A frequency comb consists of a number of equidistant discrete frequency tones. Over the last decade, optical frequency combs have become indispensable tools for molecular fingerprinting, low-phase-noise and spectrally-pure radio frequency signal synthesis, astronomical spectrograph calibration and search for exoplanets, and frequency metrology and timekeeping. While originally realized by mode-locked lasers, frequency combs have in recent years been generated in high-quality-factor Kerr-nonlinear optical resonators driven by a continuous wave (CW) laser pump. "Microcombs" hold great promise for light-weight, small-footprint, robust, and power-efficient comb sources with larger repetition rates and in frequency regimes not available to mode-locked lasers. In addition to their many practical applications, microcombs provide a rich and flexible platform for the study of nonlinear optical, quantum optical, and nonlinear dynamical phenomena. This thesis focuses on ultrashort pulse generation in Kerr-nonlinear optical microresonators, which arise when the tones of a frequency comb oscillate in synchrony. In this thesis, a new synchronization model for mode locking in parametric frequency combs is introduced, which is also applicable to mode-locked laser systems based on saturable absorbers. A novel technique for deterministic generation of chip-based ultrashort pulses (dissipative cavity Kerr solitons) based on modulating the driving CW pump is devised. The influence of higher-order dispersion on frequency comb generation and stability is investigated. Based on this study, the impact of higher-order dispersion on the transfer of pump power fluctuations to the repetition rate of generated pulses is analyzed. It is also shown that pure fourth-order dispersion supports the generation of Gaussian pulses in microresonators. An important limitation of the well-established split-step Fourier transform method for the theoretical study of microcombs is identified and explained, hinting at the advantages of the alternate approach exploiting coupled-wave equations. Finally, broadband frequency comb generation in the near-infrared regime, based on realistic dispersion-engineered crystalline resonators and with experimentally feasible pump and photodetector parameters is numerically demonstrated. This novel broadband frequency comb is spectrally wide enough for application in rubidium-based optical clocks.
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