Electromagnetic curves of the polarized light wave along the optical fiber in De-Sitter 2-space ${\mathbb{S}}_{1}^{2}$

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 De-Sitter 2-space. The optical fiber is assumed to be a one-dimensional object imbedded in the De-Sitter 2-space ${\mathbb{S}} _{1}^{2}$ along with the paper. Mathematically, De-Sitter 2-space ${\mathbb{S}}_{1}^{2}$ is described to be a 2-sphere in a Lorentzian space with positive curvature. Thus, in the De-Sitter 2-space, we have demonstrated 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 ${\mathcal{E}}$, which is considered as the direction of the state of the linearly polarized light wave traveling through the optical fiber in De-Sitter 2-space. Later, we have defined a special class of magnetic curves called by Lorentzian spherical electromagnetic curves (${\mathcal{LSEM}}$ curves), which are generated by the electric field ${\mathcal{E}}$ along with the linearly polarized monochromatic light wave propagating in the optical fiber. In this way, not only we are able to define a special class of linearly polarized point particles corresponding to ${\mathcal{LSEM}}$-curves of the electromagnetic field along with the optical fiber in the De-Sitter 2-space, but we can 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 De-Sitter 2-space.