Topological photonics in one-dimensional settings

Over the past decade, topological photonics has emerged as a vibrant field, attracting significant attention and witnessing remarkable advancements. This growth can be attributed to its fundamental appeal and the unique opportunities it offers for unconventional control of light, promising innovations in next-generation photonic devices. At the heart of topological photonics lies the one-dimensional (1D) SSH model. Originally conceived to elucidate the physics of a molecular chain of polyacetylene, this model has found widespread applications in exploring a wide range of topological phenomena in photonics and beyond. In this chapter, we aim to provide an overview of topological photonics in one-dimensional (1D) settings. After briefly introducing paradigmatic 1D models, including the SSH, Rice-Mele, and AAH models, we review recent advances in experimental studies and applications of topological photonics based on 1D platforms. Our discussion highlights demonstrated examples, such as the nonlinear tuning of topological states in both Hermitian and non-Hermitian photonic SSH lattices, as well as nonlinear harmonic generation and topological lasing in SSH-type photonic microstructures. We further discuss characteristic topological phenomena in other representative 1D settings, including Floquet systems, topological pumping, quasicrystals, and synthetic non-Hermitian systems. Finally, we examine selected examples of two-dimensional (2D) photonic topological crystalline insulators that are closely linked to the SSH model. Towards the end, we summarize the chapter and provide a list of key contributions, together with an outlook on possible future directions in 1D topological photonics. While this review focuses specifically on 1D topological photonics, it is not intended to be comprehensive or exhaustive.

💡 Research Summary

This review article provides a comprehensive overview of the rapidly evolving field of one‑dimensional (1D) topological photonics, focusing on the foundational Su‑Schrieffer‑Heeger (SSH) model and its extensions such as the Rice‑Mele and Aubry‑André‑Harper (AAH) models. The authors begin by summarizing the historical context: topological concepts originally developed for electronic quantum Hall systems have been transplanted into optics, where the SSH chain serves as the simplest yet richest platform for exploring symmetry‑protected topological phases.

In the theoretical section, the SSH Hamiltonian is presented in its tight‑binding form with intracell (v) and intercell (w) hopping amplitudes. The presence of chiral (or inversion) symmetry forces the Hamiltonian into an off‑diagonal structure, allowing the definition of a winding number ( \mathcal{W}= \frac{1}{2\pi i}\int dk,\partial_k \ln\det h(k) ). When ( v<w ) the winding number equals one, guaranteeing a pair of zero‑energy edge states under open boundary conditions; when ( v>w ) the system is topologically trivial. The review explains how, in realistic photonic implementations where perfect chiral symmetry is hard to maintain, the edge states may shift away from exact zero energy but remain mid‑gap and retain their topological character.

The Rice‑Mele model adds a staggered on‑site potential ( u ) to the SSH chain, making ( u, v, w ) time‑dependent (or propagation‑distance dependent). This introduces an extra synthetic dimension, enabling a mapping onto a two‑dimensional Chern insulator. The resulting Chern number ( C = \frac{1}{2\pi}\int dk, d\alpha, (\partial_k A_\alpha - \partial_\alpha A_k) ) quantifies the number of particles pumped across the system during one adiabatic cycle—a direct photonic analogue of Thouless pumping.

The AAH model is discussed as a quasi‑periodic system where the on‑site potential is modulated by an irrational frequency ( b ). For rational ( b ) the lattice is periodic; for irrational ( b ) it becomes a quasicrystal, exhibiting a fractal Hofstadter‑like spectrum. The off‑diagonal version of the AAH model is highlighted for its experimental convenience, especially in curved‑waveguide implementations.

Experimentally, the authors describe their own work on optically induced photonic lattices in photorefractive crystals. By writing waveguide arrays with controlled spacing, they realize SSH lattices with tunable ( v ) and ( w ). Nonlinear Kerr or photorefractive effects allow real‑time modulation of the hopping ratio, thereby driving topological phase transitions (winding number changes) and enabling dynamic control of edge states.

Non‑Hermitian extensions are explored in depth. By introducing balanced gain and loss on alternating sublattices, PT‑symmetric SSH lattices exhibit complex band structures where the winding number can still be defined in the biorthogonal sense. Experiments demonstrate topological funneling of light, asymmetric transmission, and selective amplification of edge modes.

The review also covers several advanced 1D platforms beyond the static SSH chain:

• Floquet‑engineered SSH lattices where periodic modulation in time (or propagation distance) creates effective synthetic dimensions and enables observation of Floquet topological phases.
• Topological pumping experiments based on the Rice‑Mele protocol, where the authors measure quantized transport of light across the lattice, directly confirming the Chern number.
• Synthetic‑frequency resonator networks that implement off‑diagonal AAH models in the frequency domain, allowing exploration of non‑Hermitian physics and disorder‑induced localization in a highly controllable setting.
• Time‑multiplexed resonator arrays that realize dissipatively coupled SSH models, demonstrating robust edge‑state lasing and nonlinear parametric amplification.

The authors then bridge 1D physics to two‑dimensional (2D) topological crystalline insulators. By stacking SSH chains or arranging them in a 2D lattice with appropriate symmetry, higher‑order topological insulators (HOTIs) emerge, featuring corner states protected by a combination of Zak phase and Chern number invariants.

In the concluding section, challenges such as fabrication tolerances that break chiral symmetry, scaling synthetic‑dimension platforms to large numbers of modes, and integrating topological components with conventional photonic circuitry are discussed. Future directions highlighted include: exploiting the interplay of nonlinearity, non‑Hermiticity, and topology for reconfigurable photonic switches; developing topological lasers with low threshold and high robustness; implementing topological quantum simulators using synthetic dimensions; and exploring topological protection in emerging platforms like metasurfaces and 2D materials.

Overall, the review convincingly argues that 1D topological photonics, enriched by nonlinear and non‑Hermitian effects and extended through synthetic dimensions, offers a versatile toolbox for next‑generation photonic devices, ranging from robust light routing and frequency conversion to topologically protected lasing and quantum information processing.