Fig. 1. (a) Sketch of the system configuration. Single-mode laser has central lasing frequency ωd. Inside the Si3N4 chip with microring resonator there are both forward A+ and backward A− waves. The microresonator is assumed to be single-mode with its eigenfrequency ωm. (b) Schematic illustration of the effect. The part of the laser radiation closest to microresonator eigenfrequency excites its mode. It provides fast optical feedback gradually increasing with laser frequency shift. At certain laser detuning and backreflection phase four-wave mixing in the laser (LFWM) appears manifesting itself in additional spectral components.

Fig. 2. (a) Evolution of the laser spectrum with the change of the laser frequency detuning ξ at ψ0=ω0τs=0.20π, κdo/κm=53.31, and β=2.91. The selection of parameters is determined by the presence of additional harmonic generation regime due to laser four-wave mixing (LFWM). In the bottom panels spectra calculated at different values of laser-microresonator detuning are presented: (b) ξ=42.62, (c) ξ=20.22, and (d) ξ=5.08 demonstrating different spectra in the process of LFWM. Laser frequency detuning ξ between the hot resonance of the laser cavity ωd(τ) and the microresonator eigenfrequency ωm is defined as ξ(τ)=ξ0+νωτ. Generation detuning from the same resonance in the microresonator ζ=2(ω−ωm)/κm, where laser generation frequency ω is defined as ω=ωm+ddτarg A˜.

Fig. 3. Locking phase sweep from the appearance to the disappearance of microwave oscillations in laser gain medium. Other parameters are fixed (κdo/κm=24.39,  β=0.85). The red dashed line corresponds to the static self-injection locking model.

Fig. 4. Experimental investigation of the LFWM phenomena. (a) Sketch of the experimental setup: LD—semiconductor laser diode; SiN—photonic chip with high-Q silicon nitride microresonator; TC and CC—temperature and current controllers; PD and FPD—slow and fast photo-detectors; OSC—oscilloscope; ESA—electrical spectrum analyzer; OSA—optical spectrum analyzer. (b) Photograph of the laser diode and lensed fiber butt-coupled to silicon nitride photonic chip with 998-GHz-FSR high-Q microring resonators. (c) Experimentally measured resonance in the linear regime using an optically isolated laser (blue line) and Lorentzian fit of the loaded mode profile (pink line) providing information about intrinsic loss coefficient (κ0/2π), coupling rate (κc/2π), and backward-wave coupling rate (γ/2π). (d)–(f) Evolution of the signal transmitted through the photonic chip during the slow tuning of the DFB laser diode frequency in the vicinity of the microresonator eigenfrequency by adjusting the diode’s current. The oscilloscope trace (d), microwave spectrogram (e), and optical spectrogram (f) of the output signal are simultaneously measured using OSC, ESA, and OSA, respectively. Along the scan three different laser diode states are observed: free-running state (Free Run), state with four-wave mixing induced by the feedback from the microresonator (LFWM), and self-injection locked state (SIL).

Fig. 5. Spectral characteristics of the DFB laser diode in the LFWM state. (a) Optical spectrum measured with optical spectrum analyzer. (b) Microwave spectrum of the output signal (beatnote between optical components) measured with a fast photodetector and electrical spectrum analyzer. (c) Measured microwave signal (blue points) linewidth estimation by the Lorentzian fitting (red line).