# Reduced Dimensionality Multiphysics Model for Efficient VCSEL Optimization
## 3 Examples of Application  
We applied the calibrated D1 ANA suite to investigate the cavity detuning (\(\Delta\lambda\)) of this sample, in view of optimizing high-temperature operation, which is of crucial importance  
Figure 9: L-I characteristics at three different ambient temperatures (\(20,50,80~{}^{\circ}\text{C}\)). Open circles refer to experimental results, solid lines to the proposed calibrated 1D model, Equation (5), and dashed lines to a constant \(\beta_{T}=1.2\). for VCSELs working in datacenters. The cavity detuning is defined as the wavelength difference between the maximum of the gain spectrum and the emission wavelength, as sketched in Figure 3.  
This is a very important parameter for optimized operation, because the gain spectrum red-shifts much faster than the emission wavelength [58], by about a factor of four. For example, if one looks at Figure 6, the wavelength shifts by 4 nm for about a 60 K temperature increase inside the VCSEL. At the same increase, there corresponds a shift of the gain spectrum by 15 nm (blue and yellow gain spectra in Figure 3). Figure 10a shows that the measured device is optimized for room-temperature operation, with a cavity detuning of +3 nm. At 80 \({}^{\circ}\)C (Figure 10b) an 18 nm cavity detuning appears to be beneficial, since it would increase the maximum optical output power by 1 mW (from 1.5 to 2.5 mW), by almost leaving the threshold current unchanged. This can be understood from Figure 3, by comparing the different interactions between the optical line with the gain spectrum in the two cases of the nominal design (\(\lambda_{nom}\) in Figure 3), and of the optimized one (\(\lambda_{opt}\)). A higher detuning would only increase the threshold current, leaving the maximum power about the same. Another detail that strongly impacts on the output power and threshold current is the DBR radiation loss. The top-DBR radiation losses are \"useful losses\", i.e., the optical output power (instead, the radiation in the substrate is lost). However, one must not think that the stronger the bottom-DBR reflection, the better. In fact, DBRs also induce absorption losses and, in addition, the bottom DBR is the main barrier for heat toward the heat-sink. This is due to the worse thermal conductivity to the ternary (AlGaAs) DBRs compared to the binary GaAs substrate, by a factor of 3. In Figure 11, we study the effect of modifying the DBRs at the two ambient temperatures of 20 and 80 \({}^{\circ}\)C. At room-temperature (Figure 11a), we find an improvement by reducing the number of output pairs to 4. At this value, we achieve 5.5 mW of maximum optical power, instead of the 3.3 mW of the nominal VCSEL. This occurs at 10 mA, the same current for which the experimental maximum power is detected at room temperature. From our simulations, a further reduction of the output DBR pairs causes an increase in the threshold current and the consequent higher heating in the device deteriorates the performance (violet curves), also in terms of maximum output power, which is reached at a lower current (8 mA). On the same plots, with dashed lines, we also report the case of a thinner bottom DBR, by 5 pairs, which corresponds to a VCSEL that is about 650 nm shorter. The results are fully in line with our previous comments. The thinner bottom DBR slightly increases the optical losses, and therefore the threshold slightly increases, and L-I  
Figure 10: L-I characteristics at different cavity detuning values at 20 (**a**) and 80 \({}^{\circ}\)C (**b**). curves are shifted to the right compared to the thicker mirror. However, the better thermal performance results in a later thermal rollover, allowing a marginal improvement of the maximum output power. As shown in Figure 3, at high temperature the available gain more than halves, moving from 300 to 390 K at the same carrier density level, due to the larger broadening of the Fermi distributions. This is the main reason that eventually prevents VCSEL operation at a certain temperature.