Fig. 1. Degenerate cavity designed to form the OAM synthetic lattice and the topology of the bulk. (a) The degenerate cavity contains diverse OAM modes. A q-plate (q=1) and a waveplate are settled for coupling polarized OAM modes. WP, waveplate. (b) The WP with a centered hole in the cavity, where only optical modes with topological charge |m|≥2 pass through the WP and their polarization can be coupled. (c) The schematic of the synthetic lattice formed by OAM modes with different spins [red for left circularly polarized modes (↺), and blue for right circularly polarized modes (↻)]; η and δ are coupling strengths of the waveplate and q-plate, respectively. The 1-D chain in synthetic space is cut off between the different polarized fundamental Gaussian modes (m=0). The unit cells labeled by index l are indicated with dashed boxes. (d) The diagram of the topological phase (ν0,νπ) of our cavity system. The dashed line corresponds to η=π/2.

Fig. 2. Experimental observation of the energy band structures of the edge states. (a) The theoretical energy spectrum with edge states (I and II). (b) Distributions of the edge states with 0 (I) or ±π (II) energy when δ=0 or ±π. (c), (e) The schemes of the OAM lattice model when exciting different sites. Red for left circularly polarized modes (↺), and blue for right circularly polarized modes (↻). (d), (f) The normalized transmission intensity spectra with edge state I or II when pumping the cavity with a left circularly polarized fundamental Gaussian mode (d) or right circularly polarized m=2 OAM mode (f). ω−ω0 is the frequency detuning, and Ω=375  MHz represents the free spectral range (FSR).

Fig. 3. Experimentally measured edge state behaviors with edge perturbations and the spectrum discretization. (a) The schemes of the OAM lattice model when modulating the phase ϕ of the states on the edge. Red for left circularly polarized modes (↺), and blue for right circularly polarized modes (↻). (b) The detected transmission spectra with different ϕ when δ=−π/3. The edge state (indicated by the arrow) moves from ω=ω0 into bulk states as ϕ varies from 0 to 0.2π. ω−ω0 is the frequency detuning, and Ω=375  MHz represents the FSR.

Fig. 4. Detected spectrum discretization. (a) Left panel: an emitter near the surface. Right panel: oscillation of energy distribution |Ψ(L)|2 caused by the interference. (b) Schematic of the lattice dynamics when exciting the left circularly polarized OAM mode with topological charge m=6. Photons travel along the purple arrow, and the black cross indicates that the edge causes the reflection of the left-traveling photons. Red for left circularly polarized modes (↺), and blue for right circularly polarized modes (↻). (c), (e) The experimental (c) and numerically simulated (e) transmission intensity spectra. (d), (f) The experimental (d) and numerically simulated (f) transmission intensity spectra when δ=0.35π, corresponding to the dotted lines in (c) and (e). ω−ω0 is the frequency detuning, and Ω=375  MHz represents the FSR.

Fig. 5. Calculated eigen-energies and eigenstates when η=π/2. (a) The eigen-energies at different δ. (b) The eigenstates distribution as δ=9π/40, corresponding to the pink points in (a). The orange bars correspond to the zero energy edge states, while the blue bars correspond to bulk states. (c) The eigenstates distribution as δ=3π/4. The orange bars correspond to the π energy edge states, while the blue bars correspond to bulk states.

Fig. 6. Simulation of transmission spectra with (red)/without (blue) disorders when δ=π/4.

Fig. 7. (a) OAM distributions of the light circulating in the cavity. (b) Normalized transmission intensity spectra versus momentum k. Top and right panels: total normalized transmission intensity spectra.

Fig. 8. Experimental setup. SLM: spatial light modulator; PBS: polarizing beam splitter; AFG: arbitrary function generator; PD: photon detector; QWP: quarter-waveplate; WP: waveplate.

Fig. 9. (a) The intensity distributions of different cavity OAM modes. The dotted lines show the size of the pinhole. (b), (c) Inserts: the diffraction patterns of the OAM modes with different topological charges m=0 (b) and m=2 (c). Curves: normalized intensity profiles along the white dotted line.

Fig. 10. (a) The diagram of the topological phase. (b) The detected winding of the unit vector n(k) when only a q-plate is placed in the degenerate cavity. (c) The detected winding number when the q-plate is deposited after a half-waveplate in the cavity.