## 2 Materials and Methods  
A single crystal of LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) was grown in air from a melt using the Czochralski method on a Kristall 2 (ZavodKristall Ltd., Likino village, Russia) induction growth setup with automatic control of the crystal diameter (Figure 1) in a platinum crucible 75 mm in diameter with a small (2.5 deg/cm) axial gradient in the direction of the polar axis (z-cut, (001)). The single-crystal boule rotation speed was \(\sim\)16-18 rpm, and the crystal pulling speed was \(\sim\)0.7 mm/h. In this case, the crystal growth rate was \(\sim\)1.02-1.04 mm/h. The technological parameters of growth corresponded to the condition of a flat crystallization front. The direct alloying of the congruent melt ([Li]/[Nb] = 0.946) with MgO and Gd\({}_{2}\)O\({}_{3}\) was carried out. To relieve thermoelastic stresses, the grown crystal was subjected to heat treatment in an air atmosphere at 1500 K for 15 h in a high-temperature furnace, Lantan (Voroshilovogradsky zavod electronnogo mashinostroeniya, Voroshilovograd, USSR). Growing optically and compositionally uniform doubly doped lithium niobate crystals is a non-trivial technological problem. Doping additives, as a rule, have impurity distribution coefficients that differ (in our case, strongly differ) in magnitude: K\({}_{\rm D}\)(Mg) \(\approx\) 1.1, K\({}_{\rm D}\)(Gd) \(\approx\) 0.25 [33]. Consequently, the melt composition near the solidification front during crystal growth can be simultaneously enriched in one alloying component and depleted in another. Thus, the composition of a doped crystal during growth can change significantly from the cone to the end of the crystal, which usually leads to a decrease in its compositional and optical homogeneity. In this case, the characteristics of the crystal can significantly vary within its different parts. To minimize such effects, it is necessary to apply changes in parameters that are natural for the process of crystal growth. These are the speed of rotation and movement of the crystal, temperature gradients in the melt and growth zone, and different conjunctions of these parameters. In our case, such technological methods, including the use of a special design of the thermal unit, which creates small temperature gradients at the crystallization front, the use of low crystallization rates, particular melt preparations before crystal growth, long postgrowth annealing, and suitable conditions for the electrothermal treatment of the crystal, are effective. When growing a single crystal of double-doped LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%), we used the design of a thermal unit with double \"warming\", which allows for the creation of an isothermal zone in the volume of the platinum screen for the postgrowth annealing of the crystal, and the growth of the crystal under the conditions of a small temperature gradient at the crystallization front. The design of the thermal unit was similar to that used in [34]. Figure 1b shows the grown LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal. The crystal growth process was finished when the LiNbO\({}_{3}\):Gd:Mg crystal's mass of less than 275 g was reached. In this case, about \(\sim\)25% of the total mass of the melt crystallized. The parameters of the LiNbO\({}_{3}\):Gd:Mg crystal growth process (pulling speed, rod rotation speed, and temperature gradient at the crystallization front) were selected experimentally, based on the need to obtain a flat crystallization front, which should ensure a sufficiently high structural perfection of the crystal. The estimated mass during the growth of the LiNbO\({}_{3}\):Gd\({}^{3+}\)(0.003):Mg\({}^{2+}\)(0.65 wt.%) crystal was determined from the readings of the weight sensor of the growth setup; the accurate mass of the crystal was determined by weighing the grown crystal after separating the seed from the crystal boule.