Abstract

The trajectory of the polarization state of a monochromatic light beam after it passes through a fixed linear polarizer and a rotating linear retarder of arbitrary retardance Δ is determined on the Poincaré sphere. The three-dimensional figure-8 contour is shown to be the line of intersection of a right-circular cylinder with the sphere. The cylinder is parallel to the polar (s3) axis, touches the sphere at the equator (at the point that represents the linear polarization transmitted by the fixed polarizer), and has a radius r=sin2(Δ/2). Projections of the trajectory in the coordinate planes of the normalized Stokes parameter space (s1, s2, s3) are also determined.

© 2000 Optical Society of America

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References

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  1. D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).
  2. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  3. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  4. P. S. Hauge, “Recent development in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
    [CrossRef]
  5. H. Poincaré, Theorie Mathématique de la Lumière (Gauthiers-Villars, Paris, 1892), Vol. II, Chap. 12.
  6. H. G. Jerrard, “Transmission of light through birefringent and optically active media: the Poincaré sphere,” J. Opt. Soc. Am. 44, 634–640 (1954).
    [CrossRef]
  7. J. E. Bigelow, R. A. Kashnow, “Poincaré sphere analysis of liquid crystal optics,” Appl. Opt. 16, 2090–2096 (1977).
    [CrossRef] [PubMed]
  8. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
    [CrossRef]
  9. S. R. Rajagopalan, S. Ramaseshan, “Rotating elliptic analysers for the automatic analysis of polarised light—part I,” Proc. Indian Acad. Sci. Sect. A 60, 297–312 (1964).
  10. R. M. A. Azzam, T. L. Bundy, N. M. Bashara, “The fixed-polarizer nulling scheme in generalized ellipsometry,” Opt. Commun. 7, 110–115 (1973).
    [CrossRef]

2000 (1)

1980 (1)

P. S. Hauge, “Recent development in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

1978 (1)

1977 (1)

1973 (1)

R. M. A. Azzam, T. L. Bundy, N. M. Bashara, “The fixed-polarizer nulling scheme in generalized ellipsometry,” Opt. Commun. 7, 110–115 (1973).
[CrossRef]

1964 (1)

S. R. Rajagopalan, S. Ramaseshan, “Rotating elliptic analysers for the automatic analysis of polarised light—part I,” Proc. Indian Acad. Sci. Sect. A 60, 297–312 (1964).

1954 (1)

Azzam, R. M. A.

R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
[CrossRef] [PubMed]

R. M. A. Azzam, T. L. Bundy, N. M. Bashara, “The fixed-polarizer nulling scheme in generalized ellipsometry,” Opt. Commun. 7, 110–115 (1973).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Bashara, N. M.

R. M. A. Azzam, T. L. Bundy, N. M. Bashara, “The fixed-polarizer nulling scheme in generalized ellipsometry,” Opt. Commun. 7, 110–115 (1973).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Bigelow, J. E.

Bundy, T. L.

R. M. A. Azzam, T. L. Bundy, N. M. Bashara, “The fixed-polarizer nulling scheme in generalized ellipsometry,” Opt. Commun. 7, 110–115 (1973).
[CrossRef]

Clarke, D.

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

Dereniak, E. L.

Descour, M. R.

Grainger, J. F.

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

Hauge, P. S.

P. S. Hauge, “Recent development in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Jerrard, H. G.

Kashnow, R. A.

Kemme, S. A.

Phipps, G. S.

Poincaré, H.

H. Poincaré, Theorie Mathématique de la Lumière (Gauthiers-Villars, Paris, 1892), Vol. II, Chap. 12.

Rajagopalan, S. R.

S. R. Rajagopalan, S. Ramaseshan, “Rotating elliptic analysers for the automatic analysis of polarised light—part I,” Proc. Indian Acad. Sci. Sect. A 60, 297–312 (1964).

Ramaseshan, S.

S. R. Rajagopalan, S. Ramaseshan, “Rotating elliptic analysers for the automatic analysis of polarised light—part I,” Proc. Indian Acad. Sci. Sect. A 60, 297–312 (1964).

Sabatke, D. S.

Sweatt, W. C.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

R. M. A. Azzam, T. L. Bundy, N. M. Bashara, “The fixed-polarizer nulling scheme in generalized ellipsometry,” Opt. Commun. 7, 110–115 (1973).
[CrossRef]

Opt. Lett. (2)

Proc. Indian Acad. Sci. Sect. A (1)

S. R. Rajagopalan, S. Ramaseshan, “Rotating elliptic analysers for the automatic analysis of polarised light—part I,” Proc. Indian Acad. Sci. Sect. A 60, 297–312 (1964).

Surf. Sci. (1)

P. S. Hauge, “Recent development in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Other (3)

H. Poincaré, Theorie Mathématique de la Lumière (Gauthiers-Villars, Paris, 1892), Vol. II, Chap. 12.

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

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Figures (5)

Fig. 1
Fig. 1

Polarization-state generator that consists of a linear polarizer and linear retarder. P and C are the orientation angles of the transmission axis of the polarizer and the fast axis of the retarder, respectively, relative to the reference x direction.

Fig. 2
Fig. 2

Locus of the polarization state of light leaving a fixed-polarizer rotating-retarder system (Fig. 1) is represented by the line of intersection of a right-circular cylinder with the Poincaré sphere. The diameter of the cylinder is half the diameter of the sphere for QWR (Δ=90°).

Fig. 3
Fig. 3

Normal projection of Fig. 2 on the (s1, s2) coordinate plane, where (s1, s2, s3) are the normalized Stokes parameters.

Fig. 4
Fig. 4

Normal projection of Fig. 2 on the (s1, s3) coordinate plane. The trajectory of the polarization states shown in Fig. 2 also lies on a parabolic cylinder parallel to the s2 axis.

Fig. 5
Fig. 5

Normal projection of Fig. 2 on the (s2, s3) coordinate plane showing a head-on view of the figure-8 contour.

Equations (9)

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s0=1,s1=cos2 2C+sin2 2C cos Δ,
s2=sin 2C cos 2C(1-cos Δ),s3=sin 2C sin Δ.
s1=cos2(Δ/2)+sin2(Δ/2)cos 4C,
s2=sin2(Δ/2)sin 4C.
[s1-cos2(Δ/2)]2+s22=sin4(Δ/2).
(s1, s2)=[cos2(Δ/2), 0],
r=sin2(Δ/2).
s32=[2 cos2(Δ/2)](1-s1),
s34-(sin2 Δ)s32+[4 cos2(Δ/2)]s22=0.

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