A kinematic model of the Rytov's law in the optical fiber via split quaternions: application to electromagnetic theory

Abstract

In this paper, we analyze the homothetic motion of the polarization plane traveling along the linearly polarized light wave ((LPL)-wave) in the optical fiber on the condition that the angle between the polarization vector epsilon and the Frenet vector t (resp. n and b) is constant in 3D semi-Riemannian manifolds. Moreover, we present the relation of the homothetic motion and the Fermi-Walker parallel transportation law in 3D semi-Riemannian manifolds. The main technique for investigation of the homothetic motion is to use split quaternions. The parametric equations of the related Rytov curves are calculated through one-parameter homothetic motion and the split quaternion product. Thus, we give some theorems and corollaries which show the relationship between the Rytov curves and the split quaternions in 3D semi-Riemannian manifolds. Moreover, using the variational vector field, we obtain the magnetic curves (epsilon M-curves) connected with the polarization vector E obtained by the homothetic motion in the optical fiber. Also, we illustrate some examples related to the theoretical results.