Abstract

Different properties of partially polarized light are discussed using the Kullback relative entropy, which provides a physically meaningful measure of proximity between probability density functions (PDFs). For optical waves with a Gaussian PDF, the standard degree of polarization is a simple function of the Kullback relative entropy between the considered optical light and a totally depolarized light of the same intensity. It is shown that the Kullback relative entropies between different PDFs allow one to define other properties such as a degree of anisotropy and a degree of non-Gaussianity. It is also demonstrated that, in dimension three, the Kullback relative entropy between a partially polarized light and a totally depolarized light can lead to natural definitions of two degrees of polarization needed to characterize the polarization state. These analyses enlighten the physical meaning of partial polarization of light waves in terms of a measure of disorder provided by the Shannon entropy.

© 2006 Optical Society of America

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  1. M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, 'Polarimetric considerations to optimize lidar detection of immersed targets,' Pure Appl. Opt. 7, 1327-1340 (1998).
    [CrossRef]
  2. S. Breugnot and Ph. Clémenceau, 'Modeling and performances of a polarization active imager at lambda=806 nm,' in Laser Radar Technology and Applications IV, G.W.Kamerman and C.Werner, ed., Proc. SPIE 3707, 449-460 (1999).
  3. A. Gleckler, 'Multiple-slit streak tube imaging lidar (MS-STIL) applications,' in Laser Radar Technology and Applications V, G.W.Kamerman, U.N.Singh, C.H.Werner, and V.V.Molebny, eds., Proc. SPIE 4035, 266-278 (2000).
  4. J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, 'Target detection in optical scattering media by polarization-difference imaging,' Appl. Opt. 35, 1855-1870 (1996).
    [CrossRef] [PubMed]
  5. T. Setälä, M. Kaivola, and A. T. Friberg, 'Degree of polarization in near fields of thermal sources: Effects of surface waves,' Phys. Rev. Lett. 88, 123902 (2002).
    [CrossRef] [PubMed]
  6. 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]
  7. A. Picozzi, 'Entropy and degree of polarization for nonlinear optical waves,' Opt. Lett. 29, 1653-1655 (2004).
    [CrossRef] [PubMed]
  8. Ph. Réfrégier, 'Polarization degree of optical waves with non-Gaussian probability density functions: Kullback relative entropy-based approach,' Opt. Lett. 30, 1090-1092 (2005).
    [CrossRef] [PubMed]
  9. C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).
  10. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
    [CrossRef]
  11. Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004).
  12. J. W. Goodman, 'Some problems involving high-order coherence,' in Statistical Optics (Wiley, 1985), pp. 237-285.
  13. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).
  14. J. W. Goodman, 'Some first-order properties of light waves,' in Statistical Optics (Wiley, 1985), pp. 116-156.
  15. Ph. Réfrégier, 'Fluctuations and covariance,' in Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004), pp. 28-32.
  16. T. M. Cover and J. A. Thomas, 'Information theory and statistics,' in Elements of Information Theory (Wiley, 1991), pp. 279-335.
    [CrossRef]
  17. T. M. Cover and J. A. Thomas, 'Maximum entropy and spectral estimation,' in Elements of Information Theory (Wiley, 1991), pp. 266-278.
    [CrossRef]
  18. S. Huard, 'Propagation of states of polarization in optical devices,' in Polarization of Light (Wiley, 1997), pp. 86-130.
  19. R. D. Richtmyer, 'Group representations II,' in Principles of Advanced Mathematical Physics (Springer-Verlag, 1985), Vol. 2, pp. 67-71.
  20. R. S. Cloude and E. Pottier, 'Concept of polarization entropy in optical scattering,' Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
    [CrossRef]
  21. J. C. Samson, 'Descriptions of the polarization states of vector processes: Applications to ULF magnetic fields,' Geophys. J. R. Astron. Soc. 34, 403-419 (1973).
  22. R. Barakat, 'N-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,' Opt. Acta 30, 1171-1182 (1983).
    [CrossRef]
  23. R. D. Richtmyer, Principles of Advanced Mathematical Physics (Springer-Verlag, 1978), Vol. 1.

2005 (1)

2004 (2)

2002 (1)

T. Setälä, M. Kaivola, and A. T. Friberg, 'Degree of polarization in near fields of thermal sources: Effects of surface waves,' Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

1998 (1)

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, 'Polarimetric considerations to optimize lidar detection of immersed targets,' Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

1996 (1)

1995 (1)

R. S. Cloude and E. Pottier, 'Concept of polarization entropy in optical scattering,' Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
[CrossRef]

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]

1973 (1)

J. C. Samson, 'Descriptions of the polarization states of vector processes: Applications to ULF magnetic fields,' Geophys. J. R. Astron. Soc. 34, 403-419 (1973).

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]

Breugnot, S.

S. Breugnot and Ph. Clémenceau, 'Modeling and performances of a polarization active imager at lambda=806 nm,' in Laser Radar Technology and Applications IV, G.W.Kamerman and C.Werner, ed., Proc. SPIE 3707, 449-460 (1999).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).

Cariou, J.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, 'Polarimetric considerations to optimize lidar detection of immersed targets,' Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Chavel, P.

Clémenceau, Ph.

S. Breugnot and Ph. Clémenceau, 'Modeling and performances of a polarization active imager at lambda=806 nm,' in Laser Radar Technology and Applications IV, G.W.Kamerman and C.Werner, ed., Proc. SPIE 3707, 449-460 (1999).

Cloude, R. S.

R. S. Cloude and E. Pottier, 'Concept of polarization entropy in optical scattering,' Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
[CrossRef]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

T. M. Cover and J. A. Thomas, 'Information theory and statistics,' in Elements of Information Theory (Wiley, 1991), pp. 279-335.
[CrossRef]

T. M. Cover and J. A. Thomas, 'Maximum entropy and spectral estimation,' in Elements of Information Theory (Wiley, 1991), pp. 266-278.
[CrossRef]

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

Engheta, N.

Floc'h, M.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, 'Polarimetric considerations to optimize lidar detection of immersed targets,' Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Friberg, A.

Friberg, A. T.

T. Setälä, M. Kaivola, and A. T. Friberg, 'Degree of polarization in near fields of thermal sources: Effects of surface waves,' Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Gleckler, A.

A. Gleckler, 'Multiple-slit streak tube imaging lidar (MS-STIL) applications,' in Laser Radar Technology and Applications V, G.W.Kamerman, U.N.Singh, C.H.Werner, and V.V.Molebny, eds., Proc. SPIE 4035, 266-278 (2000).

Goodman, J. W.

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

J. W. Goodman, 'Some problems involving high-order coherence,' in Statistical Optics (Wiley, 1985), pp. 237-285.

Goudail, F.

Huard, S.

S. Huard, 'Propagation of states of polarization in optical devices,' in Polarization of Light (Wiley, 1997), pp. 86-130.

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, 'Degree of polarization in near fields of thermal sources: Effects of surface waves,' Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Kieleck, C.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, 'Polarimetric considerations to optimize lidar detection of immersed targets,' Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Le Brun, G.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, 'Polarimetric considerations to optimize lidar detection of immersed targets,' Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Lotrian, J.

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, 'Polarimetric considerations to optimize lidar detection of immersed targets,' Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Picozzi, A.

Pottier, E.

R. S. Cloude and E. Pottier, 'Concept of polarization entropy in optical scattering,' Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
[CrossRef]

Pugh, E. N.

Réfrégier, Ph.

Richtmyer, R. D.

R. D. Richtmyer, 'Group representations II,' in Principles of Advanced Mathematical Physics (Springer-Verlag, 1985), Vol. 2, pp. 67-71.

R. D. Richtmyer, Principles of Advanced Mathematical Physics (Springer-Verlag, 1978), Vol. 1.

Rowe, M. P.

Samson, J. C.

J. C. Samson, 'Descriptions of the polarization states of vector processes: Applications to ULF magnetic fields,' Geophys. J. R. Astron. Soc. 34, 403-419 (1973).

Setälä, T.

T. Setälä, M. Kaivola, and A. T. Friberg, 'Degree of polarization in near fields of thermal sources: Effects of surface waves,' Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Thomas, J. A.

T. M. Cover and J. A. Thomas, 'Maximum entropy and spectral estimation,' in Elements of Information Theory (Wiley, 1991), pp. 266-278.
[CrossRef]

T. M. Cover and J. A. Thomas, 'Information theory and statistics,' in Elements of Information Theory (Wiley, 1991), pp. 279-335.
[CrossRef]

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Tyo, J. S.

Appl. Opt. (1)

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

J. C. Samson, 'Descriptions of the polarization states of vector processes: Applications to ULF magnetic fields,' Geophys. J. R. Astron. Soc. 34, 403-419 (1973).

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

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. Eng. (Bellingham) (1)

R. S. Cloude and E. Pottier, 'Concept of polarization entropy in optical scattering,' Opt. Eng. (Bellingham) 34, 1599-1610 (1995).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

T. Setälä, M. Kaivola, and A. T. Friberg, 'Degree of polarization in near fields of thermal sources: Effects of surface waves,' Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

M. Floc'h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, 'Polarimetric considerations to optimize lidar detection of immersed targets,' Pure Appl. Opt. 7, 1327-1340 (1998).
[CrossRef]

Other (14)

S. Breugnot and Ph. Clémenceau, 'Modeling and performances of a polarization active imager at lambda=806 nm,' in Laser Radar Technology and Applications IV, G.W.Kamerman and C.Werner, ed., Proc. SPIE 3707, 449-460 (1999).

A. Gleckler, 'Multiple-slit streak tube imaging lidar (MS-STIL) applications,' in Laser Radar Technology and Applications V, G.W.Kamerman, U.N.Singh, C.H.Werner, and V.V.Molebny, eds., Proc. SPIE 4035, 266-278 (2000).

R. D. Richtmyer, Principles of Advanced Mathematical Physics (Springer-Verlag, 1978), Vol. 1.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

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

J. W. Goodman, 'Some problems involving high-order coherence,' in Statistical Optics (Wiley, 1985), pp. 237-285.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

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

Ph. Réfrégier, 'Fluctuations and covariance,' in Noise Theory and Application to Physics: From Fluctuations to Information (Springer, 2004), pp. 28-32.

T. M. Cover and J. A. Thomas, 'Information theory and statistics,' in Elements of Information Theory (Wiley, 1991), pp. 279-335.
[CrossRef]

T. M. Cover and J. A. Thomas, 'Maximum entropy and spectral estimation,' in Elements of Information Theory (Wiley, 1991), pp. 266-278.
[CrossRef]

S. Huard, 'Propagation of states of polarization in optical devices,' in Polarization of Light (Wiley, 1997), pp. 86-130.

R. D. Richtmyer, 'Group representations II,' in Principles of Advanced Mathematical Physics (Springer-Verlag, 1985), Vol. 2, pp. 67-71.

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Figures (4)

Fig. 1
Fig. 1

Schematic illustrations of the consequences of (a) property A and (b) property B.

Fig. 2
Fig. 2

Illustration of the particular Poincaré sphere where the radius corresponds to the intensity.

Fig. 3
Fig. 3

Illustration of the effect of R ω ¯ and of the definition of P a iso . One considers a particular Poincaré sphere where the radius corresponds to the intensity. (a) The cloud of points is a schematic representation of the PDF of the optical electric field. (b) and (c) PDFs obtained after the action of two different rotations R ω ¯ 1 and R ω ¯ 2 . (d) A schematic representation of the PDF P a iso .

Fig. 4
Fig. 4

Schematic illustrations of the consequences of (a) property D and (b) property E.

Tables (2)

Tables Icon

Table 1 Definition of Different Degrees

Tables Icon

Table 2 Values of the Different Degrees for Some Examples of Polarization Statistics

Equations (47)

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Γ = [ E 1 2 E 1 E 2 * E 1 E 1 * E 2 2 ] = [ μ 1 ρ ρ * μ 2 ] ,
P 2 = 1 4 det ( Γ ) tr ( Γ ) 2 ,
P Γ ( E ) = 1 π 2 det ( Γ ) exp ( E Γ 1 E ) ,
K ( P a P b ) = P a ( E ) log [ P a ( E ) P b ( E ) ] d E ,
P a 2 = 1 exp [ K ( P Γ a P Γ 0 ) ] ,
Π N ( P Γ a P Γ 0 ) ( 1 P a 2 ) N .
( P a N L ) 2 = 1 exp [ K ( P a P Γ 0 ) ] .
K ( P a P Γ ) = K ( P a P Γ a ) + K ( P Γ a P Γ ) .
K ( P a P Γ 0 ) = K ( P a P Γ a ) + K ( P Γ a P Γ 0 ) .
S ( P ) = P ( E ) log [ P ( E ) ] d E .
K ( P a P Γ a ) = S ( P Γ a ) S ( P a ) ,
K ( P a P Γ 0 ) = S ( P Γ 0 ) S ( P a ) ,
K ( P Γ a P Γ 0 ) = S ( P Γ 0 ) S ( P Γ a ) ,
S ( P a ) S ( P Γ a ) S ( P Γ 0 ) .
( N a G ) 2 = 1 β N L 2 = 1 exp [ K ( P a P Γ a ) ] .
( A a ) 2 = 1 exp [ K ( P a P a iso ) ] ,
P a iso ( E ) = 1 A S S P a , ω ¯ ( E ) d ω ¯ ,
K ( P a P Γ 0 ) = K ( P a P a iso ) + K ( P a iso P Γ 0 ) .
K ( P a P a iso ) = S ( P a iso ) S ( P a ) .
S ( P a ) S ( P a iso ) S ( P Γ 0 ) .
S ( P Γ a ) = 3 log ( π e ) + log [ det ( Γ a ) ] ,
K ( P Γ a P Γ 0 ) = log ( I 0 3 27 ) log ( λ 1 λ 2 λ 3 ) .
P a 2 = 1 27 λ 1 λ 2 λ 3 ( λ 1 + λ 2 + λ 3 ) 3 .
S ( P Γ a R k ) = 3 log ( π e ) + log [ ( λ i + λ j ) 2 4 λ k ] .
S ( P Γ 0 ) S ( P Γ a R k ) = log [ ( λ i + λ j + λ k ) 3 27 ] log [ ( λ i + λ j ) 2 4 λ k ] ,
( P a R k ) 2 = 1 exp [ S ( P Γ a R k ) S ( P Γ 0 ) ] = 1 27 4 ( λ i + λ j ) 2 λ k ( λ i + λ j + λ k ) 3 .
K ( P Γ a P Γ a R k ) = S ( P Γ a R k ) S ( P Γ a ) .
K ( P Γ a P Γ 0 ) = S ( P Γ 0 ) S ( P Γ a ) ,
K ( P Γ a P Γ 0 ) = S ( P Γ 0 ) S ( P Γ a R k ) + S ( P Γ a R k ) S ( P Γ a ) ,
K ( P Γ a P Γ 0 ) = K ( P Γ a P Γ a R k ) + K ( P Γ a R k P Γ 0 ) .
K ( P Γ a P Γ a R k ) = log [ ( λ i + λ j ) 2 λ k 4 ] log ( λ i λ j λ k ) = log [ ( λ i + λ j ) 2 4 λ i λ j ] .
( P a i , j ) 2 = 1 4 λ i λ j ( λ i + λ j ) 2 .
( P a i , j ) 2 = 1 exp [ K ( P Γ a P Γ a R k ) ] .
1 ( P a ) 2 = [ 1 ( P a R k ) 2 ] [ 1 ( P a i , j ) 2 ] .
( P a R 3 ) 2 = 1 27 4 μ 3 ( 1 μ 3 ) 2 ,
K ( P a P Γ 0 ) = P a ( E ) { log [ P a ( E ) P a iso ( E ) ] + log [ P a iso ( E ) P Γ 0 ( E ) ] } d E .
P a ( E ) log [ P a iso ( E ) P Γ 0 ( E ) ] d E = P a iso ( E ) log [ P a iso ( E ) P Γ 0 ( E ) ] d E ,
P a ( I , ψ , χ , φ ) = 1 2 δ ( I I 0 ) [ δ ( χ ) + δ ( χ π 2 ) ] δ ( ψ ) u ( φ ) ,
f σ ( x ) = 1 2 π σ exp ( x 2 2 σ 2 ) .
P a σ ¯ ( I , ψ , χ , φ ) = f σ 1 ( I I 0 ) 1 2 [ f σ 2 ( χ ) + f σ 2 ( χ π 2 ) ] f σ 3 ( ψ ) u ( φ )
S ( p ) = p ( χ ) log p ( χ ) d χ ,
1 2 f σ ( χ ) log f σ ( χ ) d χ 1 2 f α ( χ π 2 ) log f σ ( χ π 2 ) d χ ,
f σ ( χ ) log f σ ( χ ) d χ = S ( f σ ) .
P b ( I , ψ , χ , φ ) = δ ( I I 0 ) g ( ψ , χ , φ ) ,
K ( P Γ a P Γ a R k ) = P Γ a ( E ) log [ P Γ a ( E ) P Γ a R k ( E ) ] d E ,
K ( P Γ a P Γ a R k ) = P Γ a ( E ) log [ P Γ a ( E ) ] d E P Γ a ( E ) log [ P Γ a R k ( E ) ] d E .
P Γ a ( E ) log [ P Γ a R k ( E ) ] d E = 3 log ( π e ) + log [ det ( Γ a R k ) ] ,

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