Multicolor interband solitons in microcombs

Theory of multicolor interband solitons generation

Here, the coupled rings are effectively replaced with a single cavity, and the three supermode families in Fig. 1g are viewed as independent transverse mode families. The assumption is validated in the Supplementary Information. The soliton dynamics in presence of parametric interaction is governed by^14,24,

Here, the slow-varying electric fields E_k (k = p, s, i for primary soliton, secondary soliton and idler sideband respectively, * denotes complex conjugate) are defined in the co-rotating frame of the primary soliton and normalized to photon number. The carrier angular frequencies are denoted by ω_k (k = p, s, i), respectively. ω_p equals the pump angular frequency. The choice of ω_s and ω_i is not deterministic, but to eliminate the phase factor in the four-wave-mixing (FWM) terms, it is forced that

We further define detuning δω_k = ω_k,c − ω_k (ω_k,c is the corresponding cavity mode angular frequency, k = p, s, i), where

Since the detunings need to be small for resonant excitation, a requirement for mode frequencies eqn. (1) arises. Furthermore, κ_k (k = p, s, i) is cavity loss rate, ΔD_1,s ≡ D_1,s − D_1,p, ΔD_1,i ≡ D_1,i − D_1,p, D_1,k, D_2,k are first- and second- order cavity dispersion parameters, g₀, g_XPM, g_FWM are effective nonlinear self-phase-modulation, cross-phase-modulation (XPM) and four-wave-mixing (FWM) coefficients respectively (for definition see Supplementary Information), and \(F=\sqrt{{\kappa }_{{\text{ext}},{\text{p}}}{P}_{{\text{in}}}/\hslash {\omega }_{{\text{p}}}}\) is the pump term, where κ_ext,p is the external coupling rate and P_in is the on-chip input power.

Analytical analysis

Several approximations are made to derive the analytical solution of eqns. (3)(4)(5). We focus on near-threshold behaviour where the power of the secondary soliton and idler sideband is much lower than the primary soliton, that \(| {E}_{\text{s}}|\),\(| {E}_{i}| \ll | {E}_{{\text{p}}}|\). The primary soliton takes the unperturbed soliton form

The dynamics of E_s and E_i, with this E_p expression inserted, yields Schrödinger-type equations in a \({sech}^{2}\) potential well with parametric gain terms. The idler sideband is approximated as a continuous wave, while the secondary soliton exhibits ground-state solution as follows:

The linear phase factor in E_s results from FSR mismatch of the primary and secondary soliton forming mode families. Δμ_s denotes a shift in secondary soliton central mode from mode ω_s,c. For γ and Δμ_s it is derived (detailed in the Supplementary Information),

Eqn. (12) indicates that the central mode of secondary soliton is shifted to where the FSR of the soliton forming mode aligns with the primary soliton.

Furthermore, a threshold behaviour is predicted. E_s and E_i compose a coupled linear system, where either exponential growth or decay can occur. The secondary soliton forms under exponential growth, when parametric gain overcomes cavity loss. By taking the inner product of both sides of eqns. (4)(5) with their respective eigenfunctions (9)(10), the equations reduce to a linear set of ordinary differential equations governing the evolution of E_s and E_i amplitudes, and the threshold condition is readily obtained. Setting ΔD_1,s = 0 for simplicity of expression, its threshold condition is calculated to be

where \(\Pi (t)\equiv {\int }_{-\infty }^{\infty }{{\text{sech}}}^{{\text{t}}}{\text{xdx}}\).

Numerical simulation

To confirm that the proposed mechanism enables multicolor interband solitons, numerical simulations are performed based on full coupled LLEs eqns. (3)–(5) using split-step Fourier transform method. For each dispersion band, 1024 modes are involved in the model.

In the simulation, the system is seeded by a single primary soliton. The results are summarized in Fig. 4. Simulated spectrum in Fig. 4a displays good similarity to experimental data in Fig. 1. The conclusions drawn from analytical model are also validated. To verify eqn. (11), dispersion parameter D_2,s is tuned and exponent γ is determined by spectrum fitting at each D_2,s. The analytical and numerical results are consistent (Fig. 4b). For eqn. (12), the central mode shift of the secondary soliton Δμ_s obtained from simulation and eqn. (12) are plotted together at different FSR mismatches ΔD_1,s in Fig. 4c, also showing good agreement. Besides, it is numerically verified that XPM is essential to stable mode-locking of the secondary soliton (see Supplementary Information).

Simulation parameters are listed as below. For Fig. 4a, ω_p = 2π × 191.68 THz, Q_int = 75 × 10⁶, Q_ext = 200 × 10⁶, δω_p = 22.5κ_p, δω_s = δω_i = 12.5κ_p, ΔD_1,s = 0, ΔD_1,i = 2π × 31.8 MHz, D_2,p = 2π × 353 kHz, D_2,s = 2π × 159 kHz, D_2,i = − 2π × 154 kHz, g₀/2π = 4.33 × 10⁻³ Hz, g_XPM/2π = 1.73 × 10⁻³ Hz, g_FWM/2π = 1.73 × 10⁻³ Hz, P_in = 300 mW. Loss rates are derived by κ_ext,p = ω_p/Q_ext, κ_k = κ_int + κ_ext = ω_p/Q_int + ω_p/Q_ext (k = p, s, i). For Fig. 4b, ΔD_1,s is fixed at 0. For Fig. 4c,d, D_2,s is fixed at 2π × 159 kHz.

Threshold behaviour of the secondary soliton generation

Threshold behaviour is observed both experimentally and numerically, which is typical for parametric processes and predicted by the theory. When the pump laser scans across the mode from blue-detuned regime to red-detuned regime, a single primary soliton is generated at first, and the secondary soliton and idler sideband emerge when pump detuning reaches a certain threshold. Optical spectra below and above threshold detuning are shown in Fig. 5b.

Measured and simulated soliton steps of primary and secondary solitons are also shown separately in Fig. 5d. The measurement setup is detailed in Fig. 5a. The comb output is amplified by an erbium-doped fiber amplifier (EDFA) and equally split into two routes. Each route is directed to an optical waveshaper to filter out primary/secondary soliton only, and then detected by a photodetector (PD). The PD signals are received by an oscilloscope to monitor soliton power. Analogous to simulation result, the measurement also verifies the sequenced generation of primary and secondary solitons, indicating the existence of threshold detuning. In the numerical model, when pump detuning is slowly ramped, existence of threshold is observed explicitly (Fig. 5c). Normalized threshold detuning 2δω_p/κ_p calculated from eqn. (13) with simulation parameters is 33.8, close to simulation value 35.7.

Furthermore, threshold behaviour in presence of FSR mismatch ΔD_1,s/2π is studied. Comb spectra are simulated under different FSR mismatches and pump detunings δω_p, and secondary soliton powers with respect to these parameters are plotted in Fig. 4d. Secondary soliton existence range is nearly symmetric with respect to ΔD_1,s/2π = 0, and threshold pump detuning increases with FSR mismatch. The primary and secondary solitons simultaneously vanish when pump detuning exceeds the primary soliton existence limit.

Experimental details

In the autocorrelation measurement, the comb output from the cavity is firstly amplified to 70 mW by an erbium-doped fiber amplifier (EDFA), and then directed to a waveshaper. The waveshaper is programmed as a band-pass filter that filters out either the primary or the secondary soliton, and in its passband, a quadratic dispersion is applied to compensate fiber dispersion. After filtering, the comb is again amplified to 300 mW by a second EDFA before sent into an autocorrelator. The data in Fig. 1d,e is measured when dispersion compensation is optimized so that the pulses display the smallest temporal widths.

Full phase stabilization is achieved by simultaneous locking of f_rep (by servo locking) and f_beat (by disciplining to a stable microwave synthesizer²⁵). It can be characterized by another inter-soliton beatnote with frequency f_rep − f_beat. The noise of this beatnote will be significantly suppressed only when f_rep and f_beat are simultaneously stabilized. RF spectrum and phase noise data for the f_rep − f_beat beatnote is plotted in Fig. 6. The phase noise of locked f_rep − f_beat beatnote closely follows that of locked f_beat beatnote, while the noise of locked f_rep is below this level, indicating successful full phase stabilization. The two solitons form a coherent set of frequency comb with broader spectral range.

In Fig. 3b and Fig. 4b, consistent with the proposed theory, the secondary soliton spectra are fitted by an envelope with form

where FT denotes Fourier transform, P is power, ν is frequency, t is time, and a, b, γ, ν_s are fitting parameters.