## 1 Introduction  
Light emitting diodes (LEDs) and laser diodes are widely used optoelectronic devices, but many important problems still require solving. One of them is the strong lattice-mismatch during GaN growth on alien substrates [1; 2]. In the case of the growth on conventional sapphire substrate, the threading dislocation (TD) density is as high as \(10^{8}\)-\(10^{10}\) cm\({}^{-2}\)[3]. As an option to decrease the TD density and increase the internal quantum efficiency (IQE), epitaxial lateral overgrowth (ELO) has been studied [4]. Another inherent problem of GaN-based LEDs is the low light extraction efficiency (LEE) due to the relatively low angle of total internal reflection (TIR) caused by the large difference between the refractive indices of air and GaN. This limits the extraction of light generated from the multi quantum well (MQW) of LEDs so that only a small portion of the light escapes towards the outside [5]. To enhance LEE, air void structures embedded in GaN-based LEDs have been studied, which is also useful to increase IQE and thus the external quantum efficiency (EQE) as a result [6; 7; 8]. IQE in InGaN/GaN blue LEDs has been remarkably improved and has been steadily approaching 100%, so that efforts on further improvement of EQE studies have to be focused on enhancing the LEE [9]. As an approach applicable to mass production, epoxy or silicone encapsulating has been preferred because the process is easy and simple, and readily meets the yield and cost requirements. When the encapsulation is applied on the outside of GaN devices which contact air, it is possible to make the refractive indices gap between the air and GaN smaller; and increase LEE because the refractive index of epoxy or silicone is between that of GaN and air [8]. Another approach to enhancing the LEE is by using textured surfaces, thus decreasing the total internal refraction (TIR) [10]. However, when the encapsulation is applied after texturing, the effect of roughening is considerably smothered. Thus, the combined effects of encapsulation and texturing do not simply add up, and the LEE enhancement produced by texturing is substantially handicapped by subsequent encapsulation [11; 12]. Accordingly, adding an encapsulating layer after texturing in sequence is not an option to increase LEE and EQE. As an alternative to processing the outward surface of LED, one can apply the texturing to inner LED layers. This will prevent negative interference between texturing and encapsulation by leaving the textured surface intact and not planarized by encapsulation applied to the front surface. As an example of successfully combining the two approaches, one can quote growth of GaN-based LED structures on patterned sapphire substrate (PSS) resulting, if properly designed, in about 60% increase of light output [13]. Slanted sides of the patterns in PSS deflect TDs laterally, resulting in bending out of TDs, decreasing the TD density toward the surface, and increasing IQE [14]. In addition, the slanted sides of patterns increase the light escape angle and thus decrease the effect of TIR. By this mechanism, photons, including photons which come out from the lower side of the MQW region, proceed toward the substrate and are reflected in a wider cone angle than the one observed for conventional sapphire substrate (CSS), which gives rise to increased LEE [15]. Thus, this paper adopts PSS as a platform for designing an epitaxial structure to maximize efficiency. As a candidate of epitaxial structure to increase LEE, the air-void with hollow three-sided prism and cone shapes on CSS were demonstrated, resulting in scattering photons at the substrate and increasing LEE [16]. To form the air-void structures, SiO\({}_{2}\) deposition on a substrate, patterning, ELO and wet etching were applied in sequence. This means that if PSS is selected as a substrate, the air-void shares the same location, the upper side of the substrate, with patterns of PSS. Because their dimension and shape are different, uniform selective ELO is hard to achieve.