Recently, higher-order topological phases that do not obey the usual bulk-edge correspondence principle have been introduced in electronic insulators and brought into classical systems, featuring with in-gap corner/hinge states. In this letter, using near-field scanning measurements, we show the direct observation of corner states in second-order topological photonic crystal slabs consisting of periodic dielectric rods on a perfect electric conductor. Based on the generalized two-dimensional Su-Schrieffer-Heeger model, we show that the emergence of corner states roots in the nonzero edge dipolar polarization instead of the nonzero bulk quadrupole polarization.We demonstrate the topological transition of two-dimensional Zak phases of PC slabs by tuning intra-cell distances between two neighboring rods. We also directly observe in-gap onedimensional edge states and zero-dimensional corner states in the microwave regime. Our work presents that the PC slab is a powerful platform to directly observe topological states, and paves the way to study higher-order photonic topological insulators.

Recent research in topological photonics has not only proposed and realized novel topological phenomena such as one-way broadband propagation and robust transport of light, but also designed and fabricated photonic devices with high-performance indexes, which are immune to fabrication errors such as defects or disorders. Photonic crystals, which are periodic optical structures with the advantages of good light field confinement and multiple adjusting degrees of freedom, provide a powerful platform to control the flow of light. With the topology defined in the reciprocal space, photonic crystals have been widely used to reveal different topological phases of light and demonstrate topological photonic functionalities. This review presents the physics of topological photonic crystals with different dimensions, models, and topological phases. The design methods of topological photonic crystals are introduced. Furthermore, the applications of topological photonic crystals in passive and active photonics are reviewed. These studies pave the way for applying topological photonic crystals in practical photonic devices.

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Topological photonics is an emerging field that attracts enormous interest for its novel ways to engineer the flow of light. With the help of topological protection, the surface modes of topological photonic systems have intriguing properties, such as the unidirectional propagation, robust transmission against defects and disorders, which meet the rapidly growing demands for information processing. Valley photonic crystals, as one kind of topological photonic systems, not only support protected surface modes, but also are friendly to micro-nano fabrication. These advantages show that it has broad prospects in constructing high-performance photonic devices or even photonic integrated circuits. Here, we review the properties and development of valley photonic crystals. Firstly, the theory and structure are briefly introduced and then the discussion of robust transmission will be followed. Furthermore, prototypes of on-chip devices based on valley photonic crystals are reviewed. As a perspective in photonics, valley photonic crystal is expected to become a good platform to study nanophotonics and realize advancing integrated photonics devices.

Edge states of photonic crystals have attracted much attention for the potential applications such as high transmission waveguide bends, spin dependent splitters and one-way photonic circuits. Here, we theoretically discuss and experimentally observe the deterministic edge states in checkerboard photonic crystals. Due to the self-complementarity of checkerboard photonic crystals, a common band gap is structurally protected between two photonic crystals with different unit cells.Deterministic edge states are found inside the common band gap by exploiting the Zak phase analysis and surface impedance calculation. These edge states are also confirmed by a microwave experiment.

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The recent exploration of the valley degree of freedom in photonic systems has enriched the topological phases of light and brought the robust transport of edge states around sharp bends. The two and more simultaneous band gaps in valley-Hall systems have attracted researchers' attention for enlarging the working bandwidth. However, band gaps with frequency-dependent topologies were not reported and the demonstrated flow of electromagnetic waves is limited to the robust transport of edge states. Here, the frequency degree of freedom is introduced into valley photonic crystals with dual band gaps. Based on the high-order plane wave expansion model, we derive an effective Hamiltonian which characterizes dual band gaps. Metallic valley photonic crystals are demonstrated as examples in which all four topological phases are found. At the domain walls between topologically distinct valley photonic crystals, frequency-dependent edge states are demonstrated and a broadband photonic detouring is proposed. Our findings provide the guidance for designing the frequency-dependent property of topological structures and show its potential applications in wavelength division multiplexers.

Valley photonic crystal is one type of photonic topological insulator, whose realization only needs P-symmetry breaking. The domain wall between two valley-contrasting photonic crystals support robust edge states which can wrap around sharp corners without backscattering. Using the robust edge states, one can achieve the pulse transmission. Here, using time-domain measurement in the microwave regime, we show distortionless pulse transmission in a sharply bended waveguide. An Ω-shaped waveguide with four 120° bends is constructed with the domain wall between two valley photonic crystal slabs. Experimental results show the progress of Gaussian pulse transmission without distortion, and the full width at half maximum of the output signal was changed slightly in the Ω-shaped waveguide. By measuring steady state electric field distribution, we also confirmed the confined edge states without out-of-plane radiation which benefits from the dispersion below the light line. Our work provides a way for high-fidelity optical pulse signal transmission and develop high-performance optical elements such as photonic circuits or optical delay lines.