Introduction

Photonic topological insulators1,2,3,4,5, extension of topological insulators from electronic materials to their photonic counterparts, have been booming development for recent decades. By analogy to multiple Hall effects in condensed matter physics 6,7, researchers have theoretically proposed and experimentally demonstrated varioustopological photonic states, such as the photonic quantum Hall (QH) states8,9,10,11,12, photonic quantum spin Hall (QSH) states13,14,15,16,17,18, photonic quantum valley Hall (QVH) states19,20,21,22,23, higher-order topological corner states24,25,26,27. The topological photonic states with backscattering-immune, unidirectional transport and robustness against defects or imperfections have been widely developed and applied in integrated waveguide-based photonic devices and circuits, such as topological quantum interfaces28, topological light sources29, topological splitters30,31 and topological lasers32,33,34,35. Additionally, based on extension of topological phase, a great many interesting physics phenomena or relative fields have been developed and studied, such as nonlinear topological photonics36,37, topological quantum38, topological bound states in the continuum39,40,41, topological rainbow42, and topological circuits43,44. Recently, some effects from other disciplines have been explored and expanded to topological photonics, hence there are several more advanced topics and frontier technologies reported by in the literature, such as antichiral topological edge states45,46,47,48, non-Hermitian skin effect49,50,51,52,53,54, and so forth, which vastly enriches researches and applications in topological photonics.

As we know, the topological edge states have been regarded as a kind of excellent information carriers in the information processing and integrated photonic circuits. Generally, manipulation for electromagnetic wave is dependent on the degree of freedom of light field, such as frequency18, polarization15,20, phase55, even the width of interface between different topological domains11,15,16,56. However, it is difficult for these states with intrinsic and local chirality15,16,17,21, such as valley-Hall kink states and pseudospin helical edge states, to achieve phase modulation. Fortunately, many compatible topological photonic systems with distinct topological classes have been presented recently, such as coexistence of photonic pseudospin-Hall and valley-Hall-like phase57,58, multiple topological phase transitions between photonic QH phase and pseudospin-Hall phase59,60,61, and coexistence of photonic QH and valley-Hall-like phase62. Such a compatible photonic system can share more degree of freedom in manipulating electromagnetic wave, in other words, the hybrid topological photonics states share individual topology property from distinct topological classes respectively, which enriches more degree of freedom to steer topological waveguide states. Therefore, phase modulation for topological photonic states (valley-Hall kink states or pseudospin helical edge states) would be possible realized in a compatible topological photonic system. So far, such manipulation and application in waveguide-based on-chip communication have been rarely reported. Steering of hybrid topological waveguide states will provide a promising platform for on-chip communication and application, such as multi-function, multi-degree of freedom, and multiple topology bands.

In the work, we theoretically design and study a compatible topological photonic system with coexistence of the photonic QH phase and the pseudospin-Hall phase. A typical kind of hybrid topological waveguide states with pseudospin-characteristic, magnetic field-dependent, and strong robustness against backscattering and perfect electric conductor (PEC) is realized in the system composed of gyromagnetic photonic crystals (GPCs) with a deformed honeycomb lattice containing six cylinders. Subsequently, we achieve the splitting for the hybrid topological waveguide states in a structure with intersection-liked based on the phase difference modulation. Robustness of the hybrid topological waveguide states in splitting has been demonstrated. Additionally, researches show that perfect electric conductor (PEC) inserted in transport channel brings optical path difference in waveguide transmission, which would influence distribution of electromagnetic wave energy in splitting. Based on the phase modulation, we achieve splitting of hybrid topological waveguide states in compatible topological photonic systems, which has potential in various applications such as sensing, signal processing, and photonics communications.

Photonic pseudospin quantum Hall states

We consider a kind of GPCs made of cylindrical yttrium iron garnet (YIG) rods arranged in a deformed honeycomb lattice with six cylinders in each unit cell (the lattice constant a0 is 12 mm, diameter of cylinder is 0.24a0), as shown Fig. 1a. All YIG rods (relative permittivity13.8) are embedded in the air (\({\varepsilon_0},{\mu_0} \)). As we know, the magnetic permeability of YIG rods is \({\mu_0} \) without applying external bias magnetic fields, which could be regarded as all-dielectric materials. Under applying external bias magnetic field along the + z axis (out-of-plane), the magnetic permeability of the YIG rods shows a second-rank tensor form11,12, which could be written as,

$$ \overline \mu = \left[ {\begin{array}{*{20}{c}} {\mu_r}&{j\kappa }&0 \\ { - j\kappa }&{\mu_r}&0 \\ 0&0&1 \end{array}} \right]\left( { + zbias} \right).$$

where \({\mu_r} = 1 + \frac{{\omega_0^{\prime}{\omega_m}}}{{\omega {{_0^{\prime}}^2} - {\omega^2}}} \),\( \kappa = \frac{{\omega {\omega_m}}}{{\omega {{_0^{\prime}}^2} - {\omega^2}}} \),\( \omega_0^{\prime} = {\omega_0} + i\alpha \omega \) with the damping factor \( \alpha \) and the operating angular frequency \( \omega \). The \( {\omega_0} = \gamma {H_0} \) is the resonance frequency, where \( \gamma = {{2.8{\rm M}{\rm H}z} \mathord{\left/ {\vphantom {{2.8{\rm M}{\rm H}z} {\rm O}}} \right. \kern-0pt} {\rm O}}e \) is gyromagnetic ratio. The characteristic circular frequency is \({\omega_m} = 4\pi \gamma {M_S} \) with the saturation magnetization \( 4\pi \gamma \) = 1850G (Gauss). In this work,\( \alpha = 0.0002 \),\( {H_0} = 2500{\rm O}e \), ω = 12.3 GHz,\( {\mu_{\text{r}}}{ = 0}{\text{.64552}} \),\( \kappa = - 0.62288 \).

Figure 1
figure 1

(a) Schematic of gyromagnetic photonic crystals (GPCs) arranged in a deformed honeycomb lattice with six cylinders in each unit cell, where vectors \( {\vec a_1} \), \( {\vec a_2} \) denote the translation vectors. Labeled ‘B’ denote applied external bias magnetic field along z axis. Right panel is enlarged view of a unit cell. (b) Photonic bulk band diagram of the GPCs unit cell in the FBZ. The right panels are electric filed (EZ) profiles of the photonic eigenmodes with pseudospin states (\( {d_- } \),\( {p_+ } \),\( {d_+ } \),\( {p_- } \)) at \( \Gamma \) point, where the pink arrows denote Poynting vectors (energy power flow).

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To simplify interaction between the next-next-nearest (NNN) sites63, we find that fixing the positions of the 1, 3 and 5 cylinders at distance \( {R_1}{ = }{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}{a_0} \) from the unit cell center whereas changing \( {R_2} \) of the 2, 4 and 6 cylinders is sufficient to induce a topological phase transition of pseudospin photonics states15. Note that we only focus on the transverse magnetic (TM) modes with nonzero (EZ, HX, HY) in this work. The photonic bulk band diagram of the GPCs unit cell in the first Brillouin zone (FBZ) is shown in Fig. 1b, where \( {R_2} = {{a_0} \mathord{\left/ {\vphantom {{a_0} {2.8}}} \right. \kern-0pt} {2.8}} \) and applying external bias magnetic field is \( 2500{\rm O}e \) along + z axis. From which one can see that pseudospin photonics states exhibit order in \( {d_- },{p_+ },{d_+ },{p_- } \), which is completely different from non-trivial topological pseudospin states12 preserved by pseudospin time-reversal symmetry (TRS). Here degenerate pseudospin states (\( {p_\pm } \) or \( {d_\pm } \)) take place band-splitting and band-inversion (\( {d_+ } \) and \( {p_+ } \)), this is because the external magnetic field is modeled by an effective vector potential9. For simplicity, here the magnitude of broken TRS is larger than that of the broken spatial inversion symmetry in a compatible topological photonic system, hence it is called QH insulator and its spin Chern number is \( {C_\pm } = (0, - 1) \)64 (The multiple topological phase transitions are described in supplementary material in detail). The Right panels describe electric filed (EZ) profiles of the photonics eigenmodes with pseudospin states (\( {d_- },{p_+ },{d_+ },{p_- } \)) at \( \Gamma \) point, where the pink arrows denote Poynting vectors (energy power flow). From which one can see that energy power flows in a cell unit for \( {p_+ } \) and \( {d_+ } \) states (\( {d_- } \) and \( {p_- } \) states) are anticlockwise (clockwise) rotation corresponding to the pseudospin-up states (pseudospin-down states), indicating that the QH insulator shares pseudospin property originating from photonic QSH insulator.

Further, the photonic bulk band diagram along \( \Gamma \)-K momentum space in the FBZ is shown in Fig. 2a1. Based on numerical calculation to supercell in finite element method (FEM), it is imposed on Floquet periodic boundary condition alone x-axis and imposed on PEC at up and down boundaries of bulk. Through the bulk-edge correspondence principle65, there are two one-way edge states appearing in the projected edge band diagram, marked by green and yellow curve lines. To further study the hybrid photonics states, we check out two eigenfrequencies marked by blue and red triangles. As shown in Fig. 2a2,a3, it is noted that one-way edge states clockwise propagate along the bulk-boundary and energy power flows are clockwise rotation (pseudospin-down) in a unit cell. Meanwhile, it is can be inferred that the one-way edge states anticlockwise propagate along the bulk-boundary and energy power flows are anticlockwise rotation (pseudospin-up) in a unit cell with applying opposite bias magnetic field to GPCs. The |EZ| distributions of one-way pseudospin quantum Hall edge states at frequency 12 GHz are shown in Fig. 2b–d, from which one can see that propagation direction of one-way pseudospin quantum Hall states excited by a line source instead of chiral source depends on direction of bias magnetic field. A typical of topological states we realized is analogous to conditional helical edge states excited by chiral source13,14,15,16,17,18. Additionally, the topological waveguide states have strong robustness against sharp corner and PEC from the evident that energy power flows can bypass PEC and keep transport along the boundary, see the inset of Fig. 2d. Hence a typical kind of hybrid topological waveguide states with pseudospin-characteristic, magnetic field-dependent, and strong robustness against backscattering and perfect electric conductor (PEC) is demonstrated in the present system.

Figure 2
figure 2

(a1) Photonic bulk band diagram along \( \Gamma \)-K in in the FBZ, where green and yellow curve lines denote topological edge states of bulk bands, the light area denotes the operation width of one-way edge states, where labeled ‘H±’ denotes applied external bias magnetic field along + z/− z axis. (a2,a3) EZ profiles of the photonics eigenfrequencies marked by blue and red sharp-triangles in (a1), where black arrow denotes transport direction of edge modes with pseudospin-down. (b–d) |EZ| distribution of one-way pseudospin quantum Hall edge states at the frequency 12 GHz, red star denotes a line source, the white arrow denotes transport direction of edge states and yellow lines denote PEC boundary condition. In (d), the PEC is inserted in the boundary marked by a white dashed rectangle.

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Models and discussions

Here we propose a structure with intersection-liked composed of the GPCs, where the magnetic field distributions are marked by H+ and H−, as shown in Fig. 3a. Here d denotes distance between two different topology domains, P1 and P2 are transport channel port 1 and port 2, respectively. Figure 3b shows the supercell of the model structure.

Figure 3
figure 3

(a) Schematic of model composed of the GPCs, where d denotes the distance between two different topological domains, P1 and P2 are transport channel port 1 and port 2, respectively. (b) Supercell of the model structure.

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d = 0

The projected photonic bulk band diagram is plotted in Fig. 4a by numerical calculation to supercell with d = 0, from which one can see that there are two pairs of one-way edge states in bulk bands gap, which can use gap Chen number (Cg) to explain. As we know, when an edge is formed by joining two materials with band gaps overlapping in frequency, the number of one-way edge states is equal to the difference of gap Chern numbers65,66. Hence we can infer that the gap Chen number is 2 (Cg =|(0,− 1)–(0,1)|= 2). We check out EZ profiles of eigenmodes at the eigenfrequencies M1, M2, noting that EZ mainly distributes the interface and decays exponentially away from the interface. It is worth noting that at the interface energy power flows are clockwise (pseudospin-down)/anticlockwise (pseudospin-up) rotation around unit cell corresponding to H+/H−, marked by black circular arrows, which resembles that of conditional helical edge states13,14,15,16,17,18. The |EZ| distributions of one-way pseudospin quantum Hall states in intersection-like structure at frequency 12 GHz are shown in Fig. 4b, from which one can see that splitting for one-way pseudospin quantum Hall states is achieved in the structure, green arrow points to location of a line source.

Figure 4
figure 4

(a) Photonic bulk band diagram of the model structure with d = 0, where there are two edge modes marked by red and green curve lines. The right panels show EZ profiles of eigenmodes at eigenfrequencies M1, M2, respectively. (b) |EZ| distribution of one-way pseudospin quantum Hall edge states at frequency 12 GHz, where green arrow denotes the location of point source.

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d = 0.5a 0

For d = 0.5a0, the projected photonic bulk band diagram is plotted in Fig. 5a. From which one can see that two one-way edge states in bulk bands (marked by red and blue lines), and these edge states have more rich physics properties, i.e., two edge modes at eigenfrequencies M1, M2 have the same direction group velocity vg (\( {{\partial \omega } \mathord{\left/ {\vphantom {{\partial \omega } {\partial k}}} \right. \kern-0pt} {\partial k}} \)) and phase velocity vp (\( {\omega \mathord{\left/ {\vphantom {\omega k}} \right. \kern-0pt} k} \)), while the other ones at eigenfrequencies M3, M4 have the same direction vg and inverse direction vp, directions of group velocity and phase velocity marked by arrows in Fig. 5a. Here light shadow denotes frequency region within which the edge modes appear. We find that the edge states at eigenfrequencies M2, M3 have the same direction vg and opposite direction phase velocity, the frequency range is marked by dark shadow. Here we can draw a conclusion that we can use two light sources with a phase delay Δφ at the boundary to excite the edge states and realize splitting for one-way pseudospin quantum Hall edge states in the structure with d = 0.5a0, where dark shadow frequency region denotes operation bandwidth of the splitter in phase modulation.

Figure 5
figure 5

(a) Photonic bulk band diagrams of the model structure with d = 0.5a0, where there are two edge modes marked by red and blue curve lines. (b–e) EZ profiles, Poynting vectors and phase distributions of one-way edge modes at eigenfrequency M1, M2, M3 and M4, respectively.

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To further demonstrate these physics properties, we check out EZ profiles, Poynting vectors and phase distributions of edge modes at eigenfrequencies M1, M2, M3 and M4, respectively, as shown in Fig. 5b–e. For EZ, all EZ mainly concentrates on the interface and decay exponentially away from the interface. For Poynting vectors, all energy power flows are clockwise (anticlockwise) rotation around the cylinders in down (up) domain with H+ (H−), which indicates all vg are same direction as well. The phase winds of edge modes at eigenfrequencies M1, M2 are in accordance with the distribution of Poynting vectors, while the other ones are inverse to the distribution of Poynting vectors of edge modes at eigenfrequencies M3, M4, which also can be inferred that two edge modes at eigenfrequencies M3, M4 have the same direction vg and inverse direction vp, see black circular arrows. It is worth noting that eigenmodes M4 is a slow-light modes21 form the evidence that energy power flow does not transport to right as if it was located at the interface. Therefore, the two kinds of analysis are consistent with each other. Additionally, the projected photonic bulk band diagrams are plotted in the Supplementary Materials, where the d = a0 and d = 1.5a0. From which one can see that a one-way pseudospin quantum Hall states with large-air-modes appear in the structure with d = a0 and the waveguide states still satisfies phase modulation, while the physics property will vanish with the d = 1.5a0 (see Supplementary Materials in detail).

Results and analysis

To demonstrate our prediction above, we use two light sources with a Δφ phase delay at the boundary to excite the edge states. All results are plotted in Fig. 6a–d, from which one can see that one-way pseudospin quantum Hall states pass the port 2 and hardly enter port 1 with the Δφ = \( {{8\pi } \mathord{\left/ {\vphantom {{8\pi } {18}}} \right. \kern-0pt} {18}} \)(see Fig. 6a), electromagnetic energy of one-way pseudospin quantum Hall states almost equally is distributed into the port 1 and port 2 with the Δφ = \( {{17\pi } \mathord{\left/ {\vphantom {{17\pi } {18}}} \right. \kern-0pt} {18}} \)(see Fig. 6b), one-way pseudospin quantum Hall states pass the port 1 and hardly enter port 2 with the Δφ = \( {{26\pi } \mathord{\left/ {\vphantom {{26\pi } {18}}} \right. \kern-0pt} {18}} \)(see Fig. 6c) and one-way pseudospin quantum Hall states simultaneously pass the port 1 and port 2 with the Δφ = \( {{34\pi } \mathord{\left/ {\vphantom {{34\pi } {18}}} \right. \kern-0pt} {18}} \)(see Fig. 6d). Hence here we realize splitting for one-way pseudospin quantum Hall states in the structure with d = 0.5a0 in the phase difference modulation at frequency 12.8 GHz within dark shadow. On the other hand, we further calculate waveguide transmission spectrum and contrast transmitted ratio \( \eta \) for the splitter as a function of Δφ (phase difference) respectively. Here P1 and P2 denote the energy flows through port 1 and port 2, respectively, and the contrast transmitted ratio \( \eta \) is defined as \( \eta = {{(P1 - P2)} \mathord{\left/ {\vphantom {{(P1 - P2)} {(P1 + P2}}} \right. \kern-0pt} {(P1 + P2}}) \). All results are described in Fig. 6e, it is worth noting that the Δφ takes place offset around 0,\( {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0pt} 2} \),\( \pi \),\( {3 \mathord{\left/ {\vphantom {3 {2\pi }}} \right. \kern-0pt} {2\pi }} \). That is because the local offset of two light sources induces light path difference, in other words, the light path difference will result in phase compensation, and appear phase offset. Certainly, we can also eliminate the light path difference and phase compensation when the locations of two light sources are perfect symmetrical with regard to the structure.

Figure 6
figure 6

(a–d) |EZ| distribution of one-way pseudospin quantum Hall states with different phase differences at frequency 12.8 GHz. (e) Energy flows and \( \eta \) as a function of Δφ, where the red (blue) line and green dashed lines represent the energy flow at port1(2) and \( \eta \), respectively.

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To demonstrate the robustness of one-way pseudospin quantum Hall edge states in the structure with d = 0.5a0, we insert a PEC (1.7a0\( \times \) 0.1a0) in the main channel in the structure. All results are shown in Fig. 7a–d, we can see that one-way pseudospin quantum Hall states exited by two light sources with Δφ = \( {{8\pi } \mathord{\left/ {\vphantom {{8\pi } {18}}} \right. \kern-0pt} {18}} \) bypass the PEC and keep continuing transport, hence one-way pseudospin quantum Hall states have strong robustness. On the other hand, the energy flows in port 1 and port 2 take place generally exchange after a PEC is inserted in the main channel. Based on phase modulation principle, with inserting a PEC, light path difference between two waveguides takes place change again, and induces phase compensation. Inversely, according to results shown in Fig. 6e, even we can infer the phase compensation induced by inserting a PEC. Certainly, the corresponding results would appear slight change at different excited frequencies. The methods we discussed above are studied in previously work as well, i.e., the transmission of a power splitter as a function of metal scatter height66, pseudospin switching by changing light path of middle finite-width waveguides67, and all-optical logic gates based on mode coupling theory68. Recently, the researchers have used mode conversion and separation to realize a splitting for chiral edge states in magneto-optical photonic crystal69. These works are proposed in a single photonic system.

Figure 7
figure 7

(a,b) |EZ| distribution of one-way pseudospin quantum Hall states without defect and with PEC at the frequency 12.7 GHz. Local enlarged view of Poynting vectors bypassing the PEC in (b). (c,d) |EZ| distribution of one-way pseudospin quantum Hall states without defect and with PEC at the frequency 12.8 GHz. Local enlarged view of Poynting vectors bypassing the PEC in (d).

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Conclusion

To conclude, a compatible topological photonic system with coexistence of the photonic quantum Hall phase and pseudospin Hall phase has been designed and studied theoretically, which is composed of GPCs with a deformed honeycomb lattice containing six cylinders. We have achieved a typical kind of hybrid topological waveguide states with pseudospin-characteristic, magnetic field-dependent, and strong robustness against backscattering and perfect electric conductor (PEC) is realized in the system. Furthermore, we have constructed a structure with intersection-liked, and achieved splitting for one-way pseudospin quantum Hall edge states by using phase difference modulation. Additionally, a PEC inserted in transport channel brings optical path difference in waveguide transmission, which would influence splitting for hybrid topological waveguide states in phase difference modulation. The work not only paves a way for manipulation of hybrid topological waveguide states in a compatible topological photonic system from distinct topological classes, but also has potential in various applications, such as all-optical logic gates or switches, signal processing, and photonics communications.

Methods

Throughout this paper, all numerical simulations are performed by COMSOL Multiphysics, 6.0a, RF module, commercial software based on finite element method (www.comsol.com).