Fig. 1. Typical examples of flatband lattices: (a) Quasi-one-dimensional (1D) rhombic lattice; (b) Lieb lattice; (c) Kagome lattice; (d) super-honeycomb lattice (sHCL). Compact localized states (CLSs) are depicted as colored sites, where zero amplitudes are denoted by gray color, and those with non-zero amplitudes of opposite phase are denoted by red and blue colors.

Fig. 2. Examples of photonic flatband structures: (a) Kagome lattice for terahertz spoof plasmons, displaying an omnidirectional minimum in the transmission at the flatband frequency (dashed line) in the right panel^[48]; (b) experimental setup exposing the zero-refractive-index all-dielectric metamaterials with a square lattice to realize cloaking inside a channel with the Dirac point, and the right panel shows corresponding three-dimensional dispersion diagrams consisting of a Dirac cone and a flatband^[50]; (c) structured microcavity forming a 1D stub lattice and its photoluminescence spectrum, revealing a flatband in the middle^[51].

Fig. 3. Examples of femtosecond laser-writing flatband photonic lattices: (a) A photonic Lieb lattice for demonstration of flatband compact localized states^[64,65]; (b) a photonic Kagome lattice established for demonstration of topological corner states^[66]; (c) a driven photonic rhombic lattice for experimental observation of Aharonov-Bohm cages^[67,68].

Fig. 4. Examples of photonic lattices created by multiple-beam optical induction method^[80,83]. Top panel shows schematic of experimental setup. PBS, polarized beam splitter; SBN, strontium barium niobite. Bottom panel shows typical examples of photonic lattices realized in experiment: (a) A “perfect” honeycomb lattice; (b) an inversion-symmetry-breaking honeycomb lattice; (c) a vortex lattice; (d) a Kagome lattice.

Fig. 5. Examples of photonic lattices created by direct cw-laser-writing technique in a nonlinear bulk crystal. Top panel shows illustration of experimental setup. SLM, spatial light modulator; BS, beam splitter; FM, Fourier mask. Bottom panel shows typical examples of photonic lattices created by direct cw-laser-writing method: (a) A photonic Lieb lattice with “bearded” edges^[89]; (b) a photonic Kagome lattice with flat boundary^[90]; (c) a photonic sHCL^[91]; (d) a driven photonic rhombic lattice with refractive index gradient parallel to the ribbon^[92].

Fig. 6. Experimental results of CLSs in flatband lattices: (a) Linear image (formed by CLSs) propagation through an optically induced Lieb photonic lattice^[85]; (b) a bound-state transmission in a Kagome photonic lattice^[104]; (c) observation of a quincunx-shaped (U = 2) compact localized state which spans over two-unit cells in a photonic rhombic lattice^[92]. From left to right: shown are the lattices, calculated band structures in the tight-binding approximation, probe beam inputs, and their in-phase and out-of-phase outputs.

Fig. 7. Demonstration of unconventional line state in photonic Lieb and super-honeycomb lattices^[89,91]: (a), (d) Schematic of flatband line states in infinite lattices, and insect in (d) shows the band structure of sHCL; (b1), (e1) out-of-phase input line beam; (b2), (e2) out-of-phase output without the lattice; (b3), (e3) out-of-phase output through the lattice; (b4), (e4) simulation results showing the out-of-phase line beam remains intact but the in-phase line deteriorates after propagating a long distance through the lattice; (b5) measured k-space spectrum of (b3) with a dashed square marking the first Brillouin zone (BZ); (c1)−(c5) the same as in (b1)−(b5), and (f1)−(f4) the same as in (e1)−(e4) except that the line beam is in in-phase excitation condition; (e5) momentum space spectrum of (e3), where the white dashed lines outline the first and second BZs.

Fig. 8. (a) Illustration of the noncontractible loop states (NLSs) in an infinitely extended Kagome lattice; (b) a torus showing two NLSs mimicking the 2D infinite lattice; (c) two robust boundary modes (RBMs) in a Kagome lattice with flat cutting edges, where black sites are of zero-amplitude, while blue and red ones distinguish non-zero sites with opposite phase; (d) schematic diagram of the Corbino-shaped Kagome lattice, where the NLS is illustrated by the orange circle; (e) from left to right (e1), (e3) experimental results of RBM1 and RBM2 under out-of-phase condition; (e2), (e4) simulation result corresponding to (e1) and (e3); (f1) experimentally established finite-sized Kagome lattice in a Corbino-geometry; (f2)－(f4) the NLS observed in (f2) experiment and (f3) simulations after propagation to 10 mm and (f4) 40 mm under out-of-phase condition. All insets are from input ring necklace of the probe beam. In-phase excitation destroys all localized states^[90].