Since the proposal of the photonic quantum Hall effect in 2008, topological photonics has attracted increasing attention [1–7]. Photonic crystal (PC) is one of the systems used to realize photonic topological states. By designing photonic bands with non-zero topological invariants, unique optical effects and transport properties in PCs can be achieved, such as topological edge states without backscattering [3,4,8–12]. In topological PCs, the transport of topological edge states is protected by the band topology, thereby ensuring that light propagation remains imperturbable in the face of large-angle bending and defects in the edge. This unique property enables the realization of waveguides that are both low-loss and capable of withstanding sharp bends [11–13].

On the other hand, optical microcavities [14,15], which enable optical localization, are also one of the important photonic devices. Traditionally, optical microcavity modes in PCs are achieved by introducing point defects [15] in the PCs. However, these modes lack topological protection, making them susceptible to fabrication errors that can significantly alter the resonance frequency [16]. This sensitivity hinders the realization of optical microcavities whose resonance frequencies are less sensitive to fabrication errors. In recent years, researchers have explored the design of optical microcavities using second-order topological corner states [17–21] or topological defects with non-trivial real-space topology [22–30]. For example, robust topological defect modes can be achieved by the introduction of a dislocation defect [26,31–35] in a single topological PC. A dislocation is characterized by the Burgers vector $\mathit{B}$. Due to the conservation of the Burgers vector, the dislocations cannot be eliminated through small local perturbations and are protected by topology [31]. In 2018, researchers observed dislocation modes in magnetic PCs in the microwave regime [26]. Due to the weak response of magnetic materials in the optical regime, this method is challenging to implement in the optical regime. Recently, dislocation modes based on two-dimensional dielectric PCs have been experimentally observed in the microwave [32] and terahertz regimes [33]. Besides, fractional topological numbers and bulk-dislocation correspondence are also observed in photonic crystals [35]. However, previous studies mostly focused on realizing dislocation modes within a single band gap, which typically supports only one dislocation mode. This limits dislocation mode applications that require two resonant modes, such as a dual-band filter and a dual-mode laser [36]. Therefore, there is an urgent need for a scheme to achieve dislocation modes in dual band gaps.

In this work, a topological PC with two topologically non-trivial band gaps is designed. By introducing a dislocation with non-trivial real-space topology into the topological PC, zero-dimensional dislocation modes emerge in both band gaps. These dislocation modes exhibit a notable degree of robustness, as their frequencies do not change significantly when the radius or position of the dielectric rod near the dislocation is modified. Furthermore, by integrating topological waveguides with two dislocations, a two-channel add-drop filter is realized. The theoretical results presented here are corroborated by experimental results in the microwave regime. Our results may promote the applications of dislocation modes in multimode microcavities and on-chip add-drop filters.

To achieve dual dislocation modes, we initially design a PC with two topologically non-trivial band gaps. Here, we consider the PC consisting of two dielectric semicylinders with a refractive index of 2.9, named as PC1, as depicted in Fig. 1(a). The lattice constant $a$ is 12.5 mm and the radius of the semicylinder $r$ is 3.75 mm. The gap between two semicylinders is denoted as $2d$. Figure 1(b) illustrates the bulk bands of the PC with $d=2.5\mathrm{mm}$. There are two photonic band gaps, as highlighted in yellow. The topological properties of each band can be determined by calculating the 2D Zak phase [37–41]: $$Z_j=\int \mathrm{d}{k}_x\mathrm{d}{k}_y\mathrm{T}\mathrm{r}[{\widehat{A}}_j],$$(1)where $j=x$ or $y$, ${\widehat{A}}_j=i⟨u(\mathit{k})|{\partial }_{k_j}|u(\mathit{k})⟩$, and $|u(\mathit{k})⟩$ denotes the periodic part of the Bloch function. For PCs exhibiting mirror symmetry along the $j$ direction, $Z_j$ is quantized to 0 and $\pi$. In addition, the Zak phase of each band can be derived from the mirror eigenvalues of the eigenmodes at high-symmetry points [40]. Figure 1(c) shows the $E_z$ fields of bulk modes of the first, second, and third bands at the $\mathrm{\Gamma },\mathrm{X}$, and $\mathrm{Y}$ points. According to the eigenmodes’ mirror symmetries with respect to the boundary of the unit cell, $Z_x$ of each bulk band is determined, as labeled on each band. Figures 1(d) and 1(e) depict the evolution of the modes at the $\mathrm{\Gamma }$ and $\mathrm{X}$ points for varying values of $d$. As $d$ increases from 0 to 2.5 mm, mode inversion occurs between band1 and band2 as well as band3 and band4 at the $\mathrm{X}$ point. In contrast, mode inversion does not appear between the adjacent bands at the $\mathrm{\Gamma }$ point. This phenomenon can be understood by mapping the 2D PC to a Su-Schrieffer-Heeger model [38,42]. The distance between two dielectric semicylinders modulates the hopping amplitude. Since we alter the distance solely along the $x$ direction, mode inversion occurs exclusively at the $\mathrm{X}$ point. Mode inversion usually results in the change of a band’s Zak phase. Figure 1(f) illustrates the evolution of $Z_x$ for each band as $d$ increases from 0 to 2.5 mm, showing that $Z_x$ of each band changes by $\pi$ after the mode inversion.

Based on the results in Fig. 1(a), we select a PC with an appropriate Zak phase to design a dislocation defect, characterized by the Burgers vector $\mathit{B}$. The number of dislocation mode is determined by the topological invariant $Q$, defined as [26,33,34] $$Q=\frac{1}{\pi }\mathit{\theta }·\mathit{B}:\mathrm{m}\mathrm{o}\mathrm{d}2,$$(2)where $\mathit{\theta }=(\frac{{\theta }_{\mathrm{\Gamma }\mathrm{Y}}}{a},\frac{{\theta }_{\mathrm{\Gamma }\mathrm{X}}}{a})$. ${\theta }_{\mathrm{\Gamma }\mathrm{Y}}$ and ${\theta }_{\mathrm{\Gamma }\mathrm{X}}$ are the Zak phases of the bulk bands along the $\mathrm{\Gamma }\mathrm{Y}$ and $\mathrm{\Gamma }\mathrm{X}$ directions with contributions from all bands below the gap. Dislocation modes emerge exclusively when the topological invariant $Q$ equals one. According to Eq. (2), a dislocation mode is achievable when $\mathit{\theta }$ is parallel to the Burgers vector $\mathit{B}$. For PC1, $\mathit{\theta }=(0,\frac{\pi }{a})$ for both gaps. To realize dislocation modes in both gaps, a dislocation defect with a Burgers vector $\mathit{B}=(0,a)$ should be introduced. As shown in Fig. 2(a), a dislocation defect is created by removing a row of unit cells and shifting the adjacent lower and upper unit cells towards the center. The red arrows form a closed loop around the dislocation and the orange arrow indicates the Burgers vector $\mathit{B}$. Note that some adjacent dielectric semicylinders are staggered. For experimental convenience, we slightly tune the positions of these adjacent dielectric semicylinders to make them form a circular dielectric cylinder, as shown in Fig. 2(b). Figure 2(c) presents the eigenfrequencies of the modes near the first bulk gap, with the yellow region highlighting the bulk band gap. The red circle indicates the dislocation mode at the frequency of 6.741 GHz and the corresponding $|E_z|$ field is strongly localized around the dislocation defect [Fig. 2(d)]. The eigenfrequencies of the modes near the second bulk gap are displayed in Fig. 2(f), also featuring a dislocation mode, highlighted by the red circle. Figure 2(g) shows the localized electric field distribution of this dislocation mode. Besides, two trivial cavity modes are present in the gap, indicated by the blue dots in Fig. 2(f). A discussion about the identification of these trivial cavity modes is provided in Appendix B.

To further confirm the localized dislocation modes, we fabricate a microwave photonic crystal whose structural parameters are the same as those descripted in Fig. 1. In the experiment, the dielectric cylinders are made of alumina and the background is filled with foam whose refractive index is 1.03. Both the alumina cylinders and foam background have a thickness of 10 mm, and are sandwiched between upper and lower metallic plates. By employing the zero-order transverse magnetic waveguide modes, which are uniform along the $z$ direction, this PC waveguide is designed to replicate the band dispersion of TM modes of a 2D PC. A source antenna is inserted through the bottom plate to excite the eigenmode, while a probe antenna is introduced from the top plate to capture the resulting electromagnetic fields. This setup enables efficient excitation and measurement of the localized modes within the photonic crystal structure (see details in Appendix D). Figure 2(e) shows the measured transmission spectrum in the first band gap by putting the source and probe antenna around the dislocation. A resonance peak near 6.777 GHz confirms the existence of the dislocation mode. In the same way, the transmission spectrum in the second band gap is also measured, as shown in Fig. 2(h). Three resonance peaks are observed. The peak observed near 12.5 GHz corresponds to the excitation of the dislocation mode within the second band gap. The other two resonance peaks are attributed to the excitation of trivial cavity modes, which lack topological protection.

Next, we investigate the robustness of these dislocation modes. In the first case, we introduce a point defect by increasing the radius of a dielectric cylinder to 5.25 mm, as highlighted by the yellow circle in Fig. 3(a). Figures 3(b) and 3(c) show the electric field of the dislocation modes at the frequency of 6.701 GHz and 12.724 GHz, respectively. For another case, we shift the positions of two dielectric cylinders with a displacement vector of ($-\mathit{a}/5$, a/5) and ($-\mathit{a}/5,-\mathit{\text{a/}}5$), as highlighted by the yellow circles in Fig. 3(d). Now, the frequencies of these two dislocation modes are 6.714 GHz [Fig. 3(e)] and 12.477 GHz [Fig. 3(f)], respectively. The dislocation modes persist in both cases and their frequencies do not change significantly, indicating the robustness of the dislocation modes.

Based on two dislocation modes, a two-channel add-drop filter is designed. The bus waveguide and drop waveguide are constructed by the edges between PC1 and another topologically trivial PC2. The unit cell of PC2 can be obtained by shifting PC1’s unit cell by 0.5a along the $x$ direction. The bulk bands of these two PCs are identical, but each bulk band’s $Z_x$ differs by $\pi$ [43]. The bulk bands and the eigenmodes at high-symmetry points of PC2 are provided in Appendix A. Due to their difference in $Z_x$, topological edge states emerge at the $y$-direction edge formed by PC1 and PC2 [39]. Figure 4(a) shows the projected band, with the inset illustrating the edge schematic. Topological edge states emerge in both band gaps, with the red (blue) line representing the edge states in the first (second) band gap. The lower panel shows the electric fields of two edge states at $k_y=0$, indicating strong confinement of electric fields around the interface. By utilizing two dislocation defects and the edge between PC1 and PC2, a two-channel add-drop filter is designed [Fig. 4(b)]. Here, compared with dislocation1, dislocation2’s center is extended by one more unit cell; see more details in Appendix C. Consequently, the modes supported by dislocation1 and dislocation2 have different frequencies. When the incident light at port A matches the dislocation mode’s frequency, the excited edge state is directed to either port C or port D through coupling to the dislocation mode. The coupling between two edge states on the right of two dislocations is blocked by PC2. To minimize the coupling between the waveguide at port A and the one at port C, the latter is strategically designed as a stepwise waveguide. Figure 4(e) shows the simulated transmission spectra at ports C and D. At the frequency of 6.708 GHz, the majority of the light is directed to port C, with the corresponding simulated electric field distribution illustrated in Fig. 4(d). In the frequency range of 13.265 GHz to 13.275 GHz, the majority of the light is routed to port D at a frequency of 13.27 GHz [Fig. 4(h)], with the corresponding simulated electric field distribution depicted in Fig. 4(g). The function of the filter is also observed in experiment. Figure 4(c) presents a photograph of the sample, with the structural parameters identical to those depicted in Fig. 4(b). Unlike the setup for measuring localized dislocation modes, here the source antenna is positioned near port A, while the probe antenna is placed at either port C or D to measure the transmitted spectra. The measured results at ports C and D within the first band gap are presented in Fig. 4(f), with the curve’s trend closely resembling the simulation results shown in Fig. 4(e). For the second band gap, a resonant peak is also observed in the measured result at port D [Fig. 4(i)]. Note that the linewidth of the resonant peak in Figs. 4(f) and 4(i) is broader than that of the simulation results. This discrepancy may arise from the fact that the top metallic plate does not tightly contact with the dielectric rods, leading to microwave leakage through the air gap.

In conclusion, we theoretically and experimentally observed the emergence of dual dislocation modes by introducing a dislocation into the PC with dual topologically non-trivial band gaps. These dislocation modes exhibit a significant degree of robustness against variations in the radius or position of the dielectric rods in their vicinity. Furthermore, we have successfully realized an add-drop filter by integrating topological waveguides with two dislocations, demonstrating the functionality of the designed device with experimental results. Our results may promote the applications of dislocation modes, such as on-chip add-drop filters. Besides, the proposed topological PC may serve as a platform for the realization of other topological defect modes within dual band gaps, such as vortex nanolasers based on dual-frequency disclination mode [24]. Note that our microwave experimental scheme is not suitable for the near-infrared band. In the future, one potential direction is studying how to realize dual-band dislocation modes in a dielectric photonic crystal slab and explore potential applications of these dislocation modes, such as on-chip dual-band filters and dual-mode lasers.