Abstract

A scheme for complete, accurate, and fast determination of the Mueller matrix M of elastic light scattering by a given sample is presented. The Stokes vector of the scattered radiation is virtually instantaneously measured using the recently described four-detector polarimeter (FDP) [Opt. Lett. 10, 309 (1985)] for various input polarizations generated by a fixed polarizer followed by a quarter-wave retarder (QWR) of adjustable fast-axis azimuth C. The output signal vector I of the FDP is recorded at four or more settings of C. If the periodic function I(C) is Fourier analyzed, a series of four terms is obtained whose coefficients determine M column by column. The QWR azimuth C can be changed manually, by a stepping or synchronous motor, or by Faraday-cell magneto-optical rotation, leading to various semiautomatic and fully automatic modes of operation with a range of speed to suit various applications.

© 1986 Optical Society of America

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References

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  1. See, for example, H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 5.
  2. G. F. Beardsley, J. Opt. Soc. Am. 58, 52 (1968).
    [CrossRef]
  3. A. C. Holland, G. Gagne, Appl. Opt. 9, 1113 (1970).
    [CrossRef] [PubMed]
  4. R. J. Perry, A. J. Hunt, D. R. Huffman, Appl. Opt. 17, 2700 (1978).
    [CrossRef] [PubMed]
  5. P. S. Hauge, Opt. Commun. 17, 74 (1976).
    [CrossRef]
  6. R. Thompson, E. Fry, J. Bottiger, Proc. Soc. Photo-Opt. Instrum. Eng. 112, 152 (1977).
  7. R. M. A. Azzam, Opt. Lett. 2, 148 (1978).
    [CrossRef] [PubMed]
  8. P. S. Hauge, J. Opt. Soc. Am. 68, 1519 (1978).
    [CrossRef]
  9. K. J. Voss, E. S. Fry, Appl. Opt. 23, 4427 (1984).
    [CrossRef] [PubMed]
  10. G. I. van Blokland, J. Opt. Soc. Am. A 2, 72 (1985).
    [CrossRef] [PubMed]
  11. R. M. A. Azzam, Opt. Lett. 10, 309 (1985).
    [CrossRef] [PubMed]
  12. An arbitrary setting for the polarizer and the use of an imperfect QWR can also be considered, as in Ref. 8, but this detracts from the purpose of this Letter.
  13. With the polarizer transmission axis in the reference scattering plane (defined, e.g., by the two arms of a standard spectrometer goniometer), and the QWR removed, the FDP is rotationally adjusted around the scattered beam as an axis until it generates the specific normalized output signal vector that corresponds to incident p-polarized light, namely, Ip = [(a00 + a01) (a10 + a11) (a20 + a21) (a30 + a31)]t/(a00 + a01), where aij are the elements of A. p and s define the linear polarization directions parallel and perpendicular to the scattering plane that coincides with the plane of incidence for reflection from the first detector.
  14. The segmented path of the light beam inside the FDP is not in one plane, notwithstanding the planar (mis)representation of Fig. 1.
  15. R. M. A. Azzam, J. Opt. Soc. Am. 68, 518 (1978).
    [CrossRef]
  16. Faraday rotations of 45° are now routinely obtained for optical-isolation applications.
  17. The reasoning is the same as is discussed in Ref. 5.
  18. This finite sweep produces the full range of modulation of the incident polarization from left- to right-circular polarization passing through the p linear state at midrange (i = 0).
  19. R. M. A. Azzam, Opt. Lett. 5, 303 (1980).
    [CrossRef] [PubMed]
  20. R. M. A. Azzam, Opt. Acta 29, 685 (1982); Opt. Acta 32, 1407 (1985).
    [CrossRef]

1985 (2)

1984 (1)

1982 (1)

R. M. A. Azzam, Opt. Acta 29, 685 (1982); Opt. Acta 32, 1407 (1985).
[CrossRef]

1980 (1)

1978 (4)

1977 (1)

R. Thompson, E. Fry, J. Bottiger, Proc. Soc. Photo-Opt. Instrum. Eng. 112, 152 (1977).

1976 (1)

P. S. Hauge, Opt. Commun. 17, 74 (1976).
[CrossRef]

1970 (1)

1968 (1)

Azzam, R. M. A.

Beardsley, G. F.

Bottiger, J.

R. Thompson, E. Fry, J. Bottiger, Proc. Soc. Photo-Opt. Instrum. Eng. 112, 152 (1977).

Fry, E.

R. Thompson, E. Fry, J. Bottiger, Proc. Soc. Photo-Opt. Instrum. Eng. 112, 152 (1977).

Fry, E. S.

Gagne, G.

Hauge, P. S.

Holland, A. C.

Huffman, D. R.

Hunt, A. J.

Perry, R. J.

Thompson, R.

R. Thompson, E. Fry, J. Bottiger, Proc. Soc. Photo-Opt. Instrum. Eng. 112, 152 (1977).

van Blokland, G. I.

van de Hulst, H. C.

See, for example, H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 5.

Voss, K. J.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

R. M. A. Azzam, Opt. Acta 29, 685 (1982); Opt. Acta 32, 1407 (1985).
[CrossRef]

Opt. Commun. (1)

P. S. Hauge, Opt. Commun. 17, 74 (1976).
[CrossRef]

Opt. Lett. (3)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. Thompson, E. Fry, J. Bottiger, Proc. Soc. Photo-Opt. Instrum. Eng. 112, 152 (1977).

Other (7)

An arbitrary setting for the polarizer and the use of an imperfect QWR can also be considered, as in Ref. 8, but this detracts from the purpose of this Letter.

With the polarizer transmission axis in the reference scattering plane (defined, e.g., by the two arms of a standard spectrometer goniometer), and the QWR removed, the FDP is rotationally adjusted around the scattered beam as an axis until it generates the specific normalized output signal vector that corresponds to incident p-polarized light, namely, Ip = [(a00 + a01) (a10 + a11) (a20 + a21) (a30 + a31)]t/(a00 + a01), where aij are the elements of A. p and s define the linear polarization directions parallel and perpendicular to the scattering plane that coincides with the plane of incidence for reflection from the first detector.

The segmented path of the light beam inside the FDP is not in one plane, notwithstanding the planar (mis)representation of Fig. 1.

Faraday rotations of 45° are now routinely obtained for optical-isolation applications.

The reasoning is the same as is discussed in Ref. 5.

This finite sweep produces the full range of modulation of the incident polarization from left- to right-circular polarization passing through the p linear state at midrange (i = 0).

See, for example, H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 5.

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Figures (1)

Fig. 1
Fig. 1

Basic instrument for MMM using the FDP.

Equations (16)

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S = MS ,
S = I 0 [ 1             ½ + ½ cos 4 C             ½ sin 4 C             sin 2 C ] t ,
I = [ i 0 i 1 i 2 i 3 ] t ,
I = AS ,
I = AMS .
J = AM S ,
M = A - 1 J S - 1 .
S ( C ) = S 0 + S 1 s sin 2 C + S 2 c cos 4 C + S 2 s sin 4 C ,
S 0 = [ 1 ½ 0 0 ] t , S 1 s = [ 0 0 0 1 ] t , S 2 c = [ 0 ½ 0 0 ] t , S 2 s = [ 0 0 ½ 0 ] t .
I ( C ) = I 0 + I 1 s sin 2 C + I 2 c cos 4 C + I 2 s sin 4 C ,
I 0 = AMS 0 , I 1 s = AMS 1 s , I 2 c = AMS 2 c , I 2 c = AMS 2 s .
M = [ C 1 M C 2 M C 3 M C 4 M ] ,
I 0 = A ( C 1 M + ½ C 2 m ) ,             I 1 s = A C 4 M , I 2 c = ½ A C 2 M ,             I 2 s = ½ A C 3 M .
C 4 M = A - 1 I 1 s , C 3 M = 2 A - 1 I 2 s , C 2 M = 2 A - 1 I 2 c , C 1 M = A - 1 I 0 - ½ C 2 M .
M = M ¯ + Δ M cos Ω t ,
I = I ¯ + Δ I cos Ω t .

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