Abstract

Mueller matrices of dense aqueous suspensions of different concentrations are measured with a phase-modulated Mueller ellipsometer as a function of the scattering angle. Different concentrations of a solution containing 404-nm-diameter polystyrene latex spheres dispersed in water were prepared. Experimental results are compared with a three-dimensional Monte Carlo simulation of the propagation of photons with the cell geometry accounted for. The Fresnel laws and the Mie theory determine the changes in direction and polarization during the propagation of the photon. Excellent agreement over the whole angular range is found between experimental and simulated Mueller matrices.

© 2001 Optical Society of America

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References

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    [CrossRef]
  6. K. Turpin, J. G. Walker, P. C. Y. Chang, K. I. Hopcraft, B. Ablitt, E. Jakeman, “The influence of particle size in active polarization imaging in scattering media,” Opt. Commun. 168, 325–335 (1999).
    [CrossRef]
  7. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  8. T. W. Mullikin, “Multiple scattering of partially polarized light,” in Transport Theory, Vol. I of SIAM-AMS Proceedings (American Mathematical Society, Providence, R.I., 1969), pp. 3–16.
  9. E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
    [CrossRef]
  10. E. Compain, B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
    [CrossRef]
  11. E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
    [CrossRef]
  12. E. Compain, S. Poirier, B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters and Mueller matrix ellipsometers,” Appl. Opt. 38, 3490–3502 (1999).
    [CrossRef]
  13. G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).
  14. I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).
  15. H. C. Van de Hulst, Light scattering by Small Particles (Dover, New York, 1981).
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  17. W. H. Press, Numerical Recipies (Cambridge University, Cambridge, 1988).
  18. R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley, New York, 1981).
    [CrossRef]
  19. D. E. Knuth, The Art of Computer Programming. 2. Seminumerical Algorithms, 2nd ed. (Addison-Wesley, Reading, Mass., 1981).
  20. B. D. Ripley, “Computer generation of random variables: a tutorial,” Int. Statist. Rev. 51, 301–319 (1983).
    [CrossRef]
  21. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  22. T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
    [CrossRef]

2000 (1)

1999 (5)

1998 (3)

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

E. Compain, B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
[CrossRef]

1997 (1)

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

1983 (1)

B. D. Ripley, “Computer generation of random variables: a tutorial,” Int. Statist. Rev. 51, 301–319 (1983).
[CrossRef]

1978 (1)

Ablitt, B.

P. C. Y. Chang, J. G. Walker, K. I. Hopcraft, B. Ablitt, E. Jakeman, “Polarization discrimination for active imaging in scattering media,” Opt. Commun. 159, 1–6 (1999).
[CrossRef]

K. Turpin, J. G. Walker, P. C. Y. Chang, K. I. Hopcraft, B. Ablitt, E. Jakeman, “The influence of particle size in active polarization imaging in scattering media,” Opt. Commun. 168, 325–335 (1999).
[CrossRef]

Bartel, S.

Bell, G. I.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chang, P. C. Y.

P. C. Y. Chang, J. G. Walker, K. I. Hopcraft, B. Ablitt, E. Jakeman, “Polarization discrimination for active imaging in scattering media,” Opt. Commun. 159, 1–6 (1999).
[CrossRef]

K. Turpin, J. G. Walker, P. C. Y. Chang, K. I. Hopcraft, B. Ablitt, E. Jakeman, “The influence of particle size in active polarization imaging in scattering media,” Opt. Commun. 168, 325–335 (1999).
[CrossRef]

Compain, E.

de Haan, J. F.

Drevillon, B.

Glasstone, S.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

Hielscher, H.

Hopcraft, K. I.

P. C. Y. Chang, J. G. Walker, K. I. Hopcraft, B. Ablitt, E. Jakeman, “Polarization discrimination for active imaging in scattering media,” Opt. Commun. 159, 1–6 (1999).
[CrossRef]

K. Turpin, J. G. Walker, P. C. Y. Chang, K. I. Hopcraft, B. Ablitt, E. Jakeman, “The influence of particle size in active polarization imaging in scattering media,” Opt. Commun. 168, 325–335 (1999).
[CrossRef]

Hovenier, J. W.

Huffman, D. R.

Hunt, A. J.

Jakeman, E.

P. C. Y. Chang, J. G. Walker, K. I. Hopcraft, B. Ablitt, E. Jakeman, “Polarization discrimination for active imaging in scattering media,” Opt. Commun. 159, 1–6 (1999).
[CrossRef]

K. Turpin, J. G. Walker, P. C. Y. Chang, K. I. Hopcraft, B. Ablitt, E. Jakeman, “The influence of particle size in active polarization imaging in scattering media,” Opt. Commun. 168, 325–335 (1999).
[CrossRef]

Jalava, J.-P.

Kaplan, B.

Knuth, D. E.

D. E. Knuth, The Art of Computer Programming. 2. Seminumerical Algorithms, 2nd ed. (Addison-Wesley, Reading, Mass., 1981).

Koblinger, L.

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

Lumme, K.

Lux, I.

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

Mullikin, T. W.

T. W. Mullikin, “Multiple scattering of partially polarized light,” in Transport Theory, Vol. I of SIAM-AMS Proceedings (American Mathematical Society, Providence, R.I., 1969), pp. 3–16.

Perry, R. J.

Poirier, S.

Press, W. H.

W. H. Press, Numerical Recipies (Cambridge University, Cambridge, 1988).

Ripley, B. D.

B. D. Ripley, “Computer generation of random variables: a tutorial,” Int. Statist. Rev. 51, 301–319 (1983).
[CrossRef]

Rubinstein, R. Y.

R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley, New York, 1981).
[CrossRef]

Turpin, K.

K. Turpin, J. G. Walker, P. C. Y. Chang, K. I. Hopcraft, B. Ablitt, E. Jakeman, “The influence of particle size in active polarization imaging in scattering media,” Opt. Commun. 168, 325–335 (1999).
[CrossRef]

Van de Hulst, H. C.

H. C. Van de Hulst, Light scattering by Small Particles (Dover, New York, 1981).

Vassen, W.

Volten, H.

Walker, J. G.

K. Turpin, J. G. Walker, P. C. Y. Chang, K. I. Hopcraft, B. Ablitt, E. Jakeman, “The influence of particle size in active polarization imaging in scattering media,” Opt. Commun. 168, 325–335 (1999).
[CrossRef]

P. C. Y. Chang, J. G. Walker, K. I. Hopcraft, B. Ablitt, E. Jakeman, “Polarization discrimination for active imaging in scattering media,” Opt. Commun. 159, 1–6 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Wriedt, T.

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

Appl. Opt. (6)

Int. Statist. Rev. (1)

B. D. Ripley, “Computer generation of random variables: a tutorial,” Int. Statist. Rev. 51, 301–319 (1983).
[CrossRef]

Opt. Commun. (2)

P. C. Y. Chang, J. G. Walker, K. I. Hopcraft, B. Ablitt, E. Jakeman, “Polarization discrimination for active imaging in scattering media,” Opt. Commun. 159, 1–6 (1999).
[CrossRef]

K. Turpin, J. G. Walker, P. C. Y. Chang, K. I. Hopcraft, B. Ablitt, E. Jakeman, “The influence of particle size in active polarization imaging in scattering media,” Opt. Commun. 168, 325–335 (1999).
[CrossRef]

Part. Part. Syst. Charact. (1)

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

Rev. Sci. Instrum. (2)

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

E. Compain, B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

Other (10)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

T. W. Mullikin, “Multiple scattering of partially polarized light,” in Transport Theory, Vol. I of SIAM-AMS Proceedings (American Mathematical Society, Providence, R.I., 1969), pp. 3–16.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

I. Lux, L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).

H. C. Van de Hulst, Light scattering by Small Particles (Dover, New York, 1981).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

W. H. Press, Numerical Recipies (Cambridge University, Cambridge, 1988).

R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley, New York, 1981).
[CrossRef]

D. E. Knuth, The Art of Computer Programming. 2. Seminumerical Algorithms, 2nd ed. (Addison-Wesley, Reading, Mass., 1981).

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Figures (6)

Fig. 1
Fig. 1

PMME general setup. PSD, polarization-state detector; PSG, polarization-state generator.

Fig. 2
Fig. 2

(a) Top and (b) side views of the cell.

Fig. 3
Fig. 3

Normalized Mueller matrices of 404-nm PSL sphere suspensions with different dilutions. The X axis is the scattering angle in degrees.

Fig. 4
Fig. 4

Dilutions 1:500, 1:200, 1:100, 1:50, and 1:20 of the initial 404-nm PSL sphere suspension: comparison between measurements and simulation. Colored circles, experimental curves; black curves, fits. Normalized Mueller matrices are plotted versus the scattering angle (degrees).

Fig. 5
Fig. 5

Simulated mean-free path as a function of dilution and linear fit.

Fig. 6
Fig. 6

Scattering-order distribution at 90° (circles) and 160° (squares) for different mean-free paths: (a) 1.4 cm, (b) 0.3 cm, (c) 0.17 cm, and (d) 0.083 cm. The distribution is defined as the proportion of photons that encounter n scattering events before reaching the detector placed in the given direction. The scale on the X axis is logarithmic.

Equations (27)

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1cSr, Ω, tt+Ω·Sr, Ω, t=4π μscarMΩΩ·Sr, Ω, tdΩ-μTrSr, Ω, t+Qextr, Ω, t.
Ψr, Ω, t=0exp-0S μTr-lΩdlχr-lΩ, Ω, t-lcds,
χr, Ω, t=μscarμTr4πMΩΩ · Ψr, Ω, tdΩ+Qextr, Ω, t,Ψr, Ω, t=μTrSr, Ω, t.
 fiαfjαdα=0  for ij.
f1α2f2α2+f3α2+f4α2.
 Sinα·Sinαtdα=D,
 M·Sinα·Sinαtdα=M·D.
Sinα, β=1, cos α, sin α cos β, sin α sin βt.
Sout=M·Sin=Im1+Qm2+Um3+Vm4.
Pcollisionx=1/L exp-x/L.
L=1μ=1NσT.
Sinc=Rϕ·S0,  exinc=cos ϕex0+sin ϕey0eyinc=-sin ϕex0+cos ϕey0ezinc=ez0.
Ssca=Mθ·Sinc=Mθ·Rϕ·S0,exsca=cos θexinc-sin θezinceysca=eyincezsca=sin θexinc+cos θezinc.
iscaθ, ϕ=½|S1θ|2I-Q cos 2ϕ-U sin 2ϕ+|S2θ|2I+Q cos 2ϕ+U sin 2ϕ,
Isca=θ=0πϕ=02π iscaθ, ϕsin θdθdϕ.
Pθ, ϕ=iscaθ, ϕIsca.
Pf<iscaθ, ϕ=Isca2π2 maxθ, ϕ iscaθ, ϕ.
Q=IW cos 2ϕ0,U=IW sin 2ϕ0.
Q cos 2ϕ+U sin 2ϕ=IW cos 2ϕ-ϕ0.
Pθ, ϕ=IIsca|S1θ|21-W+2W sin2ϕ-ϕ0+|S2θ|21-W+2W cos2ϕ-ϕ0.
t1θ, ϕ=|S1θ|21-W+2W sin2ϕ-ϕ0/2,t2θ, ϕ=|S2θ|21-W+2W cos2ϕ-ϕ0/2;T1=0π02π t1θ, ϕsin θdθdϕ=π 0π |S1θ|2 sin θdθ,T2=0π02π t2θ, ϕsin θdθdϕ=π 0π |S2θ|2sin θdθ.
Pθ, ϕ=T1T1+T2t1θ, ϕT1+T2T1+T2t2θ, ϕT2,
p1θ, ϕ=t1θ, ϕ/T1, if i0=1,
p2θ, ϕ=t2θ, ϕ/T2, if i0=2.
Pϕ=1-W12π+W sin2ϕ-ϕ0π, if i0=1;Pϕ=1-W12π+W cos2ϕ-ϕ0π,if i0=2.
1cSr, Ω, tt+Ω·Sr, Ω, t=k4π μsca,kMkΩΩ·Sr, Ω, tdΩ-μT,kSr, Ω, t+Qextr, Ω, t=4π μsca M ΩΩ·Sr, Ω, tdΩ-μTSr, Ω, t+Qextr, Ω, t.
μsca=k μsca,k,μT=k μT,k,M=k μsca,kMkk μsca,k.

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