On the behavior of degree of polarization surfaces at the limit of Poincaré sphere walls

Tiberiu Tudor

doi:10.1364/JOSAB.34.001147

On the behavior of degree of polarization surfaces at the limit of Poincaré sphere walls

Tiberiu Tudor

Journal of the Optical Society of America B
Vol. 34,
Issue 6,
pp. 1147-1155
(2017)
•https://doi.org/10.1364/JOSAB.34.001147

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Tiberiu Tudor, "On the behavior of degree of polarization surfaces at the limit of Poincaré sphere walls," J. Opt. Soc. Am. B 34, 1147-1155 (2017)

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Related Topics
Table of Contents Category
- Physical Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Relativity (350.5720)
- Physical optics (260.0260)
- Polarization (260.5430)

About this Article
History
- Original Manuscript: January 31, 2017
- Revised Manuscript: April 6, 2017
- Manuscript Accepted: April 25, 2017
- Published: May 18, 2017
Virtual Issues

July 13, 2017 Spotlight on Optics

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Open Access

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.

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References

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Year
|
Author
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Publication

J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. 40, 1–47 (2007).
[Crossref]
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).
[Crossref]
S. V. Savenkov, O. Sydoruk, and R. S. Muttiah, “Conditions for polarization elements to be dichroic and birefringent,” J. Opt. Soc. Am. A 22, 1447–1452 (2005).
[Crossref]
R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
[Crossref]
H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37–41 (1973).
[Crossref]
S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
M. Kitano and T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[Crossref]
P. Pellat-Finet, “What is common to both polarization optics and relativistic kinematics?” Optik 90, 101–106 (1992).
T. Opatrný and J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
[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]
Y. S. Kim, “Lorentz group in polarization optics,” J. Opt. B 2, R1–R5 (2000).
[Crossref]
J. A. Morales and E. Navarro, “Minkovskian decription of polarized light and polarizers,” Phys. Rev. E 67, 026605 (2003).
[Crossref]
J. Lages, R. Giust, and J. M. Vigoureux, “Composition law for polarizers,” Phys. Rev. A 78, 033810 (2008).
[Crossref]
M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
[Crossref]
B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941–4958 (2004).
[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).
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]
J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005), p. 306.
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]
T. Tudor, “Gyrovectors and degree of polarization surfaces in polarization theory,” J. Opt. Soc. Am. B 32, 2528–2535 (2015).
[Crossref]
T. Tudor, “On a quasi-relativistic formula in polarization theory,” Opt. Lett. 40, 693–696 (2015).
[Crossref]
A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,” Found. Phys. Lett. 1, 57–89 (1988).
[Crossref]
R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1977), Vol. I, Chap. 15.
T. Tudor and V. Manea, “Symmetry between partially polarized light and partial polarizers,” J. Mod. Opt. 58, 845–852 (2011).
[Crossref]
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).
J. J. Gil, R. Ossikovski, and I. San José, “Singular Mueller matrices,” J. Opt. Soc. Am. A 33, 600–609 (2016).
[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]
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. J. Monzón and L. L. Sánchez-Soto, “Lossless multilayers and Lorentz transformations: more than an analogy,” Opt. Commun. 162, 1–6 (1999).
[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]
S. Başkal and Y. S. Kim, “Wigner rotations in laser cavities,” Phys. Rev. E 66, 026604 (2002).
[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]
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]

2011 (2)

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]

2009 (1)

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).
[Crossref]

2008 (3)

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]

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

S. V. Savenkov, O. Sydoruk, and R. S. Muttiah, “Conditions for polarization elements to be dichroic and birefringent,” J. Opt. Soc. Am. A 22, 1447–1452 (2005).
[Crossref]

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)

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]

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

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)

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

M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
[Crossref]

1973 (1)

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

1963 (1)

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
[Crossref]

Angelsky, O. V.

Barakat, R.

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
[Crossref]

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.

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

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.

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

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.

S. V. Savenkov, O. Sydoruk, and R. S. Muttiah, “Conditions for polarization elements to be dichroic and birefringent,” J. Opt. Soc. Am. A 22, 1447–1452 (2005).
[Crossref]

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.

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]

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

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.

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

Savenkov, S. V.

S. V. Savenkov, O. Sydoruk, and R. S. Muttiah, “Conditions for polarization elements to be dichroic and birefringent,” J. Opt. Soc. Am. A 22, 1447–1452 (2005).
[Crossref]

Sydoruk, O.

S. V. Savenkov, O. Sydoruk, and R. S. Muttiah, “Conditions for polarization elements to be dichroic and birefringent,” J. Opt. Soc. Am. A 22, 1447–1452 (2005).
[Crossref]

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.

M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
[Crossref]

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)

M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
[Crossref]

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)

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
[Crossref]

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

S. V. Savenkov, O. Sydoruk, and R. S. Muttiah, “Conditions for polarization elements to be dichroic and birefringent,” J. Opt. Soc. Am. A 22, 1447–1452 (2005).
[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]

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

J. Opt. Soc. Am. B (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]

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Fig. 1.

Poincarè sphere notations.

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Fig. 2.

DoP sphere $Σ_{2}^{p_{i}}$ , $p_{i} = 0.7000$ , and the corresponding ellipsoid for (a) $p_{d} = 0.7000$ , (b) $p_{d} = 0.9950$ .

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Fig. 3.

Evolution of the ellipsoid at a given value of $p_{i} = 0.40$ for various values of $p_{d} = 0.20$ , 0.40, 0.68, 0.80.

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Fig. 4.

Evolution of the ellipsoid at a given value of $p_{d} = 0.80$ for various values of $p_{i} = 0.45$ , 0.50, 0.80, 0.90.

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Fig. 5.

Evolution of the ellipsoid at a higher value of $p_{i} = 0.900$ for various values of $p_{d} = 0.850$ , 0.900, 0.994, 0.997.

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Fig. 6.

Evolution of ellipsoid at a higher value of $p_{d} = 0.997$ for various values of $p_{i} = 0.925$ , 0.997, 0.999, 0.9998.

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Fig. 7.

$Δ x$ as a function of $p_{d}$ , with $p_{i}$ as parameter: (a) upper line $p_{i} = 0.1$ , lower curve $p_{i} = 0.5$ ; (b) upper curve $p_{i} = 0.95$ , lower curve $p_{i} = 0.99$ .

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Fig. 8.

$Δ x$ as a function of $p_{i}$ , with $p_{d}$ as parameter: (a) lower curve $p_{d} = 0.1$ , upper curve $p_{d} = 0.5$ ; (b) lower curve $p_{d} = 0.95$ , upper curve $p_{d} = 0.99$ .

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Fig. 9.

$Δ x$ as a function of $p_{i}$ and $p_{d}$ in the neighborhood of the limit $p_{i} \to 1$ , $p_{d} \to 1$ .

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Equations (16)

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

w = u \oplus v = \frac{u + v}{1 + u . v} + \frac{γ}{γ + 1} \frac{u \times (u \times v)}{1 + u . v} .

γ = γ_{u} = 1 / \sqrt{1 - u^{2}}

s_{o} = s_{d} \oplus s_{i} = \frac{s_{d} + s_{i}}{1 + s_{d} {. s}_{i}} + \frac{γ_{d}}{γ_{d} + 1} \frac{s_{d} (s_{d} \times s_{i})}{1 + s_{d} {. s}_{i}},

γ_{d} = 1 / \sqrt{1 - p_{d}^{2}},

v \in (R^{3}, | v | < 1)

w = u + v,

s_{o} = s_{d} {+ s}_{i}

s_{i | |} = s_{i | |} n_{d} = p_{i} n_{d} \cos ϕ,

s_{i ⊥} = s_{i ⊥} m_{d} = p_{i} m_{d} \sin ϕ,

s_{o | |} = \frac{s_{i | |} + s_{d}}{1 + s_{i | |} s_{d}} n_{d} = \frac{p_{i} \cos ϕ + p_{d}}{1 + p_{i} p_{d} \cos ϕ} n_{d},

s_{o ⊥} = \frac{s_{i ⊥}}{γ_{d} (1 + s_{i | |} s_{d})} m_{d} = \frac{p_{i} \sin ϕ \sqrt{1 - p_{d}^{2}}}{1 + p_{i} p_{d} \cos ϕ} m_{d} .

s_{o} = \frac{p_{i} \cos ϕ + p_{d}}{1 + p_{i} p_{d} \cos ϕ} n_{d} + \frac{p_{i} \sin ϕ \sqrt{1 - p_{d}^{2}}}{1 + p_{i} p_{d} \cos ϕ} m_{d} .

\frac{{[x - \frac{p_{d} (1 - p_{i}^{2})}{1 - p_{d}^{2} p_{i}^{2}}]}^{2}}{p_{i}^{2} {(\frac{1 - p_{d}^{2}}{1 - p_{d}^{2} p_{i}^{2}})}^{2}} + \frac{y^{2}}{p_{i}^{2} \frac{1 - p_{d}^{2}}{1 - p_{d}^{2} p_{i}^{2}}} + \frac{z^{2}}{p_{i}^{2} \frac{1 - p_{d}^{2}}{1 - p_{d}^{2} p_{i}^{2}}} = 1 .

a_{x} = \frac{p_{i} (1 - p_{d}^{2})}{1 - p_{i}^{2} p_{d}^{2}},

a_{y} = p_{i} {(\frac{1 - p_{d}^{2}}{1 - p_{i}^{2} p_{d}^{2}})}^{1 / 2},

Δ x = \frac{p_{d} (1 - p_{i}^{2})}{1 - p_{i}^{2} p_{d}^{2}} .

Abstract

References

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