Objective To investigate the formation, evolution, and regulation mechanisms of topological states induced by the combined effects of disclination topological defects and Floquet periodic modulation in honeycomb photonic lattices, and to explore how disclination structures with different rotational symmetries regulate the generation, localization, and propagation characteristics of vortex beams. In contrast to conventional topological photonic systems, where topological states are mainly associated with edges or boundaries, this study focuses on higher-order localized topological states bound to structural defects inside the lattice. By constructing \rmC_5 and \rmC_7 disclination honeycomb photonic lattices, this work aims to reveal the relationship between disclination-induced degenerate states, fractional charge localization, and vortex modes carrying orbital angular momentum. Furthermore, the role of Floquet periodic modulation in reconstructing the band structure, opening band gaps, and controlling localized topological states is analyzed. The study provides a theoretical basis for realizing stable topological vortex optical fields and offers guidance for the design of tunable topological photonic devices based on defect engineering and nonequilibrium periodic driving.

Methods Honeycomb photonic lattice models containing disclination structures with \rmC_5 and \rmC_7 rotational symmetries are constructed. These two types of disclination topological defects are introduced by removing or inserting a 60° wedge angle in an ideal honeycomb lattice, corresponding to fivefold and sevenfold coordination around the defect core, respectively. Such geometrical operations break the original translational and rotational symmetries of the lattice and induce effective curvature or gauge-field-like responses near the defect center. Based on Wannier center theory and fractional charge theory, the formation mechanisms of disclination states, corner states, and edge states are analyzed, and the connection between localized fractional charge distributions and higher-order topological states is discussed. For the optical propagation model, the paraxial wave equation is employed to describe the evolution of light fields in the photonic lattice. Under the linear approximation, the nonlinear term is neglected, and the problem is reduced to an eigenvalue equation similar to a two-dimensional Schrödinger equation with an effective periodic potential. The finite-difference method is then used to discretize the system and solve the eigenvalue problem numerically, from which the band structure, eigenvalue spectrum, and eigenmode distributions of the \rmC_5 and \rmC_7 disclination structures are obtained. To further introduce nonequilibrium topological control, a helical periodic potential is applied to implement Floquet periodic modulation. By transforming the system into a co-rotating reference frame, an equivalent gauge-field theoretical model is established, in which the helical modulation acts as a periodically driven effective vector potential. Finally, vortex modes carrying orbital angular momentum are constructed by linearly combining degenerate disclination states with an additional π/2 phase shift, and their propagation behavior in the periodic potential is studied through numerical simulations over long propagation distances.

Results and Discussions Numerical results show that both \rmC_5 and \rmC_7 disclination honeycomb photonic lattices can generate stable localized disclination states under appropriate structural parameters. The number of disclination states is closely related to the rotational symmetry of the system. Specifically, five disclination states are formed in the \rmC_5 structure, while seven disclination states appear in the \rmC_7 structure. Several of these states exhibit clear degeneracy in the eigenvalue spectrum, such as the degenerate pairs observed in the corresponding disclination-state modes. These results are consistent with the theoretical analysis based on Wannier center displacement and fractional charge localization, indicating that disclination defects can induce higher-order localized topological states by modifying the local topological response of the lattice. After the introduction of Floquet periodic modulation, the band structure of the system is significantly reconstructed. The helical periodic potential introduces a periodic driving along the propagation direction, causing the bands to fold in quasienergy space and leading to the opening of new band gaps near the original band-crossing or band-approaching regions. Meanwhile, some originally degenerate disclination states undergo rearrangement in their energy ordering and mode distributions. For example, in the \rmC_5 structure, the degenerate disclination states change their relative spectral positions after helical modulation, showing that Floquet periodic modulation can not only regulate the band width and band-gap size, but also affect the formation, separation, and stability of localized topological states. This demonstrates that disclination defects and Floquet modulation have a synergistic effect in controlling topological states in nonequilibrium photonic systems. The vortex modes constructed from degenerate disclination states show robust propagation behavior. For the \rmC_5 structure, degenerate disclination states are linearly combined to form a complex vortex mode with a well-defined phase-winding feature. A similar construction is applied to the \rmC_7 structure using its corresponding degenerate disclination states. During propagation, the amplitude distributions of the vortex beams remain relatively stable in both structures, with only small overall fluctuations even after 100 propagation periods. The phase distributions exhibit a continuous variation from -π to π around the vortex core, confirming that the constructed modes possess typical vortex phase structures and well-defined orbital angular momentum. These results indicate that disclination structures can effectively induce and maintain stable topological vortex beams, while Floquet periodic modulation further enhances the controllability and robustness of their propagation.

Conclusions The study demonstrated that disclination topological defects can effectively induce higher-order localized topological states in honeycomb photonic lattices, and that the number and distribution of these states are strongly determined by the rotational symmetry of the disclination structure. Floquet periodic modulation provides an additional dynamic control mechanism by reconstructing the band structure, opening band gaps, and regulating the degeneracy and spatial distribution of disclination states. Vortex modes formed by linear combinations of degenerate disclination states exhibit stable amplitude distributions, clear phase-winding characteristics, and strong propagation robustness over long propagation distances. These findings reveal the synergistic mechanism of disclination defects and nonequilibrium periodic driving in controlling topological optical fields. They provide theoretical insights into topological defect states, Floquet topological photonics, and vortex beam manipulation, and may contribute to the future design of robust topological photonic devices for optical information processing, vortex beam control, and topological light transport.