Optical frequency combs can be used to measure light frequencies and time intervals more easily and precisely than ever before, opening a large avenue for applications. Traditional frequency combs are usually associated with trains of evenly spaced, very short pulses. More recently, a new generation of comb sources has been demonstrated in compact high-Q optical microresonators with a Kerr nonlinearity pumped by continuous-wave laser light. These combs are now referred to as Kerr frequency combs and have attracted a lot of interest in the last few years. They can be modeled in a way that is strongly reminiscent of temporal cavity solitons (CSs) in nonlinear cavities. Temporal CSs have been experimentally studied in fiber resonators and their description is based on a now classical equation, the Lugiato-Lefever equation, that describes pattern formation in optical systems.
The work presented here consists of three main parts. Firstly, we perform a theoretical study of the correspondence between the CSs and patterns with frequency combs. It is known that the CSs appear in reversible systems that present bistability between a pattern and a homogeneous steady state through what it is called a homoclinic snaking structure. We study the changes of the homoclinic snaking for different parameter regimes in the Lugiato-Lefever equation and determine the stability and shape of the frequency combs through comparison with the underlying CSs and patterns. Secondly, we show that time-periodic oscillations of a 1D CS appear naturally in a broad region of parameter space. More than this oscillatory regime, we theoretically report on several kinds of chaotic dynamics. Finally, we include third order dispersion in the system and study its effect on the multistable snaking structure. For high dispersion strengths the CS structures and the corresponding Kerr frequency combs disappear.