Impact of Kerr and Raman Nonlinear Effects on the Whispering Gallery Modes of a Spherical Microcavity

Abstract

Whispering gallery modes (WGM) microcavity have played an ubiquitous role due to their high quality factor Q and small effective mode volume $V_{eff}$. They are suitable for a broad range of applications and scientific research including cavity quantum electrodynamics (c-QED), sensing, parametric oscillation, frequency comb and so forth. The major nonlinear effect in silica is the Kerr nonlinearity that arises from the dependence of refractive index on the signal intensity. In this thesis, we focus on the theoretical analysis of Kerr and Raman nonlinear effects in a silica spherical microcavity. We derive several analytical models for various nonlinear effects, including self phase modulation (SPM), cross phase modulation (XPM) and stimulated Raman scattering (SRS).

The first part of this thesis develops a theoretical framework to describe the impact of Kerr nonlinearity, especially SPM on WGM. First a mathematical formulation to express $\chi^{(3)}$ in spherical co-ordinates is developed. We define the effective mode volume $V_{eff}$ for the the first time to analyze SPM effects by taking $\chi^{(3)}$ tensor in spherical co-ordinates and it is found that the effective mode volume is always smaller than the physical volume of the microsphere. Simulation results show that whispering gallery mode undergoes a negative frequency shift proportional to the injected energy due to SPM.

Later, we extended the analysis to describe the nonlinear interaction between two WGMs. An analytical model is developed to describe the XPM effect in microsphere. Expressions for effective mode volumes and effective nonlinear coefficients to describe XPM are derived analytically. It is found that, when the effective mode volume increases, effective nonlinear coefficient becomes smaller and hence, we achieve a lower frequency shift. An analytical expression for the coupling of whispering gallery mode is derived. The resonant frequency of a weak probe mode can be shifted by a strong pump mode due to XPM and the frequency shift of the probe is proportional to the pump energy. Also, An analytical expression for describing the Raman effect in a spherical microcavity is developed by including the delayed Raman response. The results show that the signal power is amplified due to the SRS effect.