Abstract

We analyze an experimental setup in which a quasi-monochromatic spatially coherent beam of light is used to probe a paraxial optical scatterer. We discuss the effect of the spatial coherence of the probe beam on the Mueller matrix representing the scatterer. We show that, according to the degree of spatial coherence of the beam, the same scattering medium can be represented by different Mueller matrices. This result should serve as a warning for experimentalists.

© 2005 Optical Society of America

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References

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  1. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
  2. C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).
  3. A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).
    [CrossRef]
  4. A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).
    [CrossRef]
  5. E. Wolf, Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  6. Hereafter, we denote as “scattering medium” any assembly of passive, linear, optical devices, including both deterministic and random media.
  7. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).
    [CrossRef]
  8. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [CrossRef]
  9. For the physical meaning of W(rA,rB,?) and its relation with the 2×2 coherence matrix J used in elementary theory of partial polarization, see Ref. [5], p. 264.
  10. A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus,” arXiv.org e-print, archive, math-ph/0412061, December 17, 2004.
  11. The spectral density Stokes parameters were originally introduced by T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000). However, our definition is slightly different from theirs since we use the normalized Pauli matrices {?(?)} defined in Ref. [10].
    [CrossRef]
  12. O. Korotkova and E. Wolf, Opt. Lett. 30, 198 (2005). The two-point Stokes parameters introduced in our Letter differ slightly from the ones defined by Korotkova and Wolf. The difference is due to a normalization factor and to a different definition of matrix W.
    [CrossRef] [PubMed]
  13. M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, 1999).
    [CrossRef]
  14. K. Kim, L. Mandel, and E. Wolf, J. Opt. Soc. Am. A 4, 433 (1987).
    [CrossRef]
  15. J. J. Gil, J. Opt. Soc. Am. A 17, 328 (2000).
    [CrossRef]
  16. D. G.M. Anderson and R. Barakat, J. Opt. Soc. Am. A 11, 2305 (1994).
    [CrossRef]
  17. E. Wolf, Opt. Lett. 28, 1078 (2003).
    [CrossRef] [PubMed]

2005 (1)

2003 (2)

2000 (2)

The spectral density Stokes parameters were originally introduced by T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000). However, our definition is slightly different from theirs since we use the normalized Pauli matrices {?(?)} defined in Ref. [10].
[CrossRef]

J. J. Gil, J. Opt. Soc. Am. A 17, 328 (2000).
[CrossRef]

1998 (2)

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).
[CrossRef]

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).
[CrossRef]

1994 (1)

1987 (1)

1965 (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Anderson, D. G.M.

Barakat, R.

Bergman, J.

The spectral density Stokes parameters were originally introduced by T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000). However, our definition is slightly different from theirs since we use the normalized Pauli matrices {?(?)} defined in Ref. [10].
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, 1999).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

Carozzi, T.

The spectral density Stokes parameters were originally introduced by T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000). However, our definition is slightly different from theirs since we use the normalized Pauli matrices {?(?)} defined in Ref. [10].
[CrossRef]

Gil, J. J.

Karlsson, R.

The spectral density Stokes parameters were originally introduced by T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000). However, our definition is slightly different from theirs since we use the normalized Pauli matrices {?(?)} defined in Ref. [10].
[CrossRef]

Kim, K.

Kliger, D. S.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Korotkova, O.

Lewis, J. W.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Mallesh, K. S.

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).
[CrossRef]

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).
[CrossRef]

Mandel, L.

K. Kim, L. Mandel, and E. Wolf, J. Opt. Soc. Am. A 4, 433 (1987).
[CrossRef]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).
[CrossRef]

Randall, C. E.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

Rao, A. V.Gopala

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).
[CrossRef]

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).
[CrossRef]

Sudha,

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).
[CrossRef]

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).
[CrossRef]

Wolf, E.

J. Mod. Opt. (2)

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).
[CrossRef]

A. V.Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).
[CrossRef]

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

Opt. Lett. (2)

Phys. Lett. A (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Phys. Rev. E (1)

The spectral density Stokes parameters were originally introduced by T. Carozzi, R. Karlsson, and J. Bergman, Phys. Rev. E 61, 2024 (2000). However, our definition is slightly different from theirs since we use the normalized Pauli matrices {?(?)} defined in Ref. [10].
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Other (7)

For the physical meaning of W(rA,rB,?) and its relation with the 2×2 coherence matrix J used in elementary theory of partial polarization, see Ref. [5], p. 264.

A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus,” arXiv.org e-print, archive, math-ph/0412061, December 17, 2004.

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

Hereafter, we denote as “scattering medium” any assembly of passive, linear, optical devices, including both deterministic and random media.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, 1999).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Polarization tomography setup described in the text. The input ( z = z 0 ) and the output ( z = z 1 ) planes are indicated by vertical lines.

Equations (15)

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E i ( r 1 , ω ) = d 2 ρ 0 G i j ( r 1 , r 0 , ω ) E j ( r 0 , ω ) ,
W i j ( r A , r B , ω ) E i ( r A , ω ) E j * ( r B , ω ) ,
W ( r 1 , ω ) = d 2 ρ 0 d 2 ρ 0 { G ( r 1 , r 0 , ω ) × W ( r 0 , r 0 , ω ) G ( r 0 , r 1 , ω ) } ,
G l j ( r 0 , r 1 , ω ) = G j l * ( r 1 , r 0 , ω ) .
S α ( r 1 , ω ) = d 2 ρ 0 d 2 ρ 0 M α β ( r 1 , r 0 , r 0 , ω ) I β ( r 0 , r 0 , ω )
M α β ( r 1 , r 0 , r 0 , ω ) Tr { σ ( α ) G ( r 1 , r 0 , ω ) σ ( β ) G ( r 1 , r 0 , ω ) } ,
W k l ( r 0 , r 0 , ω ) = E k l w ( r 0 , r 0 , ω ) ,
w ( r 0 , r 0 , ω ) E ( r 0 , ω ) E * ( r 0 , ω )
S α ( r 1 , ω ) = d 2 ρ 0 d 2 ρ 0 w ( r 0 , r 0 , ω ) × M α β ( r 1 , r 0 , r 0 , ω ) S β ,
w ( r 0 , r 0 , ω ) = u ( r 0 , ω ) u * ( r 0 , ω ) ,
M α β ( C ) ( r 1 , ω ) Tr { σ ( α ) K ( r 1 , ω ) σ ( β ) K ( r 1 , ω ) } ,
w ( r 0 , r 0 , ω ) = w ( r 0 , ω ) δ ( 2 ) ( ρ 0 ρ 0 ) ,
M α β ( I ) ( r 1 , ω ) = d 2 ρ 0 w ( r 0 , ω ) M α β ( r 1 , r 0 , ω ) ,
M α β ( P ) ( r 1 , ω ) = d 2 ρ 0 d 2 ρ 0 w ( r 0 , r 0 , ω ) × M α β ( r 1 , r 0 , r 0 , ω ) ,
M ( n ) ( r 1 , ω ) = Tr { σ ( α ) K n ( r 1 , ω ) σ ( β ) K n ( r 1 , ω ) } ,

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