Cladding-free Fermi arc surface states and topological directional couplers in ideal photonic Weyl metamaterials

Photons can freely propagate in a vacuum, making it not a simple insulator but rather a conductor for photons. Consequently, in topological photonics, domain wall structures with opposing effective mass terms are used as cladding to confine electromagnetic waves. This approach is necessary to demonstrate topological edge/surface waves and Fermi arc surface states (FASS). Here, we show that the cladding-free FASS with high field localization at the boundary can be achieved using ideal Weyl gyromagnetic metamaterials (GMs). In these GMs, the ideal Weyl semimetal phase exists due to the dispersionless longitudinal modes. At the boundary of the GMs-vacuum system, the cladding-free FASS connects the projections of Weyl nodes with opposite chirality, thanks to the bulk-boundary correspondence principle. We further confirm that chiral boundary modes can propagate without experiencing scattering or backward reflection, i.e., they can advance seamlessly approximately various types of defects. Remarkably, various types of topological directional couplers are achieved by utilizing cladding-free FASS in an ideal gyromagnetic medium. Our theoretical analysis reveals that the underlying operational principle for accomplishing these nonreflecting directional couplers is due to the single coupling channel between the cladding-free FASS and the multi-type scatterers of the continuous media. Furthermore, the controllable propagation and topological directional coupling of cladding-free FASS can be further explored by adjusting the ideal gyromagnetic medium and boundary configurations of the continuous media system. This research offers increased flexibility for the development of cladding-free and directionally coupled topological devices.


Introduction
Topology is a branch of mathematics that describes quantities that remain unchanged under continuous deformation [1][2][3].Objects with identical topological invariants are considered topologically equivalent.The concepts and foundational principles of photonic topological materials are historically rooted in research on the topological phase of condensed matter systems [4][5][6][7].Recently, topological photonics has emerged as a promising platform for exploring topological phenomena, due to its diversity and ease of manipulation.A range of photonic topological materials has been proposed and implemented, including three-dimensional (3D) Weyl semimetals [8,9], photonic topological insulators [10,11], chiral bulk modes [12,13], non-Abelian topological charges [14,15], and photonic Weyl nodal line semimetals [16].Notably, the studied and explored Weyl semimetals have garnered significant attention in both the materials science and physics communities [17][18][19].
This research has led to the discovery of numerous striking physical phenomena, for example topologically negative refractions, chiral Landau levels, and negative magnetoresistance effects [20].
A characteristic feature of Weyl semimetals is the occurrence of open FASS, which occurs between Weyl nodes with opposite topological charge projections located at different positions [8].These FASS are robust against perturbations, and their range of stable existence is used to measure the robust strength of topological semimetals.Generally, both Weyl nodes and FASS have been extensively studied in the gyromagnetic photonic crystals and continuous media [21][22][23][24][25][26][27][28].The FASS can achieve robust transmission through arbitrary defects due to the systems breaking of time-reversal symmetry under an external magnetic field.However, unlike topological electronic systems, the vacuum state is a good conduction material for photons.Therefore, in such cases, domain walls with inverse topological properties must be used as cladding to confine electromagnetic waves and demonstrate topological FASS.This raises a significant question: is it possible to achieve cladding-free FASS in a gyromagnetic medium?If so, what intriguing physical phenomena might this lead to?These questions warrant further investigation.
Here, we investigate cladding-free FASS and achieve topological directional couplers in a vacuum-ideal Weyl GMs system.Using Maxwells equations, we derive the 3D band structures and equal frequency surfaces of the gyromagnetic medium.Specifically, we calculate the equations for the longitudinal and transverse modes by the eigenmodes of the gyromagnetic medium to confirm the existence of the ideal Weyl point and demonstrate its related physical characteristics.Notably, within the stable band gap shadow of the vacuum-gyromagnetic medium system, the boundaries do not require cladding to confine electromagnetic waves.This is because the cladding-free FASS exhibits superior field localization characteristics.Furthermore, our numerical simulations reveal that boundary modes at the interface between the gyromagnetic medium and the vacuum can advance seamlessly approximately various types of defects without backward reflection or scattering.Remarkably, the nonreflecting directional couplers of the vacuum-GMs system can be achieved through the cladding-free FASS.In the continuous media system, we uncover that the physical mechanism behind the realization of these couplers is due to the single coupling channel between the scatterers and the cladding-free FASS.Moreover, by adjusting the gyromagnetic parameters and boundary configurations in the topological directional couplers, it is possible to control the propagation and achieve topological directional coupling of the cladding-free FASS.

Ideal Weyl points and Berry fluxes of the GMs
GMs are types of electromagnetic metamaterials characterized as homogeneous continuum materials [29][30][31].The electromagnetic behaviors of the gyromagnetic medium are characterized by the effective permeability/ permittivity tensors.Notably, the effective permeability/permittivity tensors of the gyromagnetic medium can be expressed as follows . ω p is the plasma frequency, ω represents the angular frequency, and g is the gyromagnetic parameter, respectively.Moreover, μ t (μ t = 1.2) and ò t (ò t = 1) in equation (1) represent constant values, i.e., they are all frequency-independent.For experimental purposes, to attain gyromagnetic effects, magnetic materials such as Yttrium-Iron-Garnet can be incorporated during the fabrication process [32].Generally, the corresponding gyromagnetic medium in equation (1) could be achieved by employing the periodic layered structure, such as the ferrite-metal multilayer structure [30].The corresponding parameters of the gyromagnetic medium can be derived using effective medium theory [33].Fundamentally, the permeability tensor in ferrite materials varies with frequency.Nevertheless, if the average permeability over a specified frequency range is utilized, especially when there is only mild frequency correlation, the essential physics remains intact [22].As a result, it is acceptable to disregard the dispersion mode of the permeability tensor elements.Notably, in a reasonable range, the thickness and the layered materials of the ferrite-metal superlattice can be adjustable [30,[33][34][35].It provides us with the flexibility to obtain the needed equivalent parameters of the gyromagnetic medium in figure 1.
In the gyromagnetic medium, the constitutive relation can be described as Combining " × H = −iωD and " × E = iωB, Maxwell equations in the gyromagnetic medium are further recast to a 6 × 6 matrix form where I represents the identity tensor matrix and k k k k k k 0, , ; , 0, ; , , 0 For simplicity, the angular frequency ω is normalized to ω p and the wave vector k is normalized to k p (k p = ω p /c), where c is the speed of light in the vacuum state.Based on equation (3), the wave propagation of the electric field E in the gyromagnetic medium can be given by represents the master equation of the gyromagnetic medium, and we can obtain the 3D band structures (k x = 0) in figures 1(a) and (d), and the equal frequency surfaces (ω = 0.99ω p = 0.99) in figures 1(b) and (e), respectively.It is noted that the gyromagnetic medium are rotationally symmetric to the z-axis.Here, boundary modes propagation along the z-axis can be further described as E x and E y are nonzero, however, E z is zero, which represents the transverse mode of the GMs.On the other hand, for the dispersive GMs considered in equation (1), it is possible with 1 0 , and in such the condition equation (5) can support arbitrary k z , only E z corresponds to nonzero, and therefore is known as the longitudinal modes of the GMs.Moreover, based on the transverse mode in equation (5), it can be simplified as Therefore, based on equation (6), we kown that the GMs will possess four Weyl points when the gyromagnetic parameter g is less than μ t (see figure 1(a)).Otherwise, only two Weyl nodes exist in the gyromagnetic medium [see figure 1(d)].Hence, the condition g = μ t in equation ( 6) can be considered the transition point of the gyromagnetic medium (see details in figure 6 of the Appendix).Remarkably, all Weyl nodes in the gyromagnetic medium [see gray planes in figures 1(a) and (d)] are at the same angular frequency ω=ω p , and the vicinities of these Weyl nodes (red and green dots) do not include any trivial bands other than their own longitudinal and transverse modes [8].Therefore, the gyromagnetic medium in figure 1 is an ideal Weyl semimetal.In the GMs, the nontrivial topological feature of the equal frequency surfaces (Weyl points) can be identified by the Chern numbers (C).Mathematically, the Chern numbers (C) of the equifrequency surfaces and Weyl points are depicted as the surface integral of the Berry curvature [Ω(k)] in the momentum space [34]:

Cladding-free FASS in the ideal Weyl GMs
Now, we examine the cladding-free FASS maintained by the interface between the vacuum and gyromagnetic medium within the common stable band gap regions [khaki-shaded areas in figures 2(a) and (b)].We consider the 3D stratified structures along the x-axis, which are translation invariant in the yz plane, as depicted in figures 2(a) and (b).Notably, we demonstrate that the topological directional couplers, illustrated in figures 3-5 (see details in section IV), can be achieved using the cladding-free FASS in the gyromagnetic medium, as shown in figures 2(a) and (b).Specifically, the gyromagnetic metamaterials in figure 3(a) with g = 0.8 and g = 1.2, separated by the vacuum state, correspond to the regions x > 0 and x < 0, respectively.Here, we theoretically reveal that the physical principle for achieving the nonreflection directional couplers is due to the single coupling channel between cladding-free FASS and scatterers in the gyromagnetic medium-vacuum systems, as shown in figure 3(a).Therefore, to correspond with the structure in figure 3(a), we study the cladding-free FASS [see figures 2(a) and (b)] for the cases where g = 0.8 and g = 1.21 correspond to x > 0 and x < 0, respectively.Moreover, based on Maxwell curl equations, the eigen states on either side of the edge (x = 0) can be determined by finding the nontrivial resolutions for the magnetic field H and electric field E of the vacuum-gyromagnetic medium system.In the vacuum state, the two independent eigenmodes can be described as follows: is the attenuation constant inside the vacuum state.On the other hand, the two independent eigenmodes of the gyromagnetic medium can be expressed as follows:  In the vacuum-GMs system, the penetration depth of the cladding-free FASS is a key parameter that describes the restriction degree of the surface modes.On the vacuum side, the skin depths are specifically calculated by d/λ =1/Im[ 0.99 p 2 ( ) w k y 2 k z 2 ], which are shown in figures 2(c) and (d).In the regions k z ä [0.995, 1.4] and k z ä [0.995, 1.5], we observe that the skin depths of the vacuum-gyromagnetic medium system decrease with increasing k z , as depicted in figures 2(c) and (d).This indicates that the cladding-free FASS in the common

Dispersion relations and topological directional couplers
The cladding-free FASS discussed in figures 2(a) and (b) can be instrumental in engineering advanced topological devices based on the vacuum-ideal Weyl GMs system.In this study, we constructed a three layered vacuum-GMs system featuring two types of topological directional couplers with either identical or opposite gyromagnetic directions, as illustrated in figures 3-5.These electromagnetic waves can seamlessly navigate around sharp step-type defects and deliver output signals at various ports, achieving topological directional coupling.The mechanism behind the operation of these topological directional couplers is a single coupling channel that exists between the cladding-free FASS and the scatterers within the vacuum-gyromagnetic medium system [refer to figures 3(b) and 5(b)].Now, we will explore the non-reflection directional couplers in the vacuum-gyromagnetic medium system.Using figure 3(c) as an example, we specifically analyze non-reflection directional coupling in this media system (see full-wave simulation details in the Methods).Based on the bulk-edge correspondence, the regions x > d (x < 0) and 0 x d are the non-reflection with g = 0.8 (g = 1.21) and the vacuum state, respectively.There are cladding-free FASS propagating in the negative and positive directions along the y-axis, represented by states I and II in figure 3 .Furthermore, the topological directional couplers can achieve reconfigurable directionality by manipulating the gyromagnetic directions in the three-layered vacuum-ideal Weyl GMs system, as illustrated in figures 3-5.

Conclusions
In conclusion, we have explored the cladding-free FASS and successfully implemented non-reflection directional couplers in the vacuum-gyromagnetic medium system.By applying Maxwells equations, we derived the 3D band structures and equal frequency of the gyromagnetic medium.To validate the presence of the ideal Weyl node and its associated physical characteristics, we calculated the equations for both longitudinal and transverse modes by the eigenmodes of the gyromagnetic medium.Notably, in this system, domain walls do not require cladding to confine electromagnetic waves, as the cladding-free FASS within the common stable band gap exhibits superior field localization properties.Based on numerical modeling results, we show that FASS at the boundary of the vacuum-gyromagnetic medium system can robustly transmit against various types of defects without scattering or backward reflection.Additionally, we have designed and implemented topological directional couplers in the vacuum-gyromagnetic medium system, leveraging the topological surface waves.The functionality of these topological directional couplers can be attributed to a single coupling channel between the FASS and the scatterers within the vacuum-gyromagnetic medium system.Notably, the topological and nonreflection directional couplers based on cladding-free FASS in the vacuum-gyromagnetic medium system exhibit robust characteristics against defects and structural perturbations, enabling controllable transmission direction and topological directional coupling effects.Additionally, these couplers can achieve directionally reconfigurable operation by manipulating the gyromagnetic directions within the vacuum-GMs system.This work demonstrates the potential for realizing direction reconfigurability through gyromagnetic manipulation and has significant implications for the development of topological optical devices in a homogeneous ideal Weyl media system.

Methods
In the full-wave simulations (see figures 2-5 and figure 7), we software to carry out the FASS simulations by utilizing the finite element analysis solver of COMSOL Multiphysics.Under a particular off-plane wave vector k z , the chiral boundary modes (in the x − y plane) at the edge between the ideal Weyl gyromagnetic metamaterials and the vacuum state is excited by the electric-dipole source (white pentagram) situated at the edge, enabling robust surface waves and topological directional coupling.Additionally, in the numerical simulation, the boundary conditions of the ideal Weyl metamaterials and vacuum state are defined as the scattering boundary conditions.

Figure 1 .
Figure 1.3D band structure, equal frequency surfaces, and Berry curvatures.(a), (d) 3D bands (k x = 0) for the GMs with g = 0.8 and g = 1.21, respectively.All Weyl points are located on the same frequency plane (gray planes) ω = ω p = 1 (i.e., ideal Weyl system).(b), (e) The equal frequency surfaces and Chern numbers of the gyromagnetic medium with g = 0.8 and g = 1.21, respectively.(c), (f) Berry curvatures and Berry fluxes of the gyromagnetic medium (with k x = 0) when ω = 0.99ω p , corresponding to gyromagnetic parameters g = 0.8 and g = 1.21, respectively.The red and green dots indicate Weyl nodes with negative and positive chirality, respectively.The electromagnetic parameters of the GMs are ò t = 1 and μ t = 1.2, respectively.

á ñ p ,
where U(k) = [E, H] T are the eigenpolarization states of the gyromagnetic medium.In particular, the Berry curvature of the equal frequency surfaces (Weyl nodes) is specifically calculated at every node (k y − k z ) on the two-dimensional (2D) equal frequency surfaces of the gyromagnetic medium, as illustrated in figures 1(c) and (f).The black and red arrows depict the inward Berry curvatures and outward Berry curvatures, respectively.Furthermore, the arrow lengths in figures 1(c) and (f) represent the magnitudes of the Berry curvatures.For the open hyperbolic equal frequency surfaces featuring Weyl nodes, the Berry curvatures of each 2D equal frequency surface either diverge or converge in the same direction, as illustrated in figures 1(c) and (f).Thus, the equal frequency surfaces (Weyl points) have a non-zero topological charge (|C| = 1).

Figure 2 .
Figure 2. Cladding-free FASS and robust surface modes.(a), (b) Cladding-free FASS at the edge between the vacuum state and the gyromagnetic medium with g = 0.8 (x > 0) and g = 1.21 (x < 0), respectively.The pink and yellow regions are the equifrequency surfaces of the gyromagnetic medium.The black/orange and dashed green lines are the cladding-free FASS and vacuum states, respectively.The khaki shadow regions represent the common stable band gaps.(c), (d) The penetration depths d/ λ in the vacuum state side, corresponds to the khaki shadow common stable band gap regions in (a) and (d), respectively.(e) Mode profiles |H| of the cladding-free FASS of the points A and B (with k z = 1.25) in (c) and (d), respectively.(f) 2D numerical simulations (ω = 0.99 ω p ) of transmission of the robust boundary modes on the different types of defects setups, corresponding to points A and B in (c) and (d), respectively.The white pentagrams represent the electric dipoles.The electromagnetic parameters of the gyromagnetic medium are the same as in figure 1.

Equation ( 11 )
expresses the characteristic equation for the cladding-free FASS in the vacuum-gyromagnetic medium system [refer to figures 2(a) and (b)].Using equation (11), we can determine the cladding-free FASS of the vacuum-gyromagnetic medium system when ω = 0.99ω p , as illustrated by the black and orange lines [see figures 2(a) and (b)].

Figure 3 .
Figure 3. Topological directional coupler with the same gyromagnetic direction (g = 0.8 and g = 1.21).(a) A schematic of the topological directional coupler is composed of ideal Weyl gyromagnetic medium (light orange and magenta) and vacuum (gray, with a thickness of d).(b) Dispersion relations of the topological directional coupler.The purple and light blue lines represent the equifrequency surfaces of the gyromagnetic medium with g = 0.8 and g = 1.21, respectively.The black and orange lines, along with the dashed green (pink) lines, indicate the cladding-free FASS and vacuum states (with cylindrical scatterers having ò d = 2).(c) 2D numerical simulations (H z ) of the topological directional coupling.(d)-(f) Mode sketches |H| of the topological and non-reflection directional coupler, associated with the black dashed lines A1-A3 in (c).The parameters of the gyromagnetic medium are identical to those in figure 1.

Figure 4 .
Figure 4. Robust transmission properties of the topological directional coupler with the same gyromagnetic direction (g = 0.8 and g = 1.21).(a), (b) Schematic of the topological directional couplers featuring different-shaped scatterers (yellow cuboid and triangular prism).(c), (d) Directional coupling (H z ) is achieved with different-shaped scatterers based on the cladding-free FASS.(e), (f) Demonstrate the robust transmission properties of the topological directional coupling.The parameters of the gyromagnetic medium are identical to those in figure 1.
(c).The thickness d in figure 3(a) is greater than twice the skin depth [see figures 2(c) and (d)] to prevent coupling between states I and II.To achieve effective coupling, the range of the equifrequency surface of the scatterer [see the dashed pink line in figure 3(b)] must be greater than that of the cladding-free FASS (in common stable band gap regions), such as points A and B (with k z = 1.25 < ò d , scatterer with ò d = 2), as illustrated by the dashed black line in figure 3(b) (see details in figure 7 of the Appendix).This is key to realizing topological directional couplers.The scenarios in figure 4 and figure 5(a) are analogous to figure 3(a).In particular, as shown in figure 3(c), when the cladding-free FASS is excited from port 1, state I couples to state II via the circular scatterer, allowing the electromagnetic waves to propagate along both the negative and positive y -axis directions [see ports 2 and 4 in figures 3(c)-(f)].However, port 3 has no electromagnetic mode output because state II FASS can only propagate along the positive y direction [see figure 3(b)]

Figure 5 .
Figure 5. Topological directional coupler with reverse gyromagnetic direction (g = − 0.8 and g = 1.21).(a) Schematic of the topological directional coupler composed of gyromagnetic medium (light blue and magenta) and vacuum (gray).(b) Dispersion relations of the topological directional coupler.The orange and light blue lines represent the equal frequency surfaces of the gyromagnetic medium with g = − 0.8 and g = 1.21, respectively.The blue/orange lines and dashed green (pink) lines indicate the cladding-free FASS and vacuum states (scatterers with ò d = 1.5).(c)-(f) Topological directional coupling (H z ) is achieved with different-shaped scatterers.The other parameters of the gyromagnetic medium are identical to those in figure 1.

Figure 6 .
Figure 6.Topological phase diagram of the ideal Weyl gyromagnetic metamaterials.The gray-shaded regions indicate the presence of four ideal Weyl points, while the light-blue shaded regions indicate the presence of two ideal Weyl points.

Figure 7 .
Figure 7.No topological directional coupler.(a) Schematic of the no topological directional coupler composed of the gyromagnetic medium (light orange and magenta) and vacuum (gray, with thickness d).(b) Dispersion relations of the no topological directional coupler.The purple and light blue lines are the equifrequency surfaces of the gyromagnetic medium g = 0.8 and g = 1.21, respectively.The black/orange lines and dashed green (red) lines are the cladding-free FASS and vacuum states (the cylindrical scatterers with ò d = 1.1).(c) 2D numerical simulations (H z ) of the no topological directional coupling.The parameters of the gyromagnetic medium are identical to those in figure 1.