Abstract

The Stokes parameters (S0, S1, S2, and S3) of monochromatic light can be measured using the adjustable azimuth settings of a quarter-wave plate and a polarizer. When measuring the Stokes parameters of light of an arbitrary wavelength, the measurement of S3 is affected by the phase difference error Δqλi, due to the mismatch with respect to wavelength with the quarter-wave plate. In this method, Δqλi, due to such a mismatch of incident light of arbitrary wavelength, can be overcome by a judicious choice of azimuth settings of the quarter-wave plate and the use of a polarizer; however, the use of a precision quarter-wave plate is necessary. The present paper proposes a measurement method of Stokes parameters of incident light of arbitrary wavelength using a quarter-wave plate with phase difference errors.

© 2011 Optical Society of America

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References

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  1. D. Clark and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, 1971).
  2. P. S. Hauge, “Survey of methods for the complete determination of a state of polarization,” Proc. SPIE 88, 3–10 (1976).
  3. S. Yoneyama, H. Kikuta, and K. K. Moriwaki, “Instantaneous phase-stepping interferometry using polarization imaging with a micro-retarder array,” Exp. Mech. 45, 451–456 (2005).
    [CrossRef]
  4. T. Kihara, “Stokes parameters measurement of light over a wide wavelength range by judicious choice of azimuthal settings of quarter-wave plate and linear polarizer,” Optics Commun. 110, 529–532 (1994).
    [CrossRef]
  5. T. Kihara, “Measurement of Stokes parameters by quarter-wave plate and polarizer,” in Advance in Experimental Mechanics IV, J.M.Dulieu-Barton and S.Quinn, eds. (Trans Tech, 2005), pp. 235–240.
  6. L. D’Acquisto, G. Petrucci, and B. Zuccarello, “Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,” Opt. Lasers Eng. 37, 389–400(2002).
    [CrossRef]
  7. P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).
  8. T. Kihara, “An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths,” Strain 39, 65–71(2003).
    [CrossRef]
  9. M. M. Frocht, Photoelasticity (Wiley, 1948), Vol.  2, Chap. 4.

2005 (1)

S. Yoneyama, H. Kikuta, and K. K. Moriwaki, “Instantaneous phase-stepping interferometry using polarization imaging with a micro-retarder array,” Exp. Mech. 45, 451–456 (2005).
[CrossRef]

2003 (1)

T. Kihara, “An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths,” Strain 39, 65–71(2003).
[CrossRef]

2002 (1)

L. D’Acquisto, G. Petrucci, and B. Zuccarello, “Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,” Opt. Lasers Eng. 37, 389–400(2002).
[CrossRef]

1994 (1)

T. Kihara, “Stokes parameters measurement of light over a wide wavelength range by judicious choice of azimuthal settings of quarter-wave plate and linear polarizer,” Optics Commun. 110, 529–532 (1994).
[CrossRef]

1976 (1)

P. S. Hauge, “Survey of methods for the complete determination of a state of polarization,” Proc. SPIE 88, 3–10 (1976).

Clark, D.

D. Clark and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, 1971).

D’Acquisto, L.

L. D’Acquisto, G. Petrucci, and B. Zuccarello, “Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,” Opt. Lasers Eng. 37, 389–400(2002).
[CrossRef]

Frocht, M. M.

M. M. Frocht, Photoelasticity (Wiley, 1948), Vol.  2, Chap. 4.

Gdoutos, E. E.

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

Grainger, J. F.

D. Clark and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, 1971).

Hauge, P. S.

P. S. Hauge, “Survey of methods for the complete determination of a state of polarization,” Proc. SPIE 88, 3–10 (1976).

Kihara, T.

T. Kihara, “An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths,” Strain 39, 65–71(2003).
[CrossRef]

T. Kihara, “Stokes parameters measurement of light over a wide wavelength range by judicious choice of azimuthal settings of quarter-wave plate and linear polarizer,” Optics Commun. 110, 529–532 (1994).
[CrossRef]

T. Kihara, “Measurement of Stokes parameters by quarter-wave plate and polarizer,” in Advance in Experimental Mechanics IV, J.M.Dulieu-Barton and S.Quinn, eds. (Trans Tech, 2005), pp. 235–240.

Kikuta, H.

S. Yoneyama, H. Kikuta, and K. K. Moriwaki, “Instantaneous phase-stepping interferometry using polarization imaging with a micro-retarder array,” Exp. Mech. 45, 451–456 (2005).
[CrossRef]

Moriwaki, K. K.

S. Yoneyama, H. Kikuta, and K. K. Moriwaki, “Instantaneous phase-stepping interferometry using polarization imaging with a micro-retarder array,” Exp. Mech. 45, 451–456 (2005).
[CrossRef]

Petrucci, G.

L. D’Acquisto, G. Petrucci, and B. Zuccarello, “Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,” Opt. Lasers Eng. 37, 389–400(2002).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

Yoneyama, S.

S. Yoneyama, H. Kikuta, and K. K. Moriwaki, “Instantaneous phase-stepping interferometry using polarization imaging with a micro-retarder array,” Exp. Mech. 45, 451–456 (2005).
[CrossRef]

Zuccarello, B.

L. D’Acquisto, G. Petrucci, and B. Zuccarello, “Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,” Opt. Lasers Eng. 37, 389–400(2002).
[CrossRef]

Exp. Mech. (1)

S. Yoneyama, H. Kikuta, and K. K. Moriwaki, “Instantaneous phase-stepping interferometry using polarization imaging with a micro-retarder array,” Exp. Mech. 45, 451–456 (2005).
[CrossRef]

Opt. Lasers Eng. (1)

L. D’Acquisto, G. Petrucci, and B. Zuccarello, “Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,” Opt. Lasers Eng. 37, 389–400(2002).
[CrossRef]

Optics Commun. (1)

T. Kihara, “Stokes parameters measurement of light over a wide wavelength range by judicious choice of azimuthal settings of quarter-wave plate and linear polarizer,” Optics Commun. 110, 529–532 (1994).
[CrossRef]

Proc. SPIE (1)

P. S. Hauge, “Survey of methods for the complete determination of a state of polarization,” Proc. SPIE 88, 3–10 (1976).

Strain (1)

T. Kihara, “An arctangent unwrapping technique of photoelasticity using linearly polarized light at three wavelengths,” Strain 39, 65–71(2003).
[CrossRef]

Other (4)

M. M. Frocht, Photoelasticity (Wiley, 1948), Vol.  2, Chap. 4.

P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, 1979).

D. Clark and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, 1971).

T. Kihara, “Measurement of Stokes parameters by quarter-wave plate and polarizer,” in Advance in Experimental Mechanics IV, J.M.Dulieu-Barton and S.Quinn, eds. (Trans Tech, 2005), pp. 235–240.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of Stokes parameters measurement: S = ( S 0 , S 1 , S 2 , S 3 ) , polarized light; Q 2 , quarter-wave plate; P 2 , polarizer; D, diffuser.

Fig. 2
Fig. 2

Image and distribution along the horizontal position for phase difference error Δ q λ i of the quarter-wave plate with respect to λ i = 514.5 nm .

Fig. 3
Fig. 3

Images and distributions along the horizontal position for simulated intensities (a)  I ( 0 , 45 ) , (b)  I ( 0 , 135 ) .

Fig. 4
Fig. 4

Images and distributions along the horizontal position for simulated intensities (a)  I ( 0 , 45 ) ini , (b)  I ( 0 , 135 ) ini , (c)  sin Δ q λ i ( x , y ) , and (d)  cos Δ q λ i ( x , y ) .

Fig. 5
Fig. 5

Image and distribution along the horizontal diameter for (a)  ρ λ i tot and (b)  Δ ρ λ i tot , simulated using the proposed method.

Fig. 6
Fig. 6

Image and distribution along the horizontal diameter for (a)  ρ λ i tot and (b)  Δ ρ λ i tot , simulated using Eq. (8).

Equations (19)

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S = P 2 ( α 2 ) Q 2 ( β 2 , Δ q λ i ) S ,
P 2 ( α 2 ) = 1 2 [ 1 cos 2 α 2 sin 2 α 2 0 cos 2 α 2 cos 2 2 α 2 sin 2 α 2 cos 2 α 2 0 sin 2 α 2 sin 2 α 2 cos 2 α 2 sin 2 2 α 2 0 0 0 0 0 ] ,
Q 2 ( β 2 , Δ q λ i ) = [ 1 0 0 0 0 1 ( 1 cos ( π / 2 + Δ q λ i ) ) sin 2 2 β 2 ( 1 cos ( π / 2 + Δ q λ i ) ) sin 2 β 2 cos 2 β 2 sin ( π / 2 + Δ q λ i ) sin 2 β 2 0 ( 1 cos ( π / 2 + Δ q λ i ) ) sin 2 β 2 cos 2 β 2 1 ( 1 cos ( π / 2 + Δ q λ i ) ) cos 2 2 β 2 sin ( π / 2 + Δ q λ i ) cos 2 β 2 0 sin ( π / 2 + Δ q λ i ) sin 2 β 2 sin ( π / 2 + Δ q λ i ) cos 2 β 2 cos ( π / 2 + Δ q λ i ) ] ,
I ( β 2 , α 2 ) = 0.5 { S 0 + S 1 [ cos 2 β 2 cos 2 ( α 2 β 2 ) + sin Δ q λ i ( x , y ) sin 2 β 2 sin 2 ( α 2 β 2 ) ] + S 2 { sin 2 β 2 cos 2 ( α 2 β 2 ) sin Δ q λ i cos 2 β 2 sin 2 ( α 2 β 2 ) ] + S 3 cos Δ q λ i sin 2 ( α 2 β 2 ) } .
S 0 = I ( 0 , 0 ) + I ( 90 , 90 ) ,
S 1 = I ( 0 , 0 ) I ( 90 , 90 ) ,
S 2 = I ( 45 , 45 ) I ( 135 , 135 ) ,
S 3 = I ( 0 , 45 ) I ( 0 , 135 ) .
I ( 0 , 0 ) I ( 90 , 90 ) = S 1 ,
I ( 45 , 45 ) I ( 135 , 135 ) = S 2 ,
I ( 0 , 45 ) I ( 0 , 135 ) = S 2 sin Δ q λ i + S 3 cos Δ q λ i .
S 3 = [ I ( 135 , 0 ) I ( 45 , 0 ) ] / cos Δ q λ i .
S 3 = [ I ( 0 , 45 ) I ( 0 , 135 ) + S 2 sin Δ q λ i ( x , y ) ] / cos Δ q λ i ( x , y ) .
sin Δ q λ i ( x , y ) = [ I ( 0 , 135 ) ini I ( 0 , 45 ) ini ] ,
Δ q λ i ( x , y ) = sin 1 [ I ( 0 , 135 ) ini I ( 0 , 45 ) ini ] ,
δ = ( 2 π / λ i ) [ 140 + 20 cos ( 2 π x / 160 ) ] ,
Δ q λ i ( x , y ) = ( 2 π / λ i ) [ 140 + 20 cos ( 2 π x / 160 ) ] π / 2 .
ψ = 0.25 tan 1 [ ( s 12 + s 21 ) / ( s 11 s 22 ) ] = 0.25 tan 1 { [ 2 sin 2 ( ρ λ i / 2 ) sin 4 ψ ] / [ 2 sin 2 ( ρ λ i / 2 ) cos 4 ψ ] } ,
ρ λ i = tan 1 [ ( s 13 sin 2 ψ s 23 cos 2 ψ ) / ( s 11 + s 22 1 ) ] = tan 1 [ sin ρ λ i / cos ρ λ i ] ,

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