Erbium-doped fiber amplifiers revolutionized long-haul optical communications and laser technology. Erbium ions could provide a basis for efficient optical amplification in photonic integrated circuits but their use remains impractical as a result of insufficient output power. We demonstrate a photonic integrated circuit–based erbium amplifier reaching 145 milliwatts of output power and more than 30 decibels of small-signal gain—on par with commercial fiber amplifiers and surpassing state-of-the-art III-V heterogeneously integrated semiconductor amplifiers. We apply ion implantation to ultralow–loss silicon nitride (Si 3 N 4 ) photonic integrated circuits, which are able to increase the soliton microcomb output power by 100 times, achieving power requirements for low-noise photonic microwave generation and wavelength-division multiplexing optical communications. Endowing Si 3 N 4 photonic integrated circuits with gain enables the miniaturization of various fiber-based devices such as high–pulse-energy femtosecond mode-locked lasers.

Multi-mode waveguides are ubiquitously used in integrated photonics. Although interaction among different spatial waveguide eigenmodes can induce novel nonlinear phenomena, spatial mode interaction is typically undesired. Adiabatic bends, such as Euler bends, have been favoured to suppress spatial mode interaction. Here, we adapt and optimize Euler bends to build compact racetrack microresonators based on ultralow-loss, multi-mode, silicon nitride photonic integrated circuits. The racetrack microresonators feature a footprint of only 0.21 mm2 for 19.8 GHz free spectral range, suitable for tight photonic integration. We quantitatively investigate the suppression of spatial mode interaction in the racetrack microresonators with Euler bends. We show that the low optical loss rate (15.5 MHz) is preserved, on par with the mode interaction strength (25 MHz). This results in an unperturbed microresonator dispersion profile. We further generate a single dissipative Kerr soliton of 19.8 GHz repetition rate without complex laser tuning schemes or auxiliary lasers. The optimized Euler bends and racetrack microresonators can be building blocks for integrated nonlinear photonic systems, as well as linear circuits for programmable processors or photonic quantum computing.

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A photonic dimer composed of two evanescently coupled high- Q microresonators is a fundamental element of multimode soliton lattices. It has demonstrated a variety of emergent nonlinear phenomena, including supermode soliton generation and soliton hopping. Here, we present another aspect of dissipative soliton generation in coupled resonators, revealing the advantages of this system over conventional single-resonator platforms. Namely, we show that the accessibility of solitons markedly varies for symmetric and antisymmetric supermode families. Linear measurements reveal that the coupling between transverse modes, giving rise to avoided mode crossings, can be substantially suppressed. We explain the origin of this phenomenon and show its influence on the dissipative Kerr soliton generation in lattices of coupled resonators of any type. Choosing an example of the topological Su-Schrieffer-Heeger model, we demonstrate how the edge state can be protected from the interaction with higher-order modes, allowing for the formation of topological Kerr solitons.

<p style='text-indent:20px;'>This paper studies ruin probabilities of a generalized bidimensional risk model with dependent and heavy-tailed claims and additional net loss processes. When the claim sizes have long-tailed and dominated-varying-tailed distributions, precise asymptotic formulae for two kinds of finite-time ruin probabilities are derived, where the two claim-number processes from different lines of business are almost arbitrarily dependent. Under some extra conditions on the independence relation of claim inter-arrival times, the class of the claim-size distributions is extended to the subexponential distribution class. In order to verify the accuracy of the obtained theoretical result, a simulation study is performed via the crude Monte Carlo method.</p>