## 2 Mathematical Model of Plasma Spark Initiation
Assuming that the temperature at the transition across the interface changes continuously, at the borders \(S_{12}\), \(S_{14}\), \(S_{23}\), \(S_{24}\), and \(S_{34}\) of the contiguous domains of the calculation scheme, the contiguity or balance of heat fluxes conditions apply:  
\[\left(k_{i}\frac{\partial T}{\partial r}l_{r}+k_{i}\frac{\partial T}{\partial z }l_{z}\right)_{S_{ij}}=\left(k_{j}\frac{\partial T}{\partial r}l_{r}+k_{j} \frac{\partial T}{\partial z}l_{z}\right)_{S_{ij}} \tag{2}\]  
where \(l_{r}\) and \(l_{z}\) are the projections of the unit vector of the normal to the corresponding surfaces on the coordinate axes \(z\) and \(r\). The outer boundary of the computational domain \(S\) was assumed to be perfectly conductive, and the ambient temperature was set on it at \(T_{0}=293\) K.  
The last term of the thermal conductivity equation \(Q\) describes the heat loss that occurs due to the interaction of optical radiation propagating along the fiber with the contact boundary between the end of the fiber and the metal plate. The contact between the fiber face and the plate is assumed to be ideal. After reflection, a standing wave is generated in the core, the energy of which is absorbed by the substance near the contact boundary. The intensity of this heat source depends on the optical power and the wavelength of radiation entering the optical fiber and is calculated using the following formula:  
\[Q(r,z)=\alpha(r,\ z)\frac{P}{A_{eff}}\cdot\Gamma(r,z) \tag{3}\]  
where \(\alpha=(\alpha_{0}+\alpha_{e})\) is fiber absorption coefficient [m\({}^{-1}\)], \(P\) is input power [W], \(\Gamma\) is standard Gaussian distribution \(\Gamma(x)=\frac{1}{\sqrt{2\pi}}\cdot e^{-\frac{x^{2}}{2}}\), and \(A_{eff}\) is effective area of the mode spot [m\({}^{-2}\)]. Because the problem is solved as a thin 2D layer, the standard Gaussian distribution takes a 2D form as follows:  
\[\Gamma(r,z)=\Gamma(r)\cdot\Gamma(z) \tag{4}\]  
The absorption coefficient \(\alpha\) is the sum of the absorption coefficient under normal conditions (at temperatures close to 293 K) \(\alpha_{0}\) and the absorption coefficient of electron gas \(\alpha_{e}\) appearing at temperatures above 1273 K. The coefficient \(\alpha_{0}\sim 1\) m\({}^{-1}\)[7] describes nonlinear absorption of radiation on chaotically located in the fiber defects like SiE\({}^{\prime}\) or GeE\({}^{\prime}\). Such defects or XE\({}^{\prime}\)-centers (Ge germanium or Si silicon can be instead of X) are connected with formation of oxygen holes in bonds: Si-X \(\rightarrow\) Si-O-X. Defects appear in the core of the fiber as a result of chemical reactions during its manufacture [20; 21]. When heated, the concentration of GeE\({}^{\prime}\) centers remains constant, while at temperatures exceeding 1000 K, the concentration of SiE\({}^{\prime}\) decreases to zero [11]. When the core temperature reaches 1273 K, thermal ionization of GeE\({}^{\prime}\) centers begins [21]. As a result, electron gas, i.e., a large number of free electrons, is formed in the core of the fiber. The concentration of the released electrons coincides with the concentration of GeE\({}^{\prime}\)-centers ions. According to [22], in a fiber doped with 4 mol% germanium, the concentration of GeE\({}^{\prime}\)-centers ions is calculated as follows:  
\[n_{\text{GeE}^{\prime}}=n_{p}\exp\left(-\frac{\text{E}_{\text{f}}}{k_{B}\text{T} }\right), \tag{5}\]  
where E\({}_{\text{f}}\) = 2.5 eV is formation energy GeE\({}^{\prime}\), \(n_{p}\) = 1.72 \(\times\) 10\({}^{15}\) m\({}^{-3}\) is concentration GeE\({}^{\prime}\)centers at normal temperature, and \(k_{B}\) is Boltzmann constant. The conductivity of quartz due to free electrons will be expressed as follows:  
\[\sigma=\text{e}\mu_{e}n_{e}=\text{e}\mu_{e}n_{p}\exp\left(-\frac{\text{E}_{ \text{f}}}{k_{B}T}\right), \tag{6}\]  
where \(e\) is electron charge modulus, and \(\mu_{e}\) is the drift mobility of electrons.