## 4 The Complex Dielectric Permittivity
The second terms in the right parts of (36), (37), respectively, reflect the influence of the polarizing field on the CDP spectra, in the regions of both strong fields (10-1000 MV/m) and ultra-high temperatures (550-1500 K)--high-temperature volumetric-charge polarization; and weak fields (100 kV/m-1000 kV/m) and ultra-low temperatures (1-10 K)--low-temperature quantum tunnel polarization. The functions \(\Gamma_{1,1}^{(\omega 2\omega;3\omega)}\), \(\Gamma_{1,2}^{(\omega;2\omega;3\omega)}\) reflect the non-linear effects of the third order on the field frequency associated with the interaction of the relaxation modes generated at frequencies \(\omega\), \(2\omega\), \(3\omega\) with the corresponding relaxation times \(T_{\rm s}T_{\rm m}T_{\rm p}\). An additional relaxation mode generated at the frequency of the \(3\omega\) field with relaxation time \(\tau_{\rm p}\) appears in functions \(\Gamma_{2,1}^{(\omega;2\omega;3\omega)}\), \(\Gamma_{2,2}^{(\omega;2\omega;3\omega)}\). At the same time, the parameters \(\Gamma_{1}^{(\omega)}\), \(\Gamma_{2}^{(\omega)}\) reflect the non-linear effects of the first order in frequency, with relaxation times \(\tau_{\rm n}\). Parameters \(\Lambda_{1}\), \(\Lambda_{2}\) of (18h), (23) are a measure of the effect of the non-linear effects of the third order (interaction between the relaxation modes at frequencies \(\omega\), \(2\omega\), \(3\omega\) and corresponding relaxation times \(\tau_{\rm s}\tau_{\rm m}\tau_{\rm p}\)) on the polarization. So, provided that the parameters \(\Xi_{0}==\frac{8n_{0}\varphi_{1}\Lambda_{0}\gamma}{\pi^{2}}<<\)1, \(\Xi_{1}=\frac{8n_{0}\varphi_{1}\Lambda_{1}\gamma}{\pi^{2}}<<\)1, \(\Xi_{2}=\frac{8n_{0}\varphi_{1}\Lambda_{2}\gamma}{\pi^{2}}<<\)1 are small, the expression (23) is determined by the relaxation mode \(\rho_{3}^{(3\omega)}(\xi,\tau)\), corresponding to the third order of the perturbation theory (k = 3). This is the simplest (by the structure of complex amplitudes \(\Re_{\rm k}^{(3\omega)}(n,\tau)\)) of the set of modes \(\rho_{\rm k}^{(3\omega)}(\xi,\tau)\) generated by the frequency \((3\omega)\), which is consistent with ([3], p. 17). Numerical calculation according to the formulas (36), (37) will allow analysis in more detail of the influence of the non-linear effects of the third order on the field frequency, on polarization, depending on the values of the parameters U\({}_{0}\), \(n_{0}\), \(\
u_{0}\), \(\xi_{0}\) ([4], pp. 150-152) and the thickness of the dielectric. The mathematical model developed in this paper is suitable for theoretical studies, previously experimentally installed in crystals of zirconia alumina ceramics (ZAC) (ZrO\({}_{2}\)-Y\({}_{2}\)O\({}_{3}\))-n (Al\({}_{2}\)O\({}_{3}\)), at the temperature of T = 1250 K and frequency of 1 kHz, abnormally high dielectric permittivity \(\varepsilon=5\cdot 10^{6}\)[5]. The nonlinearities of the first and third order can be considered as the reason for the semi-ferroelectric effect [6] associated with the rearrangement of the oxygen subarray in the ZAC, near the phase transition point. The areas closest to electrical and radio engineering in the practical application of this model are laser and fiber optic technology: femtosecond lasers, optical light manipulators, incident radiation direction detectors, differential phase and frequency analyzers, etc. [8; 9; 10; 11].