Photonic Dirac cavity on a gradient dislocation

Abstract

Recent studies on the interplay between band topology and topological defects in real space offer an unprecedented opportunity to design photonic cavities. Here, we propose a concept of gradient dislocation by placing two chunks of square-lattice photonic crystals with the same width but differing by one period together, which is described by a one-dimensional Dirac equation with a position-dependent and sign-switching mass distribution. A photonic bound state, dubbed the Dirac cavity mode, localized at the center of the gradient dislocation is demonstrated. Compared to higher-order Dirac cavity modes, the Dirac cavity mode exhibits the smallest modal area and a relatively high Q-factor, with its frequency demonstrating robustness against certain perturbations. We also discuss a potential application in index sensing by designing a coupled cavity-waveguide system based on the photonic Dirac cavity. Our work presents an ultracompact photonic cavity design and paves the way toward future photonic integrated circuits.

Graphical abstract

Keywords

photonic cavity / dislocation / Dirac equation / photonic crystals

1 Introduction

Photonic crystal (PC) cavities that can effectively confine light within small volume for extended durations have attracted significant attention owing to its potential applications in photonic integrated circuits [1−6]. Recent progress on topological photonics opens new routes for designing photonic cavity with strong robustness [7−13]. For example, by introducing the topological one-way edge states in a closed concave boundary, whispering-gallery-mode cavities featuring nonreciprocity and geometry independence are designed to produce nonreciprocal lasing [14−17]. However, the requirement of the time-reversal symmetry breaking hinders its extension in optical application. To overcome this limitation, researchers have leveraged the photonic pseudospin (such as orbital and valley) degree of freedoms to preserve the time-reversal symmetry. This approach has led to the discovery of the photonic topological insulator based whispering-gallery-mode cavities [18−20] and topology-controlled photonic cavities [21−23]. In addition, the higher-order band topology [9, 24−26], which goes beyond the conventional bulk-edge correspondences, enables the manipulation of the electromagnetic wave in lower dimensions, resulting in the proposal of the higher-order topology induced cavity [27−31].

On the other hand, photonic bound states, arising from the interplay between the topological defects (e.g., dislocation [32, 33] and disclination [34−36]) and topological photonics are also considered promising candidates for photonic cavity [37]. Generally, the underlying physics of these photonic bound states can be understood by solving the celebrated Dirac equations, where the topological defect acts as the Dirac mass term [38, 39]. Depending on the nature of the spatially varying Dirac mass, the photonic bound states can be further classified as Jackiw−Rebbi and Jackiw−Rossi modes [40, 41], corresponding to the kink (or monopole) and vortex solutions to the Dirac equations, respectively. For example, a photonic bound state induced by dislocation was demonstrated in a topological PC, fully explained by a model of sign switch of the mass term of the one-dimensional (1D) Dirac equation [32]. Moreover, employing the strategy of Kekulé modulation, the Dirac vortex cavities are experimentally demonstrated in systems with spatial varying mass distribution [42, 43], finding potential application on topological cavity surface emitting laser [44, 45]. Nevertheless, most of existing proposals rely on the complex geometric designs or multiple-parameter fine-tuning.

To address the need for compact cavity design, we propose a concept of gradient dislocation, formed by placing two chunks of square-lattice PCs with same width but differ by a periodic number together. The gradient distribution of the relative displacement between upper and lower PCs is described by a 1D position-dependent mass system. Crucially, the signs of the Dirac mass on the two sides are opposite, giving rise to a mass domain-wall at the gradient dislocation. We demonstrate the Dirac cavity mode, as a zero-energy solution to the 1D Dirac equation, exhibiting the smallest modal area and a relatively high Q-factor. The robustness of the Dirac cavity mode and its possible application on the index sensing are also discussed.

2 Results and discussion

We start by introducing the concept of gradient dislocation. Considering a finite square-lattice PCs, a column of the PCs is removed from the upper-half of the square-lattice PCs. The missing lattice constant is then uniformly distributed to each column of the upper-half PCs, resulting in a gradient dislocation. From another perspective, the gradient dislocation is formed by adjoining two finite PCs with identical width but differing one period along the x-direction. As shown in Fig.1(a), the upper-half PCs has (N-1) periods along x-direction (lattice constant is denoted as a^{\prime }) while the lower-half PCs has N period along x-direction. To maintain the same width, the lattice constant of the lower-half PCs gives a=\frac{N-1}{N}{a}^{\prime }. Obviously, the difference between upper-half and lower-half PCs can be almost ignored when the lattice periodic number N is sufficiently large.

To elucidate the formation of Dirac cavity mode in a gradient dislocation, we select three interface configurations at left-/right-side, and middle of the gradient dislocation for consideration. For the middle area of the gradient dislocation, glide symmetry guarantees the emergence of a pair of gapless interface states [33]. However, away from the middle area, the breaking of the glide symmetry leads to a pair of gapped interface states. Since the gradient dislocation is symmetric with respect to the center line, the gapped interface states exhibit opposite Dirac mass terms at left- and right-sides of the gradient dislocation. Consequently, a mass domain-wall forms at the center of the gradient dislocation [see Fig.1(b)]. According to the Jackiw−Rebbi theory, there will be a photonic bound states localized at the gradient dislocation center.

To realize the above scenarios, we design a square-lattice PCs comprising a single cylinder rod with permittivity ϵ=5.29 in each unit cell. The rod diameter is 0.48a, where a refer to the lattice constant. As shown in Fig.2(a), we consider two primitive cell configurations: UC1, where the rod is split into two halves positioned at the upper and lower edges, and UC2, where the rod is split into four quarters positioned at the four corners. Throughout this work, we consider 2D transverse-magnetic harmonic modes and all the numerical simulations are carried out using the commercial software COMSOL Multiphysics based on finite element method. The photonic bands for UC1 and UC2 are shown in Fig.2(b). Here, we focus on the band gap between the first and the second bands, taking the bulk polarization (or the Wannier center) as the bulk topological invariant. Intuitively, the Wannier center for UC1 resides midway between the upper and lower edges, while for UC2 it resides at the corners of the unit cell (see Appendix A for details). We then consider an interface formed by adjoining UC1 and UC2, as depicted in Fig.2(c). In this configuration, the Wannier center in the lower half PC is shifted by half a lattice constant relative to that in the upper half PC. Eigenmode calculations in Fig.2(d) indicate that the interface formed by UC1 and UC2 supports a pair of gapless interface states, as a consequence of a synthetic Kramers theorem induced by the combination of the glide symmetry and time-reversal symmetry. From another perspective, a relative displacement parameter g, i.e., g=\pm 0.5a, between upper and lower interface generates the glided-symmetric interface. Deviations from g=\pm 0.5a break glide symmetry, resulting in the gapped interface states [see the projected band structure of the interface states with g=\pm 0.3a in Fig.2(d)]. We emphasize that the Wannier center is ill-defined when the displacement parameter is not integer times of the half lattice constant. To this end, we employ a Dirac mass, which defined as the frequency gap between two projected interface bands and is proportional to the overlapping integral of the electromagnetic field of the two interface states at k_x=\pi /a, to characterize the topological nature of the interface state gap. For example, the right panel of Fig.2(d) shows that the electric field patterns of the interface states at k_x=\pi /a for the upper and lower bands with g=0.3a and g=-0.3a. The overlapping integral results indicate a sign flip in the Dirac mass despite their identical projected band structure.

We then construct a finite square-lattice PCs with a size of 20a\times 51a, which is bounded with perfect matched layers, and remove a column of the upper-half PCs and distribute the missing lattice constant to each column of upper-half PCs uniformly [see Fig.2(e)]. For the nth column of the upper-half PCs, the accumulated relative displacement PCs is g=n{g}_0, where g_0=\frac{a_0}{50} refer to the relative displacement per column. Fig.2(f) shows the evolution of the band edges of the interface states versus relative displacement in Fig.2(f). It is seen that interface states exchange its frequency ordering upon crossing g=0.5a, indicating that the interface configured with g>0.5a and are of opposite Dirac masses. Furthermore, Fig.2(g) plots the eigen spectra of the interface states for five representative cases, namely, n=17,22,26,30, and 35. The frequency gap first decreases from n=17 to n=26, and then increase from n=26 to n=35. The underlying physics of the gradient dislocation can be described by the 1D Dirac equation with a position dependent and sign switch mass distribution (see details in Appendix B). Notably, the Dirac mass has opposite signs on the left and right sides of the gradient dislocation center. According the Jackiw-Rebbi theory, this suggests an emergent photonic bound state localize at the center of the gradient dislocations. As expected, the eigen calculation reveal a photonic bound state — the Dirac cavity mode — with even-symmetry localized at the center of the gradient dislocation [see Fig.2(h)]. In addition to the Dirac cavity mode, we notice that there exist other bound modes in the bulk band gap, which are also the solutions of the Dirac equation and can be interpreted as standing wave solutions [also see in Fig.3(a)]. Furthermore, we also implement an eigen calculation for a gradient dislocation with 52 periods along x-direction. Interestingly, the eigen mode in Fig.2(i) reveals a photonic bound states with odd symmetry emerging at the center of the gradient dislocation.

Since a larger periodic number N corresponds to a larger gradient structure with smaller relative displacement per column, the periodic number N plays a vital role in realizing high-performance Dirac cavity mode. Fig.3(a) shows the eigen spectra of the finite gradient dislocation versus periodic number. It is observed that the frequency of the Dirac cavity mode gradually increases and towards to pin at 0.4\frac{c}{a}, which corresponding to the frequency of degenerated interface states with g=0.5a. Additionally, we also notice that other bound states within the bulk band gap correspond to standing wave solutions of the 1D Dirac equation and are identified as higher-order Dirac cavity modes (see details in Appendix B and Appendix C). We then pay special attention to the performance of Dirac cavity by studying its Q-factor and modal area. As shown in Fig.3(b), the Q-factor increase rapidly for N<40, and nearly saturates for N>40. The reason is that there are two decay mechanisms for the Dirac cavity mode: it can decay into the upper and lower PCs or it can decay along the gradient dislocation, which are characterized by the two different decay rates 1/Q_y and 1/Q_x [1]. Since that Q_x increases exponentially with N, while Q_y is roughly N-independent, the overall Q-factor Q=\frac{1}{1/Q_x+1/Q_y} increases exponentially initially with N and then saturated at a certain N. For the Dirac cavity mode, the Q-factor saturated at nearly {10}^9 around N=40, while the higher-order Dirac mode I and II exhibit lower and higher saturated Q-factor, respectively. For comparison, we calculate both the Q-factor and the modal area of all eigen modes within the bulk band gap for N=52. The results in Fig.3(c) show that the Dirac cavity mode possesses a relatively high Q-factor with lowest modal area, indicating superior performance for applications. Alternatively, we compute the distribution of topological cavity mode in momentum space by utilizing the Fourier transformation. Fig.3(d) shows that the Dirac cavity mode is mainly localized at the edge of the Brillouin zone (k_x=\pm \frac{\pi }{a}) and lies outside of the line cone indicated by the dashed circle.

To further distinguish the Dirac cavity mode from higher-order Dirac mode, we consider a perturbed gradient dislocation by inserting a perfect electric conductor (PEC) into different positions of the gradient dislocation [see Fig.4(a)]. The distance between the PEC and the gradient dislocation center is denoted as d. We implement the eigen calculation of the perturbed gradient dislocation with d ranging from 0 to 12a, and the results are presented in Fig.4(b). When the PEC is inserted at the center (d=0), the persevered mirror symmetry of gradient dislocation results in two degenerate Dirac cavity modes with odd and even symmetry, respectively [see Fig.4(c)]. Inserting the PEC away from the center (e.g., d=a,4a,7a,10a) breaks mirror symmetry and significantly distorts the Dirac cavity mode [see the lower panel of Fig.4(b)]. Remarkably, the eigenfrequency of Dirac cavity mode remains unchanged [see the red points in Fig.4(b)]. In contrast, the eigenfrequencies of higher-order Dirac cavity modes shift, and their field distributions are disrupted. Fig.4(d) presents the Q-factor and the modal area of the Dirac cavity mode versus PEC position d. It is seen that both the Q-factor and the modal area exhibit a sudden decrease when d=a. This occurs because that the field profile of the Dirac cavity mode is largely distorted. As the PEC moves away from the center of gradient dislocation, both Q-factor and the modal area recover to their original levels as the Dirac cavity field profile restores.

Finally, we propose a coupled cavity-waveguide system by combining a gradient dislocation with a glide-symmetric waveguide. As shown in Fig.5(a), the gradient dislocation is formed by adjoining two finite PCs (orange and blue) of identical width but differing by one period. The boundary is surrounded by perfect matched layers. For convenience, we set the lattice constant a=740 nm. A glide-symmetric waveguide is introduced by shifting the lower PCs (green area) relative to the middle PCs (blue area) by half a lattice constant. The distance between the glide-symmetric waveguide and the gradient dislocation is denoted as d. By tuning the distance d, the coupling strength between the Dirac cavity and glide-symmetric waveguide can be effectively controlled. As shown in Fig.5(b), transmission dips confirm that the propagation of the interface states is strongly perturbed near the cavity eigenfrequencies. Utilizing the coupled cavity waveguide system, we further explore its potential application on refractive index sensing. Fig.5(c) presents the transmission spectra for refractive index change as the refractive index of the surrounding medium increases from n=1.000 to n=1.005 at an interval of {10}^{-3}. It is observed that the resonance dip redshifts with increasing of n, indicating a strong dependence of the Dirac cavity mode frequency on the surrounding refractive index. Fig.5(d) plots the resonance wavelength shift (\mathrm{\Delta }\lambda) versus refractive index change (\mathrm{\Delta }n) for different coupling strengths (controlled by d) between the Dirac cavity and the glide-symmetric waveguide. Linear fitting yields a refractive index sensitivity S=\frac{\mathrm{\Delta }\lambda }{\mathrm{\Delta }n}=521.44\mathrm{n}\mathrm{m}/\mathrm{R}\mathrm{I}\mathrm{U} for d=5a, demonstrating the potential of the Dirac cavity for index sensing application.

3 Conclusion

In conclusion, we propose an ultracompact photonic cavity design based on gradient dislocation. The underlying physics is well described by 1D Dirac equation with a position-dependent mass term that switches sign. The resultant photonic bound states — Dirac cavity mode — emerges a kink solution to the Dirac equation, localized at the center of gradient dislocation. We demonstrate that the Dirac cavity mode exhibits an ultrasmall modal area with relatively high Q-factor compared with other higher-order Dirac cavity modes. We also discuss the robustness of Dirac cavity and its potential application in refractive index sensing. Our ultracompact cavity design is valuable for scalable and tunable integrated photonic devices and can be generalized to other classical wave systems.

References

[1] J. D. Joannopoulos (Ed.), Photonic Crystals: Molding the Flow of Light, 2nd Ed., Princeton University Press, 2008

[2] S. Noda, A. Chutinan, and M. Imada, Trapping and emission of photons by a single defect in a photonic bandgap structure, Nature 407(6804), 608 (2000)

[3] Y. Akahane, T. Asano, B. S. Song, and S. Noda, High-Q photonic nanocavity in a two-dimensional photonic crystal, Nature 425(6961), 944 (2003)

[4] B. S. Song, S. Noda, T. Asano, and Y. Akahane, Ultra-high-Q photonic double-heterostructure nanocavity, Nat. Mater. 4(3), 207 (2005)

[5] T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity, Nat. Photonics 1(1), 49 (2007)

[6] M. Albrechtsen, B. Vosoughi Lahijani, R. E. Christiansen, V. T. H. Nguyen, L. N. Casses, S. E. Hansen, N. Stenger, O. Sigmund, H. Jansen, J. Mørk, and S. Stobbe, Nanometer-scale photon confinement in topology-optimized dielectric cavities, Nat. Commun. 13(1), 6281 (2022)

[7] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91(1), 015006 (2019)

[8] M. Kim, Z. Jacob, and J. Rho, Recent advances in 2D, 3D and higher-order topological photonics, Light Sci. Appl. 9(1), 130 (2020)

[9] B. Xie, H. X. Wang, X. Zhang, P. Zhan, J. H. Jiang, M. Lu, and Y. Chen, Higher-order band topology, Nat. Rev. Phys. 3(7), 520 (2021)

[10] G. Tang, X. He, F. Shi, J. Liu, X. Chen, and J. Dong, Topological photonic crystals: Physics, designs, and applications, Laser Photonics Rev. 16(4), 2100300 (2022)

[11] H. X. Wang,J. H. Jiang, A short review of all-dielectric topological photonic crystals, Front. Phys. (Lausanne) 10, 866552 (2022)

[12] X. Zhang,F. Zangeneh-Nejad,Z. G. Chen,M. H. Lu,J. Christensen, A second wave of topological phenomena in photonics and acoustics, Nature 618(7966), 687 (2023)

[13] R. Lin, T. Tai, L. Li, and C. H. Lee, Topological non-Hermitian skin effect, Front. Phys. (Beijing) 18(5), 53605 (2023)

[14] B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, Nonreciprocal lasing in topological cavities of arbitrary geometries, Science 358(6363), 636 (2017)

[15] Y. Chen, N. Gao, G. Zhu, and Y. Fang, Chiral topological whispering gallery modes formed by gyromagnetic photonic crystals, Phys. Rev. B 108(19), 195423 (2023)

[16] Z. Wu, Z. Wang, Y. Meng, J. Chen, X. Xi, P. P. Shum, and Z. Gao, Realization of a chiral topological whispering-gallery-mode cavity in gyromagnetic photonic crystals, Laser Photonics Rev. 19(6), 2401451 (2024)

[17] H. Mo, Z. Gao, X. Chen, and J. Dong, Realization of chiral whispering gallery mode cavities enabled by photonic Chern insulators, Laser Photonics Rev. 19(10), 2401450 (2024)

[18] Y. Yang and Z. H. Hang, Topological whispering gallery modes in two-dimensional photonic crystal cavities, Opt. Express 26(16), 21235 (2018)

[19] Y. Zeng, U. Chattopadhyay, B. Zhu, B. Qiang, J. Li, Y. Jin, L. Li, A. G. Davies, E. H. Linfield, B. Zhang, Y. Chong, and Q. J. Wang, Electrically pumped topological laser with valley edge modes, Nature 578(7794), 246 (2020)

[20] R. Barczyk, N. Parappurath, S. Arora, T. Bauer, L. Kuipers, and E. Verhagen, Interplay of leakage radiation and protection in topological photonic crystal cavities, Laser Photonics Rev. 16(9), 2200071 (2022)

[21] Y. Li, Y. Yu, F. Liu, B. Zhang, and G. Shvets, Topology-controlled photonic cavity based on the near-conservation of the valley degree of freedom, Phys. Rev. Lett. 125(21), 213902 (2020)

[22] B. Yan, B. Liao, F. Shi, X. Xi, Y. Cao, K. Xiang, Y. Meng, L. Yang, Z. Zhu, J. Chen, X. D. Chen, G. G. Liu, B. Zhang, and Z. Gao, Realization of topology-controlled photonic cavities in a valley photonic crystal, Phys. Rev. Lett. 134(3), 033803 (2025)

[23] D. Muis, Y. Li, R. Barczyk, S. Arora, L. Kuipers, G. Shvets, and E. Verhagen, Broadband localization of light at the termination of a topological photonic waveguide, Sci. Adv. 11(16), eadr9569 (2025)

[24] H. X. Wang, L. Liang, B. Jiang, J. Hu, X. Lu, and J. H. Jiang, Higher-order topological phases in tunable C₃ symmetric photonic crystals, Photon. Res. 9(9), 1854 (2021)

[25] L. Bao, A. Shi, Y. Peng, P. Peng, J. Ning, Z. Wang, C. Peng, and J. Liu, Non-Abelian permutations meet D₄ group in a four-state non-Hermitian system, Phys. Rev. B 110(2), L020104 (2024)

[26] A. Shi, L. Bao, P. Peng, J. Ning, Z. Wang, and J. Liu, Non-Hermitian Floquet higher-order topological states in two-dimensional quasicrystals, Phys. Rev. B 111(9), 094109 (2025)

[27] J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, Topological protection of photonic mid-gap defect modes, Nat. Photonics 12(7), 408 (2018)

[28] Y. Ota, F. Liu, R. Katsumi, K. Watanabe, K. Wakabayashi, Y. Arakawa, and S. Iwamoto, Photonic crystal nanocavity based on a topological corner state, Optica 6(6), 786 (2019)

[29] W. Zhang, X. Xie, H. Hao, J. Dang, S. Xiao, S. Shi, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, Low-threshold topological nanolasers based on the second-order corner state, Light Sci. Appl. 9(1), 109 (2020)

[30] X. Xie, W. Zhang, X. He, S. Wu, J. Dang, K. Peng, F. Song, L. Yang, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, Cavity quantum electrodynamics with second-order topological corner state, Laser Photonics Rev. 14(8), 1900425 (2020)

[31] Z. Li, S. Li, B. Yan, H. C. Chan, J. Li, J. Guan, W. Bi, Y. Xiang, Z. Gao, S. Zhang, P. Zhan, Z. Wang, and B. Xie, Symmetry-related large-area corner mode with a tunable mode area and stable frequency, Phys. Rev. Lett. 134(11), 116607 (2025)

[32] F. F. Li, H. X. Wang, Z. Xiong, Q. Lou, P. Chen, R. X. Wu, Y. Poo, J. H. Jiang, and S. John, Topological light-trapping on a dislocation, Nat. Commun. 9(1), 2462 (2018)

[33] Z. M. Zhan, P. Y. Guo, W. Li, H. X. Wang, and J. H. Jiang, Topological light guiding and trapping via shifted photonic crystal interfaces, Appl. Phys. Lett. 123(25), 251107 (2023)

[34] Q. Wang, H. Xue, B. Zhang, and Y. D. Chong, Observation of protected photonic edge states induced by real-space topological lattice defects, Phys. Rev. Lett. 124(24), 243602 (2020)

[35] Y. Liu, S. Leung, F. F. Li, Z. K. Lin, X. Tao, Y. Poo, and J. H. Jiang, Bulk–disclination correspondence in topological crystalline insulators, Nature 589(7842), 381 (2021)

[36] M. S. Hwang, H. R. Kim, J. Kim, B. J. Yang, Y. Kivshar, and H. G. Park, Vortex nanolaser based on a photonic disclination cavity, Nat. Photonics 18(3), 286 (2024)

[37] Z. K. Lin, Q. Wang, Y. Liu, H. Xue, B. Zhang, Y. Chong, and J. H. Jiang, Topological phenomena at defects in acoustic, photonic and solid-state lattices, Nat. Rev. Phys. 5(8), 483 (2023)

[38] S. Q. Shen, Topological Insulators: Dirac Equation in Condensed Matter, 2nd Ed., Springer Series in Solid-State Sciences No. 187, Springer, 2017

[39] F. Schindler, Dirac equation perspective on higher-order topological insulators, J. Appl. Phys. 128(22), 221102 (2020)

[40] R. Jackiw and P. Rossi, Zero modes of the vortex-fermion system, Nucl. Phys. B 190(4), 681 (1981)

[41] R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Phys. Rev. D 13(12), 3398 (1976)

[42] X. Gao, L. Yang, H. Lin, L. Zhang, J. Li, F. Bo, Z. Wang, and L. Lu, Dirac-vortex topological cavities, Nat. Nanotechnol. 15(12), 1012 (2020)

[43] K. Chen, F. Komissarenko, D. Smirnova, A. Vakulenko, S. Kiriushechkina, I. Volkovskaya, S. Guddala, V. Menon, A. Alù, and A. B. Khanikaev, Photonic Dirac cavities with spatially varying mass term, Sci. Adv. 9(12), eabq4243 (2023)

[44] L. Yang, G. Li, X. Gao, and L. Lu, Topological-cavity surface-emitting laser, Nat. Photonics 16(4), 279 (2022)

[45] J. Ma, T. Zhou, M. Tang, H. Li, Z. Zhang, X. Xi, M. Martin, T. Baron, H. Liu, Z. Zhang, S. Chen, and X. Sun, Room-temperature continuous-wave topological Dirac-vortex microcavity lasers on silicon, Light Sci. Appl. 12(1), 255 (2023)

[46] P. Delplace, Berry−Chern monopoles and spectral flows, SciPost Phys. Lect. Notes 39 (2022)

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