## 2 Enhancement of Differential Quantum Efficiency
Both n- and p-electrodes were fabricated in each structure, followed by polishing of the GaN substrate and coating of the substrate's back surface with an anti-reflection coating comprising four layers of a SiO\({}_{2}\)/Nb\({}_{2}\)O\({}_{5}\) dielectric. Note that the substrate side is defined herein as the front of the device (see Figure 2). The reflectivity values for the 46-pair AlInN/GaN and 10.5-pair SiO\({}_{2}\)/Nb\({}_{2}\)O\({}_{5}\) DBRs at 445 nm were determined to be 99.8% and 99.98%, respectively. By way of comparison, a previous study produced a VCSEL having a 40-pair AlInN/GaN DBR and a 410 nm band that showed 99.7% reflectivity [11]. To obtain the same level of reflectivity in a VCSEL with a 445-nm band, the number of AlInN/GaN pairs must be increased, because the refractive index difference between AlInN and GaN is relatively small at longer wavelengths. In this work, the number of AlInN/GaN DBR pairs was also varied, over the range from 42 to 46, so as to tune the reflectivity. The characteristics of the structures in Figure 2a,b were compared. Figure 2: Schematics of the fabricated blue VCSELs with (**a**) simple SiO\({}_{2}\) aperture and (**b**) SiO\({}_{2}\)-buried lateral optical confinement structures. The AlInN/GaN DBR reflectivity value was calculated based on the front slope efficiency ratioed to the rear slope efficiency, in conjunction with pulsed operation [25]. A reflectivity of 99.98% was determined for the 10.5-pair SiO\({}_{2}\)/Nb\({}_{2}\)O\({}_{5}\) DBR. The slope efficiency ratio can be expressed as:  
\[\frac{\eta_{s}^{F}}{\eta_{s}^{R}}=\frac{\Big{(}1-R_{f}\Big{)}\sqrt{R_{r}}}{(1-R_ {r})\sqrt{R_{f}}}, \tag{2}\]  
where \(\eta_{s}^{F}\) and \(\eta_{s}^{R}\) are the front and rear slope efficiencies, and \(R_{f}\) and \(R_{r}\) are the front and rear reflectivity, respectively. In order to prevent thermal effects on the slope efficiency, these data were acquired with a pulse width of 300 ns and a 0.5% duty ratio. The time-resolved waveform of the current signal was obtained using a sampling oscilloscope (LeCroy 9374 TM 1 GHz bandwidth) and by measuring the average light output power with an optical power meter (ADCMT 8230). Figure 3a,b show the reflectivity and slope efficiency from the front side as functions of the number of AlInN/GaN DBR pairs. These VCSELs exhibited the lasing wavelength ranged from 440 nm to 443 nm. From Figure 3a, it is evident that the reflectivity decreases as the number of DBR pairs decreases from 46 to 42, corresponding to a change in the reflectivity from 99.8% to 99.5%. The slight difference in reflectivity between the VCSELs with and without the LOC structure may be due to the In-plane distribution. This drop in reflectivity from 99.8% to 99.5% approximately doubles \(\alpha_{m}^{F}\) and therefore, based on Equation (1), gives a higher slope efficiency. Compared to the efficiency of a 46-pair AlInN/GaN DBR VCSEL with a conventional SiO\({}_{2}\) aperture (0.13 W/A), the efficiency increased by a factor of 4.7 (to 0.61 W/A) upon introducing the LOC structure, and was further improved, to a total factor of 6.7 (to 0.87 W/A), by additionally tuning the front cavity mirror. This represents the highest slope efficiency yet reported for a GaN-based VCSEL. To evaluate the impact of reducing the internal loss by introducing the LOC structure, we analyzed the relationship between the mirror loss and the total external differential quantum efficiency  
Figure 3: (**a**) Reflectivity of front mirror as a function of the number of AlInN/GaN DBR pairs. (**b**) Slope efficiency under pulsed conditions as a function of the number of AlInN/GaN DBR pairs. The open and closed symbols indicate the results obtained from VCSELs with a conventional SiO\({}_{2}\) aperture and a SiO\({}_{2}\)-buried lateral optical confinement structure, respectively. (\(\eta_{d}\)_total_) [26; 27]. \(\eta_{d}\)_total_ was calculated by determining both the front and rear side slope efficiencies.