takes place. Unlike the conventional spherical and ellipsoidal equifrequency surfaces typically observed in natural materials, photonic metamaterials enable the engineering of singular equifrequency surfaces with tailored geometries, such as type-I and type-II hyperbolic equifrequency surfaces.30^–32 This capability allows for unprecedented control over wave propagation and dispersion characteristics, expanding the potential for novel optical phenomena and applications. Recently, in 3D topological photonics, the topological transitions of multitype 3D equifrequency surfaces are reported separately for the biaxial gyroelectromagnetic medium,30 magnetized plasma,31 and gyromagnetic metamaterial.32 However, the realization of these magnetic systems that lack time-reversal symmetry usually relies on magneto-optical effects induced by a strong static magnetic field, posing significant challenges for practical experimental implementation.

Unlike homogeneous materials that require broken time-reversal symmetry to achieve magneto-optical effects, chiral metamaterials can exhibit distinctive electromagnetic properties without relying on an applied magnetic field.33 On the other hand, the magneto-optical effects of magnetic metamaterials are usually weak at the optical frequency regime. On the contrary, the chiral metamaterials retain time-reversal symmetry while lacking spatial inversion symmetry, a consequence of the intrinsic magnetoelectric coupling between their magnetic and electric field components.34^–36 Specifically, magnetoelectric effects can be found in various natural materials, and advancements in metamaterial engineering have enabled the fabrication of strong chiral media with enhanced electromagnetic responses. Thus, the chiral metamaterials could allow the exploration of the characteristics of 3D photonic topological phases within a frequency range. Moreover, to achieve strong chiral coupling, a common approach is the design of chiral metamaterials featuring unit cells constructed from metallic helical structures.37^–40 In recent years, chiral photonic metamaterials have been widely studied in 3D photonic topological phases, for example, photonic nodal surface,41 topological triple degeneracy point,42 Fermi arc surface states,43 and Weyl points (or exceptional contours).44 However, in chiral metamaterials, lots of recent works37^,42^,43^,45 mainly focused on the single-type equifrequency surfaces and topological surface modes at a specific angular frequency ω. Hence, two fundamental research questions arise. Can multiple topological transitions and multitype 3D equifrequency surfaces exist in chiral metamaterials? If so, what unique physical manifestations might arise as a result?

In this work, we study the multitype 3D equifrequency surface and multiple topological transitions in chiral photonic metamaterials. The dual-frequency and direction-dependent photonic Weyl points appear when the electrical/magnetic dispersions of the chiral metamaterials are simultaneously considered. Utilizing the framework of effective medium theory combined with topological band theory, we conduct a systematic investigation and establish complete topological phase diagrams corresponding to various 3D equifrequency surfaces in chiral media. Our theoretical analysis further reveals that both the resonant frequency ω0 and dual-frequency Weyl points serve as fundamental determinants governing these topological transition processes. Remarkably, when the vacuum state interacts with either phase I or phase III chiral metamaterials of the topological diagram, highly localized topological Fermi arc surface states arise, and their transmission directions exhibit strong dependence on both frequency and chirality. Our analysis demonstrates that the parameter ω serves as a degree of freedom, allowing precise control over the bandwidth of these topological Fermi arc surface states. Furthermore, by leveraging the chirality-dependent Fermi arc surface states, various forms of multichannel and directional topological photonic routings are realized. Through theoretical analysis, we clarify that the underlying physical mechanism enabling these multichannel topological photonic routings originates from the distinct interfacial characteristics among nonidentity material phases.

This paper is organized as follows. In Sec. 2, the existence of dual-frequency band photonic Weyl points in chiral metamaterials is studied. In Sec. 3, comprehensive topological phase diagrams and multiple topological transitions are investigated. In Sec. 4, multitype equifrequency lines and frequency chirality–dependent surface waves are studied. The physical origin and bandwidth tunability of photonic Fermi arc surface states are analyzed in Sec. 5. Multichannel and directional topological photonic routings are shown in Sec. 6. Finally, the conclusions are presented in Sec. 7.

Recently, based on effective medium and topological band theories, the 3D topological band structures,36 non-Hermitian topological degeneracies,16 copropagating photonic Fermi arc,45 and triple degenerate points42 in chiral metamaterials have been widely studied. Here, we consider a type of anisotropic chiral photonic metamaterials, which with the relative permittivity, permeability, and chirality tensors can be described as ε=diag(εx,εy,εz),μ=diag(μx,μy,μz),γ=diag(0,γ,0),(1)where εy=εt+α/(ω02−ω2), εz=1−ωp2/ω2, μy=1+αω2/(ω02−ω2), and γ=αω/(ω02−ω2), respectively. In particular, ω represents the angular frequency, ωp represents the plasma frequency, and ω0 is the resonance frequency, respectively. The permittivity, permeability, and chirality formulas we have chosen in Eq. (1) to realize the noncoaxial low-frequency (ky-axis) and high-frequency (kz-axis) Weyl points [see Fig. 1(a)], which is a prerequisite for studying the multitype 3D equifrequency surface and multiple topological transition processes. Notably, the permittivity/permeability/chirality tensors choice in Eq. (1) is merely a prototype of the existence of such noncoaxial low-frequency and high-frequency Weyl points, not a constraint condition. Particularly, Yang et al.35, using a realistic layer structure, have obtained the specific electromagnetic tensors of the chiral metamaterials with noncoaxial low-frequency and high-frequency Weyl points. Moreover, the electromagnetic response of chiral metamaterials, as we discussed in Eq. (1), physically belongs to the same type as those in Yang et al.’s work, with only differences in the dispersion directions of chiral/permittivity/permeability. For such anisotropic chiral photonic metamaterials, the metallic helical element is a good candidate to generate the magnetoelectric coupling effect in the x−y plane, and the metal wire is used to produce the z-direction plasma mode. Following the idea of Ref. 35, regarding the aspect of structural design, we show a possible structural schematic (as a prototype) of such a chiral metamaterial supporting dual-frequency noncoaxial Weyl points (see more details of such chiral photonic metamaterial design in the Supplementary Material Note 1). This idea and strategy have been mentioned in other recent works.16^,45

The constitutive relations of the chiral metamaterials can be given by D=ε0·ε·E+iγ·H/c,B=μ0·μ·H−iγ·E/c,(2)where ε0, c, and μ0 represent the permittivity, the speed of light, and the permeability within the vacuum state, respectively. Then, combining ∇×E=iωB and ∇×H=−iωD, the Maxwell equations of the chiral metamaterials can be rewritten to the following 6-by-6 matrix form: A·(EH)=(ω·εκ+iω·γ−iω·γ−κω·μ)·(EH)=0,(3)where κ=[0,−kz,ky;kz,0,−kx;−ky,kx,0] is the antisymmetric tensor. For simplicity, the angular frequency ω and resonance frequencies ω0 are normalized plasma frequency ωp. The wave vector k is normalized to kp(kp=ωp/c). Moreover, we set ε0=c=μ0=1 in the present analysis to simplify the derivation.

Based on Det[A]=0 in Eq. (3), the characteristic equation of the chiral photonic metamaterials can be deduced analytically ky4a1+εz(a2a3−γ2ω4εxμx)μz+ky2[εz(kz2μy−ω2μxa1)+(kz2εy−ω2εxa1)μz]=0,(4)where a1=(εyμy−γ2), a2=(kz2−ω2εyμx), and a3=(kz2−ω2εxμy). Based on Eq. (4), we give the 3D band structures, dual-frequency photonic Weyl points, and nodal lines of the chiral metamaterials, as illustrated in Fig. 1(a). Different from the traditional single-frequency Weyl point in photonic metamaterials, these Weyl points with opposite topological charges are distributed in two different frequency planes, as shown by the gray planes in Fig. 1(a). They correspond to critical points of multiple topological transitions (as we will discuss later).

In Fig. 1(a), the dual-frequency Weyl points and nodal lines within the ky−kz plane (kx=0) originate from the degeneracies of the 3D orange/blue bands and blue/cyan bands, which correspond to three different eigenvalues E2, E3, and E4, respectively. In particular, the two-dimensional (2D) density distributions of |E3-E2| and |E4-E3| are shown in Figs. 1(b) and 1(c). Here, the minimum points observed in the density plots correspond to the dual-frequency Weyl points and nodal lines, respectively. Moreover, in contrast to the photonic nodal lines, which exhibit one-dimensional (1D) line degeneracies [see Fig. 1(c)] in 3D space, these Weyl points in Fig. 1(b) are all isolated zero-dimensional (0D) points.

Here, to demonstrate the existence of dual-frequency photonic Weyl points in the chiral metamaterials, based on electromagnetic tensors and constitutive relations, we specifically calculated the expressions of the longitudinal modes (LMs) and transverse modes (TMs) of the 0D photonic Weyl points [see Fig. 1(a)] along the y- and z-directions, respectively. In particular, based on Eq. (4), along the y-axis, i.e., kx=kz=0, the LM and TM of the low-frequency Weyl points [see the blue dots in Fig. 1(a)] can be given as γ2−εyμy=0,ky2−ω2εxμz=0.(5)

Then, based on Eqs. (1) and (5), the positions ky and ω1 of the low-frequency Weyl point are ky=ω1=−(α+ω02εt)/(α−1)εt.(6)

Similarly, along the z-axis (kx=ky=0), the LM and TM of the high-frequency photonic Weyl points in Fig. 1(a) can be further given as 1−ωp2/ω2=0,a2a3−γ2ω4εxμx=0.(7)

Based on Eq. (7), the positions kz and ω2 of the high-frequency Weyl points are kz=ω2(εyμx+εxμy+εy2μx2+εx2μy2+2εxμx(2γ2−εyμy))/2,ω2=ωp.(8)

Based on Eqs. (6) and (8), to show a clearer representation of the dual-frequency photonic Weyl point crossings among 2D branches at various kz and ky values, we give the dispersion relations ω(ky) and ω(kz) in Figs. 1(d) and 1(e), respectively. These results reveal that the 2D branch crossings associated with the low-frequency and high-frequency Weyl points occur only when kz=0 and ky=0, respectively, as indicated by the green and orange lines (see more details of the low-frequency Weyl points in the Supplementary Material Note 2). Notably, the positions of the minimum points shown in the 2D density plots in Fig. 1(b) exhibit exact correspondences with the analytic calculation results in Figs. 1(d) and 1(e).

Then, using Eqs. (2) and (3), the wave transmission of the magnetic field H=(HxHyHz)T in the chiral photonic metamaterials in Eq. (1) can be described as [(κ−iω·γ)·(ω·ε)−1·(κ−iω·γ)+ω·μ]H=0.(9)

By utilizing Eqs. (1) and (9), the analytical expression for the specific eigen magnetic field in chiral photonic metamaterials can be derived. Here, the components of Hx(y,z) in Eq. (9) take the following forms: Hx=γkxky2kz−iωky3a1−γω2kxkzεxμz−iky[ωkx2εxμy+ω(kz2εy−ω2εxa1)μz],(10)Hy=γky3kz+iωkxky2εyμx−γω2kykzεxμz−iωkxεx(−kx2μx−a2μz),(11)Hz=γky2kz2+γω2kx2εxμx+iωkxkykz(εyμx−εxμy).(12)

On the other hand, utilizing Eqs. (3) and (10)–(12), the component expressions for the electric field Ex(y,z) can be determined using the following equation: (ExEyEz)T=−(ω·ε)−1·(κ−iω·γ)·(HxHyHz)T.(13)

In the chiral metamaterials, by calculating the nonzero Berry phases γ, we demonstrate the nontrivial topological properties and topological charges of the dual-frequency photonic Weyl points in Fig. 1(a). Particularly, based on Eqs. (1) and (10)–(13), under certain angular frequency ω (near the Weyl point frequencies ω1 and ω2), by the surface integral of Berry curvatures Ω(k) at equifrequency surfaces in momentum space, the Berry fluxes can be numerically solved γ=∬Ω(k)·ds=∬∇k×⟨U(k)|i∇k|U(k)⟩·ds,(14)where U(k)=[E,H]T represents the eigen electric/magnetic field modes of the chiral photonic metamaterials. In Figs. 1(f) and 1(g), for the 2D equifrequency lines, the nonzero Berry curvatures are all predominantly distributed near the dual-frequency Weyl points. As the wave vectors approach infinity, these calculations indicate that the Berry curvatures diminish to negligible values. As the wave vectors tend to infinity, the results reveal that the Berry curvatures gradually decrease to negligible values. Moreover, based on Eq. (14), by integrating the nonzero Berry curvatures over the 3D equifrequency surfaces of the chiral metamaterials, the values of −1 [see Fig. 1(f)] and +1 [see Fig. 1(g)] are obtained, corresponding to the negative and positive chirality Weyl points in Fig. 1(a). Notably, the positive and negative topological charge Weyl points are the critical transition points of the multitype 3D equifrequency surfaces and multiple topological transitions (as we will discuss later).

In general, multifarious topology forms of 3D equifrequency surfaces stem from the different relations among the electromagnetic tensor components of the photonic metamaterials. Therefore, varying the angular frequency ω can essentially induce topological transitions. In the chiral photonic metamaterials, we give the dispersion properties varying with angular frequency of the electromagnetic parameters εy, εz, μy, and γ in Eq. (1), as shown in Fig. 2(a). Specifically, based on the diverse electromagnetic parameter combinations and their positive/negative responses, the comprehensive topological phase diagram can be divided into four different regions, i.e., Roman numerals I–IV. In the phase I region, εy<0, εz<0, μy>0, and γ>0; in the phase II region, |εy|>0, εz<0, |μy|>0, and γ<0; in the phase III region, εy>0, εz<0, μy>0, and γ<0; and εy>0, εz>0, μy>0, and γ<0 in the phase IV region. Thus, the resonance frequency ω0 and plasma ωp frequency are the critical points of sign reversal of the parameters γ (μy) are εz, respectively. On the other hand, the other parameters εx, μx, and μz in Eq. (1) do not change throughout the whole topological phase diagram [see Fig. 2(a)] because the dispersion responses are not considered.

Now, we study multitype 3D equifrequency surfaces and multiple topological transitions in chiral metamaterials based on the comprehensive topological diagram in Fig. 2(a). When the angular frequency ω approaches 0, the parameters εy and εz in Eq. (1) approach infinity. Here, the radii of the 3D equifrequency surfaces (green planes) of the chiral metamaterials are extremely large. Thus, in 3D momentum space, the two separate equifrequency surfaces are all flat planes, as shown in Fig. 2(b). In the phase I region [see Fig. 2(a)], due to the introduction of magnetoelectric coupling effect (γ is nonzero), there is mixed-type dispersion containing an ellipsoid (pink) and twofold type-I hyperboloid (orange) 3D equifrequency surfaces along the z-direction of the chiral metamaterials, as illustrated in Fig. 2(c) (see more details of 3D equifrequency surfaces under different angular frequencies ω in phase I region in the Supplementary Material Note 3). When the angular frequency ω=ω0 is at the critical point between phases I and II, a twofold type-I hyperboloid (cyan) 3D equifrequency surface appears along the y-direction in Fig. 2(d). Further increasing the angular frequency ω when it is in the phase II region, the type-I hyperboloid form in Fig. 2(d) turns into a type-II hyperboloid 3D equifrequency surface (orange) [see Fig. 2(e)].

When the angular frequency ω happens to reach the low-frequency Weyl frequency ω1 in Eq. (6), the symmetric Weyl points (blue dots), twofold type-I hyperboloid (pink), and ellipsoid (orange) equifrequency surfaces can coexist because of the 3D space isolation of these Weyl points, as shown in Fig. 2(f). In the phase III region, similar to the phase I case [see Fig. 2(c)], the 3D equifrequency surfaces in Fig. 2(g) get back to the ellipsoid (orange) and twofold type-I hyperboloid (cyan) mixed type. However, these equifrequency surfaces of phases I and III have opposite topological properties due to the sign inversion of the magnetoelectric coupling parameter γ (as we will discuss later). When the angular frequency ω is at the high-frequency Weyl frequency ω2=ωp in Eq. (8), i.e., the critical point between phases III and IV, the twofold type-I hyperboloid equifrequency plane in Fig. 2(g) becomes two 3D isolated Weyl points with the same topological charges, as illustrated in Fig. 2(h). Moreover, in the phase IV region, two ellipsoid 3D equifrequency surfaces with different sizes in Fig. 2(i) arise owing to the parameter εz>0 [see Fig. 2(a)]. Hence, in chiral photonic metamaterials, multitype 3D equifrequency surfaces and multiple topological transitions can be achieved by only changing the parameters ω.

Now, to better visualize the essential differences and connections among different 3D equifrequency surfaces in Fig. 2, we study the 2D multiple equifrequency lines and different cross-section views of these 3D equifrequency surfaces. For the phase I region, the equifrequency lines located in kx−kz and ky−kz planes have a complete band gap along the kz direction, corresponding to the gapped phase case, as shown in Fig. 3(a). It is one of the research focuses of our work. At the critical point (resonance frequency ω0) of phases I and II, the different 2D equifrequency lines have only one band gap in the ky axis [see Fig. 3(b)]. On the other hand, once the angular frequency ω is located in the phase II region, the complete band gap in the z-axis direction in Fig. 3(a) disappears, as illustrated in Fig. 3(c). Thus, in the chiral photonic metamaterials, when the parameter ω passes the resonance frequency ω0, the system changes from a gapped phase to a gapless phase in momentum space.

For phases II and III, the low-frequency Weyl frequency ω1 as a phase transition critical point of the multiple equifrequency lines, where the chiral media system changes from a gapless phase [see Fig. 3(d)] to a gapped phase [see Fig. 3(f)]. Notably, there is a self-intersection point in kx−kz plane, as shown in Fig. 3(e). Now, we analyze in detail the physical origin of such self-intersection points. Along the kx axis (ky=kz=0), the characteristic equation in Eq. (4) can be further deduced analytically kx4+ω4εz(εyμy−γ2)μz−ω2kx2(εzμy+εyμz)=0.(15)

Then, by solving Eq. (15), the expressions kx1 and kx2 of the 2D equifrequency lines are kx1=ω2[(εzμy+εyμz)−a4]/2,kx2=ω2[(εzμy+εyμz)+a4]/2,(16)where a4=εz2μy2+2εz(2γ2−εyμy)μz+εy2μz2. Notably, we only give the case of the kx>0 because it has axial symmetry. Based on Eqs. (1) and (16), in the case of kx=0 [i.e., the self-intersection point in Fig. 3(e)], we get the following concrete expression: αω/(ω02−ω2)+1+αω2/(ω02−ω2)εt+α/(ω02−ω2).(17)

By solving Eq. (17), we know that a solution exists only if the angular frequency ω is equal to the low-frequency Weyl frequency ω1. Similarly, when the parameter ω passes the high-frequency Weyl frequency ωp [see Fig. 3(h)], the 2D equifrequency lines change from a gapped phase [see Fig. 3(g)] to a gapless phase [see Fig. 3(i)].

Based on Figs. 2 and 3, we give the complete topological phase diagram describing the band gap in the kz direction, as shown in Fig. 4(a). Here, Roman numerals I–IV represent four different topological phases, and only phases I and III have complete kz band gaps. In general, the topological properties of chiral metamaterials are described by topological invariants (such as Berry fluxes) owing to the nonzero magnetoelectric coupling effect. Notably, the distribution directions of nonzero Berry fluxes are closely related to the signs of magnetoelectric coupling parameters. Here, based on Eqs. (4) and (10)–(13), we give the 2D equifrequency lines and Berry fluxes (Chern numbers) distributions for the angular frequencies ω=0.2 and ω=0.8, corresponding to phases I and III regions in Fig. 4(a), respectively, as illustrated in Figs. 4(b) and 4(c). Notably, the opposite Berry flux distributions in Figs. 4(b) and 4(c) are caused by the opposite signs of magnetoelectric coupling effect γ in Eq. (1) [see Fig. 2(a)]. It is the prerequisite for the generation of frequency chirality-dependent surface waves.

For photonic topological systems, a defining property is the existence of chiral surface states. Now, we specifically analyze the surface states supported by the boundary between the vacuum state and chiral photonic metamaterials. In particular, we consider a 3D system stratified structure along the y-axis, as shown in the inset panel of Figs. 4(d) and 4(h). Then, the eigenmodes on either side of the interface at y=0 can be calculated by applying Maxwell’s equations. In the vacuum state, the two independent eigen electric/magnetic fields are expressed as E1=(−ikxky1,kx2+kz2,−iky1kz),H1=(−kzω,0,ωkx),(18)E2=(kxkz,iky1kz,kz2−ω2),H2=(−iky1ω,ωkx,0),(19)where ky1=kx2+kz2−ω2 represents the decay constant in the vacuum state.

Similarly, the eigen electric/magnetic fields in the chiral metamaterials are represented as E3=(E3x,E3y,E3z),H3=(H3x,H3y,H3z),(20)E4=(E4x,E4y,E4z),H4=(H4x,H4y,H4z).(21)

Enforcing Maxwell’s boundary conditions at y=0 requires the continuity of the tangential components of both electric and magnetic fields. In this case, Eqs. (18)–(21) can be simplified into the nontrivial determinant calculation constraint matrix M, Det[M]=|E1xE2xE3xE4xE1zE2zE3zE4zH1xH2xH3xH4xH1zH2zH3zH4z|=0.(22)

Based on Eq. (22), we give the distributions of the frequency chirality-dependent surface states, as shown in Figs. 4(d) and 4(h). Notably, in these two distinct configurations, the shadow gap regions’ surface states exhibit opposite gap Chern numbers (Cgap) and group velocity directions.

Now, we study the spatial field confinement of the frequency chirality–dependent surface states. In Figs. 4(d) and 4(h), the extent of localization of these gap surface states is rigorously evaluated by their skin depths (y/λ). In mathematics, the skin depth in the vacuum state can be defined as y/λ=1/Im[ky]=1/Im[ω2−kx2−kz2].(23)

Using Eqs. (22) and (23), we specifically calculate the skin depths of the frequency chirality–dependent surface states, as shown in Figs. 4(e) and 4(i). The spatial distributions of the electric field magnitude |E| for these surface states are illustrated in Figs. 4(f) and 4(j), which correspond to the points A and B in Figs. 4(e) and 4(i), respectively. Notably, the intensity of these surface modes diminishes sharply as they extend away from the interface, indicating strong localization at the boundary, as shown in Figs. 4(f) and 4(j). Moreover, the frequency chirality-dependent transmissions of the unidirectional surface waves are demonstrated by numerical simulation in Figs. 4(g) and 4(k).

Now, we study the physical origin and bandwidth tunability of photonic Fermi arc surface states. Similar to the calculation method in Figs. 4(d) and 4(h), we give the surface states of 2D equifrequency lines and Fermi arc surface states at different angular frequencies ω, as shown in Figs. 5(a)–5(h). In these cases, the types of equifrequency lines of the chiral metamaterials change dramatically, as shown by the orange lines. In particular, there is no complete kz band gap that exists in Figs. 5(a) and 5(h), whereas the common kz band gap can arise in Figs. 5(b)–5(g). By contrast, the red and black Fermi arc surface states remain stable in all cases, and their group velocities do not vary within each other. These results prove that the photonic Fermi arc surface states are robust to changes in electromagnetic parameters.

The number of surface states corresponds to the topological charge of the dual-frequency Weyl point. In Figs. 5(b)–5(g), the red and black Fermi arc surface states can span the whole band gap shadow regions. Specifically, at the low-frequency Weyl frequency ω1, the surface states originate from the negative chirality Weyl points (blue dots) and terminate at the bulk states (orange lines) of the chiral metamaterials in Fig. 5(b). In the phase III region, the bulk states (orange lines) and vacuum states (gray dashed lines) are connected by the Fermi arc surface states, as shown in Figs. 5(c)–5(f). When the angular frequency reaches high-frequency Weyl frequency ωp, the Fermi arc surface states originate from the positive chirality Weyl points (red dots) and also terminate at the bulk states (orange lines), as illustrated in Fig. 5(g). Moreover, in Fig. 5(i), we give the phase diagram of the evolution of the kz gap size varying with the parameter ω, corresponding to the cases in Figs. 5(b)–5(g). The sizes of these band gaps are closely related to the angular frequency ω (see more details in the Supplementary Material Note 4). These results reveal that the parameter ω can be used as a degree of freedom to regulate the bandwidth of the photonic Fermi arc surface states.

The frequency chirality-dependent Fermi arc surface states, as we study in Figs. 4 and 5, hold significant potential for fostering innovation in the development and realization of photonic topological functional devices. Here, Fig. 6 shows the design of multichannel and directional topological photonic routings, realized by the gap Chern number–locked Femi arc surface states.

In Figs. 6(a)–6(d), the negative and positive y-direction topological surface waves are excited at the interface between the vacuum state and chiral metamaterials (see more details of the mechanism of implementing topological surface waves without cladding structures in the Supplementary Material Note 5). Here, by only changing the signs of parameter kz, we can realize the topologically mode-selective output of different ports. In particular, the mode field distributions of these surface waves are almost the same before and after passing through the defect, as shown in Fig. 6(c). Moreover, a sharp defect does not cause mode coupling or scattering of the topological surface waves [see Fig. 6(d)].

On the other hand, based on the frequency chirality–dependent Fermi arc surface states, we also design multitype and multichannel directional topological photonic routings, as illustrated in Figs. 6(e)–6(k). In the chiral media–vacuum system, the physical mechanism of realizing these multichannel topological routings is caused by the group velocity direction and gap Chern number–locking photonic Fermi arc surface states [see Fig. 5(e)]. Moreover, to better exhibit the different (or multi) port mode output characteristics, we give the 1D normalization electric field profiles |E| of the harpoon-like and +-type multichannel topological photonic routings, as shown in Figs. 6(h), 6(i), and 6(k).

In conclusion, for the topological chiral photonic metamaterials, we study the multitype 3D equifrequency surfaces and multiple topological transitions. Utilizing effective medium theory in conjunction with topological band theory, we conduct a systematic analysis and construct detailed topological phase diagrams that illustrate various 3D equifrequency surfaces of nonmagnetic photonic systems. Dual-frequency band and direction-dependent photonic Weyl points emerge because of the simultaneous incorporation of the electric and magnetic Drude dispersion effects within the chiral metamaterials. Moreover, the resonance frequency ω0 and dual-frequency Weyl points are demonstrated as the critical points of multiple topological transition processes. Based on topological diagrams, the highly localized and robust Fermi arc surface states can appear when the vacuum state directly contacts phase I and III chiral photonic metamaterials. These surface states exhibit strong frequency chirality–dependent propagation characteristics. Remarkably, we further reveal that the angular frequency ω serves as a degree of freedom to control the bandwidth of the photonic Fermi arc surface states. By leveraging the charity-dependent Fermi arc surface states, we design various forms of multichannel and directional topological photonic routings. Through rigorous theoretical investigation, we elucidate that the underlying physical mechanism behind the emergence of multichannel topological photonic routings stems from the unique interfacial properties arising at the junctions of distinct material phases. This research has the potential to introduce fresh insights into 3D photonic topological semimetal systems while enhancing the design flexibility of multichannel devices within nonmagnetic continuum media.

Biographies of the authors are not available.