Efficient Kerr soliton comb generation in micro-resonator with interferometric back-coupling

The study addresses the long-standing problem of low pump-to-comb conversion efficiency in dissipative Kerr soliton (DKS) microresonator frequency combs, where typical efficiencies are only a few percent due to red-detuned pumping, limited coupling, and gain saturation. The authors propose and investigate a hybrid Mach-Zehnder micro-ring resonator where a Kerr-active ring is embedded in a feedback cavity with twice the optical path length. This architecture enables interferometric back-coupling and phase-controlled interference of fields to deplete the pump at the output while maintaining sufficient circulating power to sustain solitons. The purpose is to dramatically enhance conversion efficiency and reduce required pump power, enabling robust access to diverse DKS states. The work is important for advancing integrated microcomb sources for applications in telecommunications, metrology, astronomy, and spectroscopy.

Prior work established CMOS-compatible microresonator frequency combs and DKS dynamics modeled by the Lugiato-Lefever equation. Bright-soliton combs in anomalous dispersion have low efficiency (<1–5%), while dark-soliton combs can reach ~30% but with narrow bandwidth and limited existence domain. Architectures with coupled resonators introduced new DKS dynamics including resonance splitting and near-complete pump recycling by storing pump in one cavity and generating DKS in another. Hybrid Mach-Zehnder micro-rings have been explored for tuning and for managing Raman competition, but their potential for enhancing conversion efficiency via interferometric back-coupling had not been systematically investigated. The present work builds on these developments by embedding a ring in a twice-length feedback cavity to exploit controlled interference to deplete the pump and enhance comb power.

Device architecture: A twisted bus waveguide couples to a Si_xN_y micro-ring resonator at two locations (nodes 1 and 2), forming two coupled cavities: (i) the ring, and (ii) a feedback cavity comprising the bus and half of the ring, with twice the optical path length. Power coupling coefficients at nodes 1 and 2 (β1, β2) are set by different gaps; microheaters atop the feedback waveguide tune the feedback phase (Δ) over >2π to control the relative phase (detuning offset δ20). The design targets very low propagation loss with intrinsic Q > 10^6. Fabrication and devices: Low-loss silicon nitride (Si_xN_y) platform with anomalous dispersion at ~1560 nm. Two resonators (A, B) on chips: ring radius R = 800 µm; gap1 = 650 nm; gap2 = 410 nm (A) or 460 nm (B). Measured/simulated couplings (at 1560 nm): γ1 ≈ 4.1×10^-3 (650 nm), γ2 ≈ 2.38×10^-2 (460 nm), γ3 ≈ 3.77×10^-2 (410 nm). Intrinsic loss ~0.1 dB/cm; fractional transmissions t1 ≈ 0.997 (ring), t2 ≈ 0.9914 (feedback). Material indices: n ≈ 2.1, n2 ≈ 2.4×10^-19 m^2/W; A_eff ≈ 0.98 µm^2; V_eff ≈ 4.92×10^-15 m^3; nonlinear gain g ≈ 0.5 Hz. Experimental setup: A tunable CW laser is EDFA-amplified; polarization adjusted to excite quasi-TE mode. Coupling via objective lens or lensed fiber into inverse-taper bus waveguides (~2.5 dB per facet). Output monitored by OSA. For repetition-rate measurements, pump is rejected with a tunable Bragg grating and detected on a fast photodiode with ESA; stability assessed by mixing with a 28.5 GHz reference and counted with a rubidium-disciplined counter. Effective pump-resonance detuning measured via a weak counter-propagating probe scanned at 10 Hz; beat with back-reflected pump identifies pump position relative to resonance. Pump scans are performed from blue to red detuning. Characterization: FSR dispersion measured by scanning a low-power laser from 1550–1630 nm. Loaded Q factors measured (e.g., 0.625×10^6 at 1569.3 nm for one resonance). Resonance linewidth variation versus heater power used to retrieve δ20 via analytical expression for Δω as a function of δ20, correlating measurements with Eq. (1). Soliton generation: Comb states generated by scanning pump across resonance at input powers from ~50 mW to 300 mW. PSCs and soliton crystals with defects are identified by spectral features and red-detuned pump location via the probe method; repetition-rate coherence measured. Modeling: An Ikeda map with temporal delay models the coupled ring-feedback system. Fields at nodes 1 and 2 are coupled with coefficients and detunings; propagation in ring and feedback arms governed by nonlinear Schrödinger equations including loss α1, GVD β2, and nonlinearity γ; timing accounts for round-trip delays (feedback contribution from previous round-trip). Output field derived from node-2 coupling. Simulations use measured/extracted device parameters; detuning sweeps explore transitions from modulation instability to soliton crystals. Evolutionary optimization is used to maximize conversion efficiency by varying θ2, δ1, δ20, and soliton number Ns at fixed P_in.

• Principle of operation: Interference between the pump leaking from the ring and the field from the feedback arm can yield equal-amplitude, opposite-phase pump components at the output, enabling extreme pump depletion and high conversion efficiency.
• Experimental efficiency: Achieved pump-to-comb conversion efficiencies up to ~55% of input power for perfect soliton crystal (PSC) states. Example: PSC with 17.8 dB pump depletion (only ~1.7% of input pump at output line) yields ~55% comb power fraction; remaining ~45% circulates in the ring to sustain solitons.
• Additional high depletion: Observed spectra with up to 22 dB pump depletion at 300 mW input, accompanied by coherent repetition-rate signals; possible breathing behavior inferred from multi-pronged repetition-rate profile over 200 kHz bandwidth with ~1 ms periodicity.
• Comparison to isolated rings: For similar ring size, isolated rings show pump ~17 dB stronger than adjacent sidebands, whereas in this architecture pump can be ~11 dB weaker than nearby comb lines due to interferometric depletion.
• Access to DKS states at lower power: Soliton crystals with defects generated at P_in ~70 mW (normalized amplitude f ~1.5), about four times lower pump power than an equivalent isolated resonator; chaotic regimes shift: STC observed near f ~2, TC near f ~2.8 (vs ~3 and ~4 in isolated rings).
• Detuning dependence: Simulations and experiments show a detuning window (δ1 ~0.03–0.035) with complete pump depletion and conversion efficiency ε ~55%; simulated pump depletion up to 25 dB at output; ring vs feedback spectra show pump amplitudes matched with π phase difference, while comb lines have smaller phase differences.
• Diversity and coherence: Wide variety of coherent red-detuned combs (PSCs and crystals with defects) generated robustly; repetition-rate signals confirm coherence; feedback does not induce chaos but reinforces stable DKS operation.
• Q-factor and phase tuning: Loaded Q varies with feedback phase δ20 as predicted by analytic model; moderately high conversion efficiencies obtained across a broad δ20 range; highest efficiencies near lowest loaded Q (δ20 ~0) but optimization over δ1 enables good performance elsewhere.
• Numerical optimization limits: Evolutionary-search simulations indicate achievable comb power fractions of ~56.8% (0.18 W), ~73.2% (0.5 W), and ~80.45% (1.0 W) with correspondingly small residual pump fractions (1.01%, 0.39%, 0.18%); optimal θ2 increases with P_in (from 3.73×10^-2 to 7.27×10^-2). Saturation suggests diminishing returns at higher powers.
• Regime maps: Simulated detuning-power sweeps (δ20 = 2.0) show MI threshold on left boundary and CW transition on right; chaotic regions (STC/TC) appear for P_in ≳135 mW and f ≳2.0; δ20 = 3.0 further lowers instability threshold but is unsuitable for solitons.

The proposed interferometric back-coupling micro-ring architecture directly addresses the low-efficiency bottleneck of DKS microcombs by engineering the interference of the pump fields at the output. By tuning the relative phase (δ20) and coupling coefficients, the pump leaking from the ring can be made equal in amplitude and opposite in phase to the feedback pump, resulting in near-complete pump cancellation at the output while maintaining sufficient intra-ring power to sustain solitons. This mechanism transforms the distribution of input power into predominantly comb lines at the output, raising conversion efficiency to ~55% experimentally and up to ~80% in simulations. The architecture also reuses pump power via feedback, reducing the required input power to access DKS states and shifting the onset of chaotic regimes, thereby enabling deterministic or robust access to PSCs and a range of soliton-crystal states at lower f than in isolated rings. Compared to two-cavity coupled-ring schemes, the present design offers less dispersion management but leverages phase-controlled interference as the primary tuning knob. The demonstrated stability and coherence of the resulting combs, together with efficient pump depletion, make the approach attractive for applications requiring high optical power per comb line and low residual carrier, such as telecommunications, astrocombs, and spectroscopy. The findings establish a platform to explore new regimes of coherent soliton dynamics in resonators with engineered feedback.

This work introduces and validates a hybrid Mach-Zehnder micro-ring resonator with a feedback cavity of twice the ring length that enables interferometric back-coupling. The architecture delivers unprecedented control of pump depletion, achieving up to ~55% experimental pump-to-comb conversion efficiency for soliton crystal states and predicting up to ~80% in simulations by optimizing coupling, detuning parameters, and soliton number. The feedback-enhanced reuse of pump power allows access to diverse DKS regimes at significantly reduced input power relative to isolated rings, while maintaining coherent, robust operation. These ultra-efficient soliton crystals are well suited for telecommunication, astronomical, and spectroscopic applications. Future work can optimize parameter sets (coupling θ2, detunings δ1 and δ20, soliton number Ns, and input power) for different operating regimes, extend to other material platforms, and further explore complex nonlinear dynamics enabled by interferometric feedback.

• Conversion efficiency is fundamentally limited by the pump power that must circulate in the ring to sustain solitons; even at maximum output depletion, a substantial fraction remains intra-cavity.
• Performance depends on precise phase control of the feedback cavity (δ20) and appropriate coupling coefficients; some δ20 values (e.g., ~3.0) may lower MI thresholds but are not suitable for soliton generation.
• The architecture offers limited dispersion management compared to dual-ring schemes; interference is the primary control mechanism.
• Experimental measurements show no clear monotonic correlation between effective red detuning magnitude and pump depletion level across states.
• At higher input powers, efficiency gains exhibit saturation.
• Accurate determination of normalized pump amplitude f is sensitive to uncertainties in nonlinear index and in-chip pump power (≈1 dB uncertainty).