## 3 Discussion of the Obtained Results
Based on the results obtained, it can be argued that significant losses occur in a particular location with a length of 5 mm (area C). That is, the effect of channeling suppression is not point-wise but distributed. At the same time, the losses are due to an increase in the backscattered signal in this area, which can be associated with a change in the refractive index. Most clearly, this increase occurs during rapid heating at sufficiently low temperatures. Additionally, from Figure 9, we can notice the shift of all peaks during the thermal cycle. The analysis of the trace dynamics shows that all peaks translocate in proportion to the sample temperature, shifting to the right with increasing temperature and to the left with decreasing temperature. As can be seen from the trace diagrams, both prominent DUT peaks--both input and output shift as the temperature changes along the coordinate (Figure 11). It can be assumed that the principal shift of the coordinates of the peaks is due to a change in the refractive index of the light-feeding optical fiber, the length of which is much longer than the length of the modulator. Then, to study the processes occurring in the modulator waveguides, it is necessary to study the difference in coordinates of input and output peaks as a function of temperature. It is this difference that will characterize the thermo-optic effect and the linear thermal expansion of the crystal. It should be noted that, in this consideration, we neglect the minor effects due to the temperature gradient inside the modulator chip when it is heated unevenly. We are entitled to make such neglect due to the small volume and mass of the crystal compared to the volume of the thermal chamber, the relatively low heat capacity of lithium niobate, and the low rate of temperature change. When changing the modulator temperature, several phenomena simultaneously contribute to the shift of peak coordinates: a change in the crystal's refractive index due to the thermo-optical effect \(\Delta\mathrm{l_{w}}\) and a change in the physical length of the crystal due to linear thermal expansion \(\Delta\mathrm{l_{TE}}\). Then \(\Delta\mathrm{l_{out}}-\Delta\mathrm{l_{in}}=\Delta\mathrm{l_{w}}+\Delta \mathrm{l_{TE}}\). Let us evaluate each of the above contributions for the temperature change \(\Delta\mathrm{T}=120\)\({}^{\circ}\)C. As indicated in [16], the coefficient of thermal expansion of LN for X- and Y-cuts depends little on the crystal composition and has insignificant changes in the temperature range from 220 to 553 K, which includes the range of temperature changes of the sample studied. For linear thermal expansion, we can use the relation \(\Delta\mathrm{l_{TE}}=\alpha\cdot l\cdot\Delta\mathrm{T}\), where \(\alpha\) is the coefficient of linear thermal expansion for a given direction, \(\Delta\mathrm{T}\) is the change in the crystal temperature. For the investigated crystal cut \(\alpha=13.4\times 10^{-6}\)\(1/\)K [17], as a result, the change in physical length can be estimated as \(\Delta\mathrm{l_{TE}}=\alpha\cdot l\cdot\Delta\mathrm{T}=61\)\(\upmu\)m. The change of optical path is 134 \(\upmu\)m. The value of the average thermo-optical coefficient for the target temperature range is \(\beta=3.35\times 10^{-5}\)\(1/\)K for the extraordinary refractive index and the emission wavelength of 1523 nm [18,19]. The change in length associated with the thermo-optic effect can be calculated by the formula \(\Delta\mathrm{l_{w}}=l\cdot\Delta\mathrm{n}=l\cdot(\beta\cdot\Delta\mathrm{ T})=152\)\(\upmu\)m. Thus, the total change in the optical path is 286 \(\upmu\)m.  
The experimental shift of peaks in the trace was associated with the sample temperature change (Figure 11) \(\Delta l=(4.195-3.545)\cdot\Delta\)T = 78 \(\upmu\)m. The change in the optical path makes 171 \(\upmu\)m.  
The difference between the experimental results and calculation can be explained by shielding the waveguides of mobile charges from crystal lattice and decreasing of pyroelectric effect.