Polarization ellipse and Stokes parameters in geometric algebra

Adler G. Santos, Quirino M. Sugon, Daniel J. McNamara

Research output: Contribution to journalArticle

3 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)89-98
Number of pages10
JournalJournal of the Optical Society of America A: Optics and Image Science, and Vision
Volume29
Issue number1
DOIs
Publication statusPublished - Jan 1 2012

Fingerprint

ellipses
Algebra
algebra
Polarization
polarization
Wave functions
Electric fields
wave functions
Magnetic fields
scalars
electric fields
Maxwell equations
Light polarization
magnetic fields
Maxwell equation
polarized light
plane waves
products

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Atomic and Molecular Physics, and Optics
  • Computer Vision and Pattern Recognition

Cite this

Polarization ellipse and Stokes parameters in geometric algebra. / Santos, Adler G.; Sugon, Quirino M.; McNamara, Daniel J.

In: Journal of the Optical Society of America A: Optics and Image Science, and Vision, Vol. 29, No. 1, 01.01.2012, p. 89-98.

Research output: Contribution to journalArticle

Santos, Adler G. ; Sugon, Quirino M. ; McNamara, Daniel J. / Polarization ellipse and Stokes parameters in geometric algebra. In: Journal of the Optical Society of America A: Optics and Image Science, and Vision. 2012 ; Vol. 29, No. 1. pp. 89-98.
@article{455ac28decfa4245b6ca1174377689e9,
title = "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 Poincar{\'e} sphere equation.",
author = "Santos, {Adler G.} and Sugon, {Quirino M.} and McNamara, {Daniel J.}",
year = "2012",
month = "1",
day = "1",
doi = "10.1364/JOSAA.29.000089",
language = "English",
volume = "29",
pages = "89--98",
journal = "Journal of the Optical Society of America A: Optics and Image Science, and Vision",
issn = "1084-7529",
publisher = "The Optical Society",
number = "1",

}

TY - JOUR

T1 - Polarization ellipse and Stokes parameters in geometric algebra

AU - Santos, Adler G.

AU - Sugon, Quirino M.

AU - McNamara, Daniel J.

PY - 2012/1/1

Y1 - 2012/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84855383407&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855383407&partnerID=8YFLogxK

U2 - 10.1364/JOSAA.29.000089

DO - 10.1364/JOSAA.29.000089

M3 - Article

C2 - 22218355

AN - SCOPUS:84855383407

VL - 29

SP - 89

EP - 98

JO - Journal of the Optical Society of America A: Optics and Image Science, and Vision

JF - Journal of the Optical Society of America A: Optics and Image Science, and Vision

SN - 1084-7529

IS - 1

ER -