## 5 Conclusions  
This paper presents theoretical calculations to solve the problem of calibrating absorption measurements using the TPCI scheme under conditions of developed heat diffusion in the tested sample. Within the framework of the Gaussian approximation, analytical formulas were obtained for the time dependence of the power of the time-varying component of the probe beam field. To measure absorption in a synthetic crystalline quartz sample, a correction factor was calculated to take into account the heat diffusion effect. The calculated coefficient coincided with that measured earlier in the experiment. A good agreement was also obtained for the power decay times of the time-varying component of the probe beam upon cooling of the samples. Thus, the presented theoretical calculations make it possible to improve the accuracy of the calibration of measurements in the TPCI scheme with various beam configurations under conditions of developed heat diffusion in the sample. The resulting formulas significantly expand the capabilities of the calibration procedure for long heating pulse durations, allowing the use of calibration samples with different heat diffusion times. Finally, this permits one to increase the energy of the heating radiation by increasing the duration of the heating pulse and, as a consequence, increase the sensitivity of measurements. For quartz glasses, the implementation of a heating duration of 5 ms (corresponding to the condition \(\tau_{\rm pulse}\approx 2\tau_{\rm diff}\)) increases the sensitivity of the scheme to 2 \(\times\) 10\({}^{-9}\) cm\({}^{-1}\), which is significantly less than the theoretical estimate of the fundamental absorption of pure quartz glass material. This sensitivity opens the principal possibility of measuring absorption deviations from fundamental absorption related to low concentrations of impurities in order to determine their concentrations at a level of \(\sim\)1 ppt in ultrapure quartz glasses.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & \(\tau_{\rm diff}\), \(\mu\)s & \(\tau_{\rm 1/2\_exp}\)\(\mu\)s & \(\tau_{\rm 1/2\_exp}\)/\(\tau_{\rm diff}\) & \(\tau_{\rm 1/2\_calc}\)/\(\tau_{\rm diff}\) \\ \hline Suprasil 311 & 2563 & 1900 & 0.74 & 0.70 \\ SCQ & 492 & 450 & 0.91 & 0.84 \\ Air & 109 & 190 & 1.74 & 1.69 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The values of the calculated and measured decay times of the amplitude of the time-varying component (at the half-maximum signal amplitude level) after the end of the heating pulse for Suprasil 311 UQG and SCQ samples and ambient air. The signal amplitude decay time was measured using digital oscillograms averaged over 10\({}^{4}\) pulses (the level of the noise component did not exceed 1/100). The uncertainty of the obtained values did not exceed \(\pm\)1%.  
**Author Contributions:** Conceptualization, K.V. and A.M.; methodology, K.V. and A.M.; validation, K.V. and A.M.; formal analysis, A.M.; investigation, K.V. and A.M.; resources, N.A.; writing--original draft preparation, K.V. and A.M.; writing--review and editing, K.V., A.M., and N.A.; supervision, project administration, and funding acquisition, N.A. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was supported by the Center of Excellence \"Center of Photonics\" funded by The Ministry of Science and Higher Education of the Russian Federation (contract No. 075-15-2022-316).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available upon request from the first author, K.V.  
**Conflicts of Interest:** The authors declare no conflict of interest.