High quality single crystal sapphire optical fiber is important not only for its capacity for high laser power delivery, but also for applications in harsh environment sensing. Improving the quality of Laser Heated Pedestal Growth (LHPG) fabricated single crystal fiber has been a long-term effort for decades. The equilibrium state during crystal growth and defect formation rate are the two most important factors in single crystal fiber fabrication. In this paper, we study the theory governing the molten zone profile and verify the theoretical predictions with a high-resolution CCD camera. We also study defect formation during the crystal growth process and observed dislocation defects with transmission electron microscopy (TEM). These analyses will help to guide high quality single crystal fiber fabrication and hopefully will lead to the production of better fibers for harsh-environment sensing applications.