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In order to elaborate the magneto-optical modulation process in detail and clarify the distribution of the magnetic field in the solenoid. In this paper, the magnetic field model of the solenoid with sine wave is established by Maxwell’s equation. The boundary condition is determined by the law of Ampere loop and the law of electromagnetic induction. The exact function expression of the magnetic field is obtained and the relevant factors are analyzed in detail. The simulation results show that the magnetic field in the solenoid with sine wave has good sine characteristic and the magnetic field is closely related to the factors such as the frequency of the driving signal and the axial magnetic field at any point inside the solenoid is much larger than the circumference magnetic field at that point. So, when the magnetic field at a point inside the solenoid is analyzed, it is possible to ignore the circumferential magnetic field at that point. But outside the solenoid, both directions of magnetic field need to be considered. In this paper, the method of studying the magnetic field of the solenoid provides a reference for the detailed analysis of the magnetic field inside and outside the solenoid under different signal driving.

The Verdet constant is one of the key parameters to characterize the material magneto-optical properties. The quantum theory is usually used to study magneto-optical properties and calculate the Verdet constant of paramagnetic material. However, the traditional quantum theory only takes into account the influence of the electron transition dipole moments caused by the particle property of light, which therefore cannot formulate the Verdet constant of magneto-optical material accurately. In view of the shortcomings of the existing theory, in this paper we propose is a wave-transition model of the Verdet constant. Due to the special wave-particle duality of light, the contribution of the non-transition dipole moment to the Verdet constant, caused by the electric field of light wave, should not be ignored. According to the basic theory of magneto-optical effect, in this paper we first explore the intrinsic mechanism of the paramagnetic material’s Verdet constant at a microscopic level and analyze the deficiency of traditional quantum theory. Furthermore, the classical electronic dynamic theory and quantum theory are used to reveal the contribution of volatility and transition of the light to the electric dipole moment. The density operator and statistical algorithm are introduced to derive the polarizability tensor of the paramagnetic magneto-optical material, thus obtaining the Verdet constant expression of the paramagnetic magneto-optical material, from which the Verdet constant is formulated. Taking the paramagnetic magneto-optical material TGG for example, the splitting energy levels and wave function of Tb3+ ions in the spin-orbit coupling, crystal field and effective field are calculated by the quantum method, and finally the Verdet constants under the traditional quantum theory and the volatility transition contribution model are obtained quantitatively. The comparative analysis shows that the results calculated by the wave-transition contribution model are more consistent with the experimental data and more accurate than the results calculated through the traditional quantum theory. The idea and method put forward in this paper will provide reference for further exploring the magneto-optical effect mechanism of paramagnetic magneto-optical materials.