Stability and accessibility of Turing rolls, soliton crystals, and single solitons in microresonators

Abstract

Broadband optical frequency combs generated in externally pumped, high-quality (Q) microresonators with the Kerr nonlinearity have important applications to metrology, high-resolution spectroscopy, and microwave photonics. In the time domain these frequency combs correspond to cnoidal waves, which are spatially and temporally stable periodic structures azimuthally propagating in the microresonator. Turing rolls, perfect solitons crystals, and single solitons in microresonators are all special cases of cnoidal waves. The types of cnoidal waves depend on experimental parameters, and each type has its own properties and applications. Determining the stability and accessibility of different types of cnoidal waves is the object of our dissertations research. In Chapter 1, we briefly introduce frequency combs and cnoidal waves. We mathematically represent the system using the Lugiato-Lefever equation (LLE) with two different normalizations. In Chapter 2, we analytically and numerically study the family of cnoidal wave solutions to the LLE normalized with respect to the frequency detuning. In the lossless case, we analytically obtain cnoidal wave solutions, and we then extend the solutions numerically to the case with loss. In Chapter 3, we investigate the stability and accessibility of cnoidal waves that correspond to Turing rolls or soliton crystals in microresonators. We apply highly-efficient dynamical methods to comprehensively explore the three-dimensional parameter space consisting of detuning, pump amplitude, and mode circumference to determine where stable solutions exist. In Chapter 4, we extend our previous study to consider thermal effects, which are present in real devices. Finally, in Chapter 5, we study the stability and accessibility of cnoidal waves in microresonators with avoided crossings. We find that deterministic generation of solitons is correlated with an enhanced region of stability for the single soliton. Varying both the offset and strength of an avoided crossing, we did not find simple rules that characterize when a single soliton can be deterministically accessed. However, we anticipate that our results are an important first step towards finding them.