Abstract

A two-frequency beam from a Zeeman laser scatters elastically from an isotropic medium, such as randomly oriented viruses or other particles suspended in water. The Zeeman effect splits the laser line by 250 kHz, and beats can be seen electronically in the signal from a phototube that views the scattered light. There are independently rotatable half-wave and quarter-wave retardation plates in the incident beam and a similar pair in the observed scattered beam, plus a fixed linear polarizer directly in front of the detector. Each of the four retarders has two angular positions, providing a total of 16 possible polarization cases. For each of the 16 cases, there are three data to be collected: (1) the average total intensity of the scattered light, (2) the amplitude of the beats in the scattered light, and (3) the phase shift between the beats of the scattered light and those of a reference signal from the laser. When a singular value decomposition technique is used, these threefold redundant data are rapidly transformed into a best-fit 4 × 4 Mueller scattering matrix. We discuss several different measurement strategies and their systematic and statistical errors. We present experimental results for two kinds of particle of wavelength size: polystyrene spheres and tobacco mosaic virus. In both cases the achiral retardation element M 34 of the Mueller matrix is easily measurable.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
    [CrossRef]
  2. H. Mueller, “The foundation of optics,” J. Opt. Soc. Am. 38, 661 (1948).
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 1.
  4. F. Allen, C. Bustamante, eds., Applications of Circularly Polarized Radiation Using Synchrotron and Ordinary Sources (Plenum, New York, 1984).
  5. Y. Shi, W. M. McClain, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. I. Dependence on wavelength and scattering angle,” J. Chem. Phys. 93, 5605–5615 (1990); Yaoming Shi, W. M. McClain, D. Tian, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. II. Molecular parameters and Perrin symmetry,” J. Chem. Phys. 94, 4726–4740 (1991).
    [CrossRef]
  6. C. Bustamante, I. Tinoco, M. F. Maestre, “Circular intensity differential scattering of light,” J. Chem. Phys. 76, 3440–3446 (1982), and references therein.
    [CrossRef]
  7. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, England, 1982).
  8. W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
    [CrossRef] [PubMed]
  9. W. M. McClain, W. A. Ghoul, “Shape effects in elastic scattering by random ensembles of model viruses,” Biopolymers 26, 2027–2040 (1987).
    [CrossRef] [PubMed]
  10. W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
    [CrossRef]
  11. D. Tian, W. M. McClain, “Nondipole light scattering by partially oriented ensembles. I. Numerical calculations and symmetries,” J. Chem. Phys. 90, 4783–4794 (1989); “Nondipole light scattering by partially oriented ensembles. II. Analytic algorithm,” J. Chem. Phys. 90, 6956–6964 (1989); “Nondipole light scattering by partially oriented ensembles. III. The Mueller pattern for achiral macromolecules,” J. Chem. Phys. 91, 4435–4439 (1989).
    [CrossRef]
  12. W. S. Bickel, “Optical system for light scattering experiments,” Appl. Opt. 18, 1707–1709 (1979).
    [CrossRef] [PubMed]
  13. W. S. Bickel, M. E. Stafford, “Biological particles as irregularly shaped scatterers,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 299–305.
    [CrossRef]
  14. W. S. Bickel, “The Mueller scattering matrix elements for Rayleigh spheres,” in Applications of Circularly Polarized Radiation Using Synchrotron and Ordinary Sources, F. Allen, C. Bustamante, eds. (Plenum, New York, 1985), pp. 69–76.
  15. R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
    [CrossRef] [PubMed]
  16. J. R. Bottiger, E. S. Fry, R. C. Thomson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 283–290.
    [CrossRef]
  17. D. H. Goldstein, “Mueller matrix dual rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  18. A. J. Hunt, D. R. Huffman, “A new polarization modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
    [CrossRef]
  19. Optra, Inc., West Peabody Office Park, 83 Pine Street, Peabody, Mass. 01960.
  20. D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, New York, 1990).
  21. mathematica, sold by Wolfram Research, Inc., 100 Trade Center Drive, Champaign, Ill. 61820-7237 (info@wri.com by e-mail).
  22. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1986), Sec. 2.9.
  23. M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
    [CrossRef]
  24. H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton N.J., 1961), Sec. 17.3.
  25. R. G. Johnston, S. B. Singham, G. C. Salzman, “Polarized light scattering,” Comments. Mol. Cell Biophys. 5, 171–192 (1988).
  26. Duke Scientific Corporation, 1135D San Antonio Road, Palo Alto, Calif. 94303.
  27. W. S. Bickel, M. E. Stafford, “Polarized light scattering from biological systems: a technique for cell differentiation,” J. Biol. Phys. 9,53–66 (1981).
    [CrossRef]
  28. H. Boedtke, N. S. Simmons, “The preparation and characterization of essentially uniform tobacco mosaic virus particles,” J. Am. Chem. Soc. 80, 2550–2557 (1957).
    [CrossRef]
  29. L. Stryer, Biochemistry, 3rd ed. (Freeman, New York, 1988).
  30. R. Oldenbourg, X. Wen, R. B. Meyer, D. L. D. Caspar, “Orientational distribution function in nematic tobacco-mosaic-virus liquid crystals measured by x-ray diffraction,” Phys. Rev. Lett. 61, 1851–1854 (1988).
    [CrossRef] [PubMed]
  31. Y. Shi, W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders,” J. Chem. Phys. 98, 1695–1711 (1993).
    [CrossRef]

1993 (1)

Y. Shi, W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders,” J. Chem. Phys. 98, 1695–1711 (1993).
[CrossRef]

1992 (2)

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

D. H. Goldstein, “Mueller matrix dual rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
[CrossRef] [PubMed]

1990 (1)

Y. Shi, W. M. McClain, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. I. Dependence on wavelength and scattering angle,” J. Chem. Phys. 93, 5605–5615 (1990); Yaoming Shi, W. M. McClain, D. Tian, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. II. Molecular parameters and Perrin symmetry,” J. Chem. Phys. 94, 4726–4740 (1991).
[CrossRef]

1989 (1)

D. Tian, W. M. McClain, “Nondipole light scattering by partially oriented ensembles. I. Numerical calculations and symmetries,” J. Chem. Phys. 90, 4783–4794 (1989); “Nondipole light scattering by partially oriented ensembles. II. Analytic algorithm,” J. Chem. Phys. 90, 6956–6964 (1989); “Nondipole light scattering by partially oriented ensembles. III. The Mueller pattern for achiral macromolecules,” J. Chem. Phys. 91, 4435–4439 (1989).
[CrossRef]

1988 (2)

R. Oldenbourg, X. Wen, R. B. Meyer, D. L. D. Caspar, “Orientational distribution function in nematic tobacco-mosaic-virus liquid crystals measured by x-ray diffraction,” Phys. Rev. Lett. 61, 1851–1854 (1988).
[CrossRef] [PubMed]

R. G. Johnston, S. B. Singham, G. C. Salzman, “Polarized light scattering,” Comments. Mol. Cell Biophys. 5, 171–192 (1988).

1987 (1)

W. M. McClain, W. A. Ghoul, “Shape effects in elastic scattering by random ensembles of model viruses,” Biopolymers 26, 2027–2040 (1987).
[CrossRef] [PubMed]

1986 (1)

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

1982 (1)

C. Bustamante, I. Tinoco, M. F. Maestre, “Circular intensity differential scattering of light,” J. Chem. Phys. 76, 3440–3446 (1982), and references therein.
[CrossRef]

1981 (1)

W. S. Bickel, M. E. Stafford, “Polarized light scattering from biological systems: a technique for cell differentiation,” J. Biol. Phys. 9,53–66 (1981).
[CrossRef]

1980 (1)

1979 (1)

1976 (1)

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[CrossRef] [PubMed]

1973 (1)

A. J. Hunt, D. R. Huffman, “A new polarization modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

1957 (1)

H. Boedtke, N. S. Simmons, “The preparation and characterization of essentially uniform tobacco mosaic virus particles,” J. Am. Chem. Soc. 80, 2550–2557 (1957).
[CrossRef]

1948 (1)

H. Mueller, “The foundation of optics,” J. Opt. Soc. Am. 38, 661 (1948).

1942 (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, England, 1982).

Bickel, W. S.

W. S. Bickel, M. E. Stafford, “Polarized light scattering from biological systems: a technique for cell differentiation,” J. Biol. Phys. 9,53–66 (1981).
[CrossRef]

W. S. Bickel, “Optical system for light scattering experiments,” Appl. Opt. 18, 1707–1709 (1979).
[CrossRef] [PubMed]

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[CrossRef] [PubMed]

W. S. Bickel, M. E. Stafford, “Biological particles as irregularly shaped scatterers,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 299–305.
[CrossRef]

W. S. Bickel, “The Mueller scattering matrix elements for Rayleigh spheres,” in Applications of Circularly Polarized Radiation Using Synchrotron and Ordinary Sources, F. Allen, C. Bustamante, eds. (Plenum, New York, 1985), pp. 69–76.

Boedtke, H.

H. Boedtke, N. S. Simmons, “The preparation and characterization of essentially uniform tobacco mosaic virus particles,” J. Am. Chem. Soc. 80, 2550–2557 (1957).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 1.

Bottiger, J. R.

R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
[CrossRef] [PubMed]

J. R. Bottiger, E. S. Fry, R. C. Thomson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 283–290.
[CrossRef]

Bustamante, C.

C. Bustamante, I. Tinoco, M. F. Maestre, “Circular intensity differential scattering of light,” J. Chem. Phys. 76, 3440–3446 (1982), and references therein.
[CrossRef]

Caspar, D. L. D.

R. Oldenbourg, X. Wen, R. B. Meyer, D. L. D. Caspar, “Orientational distribution function in nematic tobacco-mosaic-virus liquid crystals measured by x-ray diffraction,” Phys. Rev. Lett. 61, 1851–1854 (1988).
[CrossRef] [PubMed]

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton N.J., 1961), Sec. 17.3.

Davidson, J. F.

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1986), Sec. 2.9.

Fry, E. S.

R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
[CrossRef] [PubMed]

J. R. Bottiger, E. S. Fry, R. C. Thomson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 283–290.
[CrossRef]

Ghoul, W. A.

W. M. McClain, W. A. Ghoul, “Shape effects in elastic scattering by random ensembles of model viruses,” Biopolymers 26, 2027–2040 (1987).
[CrossRef] [PubMed]

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

Goldstein, D. H.

Huffman, D. R.

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[CrossRef] [PubMed]

A. J. Hunt, D. R. Huffman, “A new polarization modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 1.

Hunt, A. J.

A. J. Hunt, D. R. Huffman, “A new polarization modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

Johnston, R. G.

R. G. Johnston, S. B. Singham, G. C. Salzman, “Polarized light scattering,” Comments. Mol. Cell Biophys. 5, 171–192 (1988).

Kilkson, R.

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[CrossRef] [PubMed]

Kliger, D. S.

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

Kumar, M. S.

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

Lewis, J. W.

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

Maestre, M. F.

C. Bustamante, I. Tinoco, M. F. Maestre, “Circular intensity differential scattering of light,” J. Chem. Phys. 76, 3440–3446 (1982), and references therein.
[CrossRef]

McClain, W. M.

Y. Shi, W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders,” J. Chem. Phys. 98, 1695–1711 (1993).
[CrossRef]

Y. Shi, W. M. McClain, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. I. Dependence on wavelength and scattering angle,” J. Chem. Phys. 93, 5605–5615 (1990); Yaoming Shi, W. M. McClain, D. Tian, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. II. Molecular parameters and Perrin symmetry,” J. Chem. Phys. 94, 4726–4740 (1991).
[CrossRef]

D. Tian, W. M. McClain, “Nondipole light scattering by partially oriented ensembles. I. Numerical calculations and symmetries,” J. Chem. Phys. 90, 4783–4794 (1989); “Nondipole light scattering by partially oriented ensembles. II. Analytic algorithm,” J. Chem. Phys. 90, 6956–6964 (1989); “Nondipole light scattering by partially oriented ensembles. III. The Mueller pattern for achiral macromolecules,” J. Chem. Phys. 91, 4435–4439 (1989).
[CrossRef]

W. M. McClain, W. A. Ghoul, “Shape effects in elastic scattering by random ensembles of model viruses,” Biopolymers 26, 2027–2040 (1987).
[CrossRef] [PubMed]

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

Meyer, R. B.

R. Oldenbourg, X. Wen, R. B. Meyer, D. L. D. Caspar, “Orientational distribution function in nematic tobacco-mosaic-virus liquid crystals measured by x-ray diffraction,” Phys. Rev. Lett. 61, 1851–1854 (1988).
[CrossRef] [PubMed]

Mueller, H.

H. Mueller, “The foundation of optics,” J. Opt. Soc. Am. 38, 661 (1948).

Oldenbourg, R.

R. Oldenbourg, X. Wen, R. B. Meyer, D. L. D. Caspar, “Orientational distribution function in nematic tobacco-mosaic-virus liquid crystals measured by x-ray diffraction,” Phys. Rev. Lett. 61, 1851–1854 (1988).
[CrossRef] [PubMed]

Perrin, F.

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1986), Sec. 2.9.

Randall, C. E.

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

Salzman, G. C.

R. G. Johnston, S. B. Singham, G. C. Salzman, “Polarized light scattering,” Comments. Mol. Cell Biophys. 5, 171–192 (1988).

Shi, Y.

Y. Shi, W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders,” J. Chem. Phys. 98, 1695–1711 (1993).
[CrossRef]

Y. Shi, W. M. McClain, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. I. Dependence on wavelength and scattering angle,” J. Chem. Phys. 93, 5605–5615 (1990); Yaoming Shi, W. M. McClain, D. Tian, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. II. Molecular parameters and Perrin symmetry,” J. Chem. Phys. 94, 4726–4740 (1991).
[CrossRef]

Simmons, N. S.

H. Boedtke, N. S. Simmons, “The preparation and characterization of essentially uniform tobacco mosaic virus particles,” J. Am. Chem. Soc. 80, 2550–2557 (1957).
[CrossRef]

Simon, R.

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

Singham, S. B.

R. G. Johnston, S. B. Singham, G. C. Salzman, “Polarized light scattering,” Comments. Mol. Cell Biophys. 5, 171–192 (1988).

Stafford, M. E.

W. S. Bickel, M. E. Stafford, “Polarized light scattering from biological systems: a technique for cell differentiation,” J. Biol. Phys. 9,53–66 (1981).
[CrossRef]

W. S. Bickel, M. E. Stafford, “Biological particles as irregularly shaped scatterers,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 299–305.
[CrossRef]

Stryer, L.

L. Stryer, Biochemistry, 3rd ed. (Freeman, New York, 1988).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1986), Sec. 2.9.

Thompson, R. C.

Thomson, R. C.

J. R. Bottiger, E. S. Fry, R. C. Thomson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 283–290.
[CrossRef]

Tian, D.

D. Tian, W. M. McClain, “Nondipole light scattering by partially oriented ensembles. I. Numerical calculations and symmetries,” J. Chem. Phys. 90, 4783–4794 (1989); “Nondipole light scattering by partially oriented ensembles. II. Analytic algorithm,” J. Chem. Phys. 90, 6956–6964 (1989); “Nondipole light scattering by partially oriented ensembles. III. The Mueller pattern for achiral macromolecules,” J. Chem. Phys. 91, 4435–4439 (1989).
[CrossRef]

Tinoco, I.

C. Bustamante, I. Tinoco, M. F. Maestre, “Circular intensity differential scattering of light,” J. Chem. Phys. 76, 3440–3446 (1982), and references therein.
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1986), Sec. 2.9.

Wen, X.

R. Oldenbourg, X. Wen, R. B. Meyer, D. L. D. Caspar, “Orientational distribution function in nematic tobacco-mosaic-virus liquid crystals measured by x-ray diffraction,” Phys. Rev. Lett. 61, 1851–1854 (1988).
[CrossRef] [PubMed]

Appl. Opt. (3)

Biopolymers (1)

W. M. McClain, W. A. Ghoul, “Shape effects in elastic scattering by random ensembles of model viruses,” Biopolymers 26, 2027–2040 (1987).
[CrossRef] [PubMed]

Comments. Mol. Cell Biophys. (1)

R. G. Johnston, S. B. Singham, G. C. Salzman, “Polarized light scattering,” Comments. Mol. Cell Biophys. 5, 171–192 (1988).

J. Am. Chem. Soc. (1)

H. Boedtke, N. S. Simmons, “The preparation and characterization of essentially uniform tobacco mosaic virus particles,” J. Am. Chem. Soc. 80, 2550–2557 (1957).
[CrossRef]

J. Biol. Phys. (1)

W. S. Bickel, M. E. Stafford, “Polarized light scattering from biological systems: a technique for cell differentiation,” J. Biol. Phys. 9,53–66 (1981).
[CrossRef]

J. Chem. Phys. (6)

Y. Shi, W. M. McClain, “Closed-form Mueller scattering matrix for a random ensemble of long, thin cylinders,” J. Chem. Phys. 98, 1695–1711 (1993).
[CrossRef]

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

D. Tian, W. M. McClain, “Nondipole light scattering by partially oriented ensembles. I. Numerical calculations and symmetries,” J. Chem. Phys. 90, 4783–4794 (1989); “Nondipole light scattering by partially oriented ensembles. II. Analytic algorithm,” J. Chem. Phys. 90, 6956–6964 (1989); “Nondipole light scattering by partially oriented ensembles. III. The Mueller pattern for achiral macromolecules,” J. Chem. Phys. 91, 4435–4439 (1989).
[CrossRef]

Y. Shi, W. M. McClain, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. I. Dependence on wavelength and scattering angle,” J. Chem. Phys. 93, 5605–5615 (1990); Yaoming Shi, W. M. McClain, D. Tian, “Longwave properties of the orientation averaged Mueller scattering matrix for particles of arbitrary shape. II. Molecular parameters and Perrin symmetry,” J. Chem. Phys. 94, 4726–4740 (1991).
[CrossRef]

C. Bustamante, I. Tinoco, M. F. Maestre, “Circular intensity differential scattering of light,” J. Chem. Phys. 76, 3440–3446 (1982), and references therein.
[CrossRef]

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[CrossRef]

J. Opt. Soc. Am. (1)

H. Mueller, “The foundation of optics,” J. Opt. Soc. Am. 38, 661 (1948).

Opt. Commun. (1)

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

Phys. Rev. Lett. (1)

R. Oldenbourg, X. Wen, R. B. Meyer, D. L. D. Caspar, “Orientational distribution function in nematic tobacco-mosaic-virus liquid crystals measured by x-ray diffraction,” Phys. Rev. Lett. 61, 1851–1854 (1988).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, “Application of polarization effects in light scattering: a new biophysical tool,” Proc. Natl. Acad. Sci. USA 73, 486–490 (1976).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

A. J. Hunt, D. R. Huffman, “A new polarization modulated light scattering instrument,” Rev. Sci. Instrum. 44, 1753–1762 (1973).
[CrossRef]

Other (13)

Optra, Inc., West Peabody Office Park, 83 Pine Street, Peabody, Mass. 01960.

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

mathematica, sold by Wolfram Research, Inc., 100 Trade Center Drive, Champaign, Ill. 61820-7237 (info@wri.com by e-mail).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1986), Sec. 2.9.

J. R. Bottiger, E. S. Fry, R. C. Thomson, “Phase matrix measurements for electromagnetic scattering by sphere aggregates,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 283–290.
[CrossRef]

W. S. Bickel, M. E. Stafford, “Biological particles as irregularly shaped scatterers,” in Light Scattering by Irregularly Shaped Particles, D. W. Scheurman, ed. (Plenum, New York, 1980), pp. 299–305.
[CrossRef]

W. S. Bickel, “The Mueller scattering matrix elements for Rayleigh spheres,” in Applications of Circularly Polarized Radiation Using Synchrotron and Ordinary Sources, F. Allen, C. Bustamante, eds. (Plenum, New York, 1985), pp. 69–76.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 1.

F. Allen, C. Bustamante, eds., Applications of Circularly Polarized Radiation Using Synchrotron and Ordinary Sources (Plenum, New York, 1984).

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, Cambridge, England, 1982).

L. Stryer, Biochemistry, 3rd ed. (Freeman, New York, 1988).

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton N.J., 1961), Sec. 17.3.

Duke Scientific Corporation, 1135D San Antonio Road, Palo Alto, Calif. 94303.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Apparatus for complete determination of the Mueller scattering matrix: H, rotatable half-wave retarder, Q, rotatable quarter-wave retarders; P fixed polarizer; PM, photomultiplier. Each rotator has only two positions and is controlled by a computer. The laser line is Zeeman split by 250 kHz.

Fig. 2
Fig. 2

Experimentally measured Mueller matrix versus scattering angle for polystyrene spheres in water. Sphere diameter is 1.635 μm and the wavelength is 6328 Å. Element M 11 is shown on a decadic log scale; the other elements are normalized by M 11 at the same angle and are shown on a linear scale. Each Mueller element is measured independently of its Perrin symmetric partner so that random noise will not be mistaken for meaningful structure.

Fig. 3
Fig. 3

Experimentally measured values of M 34/M 11 from a run similar to that of Fig. 2, but with points every 2° (dotted curve) and the prediction of Mie theory (solid curve), using λvacuum = 0.6328 μm, d sphere = 1.635 μm, n sphere = 1.56, and n water = 1.33.

Fig. 4
Fig. 4

Experimentally measured Mueller matrix for TMV in water of very low ionic strength. The concentration was 5.6 μg/mL, and ~95% of the virus particles were unbroken, as judged from electron microscopy. Everything else is identical to Fig. 2.

Tables (6)

Tables Icon

Table 1 Angles Used in the Sparse Strategy

Tables Icon

Table 2 Example Mueller Matrix on Which the Error Calculations Are Based, Showing the Perrin Symmetry

Tables Icon

Table 3 Data Expected When Measuring the Mueller Matrix of Table 2 by Using the Measurement Strategy of Table 1

Tables Icon

Table 4 Tabulated quantity (1/10)(d/dθ)log(M ij ) for the Sparse Strategya

Tables Icon

Table 5 Specification of the Diagonal Strategy, with a D+ Polarizer Preceding Retarder 1 and Following Retarder 4

Tables Icon

Table 6 Tabulated Quantity (1/10)(d/θ)log(Mij) for the the Diagonal Strategy

Equations (46)

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

s α = M α β s β .
M orientation 1 λ 4 [ 1 1 ( d / λ ) 4 ( d / λ ) 1 1 ( d / λ ) 4 ( d / λ ) ( d / λ ) 4 ( d / λ ) 4 1 ( d / λ ) 5 ( d / λ ) ( d / λ ) ( d / λ ) 5 1 ] .
E i = P i j H j k ( ρ 4 ) Q k m ( ρ 3 ) J m N Q N P ( ρ 2 ) H P Q ( ρ 1 ) E Q .
D i m ( ρ 3 , ρ 4 ) = P i j H j k ( ρ 4 ) Q k m ( ρ 3 ) ,
A N Q ( ρ 1 , ρ 2 ) = Q N P ( ρ 2 ) H P Q ( ρ 1 ) ,
E i = D i m ( ρ 3 , ρ 4 ) J m N A N Q ( ρ 1 , ρ 2 ) E Q .
E i * = D i r * ( ρ 3 , ρ 4 ) J r S * A S T * ( ρ 1 , ρ 2 ) E T * .
E i E i * = D i m D i r * J m N J r S * A N Q A S T * E Q E T * ,
E i E i * = D m i T D i r * J m N J r S * A N Q A S T * E Q E T * ,
V = η E i E i * ,
V = η ( D ( T ) · D * ) m r J J * m r , N S ( A A * ) N S , Q T ( E E * ) Q T .
( C ) α , j m = 1 2 [ 1 0 0 1 1 0 0 - 1 0 1 1 0 0 i - i 0 ] ,
V = η [ ( D ( T ) · D * ) m r C m r , ρ - 1 ] [ C ρ , m r J J * m r , N S C N S , σ - 1 ] [ C σ , N S ( A A * ) N S , Q T C Q T , τ - 1 ] , [ C τ , Q T ( E E * ) Q T ] .
V = η L ρ M ρ , σ K σ , τ s τ ,
L ρ = ( D ( T ) · D * ) j k C j k , ρ - 1 ,
K σ , τ = C σ , S T ( A A * ) S T , Q R C Q R , τ - 1 .
V = η M ρ σ [ L ρ K σ τ s τ ] = η ( M ) ρ σ ( L ) ρ ( K · s ) σ = η ( M ) ρ σ ( LK · s ) ρ σ .
V = η ( M ) Ω ( LK · s ) Ω .
R ( ρ , δ ) = [ cos ( 2 ρ ) sin ( δ / 2 ) - i cos ( δ / 2 ) sin ( 2 ρ ) sin ( δ / 2 ) sin ( 2 ρ ) sin ( δ / 2 ) - cos ( 2 ρ ) sin ( δ / 2 ) - i cos ( δ / 2 ) ] ,
H ( ρ ) = [ cos ( 2 ρ ) sin ( 2 ρ ) sin ( 2 ρ ) - cos ( 2 ρ ) ] .
Q ( ρ ) = 1 2 [ 1 + i cos ( 2 ρ ) i sin ( 2 ρ ) i sin ( 2 ρ ) 1 - i cos ( 2 ρ ) ]
P = [ 0 0 0 1 ] .
K = [ 1 0 0 0 0 c 1 c 2 s 1 c 2 s 2 0 c 1 s 2 s 1 s 2 - c 2 0 - s 1 c 1 0 ] ,
{ c 1 s 1 } = { cos sin } ( 4 ρ 1 - 2 ρ 2 ) ,             { c 2 s 2 } = { cos sin } ( 2 ρ 2 ) ,
L = [ 1 , - c 4 c 3 , - c 4 , s 3 , s 4 ] / 2 ,
{ c 3 s 3 } = { cos sin } ( 2 ρ 3 ) ,             { c 4 s 4 } = { cos sin } ( 2 ρ 3 - 4 ρ 4 ) .
V = η S · M .
M = 1 η S ( - 1 ) · V .
s = { 1 , cos ( 2 ω t ) , sin ( 2 ω t ) , 0 } .
LK · s = [ LK · s ] 0 + [ LK · s ] in cos ( 2 ω t ) + [ LK · s ] out sin ( 2 ω t ) = [ 1 0 0 0 - c 4 c 3 0 0 0 - c 4 s 3 0 0 0 s 4 0 0 0 ] + [ 0 ( c 1 c 2 ) ( c 1 s 2 ) - s 1 0 ( c 1 c 2 ) ( - c 4 c 3 ) ( c 1 s 2 ) ( - c 4 c 3 ) s 1 ( c 4 c 3 ) 0 ( c 1 c 2 ) ( - c 4 s 3 ) ( c 1 s 2 ) ( - c 4 s 3 ) s 1 ( c 4 s 3 ) 0 ( c 1 c 2 ) s 4 ( c 1 s 2 ) s 4 - s 1 s 4 ] cos ( 2 ω t ) + [ 0 ( c 2 s 1 ) ( s 1 s 2 ) c 1 0 ( c 2 s 1 ) ( - c 4 c 3 ) ( s 1 s 2 ) ( - c 4 c 3 ) c 1 ( - c 4 c 3 ) 0 ( c 2 s 1 ) ( - c 4 s 3 ) ( s 1 s 2 ) ( - c 4 s 3 ) c 1 ( - c 4 s 3 ) 0 ( c 2 s 1 ) s 4 ( s 1 s 2 ) s 4 c 1 s 4 ] sin ( 2 ω t ) .
V ( t ) = V 0 + V in cos ( 2 ω t ) + V out sin ( 2 ω t ) .
[ V 0 V in V out ] = η [ ( LK · s ) Ω 0 ( LK · s ) Ω in ( LK · s ) Ω out ] M Ω . ( 3 ) = ( 3 × 16 ) ( 16 )
V = η S · M , ( 48 ) ( 48 × 16 ) ( 16 )
( S ) Ω , N ( - 1 ) ( S ) N , Ξ = δ Ω , Ξ . ( 16 × 48 ) ( 48 × 16 ) = ( 16 × 16 )
( M ¯ ) Ω = 1 η ( S ( - 1 ) ) Ω , N ( V ) N .
R = V = η S · M ¯ .
( S ) Ω , N = ( U ) Ω , Ξ ( W ) Ξ , Γ ( X T ) Γ , N , ( 48 × 16 ) = ( 48 × 16 ) ( 16 × 16 ) ( 16 × 16 )
[ X T M ] = 1 η W ( - 1 ) [ U T V ] . 16 = ( 16 × 16 ) 16
{ 4.27 , 1.32 , 3.70 ; 3.70 ; 3.02 , 2.83 , 2.00 , 2.00 ; 0.94 , 1.15 , 1.15 , 1.41 ; 1.73 , 1.73 , 2.45 , 2.45 } ,
S ( - 1 ) = X W ( - 1 ) U T .
μ = Σ a i μ i ,             σ 2 = Σ a i 2 σ i 2 .
100 σ ( M i j ) μ ( M i j ) = [ 0.045 0.055 0.48 0.51 0.056 0.047 0.36 0.29 2.99 2.00 0.08 0.17 4.21 2.26 0.14 0.06 ] .
d M d θ = lim δ θ 0 { [ S ( - 1 ) ( θ ) · S ( θ + δ θ ) · M ] - M δ θ } .
Err ( θ ) = lim δ θ 0 { [ S ( - 1 ) ( θ ) · S ( θ + δ θ ) - 1 ] / δ θ } ,
d M d θ = Err ( θ ) · M .
100 σ ( M α β ) μ ( M α β ) = [ 0.23 0.43 9.79 5.41 0.49 0.63 12.3 5.05 9.00 10.47 1.01 0.67 8.66 7.76 1.19 0.20 ] .

Metrics