## 2 Materials and Methods
### Experimental Research Methods and Materials Used
#### 2.2.3 Effect of Fiber Optic Element Deformations on the Physical and Mathematical Model  
A presentation of mathematical methods for taking into account the influence of the deformation parameters of a fiber-optic element on its refractive index and, in general, on the experimentally measured value of the volumetric power density of an electromagnetic wave (optical signal recorded on a photographic plate) will be given below. The proposed methods for analyzing the influence of elastically deformed states of a fiber-optic element (both from the point of view of longitudinal and transverse deformations) on its optical characteristics are not fundamentally new, but are limited to known methods of the theory of elasticity and are based on calculations of dependencies of the form \(n_{def;\infty}(\zeta_{0};E_{0},H_{0},\omega;\mathrm{T};\theta;\mathrm{b};\alpha_{ 0};l;R)\), \(n_{non-def;\infty}(\zeta_{0};E_{0},H_{0},\omega;T;\theta;\mathrm{b}_{0};\alpha_ {0};l_{0};R_{0})\), where are the parameters in the model \((l_{0};R_{0})\), \((l;R)\), where \(R_{0}=\frac{\omega_{0}}{2}\), \(R=\frac{\omega_{0}}{2}\), are, respectively, the values of the length and radius of the cross section of a horizontal cylindrical beam (loaded with external force influences) inside which, in the direction of the e axis, a fiber optic cable is stretched, experiencing the corresponding deformation effects. A more detailed mathematical description of the relations between the longitudinal and transverse deformation parameters of the modeled OM element is given in Appendix B. Comparison of the refractive index in the non-deformed state and, considering deformations, by expanding the analytical function:  
\[ \begin{array}{c}\Delta n_{def;\infty}(\Omega_{3}\Omega_{0})(\alpha_{0})=n_{ def;\infty}(\zeta_{0};E_{0},H_{0},\omega;\mathrm{T};\theta;\mathrm{b}; \alpha_{0};l;R)\\ -n_{non-def;\infty}(\zeta_{0};E_{0},H_{0},\omega;\mathrm{T};\theta;\mathrm{b}_{ 0};\alpha_{0};l_{0};R_{0})\end{array}  \tag{5}\]  
into an infinite power series in dimensionless small parameters \(\Delta_{d\perp}=|R-R_{0}|/R_{0}\), \(\Delta_{d\parallel}=|\Delta l|/l_{0}\)  
\[ \begin{array}{c}\Delta n_{def;\infty}(\Omega_{3}\Omega_{0})(\alpha_{0})=\sum _{f_{1}=0}^{\infty}\sum_{f_{2}=0}^{\infty}\left[\Delta_{d\parallel}^{f_{1}} \Delta_{d\perp}^{f_{2}}\ \times\Delta n_{def;\infty}^{(\Omega_{3}\Omega_{0});(f_{1}f_{2})}(\alpha_{0})\right] \end{array}  \tag{6}\]  
This will allow, after substituting (6) into the kinetic equations and the d'Alembert equation, us to study the previously indicated effects of the fiber material deformed state in various approximations of perturbation theory in small parameters \(\Delta_{d\perp}\), \(\Delta_{d\parallel}\) starting from the \"zero\" approximation (in the absence of deformation) and, further, up to higher orders of smallness, to reveal the effects of the orders \((f_{1};f_{2})\) in function (6) on nonlinearities caused by the deformed state of the sample (effects of photoelasticity and thermoelasticity). The effect of the temperature range and the parameters of the electromagnetic field of the wave \(E_{0},H_{0},\omega\) is manifested in nonlinear optoelectrical effects, depending, at the mathematical level, on the ratio of the energies of the thermal motion of the relaxers and the energies of the relaxers in the field of the electromagnetic wave. The parameters of the electromagnetic field of the wave may impact the effects of the deformed state, and vice versa. The purpose of the theoretical part of this work is not reduced to the detailed study of the physical mechanisms of the photoelasticity and thermoelasticity effects and related solutions to the equations of the phenomenological model, but is limited, at this stage, to a central descriptive approach, based on which, in subsequent developments, an extended scheme of analytical description of nonlinear electronic optical effects in deformable samples of fiber-optic elements in a wide range of temperatures and parameters of the electromagnetic field of the wave will be disclosed.