A single or multi-color light emitting diode (LED) with high extraction efficiency is comprised of a substrate, a buffer layer formed on the substrate, one or more patterned layers deposited on top of the buffer layer, and one or more active layers formed on or between the patterned layers, for example by Lateral Epitaxial Overgrowth (LEO), and including one or more light emitting species, such as quantum wells. The patterned layers include a patterned, perforated or pierced mask made of insulating, semiconducting or metallic material, and materials filling holes in the mask. The patterned layer acts as an optical confining layer due to a contrast of a refractive index with the active layer and/or as a buried diffraction grating due to variation of a refractive index between the mask and the material filling the holes in the mask.