Compared with those materials with superior magneto-optical properties, such as YIG, Ce:YIG and Ba₃Tb(PO₄)₃, pure terbium gallium garnet (TGG) crystal has comparative low Verdet constant and cannot meet the requirements of some high-power devices. Doping Pr³⁺ ions in TGG crystal can remarkably enhance its magneto-optical properties and expand its application scope, but there are still lack of systematic theoretical calculations to clarify this phenomenon. Based on the quantum theory, this paper presents the influence of doping Pr³⁺ ions on the magneto-optical performance and the corresponding quantitative calculation results. Firstly, taking various effects on Tb³⁺ ions and Pr³⁺ ions in the crystal into consideration, the Hamiltonian is modeled and discussed in detail. The secular equations are solved by applying the perturbation method, and then the energy level shifts and wave functions of the Tb³⁺ ions and Pr³⁺ ions are worked out, where the spin-orbit coupling, crystal field, effective field and super-exchange interaction between the two types of ions are considered. Furthermore, the transition dipole moments of Tb³⁺ ions and Pr³⁺ ions from the 4f ground state to higher level 5d, together with the distribution probability at each energy level and the average magnetic moment, are resolved. Finally, the Verdet constants and magnetic susceptibilities of pure TGG crystal and Pr:TGG crystal are calculated and compared with each other. Moreover, the relationship between the Verdet constant of Pr:TGG crystal and the Pr³⁺-doping amount is derived. The results show that the Faraday rotation angle caused by Pr³⁺ ions is larger than that of Tb³⁺ ions, meanwhile, the strong super-exchange between Tb³⁺ ions and Pr³⁺ ions causes further splitting of the 4f energy level, resulting in a significant increasement of the Verdet constant of the Pr:TGG crystal, which reaches 313.4 rad/m·T, 191.2 rad/m·T and 60.4 rad/m·T at the wavelengths of 532 nm, 632.8 nm and 1064 nm, respectively. In addition, doping Pr³⁺ ions inside the crystal improves the internal effective magnetic moment, which can reach 9.92 μ_B at 10 K. At the same time, the magnetic susceptibility increases, while the temperature interdependency decreases. The linear relationship between the reciprocal of magnetic susceptibility and temperature reduces from 4.41/K to 3.92/K.

The Verdet constant of the Pr:TGG crystal is linear with the amount of Pr³⁺ ions doping. When the contents of Tb³⁺ ions and Pr³⁺ ions inside the crystal are equal, the maximum value is reached, which is about 2913.4 rad/m·T. The calculation results in this paper are in good agreement with the existing experimental data.