Introduction

Topological insulators can explain a wide range of materials with a variety of symmetries, including time-reversal symmetry, spatial symmetries such as reflection and inversion, and crystalline symmetry1. The topological materials have distinct band structures. The topology of the bulk bands leads to a protected edge state, according to the topological bulk-edge correspondence. In addition, higher-order topological insulators (HOTIs) increase the range of topological materials and enable a better understanding of band topology2,3,4,5,6,7. Bulk topologies have been expanded to include protected modes even at corners, resulting in bulk-corner correspondence.

A topological insulator with D dimensions can support gapless states in D − 1 dimensions. This means that in a 2D topological insulator, for example, we find edge states that are gapless and confined to 1D boundaries1. In higher-order topological systems, we see the appearance of lower-dimensional states such as 0D corner states in the 2D system and 1D hinge states in the 3D system. This concept can be extended to the multipole moment2,8,9, Su–Schrieffer–Heeger (SSH)10,11,12, Lieb13,14,15,16, Kagome17,18,19, honeycomb20,21,22, and valley-hall lattice23,24,25 types, leading to new topological phases with lower-dimensional boundary states10,17,26. A higher-order band topology is protected inside the topological lattice type through reflection4, rotation6, and inversion symmetry27. Lower-dimensional topological states such as corner and hinge states remain immune to surface perturbations if the protective symmetry is preserved7.

A topological deformation is the next stage. The deformation of the host lattice characterized by real-space topology cannot be eliminated by local continuous transformation28. Dislocation29,30,31,32, disclination33,34,35,36,37,38, moiré lattice39,40,41,42, and Dirac vortex43,44,45,46,47,48,49,50,51,52 structures are representative topological deformations of the host lattice (Fig. 1). Topological deformations exhibit extensive lattice characteristics defined by real-space topological invariants, rendering them resilient to local deformations. Consequently, topological deformations enabled the successful demonstration of unique photonic devices, showing vector vortex lasing in a photonic disclination nanostructure33, polarization vortex modes in a Dirac vortex structure48, and magic-angle lasing in a moiré photonic lattice39 (Figs. 2 and 3).

Fig. 1: Design strategies for topologically deformed lattices.
figure 1

a Construction of an edge dislocation. Reprinted by permission from ref. 53, copyright 2021. b Volterra process for designing a topological disclination structure. Reprinted by permission from ref. 33, copyright 2024. c Honeycomb supercell of the generalized Kekulé modulation (left). Illustration of the Dirac-vortex cavity (right). Reprinted by permission from ref. 48, copyright 2020. d Generation of a twist bilayer moiré photonic crystal. Reprinted by permission from ref. 41, copyright 2022.

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Fig. 2: Photonic structures with topological dislocation and disclination.
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a Illustration of the dislocation defect with the nontrivial unit cell (left). Normalized s-SNOM amplitude image (top) and the simulated Ez fields (bottom). Reprinted by permission from ref. 31, copyright 2021. b Two topological defects connected by a disclination forming an open arc (left). Measured Ez field for a curved topological waveguide (right). Reprinted by permission from ref. 37, copyright 2020. c Spectral charge distribution in a finite disclination structure (top). Ez field profiles of disclination states. Reprinted by permission from ref. 36, copyright 2021. d SEM images of fabricated photonic disclination cavities (left). Measured polarization-resolved lasing images (right). Reprinted by permission from ref. 33, copyright 2024.

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Fig. 3: Other types of topologically deformed lattices.
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a Schematic of a topological junction consisting of two different gratings (top). Dirac mass distribution and associated Jackiw-Rebbi state intensity profiles (bottom). Reprinted by permission from ref. 58, copyright 2022. b Spectra and far fields of Dirac vortex cavity modes of different winding numbers (left). SEM image of a Dirac vortex cavity (right). Reprinted by permission from ref. 48, copyright 2020. c Calculated band diagram of a magic-angle laser (left). SEM image and single-cell lasing pattern of a magic-angle laser with a twist angle of 4.41° (right). Reprinted by permission from ref. 39, copyright 2021. d Illustration of a Floquet fractal topological insulator (left) and its eigenvalue spectrum (right). Reprinted by permission from ref. 61, copyright 2022.

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In this Perspective, we review advancements in various topologically deformed lattices and their photonic applications, including dislocation, disclination, Dirac vortex, moiré lattice, and fractals. Each structure exhibits various deformation characteristics. For example, dislocations and disclinations are lattice defects caused by bulk distortions, and moiré patterns are formed by rotation and overlapping. In addition, Dirac vortices are formed by the gradual spatial distribution of Dirac mass, and fractals form through lattice partitioning with the central area remaining void. We investigate the underlying physical principles and discuss how low-threshold lasers and efficient photonic devices can be realized in a variety of topological structures. We also provide an outlook on the challenges and potential future developments and their impact on nanophotonics.

Dislocation

A topological lattice defect occurs as a result of a deformation of the hosting lattice, which is protected by the nontrivial bulk band topology. A dislocation is a representative example of a topological defect. A dislocation within a photonic crystal is essentially a structural anomaly characterized by a Burgers vector that encapsulates the mismatch in lattice translation around a loop enclosing the dislocation (Fig. 1a). To create this dislocation in a 2D lattice, the following process is employed53: a line of atoms is removed, ending at a site known as its center or core, and the sites across the missing line of atoms are subsequently joined, restoring translational symmetry everywhere in the system except near the core of the defect. A closed loop around the dislocation center exhibits a missing translation by the Burgers vector b. The weak topological invariant Q, which indicates the number of localized modes, incorporates elements from the crystal’s band structure topology and its dislocation30:

$$Q=\frac{1}{\pi }\,{\boldsymbol{\theta }}\,\cdot\, {\boldsymbol{b}}\,\mathrm{mod}\,2,$$

where \({\boldsymbol{\theta }}\,{\boldsymbol{=}}\left(\frac{{\theta }_{{XM}}}{P}{\boldsymbol{,}}\frac{{\theta }_{{YM}}}{P}\right)\) denotes the Zak phase below the bandgap along the XM and YM directions, while b characterizes the edge dislocation in real space. The value of Q is constrained to either 0, indicating the absence of a localized mode, or 1, signifying the presence of one.

Lu et al. used scattering-type scanning near-field optical microscopy (s-SNOM) to experimentally verify the presence of localized 0D dislocation modes at approximately 100 THz within a topological photonic crystal31 (Fig. 2a). The topological photonic crystal was designed with a nontrivial Zak phase and edge dislocation, which theoretically imply strong light localization. The experimental setup comprised measuring optical near fields on top of an array of amorphous silicon nanopillars placed on a chromium film at various frequencies around 100 THz. The localized mode observed at the dislocation center within the bandgap indicated light confinement to subwavelengths. This extensive experimental analysis revealed the mid-bandgap light localization with nontrivial Zak phase and an edge dislocation, highlighting the potential of weak topology for precisely manipulating light at the nanoscale. A similar finding of topological light-trapping within a dislocation in a 2D photonic crystal was experimentally attained using a dual-topology process, revealing a unique way for robust, subwavelength-scale cavity mode localization30. Furthermore, the 2D dislocation structure can be extended into 3D in nanophotonics. It has the possibility to explore the interplay of dislocations and 3D Weyl crystals54,55.

Disclination

When a cylinder is cut along the long side to expose two edges and the edges are rotated relative to one another, rotational defects generate disclinations. More systematically, disclination geometries with Cn symmetry can be constructed via the Volterra process applying the Frank angle Ω (=π/2) to the 2D lattice (Fig. 1b). For example, a C5 symmetric disclination structure is created using the following process: the lattice is cut into four identical sectors, as is always feasible due to C4 symmetry, and a disclination defect with a positive Frank angle is formed by adding one sector and gluing the remaining sectors back together, resulting in a nominally pentagonal shape35.

Fractional numbers of charges localized at corners provide an additional method for identifying HOTIs. Disclinations can also trap fractional charges, allowing for the emergence of localized states with topological properties. The bulk-disclination correspondence highlights the value of fractional charges in analyzing the crystalline topology of such systems. In rotationally symmetric topological crystalline insulators, the fractional charge q, is defined by the relationship:

$$q=\,\frac{\Omega }{2\pi }\eta +\,\sum _{i,j=1,2}{\varepsilon }_{{ij}}{B}_{i}{P}_{j}\,\mathrm{mod}\,1,$$

where the Frank angle (Ω) and Burgers vector (B) indicate the topological class of the disclination defect. The band structure topology is reflected by the polarization P and Wannier representation index η, and the Levi–Civita symbol \({\varepsilon }_{{ij}}\) indexes the dimensions. Higher-order topological states are typically observed at the boundaries of the structure’s disclination core.

Wang et al. combined topological photonics and crystal lattice defects, and introduced topological deformations in valley photonic crystals, providing a method for generating robust photonic edge states37 (Fig. 2b). The manipulation of these topological defects to generate disclinations that functioned as conduits for photonic topological edge states differed from conventional techniques that relied on the band structure’s nontrivial topology. The edge states allowed the construction of waveguides and resonators in large-scale photonic structures without the use of external boundaries. The theoretical analysis and experimental results demonstrated that such disclination-induced edge states were able to traverse curved paths and sharp bends with minimal loss, underlining their potential for designing complex photonic systems.

Liu et al. investigated the concept of bulk-disclination correspondence within topological crystalline insulators, using reconfigurable photonic crystals to reveal the interaction between the crystalline structure’s bulk properties and topological disclinations36 (Fig. 2c). The topological defects are inherent in crystalline materials and manifest fractional spectral charge and robust bound states, providing a direct probe into the crystalline topology. This work showed how disclination-induced fractional charge developed from symmetry-protected bulk charge patterns. The emergence of localized disclination states was observed using pump-probe and near-field detection measurements. These findings established a foundational understanding of the interaction between topology and crystallography in photonic systems.

More recently, Hwang et al. successfully demonstrated wavelength-scale vortex and anti-vortex nanolasers built within C5 symmetric optical cavities with topological disclinations33 (Fig. 2d). By correlating tight-binding models with optical simulations, disclination geometries were converted into optical nanocavities, resulting in resonant modes with strong confinement and preserved topological charges. The experimental verification of these vector vortices included precise measurements of polarization-resolved images, Stokes parameters, and self-interference patterns, which clearly showed the unique lasing features. This work pioneered a strategy for developing low-threshold, high-quality vortex nanolasers, representing a significant advancement in nanophotonic devices capable of controlling light’s angular momentum for improved data transmission channels in optical communication.

These recent insights into disclinations have highlighted the importance of topology in a variety of physical systems, including those involving matter and light. Such topologically defined nanostructures may be crucial in incorporating orbital angular momentum (OAM) into on-chip photonic devices. OAM from topological nanostructures will be employed in next-generation optical communications to take advantage of its massive encoding capacity, improving data rates while also introducing a novel approach for producing coherent structured light from subwavelength devices.

Dirac vortex

HOTIs are seen as boundary states governed by massive Dirac Hamiltonians influenced by the signs of the Dirac masses1. In symmetry-preserving systems, flipping these mass signs on adjacent surfaces results in gapless topological states. The Jackiw-Rebbi state, defined in terms of a 1D Dirac field coupled to a static background soliton field, has a localized zero-energy mode near the domain wall where the Dirac mass term changes sign56. This model, one of the earliest descriptions of a topological insulator, highlights the trapping of a state caused by the mass term’s sign flip at the boundary, which resembles a metallic boundary between two gapped regions57. On the other hand, the Jackiw-Rossi state emerges in 2D systems with a Dirac mass structure that resembles a vortex. This configuration is exemplified by a Kekulé modulation, which alters the phase of the Dirac mass term, thereby creating a Dirac vortex. Furthermore, the mass configuration causes a topological deformation that traps modes, thereby enhancing the high-power performance of surface-emitting lasers and enabling vector beams in quantum cascade lasers49,52.

Recent research investigates the design of a nanophotonic light emitter using a topological junction produced by two guided-mode resonance gratings58 (Fig. 3a). This structure uses a leaky Jackiw-Rebbi state resonance to confine light in-plane via a funnel-like energy flow, considerably increasing the probability of emission. The emitter’s design facilitates highly efficient, narrow beam light emission with a small divergence angle. Additionally, the emitter allows for customizable beam shaping by adjusting the Dirac mass distribution inside the lattice geometry. The topological phase of the Dirac mass distribution can be manipulated to produce beams with tailored profiles, supporting applications in telecommunications.

Advances in photonics technology have led to the development of Dirac-vortex topological cavities, utilizing photonic crystals with a honeycomb structure to improve single-mode operation in semiconductor lasers48 (Fig. 3b). This work provides scalable mode areas, arbitrary mode degeneracies, and vector-beam vertical emission, enhancing their applicability on high-index substrates. Notably, the design leverages generalized Kekulé modulations to facilitate the integration of complicated lasing modes with a large free spectral range, which is required for steady single-mode operation over a wide spectrum range. The Dirac-vortex cavity showed a single mid-gap mode with a tunable modal diameter, exploring the 2D upgrade of the 1D feedback structure in the distributed feedback and vertical-cavity surface-emitting lasers.

Moiré lattice

Moiré superlattices form when two periodic arrays overlap with either a difference in lattice constants or a shift in spatial orientation, resulting in modified rotational symmetry and pattern periodicity40. Real-space periodicity in moiré superlattices produces reciprocal lattice vectors, allowing Bloch modes with different momenta to be coupled across twisted layers. This interaction encourages the creation of moiré Bloch modes with unique wavevectors, which results in localized wavefunctions and flatbands in the energy diagram. The moiré structure is particularly different from conventional photonic crystals, where Bloch modes remain delocalized42.

In 2021, Mao et al. introduced and experimentally demonstrated magic-angle lasers that use twisted photonic graphene superlattices within semiconductor membranes, deviating from the requirement for a full photonic bandgap for light localization39 (Fig. 3c). By adjusting the twist angle between two photonic graphene lattices, a flatband condition was obtained showing strong field localization and high-quality lasing without a full bandgap. Mode volumes are smaller than (λ/n)3 and quality factors are over 400,000. This unique interlayer coupling between the twisted lattices leads to the emergence of the flatband with a narrow frequency range and a boosted density of states. In addition, all these magic-angle superlattices have dominant lasing mode in the flatband despite the distinct difference in band diagrams. They provide a simple yet effective method for constructing compact and reconfigurable nanolaser arrays with low thresholds and narrow linewidths. Nanolaser arrays based on optical flatbands within twisted photonic graphene structures were effectively demonstrated, achieving synchronized lasing from individual to collective arrangements with exceptional control over spatial and spectral coherence42.

Fractal

Fractals, first proposed by Mandelbrot in 1967 to address the coastline paradox, are complex structures in which smaller sections reflect the whole59. These mathematical structures have had a substantial impact on a variety of scientific domains, appearing in natural phenomena such as polymer forms and aggregation processes, as well as crucial roles in nonlinear photonic applications and chaotic systems. For example, the Sierpinski gasket60 has emerged as a key structure in the study of complicated topological processes. This fractal is distinguished by its self-similar design. It is created by recursively subdividing a triangle into smaller triangles, deleting the central one at each step. This iterative method produces a highly fractured structure that challenges traditional views on spatial continuity and dimensional integrity61.

Recently, the features of fractal geometries within the framework of topological insulators using photonic lattices, specifically the Sierpinski gasket design, were explored. This study considerably contradicts standard topological theories that rely primarily on the existence of a bulk to provide topologically protected states61. Despite the absence of actual bulk in fractal lattices, these systems can maintain topologically protected chiral edge states, which are obtained by using helically driven photonic lattices of coupled waveguides. Under periodic driving, topologically trivial structures transition to nontrivial topological properties. Fractal topological insulators not only preserve but significantly improve transport features along their edges, exhibiting higher velocities than comparable non-fractal systems and providing fresh insights into the mobility and robustness of topological edge states against disruptions. This work broadens the understanding of topological insulators beyond traditional boundaries, implying that fractal geometries may change the design and deployment of more efficient photonic devices by using their distinct topological properties.

Perspectives

We examined recent achievements and innovations in photonic structures with topologically deformed lattices. Topological defects in photonic crystals, such as dislocations, disclinations, Dirac vortices, moiré lattices, and fractals, play a pivotal role in shaping the properties and applications of optical devices. Dislocations induce localized modes within a crystal, increasing the potential for nanoscale light manipulation. Disclinations, characterized by rotational defects and fractional charges, facilitate the emergence of robust and localized topological states, considerably improving light manipulation and data transmission. Dirac vortex structures and topological cavities lead to advances in photonics by enhancing light emission and confinement, thus boosting the efficiency and customizability of devices for telecommunications and optical computing. Moiré lattices, formed by overlapping periodic arrays, alter rotational symmetry and periodicity, enabling unique interactions that trap light and change its flow within the material. Last, fractals, such as the Sierpinski gasket, display self-similarity and maintain topologically protected states even without a conventional bulk, suggesting new pathways for photonic device development.

Deformations should be considered in the context of their interactions with more complex topological systems. First, higher-order topological deformations based on multipole moments, such as octupoles, are a promising research topic alongside dipole and quadrupole moments2,62. Second, there exist topological states induced by non-Hermitian modulation, which are viewed as a type of deformed lattice in imaginary space. For example, the Jackiw-Rebbi model with imaginary mass transition can lead to a deformed lattice supporting non-Hermitian interface states without the Hermitian counterpart. These approaches can provide unique strategies to control the topologically protected states63,64. Third, inhomogeneous strains at optical frequencies in a photonic lattice generate pseudomagnetic fields. They can achieve nearly flatten Landau levels by developing a Landau gauge vector potential for a pseudomagnetic field65,66,67.

These topological properties in photonic structures, notably those with topologically deformed lattices, contribute to the creation of high-efficiency photonic systems. Such advancements not only promise substantial energy savings but also enhance the performance and scalability of photonic systems, paving the way for more sustainable technological solutions in the field of optical communications and beyond. Furthermore, these structures provide a toolbox for stabilizing information and energy transport and manipulation, enabling topologically protected quantum/integrated photonics, topological lasing, topological switching, and topological waveguiding. We believe that the intersection of topological physics and device engineering will herald in a new era of environmentally friendly and highly effective photonic technologies.