# Fiber-Optic System for Monitoring Pit Collapse Prevention
## 2 Materials and Methods
The mathematical model of the processes of interaction of an optical signal with the substance of an OF will be built using the expression for the bias current density \(\overset{\rightarrow}{J}(t)=\frac{\partial\overset{\rightarrow}{D}}{\partial t }=i\omega\varepsilon_{0}\varepsilon_{\infty}\overset{\rightarrow}{E}(t)+ \frac{\partial\overset{\rightarrow}{P_{T}}(t)}{\partial t}\)[30], where \(\varepsilon_{\infty}\) is an inductive (high-frequency) component of the complex permittivity (CDP), neglecting the relaxation (low-frequency) polarization of the sample \(\frac{\partial\overset{\rightarrow}{P_{T}}(t)}{\partial t}\to 0\)[30;31], and calculating the volumetric density of the electromagnetic wave in the optic fiber in the form [32]:  
\[P_{Vt}=2<Re(\overset{\rightarrow}{E}(t))\times Im(\overset{\rightarrow}{J}(t ))>\approx\omega\varepsilon_{0}\varepsilon_{\infty}\overset{\rightarrow}{E}_{0}. \tag{4}\]  
The equation \(\overset{\rightarrow}{\mu_{0}H_{0}^{2}}=\varepsilon_{0}\varepsilon_{\infty} \overset{\rightarrow}{E}_{0}^{2}\) is used for the electromagnetic wave on the condition \(\mu_{\infty}=1\) (for non-magnetic material). The value of the refractive index of a light wave in this model (in the optical and wavelength range) is determined by the high-frequency component of the complex dielectric constant \(n_{\infty}=\sqrt{\varepsilon_{\infty}}\). The refractive index of OF, in general, can be represented as a multidimensional functional \(n_{\infty}(\varsigma_{0};\omega;E_{0};T;d;l;\theta;\alpha_{0})\), defined on a set of variables representing a set of physical parameters of various types reflecting the properties of the structure and geometry of the experimental sample (fiber-optic element). First of all, these parameters include:  
1. Microscopic characteristics of the structure or molecular characteristics presented in the form of a set of variables of the model under study \(\varsigma_{0}=n_{\infty}(U_{0};v_{0};\delta_{0};n_{0};a)\)[1], where \(U_{0}\) is the activation energy of relaxators (centers of interaction of an electromagnetic wave (optical signal) with matter); \(v_{0}\) is the natural oscillation frequency of the relaxator (molecule, ion, cluster) relative to the equilibrium position; \(\delta_{0}\) is the width of the potential barrier overcome by the relaxer due to thermal activation (in this case, as a result of thermal fluctuations that significantly affect optical processes in the fiber in the field of thermoelastic deformations); \(n_{0}\) is the equilibrium concentration of relaxers (in the absence of external disturbances (electromagnetic fields) in OF); \(a\) is the lattice constant (in the case of a crystal structure) [10]. For the first time, the application of the method of minimizing the comparison function in relation to theoretical studies of the kinetics of electro-optical and electrophysical processes in materials of various crystal structure complexity (in particular, in fiber-optic samples) was proposed in [10; 11; 12; 13; 14]. 2. Field characteristics: amplitude \(E_{0}\) and frequency \(\omega\) of the electromagnetic wave (optical signal) \(\overset{\rightarrow}{E}=\overset{\rightarrow}{E}_{0}\cdot\exp(i\omega t-( \overset{\rightarrow}{k}\overset{\rightarrow}{r}))\). In this model, the electric field strength vector of a wave propagating in space with a circular frequency \(\omega=\frac{2\pi\
u}{\lambda}\)\(\omega=\frac{2\pi\
u}{\lambda}\) and a wave vector \(k=\frac{2\pi}{\lambda}\) is recorded at an arbitrary point in space \(\overset{\rightarrow}{r}\) at time \(t\), according to a generalized solution of the classical Dalember wave equation \(\overset{\rightarrow}{\bigtriangledown}^{2}\overset{\rightarrow}{E}- \frac{1}{\
u^{2}}\frac{\partial^{2}E}{\partial t^{2}}=0\) for a free electromagnetic field (away from the source). 3.