Abstract

In the last decade we have elaborated a mathematical tool for the description of the interaction of polarized light with polarization devices, alternative to the standard matrix (Jones and Mueller) formalisms, namely a vectorial pure operatorial Pauli algebraic one. After a brief, coherent survey of this formalism, we present some applicative results obtained in this frame, referring to the gain and the modification of the state of polarization at the interaction of the polarized light with deterministic devices. Due to an adequate parameterization of the problem, specific to this method, symmetric expressions of the gain and of the generalized Malus’ law are obtained. On the other hand, the equation of the ellipsoid in which a Poincaré sphere of a given degree of polarization is mapped by such a device can be established.

© 2012 Optical Society of America

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  1. P. Lancaster and M. Tismenetsky, The Theory of Matrices with Applications (Academic, 1985).
  2. H. F. Jones, Groups, Representations and Physics (Taylor & Francis, 1998).
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    [CrossRef]
  4. T. Tudor, “Pauli algebraic analysis of polarized light modulation,” Appl. Opt. 47, 2721–2728 (2008).
    [CrossRef]
  5. C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).
  6. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Deckker, 1993).
  7. R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1996).
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    [CrossRef]
  9. W. A. Shurcliff, Polarized Light (Harvard University, 1962).
  10. U. Fano, “Remarks on the classical and quantum-mechanical treatment of partial polarization,” J. Opt. Soc. Am. 39, 859–863 (1949).
    [CrossRef]
  11. U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
    [CrossRef]
  12. R. W. Schmieder, “Stokes-algebra formalism,” J. Opt. Soc. Am. 59, 297–302 (1969).
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  15. P. K. Aravind, “Simulating the Wigner angle with a parametric amplifier,” Phys. Rev. A 42, 4077–4084 (1990).
    [CrossRef]
  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. L. C. Biedenharn, J. D. Louck, and P. A. Carruthers, Angular Momentum in Quantum Physics: Theory and Applications, Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, 1981).
  18. B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941–4958 (2004).
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  19. S.-Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
    [CrossRef]
  20. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical systems from polar decomposition of its Mueller matrix,” Optik 76, 67–71 (1987).
  21. T. Tudor and V. Manea, “The ellipsoid of the polarization degree: a vectorial, pure operatorial Pauli algebraic approach,” J. Opt. Soc. Am. B 28, 596–601 (2011).
    [CrossRef]
  22. R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
    [CrossRef]
  23. G. N. Ramachandran and S. Ramaseshan, Crystal Optics, Handbuch der Physik XXV, S. Flügge, ed. (Springer Verlag, 1961).
  24. L. Dettwiller, “Interpretation and generalization of polarized light interferences by means of the Poincaré sphere,” Eur. J. Phys. 22, 575–586 (2001).
    [CrossRef]
  25. P. V. Polyanskii, “Complex degree of mutual polarization, generalized Malus law and optics of observable quantities,” Proc. SPIE 6254, 625405 (2006).
    [CrossRef]
  26. J. Lages, R. Giust, and J. M. Vigoureux, “Composition law for polarizers,” Phys. Rev. A 78, 033810 (2008).
    [CrossRef]
  27. T. Tudor and V. Manea, “Symmetry between partially polarised light and partial polarisers in the vectorial Pauli algebraic formalism,” J. Mod. Opt. 58, 845–852 (2011).
    [CrossRef]
  28. O. V. Angelsky, S. B. Yermolenko, C. Yu. Zenkova, and A. O. Angelskaya, “Polarization manifestations of correlation (intrinsic coherence) of optical fields,” Appl. Opt. 47, 5492–5499 (2008).
    [CrossRef]
  29. O. V. Angelsky, S. G. Hanson, C. Yu. 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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    [CrossRef]
  31. T. Tudor, “Vectorial pure operatorial Pauli algebraic approach in polarization optics: a theoretical survey and some applications,” Proc. SPIE 8338, 833804 (2011).
    [CrossRef]

2011

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

T. Tudor, “Vectorial pure operatorial Pauli algebraic approach in polarization optics: a theoretical survey and some applications,” Proc. SPIE 8338, 833804 (2011).
[CrossRef]

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

2009

2008

T. Tudor, “Pauli algebraic analysis of polarized light modulation,” Appl. Opt. 47, 2721–2728 (2008).
[CrossRef]

O. V. Angelsky, S. B. Yermolenko, C. Yu. Zenkova, and A. O. Angelskaya, “Polarization manifestations of correlation (intrinsic coherence) of optical fields,” Appl. Opt. 47, 5492–5499 (2008).
[CrossRef]

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 the polarization devices: a vectorial Pauli algebraic approach,” J. Phys. A 41, 4153031 (2008).
[CrossRef]

2007

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

2006

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

2005

2004

2001

L. Dettwiller, “Interpretation and generalization of polarized light interferences by means of the Poincaré sphere,” Eur. J. Phys. 22, 575–586 (2001).
[CrossRef]

2000

1998

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

1990

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

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

1987

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical systems from polar decomposition of its Mueller matrix,” Optik 76, 67–71 (1987).

1971

1969

1965

1957

U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
[CrossRef]

1949

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]

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 nondepolarizing optical systems from polar decomposition of its Mueller matrix,” Optik 76, 67–71 (1987).

Biedenharn, L. C.

L. C. Biedenharn, J. D. Louck, and P. A. Carruthers, Angular Momentum in Quantum Physics: Theory and Applications, Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, 1981).

Brosseau, C.

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

Carruthers, P. A.

L. C. Biedenharn, J. D. Louck, and P. A. Carruthers, Angular Momentum in Quantum Physics: Theory and Applications, Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, 1981).

Chipman, R.

Chipman, R. A.

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

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Deckker, 1993).

DeBoo, B.

Dettwiller, L.

L. Dettwiller, “Interpretation and generalization of polarized light interferences by means of the Poincaré sphere,” Eur. J. Phys. 22, 575–586 (2001).
[CrossRef]

Fano, U.

U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
[CrossRef]

U. Fano, “Remarks on the classical and quantum-mechanical treatment of partial polarization,” J. Opt. Soc. Am. 39, 859–863 (1949).
[CrossRef]

Gil, J. J.

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

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical systems from polar decomposition of its Mueller matrix,” Optik 76, 67–71 (1987).

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.

Hanson, S. G.

Jones, H. F.

H. F. Jones, Groups, Representations and Physics (Taylor & Francis, 1998).

Lages, J.

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

Lancaster, P.

P. Lancaster and M. Tismenetsky, The Theory of Matrices with Applications (Academic, 1985).

Louck, J. D.

L. C. Biedenharn, J. D. Louck, and P. A. Carruthers, Angular Momentum in Quantum Physics: Theory and Applications, Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, 1981).

Lu, S.-Y.

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

Manea, V.

T. Tudor and V. Manea, “The 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 polarised light and partial polarisers in the vectorial Pauli algebraic formalism,” J. Mod. Opt. 58, 845–852 (2011).
[CrossRef]

Marathay, A. S.

Muttiah, R. S.

Polyanskii, P. V.

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

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, Crystal Optics, Handbuch der Physik XXV, S. Flügge, ed. (Springer Verlag, 1961).

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, Crystal Optics, Handbuch der Physik XXV, S. Flügge, ed. (Springer Verlag, 1961).

Sasian, J.

Savenkov, S. V.

Schmieder, R. W.

Segre, S. E.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard University, 1962).

Simon, R.

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

Sydoruk, O.

Tismenetsky, M.

P. Lancaster and M. Tismenetsky, The Theory of Matrices with Applications (Academic, 1985).

Tudor, T.

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

T. Tudor, “Vectorial pure operatorial Pauli algebraic approach in polarization optics: a theoretical survey and some applications,” Proc. SPIE 8338, 833804 (2011).
[CrossRef]

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

T. Tudor, “Pauli algebraic analysis of polarized light modulation,” Appl. Opt. 47, 2721–2728 (2008).
[CrossRef]

T. Tudor, “Interaction of light with the polarization devices: a vectorial Pauli algebraic approach,” J. Phys. A 41, 4153031 (2008).
[CrossRef]

Vigoureux, J. M.

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

Whitney, C.

Yermolenko, S. B.

Zenkova, C. Yu.

Appl. Opt.

Eur. J. Phys.

L. Dettwiller, “Interpretation and generalization of polarized light interferences by means of the Poincaré sphere,” Eur. J. Phys. 22, 575–586 (2001).
[CrossRef]

Eur. Phys. J.

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

J. Mod. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. A

T. Tudor, “Interaction of light with the polarization devices: a vectorial Pauli algebraic approach,” J. Phys. A 41, 4153031 (2008).
[CrossRef]

Opt. Commun.

S.-Y. 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

Optik

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical systems from polar decomposition of its Mueller matrix,” Optik 76, 67–71 (1987).

Phys. Rev. A

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

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

Proc. SPIE

T. Tudor, “Vectorial pure operatorial Pauli algebraic approach in polarization optics: a theoretical survey and some applications,” Proc. SPIE 8338, 833804 (2011).
[CrossRef]

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

Rev. Mod. Phys.

U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
[CrossRef]

Other

L. C. Biedenharn, J. D. Louck, and P. A. Carruthers, Angular Momentum in Quantum Physics: Theory and Applications, Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, 1981).

W. A. Shurcliff, Polarized Light (Harvard University, 1962).

P. Lancaster and M. Tismenetsky, The Theory of Matrices with Applications (Academic, 1985).

H. F. Jones, Groups, Representations and Physics (Taylor & Francis, 1998).

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

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Deckker, 1993).

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

G. N. Ramachandran and S. Ramaseshan, Crystal Optics, Handbuch der Physik XXV, S. Flügge, ed. (Springer Verlag, 1961).

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Figures (3)

Fig. 1.
Fig. 1.

Poincaré unit vector of the incident light and vectors used in the calculations.

Fig. 2.
Fig. 2.

Ellipsoids of the output SOPs and Poincaré spheres of the corresponding input SOPs for two different cases: (a) pi=0.7, η=0.5; (b) pi=0.5, η=1.5.

Fig. 3.
Fig. 3.

Variation of the gain with the angle α for partially polarized incident light when pi and τ¯ are constant. The dashed line represents the contribution of the unpolarized component (pi=0.7, τM=0.9, τm=0.1).

Equations (50)

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

Un(δ)=eiαoeiδ2n·σ=eiαo(σ0cosδ2in·σsinδ2),
Qn(ρ,η)=eρeη2n·σ=eρ(σ0coshη2+n·σsinhη2),
eρ=eη1+η22,eη=eη1η2,
Pn=12(σ0+n·σ).
J=12(σ0+s·σ),
J=I2(σ0+ps·σ),
Jo=DJiD.
Ji=12(σ0+pisi·σ),
Jo=12(σ0cosδ2in·σsinδ2)(σ0+pisi·σ)(σ0cosδ2+in·σsinδ2)=12{σ0+pi[si·σcosδ+(n×si)·σsinδ+2n.si(n·σ)sin2δ2]}.
Jo=12(σ0+poso·σ),
po=pi.
so=sicosδ+n×sisinδ+2(n·si)nsin2δ2.
so=Rn(δ)si,
Rn(δ)=cosδ+(1cosδ)n(n.)+sinδ(n×).
Jo=12e2ρ(σ0coshη2+n·σsinhη2)(σ0+pisi·σ)(σ0coshη2+n·σsinhη2)=12e2ρ{σ0(coshη+pisi·nsinhη)+[pisi+nsinhη+2pi(n.si)nsinh2η2]·σ}.
Jo=12g(σ0+poso·σ),
g=e2ρ(cosη+picosαsinhη)=e2η11+picosα2+e2η21picosα2.
poso=pisi+nsinhη+2pincosαsinh2η2coshη+picosαsinhη=pisi+nsinhη+pincosα(coshη1)coshη+picosαsinhη,
i|Ψ(t)t=H|Ψ(t),
|Ψ(t)=U(t,0)|Ψ(0),
iUt=H(t)U(t).
iJ(t)t=[H(t),J(t)].
J=n=03sjσj=s0σ0+s·σ,
H=[H11H12H21H22]=n=03χkσk=χ0σ0+Hχ·σ,
AB=(a0b0+a·b)σ0+(b0a+a0b)·σ+i(a×b)·σ,
J(t)t=(s0t+st·σ),
[H(t),J(t)]=(χ0s0+Hχ·s)σ0+(s0Hχ+χ0s)·σ+i(Hχ×s)·σ(s0χ0+s·Hχ)σ0(χ0s+s0Hχ)·σi(s×Hχ)·σ=2i(Hχ×s)·σ,
J(t)t=s0t+st·σ=2H(χ×s)·σ.
s˙=H×s,
H=2Hχ
s˙(t)=H(t)×s(t),
poso=pi[sincosα]+nsinhη+pincosαcoshηcoshη+picosαsinhη.
n.(sincosα)=0,
|sincosα|=12n.sicosα+cos2α=1cos2α=sinα.
poso=mpisinαcoshη+picosαsinhη+nsinhη+picosαcoshηcoshη+picosαsinhη.
x=sinhη+picosαcoshηcoshη+picosαsinhη,
y=pisinαcoshη+picosαsinhη.
x2(cosh2ηpi2sinh2η)2xsinhηcoshη(1pi2)+y2pi2cosh2η+sinh2η=0,
xsinhηcoshη(1pi2)cosh2ηpi2sinh2ηX,yY.
X2pi2(cosh2ηpi2sinh2η)2+Y2pi2cosh2ηpi2sinh2η=1.
X2pi2(cosh2ηpi2sinh2η)2+Y2pi2cosh2ηpi2sinh2η+Z2pi2cosh2ηpi2sinh2η=1,
gM=e2ρ(coshη+pisinhη),
gm=e2ρ(coshηpisinhη).
γ=gMgmgM+gm,
γ=pitanhη.
γ=gMgmgM+gm=pipd.
g=e2η11+picosα2+e2η21picosα2=τM+τm2+pin.siτMτm2g=τ¯(1+pdn.pisi).
gup=τM+τm2=τ¯.
gtp=τ¯(1+pdsi·n)=τM1+si·n2+τm1si·n2=τMcos2α2+τmsin2α2,
g=(1pi)τ¯+pi(τMcos2α2+τmsin2α2).

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