Polarization ellipse and Stokes parameters in geometric algebra

Abstract

In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell's equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.

Generating Optical Birefringence and Chirality in Silicon Nanowire Dimers

“…The shape of the ellipse is defined by the magnitudes of the E x and E y components and the relative phase between them. For linearly polarized light a straight line is obtained, while completely circularly polarized light results in a circle . The simulated polarization ellipses and the orientation angles together confirm that with increasing length of the nanowire dimer the polarization state of the reflected light transforms from elliptical to circular.…”

“…For linearly polarized light a straight line is obtained, while completely circularly polarized light results in a circle. 40 The simulated polarization ellipses and the orientation angles together confirm that with increasing length of the nanowire dimer the polarization state of the reflected light transforms from elliptical to circular. The numerical simulations predict that the Si nanowire dimer D 2μm,70nm,60nm achieves a phase shift of γ = π/2.…”

Geometric Algebra: A natural representation of three-space

“…The development of GA is now expanding rapidly, with benefits being found in research into quantum field theory 16 , quantum tunneling 17 , quantum computing 18 , spacetime 11,19,20 , general relativity and cosmology [21][22][23] , computer vision 24 , protein folding 25 , optics and metamaterials [26][27][28] , conformal algebra 29 , electrodynamics 30 , electrical circuit analysis 31 and EPR-Bell experiments 32 .…”

Phase and polarization changes of pulsed Gaussian beams during focusing and propagation

“…There are several approaches to determine the properties of the polarization ellipse using the attributes of the circular basis vectors (2.18) and (2.19). For example, introducing phasors can yield the relationships in question [127], or Clifford (geometric) algebra can be used to find them [128]. Of course, simple algebraic manipulations can also lead to the expressions giving the properties of the polarization ellipse (see them in Appendix A.2.2).…”