## 2 Design, Modeling, and STOP Simulation of an Optical Laser Head for LAAM
### Structural-Thermal-Optical-Performance (STOP) Modeling  
After simplifying the geometry model, a meshed model of the whole optical laser head was created in COMSOL, as shown in Figure 3. The laser-irradiated areas such as the lenses and mirrors were modelled with fine meshes (minimum element size is 10 \(\upmu\)m), while the other zones were modelled with coarse meshes to maintain computation accuracy as well as to reduce simulation time. The whole model consists of three physics-based models: a thermal model, a structural model, and a ray-tracing optical model. The total number of the elements is about 430 k. The output and input data interfaces were implemented for the STOP analysis. In this study, the former two models were sequentially coupled and simulated in a stationary domain, while the ray tracing optics model was simulated in the time-dependent field. Figure 1: Geometric model of the developed optical laser head. Figure 2: Optical delivery path design. For the thermal model, the partial differential equation (PDE) of heat conduction (Fourier's Law) is given by  
\[k_{x}\frac{\partial^{2}T}{\partial x^{2}}+k_{y}\frac{\partial^{2}T}{\partial y^{2 }}+k_{z}\frac{\partial^{2}T}{\partial z^{2}}+Q_{r}=0 \tag{1}\]  
where \(k_{x},k_{y}\), and \(k_{z}\) are thermal conductivities (W/mK) in \(x\), \(y\), and \(z\) directions, respectively; \(T\) is the temperature (K); and \(Q_{r}\) is the input laser energy density (W/m\({}^{3}\)) due to absorption and reflection by laser beam irradiation. The thermal boundary condition is governed by  
\[q=h(T-T_{0}) \tag{2}\]  
where \(h\) is the convection coefficient. Equation (1) was used to obtain the steady-state temperatures. The temperatures were applied to the structural model as the thermal loading. The thermal-induced displacements and stresses were calculated by the following elasticity equilibrium equation (Hooke's Law) with the thermal expansion equation:  
\[\sigma=E\cdot\varepsilon \tag{3}\]  
\[\varepsilon_{th}=\alpha\cdot(T-T_{0}) \tag{4}\]  
where \(\sigma\) is stress, \(E\) is elastic modulus, \(\varepsilon\) is strain, \(\sigma\) is thermal expansion coefficient, \(T_{0}\) is temperature at initial time, and \(\varepsilon_{th}\) is thermal strain. The optical model adopted ray-tracing approach, which computed the angles of each ray emitting or refraction based on Snell's Law equation. However, ray paths could be changed due to the thermally deformed geometry and temperature-dependent refractive index. The ray heating source is represented by  
\[\frac{\partial Q_{r}}{\partial t}=-\sum_{j=1}^{N_{r}}\frac{\partial Q_{j}}{ \partial t}\mu_{j} \tag{5}\]  
where \(Q_{j}\) (W/m\({}^{3}\)) is the laser power transmitted by the \(j\)th ray, \(N_{r}\) is the total number of rays, and \(\mu_{j}\) is ray power coefficient in the \(j\)th ray. Since the optical components such as lenses focusing the laser beam are almost completely transparent, the amount of heat generated in these optical components can degrade the ability of the system to correctly focus the beam. In this study, the laser power in an absorbing medium (lens) is assumed to decay exponentially as the wave propagates, assuming the absorption coefficient remains constant:  
\[Q_{j}=Q_{0}\exp\left(-\frac{4\pi\kappa ct}{\lambda_{0}n}\right) \tag{6}\]  
Figure 3: Meshed model of the optical laser head for the structural-thermal-optical-performance (STOP) analysis. where \(Q_{0}\) is the laser source power, \(\lambda_{0}\) is the free-space wavelength, \(c\) is the speed of light in a vacuum, and \(t\) is the current time. The complex-valued refractive index is expressed as \(n-\kappa_{i}\), where \(n\) and \(\kappa\) are dimensionless real numbers. The flowchart of the STOP simulation procedure is depicted in Figure 1:0 Firstly, the laser beam parameters and laser power, material properties, and structural and thermal boundary conditions were defined in the geometry model. The laser heating flow on the surfaces of optical lenses was initiated as an absorbent heat source.