## 2 Materials and Methods
The grown crystals had a flat crystallization front and geometric dimensions: diameter (\(\O\)) \(\approx\) 38 mm,  
Figure 1: Kristall 2 growth setup for growing lithium niobate crystals using the Czochralski method—(**a**); crystal LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\) (0.65 wt.%) **(b**). Figure is reproduced with the permission of Elsevier from the paper [34]. length of the cylindrical part L\({}_{\rm c}\approx\) 40 mm. Impurities were introduced into the charge in the form of high-purity oxides of MgO and Gd\({}_{2}\)O\({}_{3}\) with a concentration of foreign admixtures of no greater than 3 \(\times\) 10\({}^{-4}\) wt.%, followed by thorough mixing. The melt before the start of crystal growth was kept for 8-11 h under superheating conditions by 180-200 \({}^{\circ}\)C relative to the melting temperature (T\({}_{\rm melt}\) = 1263 \({}^{\circ}\)C) of lithium niobate, to homogenize the impurities in the melt. After growth, the Mg crystal was annealed at 1200 \({}^{\circ}\)C in a growth setup for 10 h, and then cooled at a rate of \(\sim\)50 deg/h. Long-term post-growth annealing in the isothermal zone (same as in [34]) under a platinum screen is required to homogenize the composition of the doped crystal, and to remove the thermal and mechanical stresses. The monodomainization of the LiNbO\({}_{3}\):Gd:Mg crystal was carried out via high-temperature electrodiffusion annealing, namely by applying a constant electric voltage to the polar cuts of the crystal during its cooling, at a rate of 20 deg/h in the temperature range \(\sim\)1230-870 \({}^{\circ}\)C.  
For the synthesis of the lithium niobate charge, Nb\({}_{2}\)O\({}_{5}\) grade A was used, and produced using Technical Specifications No. 1763-025-00545484-2000 at Solikamsk magnezium works (Solikamsk, Russia), and with Li\({}_{2}\)CO\({}_{3}\) of high purity, with an amount of foreign admixtures of no greater than 3 \(\times\) 10\({}^{-4}\) wt.%. From these initial components, a granular charge of a congruent composition ([Li\({}_{2}\)O] = 48.6 mol.%) with a high bulk density (\(\sim\)3.4 g/cm\({}^{3}\)) was obtained, using the synthesis-granulation method. The preparation of the charge is described in detail in [35]. To estimate the impurity content in the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal, and the distribution features of the change in their concentration along the length of the crystal, samples were studied from the upper (cone) and lower cylindrical parts of the boule. Inductively coupled plasma atomic emission spectroscopy (ICP-AES) on an Optima 8300 ICP-OES (Mg) (PerkinElmer, Waltham, MA, USA) spectrometer, and atomic absorption spectrometry (AAS) on a Kvant-FA instrument (Zn) (GRANAT, Saint Petersburg, Russia) were used to determine the concentration of dopants in milled samples. Table 1 shows the admixture concentrations of the granular mixture and the studied LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) single crystal. The sample for studying the Raman spectra was a parallelepiped with dimensions of 4.93 \(\times\) 5.98 \(\times\) 8.41 mm\({}^{3}\), and edges coinciding with the direction of the principal crystallographic axes. The faces of the parallelepiped were thoroughly burnished. To record the Raman spectra in the visible region, we used a BWS465-532S i-Raman Plus spectrometer (B&W Tek, Plainsboro Township, NJ, USA), which has a wavelength of 532 nm and allows for the recording of spectra in the range of 50-4000 cm\({}^{-1}\). The power of laser radiation during the registration of spectra was 30 mW. The numerical aperture was \(\approx\)0.22. To record the Raman spectra in the near-IR region, a BWS465-785H