Configurable topological beam splitting via antichiral gyromagnetic photonic crystal

Antichiral gyromagnetic photonic crystal (GPC) in a honeycomb lattice with the two interpenetrating triangular sublattices A and B magnetically biased in opposite directions can realize antichiral one-way edge states propagating along the same direction at its two parallel edges. Here, we report the construction and observation of topological beam splitting with the easily adjustable right-to-left ratio in an antichiral GPC. The splitter is compact and configurable, has high transmission efficiency, and allows for multi-channel utilization, crosstalk-proof, and robust against defects and obstacles. This magnificent performance is attributed to the peculiar property that antichiral one-way edge states exist only at zigzag edge but not at armchair edge of antichiral GPC. When we combine two rectangular antichiral GPCs holding left- and right-propagating antichiral one-way edge states respectively, bidirectionally radiating one-way edge states at two parallel zigzag edges can be achieved. Our observations can enrich the understanding of fundamental physics and expand topological photonic applications.

A prominent means to create topologically protected edge state is a gyromagnetic photonic crystal (GPC) immersed in the external magnetic field thereby breaking the time-reversal symmetry 14−17 . In 2008, Raghu and Haldane first theoretically predicted that topologically protected one-way edge state can be created by analogy to the integer quantum Hall effect in two-dimensional electron gas system 14 . Consequently, some authors 15−17 experimentally observed topologically protected one-way edge state in the square and honeycomb GPCs. In all these works, topologically protected one-way edge states display the chirality where the one-way edge states propagate along the opposite directions at two parallel edges of a bulk GPC, and they are called chiral one-way edge states.
Recently, we theoretically proposed another intriguing case where the one-way edge states at two opposite parallel zigzag edges can propagate in the same direction, and they are called antichiral one-way edge states 40 . These antichiral one-way edge states can be realized in a GPC with a honeycomb lattice consisting of two interpenetrating triangular sublattices A and B being immersed in opposite external magnetic fields respectively, and they are also robust against backscattering from defects, obstacles and sharp bends. Later on, Zhou et al. experimentally observed this antichiral one-way edge state in a similar GPC system 41 . To date, antichiral one-way edge states have been studied in various systems, such as exciton polariton strips 42 , graphene structures 43,44 , Heisenberg ferromagnets 45 , acoustic resonators 46 , and recently have been realized in photonic crystals 41 and electric circuits 47 . Very recently, Zhang et al. also theoretically realized reconfigurable light imaging in an antichiral gyromagnetic photonic crystal with both broken time-reversal and inversion symmetries 48 . Although there are many of studies focused on the demonstration of antichiral one-way transport property, little of them touch the unique properties of antichiral topological systems and novel applications.
Here, we consider a rectangular bulk GPC of a honeycomb lattice with zigzag and armchair edges. When two interpenetrating triangular sublattices A and B are immersed in opposite external magnetic fields respectively, we observe that antichiral one-way edge states can only exist at zigzag edges and will not exist at armchair edges. Utilizing this unique feature, we combine two rectangular antichiral GPCs biased by opposite external magnetic fields, so that they can support left-and right-propagating antichiral one-way edge states respectively, to construct a compound antichiral GPC, as shown in Fig. 1. We demonstrate that bidirectionally radiating one-way edge states from the sources placed at the boundary of two GPC bulks can be achieved with high-efficiency, being reflection-free, crosstalk-proof, and robust against obstacle. Based on these unique features, we realize a topological beam splitter with configurable splitting ratio. Our results will enrich the understanding of fundamental physics underlying topological photonics and help design and construct novel topological photonic devices.

Antichiral one-way edge state
We consider a two-dimensional rectangular GPC with a honeycomb lattice of yttrium-ion-garnet (YIG) gyromagnetic cylinders immersed in air, as seen in Fig. 2(a) and 3(a). The lattice constant along the x direction is a=10.0 mm, and the radius of YIG cylinders is r=1.5 mm. The rectangular GPC possesses two types of edge, where the zigzag edge is along x direction and the armchair edge is along y direction. There are five layers in the entire experimental configuration, with the top (1 st ) and bottom (5 th ) layers being the honeycomb array of NdFeB permanent magnet rods, the center (3 rd ) layer being the GPC structures made from honeycomb array of YIG rods, and the 2 nd and fourth layers being the metal confinement layers. Two aluminum alloy plates with thickness h 1 =h 5 =2 mm are set in the first and fifth layers. A honeycomb array of holes are drilled into both plates, and a pair of N and S NdFeB permanent magnet rods are pressed tightly into these face-to-face holes in these two layers, so that the displacement of magnets owing to the attraction and repulsion of different magnets is completely avoided. Each NdFeB cylinder is of radius 2 mm and height 2 mm. These NdFeB cylinder pairs can apply one-to-one external magnetic field to the YIG cylinders. By flipping the biasing direction of magnets at neighboring sites, opposite magnetic fluxes are uniformly applied on the sublattices A and B in the x-y plane. The crystal of YIG cylinders (third layer with height h 3 =5 mm) is placed in an air-loaded planar waveguide sandwiched between two parallel aluminum alloy layers (i.e. second and fourth layers with height h 2 =h 4 =1 mm) that are used to completely forbid the EM waves from leaking off in the z direction. In this case, the polarization is identical to the pure TE states in two-dimensional photonic crystals. The transmission measurement is made with the Keysight P9374A vector network analyzer, which supports EM waves with frequency ranging from 300 kHz to 20 GHz. The relative permittivity and permeability of the air are ε 1 =1 and μ 1 =1, respectively. The gyromagnetic material is yttrium iron garnet (YIG), a ferrite with measured relative permittivity 14.5 and dielectric loss tangent 0.0001. Its measured saturation magnetization is M s =1950 Gauss, with a ferromagnetic resonance linewidth of 20 Oe. Typically, the magnetization will decrease to a negligible value without the presentence of the external magnetic field. The relative magnetic permeability of the YIG has the form , , is the external magnetic field, is the gyromagnetic ratio, is the damping coefficient, and ω is the operating frequency. The band structure and all the simulations are calculated by using the commercial software COMSOL MULTIPHYSICS with RF module in frequency domain, and only E polarization (where the electric field E is parallel to the z-axis direction) is considered.
We first calculate and analyze the band structures of nonmagnetized, uniformly magnetized and compound magnetized GPC to discuss the construction of antichiral one-way edge state. Figure 2(b) shows the first Brillouin zone. As shown in Fig. 2(c), when all gyromagnetic cylinders are nonmagnetized, there is no bandgap between the first and second band, because these two bands intersect with each other at two Dirac points at points K and K' in the first Brillouin zone, and the band structure is symmetric about the high symmetry point M. The frequencies of two Dirac points are at 8.85 GHz and the slope of the green dotted line connecting two Dirac points is zero, so there are no one-way edge states. Besides, when all gyromagnetic cylinders are uniformly magnetized along the +z direction, as plotted in Fig. 2(d), the time-reversal symmetry of GPC is broken, the degeneracy of Dirac points at points K and K' are lifted up, and the band structure is still symmetric about the high symmetry point M. As a result, the first and second bands separate from each other, and then a complete bandgap with a pair of chiral one-way edge states emerges (green zone). The frequency range of this bandgap (marked as the green region) extends from 9.10 to 9.40 GHz, thus the bandgap width is 0.30 GHz. Moreover, when two sublattices A and B are magnetized along the -z and +z directions respectively, as illustrated in Fig. 2(e). The first and second bands are not separated from each other and still intersect at Dirac points at points K and K'. However, the band structure tilts and loses the symmetry about point M in the first Brillouin zone because the Dirac points at points K and K' move down and up Frequency (GHz) respectively, resulting in the appearance of antichiral one-way edge states at the zigzag edges. Obviously, the slope of the green dotted line connecting two Dirac points is a positive value instead of being zero. The frequencies at points K and K' are 9.10 and 9.40 GHz respectively, and their difference is also 0.30 GHz, which is consistent with the bandgap width of Fig. 2(d).
We apply negative and positive magnetic fields to sublattices A and B respectively, which creates the sublattice symmetry breaking. Notably, the geometry used in obtaining the simulated and experimental data is a zigzag ribbon along the x direction, as shown in Fig. 3(a). Figure 3(b) shows that there exist two mutually intersecting dispersion curves degenerate with each other and possessing identical dispersion behavior around k x =1/2 (2π/a) and within 9.10~9.40 GHz. We plot the eigenmodal field profiles corresponding to the points 1, 2, 3 and 4 at 9.30 GHz in Fig. 3(c). There exist two edge states (eigenmodes 2 and 3) and two bulk states (eigenmodes 1 and 4). Because the slope for the edge states is positive while the slope for the bulk states is negative, the edge states and the bulk states will only propagate rightwards and leftwards respectively. Thus, these two one-way edge states propagate along the same direction at both zigzag edges, showing the antichirality. Besides, since these antichiral one-way edge states occur only in gapless system, they need the associated bulk states to transport in opposite direction to ensure the energy conservation 40,41 .
When we set two point sources (marked as stars) oscillating at 9.30 GHz at both zigzag edges [ Fig. 3(d)], antichiral one-way edge states are excited. These two edge states at both zigzag edges transport rightwards with high-efficiency transmission, as shown in Fig. 3(e) by simulation. Since the upper and lower zigzag edges both support the right-propagating one-way edge states, their transmission data are completely coincident (i.e. S21=S43, S12=S34). The measured transmission data at the upper and lower zigzag edges are plotted in Fig. 3(f) and 3(g). They are not exactly the same, due to the inevitable departure in two measurements induced by uncontrollable subtle experimental details, but they remain strongly nonreciprocal between 9.10 and 9.40 GHz, with a 30 -40-dB difference between the rightward and leftward transmissions. These experimental results agree well with the theoretical prediction in Fig. 3(e). It should be noted that the transmission in the experimental results is smaller than that in the numerical results, and the transmission loss mainly originates from the coupling loss at input/output end and the fabrication errors of the sample. This transmission property can be improved by using the transmission lines with lower loss, fabricating a closed waveguide cavity, and reducing the coupling loss at the input/output end. Next, we calculate the projected band structure and eigenmodal field of antichiral GPC along armchair edge. Notably, the geometry used in obtaining the simulated and experimental data is an armchair ribbon along the y direction, as shown in Fig. 4(a). Figure 4(a) shows that there exists a bandgap within 9.255~9.335 GHz (yellow region). We choose four eigenmodes 1, 2, 3 and 4 at 9.14 and 9.41 GHz and plot their eigenmodal fields [ Fig.   4(b)]. Their electric fields are dispersed uniformly in the bulk, meaning that there are no edge states localized at the armchair edge. Besides, the eigenmodal field distribution at same frequency are symmetrical, indicating that the transport of bulk states is reciprocal. We further place the perfect electric conductors adjacent to the lower zigzag and right armchair edges to form a zigzagarmchair waveguide channel [ Fig. 4(c)]. We set a point source oscillating at 9.30 GHz at lower zigzag edge. EM wave can unidirectionally transport along the lower zigzag edge, but when it meets the lower right 90° corner, it cannot bypass the corner and couple to the right armchair edge. One can see that, although there exists some energy leaking to the left of the excitation source, they will eventually transport rightwards 40 . We proceed to place a point source oscillating at 9.30 GHz at the right armchair edge, and be excited. These results indicate that the armchair edge does not support any edge states and thus the right armchair waveguide channel is closed. The simulated transmission spectra in Fig. 4(e) also show that there exists a bandgap within 9.255~9.335 GHz and there is no edge state at the armchair edge. Besides, the experimental results in Fig. 4(f) are also in full agreement with the theoretical results. The physics underlying this surprising distinctive feature is that when we apply opposite external magnetic fields to the two sublattices A and B respectively, the sublattice symmetry of GPC is broken. Then the Dirac points at K and K' valley will up-move and down-move respectively, resulting in the tilting of the band structure. As a result, the antichiral one-way edge states are only generated at two parallel zigzag edges and no antichiral one-way edge states are created at armchair edges. However, it should be emphasized that for a chiral GPC of honeycomb lattice, the time-reversal sym-metry is broken when uniformly magnetized but the sublattice symmetry is still preserved, so the chiral one-way edge states can exist at both the zigzag and armchair edges 49 .
Bidirectionally radiating one-way edge state We proceed to construct a compound antichiral GPC consisting of two rectangular GPCs biased by opposite external magnetic fields, and their interface is marked by a blue dotted line [see Fig. 5(a) for the practical experimental setup], and the other three combinations can be viewed in Supplementary information. We place two point sources oscillating at 9.30 GHz at the center of upper and lower zigzag edges to excite one-way edge states. As shown in Fig. 5 meaning that the upper and lower edge states are independent and will not couple with each other, thus, these four one-way edge states are crosstalk-proof. The measurement results illustrated in Fig. 5(d-g) also show that the four waveguide channels all support one-way edge states, where the transmission in each channel is strongly nonreciprocal at 9.255~9.335 GHz. We further replace four YIG cylinders at upper and lower edges with four metal cylinders [ Fig. 5(c)]. The two pairs of bidirectionally radiating one-way edge states can bypass the metallic obstacles and propagate forwards almost perfectly. The experimental measurement results are plotted in Fig. 5(h-k). Compared with Fig. 5(d-g), the transmission characteristics in the unperturbed and perturbed cases remain almost the same, although there exists some disturbance in the magnitude. The difference between forward and backward transmission is still large, e.g. about 35 dB in the vicinity of 9.30 GHz. We also provide the simulation result in Supplementary information that the bidirectionally radiating one-way edge states can bypass the rectangular metallic obstacle and continue forwards almost without any backscattering. Thence, the bidirectionally radiating one-way edge states at the two opposite parallel zigzag edges of a compound antichiral GPC have major advantages of high-efficiency, being reflection-free and crosstalk-proof, and robustness against obstacles. More importantly, we must note that such a high performance topological beam splitting is not possible to be realized in other betterstudied PC systems, such as trivial PC 50 , topological valley-and spin-Hall photonic crystals 51,52 , and chiral topo-logical photonic crystal 53,54 (see Supplementary information for the beam splitting based on various physical mechanisms).

Configurable topological beam splitting
We proceed to design a topological beam splitting with the configurable splitting ratio easy to be adjusted by simply changing the source excitation condition. We consider EM waves incident from the lower zigzag edge as the excitation source. The incident angle φ is defined as the angle between the incident EM wave and the lower zigzag edge [ Fig. 6(a-c)]. We calculate and show the electric field intensity pattern in right and left channels, draw the line scanning of calculated results along the lower zigzag edge, and directly superimpose these figures for the clarity to show clearly the energy ratio of left and right channels.
When φ=30°, the one-way edge states of left and right channels both are excited, but the major energy flux is transmitted rightwards, and the energy ratio of them is about 1:5 [ Fig. 6(a)]. When φ increases to 60°, more energy flux is transferred to the left channel, so that the energy ratio of them is about 3:5 [ Fig. 6(b)]. When EM wave is incident perpendicular to zigzag edge (i.e. φ=90°), the energy flux is evenly split and transmitted to two channels [ Fig. 6(c)]. When the incident angle continues to increase from 90° to 180°, the energy ratio of the left channel begins to dominate, e.g. when φ=120°a nd 150°, the energy ratios are about 5:3 and 5:1 respectively. The measured transmission data of lower channel (i.e. S21 and S31 parameters) at φ=30°, 60° , and 90° are a d   shown in Fig. 6(d-f). We further normalize S21 and S31 parameters of Fig. 6(d-f), i.e. the sum of transmission parameters S21 and S31 is 1, and the results are plotted in Fig. 6(g). At 9.255~9.335 GHz, when φ increases from 30° to 90°, the transmission of the right channel decreases gradually, while that of the left channel increases. Thus, by continuously controlling the excitation angle of EM waves, the flow of EM waves at 9.255~9.335 GHz is split into the left and right directions, and the splitting ratio is configurable in an easy way. Similarly, the upper zigzag edge can also implement the configurable topological beam splitting functionality. As a result, the compound antichiral GPC can provide a powerful path to realize high-density dual-edge four-channel configurable topological beam splitting.

Conclusion and discussion
In summary, we have realized a compact dual-edge fourchannel configurable topological beam splitting with the right-to-left splitting ratio adjustable. It is attributed to the peculiar property that antichiral one-way edge states exist only at zigzag edge but not at armchair edge. This scheme solves a challenging task for the realization of topological beam splitting, which is more compact and configurable, has high transmission efficiency, allows for multi-channel utilization, crosstalk-proof, and robust against defects and obstacles induced from manufacturing. Our experimental results show that the difference between forward and backward transmission is about 30~40-dB, and this strong nonreciprocity can well satisfy practical needs. Our results are of significance not only in the basic understanding of fundamental physics underlying topological photonics, but also in offering useful insights and routines to design novel topological photonic devices which can be used to build nonreciprocal microwave devices such as configurable topological beam splitter, compact multi-channel one-way waveguides 40 , reflection-free one-way waveguides with sharp corners 41 and even reconfigurable light imaging 48 . Besides, the magnetic field exerted on the GPC comes from tiny permanent magnets, which avoids the use of large magnetic field equipment. So they do not add volume to the microwave devices, and would not be an issue in constructing practical microwave topological devices. Finally, we can construct nonreciprocal microwave device with a more compact setup in a more economical and flexible way.
Moreover, although our work has focused on GPC, similar ideas can be more broadly generalized to not only the other systems in photonics, e.g. Floquet 20,21 , liquidcrystal 24 , and synthetic dimension 27,28 topological insulators, but also in other physical fields, e.g. exciton polariton strips 42 , graphene structures 43,44 , Heisenberg ferromagnets 45 , acoustic resonators 46 , and electrical circuits 47 .