Abstract

Recently, we described a geometric construction for determining the eigenvalues of the coherency matrix in three dimensions. We show that this leads directly to a representation of the three-dimensional degree of polarization in terms of a triangular composition plot, in which different polarization measures have simple properties and can be expressed in terms of the matrix invariants. This composition plot is an alternative to the spherical plot recently used to illustrate the degree of polarization in terms of entanglement.

© 2012 Optical Society of America

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References

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  1. J. C. Samson, Geophys. J. R. Astron. Soc. 34, 403 (1973).
    [CrossRef]
  2. R. Barakat, Opt. Commun. 23, 147 (1977).
    [CrossRef]
  3. W. A. Holm and R. M. Barnes, in Proceedings of the 1988 IEEE National Radar Conference (IEEE, 1988), pp. 249–254.
  4. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  5. C. J. R. Sheppard, J. Opt. Soc. Am. A 28, 2655 (2011).
    [CrossRef]
  6. J. Ellis and A. Dogariu, Opt. Commun. 253, 257 (2005).
    [CrossRef]
  7. J. Gil, EPJ Appl. Phys. 40, 1 (2007).
    [CrossRef]
  8. J. C. Petrucelli, N. J. Moore, and M. A. Alonso, Opt. Commun. 283, 4457 (2010).
    [CrossRef]
  9. X. F. Qian and J. H. Eberly, Opt. Lett. 36, 4110 (2011).
    [CrossRef]
  10. T. Saastamoinen and J. Tervo, J. Mod. Opt. 51, 2039 (2004).
    [CrossRef]
  11. C. Brosseau and A. Dogariu, in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), Vol. 49, pp. 315–380.

2011 (2)

2010 (1)

J. C. Petrucelli, N. J. Moore, and M. A. Alonso, Opt. Commun. 283, 4457 (2010).
[CrossRef]

2007 (1)

J. Gil, EPJ Appl. Phys. 40, 1 (2007).
[CrossRef]

2005 (1)

J. Ellis and A. Dogariu, Opt. Commun. 253, 257 (2005).
[CrossRef]

2004 (1)

T. Saastamoinen and J. Tervo, J. Mod. Opt. 51, 2039 (2004).
[CrossRef]

2002 (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

1977 (1)

R. Barakat, Opt. Commun. 23, 147 (1977).
[CrossRef]

1973 (1)

J. C. Samson, Geophys. J. R. Astron. Soc. 34, 403 (1973).
[CrossRef]

Alonso, M. A.

J. C. Petrucelli, N. J. Moore, and M. A. Alonso, Opt. Commun. 283, 4457 (2010).
[CrossRef]

Barakat, R.

R. Barakat, Opt. Commun. 23, 147 (1977).
[CrossRef]

Barnes, R. M.

W. A. Holm and R. M. Barnes, in Proceedings of the 1988 IEEE National Radar Conference (IEEE, 1988), pp. 249–254.

Brosseau, C.

C. Brosseau and A. Dogariu, in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), Vol. 49, pp. 315–380.

Dogariu, A.

J. Ellis and A. Dogariu, Opt. Commun. 253, 257 (2005).
[CrossRef]

C. Brosseau and A. Dogariu, in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), Vol. 49, pp. 315–380.

Eberly, J. H.

Ellis, J.

J. Ellis and A. Dogariu, Opt. Commun. 253, 257 (2005).
[CrossRef]

Friberg, A. T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Gil, J.

J. Gil, EPJ Appl. Phys. 40, 1 (2007).
[CrossRef]

Holm, W. A.

W. A. Holm and R. M. Barnes, in Proceedings of the 1988 IEEE National Radar Conference (IEEE, 1988), pp. 249–254.

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Moore, N. J.

J. C. Petrucelli, N. J. Moore, and M. A. Alonso, Opt. Commun. 283, 4457 (2010).
[CrossRef]

Petrucelli, J. C.

J. C. Petrucelli, N. J. Moore, and M. A. Alonso, Opt. Commun. 283, 4457 (2010).
[CrossRef]

Qian, X. F.

Saastamoinen, T.

T. Saastamoinen and J. Tervo, J. Mod. Opt. 51, 2039 (2004).
[CrossRef]

Samson, J. C.

J. C. Samson, Geophys. J. R. Astron. Soc. 34, 403 (1973).
[CrossRef]

Setälä, T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Sheppard, C. J. R.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Tervo, J.

T. Saastamoinen and J. Tervo, J. Mod. Opt. 51, 2039 (2004).
[CrossRef]

EPJ Appl. Phys. (1)

J. Gil, EPJ Appl. Phys. 40, 1 (2007).
[CrossRef]

Geophys. J. R. Astron. Soc. (1)

J. C. Samson, Geophys. J. R. Astron. Soc. 34, 403 (1973).
[CrossRef]

J. Mod. Opt. (1)

T. Saastamoinen and J. Tervo, J. Mod. Opt. 51, 2039 (2004).
[CrossRef]

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

Opt. Commun. (3)

J. Ellis and A. Dogariu, Opt. Commun. 253, 257 (2005).
[CrossRef]

R. Barakat, Opt. Commun. 23, 147 (1977).
[CrossRef]

J. C. Petrucelli, N. J. Moore, and M. A. Alonso, Opt. Commun. 283, 4457 (2010).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Other (2)

C. Brosseau and A. Dogariu, in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), Vol. 49, pp. 315–380.

W. A. Holm and R. M. Barnes, in Proceedings of the 1988 IEEE National Radar Conference (IEEE, 1988), pp. 249–254.

Supplementary Material (2)

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Figures (3)

Fig. 1.
Fig. 1.

Different polarization measures plotted on the surface of a sphere. Top row: P, ψ, PB. Second row: PPP, PNU, PPU. (Media 1): Parameters Q, Σ/Λ02, 27Δ/Λ03.

Fig. 2.
Fig. 2.

Triangle composition plot. The three pure polarization states are represented by the vertices of the equilateral triangle. The point U corresponds to the fully unpolarized condition. The triangle (U, PU, PP) satisfies the condition 1λ1λ2λ30. PU represents unpolarized within a plane, and PP represents pure polarized. The line PU-PP corresponds to beams. The quantities 2P/3 and ψ are polar coordinates, and (X,Y) and (A,B) are pairs of Cartesian coordinates. Contours of constant P are shown.

Fig. 3.
Fig. 3.

Different polarization measures plotted on the scalar composition plot. (a)–(f) P, PB, Q, PPP, PNU, and PPU, respectively. (Media 2): Parameters (g) Σ/Λ02 and (h) 27Δ/Λ03.

Equations (12)

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C=13[Λ0+Λ3+Λ83Λ1iΛ2Λ4iΛ5Λ1+iΛ2Λ0Λ3+Λ83Λ6iΛ7Λ4+iΛ5Λ6+iΛ7Λ02Λ83].
λ1+λ2+λ3=Λ0,λ12+λ22+λ32=S2=Σ/3,λ13+λ23+λ33=S3=[6Δ+Λ0(ΣΛ02)]/2.
P2=12(3S2Λ021)=12[(Σ/Λ02)1],Q3=18(9S3Λ031)=2(27Δ/Λ03)+9(Σ/Λ02)1116,
λjΛ0=13[1+2Pcos(ψ+2πj3)],j=1,2,3,
ψ=13arccos(3P24Q3P3)=13arccos{52(27Δ/Λ03)3(Σ/Λ02)2[(Σ/Λ02)1)]3/2}.
PB=127Δ/Λ03=P3+2Pcos3ψ.
PPP=λ1λ2=23Psinψ,PU=3λ3=12Pcosψ,PNU=1PU=13λ3=2Pcosψ,PPU=2(λ2λ3)=2P(cosψ13sinψ).
X=2P3cosψ=PNU3,Y=2P3sinψ=PPP3,
PPU=3X3Y,
λ1Λ0=13+X2+3Y2,λ2Λ0=13+X23Y2,λ3Λ0=13X.
P2=94(X2+Y2),PB2=274(X2+X33XY2+Y2),Q3=2732(2X2X3+3XY2+2Y2),ΣΛ02=1+92(X2+Y2),27ΔΛ03=(13X)(1+3X+94X2274Y2).
PPU=23A,P2=94(A2+B2),PB2=274(A2+3A2B+B2B3),Q3=2732(2A23A2B+2B2+B3).

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