## 2 Mathematical Model of Plasma Spark Initiation  
We study an anisotropic optical fiber, which has a cylindrical shape, so cylindrical coordinates are used for modeling, where the z-axis is directed along the fiber axis and the r-axis along its radius. The computational domain is presented in Figure 3. The problem was solved in the axisymmetric formulation, so all calculations were performed in the z-r plane in half of the fiber and metal plate. At the fiber symmetry axis (r = 0), the solution symmetry conditions were set. The temperature field independent of the polar angle was determined within the model. The computational domain in Figure 3 consists of five  
Figure 2: The moment of plasma spark initiation in the fiber. elements with different thermophysical properties. At the interfaces of the elements, the thermal balance conditions are satisfied. They will be discussed below. A section of a single-mode fiber with a stepped refractive index profile SMF-28e (see Figure 3), consisting of a core (1), a shell (2), and a protective coating (3), is studied. The fiber is tightly pressed against the initiator, in this case, a duralumin plate (4). The initial and ambient temperatures of the fiber and the metal plate are assumed to be \(T_{0}\) = 293 K.  
In the calculations, the physical properties of the optical fiber were set according to the specifications of the fiber SMF-28e. Duralumin plate D16 was used according to the 2024 International Organization for Standardization (ISO). The materials and dimensions of domains of the calculation area are presented in Table 1. Thermal and optical parameters of materials were set for each domain of the computational area and selected from the COMSOL Multiphysics library. The fiber core (1), its shell (2), the protective strengthening coating (3), and the duralumin plate (4) in Figure 3 have contact boundaries between them at \(S_{12}\), \(S_{14}\), \(S_{23}\), \(S_{24}\), and \(S_{34}\). The outer boundary of the fiber and the metal plate with the environment is denoted by \(S\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Domain** & **Material** & **z, μm** & **r, μm** \\ \hline \multirow{2}{*}{A fiber core} & Ge- doped SiO\({}_{2}\) (fused quartz) & \multirow{2}{*}{250} & \multirow{2}{*}{4.1} \\  & [solid, average] & & \\ \multirow{2}{*}{A fiber shell} & SiO\({}_{2}\) (fused quartz) & & \\  & [solid, average] & & \\ \multirow{2}{*}{A protective coating} & PMMA—Polymethyl methacrylate & \multirow{2}{*}{250} & \multirow{2}{*}{62.5} \\ \multirow{2}{*}{A duralumin plate} & duralumin 2024 [solid] & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Materials and dimensions of elements of the calculation area. Figure 3: Schematic diagram of the computational domain in axisymmetric formulation, consisting of a core (1), a shell (2), a protective coating of optical fiber (3), and a metal plate—the initiator of optical breakdown (4). Within the framework of the mathematical model, the distribution of the temperature field T (\(r\), \(z\), \(t\)) in the entire computational domain (Figure 3) is described via the following heat conduction equation [20]:  
\[\rho_{i}C_{Pi}\frac{\partial T}{\partial t}=k_{i}\left(\frac{\partial^{2}T}{ \partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial^{2}T }{\partial z^{2}}\right)+Q, \tag{1}\]  
where \(i=1,2,3,4\) is the number of the domain in the computational area (Figure 3), \(\rho_{i}\) is densities [kg/m\({}^{3}\)], \(C_{Pi}\) is specific heat capacities [J/(kg\(\cdot\)K)], \(k_{i}\) is thermal conductivity of materials [W/(m\(\cdot\)K)], and \(Q\) is heat source, the intensity of which is determined via the optical radiation [W/m\({}^{3}\)]. These functions have a complex form depending on the temperature and are shown in Appendix A, Table A1. As mentioned above, all thermophysical properties of materials are taken from the COMSOL Multiphysics library.