Abstract

In the frame of a vectorial, pure operatorial (nonmatrix) Pauli algebraic approach to the action of the polarization devices on the polarized incident light, we obtain the analytic equation of the ellipsoid into which a deterministic device deforms any Poincaré sphere corresponding to incident light of a given degree of polarization. On the basis of this equation, some graphical representations of the ellipsoid of the output polarization states are given, offering a better characterization of the global action of deterministic devices.

© 2011 Optical Society of America

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References

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  1. W. A. Shurcliff, Polarized Light (Harvard University, 1969).
  2. R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).
  3. C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).
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    [CrossRef]
  5. S. R. Cloude, “Group theory and polarization algebra,” Optik (Jena) 75, 26–36 (1986).
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  8. C. Brosseau, “Mueller matrix analysis of light depolarization by linear optical medium,” Opt. Commun. 131, 229–235 (1996).
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    [CrossRef]
  10. Shih-Yau Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
    [CrossRef]
  11. J. W. Hovenier and C. V. M. van der Mee, “Bounds for the degree of polarization,” Opt. Lett. 20, 2454–2456 (1995).
    [CrossRef] [PubMed]
  12. R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
    [CrossRef]
  13. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67–71 (1987).
  14. S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  15. P. K. Aravind, “Simulating the Wigner angle with a parametric amplifier,” Phys. Rev. A 42, 4077–4084 (1990).
    [CrossRef] [PubMed]
  16. 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]
  17. T. Opatrnỳ and J. Peřina, “Non-image-forming polarization optical devices and Lorentz transformations—an analogy,” Phys. Lett. A 181, 199–202 (1993).
    [CrossRef]
  18. A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A Pure Appl. Optics 11, 094010 (2009).
    [CrossRef]
  19. P. V. Polyanskii, “Complex degree of mutual polarization, generalized Malus law, and optics of observable quantities,” Proc. SPIE 6254, 625405 (2006).
    [CrossRef]
  20. O. V. Angelsky, S. G. Hanson, C. Y.Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17, 15623–15634 (2009).
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  21. O. V. Angelsky, S. B. Yermolenko, C. Y.Zenkova, and A. O. Angelskaya, “Polarization manifestations of correlation (intrinsic coherence) of optical fields,” Appl. Opt. 47, 5492–5499 (2008).
    [PubMed]

2009 (2)

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A Pure Appl. Optics 11, 094010 (2009).
[CrossRef]

O. V. Angelsky, S. G. Hanson, C. Y.Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17, 15623–15634 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (1)

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

2006 (1)

P. V. Polyanskii, “Complex degree of mutual polarization, generalized Malus law, and optics of observable quantities,” Proc. SPIE 6254, 625405 (2006).
[CrossRef]

2005 (1)

2004 (2)

1998 (1)

Shih-Yau Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[CrossRef]

1996 (1)

C. Brosseau, “Mueller matrix analysis of light depolarization by linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

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

1990 (2)

P. K. Aravind, “Simulating the Wigner angle with a parametric amplifier,” Phys. Rev. A 42, 4077–4084 (1990).
[CrossRef] [PubMed]

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

1987 (1)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67–71 (1987).

1986 (2)

Angelskaya, A. O.

Angelsky, O. V.

Aravind, P. K.

P. K. Aravind, “Simulating the Wigner angle with a parametric amplifier,” Phys. Rev. A 42, 4077–4084 (1990).
[CrossRef] [PubMed]

Azzam, R. M.

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

Bashara, N. M.

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

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67–71 (1987).

Bogatyryova, H. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A Pure Appl. Optics 11, 094010 (2009).
[CrossRef]

Brosseau, C.

C. Brosseau, “Mueller matrix analysis of light depolarization by linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

Chernyshov, A. A.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A Pure Appl. Optics 11, 094010 (2009).
[CrossRef]

Chipman, R.

Chipman, R. A.

Shih-Yau Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[CrossRef]

S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
[CrossRef]

Cloude, S. R.

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

DeBoo, B.

Dogariu, A.

Ellis, J.

Felde, Ch. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A Pure Appl. Optics 11, 094010 (2009).
[CrossRef]

Gil, J. J.

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

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67–71 (1987).

Gorodyns’ka, N. V.

Gorsky, M. P.

Hanson, S. G.

Hovenier, J. W.

Lu, S.-Y.

Lu, Shih-Yau

Shih-Yau Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[CrossRef]

Muttiah, R. S.

Opatrn?, T.

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

Perina, J.

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

Polyanskii, P. V.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A Pure Appl. Optics 11, 094010 (2009).
[CrossRef]

P. V. Polyanskii, “Complex degree of mutual polarization, generalized Malus law, and optics of observable quantities,” Proc. SPIE 6254, 625405 (2006).
[CrossRef]

Sasian, J.

Savenkov, S. V.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard University, 1969).

Simon, R.

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

Soskin, M. S.

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A Pure Appl. Optics 11, 094010 (2009).
[CrossRef]

Sydoruk, O.

van der Mee, C. V. M.

Williams, M. W.

Yermolenko, S. B.

Zenkova, C. Y.

Appl. Opt. (2)

Eur. Phys. J. Appl. Phys. (1)

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

J. Opt. A Pure Appl. Optics (1)

A. A. Chernyshov, Ch. V. Felde, H. V. Bogatyryova, P. V. Polyanskii, and M. S. Soskin, “Vector singularities of the combined beams assembled from mutually incoherent orthogonally polarized components,” J. Opt. A Pure Appl. Optics 11, 094010 (2009).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

C. Brosseau, “Mueller matrix analysis of light depolarization by linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

Shih-Yau Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[CrossRef]

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Optik (Jena) (2)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67–71 (1987).

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

Phys. Lett. A (1)

T. Opatrnỳ 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 (1)

P. K. Aravind, “Simulating the Wigner angle with a parametric amplifier,” Phys. Rev. A 42, 4077–4084 (1990).
[CrossRef] [PubMed]

Proc. SPIE (1)

P. V. Polyanskii, “Complex degree of mutual polarization, generalized Malus law, and optics of observable quantities,” Proc. SPIE 6254, 625405 (2006).
[CrossRef]

Other (3)

W. A. Shurcliff, Polarized Light (Harvard University, 1969).

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

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

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Figures (3)

Fig. 1
Fig. 1

Poincaré unit vector of the incident light and axes implied in the calculations.

Fig. 2
Fig. 2

Ellipsoid of the output SOPs and Poincaré sphere of the corresponding input SOPs: (a)  p i = 0.7 , η = 0.5 ; (b)  p i = 0.5 , η = 1.5 .

Fig. 3
Fig. 3

Ellipsoid of the output SOPs and Poincaré sphere of the corresponding input SOPs for the case in which unpolarized light can be obtained at the output.

Equations (24)

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H n ( ρ , η ) = e ρ e η 2 n · σ = e ρ ( σ 0 cosh η 2 + n · σ sinh η 2 ) ,
e ρ = e η 1 + η 2 2 ,
e η = e η 1 η 2 ,
J = I 2 ( σ 0 + p s · σ ) ,
J i = 1 2 ( σ 0 + p i s i · σ ) ,
J o = g 2 ( σ 0 + p o s o · σ ) ,
J o = H n ( ρ , η ) J i H n ( ρ , η ) .
J o = 1 2 e 2 ρ ( σ o cosh η 2 + n · σ sinh η 2 ) ( σ o + p i s i · σ ) ( σ o cosh η 2 + n · σ sinh η 2 ) = 1 2 e 2 ρ { σ o ( cosh η + p i s i · n sinh η ) + [ p i s i + n sinh η + 2 p i ( n · s i ) n sinh 2 η 2 ] · σ } ,
g = e 2 ρ ( cosh η + p i s i · n sinh η ) = e 2 η 1 1 + p i cos α 2 + e 2 η 2 1 p i cos α 2 ,
p o s o = p i s i + n sinh η + 2 p i ( n · s i ) n sinh 2 η 2 cosh η + p i ( n · s i ) sinh η = p i s i + n sinh η + p i n cos α ( cosh η 1 ) cosh η + p i cos α sinh η .
p o s o = p i [ s i n cos α ] + n sinh η + p i n cos α cosh η cosh η + p i cos α sinh η .
n · ( s i n cos α ) = 0 ,
| s i n cos α | = 1 2 n · s i cos α + cos 2 α = 1 cos 2 α = sin α .
p o s o = m p i sin α cosh η + p i cos α sinh η + n sinh η + p i cos α cosh η cosh η + p i cos α sinh η .
x = sinh η + p i cos α cosh η cosh η + p i cos α sinh η ,
y = p i sin α cosh η + p i cos α sinh η .
( x cosh η sinh η ) 2 p i 2 ( cosh η x sinh η ) 2 + y 2 = 0 , x 2 ( cosh 2 η p i 2 sinh 2 η ) 2 x sinh η cosh η ( 1 p i 2 ) + y 2 p i 2 cosh 2 η + sinh 2 η = 0 ,
x 2 2 x sinh η cosh η ( 1 p i 2 ) cosh 2 η p i 2 sinh 2 η + y 2 cosh 2 η p i 2 sinh 2 η p i 2 cosh 2 η sinh 2 η cosh 2 η p i 2 sinh 2 η = 0 , [ x sinh η cosh η ( 1 p i 2 ) cosh 2 η p i 2 sinh 2 η ] 2 + y 2 cosh 2 η p i 2 sinh 2 η = p i 2 ( cosh 2 η p i 2 sinh 2 η ) 2 .
Δ x = sinh η cosh η ( 1 p i 2 ) cosh 2 η p i 2 sinh 2 η .
x sinh η cosh η ( 1 p i 2 ) cosh 2 η p i 2 sinh 2 η X y Y ,
X 2 p i 2 ( cosh 2 η p i 2 sinh 2 η ) 2 + Y 2 p i 2 cosh 2 η p i 2 sinh 2 η = 1.
X 2 p i 2 ( cosh 2 η p i 2 sinh 2 η ) 2 + Y 2 p i 2 cosh 2 η p i 2 sinh 2 η + Z 2 p i 2 cosh 2 η p i 2 sinh 2 η = 1 ,
ε = cosh 2 η p i 2 sinh 2 η 1 ,
X 2 + Y 2 + Z 2 = 1 ,

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