Abstract

The interaction of arbitrary three-dimensional light beams with optical elements is described by the generalized Jones calculus, which has been formally proposed recently [Azzam, J. Opt. Soc. Am. A 28, 2279 (2011)]. In this work we obtain the parametric expression of the 3×3 differential generalized Jones matrix (dGJM) for arbitrary optical media assuming transverse light waves. The dGJM is intimately connected to the Gell-Mann matrices, and we show that it provides a versatile method for obtaining the macroscopic GJM of media with either sequential or simultaneous anisotropic effects. Explicit parametric expressions of the GJM for some relevant optical elements are provided.

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References

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  1. K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys.40(1), 1–47 (2007).
    [CrossRef]
  5. R. M. A. Azzam, “Three-dimensional polarization states of monochromatic light fields,” J. Opt. Soc. Am. A28(11), 2279–2283 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
  7. R. Barakat, “Exponential versions of the Jones and Mueller-Jones polarization matrices,” J. Opt. Soc. Am. A13(1), 158–163 (1996).
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  8. C. Brosseau, Fundamentals of polarized light. A statistical optics approach (Wiley, 1998).
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  10. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus,” J. Opt. Soc. Am.68(12), 1756–1767 (1978).
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  12. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

2012 (1)

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A85(6), 063825 (2012).
[CrossRef]

2011 (2)

2010 (1)

2007 (2)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
[CrossRef]

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

1997 (1)

1996 (1)

1978 (1)

1948 (1)

Arce-Diego, J. L.

Azzam, R. M. A.

Banzer, P.

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A85(6), 063825 (2012).
[CrossRef]

Barakat, R.

Bauer, T.

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A85(6), 063825 (2012).
[CrossRef]

Friberg, A. T.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
[CrossRef]

Gil, J. J.

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

Gu, M.

Han, D.

Jia, B.

Jones, C. R.

Kaivola, M.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
[CrossRef]

Kang, H.

Kim, Y. S.

Lindfors, K.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
[CrossRef]

Noz, M. E.

Orlov, S.

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A85(6), 063825 (2012).
[CrossRef]

Ortega-Quijano, N.

Peschel, U.

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A85(6), 063825 (2012).
[CrossRef]

Priimagi, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
[CrossRef]

Setälä, T.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
[CrossRef]

Shevchenko, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
[CrossRef]

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

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

J. Opt. Soc. Am. (2)

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

Nat. Photonics (1)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics1(4), 228–231 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering,” Phys. Rev. A85(6), 063825 (2012).
[CrossRef]

Other (2)

C. Brosseau, Fundamentals of polarized light. A statistical optics approach (Wiley, 1998).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

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Figures (1)

Fig. 1
Fig. 1

Index ellipsoid of a uniaxial medium with the optic axis parallel to the laboratory z axis (left) and aligned with the direction z ( θ b , ϕ b ) of local coordinate system x y z (right).

Equations (25)

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E o =G E i ,
j= 1 2 l=0 3 i f l σ l ,
f 0 =2 γ i =2( η i i κ i ),
f 1 = γ q = η q i κ q ,
f 2 = γ u = η u i κ u ,
f 3 = γ v = η v i κ v ,
j= 1 2 [ 2 κ i + κ q +i( 2 η i + η q ) η v + κ u +i( η u κ v ) η v + κ u +i( η u + κ v ) 2 κ i κ q +i( 2 η i η q ) ].
O 0 =[ 1 0 0 0 1 0 0 0 1 ] O 1 =[ 1 0 0 0 1 0 0 0 0 ] O 2 =[ 0 1 0 1 0 0 0 0 0 ] O 3 =[ 0 i 0 i 0 0 0 0 0 ] O 4 = 1 3 [ 1 0 0 0 1 0 0 0 2 ] O 5 =[ 0 0 1 0 0 0 1 0 0 ] O 6 =[ 0 0 i 0 0 0 i 0 0 ] O 7 =[ 0 0 0 0 0 1 0 1 0 ] O 8 =[ 0 0 0 0 0 i 0 i 0 ] .
g= 1 2 m=0 8 i p m O m ,
{ p 0... 8 }={ 2 γ i , γ q xy , γ u xy , γ v xy ,2/ 3 γ q z , γ u xz , γ v xz , γ u yz , γ v xz }.
g= 1 2 [ 2 κ i + κ q xy + 2 κ q z /3 +i( 2 η i + η q xy + 2 η q z /3 ) η v xy + κ u xy +i( η u xy κ v xy ) η v xz + κ u xz +i( η u xz κ v xz ) η v xy + κ u xy +i( η u xy + κ v xy ) 2 κ i κ q xy + 2 κ q z /3 +i( 2 η i η q xy + 2 η q z /3 ) η v yz + κ u yz +i( η u yz κ v yz ) η v xz + κ u xz +i( η u xz + κ v xz ) η v yz + κ u yz +i( η u yz + κ v yz ) 2 κ i 4 κ q z /3 +i( 2 η i 4 η q z /3 ) ].
d E / dl =g E ,
G=exp( gl ).
G= i exp( g i l ) ,
E x y z =C( θ,ϕ ) E xyz ,
C( θ,ϕ )=[ sinϕ cosθcosϕ sinθcosϕ cosϕ cosθsinϕ sinθsinϕ 0 sinθ cosθ ].
g LR | xyz z = i 3 η q z O 4 =diag( i η q z /3 , i η q z /3 , i2 η q z /3 ),
G LR | xyz z =diag( e iδ /3 , e iδ /3 , e i2δ /3 ).
G LR =C( θ b , ϕ b ) G LR | xyz z C 1 ( θ b , ϕ b ).
G LR | θ b =π/2 =[ e i2δ /3 cos 2 ϕ b + e iδ /3 sin 2 ϕ b ( e i2δ /3 e iδ /3 )cos ϕ b sin ϕ b 0 ( e i2δ /3 e iδ /3 )cos ϕ b sin ϕ b e i2δ /3 sin 2 ϕ b + e iδ /3 cos 2 ϕ b 0 0 0 e iδ /3 ].
g LD | xyz z = 1 3 κ q z O 4 =diag( κ q z /3 , κ q z /3 , 2 κ q z /3 ),
G LD | x y z z =diag( e α/3 , e α/3 , e 2α /3 ).
G LD | θ d =π/2 =[ e 2α /3 cos 2 ϕ d + e α/3 sin 2 ϕ d ( e 2α /3 e α/3 )cos ϕ d sin ϕ d 0 ( e 2α /3 e α/3 )cos ϕ d sin ϕ d e 2α /3 sin 2 ϕ d + e α/3 cos 2 ϕ d 0 0 0 e α/3 ].
g OR = 1 2 i η v ( O 3 + O 6 + O 8 )= 1 2 [ 0 η v η v η v 0 η v η v η v 0 ].
G OR = 1 3 [ 2cosψ+1 cosψ+ 3 sinψ1 cosψ+ 3 sinψ+1 cosψ 3 sinψ1 2cosψ+1 cosψ+ 3 sinψ1 cosψ 3 sinψ+1 cosψ 3 sinψ1 2cosψ+1 ].

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