# Reduced Dimensionality Multiphysics Model for Efficient VCSEL Optimization
## 1 Introduction  
Since their introduction into the laser market, vertical-cavity surface-emitting lasers (VCSELs) are having increasing success, thanks to low production costs, array-oriented manufacturability, great efficiency (and therefore low power consumption), and temperature stability. Nowadays, VCSELs are widely included in several applications [1, 2, 3, 4], ranging from laser mice to spectroscopy. For example, VCSELs can be used as single-mode sources for activating Cs and Rb in atomic clocks, needed for GPS. Other interesting applications involve VCSELs for 3D gesture and face recognition or for automotive purposes, such as LiDAR (Laser Imaging Detection and Ranging) for autonomous driving [5]. In particular, VCSELs are already the dominant optical sources for optical links in datacenters and high-performance computers [6, 7, 8, 9].  
In this context, 850-980 nm VCSELs appear as key technology enablers for intra-datacenter links within 1 km. From this viewpoint, our research group developed the in-house three-dimensional solver VENUS (Vesel Electro-optho-thermal NUMerical Simulator), which is able to simulate in a self-consistent fashion the electrical, optical, and thermal problems at steady state [10, 11]. In [11] we performed a parameter calibration that allowed us to achieve very good agreement between numerical and experimental results of all the observable macroscopic relevant quantities, such as current-voltage (I-V) and light-current (L-I) characteristics and optical modal features (wavelength red-shift and side-mode suppression ratio). The main goal was to extend the agreement from room temperature to higher temperatures, up to 110 \({}^{\circ}\)C, which is of the highest importance in applications like miniaturized atomic Cs-clocks [12, 13] or datacenters [2, 14]. As a matter of fact, many hyperscale datacenter operators need to adopt high-speed connections to handle data traffic in environments reaching temperatures of 85 \({}^{\circ}\)C [15].  
A quantitative description of VCSEL operation requires multiscale and multiphysics solvers, which are able to deal self-consistently with optical, electric, and thermal problems [16; 17; 18; 19; 20; 21; 22; 23]. The computational burden entailed by such 3D solvers considerably limits their application in large optimization campaigns, which are of the maximum interest. In view of such a task, in this paper we investigate the possibility of applying a reduced dimensionality multiphysics approach to study the entangled opto-electro-thermal VCSEL operation. Namely, to test how a one-dimensional version of VENUS performs, as shown schematically in Figure 1.  
To this end, we have extended our in-house 1D quantum-corrected drift-diffusion code D1 ANA (Drift-diffusion 1-d ANAlysis) [24], by coupling it to the optical and thermal solvers [10]. Therefore, all the subproblems governing the VCSEL operation have been reduced to 1D problems in the longitudinal direction, which is the main direction of the current flow, of the heat flow, and of the photon emission. In this scenario, D1 ANA could be thought as an intermediate model between fully phenomenological rate equations [25] and entirely physics-based 3D descriptions [10; 26; 27], efficient enough to be applied to preliminary extended parametric campaigns and/or in optimization loops.  
In Section 2 we describe the steps needed to switch from a 3D to a 1D framework, the level of agreement one might expect and, at the end, if dimensionality downscaling is advisable.  
After a positive outcome, in Section 3, we apply our calibrated D1 ANA to the optimization of an actual device investigated in [11], with a special focus on high-temperature operation.