Theory of Soltions in Microresonators

Abstract

Recent developments in resonator technologies has allowed the expansion of the frequency comb generation in microresonators with ultra-high quality factors. Extensive research in this area has shown that the microresonator frequency combs are often associated with soliton formation. In this thesis, we develop a theory that describes the generation of solitons and other mode-locked combs in microresonators with quadratic ($\chi^{(2)}$) and cubic (Kerr) nonlinearities.

We develop theory of the four-wave mixing thresholds in Kerr microresonators, which are the precursors of solitons. We show that these new threshold conditions break the pump laser parameter space into a sequence of narrow in frequency and broad in power Arnold tongues. Instability tongues become a dominant feature in resonators with finesse dispersion parameter close to or above one. Utilising Bloch theory we show that these instabilities give birth to the weakly-modulated Turing pattern solutions which are connected to the so-called soliton crystals. We report the instabilities leading to the decrease of the soliton number inside the cavity. We build the soliton crystal stability diagram. We demonstrate the existence of high-order dispersion induced soliton breathers which lead to the formation of strong sub-combs and multiple frequency offsets.

We start our study of microresonators with $\chi^{(2)}$ nonlinearity from the second-harmonic generation regime. First, we show the generation and interaction of the photon-photon polaritons in a strong coupling regime using a dressed resonator framework. We demonstrate the existence of parametric instability tongues and Turing patterns. We show that high frequency mismatch can mitigate the group velocity walk-off to facilitate the generation of solitons. We discuss the generation of bright solitons when dispersions at both harmonics are normal. Then we show the generation of bright soliton in the presence of both $\chi^{(2)}$ and Kerr nonlinearities when the pump and signal harmonics have anomalous and normal dispersions, respectively. We finally present bright solitons in the case of dominant Kerr nonlinearity in the anomalous-anomalous dispersion case.

In the half-harmonic generation regime we show that large group velocity offset can be utilised to develop a slowly varying amplitude theory. We show that developed analytical optical parametric oscillation (OPO) solutions and their stability region match the numerical data. We show that OPO solutions can be destabilised through the Eckhaus instability scenario.

Date of Award	22 Feb 2023
Original language	English
Awarding Institution	University of Bath
Supervisor	Dmitry Skryabin (Supervisor) & Andriy Gorbach (Supervisor)