Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D Riemannian manifold with Bishop equations

Abstract

We review the geometric evolution of a linearly polarized light wave coupling into an optical fiber and the rotation of the polarization plane in a three-dimensional (3D) Riemannian manifold. The optical fiber is assumed to be a one-dimensional object imbedded in the 3D Riemannian manifold along the paper. Thus, in the 3D Riemannian manifold, we demonstrate that the evolution of a linearly polarized light wave is associated with the Berry phase or more commonly known as the geometric phase. The ordinary condition for parallel transportation is defined by the Fermi–Walker parallelism law. We define other Fermi–Walker parallel transportation laws and connect them with the famous Rytov parallel transportation law for an electric field E, which is considered as the direction of the state of the linearly polarized light wave in the optical fiber in the 3D Riemannian manifold. Later, we define a special class of magnetic curves called by Bishop electromagnetic curves (BEM-curves), which are generated by the electric field E along the linearly polarized monochromatic light wave propagating in the optical fiber. In this way, not only we define a special class of linearly polarized point-particles corresponding to BEM-curves of the electromagnetic field along with the optical fiber in the 3D Riemannian manifold, but we also calculate, both numerically and analytically, the electromagnetic force, Poynting vector, energy-exchanges rate, optical angular and linear momentum, and optical magnetic-torque experienced by the linearly polarizable point-particles along with the optical fiber in the 3D Riemannian manifold.