In the last 15 years, an immense amount of research has gone into developing high efficiency Gallium Nitride based light emitting diodes (LED). These devices have become increasingly popular in LED displays and solid state lighting. Due to the large difference in refractive index between GaN and Air, a significant amount of light reflects at the boundary and does not escape the device. This drawback decreases external quantum efficiency (EQE) by minimizing light extraction. Scientists and engineers continue to develop creative solutions to enhance light extraction. Some solutions include surface roughening, patterned sapphire substrates, and reflective layers. This study proposes to increase external quantum efficiency and optimize light extraction efficiency of several LED structures using finite difference time domain analysis (FDTD). The structures under investigation include GaN based LEDs with nanoscale top gratings, patterned sapphire substrates in combination with SiO2 nanorod arrays, and reflective surfaces below and above the sapphire substrate. First, we optimize GaN based nanoscale top gratings and increase light extraction by 17.8%. Next, we simulate ITO based top gratings and enhance light extraction by 40%. Third, we optimize patterned sapphire substrate period and width and the vertical position of a SiO2 nanorod array. We achieve as high as 51.8% improvement in light extraction. Finally, we increase light extraction by 160% with the use of a silver reflection layer.