## 2 Enhancement of Differential Quantum Efficiency
The mirror loss in a VCSEL, \(\alpha_{m}\), can be expressed as:  
\[\alpha_{m}=\frac{1}{2L_{eff}}\ln\left(\frac{1}{R_{f}R_{r}}\right), \tag{3}\]  
where \(L_{eff}\) is the total effective cavity length (equal to the physical length of the device plus the penetration depths into the front and rear mirrors). In this study, \(L_{eff}\) was calculated to be 1002 nm for our device, based on coupled-mode theory [25; 28]. We used the calculated reflectivity value of 99.98% for the 10.5-pair SiO\({}_{2}\)/Nb\({}_{2}\)O\({}_{5}\) DBR as \(R_{r}\) and the measured reflectivity of the front cavity mirror as \(R_{f}\). Figure 4 shows the inverse of \(\eta_{d}\)_total_ as a function of the inverse of the mirror loss. The inverse of the internal quantum efficiency can be obtained from this plot as the intercept of a linear fit to the data, while the internal loss can be obtained from the slope of the line. From these data, the standard GaN-based VCSEL was determined to have an internal quantum efficiency of 0.5 together with an internal loss of 73 cm\({}^{-1}\) in the absence of an LOC structure. As anticipated based on theoretical studies, the internal loss associated with this device was exceptionally high, at 73 cm\({}^{-1}\)[23]. In contrast, the internal loss of the VCSEL incorporating the LOC structure was significantly reduced, to 23 cm\({}^{-1}\). Thus, it was experimentally found that the impact on the transverse radiation loss was extremely high, namely a reduction of 50 cm\({}^{-1}\), more than 2/3 of the whole internal loss in the conventional VCSELs. In addition, a maximum \(\eta_{d}\)_total_ of 32% was obtained from the SiO\({}_{2}\)-buried VCSEL with a 42-pair AlInN/GaN DBR under pulsed operation. This is also the highest efficiency ever reported for a GaN-based VCSEL. From Figure 4, the internal quantum efficiency of the VCSEL with the LOC structure was found to be 0.6. This value was also improved compared to that of the conventional VCSEL structure possibly due to the lower losses and threshold current densities. However, there is still room to improve the internal quantum efficiency as well as the internal loss value of blue GaN-based VCSELs. In subsequent trials, VCSEL chips having LOC structures and 8 \(\upmu\)m apertures were assessed during CW operation after being placed on heat sinks made of Cu with their p-sides facing downwards (equivalent to flip-chip bonding). The front side light output power and voltage during room temperature CW operation are plotted against the injection current in Figure 5a. The threshold current, threshold current density and threshold voltage for the VCSEL with 42-pair AlInN/GaN DBR were determined to be 3 mA, 6.0 kA/cm\({}^{2}\) and 4.7 V, respectively. Although this device had the highest mirror loss among the fabricated VCSELs in this study, the threshold current and current density were relatively low owing to the low internal loss of 23 cm\({}^{-1}\). A maximum output power of  
Figure 4: Inverse of external differential quantum efficiency as a function of the inverse of the mirror loss. The open and closed circles indicate the results obtained from VCSELs having a conventional SiO\({}_{2}\) aperture and a SiO\({}_{2}\)-buried lateral optical confinement structure, respectively. 6 mW was obtained. In the figure, it can be seen that the L-I curve exhibits kinks due to multimode lasing. The 8-um aperture was relatively large and hence higher-order modes were possible even with an extremely low \(\Delta n_{eff}/n\) value of \(2.6\times 10^{-3}\)[14]. Multimode spectra were also acquired at a lasing wavelength of 441 nm using an optical spectrum analyzer (ANDO AQ-6315B), as shown in Figure 4(b). The slope efficiency and the external differential efficiency for the front side were 0.64 W/A and 23% in the vicinity of the threshold current under CW operation. Comparing these values to the results obtained under pulsed operation, the efficiencies were decreased due to thermal effects. From Figure 4(a), the maximum power under CW operation when the number of DBR pairs was reduced.