Linear and nonlinear susceptibilities for magnetized plasmas: General formalism

This is the third in a series of papers [Yoon, Phys. Plasmas 31, 032309 (2024); AIP Adv. 14, 045240 (2024)] where the linear and nonlinear susceptibilities of magnetized plasma are discussed. The linear plasma susceptibility is central to the theory of plasma waves and instabilities, and is well known. The nonlinear susceptibility is important for nonlinear processes in plasmas, which includes turbulence and coherent nonlinear processes, but unlike the linear susceptibility or its counterpart in unmagnetized plasmas, the nonlinear susceptibility for magnetized plasmas found in the literature is unwieldy for the inherent multiple Bessel function/cyclotron harmonic series. The previous two papers managed to simplify the existing expressions substantially by invoking the Bessel function addition theorem and applied the concept first under the simplifying electrostatic approximation [Yoon, Phys. Plasmas 31, 032309 (2024) and later generalizing to fully electromagnetic formalism [Yoon, AIP Adv. 14, 045240 (2024)]. However, the formalism found in these works was carried out under an additional assumption of dynamical processes taking place in a two-dimensional plane. The present paper removes such an assumption and formulates these susceptibilities for a fully three-dimensional configuration, thus accomplishing the goal of a full generalization.