## 2 Enhancement of Differential Quantum Efficiency  
The \(\eta_{d}{}^{F}\) value obtained from the front side mirror for a VCSEL can be expressed as:  
\[\eta_{d}^{F}=\eta_{i}\times\frac{\alpha_{m}^{F}}{\alpha_{i}+\alpha_{m}^{F}+ \alpha_{m}^{R}} \tag{1}\]  
where \(\alpha_{i}\) is the internal loss, \(\eta_{i}\) is the internal quantum efficiency, and \(\alpha_{m}{}^{F}\) and \(\alpha_{m}{}^{R}\) are the front and rear mirror losses, respectively [22]. Reducing the internal loss is important for enhancing \(\eta_{d}{}^{F}\). One potential reason for a low \(\eta_{d}{}^{F}\) is a high internal loss in the cavity. Most VCSELs based on GaN that have been reported in the literature thus far have achieved homogeneous current injection and current confinement by employing SiO\({}_{2}\) apertures and intra-cavity contacts made of indium tin oxide (ITO), despite the anti-guiding that can result from such features [23]. [PERSON] et al. [23] performed a theoretical analysis of the transverse radiation loss caused by a lateral anti-guiding structure and determined that the total internal loss could range from 50 to 70 cm\({}^{-1}\) in a GaN-based VCSEL with a simple SiO\({}_{2}\) aperture. They proposed several lateral optical confinement (LOC) structures to reduce the internal loss. Equation (1) shows that a low \(\eta_{d}{}^{F}\) can also result from the cavity mirror design. Thus, the mirror reflectivity should be adjusted in accordance with variations of the optical gain and loss parameters. Reducing \(\alpha_{i}\) allows more range in the mirror design and reduces the front mirror reflectivity. The present section describes a blue VCSEL based on GaN exhibiting greatly improved differential quantum efficiency, as a result of lowering the cavity mirror reflectivity and incorporating a SiO\({}_{2}\)-buried LOC structure. Figure 1: Light output power as a function of differential quantum efficiency for reported GaN-based VCSELs [7; 8; 12; 13; 15; 16; 17; 18; 19; 20; 21]. The open and closed circles indicate pulsed operation and CW operation, respectively. Figure 2 presents diagrams of a SiO\({}_{2}\)-buried VCSEL and a conventional SiO\({}_{2}\) aperture, both of which were fabricated using metalorganic vapor phase epitaxy in conjunction with GaN substrates. Each unit comprised a 10.5-pair SiO\({}_{2}\)/Nb\({}_{2}\)O\({}_{5}\) distributed Bragg reflector (DBR), a 4.5\(\upmu\)cavity and a 46-pair AlInN/GaN DBR. A 75 nm p-GaN layer, 20 nm p-Al\({}_{0.15}\)Ga\({}_{0.85}\)N electron blocking layer, 3 nm 5-pair GaInN/4 nm GaN multiple quantum well and 570 nm n-GaN layer were situated between the DBRs. In Figure 2a, the thickness of the SiO\({}_{2}\) layer was 100 nm with an aperture diameter of 8 \(\upmu\)m. A 20 nm ITO contact layer was deposited following fabrication of the conventional SiO\({}_{2}\) aperture, following which a 10.5-pair SiO\({}_{2}\)/Nb\({}_{2}\)O\({}_{5}\) DBR and a 38 nm Nb\({}_{2}\)O\({}_{5}\) spacer layer were applied to the ITO such that a VCSEL cavity was obtained [11]. Assuming that a local shift of the resonance wavelength produces a corresponding change in the effective index [24], a calculation of the resonance wavelength difference gave a normalized effective index difference (\(\Delta n_{\mathit{eff}}/n\)) of \(-\)0.025 for this structure, thus showing anti-guiding. Figure 2b shows a buried SiO\({}_{2}\) structure that was fabricated from an 8 \(\upmu\)m diameter region of the p-GaN layer by reactive ion etching to a depth of 20 nm. Following this, magnetron sputtering was employed to deposit a thin (20 nm) SiO\({}_{2}\) film that was subsequently lifted off in a self-aligned manner. A 20 nm ITO contact layer was applied to the buried SiO\({}_{2}\) and central current injection regions, followed by the deposition of a 38 nm Nb\({}_{2}\)O\({}_{5}\) spacer layer and a 10.5-pair SiO\({}_{2}\)/Nb\({}_{2}\)O\({}_{5}\) DBR. The \(\Delta n_{\mathit{eff}}/n\) value for this device was determined to be 2.6 \(\times\) 10\({}^{-3}\), confirming lateral optical guiding.