| Self-injection locking with synthetic reflection. a,

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... synthetic reflection. The reflection is controlled by periodic nano-patterned corrugations of the ring-resonators' inner walls. The angular corrugation period is θ 0 = 2π/(2m 0 ), where m 0 is the angular (azimuthal) mode number, for which a deliberate coupling between forward and backward propagating waves with a coupling rate γ is induced (see Fig. 1a). Besides inducing the desired synthetic reflection, the coupling leads to mode hybridization resulting in a split resonance lineshape (frequency splitting 2γ) in both transmission and reflection (see Fig. 1b). Here, we only consider the lower frequency hybrid mode for pumping, as it corresponds to strong (spectrally local) anomalous ...

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... mode number, for which a deliberate coupling between forward and backward propagating waves with a coupling rate γ is induced (see Fig. 1a). Besides inducing the desired synthetic reflection, the coupling leads to mode hybridization resulting in a split resonance lineshape (frequency splitting 2γ) in both transmission and reflection (see Fig. 1b). Here, we only consider the lower frequency hybrid mode for pumping, as it corresponds to strong (spectrally local) anomalous dispersion, which prevents high-noise comb states [47]. For choosing γ we balance multiple ...

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... conventional resonators the normalized forward-backward coupling is usually small 2γ/κ < 1 and the intersection between SIL and DKS ranges is limited to small detunings, complicating access to DKS states. In contrast, strong forward-backward coupling could enable robust access to DKS states over a wide range of detunings. This is exemplified in Fig. 1c, where the SIL range [37] is shown along with the conventional analytic DKS existence range (valid for small γ) and the numerically computed DKS existence range for large γ, obtained through numeric integration of the coupled mode equations (cf. Methods). Note, that in a resonator with a shifted pump mode [49], the existence range of ...

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... computed DKS existence range for large γ, obtained through numeric integration of the coupled mode equations (cf. Methods). Note, that in a resonator with a shifted pump mode [49], the existence range of DKS deviates strongly from that known from resonators without a shifted pump mode [48] and can currently only be obtained numerically (Fig. ...

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... comb initiation and deterministic generation of single DKS. Based on these considerations we choose a PhCR with a synthetic coupling for the pump mode at 1557 nm of 2γ/κ ≈ 4.2 ( γ 2π ≈ 250 MHz), within the ideal range. This PhCR is critically coupled and exhibits anomalous group velocity dispersion (D 2 ≈ 8 MHz). As shown for those values in Fig. 1c, numeric simulation confirms that the DKS existence and SIL ranges have significant overlap. We note that another band of DKS existence may exist [49], however, it is inaccessible for spontaneous MI-assisted comb initiation and not considered here. The DFB pump laser diode is mounted on a piezo translation stage to adjust the injection ...

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... SIL regime when increasing (decreasing) the DFB pump current follows a nontrivial behavior that may include non-monotonic sections [37] by pronounced dip of the transmitted power after optimization of the injection phase), we observe at first only the single optical frequency of the SIL pump laser, as in the lower power experiment before (Fig. 3a 1 ). Continuing the scan we next observe an abrupt transition into a single-DKS microcomb state (Fig. 3a 2 ). Such single-DKS states are characterized by a smooth squared hyperbolic-secant amplitude and a pulse repetition rate that corresponds to the resonator's FSR; these properties are highly-desirable for applications. Further ...

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... pulse repetition rate that corresponds to the resonator's FSR; these properties are highly-desirable for applications. Further continuing the scan induces a surprising second abrupt transition into a different single-DKS state (Fig. 3a 3 ). Scanning even further causes the DKS to disappear, with the system returning to CW SIL (spectrum similar to Fig. 3a 1 ), before eventually exiting the SIL regime entirely. When repeated, each scan shows the same SIL dynamics, including deterministic single-DKS generation. Reversing the scan direction qualitatively yields the same phenomena in reversed order. Turing patterns, noisy comb-states and multi-DKS regimes are absent in stark contrast to ...

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... Fig. 3e shows a similar transmission and filtered power trace obtained with a non-PhCR microresonator of the same FSR. To achieve SIL-based DKS in a non-PhCR, we screened a large number of resonators to find one with large random reflection (2γ/κ ≈ 0.6) and validated numerically that it can meet the criteria for SILbased DKS (similar to Fig. 1c). In contrast to the PhCR, it does however not respect the criterion for deterministic single-DKS. Indeed, different from the SIL dynamics in the PhCR, noisy comb states before (after) DKS generation are observable in the laser scan and the DKS-regime is dominated by less desirable multi-DKS states, showing the characteristic multi-step ...