Fig. 2: Symmetric and asymmetric satellite comb phase-matching: comparison between measurements and modeling.

a Modeled group velocity dispersion (GVD; green solid curve) and third-order dispersion (TOD; blue dashed curve). Zero dispersion is plotted in green dashed line. Inset: measured GVD by swept wavelength interferometry (red dot) with fitting uncertainty plotted as error bar, along with numerical modeling support (blue solid curve). b Analytical cavity phase mismatch as a function of azimuthal mode number m, \({\mathrm{\Delta }}\omega _m = \omega _m + \omega _{ - m} - 2\omega _0\), with symmetric satellite comb phase-matching highlighted in black box and asymmetric highlighted in orange box. Highlighted stars are experimentally measured comb number of the satellite centroids pumped at 1576.615 nm (blue), 1581.881 nm (magenta) and 1584.704 nm (green) with coupled powers of 30.5 dBm. Due to the large positive fourth-order dispersion (FOD, β₄), phase-matching occurs simultaneously at multiple spectral ranges, leading to the satellite comb families at O-band and ≈ 2 μm as shown in Fig. 1a–c. With the large FOD, the residual dispersion folds backwards, leading to the formation of symmetric satellite combs. If the residual dispersion equals one cavity FSR, parametric oscillation may generate comb mode at (m + 1), (m + 2), or even the (m + n)th cavity mode, leading to the potential formation of the asymmetric satellite combs. c Zoom-in view of black box in panel (b). Theoretically calculated phase-matching curves for symmetric satellites, along with the measured satellite centroid position illustrated as datapoints (stars with the same colors).