Inspired by the rapid advancements in topological physics within condensed matter systems [1–3], photonic topological insulators have emerged as a vibrant research frontier, enabling robust control of electromagnetic (EM) waves. This capability is realized through photonic quantum Hall states in systems with broken time-reversal symmetry (TRS) [4–8], as well as spin Hall states [9–11] and valley Hall states [12–19] in systems preserving TRS. Notably, topological photonic systems with intact TRS, despite sacrificing partial absolute robustness, have garnered significant attention due to their enhanced compatibility with semiconductor-based electronic and optical devices. To date, topological pseudospin and valley states have been extensively utilized in the design of topological devices for on-chip communication and signal processing systems, owing to their practical implementation methods and intrinsic wave-manipulation properties [20–27].

While these topological edge states have demonstrated significant potential in guiding waves, they are not without limitations. A common drawback is that energy collection and transfer are typically confined to a zero-width domain wall at the boundary, which restricts high-capacity robust energy transmission to a relatively low level. Very recently, a three-layer heterostructure based on valley sonic crystals was proposed to enable large-area waveguide states [28], presenting a viable approach for high-capacity energy transmission. Following this pioneering work, large-area topological waveguide states have been realized based on the photonic quantum Hall systems [29,30], the photonic quantum valley-Hall systems [31,32], and the photonic quantum spin Hall systems [33–46] in heterostructures of topological photonic crystals (PhCs). These advancements have significantly advanced the manipulation of electromagnetic waves [28–43]. However, the operational bandwidth of these topological waveguides depends on the bandgap size and waveguide width. Notably, as the waveguide width increases, the bandwidth rapidly decreases [28–43]. Consequently, in photonic topological heterostructures, large-width waveguide states seem to lack large operating bandwidths, limiting their application in broadband photonic devices.

On the other hand, due to the bosonic nature of photons, photonic topological phenomena can involve multiple bandgaps, leading to multi-band topologies with the same topological origin [44–47] and different topological origins [48–52]. Most recently, Guo et al. theoretically realized the dual-band topological large-area waveguide states in photonic heterostructures [37]. However, although this method can effectively broaden the bandwidth, the identical mode features impose difficulties for further mode decoupling in information processing and distribution. Li et al. theoretically proposed a tri-band heterostructure system based on a staggered triangular lattice to achieve different types of large-area waveguide states [40]. To increase the effective operating bandwidth while enabling signal demultiplexing and propagation for subsequent information processing paths, it is necessary to construct a topological heterostructure waveguide to support the coexistence of different types of large-area waveguide states. Meanwhile, microstrip structures loaded with lumped circuit elements are ideal for realizing topological photonics. However, up to now, only topological modes operating in a single bandgap have been experimentally demonstrated [53–55]. Therefore, it is interesting and useful to construct a planar heterostructure waveguide with lumped circuit elements that simultaneously inherits the properties of the pseudospin-locked effect in the quantum spin Hall system and the valley-locked effect in the quantum valley Hall system, which provides an efficient multifunctional application scheme to manipulate the transmission behaviors of EM waves accompanied by great flexibility and tunability.

In this work, we propose, design, and experimentally observe a planar heterostructure waveguide using topological LC circuits based on microstrips that simultaneously support large-area pseudospin waveguide states (LPWSs) and large-area valley waveguide states (LVWSs) in different frequency bandgaps. The lattice is a planar microstrip array composed of lumped element circuits with a honeycomb structure. It is worth noting that this structure is common in various electronic devices and is a typical transmission line in the microwave frequency band, consisting of a bottom metal film, an intermediate dielectric substrate, and a top patterned metal strip [56]. Different from recent studies of dual-band topological large-area waveguides with identical topological origin [37], introducing a $C_3$-symmetric texture with alternating wide and narrow metallic strips opens two nontrivial bandgaps, one of which is associated with pseudospin degrees of freedom at the $\mathrm{\Gamma }$ point and the other with valley degrees of freedom at the K point. In addition, the planar microstrip array with a three-layer heterostructure supports pseudospin- or valley-waveguide state propagation, which can flexibly enlarge the mode width with good topological properties. The simple structure of the present topological microstrip device displaying large-area pseudospin and valley waveguides in its EM modes enables easy fabrication and on-chip integration. Taking advantage of the planar and open structure of this topological microstrip device, we measure distributions of the amplitude of the out-of-plane electric field of LPWSs and LVWSs using near-field techniques. Inheriting from the topological pseudospin and valley, we experimentally demonstrate a high-energy-capacity topological channel intersection at two separated bandgaps that exhibit the pseudospin and valley locked effect and a topological beam expander with both pseudospin and valley mode width degrees of freedom. Finally, as a potential application of dual-band topological large-area waveguides with distinct topological origins, a broadband spatial demultiplexer based on their intrinsic coupling properties is also exhibited. This proposed open planar structure with multi-band and multi-mode width degrees of freedom further promotes the application of topological devices in on-chip communication and signal processing.

An original schematic of the lumped-element circuit model of the proposed planar structure is shown in Fig. 1(a), which consists of a hexagonal LC circuit for each cell of a triangular lattice structure, sharing the same lattice vector $a_1$, $a_2$, and lattice constant $a_0$. Figure 1(b) shows a schematic diagram of the lumped element LC circuit for each hexagonal cell. A capacitor on a node with uniform capacitance $C$ establishes a shunt with a common ground plane. To further obtain the band structures of the LC circuit above, the voltage drops caused when current passes through the inductive and capacitive elements are $V=L\mathrm{d}I/\mathrm{d}t$, $V=\int I\mathrm{d}t/C$, respectively. The voltage and current between neighboring node $i$ and node $j$ satisfy the following relationship: $$I_{ij}=\frac{1}{L_{ij}}{\int }_{t_0}^t(V_i-V_j)\mathrm{d}{t}_1.$$(1)According to Kirchhoff’s law in charge conservation systems, the voltage relative to the common ground plane can be expressed as $$V_i=\frac{-1}{C}{\int }_{t_0}^t\mathrm{d}{t}_1\sum _{j=1}^3\frac{1}{L_{ij}}{\int }_{t_0}^{t_1}(V_i-V_j)\mathrm{d}{t}_2,$$(2)where the system is charge neutralized at a moment $t_0$, and summation is taken over the nearest neighbor nodes. Its differential form serves as the equation of motion for the system: $${\mathrm{d}}^2{V}_i/\mathrm{d}{t}^2=\frac{-1}{C}\sum _{j=1}^3\frac{1}{L_{ij}}(V_i-V_j).$$(3)Taking the hexagonal unit cell, unit vectors, and the numbering of nodes shown in Fig. 1(a), the normal frequency modes are described by a vector of six components: $$\mathit{V}={\mathit{V}}_0\exp (i\mathit{k}·\mathit{r}-i\omega t)\equiv {[V_1{V}_2{V}_3{V}_4{V}_5{V}_6]}^{\mathit{T}}\exp (i\mathit{k}·\mathit{r}-i\omega t),$$(4)and we obtain the following secular equation: $${\omega }^2{\mathit{V}}_0=H{\mathit{V}}_0,$$(5)$$H=\left[\begin{array}{cccccc}\phi \tau & 0& 0& \frac{-L_1{e}^{i({\beta }_3)}}{C{\phi }_1}& \frac{-L_2}{C{\phi }_2}& \frac{-L_2}{C{\phi }_1}\\ 0& \phi \tau & 0& \frac{-L_2}{C{\phi }_1}& \frac{-L_1{e}^{-i{\beta }_1}}{C{\phi }_1}& \frac{-L_2}{C{\phi }_2}\\ 0& 0& \phi \tau & \frac{-L_2}{C{\phi }_2}& \frac{-L_2}{C{\phi }_1}& \frac{-L_1{e}^{i{\beta }_2}}{C{\phi }_1}\\ \frac{-L_1{e}^{i(-{\beta }_1)}}{C{\phi }_1}& \frac{-L_2}{C{\phi }_1}& \frac{-L_2}{C{\phi }_3}& \phi \tau & 0& 0\\ \frac{-L_2}{C{\phi }_2}& \frac{-L_1{e}^{i{\beta }_1}}{C{\phi }_1}& \frac{-L_2}{C{\phi }_1}& 0& \phi \tau & 0\\ \frac{-L_2}{C{\phi }_1}& \frac{-L_2}{C{\phi }_2}& \frac{-L_1{e}^{-i{\beta }_2}}{C{\phi }_1}& 0& 0& \phi \tau \end{array}\right],$$(6)where $\tau =1/C{L}_1{L}_2{L}_3$, ${\phi }_1=L_1{L}_2$, ${\phi }_2=L_2{L}_3$, ${\phi }_3=L_1{L}_3$, $\phi ={\phi }_1+{\phi }_2+{\phi }_3$, ${\beta }_1=3\sqrt{3}{a}_0{k}_x/2+3a_0{k}_y/2$, ${\beta }_2=3\sqrt{3}{a}_0{k}_x/2-3a_0{k}_y/2$, and ${\beta }_3=3a_0{k}_y$.

According to Eq. (6), the frequency band structures in the LC circuit are shown in Figs. 1(c)–1(e). It is worth noting that in current topological circuits, the nontrivial topology emerges purely from the symmetry of two-dimensional honeycomb structures [9,11]. When the inductance value of the hexagonal cell of the LC circuit is $L_1=L_2=L_3$, there exists a four-fold degeneracy at $\mathrm{\Gamma }$ point and a three-fold degeneracy at K point, which are guaranteed by the band folding effect and $C_6$ rotation symmetry; the corresponding band structure is shown in Fig. 1(d). In general, by modulating the intra- and inter-coupling of the honeycomb lattice, i.e., $L_1=L_2\ne L_3$, the quadruple degeneracy at the $\mathrm{\Gamma }$ point can be evolved into a pair of double-degeneracy points, thus forming a bandgap around the degeneracy frequency. However, the degeneracy point at the K valley still exists, due to the protection of the rotational symmetry of $C_6$ [9]. Further, by setting $L_1\ne L_2$, the rotational symmetry is broken, and, consequently, the sixfold rotation symmetry $C_6$ is reduced to $C_3$. One can see from Figs. 1(c) and 1(e) that these two degenerate points can be both lifted, resulting in two complete bandgaps. It should be pointed out that the mechanism of opening different bandgaps in this way is also applicable to other systems, such as the $C_4$ system. In Fig. 1(f), the distinct topological properties were verified by calculating the EM eigenmodes at the $\mathrm{\Gamma }$ point for the spin-Hall-type bandgap (marked with purple and red dots), and the phase distributions at K point for the valley-Hall-type bandgap (marked with yellow and green dots). The eigenmodes and phase distributions are respectively switched in these two cases. Note that the eigenmodes of spin-Hall-type bands at the $\mathrm{\Gamma }$ point are isomorphic to the $p_x/p_y$ orbitals and $d_{x^2-y^2}/d_{xy}$ orbitals originating from $C_6$ symmetry [9], but are intrinsically hybridized due to the breaking of the inversion symmetry of the unit cell. Therefore, the lumped element circuit exhibits topological phase transitions of spin Hall and valley Hall types with the pristine honeycomb structure as the transition point.

According to the topological LC circuits model, we design a planar microstrip array waveguide consisting of three domains $\mathrm{A}|{\mathrm{B}}_n|\mathrm{C}$ [see Figs. 2(a) and 2(b)], where $n$ denotes the number of layers in the B domain, and implement experimentally the above coexistence of two types of topological photonic waveguide states. Different equivalent inductance/capacitance values can be realized by etching different widths of metallic strips on the surface of the structure, and the distributed capacitance also can be a good approximation of the node capacitance. The metallic strips width of intra/inter hexagonal unit cells and the lumped capacitance on the node are taken as tuning parameters, which reproduce the theoretical discussion of the topological LC circuit model. The system was designed and fabricated using an F4B dielectric substrate with a thickness of 1.6 mm and a relative permittivity of 2.2. All the nodes were loaded with a lumped capacitor of $C=5.6\mathrm{pF}$ at all the node locations. The length of the metallic strips between the nodes in our designed planar microstrip array system is $L=10.9\mathrm{mm}$. In the upper half of the system, the metallic strip widths in the intra/inter hexagonal unit cells are 0.5, 2.5, and 3.2 mm, respectively [see the right panel of Fig. 2(b)]. In the lower half of the system, the metallic strip widths in the intra/inter hexagonal unit cells are 3.2, 2.5, and 0.5 mm, respectively. The width of the metallic strips in the middle part between the upper and lower halves of the system is 2 mm. The distributed inductance value of the microstrip structure so designed is very close to the value of the lumped inductance loaded to the LC circuit model in our theoretical discussion. In addition, the capacitance value we loaded on the nodes of the microstrip structure ($C=5.6\mathrm{pF}$) is slightly smaller than the value of the total collector capacitance loaded on the nodes of the structure in the theoretical discussion ($C=7.27\mathrm{pF}$) because the metal strips around the nodes in the actual microstrip structure also contribute a portion of the distributed capacitance.

To reveal explicitly the topological EM properties of planar microstrip arrays $\mathrm{A}|{\mathrm{B}}_2|\mathrm{C}$ with modal wide degrees of freedom, we performed numerical calculations based on a supercell (see Appendix A for more details). Due to the inclusion of the intermediate space between the two half-spaces of distinct topology, the topological frequency dispersion occurs in two different bulk frequency gaps. Figure 2(c) shows the calculated frequency band structure for the topological spin-Hall-type bandgap. A pair of waveguide modes (red line) exhibiting a typical helical feature of conventional pseudospin-Hall edge states appears, with a small gap between them. This small gap is due to the reduced rotational symmetry at the domain-wall interfaces [33,35]. In addition to the topological waveguide modes, non-topological modes also appear within the bandgap [blue line in Fig. 2(c)], due to the finite width of domain B, which allows part of the bulk state to appear in the bandgap [33]. These results indicate that our planar heterostructure waveguides support LPWSs. In Fig. 2(d), we show that the width of the topological frequency window decreases as $n$ increases in the B domain. This is because LPWSs are generated by the strong coupling and hybridization between interface states and gap-free bulk states in the structure. Thus, as the B domain width increases, the coupling and hybridization attenuation result in a decrease in the topological frequency window. In addition, the increased width of domain B alleviates the symmetry mismatch between domains A and C, and the small gap (light orange region) between the two branches of the topological waveguide mode becomes narrower. To confirm the above theoretical analysis, we perform full-wave simulation by using a finite-element time-domain solver as a part of the CST Microwave Studio. To simulate the perfect absorption boundary, we add lumped resistance between the microstrip around the sample and the bottom common grounded metal plate to achieve the perfect matching boundary condition. We add different values of lumped resistors according to the characteristic impedance of the microstrip. The metal strips of microstrips with widths of 0.5 mm, 1.5 mm, 2 mm, 2.5 mm, and 3.2 mm are added with lumped resistors with resistance values of 145 Ω, 97 Ω, 85 Ω, 75 Ω, and 66 Ω, respectively. Next, to experimentally detect the LPWSs, we launch an EM wave from a linearly polarized dipole source located in the middle of domain B with a frequency within the topological spin-Hall-type bandgap. The experimental setup and details are presented in Appendix D. The signals are generated by a vector network analyzer (Agilent PNA Network Analyzer N5222A) and then input into the sample that acts as a system source. It is worth noting that EM wave injection into the system without disturbing the bulk frequency band is a unique feature of the bosonic behavior of photons, which electrons do not have. As shown in Figs. 2(e) and 2(f), the electric field $|E_z|$ distribution obtained by full-wave simulation and experimental measurements shows that the EM wave is well confined around the B domain and decays strongly into the bulk in the transverse direction.

In the meantime, we further focus on the valley-Hall-type bandgap to examine the LVWSs. Figure 2(g) shows the calculated frequency band structure for LVWSs. A topological waveguide mode marked as a red line is observed within the bulk bandgap. Additionally, there is a non-topological waveguide mode (blue line) that delimits the frequency window for the topological waveguide mode. As shown in Fig. 2(h), when the width of domain B increases, the topological frequency window (denoted as a white area) decreases. Thus, we can predict that when $n$ in domain B is large enough, the topological waveguide modes will merge into the bulk modes. In Fig. 2(i), we excite the LVWSs through a linearly polarized dipole source that emits EM waves. The EM wave is concentrated and propagated in domain B as demonstrated by the full-wave simulations. In addition, the LVWSs are verified experimentally in Fig. 2(j). It is evident that the electric fields $|E_z|$ are centered in domain B. We further demonstrate the strong robustness of LPWSs and LVWSs against large defects in Appendix B.

The prominent features of LPWSs and LVWSs, such as pseudospin- and valley-locked properties with modal wide degrees of freedom, make them promising for high-capacity energy transmission in waveguide-based integrated photonic devices and circuits. To demonstrate the application potential, we first construct a topological channel intersection with finite waveguide width that has both pseudospin- and valley-locked properties. The configurations of the topological channel intersection are shown in Figs. 3(a)–3(e), consisting of five domains separated by the white dotted line, with the number of layers $n=2$ in the B domain. We labeled ports 1–4 at the terminals of the B domain and placed a linearly polarized source as input ports at port 1 to excite LPWSs and LVWSs. To demonstrate the pseudospin-locked effect, the out-of-plane electric field $|E_z|$ distributions of LPWSs obtained by full-wave simulation and experimental measurement are shown in Figs. 3(b) and 3(c), respectively. As can be seen, when a linearly polarized source is located at port 1, the LPWSs can only propagate to ports 3 and 4 and are suppressed in port 2. This is because port 2 does not support the topological waveguide state, due to the different chirality of the transmission channel from port 1 to port 3 or port 4 compared to the transmission channel from port 1 to port 2. Additionally, the $|E_z|$ distribution of LVWSs was obtained by full-wave simulation and experimental measurement, respectively, as shown in Figs. 3(d) and 3(e). Similar to the LPWSs, the LVWSs propagate along the channel toward ports 3 and 4 but not toward port 2 because of the opposite valley properties. These results show that the designed topological channel intersection with finite waveguide width has pseudospin- and valley-locked properties. We further demonstrate the pseudospin- and valley-momentum-locking unidirectional propagation of LPWSs and LVWSs in Appendix C.

Since the existence of LPWSs and LVWSs is independent of the number of layers $n$ in domain B, we further design a topological beam modulator that can simultaneously expand the beam width of both LPWSs and LVWSs as shown in Fig. 4. Here, the number of layers in domain B changes from one layer to five layers along the propagation path. For the beam expander of LPWSs, the out-of-plane electric field $|E_z|$ distribution obtained by full-wave simulation and experimental measurement is shown in Figs. 4(b) and 4(c), respectively. It is evident that the energy is transmitted in the narrow part of domain B and then expanded in the broader part of domain B. Meanwhile, Figs. 4(d) and 4(e) show the distribution of out-of-plane electric field $|E_z|$ of LVWSs obtained by full-wave simulation and experimental measurement for the beam expander, respectively. These topological beam expanders of LPWSs and LVWSs could be potentially used for modulating field strength or the energy transmission width.

The LPWSs and LVWSs discussed above could be useful for designing functional EM devices. To demonstrate this, we further reveal two types of large-area waveguide modes coupled to the background space for further wave manipulations. The LPWSs and LVWSs exhibit unique topological origins, enabling their easy spatial separation due to their distinctive coupling characteristics, which could be utilized for signal demultiplexing. We analyze the coupling features in the corresponding $k$-space at the sloping termination as shown in Fig. 5(a). There is a sloping termination interface between the finite structure composed of $\mathrm{A}|{\mathrm{B}}_2|\mathrm{C}$ and the D domain of the tetragon-lattice circuit based on microstrips. We expect the direction of the outgoing beam into the D domain to depend on the type of the large-area topological waveguide mode. The LVWSs in this structure that propagate are projected from the $\mathrm{K}$ (${\mathrm{K}}^{\prime }$) valley, in which the value of the wave vector is $|\mathit{K}|=4\pi /3a_0$. In addition, the wave vector in the background space can be read from the equifrequency curve in the D domain, i.e., $|\mathit{k}|=2\pi f\sqrt{{\epsilon }_{\mathrm{eff}}}/c$. The effective permittivity of the D domain can be written by ${\epsilon }_{\mathrm{eff}}=(\epsilon +1)/2+(\epsilon +1)[{(1+12h/w)}^{-0.5}+0.04{(1-w/h)}^2]$. The dielectric permittivity, strip width, and thickness of the tetragon-lattice circuit in the D domain are represented by $\epsilon$, $w$, and $h$, respectively.

To illustrate the coupling feature with a sloping termination for large-area waveguide mode, we introduce the Brillouin zone of the lattice $\mathrm{A}|{\mathrm{B}}_2|\mathrm{C}$ and the equifrequency contour in the tetragon-lattice of the D domain [black solid regular hexagons and red dotted circles in Figs. 5(b) and 5(c)] that represent the relative values of the wave vectors of the incident and outgoing waves (denoted as $\mathit{K}$ and $\mathit{k}$). By introducing the phase-matching condition, mode coupling to the D domain requires the wave vector $\mathit{k}$ to satisfy $\mathit{k}·{\mathit{e}}_{\mathrm{ter}}=\mathit{K}·{\mathit{e}}_{\mathrm{ter}}$, where ${\mathit{e}}_{\mathrm{ter}}$ is the unit vector parallel to the termination interface. In the case of LPWSs, the excitation is dominated at $\mathrm{\Gamma }$ point (i.e.,  $\mathit{K}=0$). Two solutions are obtained by intersecting the $\mathrm{\Gamma }$ conservation line with the D domain dispersion, as shown in Fig. 5(b). The solutions show that the LPWSs are allowed to couple to the D domain toward the normal termination. As expected, the simulated electric field in Fig. 5(d) indicates that the LPWSs propagate along the normal direction, forming a coupled channel in the D domain.

Based on the above analysis, we further consider the coupling features of LVWSs at the sloping termination [see Figs. 5(c) and 5(e)]. The LVWSs in this structure that propagate are projected from the $\mathrm{K}$ (${\mathrm{K}}^{\prime }$) valley at a frequency of 0.938 GHz, in which the value of the wave vector is $|\mathit{K}|=4\pi /3a_0=128.03{\mathrm{m}}^{-1}$ while that in the D domain is $|\mathit{k}|=2\pi f\sqrt{{\epsilon }_{\mathrm{eff}}}/c=26.06{\mathrm{m}}^{-1}$. Obviously, the LVWSs cannot be scattered into the D domain due to the strong wave vector mismatch ($\mathit{k}\ll \mathit{K}$). By applying the phase-matching condition, no intersection was found between the valley conservation lines and the dispersion of the tetragon-lattice in the D domain, where the red (cyan) dashed line perpendicular to the termination line marks the ${\mathrm{K}}^{\prime }$ ($\mathrm{K}$) valley conservation as shown in Fig. 5(c). The mismatch ceases the coupling of LVWSs to the D domain. Nevertheless, the simulated result in Fig. 5(e) shows outgoing waves are well confined at the interface boundary between the $\mathrm{A}|{\mathrm{B}}_2|\mathrm{C}$ domain and the D domain, with little leakage into D domain space. This is because an alternative channel formed in the interface due to the existence of the self-localized edge states guaranteed by the distinct topological invariant of the A domain lattice (topological regime) and D domain lattice (trivial regime) [57]. The above results exhibit the potential application of dual-band topological large-area waveguides with distinct topological origins in signal demultiplexing. Because of the dual topological frequency window, the resulting signal demultiplexing is broadband in nature. It is worth noting that the above-mentioned functions can also be achieved even if the $C_6$ system remains unchanged.

In summary, we have proposed and experimentally observed the coexistence of LPWSs and LVWSs in a planar topological LC circuit based on microstrips. Both the LPWSs and LVWSs in the system possess prominent topological properties, such as pseudospin- or valley-locked propagation, and tunable mode width for high-capacity energy transport. Based on its excellent performance, a topological channel intersection and a topological beam modulator are designed with pseudospin- and valley-mode width degrees of freedom, which provides new opportunities for designing various multifunctional photonic devices. In addition, as a practical example, the signal demultiplexer and router are demonstrated through the distinct coupling properties of the LPWSs and LVWSs with dual topological frequency windows. The topological waveguide properties achieved in the planar circuit can not only be extended up to the infrared frequency regime [58], but can also be compatible with various lithographically fabricated planar devices. With extension of the lumped element circuit discussed in the present work to loading lumped variable capacitance diodes, this reconfigurable planar circuit system can also support spatiotemporal modulation of the topological frequency, enabling future experiments on dynamic topological phenomena. By incorporating great flexibility and tunability within a single topological planar circuit system, our work significantly expands the multifunctional range of advanced on-chip photonic circuits for communication and signal processing.