A nonlinear stability analysis of rhombic optical pattern formation in an atomic sodium vapor ring cavity

This dissertation contributes to the theory of optical pattern formation in a purely absorptive medium, namely a resonantly excited two-level atomic sodium vapor system in a ring cavity, by means of a rhombic-planform weakly nonlinear stability analysis applied to the governing time-evolution equation for that phenomenon. In this system, under appropriate conditions, diffraction of radiation can induce the onset of transverse patterns consisting of stripes and rhombi, in an initially uniform plane-wave configuration. This phenomenon is modeled by a Swift-Hohenberg type-equation describing the intracavity field, and defined on an unbounded spatial domain. This equation is derived from the mean-field ring cavity model of optical bi-stability, generalized to include diffraction. These are complex valued Maxwell-Bloch equations that, under appropriate conditions, can be reduced to a single nonlinear time-evolution partial differential equation for the intracavity field. Steady-state spatially homogeneous (uniform) solutions of this asymptotic equation are known. The magnitude of the uniform solution and the system's absorption coefficient are the pattern formation parameters. Linear stability analysis shows that only the real part of the solution can be unstable when the absorption coefficient exceeds a critical level. One dimensional analysis shows that supercritical stationary equilibrium patterns occur for an interval of the magnitude of the uniform solution. Two dimensional analysis shows that stripes and rhombi occur depending on the pattern formation parameters. These results are in accord with relevant experimental evidence and numerical simulations.