Fig. 1. Configuration of a synthetic photonic stub lattice. (a) Two coupled ring resonators, where the FSR of ring A is half of the FSR of ring B, i.e., 2ΩA=ΩB≡Ω. Ring A undergoes dynamic modulation by placing an EOM with the modulation frequency ΩM=Ω/2. Waveguides are connected to rings for input/output signals. (b) The system in (a) can be mapped into a photonic stub lattice along the synthetic frequency dimension (f), with An, Bn, and Cn indicating three types of lattice sites. (c) The corresponding band structures of the synthetic stub lattice in (b) with g=κ and ϕ=−0.5π.

Fig. 2. Band structure measurements for the case of Bin→Bout. (a1)–(d1) Experimentally observed band structures with different modulation amplitudes VM. (a2)–(d2) Simulation results of the projected output intensity distribution of the band structure on mode Bk, based on Eqs. (4)–(6), where g takes different values with fixed κ=0.06Ω and ϕ=−0.5π. (a3)–(d3) Measured transmission spectra from the drop port of ring B. The vertical axis represents the frequency detuning of the input laser source normalized to Ω, while the bottom horizontal axis in (a1)–(d2) represents one roundtrip time in ring B with the period of 2π/Ω.

Fig. 3. Band structure measurements for the case of Ain→Aout. (a1)–(c1) Experimentally observed band structures varied with VM. (a2)–(c2) Simulation results of the projected intensity distribution of the band structure on modes Ak and Ck, based on Eqs. (4), (5) and (7), (8), with κ=0.06Ω and ϕ=−0.5π. (a3)–(c3) Transmission spectra measured from the drop port of ring A. The bottom horizontal axis in (a1)–(c2) represents one roundtrip time in ring A with the period of 4π/Ω.

Fig. 4. Mode distributions for the case of Bin→Bout. (a) Experimentally resolved resonant mode spectra as a function of frequency detuning with VM=3V. (b) The corresponding mode distributions of two selected input frequencies in (a) located at Δω=0 and Δω=0.08Ω, respectively. (c) Simulated resonant mode spectra with g=κ=0.06Ω, and (d) the corresponding intensity distributions of the two chosen input frequencies at Δω=0 and Δω=0.08Ω, respectively. The horizontal axis represents the mode number n for the frequency ωn.

Fig. 5. Observations of flat-to-nonflat band transition for the case of Bin→Bout. (a1)–(d1) Experimentally measured band structures with different long-range modulation amplitudes VM′ and fixed VM=2V. (a2)–(d2) Simulation results of the projected intensity distribution of the band structure on mode Bk varied with g′, where g=0.03Ω, γ=0.07Ω, and ϕ=ϕ′=−0.5π.