Abstract

Recently a new class of mathematical entities—the gyrovectors—came into light in physics. The most intuitive examples of gyrovectors are the relativistic allowed velocities and the Poincaré polarization vectors. Taking advantage of the (also recently elaborated) approach of degree of polarization surfaces in polarization theory, I make an analysis of the strange behavior of these entities in the interaction between the polarized light and the orthogonal dichroic polarization devices. The same approach can be applied in the theory of relativity as well as in many other branches of optics (multilayer optics, ray optics, laser cavity physics, quantum optics) where the Lorentzian character of the transformations which govern some specific problems has been recognized.

© 2017 Optical Society of America

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References

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    [Crossref]
  2. O. V. Angelsky, S. G. Hanson, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology of the degree of coherence of optical waves,” Opt. Express 17, 15623–15634 (2009).
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    [Crossref]
  6. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
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    [Crossref]
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  9. T. Opatrný and J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
    [Crossref]
  10. D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
    [Crossref]
  11. Y. S. Kim, “Lorentz group in polarization optics,” J. Opt. B 2, R1–R5 (2000).
    [Crossref]
  12. J. A. Morales and E. Navarro, “Minkovskian decription of polarized light and polarizers,” Phys. Rev. E 67, 026605 (2003).
    [Crossref]
  13. J. Lages, R. Giust, and J. M. Vigoureux, “Composition law for polarizers,” Phys. Rev. A 78, 033810 (2008).
    [Crossref]
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    [Crossref]
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    [Crossref]
  16. C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monografías del Seminario Matemático García de Galdeano 33, 115–119 (2006).
  17. R. Ossikovski, J. J. Gil, and I. San José, “Poincaré sphere mapping by Mueller matrices,” J. Opt. Soc. Am. A 30, 2291–2305 (2013).
    [Crossref]
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  20. T. Tudor, “Interaction of light with the polarization devices: a vectorial Pauli algebraic approach,” J. Phys. A 41, 415303 (2008).
    [Crossref]
  21. T. Tudor, “Gyrovectors and degree of polarization surfaces in polarization theory,” J. Opt. Soc. Am. B 32, 2528–2535 (2015).
    [Crossref]
  22. T. Tudor, “On a quasi-relativistic formula in polarization theory,” Opt. Lett. 40, 693–696 (2015).
    [Crossref]
  23. A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,” Found. Phys. Lett. 1, 57–89 (1988).
    [Crossref]
  24. R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1977), Vol. I, Chap. 15.
  25. T. Tudor and V. Manea, “Symmetry between partially polarized light and partial polarizers,” J. Mod. Opt. 58, 845–852 (2011).
    [Crossref]
  26. A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity (World Scientific, 2008).
  27. R. D. Sard, Relativistic Mechanics (Benjamin, 1970).
  28. M. C. Møller, The Theory of Relativity (Clarendon, 1952).
  29. J. J. Gil and R. Ossikovski, Polarizerd Light and the Mueller Matrix Approach (CRC Press, 2016).
  30. J. J. Gil, R. Ossikovski, and I. San José, “Singular Mueller matrices,” J. Opt. Soc. Am. A 33, 600–609 (2016).
    [Crossref]
  31. 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]
  32. J. M. Vigoureux and Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
    [Crossref]
  33. J. J. Monzón and L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
    [Crossref]
  34. S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B 6, S455–S472 (2004).
    [Crossref]
  35. S. Başkal and Y. S. Kim, “Wigner rotations in laser cavities,” Phys. Rev. E 66, 026604 (2002).
    [Crossref]
  36. D. Han, E. E. Hardekopl, and Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
    [Crossref]
  37. J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
    [Crossref]

2016 (1)

2015 (2)

2013 (1)

2011 (2)

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,” J. Mod. Opt. 58, 845–852 (2011).
[Crossref]

2009 (1)

2008 (3)

J. Lages, R. Giust, and J. M. Vigoureux, “Composition law for polarizers,” Phys. Rev. A 78, 033810 (2008).
[Crossref]

T. Tudor, “Interaction of light with polarization devices: a vectorial Pauli algebraic approach,” Proc. SPIE 7008, 700804 (2008).

T. Tudor, “Interaction of light with the polarization devices: a vectorial Pauli algebraic approach,” J. Phys. A 41, 415303 (2008).
[Crossref]

2007 (1)

J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. 40, 1–47 (2007).
[Crossref]

2006 (1)

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monografías del Seminario Matemático García de Galdeano 33, 115–119 (2006).

2005 (1)

2004 (2)

B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941–4958 (2004).
[Crossref]

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B 6, S455–S472 (2004).
[Crossref]

2003 (1)

J. A. Morales and E. Navarro, “Minkovskian decription of polarized light and polarizers,” Phys. Rev. E 67, 026605 (2003).
[Crossref]

2002 (1)

S. Başkal and Y. S. Kim, “Wigner rotations in laser cavities,” Phys. Rev. E 66, 026604 (2002).
[Crossref]

2000 (1)

Y. S. Kim, “Lorentz group in polarization optics,” J. Opt. B 2, R1–R5 (2000).
[Crossref]

1999 (1)

J. J. Monzón and L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[Crossref]

1997 (1)

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[Crossref]

1993 (2)

T. Opatrný and J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[Crossref]

J. M. Vigoureux and Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[Crossref]

1992 (2)

P. Pellat-Finet, “What is common to both polarization optics and relativistic kinematics?” Optik 90, 101–106 (1992).

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
[Crossref]

1989 (2)

M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[Crossref]

D. Han, E. E. Hardekopl, and Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[Crossref]

1988 (1)

A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,” Found. Phys. Lett. 1, 57–89 (1988).
[Crossref]

1986 (2)

1973 (1)

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37–41 (1973).
[Crossref]

1963 (1)

Angelsky, O. V.

Barakat, R.

Baskal, S.

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B 6, S455–S472 (2004).
[Crossref]

S. Başkal and Y. S. Kim, “Wigner rotations in laser cavities,” Phys. Rev. E 66, 026604 (2002).
[Crossref]

Chipman, R.

Cloude, S. R.

S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).

Correas, J. M.

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monografías del Seminario Matemático García de Galdeano 33, 115–119 (2006).

Damask, J. N.

J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005), p. 306.

DeBoo, B.

Ferreira, C.

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monografías del Seminario Matemático García de Galdeano 33, 115–119 (2006).

Feynman, R. P.

R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1977), Vol. I, Chap. 15.

Georgieva, E.

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B 6, S455–S472 (2004).
[Crossref]

Gil, J.

J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. 40, 1–47 (2007).
[Crossref]

Gil, J. J.

J. J. Gil, R. Ossikovski, and I. San José, “Singular Mueller matrices,” J. Opt. Soc. Am. A 33, 600–609 (2016).
[Crossref]

R. Ossikovski, J. J. Gil, and I. San José, “Poincaré sphere mapping by Mueller matrices,” J. Opt. Soc. Am. A 30, 2291–2305 (2013).
[Crossref]

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monografías del Seminario Matemático García de Galdeano 33, 115–119 (2006).

J. J. Gil and R. Ossikovski, Polarizerd Light and the Mueller Matrix Approach (CRC Press, 2016).

Giust, R.

J. Lages, R. Giust, and J. M. Vigoureux, “Composition law for polarizers,” Phys. Rev. A 78, 033810 (2008).
[Crossref]

Gorodyns’ka, N. V.

Gorsky, M. P.

Grossel, Ph.

J. M. Vigoureux and Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[Crossref]

Han, D.

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[Crossref]

D. Han, E. E. Hardekopl, and Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[Crossref]

Hanson, S. G.

Hardekopl, E. E.

D. Han, E. E. Hardekopl, and Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[Crossref]

Kim, Y. S.

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B 6, S455–S472 (2004).
[Crossref]

S. Başkal and Y. S. Kim, “Wigner rotations in laser cavities,” Phys. Rev. E 66, 026604 (2002).
[Crossref]

Y. S. Kim, “Lorentz group in polarization optics,” J. Opt. B 2, R1–R5 (2000).
[Crossref]

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[Crossref]

D. Han, E. E. Hardekopl, and Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[Crossref]

Kitano, M.

M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[Crossref]

Lages, J.

J. Lages, R. Giust, and J. M. Vigoureux, “Composition law for polarizers,” Phys. Rev. A 78, 033810 (2008).
[Crossref]

Leighton, R.

R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1977), Vol. I, Chap. 15.

Manea, V.

T. Tudor and V. Manea, “Symmetry between partially polarized light and partial polarizers,” 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]

Møller, M. C.

M. C. Møller, The Theory of Relativity (Clarendon, 1952).

Monzón, J. J.

J. J. Monzón and L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[Crossref]

Morales, J. A.

J. A. Morales and E. Navarro, “Minkovskian decription of polarized light and polarizers,” Phys. Rev. E 67, 026605 (2003).
[Crossref]

Muttiah, R. S.

Navarro, E.

J. A. Morales and E. Navarro, “Minkovskian decription of polarized light and polarizers,” Phys. Rev. E 67, 026605 (2003).
[Crossref]

Noz, M. E.

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B 6, S455–S472 (2004).
[Crossref]

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[Crossref]

Opatrný, T.

T. Opatrný and J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[Crossref]

Ossikovski, R.

Pellat-Finet, P.

P. Pellat-Finet, “What is common to both polarization optics and relativistic kinematics?” Optik 90, 101–106 (1992).

Perina, J.

T. Opatrný and J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[Crossref]

San José, I.

J. J. Gil, R. Ossikovski, and I. San José, “Singular Mueller matrices,” J. Opt. Soc. Am. A 33, 600–609 (2016).
[Crossref]

R. Ossikovski, J. J. Gil, and I. San José, “Poincaré sphere mapping by Mueller matrices,” J. Opt. Soc. Am. A 30, 2291–2305 (2013).
[Crossref]

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monografías del Seminario Matemático García de Galdeano 33, 115–119 (2006).

Sánchez-Soto, L. L.

J. J. Monzón and L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[Crossref]

Sands, M.

R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1977), Vol. I, Chap. 15.

Sard, R. D.

R. D. Sard, Relativistic Mechanics (Benjamin, 1970).

Sasian, J.

Savenkov, S. V.

Sydoruk, O.

Takenaka, H.

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37–41 (1973).
[Crossref]

Tudor, T.

T. Tudor, “On a quasi-relativistic formula in polarization theory,” Opt. Lett. 40, 693–696 (2015).
[Crossref]

T. Tudor, “Gyrovectors and degree of polarization surfaces in polarization theory,” J. Opt. Soc. Am. B 32, 2528–2535 (2015).
[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]

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

T. Tudor, “Interaction of light with polarization devices: a vectorial Pauli algebraic approach,” Proc. SPIE 7008, 700804 (2008).

T. Tudor, “Interaction of light with the polarization devices: a vectorial Pauli algebraic approach,” J. Phys. A 41, 415303 (2008).
[Crossref]

Ungar, A.

A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,” Found. Phys. Lett. 1, 57–89 (1988).
[Crossref]

A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity (World Scientific, 2008).

Vigoureux, J. M.

J. Lages, R. Giust, and J. M. Vigoureux, “Composition law for polarizers,” Phys. Rev. A 78, 033810 (2008).
[Crossref]

J. M. Vigoureux and Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[Crossref]

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
[Crossref]

Williams, M. W.

Yabuzaki, T.

M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[Crossref]

Zenkova, C. Yu.

Am. J. Phys. (1)

J. M. Vigoureux and Ph. Grossel, “A relativistic-like presentation of optics in stratified planar media,” Am. J. Phys. 61, 707–712 (1993).
[Crossref]

Appl. Opt. (1)

Eur. Phys. J. (1)

J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. 40, 1–47 (2007).
[Crossref]

Found. Phys. Lett. (1)

A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,” Found. Phys. Lett. 1, 57–89 (1988).
[Crossref]

J. Mod. Opt. (1)

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

J. Opt. B (2)

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B 6, S455–S472 (2004).
[Crossref]

Y. S. Kim, “Lorentz group in polarization optics,” J. Opt. B 2, R1–R5 (2000).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

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

J. Phys. A (1)

T. Tudor, “Interaction of light with the polarization devices: a vectorial Pauli algebraic approach,” J. Phys. A 41, 415303 (2008).
[Crossref]

Monografías del Seminario Matemático García de Galdeano (1)

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monografías del Seminario Matemático García de Galdeano 33, 115–119 (2006).

Nouv. Rev. Opt. (1)

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37–41 (1973).
[Crossref]

Opt. Commun. (1)

J. J. Monzón and L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Optik (2)

P. Pellat-Finet, “What is common to both polarization optics and relativistic kinematics?” Optik 90, 101–106 (1992).

S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).

Phys. Lett. A (2)

M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[Crossref]

T. Opatrný and J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[Crossref]

Phys. Rev. A (2)

J. Lages, R. Giust, and J. M. Vigoureux, “Composition law for polarizers,” Phys. Rev. A 78, 033810 (2008).
[Crossref]

D. Han, E. E. Hardekopl, and Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269–1276 (1989).
[Crossref]

Phys. Rev. E (3)

J. A. Morales and E. Navarro, “Minkovskian decription of polarized light and polarizers,” Phys. Rev. E 67, 026605 (2003).
[Crossref]

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[Crossref]

S. Başkal and Y. S. Kim, “Wigner rotations in laser cavities,” Phys. Rev. E 66, 026604 (2002).
[Crossref]

Proc. SPIE (1)

T. Tudor, “Interaction of light with polarization devices: a vectorial Pauli algebraic approach,” Proc. SPIE 7008, 700804 (2008).

Other (6)

A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity (World Scientific, 2008).

R. D. Sard, Relativistic Mechanics (Benjamin, 1970).

M. C. Møller, The Theory of Relativity (Clarendon, 1952).

J. J. Gil and R. Ossikovski, Polarizerd Light and the Mueller Matrix Approach (CRC Press, 2016).

R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1977), Vol. I, Chap. 15.

J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005), p. 306.

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Figures (9)

Fig. 1.
Fig. 1.

Poincarè sphere notations.

Fig. 2.
Fig. 2.

DoP sphere Σ2pi, pi=0.7000, and the corresponding ellipsoid for (a) pd=0.7000, (b) pd=0.9950.

Fig. 3.
Fig. 3.

Evolution of the ellipsoid at a given value of pi=0.40 for various values of pd=0.20, 0.40, 0.68, 0.80.

Fig. 4.
Fig. 4.

Evolution of the ellipsoid at a given value of pd=0.80 for various values of pi=0.45, 0.50, 0.80, 0.90.

Fig. 5.
Fig. 5.

Evolution of the ellipsoid at a higher value of pi=0.900 for various values of pd=0.850, 0.900, 0.994, 0.997.

Fig. 6.
Fig. 6.

Evolution of ellipsoid at a higher value of pd=0.997 for various values of pi=0.925, 0.997, 0.999, 0.9998.

Fig. 7.
Fig. 7.

Δx as a function of pd, with pi as parameter: (a) upper line pi=0.1, lower curve pi=0.5; (b) upper curve pi=0.95, lower curve pi=0.99.

Fig. 8.
Fig. 8.

Δx as a function of pi, with pd as parameter: (a) lower curve pd=0.1, upper curve pd=0.5; (b) lower curve pd=0.95, upper curve pd=0.99.

Fig. 9.
Fig. 9.

Δx as a function of pi and pd in the neighborhood of the limit pi1, pd1.

Equations (16)

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w=uv=u+v1+u.v+γγ+1u×(u×v)1+u.v.
γ=γu=1/1u2
so=sdsi=sd+si1+sd.si+γdγd+1sd(sd×si)1+sd.si,
γd=1/1pd2,
v(R3,|v|<1)
w=u+v,
so=sd+si
si||=si||nd=pindcosϕ,
si=simd=pimdsinϕ,
so||=si||+sd1+si||sdnd=picosϕ+pd1+pipdcosϕnd,
so=siγd(1+si||sd)md=pisinϕ1pd21+pipdcosϕmd.
so=picosϕ+pd1+pipdcosϕnd+pisinϕ1pd21+pipdcosϕmd.
[xpd(1pi2)1pd2pi2]2pi2(1pd21pd2pi2)2+y2pi21pd21pd2pi2+z2pi21pd21pd2pi2=1.
ax=pi(1pd2)1pi2pd2,
ay=pi(1pd21pi2pd2)1/2,
Δx=pd(1pi2)1pi2pd2.

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