Abstract

The basic compensator–nalyzer–olarimeter is automated by inserting two modulated optical rotators, one in front of the compensator and the other between the compensator and the analyzer. The modulated optical rotators (e.g., two series-connected Faraday cells) are excited by the same ac source to produce sinusoidal optical rotations of the same frequency, generally of different amplitudes, that are mutally in or 180° out of phase. We show that, when the radiation leaving this generalized polarimeter (which is of constant polarization) is linearly detected, limited Fourier analysis of the resulting signal yields the four Stokes parameters of the incident radiation.

© 1977 Optical Society of America

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References

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  1. R. M. A. Azzam, “Simulation of mechanical rotation by optical rotation: application to the design of a new Fourier photopolarimeter,” J. Opt. Soc. Am. (in press).
  2. R. M. A. Azzam, “Oscillating-analyzer ellipsometer,” Rev. Sci. Instrum. 47, 624–628 (1976).
    [CrossRef]
  3. R. M. A. Azzam, “Hybrid null-photometric ellipsometer using sinusoidal optical rotation,” Optik 48, 279–288 (1977).
  4. See, for example, D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971), p. 124.
  5. M. Abramowitz, J. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 361.
  6. P. S. Hauge, “Survey of methods for the complete determination of a state of polarization,” in Polarized Light: Instruments, Devices, and Applications, W. L. Hyde, R. M. A. Azzam, eds., Proc. Soc. Photo-Opt. Instrum. Eng.88, 3–10 (1976).
  7. When r̂1=0, we have an oscillating- (linear-) analyzer ellipsometer preceded by a compensator in the path of the incident radiation. The effect of the compensator is to scramble the Stokes parameters such that, if measurements are made with the compensator in and out of the path of the incident beam, complete determination of the Stokes parameters of the incident light becomes possible. When r̂2=0, the combination of the compensator and analyzer can be considered as one unit, an elliptic analyzer, which is oscillated because of the presence of r̂1≠0. This results in an oscillating-(elliptic-) analyzer ellipsometer (partial polarimeter), which is capable of measuring S1, and S2 and a linear combination of S0 and S3. Obviously, when C = 0 (i.e., when the principal axes of the compensator and the analyzer are coincident), the compensator becomes essentially nonfunctional and the ellipsometer reduces to the oscillating- (linear-) analyzer ellipsometer.
  8. D. E. Aspnes, P. S. Hauge, “Rotating-compensator/analyzer fixed-analyzer ellipsometer: analysis and comparison to other automatic ellipsometers,” J. Opt. Soc. Am. 66, 949–954 (1976).
    [CrossRef]

1977 (1)

R. M. A. Azzam, “Hybrid null-photometric ellipsometer using sinusoidal optical rotation,” Optik 48, 279–288 (1977).

1976 (2)

Aspnes, D. E.

Azzam, R. M. A.

R. M. A. Azzam, “Hybrid null-photometric ellipsometer using sinusoidal optical rotation,” Optik 48, 279–288 (1977).

R. M. A. Azzam, “Oscillating-analyzer ellipsometer,” Rev. Sci. Instrum. 47, 624–628 (1976).
[CrossRef]

R. M. A. Azzam, “Simulation of mechanical rotation by optical rotation: application to the design of a new Fourier photopolarimeter,” J. Opt. Soc. Am. (in press).

Clarke, D.

See, for example, D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971), p. 124.

Grainger, J. F.

See, for example, D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971), p. 124.

Hauge, P. S.

D. E. Aspnes, P. S. Hauge, “Rotating-compensator/analyzer fixed-analyzer ellipsometer: analysis and comparison to other automatic ellipsometers,” J. Opt. Soc. Am. 66, 949–954 (1976).
[CrossRef]

P. S. Hauge, “Survey of methods for the complete determination of a state of polarization,” in Polarized Light: Instruments, Devices, and Applications, W. L. Hyde, R. M. A. Azzam, eds., Proc. Soc. Photo-Opt. Instrum. Eng.88, 3–10 (1976).

J. Opt. Soc. Am. (1)

Optik (1)

R. M. A. Azzam, “Hybrid null-photometric ellipsometer using sinusoidal optical rotation,” Optik 48, 279–288 (1977).

Rev. Sci. Instrum. (1)

R. M. A. Azzam, “Oscillating-analyzer ellipsometer,” Rev. Sci. Instrum. 47, 624–628 (1976).
[CrossRef]

Other (5)

R. M. A. Azzam, “Simulation of mechanical rotation by optical rotation: application to the design of a new Fourier photopolarimeter,” J. Opt. Soc. Am. (in press).

See, for example, D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971), p. 124.

M. Abramowitz, J. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 361.

P. S. Hauge, “Survey of methods for the complete determination of a state of polarization,” in Polarized Light: Instruments, Devices, and Applications, W. L. Hyde, R. M. A. Azzam, eds., Proc. Soc. Photo-Opt. Instrum. Eng.88, 3–10 (1976).

When r̂1=0, we have an oscillating- (linear-) analyzer ellipsometer preceded by a compensator in the path of the incident radiation. The effect of the compensator is to scramble the Stokes parameters such that, if measurements are made with the compensator in and out of the path of the incident beam, complete determination of the Stokes parameters of the incident light becomes possible. When r̂2=0, the combination of the compensator and analyzer can be considered as one unit, an elliptic analyzer, which is oscillated because of the presence of r̂1≠0. This results in an oscillating-(elliptic-) analyzer ellipsometer (partial polarimeter), which is capable of measuring S1, and S2 and a linear combination of S0 and S3. Obviously, when C = 0 (i.e., when the principal axes of the compensator and the analyzer are coincident), the compensator becomes essentially nonfunctional and the ellipsometer reduces to the oscillating- (linear-) analyzer ellipsometer.

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Figures (1)

Fig. 1
Fig. 1

Photopolarimeter using two optical rotators. C and A represent the (quarter-wave) compensator and (linear) analyzer with azimuth angles C and A, while R1 and R2 are two optical rotators that produce rotations r1 and r2, respectively. D is a linear photodetector that measures the light flux leaving the polarimeter, and its output signal i is analyzed to yield the Stokes parameters of the incident radiation. (x,y), (f,s), and (t,e) represent a reference coordinate system, the (fast, slow) principal axes of the compensator, and the (transmission, extinction) axes of the analyzer, respectively.

Equations (11)

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i = S 0 + ( S 1 cos 2 C + S 2 sin 2 C ) × cos ( 2 A 2 C ) + S 3 sin ( 2 A 2 C ) ,
i = S 0 + S 1 cos ( 2 C + 2 r 1 ) cos ( 2 A 2 C + 2 r 2 ) + S 2 sin ( 2 C + 2 r 1 ) cos ( 2 A 2 C + 2 r 2 ) + S 3 sin ( 2 A 2 C + 2 r 2 ) .
i = S 0 + ½ S 1 [ cos 4 C cos ( 2 r 1 2 r 2 ) sin 4 C sin ( 2 r 1 2 r 2 ) + cos ( 2 r 1 + 2 r 2 ) ] + ½ S 2 [ sin 4 C cos ( 2 r 1 2 r 2 ) + cos 4 C sin ( 2 r 1 2 r 2 ) + sin ( 2 r 1 + 2 r 2 ) ] S 3 ( sin 2 C cos 2 r 2 cos 2 C sin 2 r 2 ) .
r 1 = r ̂ 1 sin ω t , r 2 = r ̂ 2 sin ω t ,
i ̂ = M S ,
i ̂ = ( i ̂ ω i ̂ 2 ω i ̂ 3 ω ) S = ( S 1 S 2 S 3 ) ,
M = [ sin 4 C J 1 ( 2 r ̂ 1 2 r ̂ 2 ) cos 4 C J 1 ( 2 r ̂ 1 2 r ̂ 2 ) + J 1 ( 2 r ̂ 1 2 r ̂ 2 ) 2 cos 2 C J 1 ( 2 r ̂ 2 ) cos 4 C J 2 ( 2 r ̂ 1 2 r ̂ 2 ) + J 2 ( 2 r ̂ 1 + 2 r ̂ 2 ) sin 4 C J 2 ( 2 r ̂ 1 2 r ̂ 2 ) 2 sin 2 C J 2 ( 2 r ̂ 2 ) sin 4 C J 3 ( 2 r ̂ 1 2 r ̂ 2 ) cos 4 C J 3 ( 2 r ̂ 1 2 r ̂ 2 ) + J 3 ( 2 r ̂ 1 2 r ̂ 2 ) 2 cos 2 C J 3 ( 2 r ̂ 2 ) ] .
i dc = S 0 + ½ S 1 [ cos 4 C J 0 ( ( 2 r ̂ 1 2 r ̂ 2 ) J 0 ( 2 r ̂ 1 + 2 r ̂ 2 ) ] + ½ S 2 [ sin 4 C J 0 ( 2 r ̂ 1 2 r ̂ 2 ) ] S 3 [ sin 2 C J 0 ( 2 r ̂ 2 ) ] .
S = M 1 i ̂ .
det M 0 ,
det M = 0 ,

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