A new calculus for the treatment of Rytov's law in the optical fiber

In the present paper, we investigate the geometric properties of the linearly polarized light wave (LPLW) and the homothetic motion of the polarization plane traveling in optical fiber in three-dimensional Riemannian manifold. We examine the behavior of the polarized plane for the conditions that the electric field ε makes a constant angle with the Frenet vectors {e1,e2,e3} of the curve related to the optical fiber that can be considered as a space curve in Riemannian 3-space. Moreover, we give the relation between the Fermi–Walker parallel transportation laws and the homothetic motion of the polarization plane in Riemannian 3-space. The key technique here we use for examining this approach is to use quaternion algebra. We give the parametric equations of the Rytov curves that are traced curves of the polarization vector ε via quaternion product and a matrix that is similar to a Hamilton operator. By means of this matrix a new motion is defined and this motion is proven to be homothetic. For this one-parameter homothetic motion, we prove some theorems about the motion of the polarization plane traveling in optical fiber in three-dimensional Riemannian manifold. Then, we obtain the characterization of the electric field and generate the electromagnetic trajectories (εM-trajectories) along the (LPLW) in the optical fiber using the variational approach. Finally, we give various examples with Maple codes to confirms the theoretical results.