Theory and Simulations of Microcavity Dissipative Kerr Solitons in the presence of Mode Interactions with Applications to Low-Noise Microwave Generation

Optical Kerr microresonators in the last years have acquired attention from the scientific community, being from a fundamental point of view rich complex physical systems but at the same time, thanks to the late improvements of the nanofabrication technologies, feasible to fabricate in a controlled way. That has recently resulted in a plethora of cutting-edge applications spanning from spectroscopy to LIDAR and atomic clocks. The complex behaviour of the optical field within the resonator arises from being both an out-of-equilibrium and nonlinear system, leading to the emergence of bistability, hysteresis of the optical resonance, chaotic behaviour, and self-organization in coherent states. This dynamics is accurately described by the Lugiato-Lefever equation, a generalization of the Nonlinear Schrödinger equation accounting for further driven-dissipative terms. Among the different attractor state of the system, there is one essential for the majority of the applications of these devices, the so-called Dissipative Kerr Soliton (DKS), a stable, coherent and localized state circulating inside the cavity. Despite its stability, its generation in the resonator can be inhibited by the presence of interactions between different cavity mode families due to non-ideal geometry and parasitic scattering centers. That gives birth to the so-called Avoided mode crossings (AMX), that lead to the generation of dispersive wave that can destabilize or completely forbid the formation of DKS. Moreover, the presence of the interaction between the confined light and the vibrational degrees of freedom of the material, known as Raman scattering, further reduce the stability of the DKS and makes it sensitive to the driver noise. Generation of noiseless soliton is one of the essential ingredient to obtain competitive integrated photonics devices. Recent experimental studies found out that in presence of AMX, the soliton response to the driver noise can be substantially reduced. In this framework, the aim of the thesis is to deepen the understanding of the phenomenon of noise reduction, by deriving a perturbed Lugiato-Lefever model – in a form of complex nonlinear PDE - that embodies both the generation of dispersive waves, key property of AMX, and Raman scattering . The nonlinear dynamics of the model has been simulated and the properties of the generated solitons have been studied through Lagrangian perturbative approach, obtaining conditions for the minimization of the noise dependence.