Generalized Mathematical Formalism Governing Free-carrier Driven Kerr Frequency Comb in Optical Micro-cavities

Abstract: Continuous-wave pumped optical microresonators have been vastly exploited to generate frequency comb (FC) utilizing the Kerr nonlinearity. Most of the nonlinear materials used to build photonic platforms exhibit nonlinear losses such as multi-photon absorption, free-carrier absorption (FCA), and free-carrier dispersion (FCD) which can strongly affect the nonlinear characteristics of the devices made out of these materials. In this work, we model the Kerr FC based on modified Lugiato-Lefever Equation (LLE) along with the rate equation and develop analytical formulations to make quick estimations of the steady-state, modulation instability (MI) gain, bandwidth and the dynamics of Kerr Frequency-Comb (FC) in presence of nonlinear losses. Our analytical model is valid over a broad wavelength range of interest as it includes the effects of all nonlinear losses. Higher order (>3) characteristic polynomial of intra-cavity power describing the steady-state homogeneous solution of the modified LLE are discussed in detail. We derive the generalized analytical expressions for the threshold of normalized pump detuning to initiate the optical bistability which is a necessary condition for the FC generation. Free-carrier dispersion-led nonlinear cavity detuning is observed through the reverse Kerr-tilt of the resonant-peaks. We further deduce the expressions for the threshold pump intensity and the range of possible cavity detuning for the initiation of the MI when all the nonlinear losses are present. To corroborate our analytical findings, LLE along with the rate equations are solved numerically through split-step Fourier method. Our theoretical study can explain several experimental results which are previously reported and thereby is able to provide a better understanding of the comb dynamics.