Abstract

We demonstrate monotonic, irreversible effects on the entropy, the degree of polarization, and different degrees of coherence when random, energy-preserving unitary transformations are applied to vectorial electromagnetic fields.

© 2008 Optical Society of America

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References

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  1. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  2. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express 11, 1137-1142 (2003).
    [CrossRef] [PubMed]
  3. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205-2215 (2004).
    [CrossRef]
  4. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328-330 (2004).
    [CrossRef] [PubMed]
  5. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78-84 (2003).
    [CrossRef]
  6. G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, “Generation of complete coherence in Young's interference experiment with random mutually uncorrelated electromagnetic beams,” Opt. Lett. 30, 120-122 (2005).
    [CrossRef] [PubMed]
  7. Ph. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051-6060 (2005).
    [CrossRef] [PubMed]
  8. Ph. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117-3119 (2005).
    [CrossRef] [PubMed]
  9. Ph. Réfrégier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. Am. A 23, 3036-3044 (2006).
    [CrossRef]
  10. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effect of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-670 (2006).
    [CrossRef] [PubMed]
  11. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Lett. 31, 2208-2210 (2006).
    [CrossRef] [PubMed]
  12. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669-2671 (2006).
    [CrossRef] [PubMed]
  13. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young's fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588-590 (2007).
    [CrossRef] [PubMed]
  14. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063-1068 (2007).
    [CrossRef]
  15. R. Martinez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471-1473 (2007).
    [CrossRef] [PubMed]
  16. R. Martinez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504-1506 (2007).
    [CrossRef] [PubMed]
  17. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907-1916 (2004).
    [CrossRef]
  18. P. Réfrégier, “Irreversible effects of random modulation on coherence properties of partially polarized light,” Opt. Lett. 33, 636-638 (2008).
    [CrossRef] [PubMed]
  19. J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, 1985), pp. 116-156.
  20. L. Mandel and E. Wolf, “Second-order coherence theory of scalar wavefields,” in Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 160-170.
  21. R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171-1182 (1983).
    [CrossRef]
  22. A. Renyi, “On the measures of entropy and information,” in Proceedings of the 4th Berkeley Symposium on Mathematics and Staticstical Probability (University of California Press, 1961), Vol. 1, pp. 547-561.
  23. Ph. Réfrégier, F. Goudail, P. Chavel, and A. Friberg, “Entropy of partially polarized light and application to statistical processing techniques,” J. Opt. Soc. Am. A 21, 2124-2134 (2004).
    [CrossRef]
  24. T. M. Cover and J. A. Thomas, “Entropy, relative entropy and mutual information,” in Elements of Information Theory (Wiley-Interscience, 1991), pp. 18-26.
  25. A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191-2193 (2007).
    [CrossRef] [PubMed]
  26. Ph. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366-1368 (2007).
    [CrossRef] [PubMed]
  27. C. Brosseau, “Polarization and the radiation field,” in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 67-176.

2008 (1)

2007 (6)

2006 (4)

2005 (3)

2004 (4)

2003 (3)

1983 (1)

R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171-1182 (1983).
[CrossRef]

Agarwal, G. S.

Barakat, R.

R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171-1182 (1983).
[CrossRef]

Borghi, R.

Brosseau, C.

C. Brosseau, “Polarization and the radiation field,” in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 67-176.

Chavel, P.

Cover, T. M.

T. M. Cover and J. A. Thomas, “Entropy, relative entropy and mutual information,” in Elements of Information Theory (Wiley-Interscience, 1991), pp. 18-26.

Dogariu, A.

Friberg, A.

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, 1985), pp. 116-156.

Gori, F.

Goudail, F.

Guattari, G.

Luis, A.

Mandel, L.

L. Mandel and E. Wolf, “Second-order coherence theory of scalar wavefields,” in Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 160-170.

Martinez-Herrero, R.

Mejías, P. M.

Morio, J.

Piquero, G.

Réfrégier, P.

Réfrégier, Ph.

Renyi, A.

A. Renyi, “On the measures of entropy and information,” in Proceedings of the 4th Berkeley Symposium on Mathematics and Staticstical Probability (University of California Press, 1961), Vol. 1, pp. 547-561.

Roueff, A.

Santarsiero, M.

Setälä, T.

Shirai, T.

Simon, R.

Tervo, J.

Thomas, J. A.

T. M. Cover and J. A. Thomas, “Entropy, relative entropy and mutual information,” in Elements of Information Theory (Wiley-Interscience, 1991), pp. 18-26.

Visser, T. D.

Wolf, E.

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

R. Barakat, “N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171-1182 (1983).
[CrossRef]

Opt. Express (2)

Opt. Lett. (12)

Ph. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117-3119 (2005).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effect of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-670 (2006).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young's interference experiment,” Opt. Lett. 31, 2208-2210 (2006).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669-2671 (2006).
[CrossRef] [PubMed]

G. S. Agarwal, A. Dogariu, T. D. Visser, and E. Wolf, “Generation of complete coherence in Young's interference experiment with random mutually uncorrelated electromagnetic beams,” Opt. Lett. 30, 120-122 (2005).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328-330 (2004).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young's fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588-590 (2007).
[CrossRef] [PubMed]

Ph. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366-1368 (2007).
[CrossRef] [PubMed]

R. Martinez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471-1473 (2007).
[CrossRef] [PubMed]

R. Martinez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504-1506 (2007).
[CrossRef] [PubMed]

A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191-2193 (2007).
[CrossRef] [PubMed]

P. Réfrégier, “Irreversible effects of random modulation on coherence properties of partially polarized light,” Opt. Lett. 33, 636-638 (2008).
[CrossRef] [PubMed]

Phys. Lett. A (1)

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

Other (5)

J. W. Goodman, “Some first-order properties of light waves,” in Statistical Optics (Wiley, 1985), pp. 116-156.

L. Mandel and E. Wolf, “Second-order coherence theory of scalar wavefields,” in Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 160-170.

A. Renyi, “On the measures of entropy and information,” in Proceedings of the 4th Berkeley Symposium on Mathematics and Staticstical Probability (University of California Press, 1961), Vol. 1, pp. 547-561.

T. M. Cover and J. A. Thomas, “Entropy, relative entropy and mutual information,” in Elements of Information Theory (Wiley-Interscience, 1991), pp. 18-26.

C. Brosseau, “Polarization and the radiation field,” in Fundamentals of Polarized Light--A Statistical Approach (Wiley, 1998), pp. 67-176.

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Figures (4)

Fig. 1
Fig. 1

Case of correlated rotations. Intrinsic degrees of coherence (DOC) as a function of the variance σ 2 of the polarization rotation angle for different values of ρ = I E , X I E , Y . For a fixed value of ρ, the largest values correspond to μ S and the smallest to μ I . The curves are drawn for μ E , S ( τ ) = 0.9 and μ E , I ( τ ) = 0.3 : solid curves, ρ = 1 ; long-dashed curves, ρ = 10 ; short-dashed curves, ρ = 0.1 .

Fig. 2
Fig. 2

Case of correlated rotations. Partial entropy δ S (see text for details) as a function of σ 2 for different values of ρ = I E , X I E , Y . The three curves for the three values ρ = 1 , ρ = 10 , and ρ = 0.1 are superposed.

Fig. 3
Fig. 3

Case of uncorrelated rotations. Intrinsic degrees of coherence (DOC) as a function of the variance σ 2 of the polarization rotation angle for different values of ρ = I E , X I E , Y . For a fixed value of ρ, the largest values correspond to μ S and the smallest to μ I . The curves are drawn for μ E , S ( τ ) = 0.9 and μ E , I ( τ ) = 0.3 : solid curves, ρ = 1 ; long-dashed curves, ρ = 10 ; short-dashed curves, ρ = 0.1 .

Fig. 4
Fig. 4

Case of uncorrelated rotations. Partial entropy δ S (see text for details) as a function of σ 2 for different values of ρ = I E , X I E , Y . The three curves for the three values ρ = 1 , ρ = 10 , and ρ = 0.1 are superposed.

Equations (58)

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Ω ( x 1 , x 2 ) = E ( x 2 ) E ( x 1 ) = [ E X ( x 2 ) E X * ( x 1 ) E X ( x 2 ) E Y * ( x 1 ) E Y ( x 2 ) E X * ( x 1 ) E Y ( x 2 ) E Y * ( x 1 ) ] ,
Υ ( x 1 , x 2 ) = [ Γ ( x 1 ) Ω ( x 1 , x 2 ) Ω ( x 1 , x 2 ) Γ ( x 2 ) ] .
S [ P E , x 1 , x 2 ] = P E , x 1 , x 2 ( E ) ln [ P E , x 1 , x 2 ( E ) ] d E ,
S α [ P ] = 1 1 α ln { [ P ( E ) ] α d E } .
P E ( E ) = 1 π 4 det [ Υ ] exp [ E Υ 1 E ] ,
S [ P E ] = 2 ln [ π e 2 I 1 ] + 2 ln [ π e 2 I 2 ] + ln [ 1 P 1 2 ] + ln [ 1 P 2 2 ] + ln [ 1 μ S 2 ] + ln [ 1 μ I 2 ] ,
M ( x 1 , x 2 ) = Γ 1 2 ( x 2 ) Ω ( x 1 , x 2 ) Γ 1 2 ( x 1 ) .
A = U E .
S α [ P A ] S α [ P E ] ,
Υ A = [ A A ] P A ( A ) d A = [ A A ] P A U ( A U ) P U ( U ) d U d A .
Υ A = U [ E E ] U P E ( E ) P U ( U ) d U d E ,
Υ A = U Υ E U P U ( U ) d U .
det [ Υ A ] det [ Υ E ] .
i = 1 2 { 2 ln [ π e 2 I A , i ] + ln [ 1 P A , i 2 ] } + l = S , I ln [ 1 μ A , l 2 ] i = 1 2 { 2 ln [ π e 2 I E , i ] + ln [ 1 P E , i 2 ] } + l = S , I ln [ 1 μ E , l 2 ] ,
B E 2 = 1 4 4 det ( Υ E ) [ tr ( Υ E ) ] 4 ,
B A B E .
γ E 2 = 4 3 tr [ ( 1 tr Υ E Υ E 1 4 I ) 2 ] = 4 3 [ tr ( Υ E 2 ) ( tr Υ E ) 2 1 4 ] .
γ A γ E .
i = 1 2 ln [ 1 P A , i 2 ] + l = S I ln [ 1 μ A , l 2 ]
i = 1 2 ln [ 1 P E , i 2 ] + l = S I ln [ 1 μ E , l 2 ] .
P A , i P E , i .
μ A , S μ E , S .
μ TSF 2 = tr [ Ω ( x 1 , x 2 ) Ω ( x 1 , x 2 ) ] tr [ Γ ( x 1 ) ] tr [ Γ ( x 2 ) ] .
μ ̃ = Max V 1 V 2 { tr [ V 2 Ω ( x 1 , x 2 ) V 1 ] tr [ Γ ( x 1 ) ] tr [ Γ ( x 2 ) ] } ,
U 1 = U 2 = ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) ,
P ϴ ( θ ) = 1 2 π σ exp ( θ 2 2 σ 2 ) .
Γ E = ( I E , X 0 0 I E , Y ) ,
Ω E ( τ ) = ( I E , X μ E , S ( τ ) 0 0 I E , Y μ E , I ( τ ) ) ,
Ω A ( τ ) = ( I A , X μ A , S ( τ ) 0 0 I A , Y μ A , I ( τ ) )
I A , X = e σ 2 [ I E , X cosh ( σ 2 ) + I E , Y sinh ( σ 2 ) ]
I A , Y = e σ 2 [ I E , Y cosh ( σ 2 ) + I E , X sinh ( σ 2 ) ] ,
μ A , S ( τ ) = I E , X cosh ( σ 2 ) μ E , S ( τ ) + I E , Y sinh ( σ 2 ) μ E , I ( τ ) I E , X cosh ( σ 2 ) + I E , Y sinh ( σ 2 ) ,
μ A , I ( τ ) = I E , X sinh ( σ 2 ) μ E , S ( τ ) + I E , Y cosh ( σ 2 ) μ E , I ( τ ) I E , X sinh ( σ 2 ) + I E , Y cosh ( σ 2 ) .
μ A , S ( τ ) = μ A , I ( τ ) = I E , X μ E , S ( τ ) + I E , Y μ E , I ( τ ) I E , X + I E , Y ,
P E = I E , X I E , Y I E , X + I E , Y .
P A = e 2 σ 2 P E ,
μ ̃ A = I E , X μ E , S ( τ ) + I E , Y μ E , I ( τ ) I E , X + I E , Y = μ ̃ E ,
Q E = I E , X μ E , S ( τ ) I E , Y μ E , I ( τ ) I E , X μ E , S ( τ ) + I E , Y μ E , I ( τ ) .
γ A 2 = e 4 σ 2 3 [ P E 2 + μ ̃ E 2 Q E 2 ] + μ ̃ E 2 3 ,
γ A 2 = 1 3 P A 2 + 2 3 μ TSF , A 2 ,
μ TSF , A = μ E ( τ ) 2 1 + P E 2 e 4 σ 2 = μ E ( τ ) 2 1 + P A 2 ,
δ S = i = 1 2 ln [ 1 P A , i 2 ] + l = S I ln [ 1 μ A , l 2 ] ,
Γ A = ( I A , X 0 0 I A , Y ) ,
μ A , S ( τ ) = I E , X μ E , S ( τ ) I E , X cosh ( σ 2 ) + I E , Y sinh ( σ 2 ) ,
μ A , I ( τ ) = I E , Y μ E , I ( τ ) I E , X sinh ( σ 2 ) + I E , Y cosh ( σ 2 ) .
μ TSF , A = e σ 2 I E , X 2 μ E , S 2 ( τ ) + I E , Y 2 μ E , I 2 ( τ ) I E , X + I E , Y .
μ ̃ A = e σ 2 I E , X μ E , S ( τ ) + I E , Y μ E , I ( τ ) I E , X + I E , Y ,
[ P E ( U A ) P U ( U ) d U ] α d A [ P E ( U A ) ] α P U ( U ) d U d A ,
S 1 [ P A ] = S [ P A ] = H [ P E ( U A ) P U ( U ) d U ] d A
H [ P E ( U A ) P U ( U ) d U ] H [ P E ( U A ) ] P U ( U ) d U ;
ln { det [ α Υ 1 + ( 1 α ) Υ 2 ] } α ln { det [ Υ 1 ] } + ( 1 α ) ln { det [ Υ 2 ] } .
ln { det [ α Υ 1 + ( 1 α ) Υ 2 ] } = ln { det [ Υ 1 ] } + ln { det [ Δ α ] } .
tr [ Ω A Ω A ] = tr [ U 2 Ω E U 1 U 1 Ω E U 2 ] P ( U 1 , U 2 ) P ( U 1 , U 2 ) d U 1 d U 2 d U 1 d U 2 ,
tr [ Ω A Ω A ] = tr [ U 2 N 2 D N 1 U 1 U 1 N 1 D N 2 U 2 ] ,
tr [ V 2 D V 1 D ] 2 ( tr [ D 2 ] ) 2 .
μ TSF , A 2 = tr [ Ω A Ω A ] tr [ Γ A , 1 ] tr [ Γ A , 2 ] tr [ Ω E Ω E ] tr [ Γ E , 1 ] tr [ Γ E , 2 ] = μ TSF , E 2 .
μ ̃ = Max V 1 V 2 { tr [ V 2 Ω V 1 ] tr [ Γ 1 ] tr [ Γ 2 ] } ,
B = ( α β γ δ ) , D = ( λ E , 1 0 0 λ E , 2 )

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