Abstract

Phase-modulated Mueller ellipsometry (PMME) is used to probe scattering by suspensions of polystyrene latex spheres, with particle diameters ranging from 400 nm to 3 µm. PMME allows simultaneous measurement of the 16 coefficients of the Mueller matrix. Furthermore PMME measurements can easily be carried out owing to a calibration procedure implemented in a scattering configuration. The measurements performed on low concentrations show good agreement with Mie theory. Moreover size distribution could be obtained with a least-squares method based on a genetic fit algorithm. Experimental evidence of multiple scattering on PMME measurements is also presented.

© 2000 Optical Society of America

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References

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  1. H. Volten, J.-P. Jalava, K. Lumme, J. F. de Haan, H. Vassen, J. W. Hovenier, “Laboratory measurements and T-matrix calculations of the scattering matrix of rutile particles in water,” Appl. Opt. 38, 5232–5240 (1999).
    [CrossRef]
  2. M. I. Mishchenko, “Light scattering by size-shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4666 (1993).
    [CrossRef] [PubMed]
  3. S. Huard, Polarisation de la lumière (Masson, Paris, 1994).
  4. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  5. 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]
  6. E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
    [CrossRef]
  7. 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]
  8. 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]
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  10. M. I. Mishchenlo, D. W. Mackowski, “Electromagneic scattering by randomly oriented bispheres: comparison of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
    [CrossRef]
  11. L. Davis, E. Goldberg, Genetic Algorithms (Addison-Wesley, New York, 1989).
  12. W. H. Press, Numerical Recipes (Cambridge U. Press, New York, 1988).

1999 (2)

1998 (2)

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

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]

1996 (1)

M. I. Mishchenlo, D. W. Mackowski, “Electromagneic scattering by randomly oriented bispheres: comparison of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

1993 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

Compain, E.

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]

E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (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, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

Davis, L.

L. Davis, E. Goldberg, Genetic Algorithms (Addison-Wesley, New York, 1989).

de Haan, J. F.

Drevillon, B.

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]

E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (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, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

Goldberg, E.

L. Davis, E. Goldberg, Genetic Algorithms (Addison-Wesley, New York, 1989).

Hovenier, J. W.

Huard, S.

S. Huard, Polarisation de la lumière (Masson, Paris, 1994).

Jalava, J.-P.

Lumme, K.

Mackowski, D. W.

M. I. Mishchenlo, D. W. Mackowski, “Electromagneic scattering by randomly oriented bispheres: comparison of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

Mishchenko, M. I.

Mishchenlo, M. I.

M. I. Mishchenlo, D. W. Mackowski, “Electromagneic scattering by randomly oriented bispheres: comparison of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

Poirier, S.

Press, W. H.

W. H. Press, Numerical Recipes (Cambridge U. Press, New York, 1988).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Vassen, H.

Volten, H.

Appl. Opt. (4)

J. Quant. Spectrosc. Radiat. Transfer (1)

M. I. Mishchenlo, D. W. Mackowski, “Electromagneic scattering by randomly oriented bispheres: comparison of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

Rev. Sci. Instrum. (2)

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

Other (5)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

S. Huard, Polarisation de la lumière (Masson, Paris, 1994).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).

L. Davis, E. Goldberg, Genetic Algorithms (Addison-Wesley, New York, 1989).

W. H. Press, Numerical Recipes (Cambridge U. Press, New York, 1988).

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Figures (9)

Fig. 1
Fig. 1

PMME general setup. The PSG independently modulates the four components of the incident Stokes vector, and the PSD splits the scattered light into four beams whose intensities are inversely related to the Stokes vector.

Fig. 2
Fig. 2

(a) Side view and (b) top view of the receptacle. Laser path and parasitic reflections are also represented. The solution containing the particles is in the center, whereas the clean water surrounding it is to avoid total reflections.

Fig. 3
Fig. 3

Example of typical measurement errors on coefficients (a) M42* and (b) M22* for a diluted 404-nm-diameter PSL sphere suspension. M22* should be equal to 1 and M 42* should be null. The errors show two contributions: statistical noise and a systematic error that is reproducible.

Fig. 4
Fig. 4

Mueller matrix measurement of background noise as a function of the scattering angle (in degrees). The solution is clean filtered water without PSL spheres.

Fig. 5
Fig. 5

Effects of concentration on the Mueller matrix. All solutions contain only 404-nm PSL spheres but the concentration is varied. A simple scattering regime is reached for a mean free path of the order of a few centimeters. When concentration is increased, matrix symmetries are lost.

Fig. 6
Fig. 6

Simulation of the scattering cross section of a PSL sphere as a function of diameter at 515 nm.

Fig. 7
Fig. 7

Experimental (dots) and fitted (line) curves for a solution containing only 404-nm-diameter spheres. The top graph shows the size distribution by percentage of the total population versus diameter. Independent normalized Mueller parameters, M 11, M 12*, M 33*, and M 34*, are plotted versus scattering angle (in degrees).

Fig. 8
Fig. 8

Experimental (dots) and fitted (line) curves for a solution containing only 3-µm-diameter spheres. The top graph shows the size distribution by percentage of the total population versus diameter. Independent normalized Mueller parameters M 11, M 12*, M 33*, and M 34* are plotted versus scattering angle (in degrees).

Fig. 9
Fig. 9

Experimental (dots) and fitted (line) curves for a solution containing three different types of sphere: 404-nm-, 700-nm-, and 3-µm-diameter spheres. The top graph shows the size distribution by percentage of the total population versus diameter. Independent normalized Mueller parameters, M 11, M 12*, M 33*, and M 34*, are plotted versus scattering angle (in degrees).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Sout=M·Sin
Mij*=M11 if i=1 and j=1, Mij*=Mij/M11 otherwise.
i1ti2ti3ti4t=A M Wsin ωtcos 2ωtsin 3ωtcos 4ωt+higher orders.
Msphere=M111M12*00M12*10000M33*M34*00-M34*M33*×with M12*2+M33*2+M34*2=1.
Msuspension=sphere iMspherei=τ1a00a10000bc00-cb with a2+b2+c21,
Mscaθ1-1 Mscaθ2 close to 1±0.500±0.5100000±0.5000.50.

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