Abstract

The spatio-temporal properties of partially polarized light are analyzed in order to separate partial polarization and partial coherence. For that purpose we introduce useful invariance properties which allow one to characterize intrinsic properties of the optical light independently of the particular experimental conditions. This approach leads to new degrees of coherence and their relation with measurable quantities is discussed. These results are illustrated on some simple examples.

© 2005 Optical Society of America

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References

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IEEE Trans. Geosci. Remote sens. (1)

R. S. Cloude and K. P. Papathanassiou, �??Polarimetric SAR interferometry,�?? IEEE Trans. Geosci. Remote sens. 36, 1551�??1565 (1998).
[CrossRef]

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

P. Vahimaa and J. Tervo, �??Unified measures for optical fields: degree of polarization and effective degree of coherence,�?? J. Opt. A: Pure Appl. Opt. 6, 41�??44 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

E. Wolf, �??Unified theory of coherence and polarization of random electromagnetic beams,�?? Phys. Lett. A 312, 263�??267 (2003).
[CrossRef]

Phys. Rev. (1)

R. J. Glauber, �??The Quantum Theory of Optical Coherence,�?? Phys. Rev. 6, 2529-2539, (1963)
[CrossRef]

Statistical Optics (1)

J. W. Goodman, �??Some first-order properties of light waves,�?? in Statistical Optics, 116�??156 (John Wiley and Sons, Inc., New York, 1985).

Other (2)

Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, New-York, 2004).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 160�??170 (Cambridge University Press, New York, 1995).

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Figures (1)

Fig. 1.
Fig. 1.

Schematic illustration of the experimental set up used for the measurement of the degrees of coherence. U 1 and U 2 denote optical modulators with unitary Jones matrices and P 1 and P 2 denote polarizers.

Equations (41)

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Ω ( r 1 , r 2 , t 1 , t 2 ) = E ( r 2 , t 2 ) E ( r 1 , t 1 )
𝒲 ( r 1 , r 2 , ω ) = + Ω ( r 1 , r 2 , τ ) exp [ i ˜ ωτ ]
μ ̄ ( r 1 , r 2 , ω ) = tr [ 𝒲 ( r 1 , r 2 , ω ) ] tr [ 𝒲 ( r 1 , r 2 , ω ) ] tr [ 𝒲 ( r 1 , r 2 , ω ) ]
μ ̄ ( r 1 , r 2 , t 1 , t 2 ) = tr [ Ω ( r 1 , r 2 , t 1 , t 2 ) ] tr [ Ω ( r 1 , r 1 , t 1 , t 2 ) ] tr [ Ω ( r 2 , r 2 , t 1 , t 2 ) ]
μ ˜ 2 ( r 1 , r 2 , ω ) = tr [ 𝒲 ( r 1 , r 2 , ω ) 𝒲 ( r 1 , r 2 , ω ) ] tr [ 𝒲 ( r 1 , r 1 , ω ) ] tr [ 𝒲 ( r 2 , r 2 , ω ) ]
μ ˜ 2 ( r 1 , r 2 , t 1 , t 2 ) = tr [ Ω ( r 1 , r 2 , t 1 , t 2 ) Ω ( r 1 , r 2 , t 1 , t 2 ) ] tr [ Ω ( r 1 , r 1 , t 1 , t 1 ) ] tr [ Ω ( r 2 , r 2 , t 2 , t 2 ) ]
Γ ( r 1 , t 1 ) = Γ ( r 2 , t 2 ) = [ α 0 0 β ]
M ( r 1 , r 2 , t 1 , t 2 ) = Γ 1 2 ( r 2 , t 2 ) Ω ( r 1 , r 2 , t 1 , t 2 ) Γ 1 2 ( r 1 , t 1 )
M ( r 1 , r 2 , t 1 , t 2 ) = N 2 D ( r 1 , r 2 , t 1 , t 2 ) N 1
M ( r 1 , r 2 , t 1 , t 2 ) M ( r 1 , r 2 , t 1 , t 2 ) = N 2 D 2 ( r 1 , r 2 , t 1 , t 2 ) N 2
Λ a ( r i , t i ) = [ A X ( r i , t i ) 2 0 0 A Y ( r i , t i ) 2 ]
Λ a 1 2 ( r i , t i ) = [ 1 I X ( r i , t i ) 0 0 1 I Y ( r i , t i ) ]
M e ( r 1 , r 2 , t 1 , t 2 ) = U 2 M a ( r 1 , r 2 , t 1 , t 2 ) U 1
M a ( r 1 , r 2 , t 1 , t 2 ) = Λ a 1 2 ( r 2 , t 2 ) Ω a ( r 1 , r 2 , t 1 , t 2 ) Λ a 1 2 ( r 1 , t 1 )
Ω a ( r 1 , r 2 , t 1 , t 2 ) = [ A X ( r 2 , t 2 ) A X * ( r 1 , t 1 ) A X ( r 2 , t 2 ) A Y * ( r 1 , t 1 ) A Y ( r 2 , t 2 ) A X * ( r 1 , t 1 ) A Y ( r 2 , t 2 ) A Y * ( r 1 , t 1 ) ]
M a ( r 1 , r 2 , t 1 , t 2 ) = [ η XX ( r 1 , r 2 , t 1 , t 2 ) η XY ( r 1 , r 2 , t 1 , t 2 ) η YX ( r 1 , r 2 , t 1 , t 2 ) η YY ( r 1 , r 2 , t 1 , t 2 ) ]
η PQ ( r 1 , r 2 , t 1 , t 2 ) = A P ( r 2 , t 2 ) A Q * ( r 1 , t 1 ) I P ( r 2 , t 2 ) I Q ( r 1 , t 1 )
Ω a ( r 1 , r 2 , t 1 , t 2 ) = [ A X ( r 2 , t 2 ) A X * ( r 1 , t 1 ) 0 0 0 ]
Λ a 1 2 ( r i , t i ) = [ 1 I X ( r i , t i ) 0 0 0 ]
M a ( r 1 , r 2 , t 1 , t 2 ) = [ η XX ( r 1 , r 2 , t 1 , t 2 ) 0 0 0 ]
Ω ( r 1 , r 2 , t 1 , t 2 ) = Ψ ( r 2 , t 2 ) Ψ ( r 1 , t 1 )
Ω a ( r 1 , r 2 , t 1 , t 2 ) = A ( r 2 , t 2 ) A ( r 1 , t 1 ) = [ Ψ ( r 2 , t 2 ) Ψ ( r 1 , t 1 ) 0 0 0 ]
η = e 1 A U ( r 2 , t 2 ) A U ( r 1 , t 1 ) e 1 e 1 A U ( r 1 , t 1 ) A U ( r 1 , t 1 ) e 1 e 1 A U ( r 2 , t 2 ) A U ( r 2 , t 2 ) e 1
η = e 1 U 2 Ω ( r 1 , r 2 , t 1 , t 2 ) U 1 e 1 e 1 U 1 Γ ( r 1 , t 1 ) U 1 e 1 e 1 U 2 Γ ( r 2 , t 2 ) U 2 e 1
η = k 2 M ( r 1 , r 2 , t 1 , t 2 ) k 1 k 1 k 2
η = a 2 M ( r 1 , r 2 , t 1 , t 2 ) a 1
M ( r 1 , r 2 , t 1 , t 2 ) = N 2 D ( r 1 , r 2 , t 1 , t 2 ) N 1 = μ S u 2 u 1 + μ I v 2 v 1
η = μ S a 2 u 2 u 1 a 1 + μ I a 2 v 2 v 1 a 1
a u ( r 2 , t 2 ) a u * ( r 1 , t 1 ) = μ S a u ( r 2 , t 2 ) a v * ( r 1 , t 1 ) = 0
a v ( r 2 , t 2 ) a u * ( r 1 , t 1 ) = 0 a v ( r 2 , t 2 ) a v * ( r 1 , t 1 ) = μ I
M ( r 1 , r 2 ) = Ω ( r 1 , r 2 ) = R δr E ( r 1 ) E ( r 1 ) = R δ r
E ( r 2 ) = [ E X ( r 2 ) E Y ( r 2 ) ] = [ cos ( θ δ r ) sin ( θ δ r ) sin ( θ δ r ) cos ( θ δ r ) ] [ α E X ( r 1 ) + δ X ( r 1 ) β E Y ( r 1 ) + δ Y ( r 1 ) ]
E ̂ ( r 1 ) = [ α E X ( r 1 ) + δ X ( r 1 ) β E Y ( r 1 ) + δ Y ( r 1 ) ]
Γ 2 = E ( r 2 ) E ( r 2 ) = R δ r E ̂ ( r 1 ) R δ r = I d
M ( r 1 , r 2 ) = Ω ( r 1 , r 2 ) = R δ r E ̂ ( r 1 ) E ̂ ( r 1 ) = R δ r D
D = [ α 0 0 β ]
M ( r 1 , r 2 ) = [ α cos ( θ δ r ) β sin ( θ δ r ) α sin ( θ δ r ) β cos ( θ δ r ) ]
μ ̄ ( r 1 , r 2 , t 1 , t 2 ) = [ α + β ] 2 cos ( θ δ r )
μ ˜ 2 ( r 1 , r 2 , t 1 , t 2 ) = α 2 + β 2 4
η = ( μ S μ I ) a 2 u 2 u 1 a 1 + μ I a 2 ( u 2 u 1 + v 2 v 1 ) a 1
η = ( μ S μ I ) a 2 u 2 u 1 a 1 + μ I ( N 2 a 2 ) N 1 a 1

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