Topological photonics has been a rapidly growing field over a decade. Photonic topological insulators (PTIs) have wave-vector space topologies that lead to unique surface states of light. PTI edge states are immune to structural defects and disorders, and thus, can be used for the robust manipulation of light and topological lasing. Topological photonics also provides a powerful platform for experiments with topological concepts developed for condensed matter phenomena, such as the quantum Hall effect and quantum spin Hall effect. Because of fundamental differences between electrons and photons, the field has developed new fundamental topological ideas for diverse photonic platforms, such as photonic crystals, coupled ring resonators, photorefractive crystals, and ultrafast-laser direct-writing (ULDW) waveguides. ULDW enables the three-dimensional (3D) fabrication of integrated optical circuit chips for a wide range of applications, such as optical communications, data storage, sensing, topological physics, and quantum computing. Ultrafast lasers enable nonlinear absorption and material structure changes, which induce permanent refractive index changes inside transparent materials, such as glass. Compared with two-dimensional photonic crystal fabrication via planar lithography, continuous ULDW can form an arbitrary 3D waveguide geometry with high precision and speed.

The paraxial propagation equation in ULDW waveguide arrays is analogous to the Schr?dinger equation. In the past decade, various topological phenomena, including chiral edge states, higher-order topological insulators, anomalous Floquet photonic insulators, non-Hermitian topology, nonlinear topology, and nonabelian topology, have been demonstrated in ULDW waveguide arrays. ULDW-enabled topological photonic devices have applications in intrachip optical networks, optical computing, and quantum information processing, and have the potential to outperform their electronic counterparts in communication, energy consumption, and computation speed bottlenecks. Conventional photonic circuits are prone to fabrication errors, which limit their performance. Quantum computing is highly sensitive to system noise and errors. Topological optical chips fabricated using ULDW have proven to be robust against device defects and can maintain quantum entanglement.

We review photonic topological insulators engineered using ULDW in recent years and their underlying topological phases. First, we briefly discuss the background and mechanism of the ULDW waveguide and several techniques to improve the waveguiding performance, such as insertion loss and propagation loss (Fig.1). Next, we introduce the paradigmatic Su-Schrieffer-Heeger (SSH) model (Fig.2) and topologically invariant Zak phases to distinguish between the nontrivial and trivial topological phases of the SSH model. We discuss the experimental implementation of the SSH model and other various one-dimensional static topological insulators that exhibit topological edge localization (Fig.3). An adiabatic quantized Thouless pump can be achieved by slow deformation of the off-diagonal Aubry-André-Harper (AAH) model and the lopsided Rice-Mele model (Fig.4). Different experimental observations have shown that the edge of the topological edge mode in a honeycomb waveguide lattice is influenced by edge type, strained deformation, and transverse momentum (Fig.5). We summarize different high-order topological insulators (HOTIs), such as the two-dimensional SSH model, Kagome model, honeycomb with Kekulé distortion, and disclination array, demonstrating their topological corner states and disclination states (Fig.6). Floquet topological insulators can break the time-reversal symmetry via periodic z-axis (effective time axis) modulation of the two-dimensional geometry to induce a nonzero Chern number or winding number of the Floquet energy band and enable topological chiral edge modes on the geometry surface (Fig.7). Several schemes have been proposed, including helical waveguides, curved waveguides, and index modulation. Mukherjee and Maczewsky discovered an anomalous Floquet topological insulator with a zero Chern number but a nonzero winding number. Non-Hermitian (Fig.8) and nonlinear (Fig.9) topologies are beyond conventional topological concepts. Parity-time symmetry breaking phenomena in different non-Hermitian systems have been investigated using waveguide wiggling, inserted scatter points, and breaking points. Cerjan demonstrated a Weyl exceptional ring using a bipartite non-Hermitian optical helical waveguide array. Researchers have experimentally achieved nonlinearity-induced and tunable topological solitons in HOTIs, disclination-defect states, and off-diagonal AAH arrays. Jürgensen observed a fractional Thouless pump in off-diagonal AAH arrays via nonlinear tuning. Several research groups have experimentally investigated the ability of ULDW PTIs, including SSH arrays, off-diagonal AAH arrays, HOTIs, and fractal anomalous photonic insulators, to topologically protect photonic quantum entanglement via Hong-Ou-Mandel interference and quantum cross-correlation measurements (Fig.10). Nonabelian braiding, as a promising quantum computing tool, has been proposed and achieved using two-mode braiding (utilizing the geometry phases of arrays) and a Thouless pump (Fig.11).

ULDW has advantages, such as robustness, high nanofabrication precision, rapid prototyping, and 3D fabrication capabilities. The ULDW has proven to be a versatile platform for realizing various types of novel photonic topological insulators and exploring emergent topological phenomenon, such as high-dimensional, non-Hermitian, nonlinear, and nonabelian topologies. Those beyond the conventional PTIs are unclear and require further experimental and theoretical investigations. Compared with other materials, glass waveguides have disadvantages in terms of their electro-optic (EO) modulation capabilities. Glass waveguides can be thermally modulated, which is slow and power-consuming. Laser-directed wiring in other materials with improved electro-optic properties, such as lithium niobate, or integrating glass waveguides with EO modulators, may be potential solutions. More innovative ULDW PTIs designs are required to further reduce the influence of noise and protect the quantum states in quantum information applications.