## 1 Introduction  
The scheme shown in the paper is based on the photothermal effect and provides one of the possible ways of indirect measurement of the absorption coefficient. The measurement of absorption in optical materials by means of the photothermal effect does not require destruction of the sample and plays a special role among many other measurement methods due to its sensitivity. There are a variety of photothermally based techniques implemented using various optical schemes: the thermal lens technique [1; 2; 3; 4; 5; 6], photothermal common-path interferometry (PCI) [7; 3; 8; 9; 10], time-resolved photothermal common-path interferometry (TPCI) [11; 12; 13; 14; 15], the photothermal deflection technique [3; 16; 17; 18; 19] (as well as the laser induced deflection technique [3; 20; 21]), surface thermal lensing [22; 23; 24], and the phase-shift (interferometric) technique [3; 25; 26; 27; 28]. Photoacoustic spectroscopy [3; 4; 8; 29; 30; 31] is also based on the photothermal effect. These methods continue to be actively developed for several reasons. First of all, the problem of increasing the sensitivity of the methods to the level of \(\sim\)10\({}^{-8}\) cm\({}^{-1}\) remains urgent. It is necessary for their use in the task of the reliable control of the absorption coefficients of ultrapure materials (e.g., Suprasil 3001 quartz glasses [32]) with values at the level of \(\sim\)10\({}^{-7}\) cm\({}^{-1}\). To confirm this, one can refer to the Heraeus data. In [33] the results of absorption coefficient measurement projects for various glass grades are given, in particular for the most transparent glass, Suprasil 3001. It was not possible to measure the bulk absorption coefficient in this glass using the PCI method because, as the authors indicate, it is below the detection limit (<10\({}^{-6}\) cm\({}^{-1}\)). Measurements employing a cavity ring-down loss meter in conjunction with a PCI also did not give a specific result and only confirmed that the bulk absorption coefficient was below the level of 10\({}^{-6}\) cm\({}^{-1}\). Increasing the sensitivity of low absorption measurements is also necessary for the identification of impurities and determination of their concentration in ultrapure quartz materials at the level of fractions of ppb. This task is inaccessible to mass spectrometry methods for a number of common elements [34]. The problem can be solved by measuring the absorption coefficients of the material at different wavelengths of heating radiation generated by low-power diode lasers [35]. In addition, our interest in the task of increasing the measurement sensitivity is conditioned by the supposed further thorough research of the spatial inhomogeneity of absorption in hydrothermal quartz crystals and quartz glasses with low concentration (~1 ppm) of hydroxyl groups. Inhomogeneities are thin (200-300 \(\upmu\)m) regions in which the value of ultra-low absorption can differ greatly [13]. Such absorption inhomogeneity can play a role in the application of these quartz materials in power optics. According to the hypothesis expressed in our paper [36], this can be related to a different ratio of the concentrations of absorbing and low-absorbing ions of impurities, that are homogenously distributed in the material volume. The spatial heterogeneity of the ratio of the ion concentration in the material volume and the formation of a spatial structure can be a consequence of strong gradients of fictive temperature and associated high local pressures formed during the glass transition of quartz glasses in their manufacture. Variations of these parameters can lead to different ratios of impurity ion concentrations with different absorption at the measurement wavelength. The initial motivation for our work in this direction was the need to control the absorption coefficient of the optical elements used in the cascade frequency multiplication scheme of a continuous wave ytterbium laser with a relatively low power of ~100 W [37].