## 3 High-Power and Narrow Divergent Beam Performance Using a Long Cavity Structure
The apparent increase of 65% in this value suggests an approximately 1.5 fold increase in the operating current at rollover, as seen in Figure 6a,b. This effect can be demonstrated by considering the 1.45 fold increase in the rollover operating current at 20 \({}^{\circ}\)C (from 20 to 29 mA). It is therefore evident that lowering \(R_{th}\) increases the output power. Based on the theoretical evaluation of \(R_{th}\) in Ref. 35, this effect of lowering \(R_{th}\) in the present work is not  
Figure 7: **(a)** Emission spectra of the VCSEL with a 5\(\lambda\)-cavity structure at different injection currents. A narrow peak wavelength at 443.5 nm was observed above the threshold. **(b)** Emission spectra of the VCSEL with a 10-\(\lambda\) cavity structure at different injection currents. A narrow peak wavelength at 440.1 nm was observed above the threshold. unexpected, so it is apparent that increasing the n-GaN layer thickness enhances thermal dissipation within the VCSEL. Using the power dissipation and \(R_{th}\), values of 159 and 161 \({}^{\circ}\)C were estimated for the internal temperature at rollover (\(T_{i}\)_rollover_) of the devices having 5\(\lambda\)- and 10\(\lambda\)-cavity structures. The fact that these temperatures are quite close to one another demonstrates that a lower \(R_{th}\) value explains the increased output power of the 10\(\lambda\)-cavity unit. Therefore, obtaining the maximum possible lasing temperature (\(T_{hs}\)\({}^{max}\)) requires optimization of both \(R_{th}\) and \(T_{i}\)_rollover_. During lasing, the value of this parameter can be written as:  
\[T_{hs}^{\text{max}}<T_{i}^{rollover}-R_{th}\times P_{diss}^{threshold}\approx T _{i}^{rollover}-R_{th}\times I_{th}\times V_{th}. \tag{5}\]  
where, \(P_{diss}\)_threshold_ is the threshold power dissipation while \(I_{th}\) and \(V_{th}\) are the threshold current and voltage. Therefore, the effect of temperature on \(I_{th}\) is a primary factor with regard to the magnitude of \(T_{hs}\)\({}^{\text{max}}\). The threshold currents are plotted as functions of temperature in Figure 8. It is evident from these data that the effect of temperature on \(I_{th}\) was small from 293 to 333 K, and that the 10\(\lambda\)-cavity VCSEL showed a comparatively elevated characteristic temperature (\(T_{0}\) = 316 K). Conversely, the \(I_{th}\) value was increased rapidly as the temperature was increased above 330 K. This trend greatly increased the internal temperature of the VCSEL at high temperatures but below the threshold, such that the device exhibited a maximum lasing temperature of 110 \({}^{\circ}\)C. The VCSEL threshold current is determined by the degree of mismatch between the peak gain and the cavity mode wavelength. Thus, the optimal extent of lasing mode detuning as a result of temperature changes is required for each specific application [39]. The long-cavity structure strongly affects not only the thermal resistance but also the transverse mode in VCSELs because the normalized effective index difference (\(\Delta n_{eff}/n\)) decreases even for SiO\({}_{2}\)-buried layers of the same thickness in Figure 2b, and the diffraction loss of higher-order-modes increases, compared to the short-cavity structure. Furthermore, the thermal lens effect in the long-cavity structure must fade off at the same dissipated power because of the change in the thermal resistance. In general, low divergent output beams are preferred for many applications because they allow for the use of simple optics and exhibit high brightness. [PERSON] et al. have demonstrated a small full width at half maximum (FWHM) divergence angle of approximately 5\({}^{\circ}\) in a GaN-based VCSEL with a 3-\(\upmu\)m aperture Si-diffusion-defined confinement structure [40]. The FFP and I-L characteristics at low power of small aperture VCSELs were also assessed so as to examine variations in the mode patterns beam divergence between the 10\(\lambda\)- and 5\(\lambda\)-cavity VCSELs.