Abstract

In a previous publication [ J. Opt. Soc. Am. A 21, 988 ( 2004)], we examined theoretically joint probability distributions of Stokes vector elements and suggested the existence of various types of globally unpolarized light that could be discriminated through measurement of the Stokes vector element correlations. We now study the joint distribution of the degree of polarization and the three Stokes parameters as it relates to material properties in highly scattering, depolarizing random media. We describe numerical and experimental results of second-order Stokes vector element correlations, demonstrating the existence of various types of nonclassical, globally unpolarized light, and we suggest experimental means for discriminating between such field distributions. We also discuss the usefulness of the Stokes vector element correlations as an experimental tool for discriminating between different globally unpolarized fields and for verifying the assumption of Gaussian statistics usually invoked in the context of multiple light scattering.

© 2005 Optical Society of America

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References

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  1. B. Ruffing, J. Fleischer, “Spectral correlation of partially or fully developed speckle patterns generated by rough surfaces,” J. Opt. Soc. Am. A 2, 1637–1643 (1985).
    [CrossRef]
  2. J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
    [CrossRef]
  3. E. M. Ortiz, F. Gonzales, F. Moreno, “Intensity statistics of the light scattered from particulate surfaces: interacting particles and non-Gaussian effects,” Opt. Commun. 181, 231–238 (2000).
    [CrossRef]
  4. E. Jakeman, “Polarization characteristics of non-Gaussian scattering by small particles,” Waves Random Media 5, 427–442 (1995).
    [CrossRef]
  5. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.
  6. J. Uozumi, K. Uno, T. Asakura, “Statistics of Gaussian speckles with enhanced fluctuations,” Opt. Rev. 2, 174–180 (1995).
    [CrossRef]
  7. J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
    [CrossRef]
  8. J. Ellis, A. Dogariu, “Differentiation of globally unpolarized complex random fields,” J. Opt. Soc. Am. A 21, 988–993 (2004).
    [CrossRef]
  9. J. Lehner, U. Leonhardt, H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
    [CrossRef] [PubMed]
  10. A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
    [CrossRef]
  11. E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).
  12. The measurement of scattering mean free path l* was performed by using optical path-length spectroscopy.See G. Popescu, A. Dogariu, “Optical path-length spectroscopy of wave propagation in random media,” Opt. Lett. 24, 442–444 (1999).
    [CrossRef]
  13. E. Collett, Polarized Light Fundamentals and Applications (Marcel Dekker, New York, 1993).

2004 (1)

2002 (1)

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

2000 (1)

E. M. Ortiz, F. Gonzales, F. Moreno, “Intensity statistics of the light scattered from particulate surfaces: interacting particles and non-Gaussian effects,” Opt. Commun. 181, 231–238 (2000).
[CrossRef]

1999 (1)

1997 (1)

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

1996 (1)

J. Lehner, U. Leonhardt, H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[CrossRef] [PubMed]

1995 (2)

E. Jakeman, “Polarization characteristics of non-Gaussian scattering by small particles,” Waves Random Media 5, 427–442 (1995).
[CrossRef]

J. Uozumi, K. Uno, T. Asakura, “Statistics of Gaussian speckles with enhanced fluctuations,” Opt. Rev. 2, 174–180 (1995).
[CrossRef]

1985 (1)

1978 (1)

J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[CrossRef]

Agarwal, G. S.

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

Asakura, T.

J. Uozumi, K. Uno, T. Asakura, “Statistics of Gaussian speckles with enhanced fluctuations,” Opt. Rev. 2, 174–180 (1995).
[CrossRef]

J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[CrossRef]

Collett, E.

E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).

E. Collett, Polarized Light Fundamentals and Applications (Marcel Dekker, New York, 1993).

Dogariu, A.

Ellis, J.

Fleischer, J.

Gonzales, F.

E. M. Ortiz, F. Gonzales, F. Moreno, “Intensity statistics of the light scattered from particulate surfaces: interacting particles and non-Gaussian effects,” Opt. Commun. 181, 231–238 (2000).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.

Jakeman, E.

E. Jakeman, “Polarization characteristics of non-Gaussian scattering by small particles,” Waves Random Media 5, 427–442 (1995).
[CrossRef]

Lehner, J.

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

J. Lehner, U. Leonhardt, H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[CrossRef] [PubMed]

Leonhardt, U.

J. Lehner, U. Leonhardt, H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[CrossRef] [PubMed]

Luis, A.

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

Moreno, F.

E. M. Ortiz, F. Gonzales, F. Moreno, “Intensity statistics of the light scattered from particulate surfaces: interacting particles and non-Gaussian effects,” Opt. Commun. 181, 231–238 (2000).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[CrossRef]

Ortiz, E. M.

E. M. Ortiz, F. Gonzales, F. Moreno, “Intensity statistics of the light scattered from particulate surfaces: interacting particles and non-Gaussian effects,” Opt. Commun. 181, 231–238 (2000).
[CrossRef]

Paul, H.

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

J. Lehner, U. Leonhardt, H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[CrossRef] [PubMed]

Popescu, G.

Ruffing, B.

Uno, K.

J. Uozumi, K. Uno, T. Asakura, “Statistics of Gaussian speckles with enhanced fluctuations,” Opt. Rev. 2, 174–180 (1995).
[CrossRef]

Uozumi, J.

J. Uozumi, K. Uno, T. Asakura, “Statistics of Gaussian speckles with enhanced fluctuations,” Opt. Rev. 2, 174–180 (1995).
[CrossRef]

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

Opt. Commun. (3)

J. Ohtsubo, T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315–319 (1978).
[CrossRef]

E. M. Ortiz, F. Gonzales, F. Moreno, “Intensity statistics of the light scattered from particulate surfaces: interacting particles and non-Gaussian effects,” Opt. Commun. 181, 231–238 (2000).
[CrossRef]

J. Lehner, H. Paul, G. S. Agarwal, “Generation and physical properties of a new form of unpolarized light,” Opt. Commun. 139, 262–269 (1997).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

J. Uozumi, K. Uno, T. Asakura, “Statistics of Gaussian speckles with enhanced fluctuations,” Opt. Rev. 2, 174–180 (1995).
[CrossRef]

Phys. Rev. A (2)

J. Lehner, U. Leonhardt, H. Paul, “Unpolarized light: classical and quantum states,” Phys. Rev. A 53, 2727–2735 (1996).
[CrossRef] [PubMed]

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

Waves Random Media (1)

E. Jakeman, “Polarization characteristics of non-Gaussian scattering by small particles,” Waves Random Media 5, 427–442 (1995).
[CrossRef]

Other (3)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.

E. Collett, Polarized Light in Fiber Optics (PolaWave Group, Lincroft, N.J., 2003).

E. Collett, Polarized Light Fundamentals and Applications (Marcel Dekker, New York, 1993).

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Figures (8)

Fig. 1
Fig. 1

Globally unpolarized light; individual realizations generated from Gaussian-distributed underlying complex fields. The distribution is uniformly covering the OPS and, as such, is the only distribution completely independent of the reference frame and the introduction of an arbitrary retardance.

Fig. 2
Fig. 2

Globally unpolarized light; individual realizations generated from a specific non-Gaussian distribution of underlying complex fields. This is a theoretical example of type II unpolarized light. The invariance with regard to the reference frame is seen in the rotational symmetry about the S 3 axis.

Fig. 3
Fig. 3

Globally unpolarized light; individual realizations generated from a specific non-Gaussian distribution of underlying complex fields. This is an example of type III unpolarized light, which is not invariant to a change in the reference frame.

Fig. 4
Fig. 4

Results of numerical simulations (see the text for details). Solid symbols, particles with uniform orientation in three dimensions; open symbols, particles uniformly oriented in a plane. The deviation of the autocorrelations of Stokes vectors from the value 1/3 are indications of an inherent coupling of fields produced by this “last scattering layer” of confined cylindrical particles. The size of the symbols is larger than the error between different numerical runs of the same number of particles.

Fig. 5
Fig. 5

The top row depicts four distributions of states of polarization incident on the first surface of the bulk diffuser. The spread of the distributions is equivalent to the uncertainty in the state of polarization across the incident beam. The second row indicates the corresponding measured distribution across the second surface of the diffuser. The distributions are roughly uniform and there is no dependence on the incident state of polarization, confirming the assumption of high multiple scattering and diffusionlike illumination of the last surface.

Fig. 6
Fig. 6

Distribution of states of polarization after coherent illumination of the bulk diffuser and a thin layer of compressed platelike particles with a mean size of 0.2 μm. Selectivity against circular polarization is clearly visible.

Fig. 7
Fig. 7

Distribution of states of polarization after coherent illumination of the bulk diffuser and a thin layer of compressed spherical particles. This distribution approaches a uniform distribution, as expected from numerical simulation and the isotropy of the particles.

Fig. 8
Fig. 8

Experimental results of Stokes vector element correlations. The average size of the particles in the surface layer is 0.2 μm. The cross correlations of Stokes vector elements, as well as the average Stokes vector elements, were negligible for all cases and are not shown here. The dashed line indicates the expected values for Gaussian-distributed fields, while the symbols are larger than the standard deviation between measurements of similar samples.

Equations (5)

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p ( E x ,   E x ,   E y ,   E y )
= 1 4 π σ 4 exp - ( E x ) 2 + ( E x ) 2 + ( E y ) 2 + ( E y ) 2 2 σ 2 ,
p ( 2 α ,   Δ ) = 1 2 π   sin ( 2 α ) 2 ,
p ( θ ,   ϕ ) = 1 2 π exp θ - π / 2 2 σ 2   1 erf [ ( π 2 ) / 4 σ ] 2 π σ 2 ,
p ( 2 α ,   Δ ) = 1 2 π   δ 2 α - π 2 ,

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