Abstract

We describe a measuring system for determining the state of polarization of a beam of light in terms of its Stokes parameters. The technique which can be fully automated incorporates a monochromator and single photon counting detection and can thus be applied over a large wavelength range for very weak optical signals. Fourier transformation of the data by an on-line minicomputer allows immediate calculation of the Stokes parameters. We discuss special applications to light emitted from excited atomic systems with and without cylindrical symmetry.

© 1977 Optical Society of America

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  1. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941); R. Perrin, J. Chem. Phys. 10, 415 (1942) and Ref. 3.
    [CrossRef]
  2. G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399 (1852) and Mathematical and Physical Papers (Cambridge, 1901), Vol. 3.
  3. D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971), Chap. 4.
  4. J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 13.
  5. H. G. Berry, L. J. Curtis, D. G. Ellis, R. M. Schectman, Phys. Rev. Lett. 32, 751 (1974).
    [CrossRef]
  6. D. Clarke, R. N. Ibbett, J. Sci. Instrum. 1, 409 (1968).
    [CrossRef]
  7. F. Laloë, Ann. Phys. 6, 5 (1971); M. Pavlovic, F. Laloë, J. Phys. 31, 173 (1970).
    [CrossRef]
  8. C. M. McIntyre, S. E. Harris, J. Opt. Soc. Am. 58, 1575 (1968).
    [CrossRef]
  9. F. Scheid, Numerical Analysis (McGraw-Hill, New York, 1968), p. 293.
  10. R. J. Higgins, Am. J. Phys. 44, 766 (1976).
    [CrossRef]
  11. S. Chandrasekhar, Astrophys. J. 105, 424 (1947).
    [CrossRef]
  12. U. Fano, J. Opt. Soc. Am. 39, 859 (1949).
    [CrossRef]
  13. U. Fano, J. Macek, Rev. Mod. Phys. 45, 553 (1973) the polarization properties of light from collision-excited systems. Stokes parameters are introduced to describe emission from anisotropic light sources in Ref. 5, and a more general treatment and further references can be found in H. G. Berry, Rep. Prog. Phys. 40, 155 (1977).
    [CrossRef]
  14. W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, Proc. Nat. Acad. Sci. 73, 486 (1976).
    [CrossRef] [PubMed]

1976 (2)

R. J. Higgins, Am. J. Phys. 44, 766 (1976).
[CrossRef]

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, Proc. Nat. Acad. Sci. 73, 486 (1976).
[CrossRef] [PubMed]

1974 (1)

H. G. Berry, L. J. Curtis, D. G. Ellis, R. M. Schectman, Phys. Rev. Lett. 32, 751 (1974).
[CrossRef]

1973 (1)

U. Fano, J. Macek, Rev. Mod. Phys. 45, 553 (1973) the polarization properties of light from collision-excited systems. Stokes parameters are introduced to describe emission from anisotropic light sources in Ref. 5, and a more general treatment and further references can be found in H. G. Berry, Rep. Prog. Phys. 40, 155 (1977).
[CrossRef]

1971 (1)

F. Laloë, Ann. Phys. 6, 5 (1971); M. Pavlovic, F. Laloë, J. Phys. 31, 173 (1970).
[CrossRef]

1968 (2)

D. Clarke, R. N. Ibbett, J. Sci. Instrum. 1, 409 (1968).
[CrossRef]

C. M. McIntyre, S. E. Harris, J. Opt. Soc. Am. 58, 1575 (1968).
[CrossRef]

1949 (1)

1947 (1)

S. Chandrasekhar, Astrophys. J. 105, 424 (1947).
[CrossRef]

1941 (1)

1852 (1)

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399 (1852) and Mathematical and Physical Papers (Cambridge, 1901), Vol. 3.

Berry, H. G.

H. G. Berry, L. J. Curtis, D. G. Ellis, R. M. Schectman, Phys. Rev. Lett. 32, 751 (1974).
[CrossRef]

Bickel, W. S.

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, Proc. Nat. Acad. Sci. 73, 486 (1976).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Astrophys. J. 105, 424 (1947).
[CrossRef]

Clarke, D.

D. Clarke, R. N. Ibbett, J. Sci. Instrum. 1, 409 (1968).
[CrossRef]

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971), Chap. 4.

Curtis, L. J.

H. G. Berry, L. J. Curtis, D. G. Ellis, R. M. Schectman, Phys. Rev. Lett. 32, 751 (1974).
[CrossRef]

Davidson, J. F.

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, Proc. Nat. Acad. Sci. 73, 486 (1976).
[CrossRef] [PubMed]

Ellis, D. G.

H. G. Berry, L. J. Curtis, D. G. Ellis, R. M. Schectman, Phys. Rev. Lett. 32, 751 (1974).
[CrossRef]

Fano, U.

U. Fano, J. Macek, Rev. Mod. Phys. 45, 553 (1973) the polarization properties of light from collision-excited systems. Stokes parameters are introduced to describe emission from anisotropic light sources in Ref. 5, and a more general treatment and further references can be found in H. G. Berry, Rep. Prog. Phys. 40, 155 (1977).
[CrossRef]

U. Fano, J. Opt. Soc. Am. 39, 859 (1949).
[CrossRef]

Grainger, J. F.

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971), Chap. 4.

Harris, S. E.

Higgins, R. J.

R. J. Higgins, Am. J. Phys. 44, 766 (1976).
[CrossRef]

Huffman, D. R.

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, Proc. Nat. Acad. Sci. 73, 486 (1976).
[CrossRef] [PubMed]

Ibbett, R. N.

D. Clarke, R. N. Ibbett, J. Sci. Instrum. 1, 409 (1968).
[CrossRef]

Jones, R. C.

Kilkson, R.

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, Proc. Nat. Acad. Sci. 73, 486 (1976).
[CrossRef] [PubMed]

Laloë, F.

F. Laloë, Ann. Phys. 6, 5 (1971); M. Pavlovic, F. Laloë, J. Phys. 31, 173 (1970).
[CrossRef]

Macek, J.

U. Fano, J. Macek, Rev. Mod. Phys. 45, 553 (1973) the polarization properties of light from collision-excited systems. Stokes parameters are introduced to describe emission from anisotropic light sources in Ref. 5, and a more general treatment and further references can be found in H. G. Berry, Rep. Prog. Phys. 40, 155 (1977).
[CrossRef]

McIntyre, C. M.

Schectman, R. M.

H. G. Berry, L. J. Curtis, D. G. Ellis, R. M. Schectman, Phys. Rev. Lett. 32, 751 (1974).
[CrossRef]

Scheid, F.

F. Scheid, Numerical Analysis (McGraw-Hill, New York, 1968), p. 293.

Stokes, G. G.

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399 (1852) and Mathematical and Physical Papers (Cambridge, 1901), Vol. 3.

Stone, J. M.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 13.

Am. J. Phys. (1)

R. J. Higgins, Am. J. Phys. 44, 766 (1976).
[CrossRef]

Ann. Phys. (1)

F. Laloë, Ann. Phys. 6, 5 (1971); M. Pavlovic, F. Laloë, J. Phys. 31, 173 (1970).
[CrossRef]

Astrophys. J. (1)

S. Chandrasekhar, Astrophys. J. 105, 424 (1947).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Sci. Instrum. (1)

D. Clarke, R. N. Ibbett, J. Sci. Instrum. 1, 409 (1968).
[CrossRef]

Phys. Rev. Lett. (1)

H. G. Berry, L. J. Curtis, D. G. Ellis, R. M. Schectman, Phys. Rev. Lett. 32, 751 (1974).
[CrossRef]

Proc. Nat. Acad. Sci. (1)

W. S. Bickel, J. F. Davidson, D. R. Huffman, R. Kilkson, Proc. Nat. Acad. Sci. 73, 486 (1976).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

U. Fano, J. Macek, Rev. Mod. Phys. 45, 553 (1973) the polarization properties of light from collision-excited systems. Stokes parameters are introduced to describe emission from anisotropic light sources in Ref. 5, and a more general treatment and further references can be found in H. G. Berry, Rep. Prog. Phys. 40, 155 (1977).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399 (1852) and Mathematical and Physical Papers (Cambridge, 1901), Vol. 3.

Other (3)

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971), Chap. 4.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 13.

F. Scheid, Numerical Analysis (McGraw-Hill, New York, 1968), p. 293.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the polarization analysis system.

Fig. 2
Fig. 2

Wavelength dependence of the phase difference for two retardation plates. Part (a) shows the variation for a quartz retarder of order N ≈ 50 and a thickness of ~2.5 mm, and (b) shows the wavelength variation of a plastic retarder over the same wavelength region. Note the enlarged scale for the phase delay in (b).

Fig. 3
Fig. 3

Data from a linearly polarized source and the resulting Fourier series components Ci, Si (i = 0,1,…,10). The light is the Neii 3230-Å 3s2D–3p2D transition from 1-MeV Ne+ excited in a thin perpendicular carbon foil. The derived Stokes parameters are I = 6487 ± 15, M/I = 0.059 ± 0.003, C/I = −0.057 ± 0.003, S/I = −0.006 ± 0.002.

Fig. 4
Fig. 4

Experimental arrangement for measuring the Stokes parameters of partially elliptically polarized light from fast ion–solid interactions.

Fig. 5
Fig. 5

Data from a strongly circularly polarized source and the Fourier series components Ci, Si (i = . . 0,1, .,10). The light is the Arii 4610-Å 4s2D5/2–4p2F7/2 transition from 1-MeV Ar+ excited by a Cu surface at 3° grazing incidence. The derived Stokes parameters are I = 1021 ± 5, M/I = −0.063 ± 0.005, C/I = 0.012 ± 0.005, S/I = 0.787 ± 0.006.

Fig. 6
Fig. 6

Data from a source with circular and linear polarization fractions and the Fourier series components Ci, Si(i = 0,1,…,10). The derived Stokes parameters are I = 780 ± 3, M/I = 0.066 ± 0.006, C/I = −0.086 ± 0.006, S/I = 0.082 ± 0.007. The line through the data is the intensity given by Eq. (11) with the coefficients C0, C2 … taken from the results shown in the lower part of the figure.

Equations (20)

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I = I ( ) + I ( 90° ) = E x 2 + E y 2 ,
M = I ( ) - I ( 90° ) = E x 2 - E y 2 ,
C = I ( 45° ) - I ( 135° ) = R e E x E y ,
S = I RHC - I LHC = I m E x E y ,
I T ( α , β , δ ) = 1 2 { 1 + ( M cos 2 β + C sin 2 β ) cos 2 ( α - β ) + [ ( C cos 2 β - M sin 2 β ) cos δ + S sin δ ] sin 2 ( α - β ) } .
I T ( α , β , δ ) = 1 2 [ 1 + ( M 2 cos 2 α + C 2 sin 2 α ) ( 1 + cos δ ) ] + 1 2 [ S sin δ · sin ( 2 α - 2 β ) ] + 1 4 [ ( M cos 2 α - C sin 2 α ) cos 4 β + ( M sin 2 α + C cos 2 α ) sin 4 β ] ( 1 - cos δ ) .
M I ( λ ) k inc ( λ ) = I max - I min I max + I min ,
I T ( β ) = 1 + η cos ( 4 β + γ ) ,
cos δ ( λ ) = 1 - η cos 2 α - 2 η / k inc ( λ ) 1 + η cos 2 α .
δ ( λ ) = 2 π ( p + ¼ ) = 2 π d λ · [ n e ( λ ) - n o ( λ ) ] ,
I T ( β ) = C 0 + C 2 cos 2 β + C 4 cos 4 β + S 2 sin 2 β + S 4 sin 4 β ,
C ω k = 2 N · 1 1 + δ K O + δ k L · i = 1 N I T i cos ω k β i S ω k = 2 N · 1 1 + δ k O + δ k L · i = 1 N I T i sin ω k β i } ,
ω k = 2 π N · k Δ β , and β i = ( i - 1 ) · Δ β .
C 0 i = I T i - C 2 cos 2 β i - S 2 sin 2 β i - C 4 cos 4 β i - S 4 sin 4 β i ,
σ 2 = 1 N · i = 1 N ( C 0 i - C 0 ) 2 .
M = 2 1 - cos δ [ C 4 cos ( 2 α + 4 β 0 ) + S 4 sin ( 2 α + 4 β 0 ) ] , C = 2 1 - cos δ [ S 4 cos ( 2 α + 4 β 0 ) - C 4 sin ( 2 α + 4 β 0 ) ] , S = C 2 sin δ sin ( 2 α + 4 β 0 ) = - S 2 sin δ cos ( 2 α + 4 β 0 ) , S = ( C 2 2 + S 2 2 ) 1 / 2 / sin 2 δ , I = C 0 - 1 + cos δ 1 - cos δ · [ C 4 cos ( 4 α + 4 β 0 ) + S 4 sin ( 4 α + 4 β 0 ) ] .
tan ( 2 β 0 + 2 α ) = - C 2 / S 2 .
tan ( 4 β 0 + 2 α ) = S 4 / C 4 .
M = 4 1 - cos δ · ( C 4 2 + S 4 2 ) 1 / 2 .
η = I max - I min I max + I min = M ( 1 - cos δ ) 2 I + M cos 2 α ( 1 + cos δ ) .

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