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.

© 2012 Optical Society of America

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References

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  1. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Proc. Cambridge Philos. Soc. 1, 115–116 (1852); also published in Trans. Cambridge Philos. Soc. 9, 399-416 (1852).
  2. E. Collett, Field Guide to Polarization (SPIE, 2005).
    [Crossref]
  3. E. Hecht, Optics (Addison-Wesley, 2002).
  4. R. C. Jones, “A new calculus for the treatment of optical systems. i. description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [Crossref]
  5. H. Poincaré, Théorie Mathématique de la Lumière (Carre, 1892), Chap. 12.
  6. W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478.
    [Crossref]
  7. H. Takenaka, “A unified formalism for polarization optics using group theory,” Nouv. Rev. Opt. 4, 37–42.
    [Crossref]
  8. W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, “Light polarization: a geometric-algebra approach,” Am J. Phys. 61, 534–545 (1993).
    [Crossref]
  9. C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University, 2003).
  10. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  11. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, 1964).
  12. M. Klein and T. Furtak, Optics, 2nd ed. (Wiley, 1986).
  13. C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. A 61, 1207–1213 (1971).
    [Crossref]
  14. T. Tudor and V. Manea, “Symmetry between partially polarized light and partial polarizers in the vectorial Pauli algebraic formalism,” J. Mod. Opt. 58, 845–852 (2011).
    [Crossref]
  15. T. Tudor and V. Manea, “Ellipsoid of the polarization degree: a vectorial pure operatorial Pauli algebraic approach,” J. Opt. Soc. Am. B 28, 596–601 (2011).
    [Crossref]
  16. R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).
  17. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley-Interscience, 1998).
  18. J. S. R. Chisholm and A. K. Common, “Foreword,” in Clifford Algebras and Their Applications in Mathematical Physics, J.S. R.Chisholm and A.K.Common, eds. (Reidel, 1986).
  19. D. Hestenes, “Vectors, spinors, and complex numbers in classical and quantum physics,” Am. J. Phys. 39, 1013–1027(1971).
    [Crossref]
  20. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Reidel, 1984).
    [Crossref]
  21. D. Hestenes, “Oersted Medal Lecture 2002: reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121(2003).
    [Crossref]
  22. H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy (Prentice-Hall, 1989).
  23. B. Jancewicz, Multivectors and Clifford Algebras in Electrodynamics (World Scientific, 1989).
  24. W. E. Baylis, Electrodynamics: A Modern Geometric Approach (Birkhauser, 1999).
  25. B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
    [Crossref]
  26. H. G. Berry, G. Gabrielse, and A. E. Livingston, “Measurement of the Stokes parameters of light,” Appl. Opt. 16, 3200–3205 (1977).
    [Crossref] [PubMed]
  27. J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967).
  28. W. E. Baylis, “Applications of Clifford algebras in physics,” http://www.uwindsor.ca/users/b/baylis/main.nsf.
  29. M. C. Sze, Q. M. Sugon, Jr., and D. J. McNamara, “Oblique superposition of two elliptically polarized lightwaves using geometric algebra: is energy-momentum conserved?” J. Opt. Soc. Am. A 27, 2468–2479 (2010).
    [Crossref]
  30. T. Vold, “Introduction to geometric algebra and its application to electrodynamics,” Am. J. Phys. 61, 505–513 (1993).
    [Crossref]
  31. Q. M. Sugon, Jr., and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkhauser, 2002), pp. 297–306.
    [Crossref]
  32. E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
    [Crossref]
  33. R. W. Ditchburn, Light, 2nd ed. (Wiley, 1963).
  34. J. W. Simmons and M. J. Guttmann, States Waves and Photons: A Modern Introduction to Light (Addison-Wesley, 1970).
  35. J. W. Rohlf, Modern Physics from α to Z0 (Wiley, 1994).

2011 (2)

T. Tudor and V. Manea, “Symmetry between partially polarized light and partial polarizers in the vectorial Pauli algebraic formalism,” J. Mod. Opt. 58, 845–852 (2011).
[Crossref]

T. Tudor and V. Manea, “Ellipsoid of the polarization degree: a vectorial pure operatorial Pauli algebraic approach,” J. Opt. Soc. Am. B 28, 596–601 (2011).
[Crossref]

2010 (1)

2007 (1)

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[Crossref]

2003 (1)

D. Hestenes, “Oersted Medal Lecture 2002: reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121(2003).
[Crossref]

1993 (2)

T. Vold, “Introduction to geometric algebra and its application to electrodynamics,” Am. J. Phys. 61, 505–513 (1993).
[Crossref]

W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, “Light polarization: a geometric-algebra approach,” Am J. Phys. 61, 534–545 (1993).
[Crossref]

1977 (1)

1971 (2)

D. Hestenes, “Vectors, spinors, and complex numbers in classical and quantum physics,” Am. J. Phys. 39, 1013–1027(1971).
[Crossref]

C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. A 61, 1207–1213 (1971).
[Crossref]

1954 (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

1941 (1)

1852 (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Proc. Cambridge Philos. Soc. 1, 115–116 (1852); also published in Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Azzam, R. M.

R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

Bailey, W. M.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478.
[Crossref]

Barrett, D.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[Crossref]

Bashara, N. M.

R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

Baylis, W. E.

W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, “Light polarization: a geometric-algebra approach,” Am J. Phys. 61, 534–545 (1993).
[Crossref]

W. E. Baylis, Electrodynamics: A Modern Geometric Approach (Birkhauser, 1999).

W. E. Baylis, “Applications of Clifford algebras in physics,” http://www.uwindsor.ca/users/b/baylis/main.nsf.

Berry, H. G.

Bickel, W. S.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478.
[Crossref]

Bonenfant, J.

W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, “Light polarization: a geometric-algebra approach,” Am J. Phys. 61, 534–545 (1993).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, 1964).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley-Interscience, 1998).

Chisholm, J. S. R.

J. S. R. Chisholm and A. K. Common, “Foreword,” in Clifford Algebras and Their Applications in Mathematical Physics, J.S. R.Chisholm and A.K.Common, eds. (Reidel, 1986).

Collett, E.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[Crossref]

E. Collett, Field Guide to Polarization (SPIE, 2005).
[Crossref]

Common, A. K.

J. S. R. Chisholm and A. K. Common, “Foreword,” in Clifford Algebras and Their Applications in Mathematical Physics, J.S. R.Chisholm and A.K.Common, eds. (Reidel, 1986).

Derbyshire, J.

W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, “Light polarization: a geometric-algebra approach,” Am J. Phys. 61, 534–545 (1993).
[Crossref]

Ditchburn, R. W.

R. W. Ditchburn, Light, 2nd ed. (Wiley, 1963).

Doran, C.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University, 2003).

Fraher, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[Crossref]

Furtak, T.

M. Klein and T. Furtak, Optics, 2nd ed. (Wiley, 1986).

Gabrielse, G.

Guttmann, M. J.

J. W. Simmons and M. J. Guttmann, States Waves and Photons: A Modern Introduction to Light (Addison-Wesley, 1970).

Haus, H. A.

H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy (Prentice-Hall, 1989).

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Hestenes, D.

D. Hestenes, “Oersted Medal Lecture 2002: reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121(2003).
[Crossref]

D. Hestenes, “Vectors, spinors, and complex numbers in classical and quantum physics,” Am. J. Phys. 39, 1013–1027(1971).
[Crossref]

D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Reidel, 1984).
[Crossref]

Huschilt, J.

W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, “Light polarization: a geometric-algebra approach,” Am J. Phys. 61, 534–545 (1993).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Jancewicz, B.

B. Jancewicz, Multivectors and Clifford Algebras in Electrodynamics (World Scientific, 1989).

Jones, R. C.

Klein, M.

M. Klein and T. Furtak, Optics, 2nd ed. (Wiley, 1986).

Lasenby, A.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University, 2003).

Livingston, A. E.

Manea, V.

T. Tudor and V. Manea, “Ellipsoid of the polarization degree: a vectorial pure operatorial Pauli algebraic approach,” J. Opt. Soc. Am. B 28, 596–601 (2011).
[Crossref]

T. Tudor and V. Manea, “Symmetry between partially polarized light and partial polarizers in the vectorial Pauli algebraic formalism,” J. Mod. Opt. 58, 845–852 (2011).
[Crossref]

McNamara, D. J.

M. C. Sze, Q. M. Sugon, Jr., and D. J. McNamara, “Oblique superposition of two elliptically polarized lightwaves using geometric algebra: is energy-momentum conserved?” J. Opt. Soc. Am. A 27, 2468–2479 (2010).
[Crossref]

Q. M. Sugon, Jr., and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkhauser, 2002), pp. 297–306.
[Crossref]

Melcher, J. R.

H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy (Prentice-Hall, 1989).

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (Carre, 1892), Chap. 12.

Rohlf, J. W.

J. W. Rohlf, Modern Physics from α to Z0 (Wiley, 1994).

Sakurai, J. J.

J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967).

Schaefer, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[Crossref]

Simmons, J. W.

J. W. Simmons and M. J. Guttmann, States Waves and Photons: A Modern Introduction to Light (Addison-Wesley, 1970).

Smyth, R.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[Crossref]

Sobczyk, G.

D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Reidel, 1984).
[Crossref]

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Proc. Cambridge Philos. Soc. 1, 115–116 (1852); also published in Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Sugon, Q. M.

M. C. Sze, Q. M. Sugon, Jr., and D. J. McNamara, “Oblique superposition of two elliptically polarized lightwaves using geometric algebra: is energy-momentum conserved?” J. Opt. Soc. Am. A 27, 2468–2479 (2010).
[Crossref]

Q. M. Sugon, Jr., and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkhauser, 2002), pp. 297–306.
[Crossref]

Sze, M. C.

Takenaka, H.

H. Takenaka, “A unified formalism for polarization optics using group theory,” Nouv. Rev. Opt. 4, 37–42.
[Crossref]

Tudor, T.

T. Tudor and V. Manea, “Symmetry between partially polarized light and partial polarizers in the vectorial Pauli algebraic formalism,” J. Mod. Opt. 58, 845–852 (2011).
[Crossref]

T. Tudor and V. Manea, “Ellipsoid of the polarization degree: a vectorial pure operatorial Pauli algebraic approach,” J. Opt. Soc. Am. B 28, 596–601 (2011).
[Crossref]

Vold, T.

T. Vold, “Introduction to geometric algebra and its application to electrodynamics,” Am. J. Phys. 61, 505–513 (1993).
[Crossref]

Whitney, C.

C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. A 61, 1207–1213 (1971).
[Crossref]

Wolf, E.

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, 1964).

Am J. Phys. (1)

W. E. Baylis, J. Bonenfant, J. Derbyshire, and J. Huschilt, “Light polarization: a geometric-algebra approach,” Am J. Phys. 61, 534–545 (1993).
[Crossref]

Am. J. Phys. (5)

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478.
[Crossref]

D. Hestenes, “Vectors, spinors, and complex numbers in classical and quantum physics,” Am. J. Phys. 39, 1013–1027(1971).
[Crossref]

D. Hestenes, “Oersted Medal Lecture 2002: reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121(2003).
[Crossref]

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[Crossref]

T. Vold, “Introduction to geometric algebra and its application to electrodynamics,” Am. J. Phys. 61, 505–513 (1993).
[Crossref]

Appl. Opt. (1)

J. Mod. Opt. (1)

T. Tudor and V. Manea, “Symmetry between partially polarized light and partial polarizers in the vectorial Pauli algebraic formalism,” J. Mod. Opt. 58, 845–852 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Nouv. Rev. Opt. (1)

H. Takenaka, “A unified formalism for polarization optics using group theory,” Nouv. Rev. Opt. 4, 37–42.
[Crossref]

Nuovo Cimento (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Proc. Cambridge Philos. Soc. 1, 115–116 (1852); also published in Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Other (20)

E. Collett, Field Guide to Polarization (SPIE, 2005).
[Crossref]

E. Hecht, Optics (Addison-Wesley, 2002).

H. Poincaré, Théorie Mathématique de la Lumière (Carre, 1892), Chap. 12.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University, 2003).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, 1964).

M. Klein and T. Furtak, Optics, 2nd ed. (Wiley, 1986).

D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Reidel, 1984).
[Crossref]

R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley-Interscience, 1998).

J. S. R. Chisholm and A. K. Common, “Foreword,” in Clifford Algebras and Their Applications in Mathematical Physics, J.S. R.Chisholm and A.K.Common, eds. (Reidel, 1986).

R. W. Ditchburn, Light, 2nd ed. (Wiley, 1963).

J. W. Simmons and M. J. Guttmann, States Waves and Photons: A Modern Introduction to Light (Addison-Wesley, 1970).

J. W. Rohlf, Modern Physics from α to Z0 (Wiley, 1994).

Q. M. Sugon, Jr., and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkhauser, 2002), pp. 297–306.
[Crossref]

J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967).

W. E. Baylis, “Applications of Clifford algebras in physics,” http://www.uwindsor.ca/users/b/baylis/main.nsf.

H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy (Prentice-Hall, 1989).

B. Jancewicz, Multivectors and Clifford Algebras in Electrodynamics (World Scientific, 1989).

W. E. Baylis, Electrodynamics: A Modern Geometric Approach (Birkhauser, 1999).

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Figures (5)

Fig. 1
Fig. 1

Rotation of the unit vectors e 1 and e 2 counterclockwise by an angle θ about the unit vector e 3 . The new vectors are e 1 = e 1 e i e 3 θ and e 2 = e 2 e i e 3 θ . The unit vector e 3 is pointing out of the page.

Fig. 2
Fig. 2

Rotation of the electric field E and magnetic field H with zero initial phase angle. The initial field vectors E 0 and H 0 point along the e 1 and e 2 directions and are rotated by an angle ϕ = ω t k z . The wave vector k is pointing out of the page.

Fig. 3
Fig. 3

Circular basis representation of the elliptically polarized light E as the sum of two vectors rotating in opposite directions with radii a + and a and initial phase angles δ + and δ . The semimajor axis is E 1 = a + + a , the semiminor axis is E 2 = a + a , and the tilt angle is β / 2 = ( δ + + δ ) / 2 .

Fig. 4
Fig. 4

Cartesian basis representation of elliptically polarized light E δ x δ y as a sum of two orthogonal oscillations with amplitudes E x and E y , and initial phase angles δ x and δ y .

Fig. 5
Fig. 5

Rotated ellipse representation of the elliptically polarized light E δ β as a rotated ellipse with semimajor axis E 1 , semiminor axis E 2 and tilt angle β / 2 .

Equations (144)

Equations on this page are rendered with MathJax. Learn more.

a b = a · b + i a × b ,
E ^ = E + i ζ H ,
E + i ζ H = ( e 1 + i e 2 ) [ a + e i ( ω t k z + δ + ) + a e i ( ω t k z δ ) ] .
E = Re ( ( c 1 σ 1 + c 2 σ 2 ) e i k · x ) = Re ( E ) .
E = e 1 [ a + e i e 3 ( ω t k z + δ + ) + a e i e 3 ( ω t k z δ ) ] ,
ζ H = e 2 [ a + e i e 3 ( ω t k z + δ + ) + a e i e 3 ( ω t k z δ ) ] .
s 0 = E 2 + ζ 2 H 2 ,
s 1 = E 2 ζ 2 H 2 ,
i s 2 = 2 i ζ E · H = 2 i ζ | E | | H | cos α ,
s 3 = 2 ζ E × H = 2 e 3 ζ | E | | H | sin α ,
s 0 2 = s 1 2 + s 2 2 + s 3 2 .
e j e k + e k e j = 2 δ j k .
e j 2 = e k 2 = 1 ,
e j e k = e k e j , j k .
i = e 1 e 2 e 3 ,
i 2 = 1 ,
i e 1 = e 1 i = e 2 e 3 ,
i e 2 = e 2 i = e 3 e 1 ,
i e 3 = e 3 i = e 1 e 2 .
a = a 1 e 1 + a 2 e 2 + a 3 e 3 ,
b = b 1 e 1 + b 2 e 2 + b 3 e 3 .
a b = a · b + a b ,
a · b = a 1 b 1 + a 2 b 2 + a 3 b 3 ,
a b = e 1 e 2 ( a 1 b 2 a 2 b 1 ) + e 2 e 3 ( a 2 b 3 a 3 b 2 ) + e 3 e 1 ( a 3 b 1 a 1 b 3 ) .
a b = i a × b .
A ^ = a 0 + a 1 + i a 2 + i a 3 ,
A ^ = a 0 a 1 + i a 2 i a 3 ,
e ± i θ = cos θ ± i sin θ ,
cos θ = 1 2 ( e i θ + e i θ ) ,
sin θ = i 2 ( e i θ e i θ ) .
e + = e 1 + i e 2 ,
e 1 + i e 2 ± = ( e 1 + i e 2 ) e ± i θ ,
e 1 = e 1 cos θ e 2 sin θ ,
e 2 ± = ± e 1 sin θ + e 2 cos θ ,
e ± i e 3 θ = cos θ ± i e 3 sin θ = cos θ ± e 1 e 2 sin θ ,
cos θ = 1 2 ( e i e 3 θ + e i e 3 θ ) ,
sin θ = 1 2 i e 3 ( e i e 3 θ e i e 3 θ ) .
e 1 i e 3 = e 1 e 1 e 2 = e 2 ,
e 2 i e 3 = e 2 e 1 e 2 = e 1 ,
e 1 e ± i e 3 θ = e 1 cos θ ± e 2 sin θ ,
e 2 e ± i e 3 θ = e 1 sin θ + e 2 cos θ .
e 1 = e 1 e i e 3 θ ,
e 2 ± = e 2 e i e 3 θ .
( e 1 + i e 2 ) e ± i θ = ( e 1 + i e 2 ) e i e 3 θ .
e 1 e ± i e 3 θ = e i e 3 θ e 1 ,
e 2 e ± i e 3 θ = e i e 3 θ e 2 ,
e 3 e ± i e 3 θ = e ± i e 3 θ e 3 .
E ^ r ^ = 0 ,
E ^ = E + i ζ H ,
r ^ = 1 c t +
· E = 0 ,
1 c E t ζ × H = 0 ,
× E + ζ c H t = 0 ,
· H = 0 ,
E ^ ± = E ± + i ζ H ± = e + a ˜ ± ψ ˜ ± 1 ,
e + = e 1 + i e 2 ,
a ˜ ± = a ± e i δ ± ,
ψ ˜ ± 1 = e i ( ω t k z ) .
E ^ ± = E ± + i ζ H ± = ( e 1 + i e 2 ) a ± e i ( ω t k z δ ± ) .
E ± = e 1 a ± cos ( ω t k z δ ± ) ± e 2 a ± sin ( ω t k z δ ± ) ,
ζ H ± = e 2 a ± cos ( ω t k z δ ± ) e 1 a ± sin ( ω t k z δ ± ) .
E ^ ± = E ± + i ζ H ± = ( e 1 + i e 2 ) a ± e ± i e 3 ( ω t k z ± δ ± ) = e 1 a ± e ± i e 3 ( ω t k z ± δ ± ) ± i e 2 a ± e ± i e 3 ( ω t k z ± δ ± ) .
E ± = e 1 a ± e ± i e 3 ( ω t k z ± δ ± ) ,
ζ H ± = e 2 a ± e ± i e 3 ( ω t k z ± δ ± ) ,
E ˜ ± = e 1 a ± e i ( ω t k z δ ± ) .
Re ( E ˜ ± ) = e 1 a ± cos ( ω t k z δ ± ) ,
Im ( E ˜ ± ) = i e 1 a ± sin ( ω t k z δ ± ) .
E ^ ± r ^ = i ( ω c k e 3 ) ( e 1 + i e 2 ) a ˜ ± ψ ˜ ± 1 = 0 .
( 1 e 3 ) ( e 1 + i e 2 ) = ( 1 e 3 ) ( 1 + e 3 ) e 1 = 0 ,
E ^ ± r ^ = i ( ω c k ) ( E ± + i ζ H ± ) = 0 ,
k · E ± = 0 ,
ω c E ± + ζ k × H ± = 0 ,
i ω c ζ H ± i k × E ± = 0 ,
i ζ k · H ± = 0 .
E ± i ζ k | k | H ± = 0 ,
ζ H ± + i k | k | E ± = 0 ,
E ^ ± = E ± + i ζ H ± = ( 1 + k | k | ) E ± .
i ζ H ± = k | k | E ± = k | k | E ± .
E ^ = e ^ + ( a ˜ + ψ ˜ + a ˜ ψ ˜ 1 ) ,
E ^ = e ^ + ( a ^ + ψ ^ + a ^ ψ ^ 1 ) ,
a ^ ± = a ± e i e 3 δ ± ,
ψ ^ ± 1 = e ± i e 3 ( ω t k z ) .
E = e 1 ( a ^ + ψ ^ + a ^ ψ ^ 1 ) ,
ζ H = e 2 ( a ^ + ψ ^ + a ^ ψ ^ 1 ) .
E = e 1 [ a + e i e 3 ( ϕ + δ + ) + a e i e 3 ( ϕ δ ) ] ,
ϕ = ω t k z .
E = e 1 ( a + cos ( ϕ + δ + ) + a cos ( ϕ δ ) ) + e 2 ( a + sin ( ϕ + δ + ) a sin ( ϕ δ ) ) .
E = E x cos ( ϕ + δ x ) e 1 + E y cos ( ϕ + δ y ) e 2 ,
E x = [ a + 2 + a 2 + 4 a + a cos ( δ + + δ ) ] 1 2 ,
E y = [ a + 2 + a 2 4 a + a cos ( δ + + δ ) ] 1 2 ,
δ x = arctan ( a + sin δ + a sin δ a + cos δ + + a cos δ ) ,
δ y = arctan ( a cos δ a + cos δ + a + sin δ + + a sin δ ) .
a ± = 1 2 [ E x 2 2 E x E y sin ( ± ( δ x δ y ) ) + E y 2 ] 1 2 ,
δ + = arctan ( E x cos δ x E y cos δ y E x sin δ x + E y sin δ y , ) ,
δ = arctan ( E x sin δ x E y cos δ y E x cos δ x + E y sin δ y ) .
a ± = 1 2 ( E 1 ± E 2 ) ,
δ ± = ± δ + β 2 ,
E = e 1 [ 1 2 ( E 1 + E 2 ) e i e 3 ( ϕ + δ ) + 1 2 ( E 1 E 2 ) e i e 3 ( ϕ + δ ) ] e i e 3 β / 2 .
E = [ E 1 cos ( ϕ + δ ) e 1 + E 2 sin ( ϕ + δ ) e 2 ] e i e 3 β / 2 .
E 1 = a + + a ,
E 2 = a + a ,
δ = 1 2 ( δ + δ ) ,
β 2 = 1 2 ( δ + + δ ) .
E 1 = 1 2 [ ( E x 2 + 2 E x E y sin ( δ x δ y ) + E y 2 ) 1 2 + ( E x 2 2 E x E y sin ( δ x δ y ) + E y 2 ) 1 2 ] ,
E 2 = 1 2 [ ( E x 2 + 2 E x E y sin ( δ x δ y ) + E y 2 ) 1 2 ( E x 2 2 E x E y sin ( δ x δ y ) + E y 2 ) 1 2 ] ,
δ = 1 2 [ arctan ( E x cos δ x E y cos δ y E x sin δ x + E y sin δ y ) + arctan ( E x sin δ x E y cos δ y E x cos δ x + E y sin δ y ) ] ,
β 2 = 1 2 [ arctan ( E x cos δ x E y cos δ y E x sin δ x + E y sin δ y ) arctan ( E x sin δ x E y cos δ y E x cos δ x + E y sin δ y ) ] .
E x = [ E 1 2 + E 2 2 + ( E 1 2 E 2 2 ) cos β 2 ] 1 2 ,
E y = [ E 1 2 + E 2 2 ( E 1 2 E 2 2 ) cos β 2 ] 1 2 ,
δ x = arctan ( E 1 sin δ cos β / 2 + E 2 cos δ sin β / 2 E 1 cos δ cos β / 2 E 2 sin δ sin β / 2 ) ,
δ y = arctan ( E 1 sin δ sin β / 2 E 2 cos δ cos β / 2 E 1 cos δ sin β / 2 + E 2 sin δ cos β / 2 ) .
S 0 = E x 2 + E y 2 ,
S 1 = E x 2 E y 2 ,
S 2 = 2 E x E y cos ( δ x δ y ) ,
S 3 = 2 E x E y sin ( δ x δ y ) .
S 0 = E 1 2 + E 2 2 ,
S 1 = ( E 1 2 E 2 2 ) cos β ,
S 2 = ( E 1 2 E 2 2 ) sin β ,
S 3 = 2 E 1 E 2 ,
sin ( arctan θ ) = θ 1 + θ 2 ,
cos ( arctan θ ) = 1 1 + θ 2 .
S 0 = 2 ( a 2 + a + 2 ) ,
S 1 = 4 a + a cos ( δ + + δ ) ,
S 2 = 4 a + a sin ( δ + + δ ) ,
S 3 = 2 ( a 2 a + 2 ) .
E 2 = ζ 2 H 2 = a + 2 + a 2 + 2 a + a cos ( 2 ϕ + δ + + δ ) ,
ζ EH = ζ HE = i e 3 [ a + 2 + a 2 + 2 a + a cos ( 2 ϕ + δ + + δ ) ] ,
s 0 E 2 + ζ 2 H 2 = 2 [ a + 2 + a 2 + 2 a + a cos ( 2 ϕ + δ + + δ ) ] ,
s 1 E 2 ζ 2 H 2 = 0 ,
i s 2 2 i ζ E · H = 0 ,
s 3 2 ζ E × H = 2 e 3 [ a + 2 + a 2 + 2 a + a cos ( 2 ϕ + δ + + δ ) ] ,
s 0 | ϕ = 0 = E 2 + ζ 2 H 2 = S 0 + S 1 ,
s 3 | ϕ = 0 = 2 ζ E × H = e 3 ( S 0 + S 1 ) ,
s 0 = E 2 + ζ 2 H 2 = 2 ( a + 2 + a 2 ) ,
s 1 = E 2 ζ 2 H 2 = 0 ,
i s 2 = 2 i ζ E · H = 0 ,
s 3 = 2 ζ E × H = 2 e 3 ( a + 2 + a 2 ) .
E ^ = E + i ζ H ,
E ^ E ^ = s 1 + i s 2 ,
E ^ E ^ = s 1 i s 2 ,
E ^ E ^ = s 0 s 3 ,
E ^ E ^ = s 0 s 3 .
( E ^ E ^ ) ( E ^ E ^ ) = ( E ^ E ^ ) ( E ^ E ^ ) ,
s 0 2 s 3 2 = s 1 2 + s 2 2 .

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