Abstract

The application of space periodic variation of light polarization for measurement and calculation of the distribution of the phase retardation between two eigenwaves propagating inside a linearly birefringent media and the distribution of the azimuth angle of the first eigenvector is described. The measuring method proposed does not require any mechanical movements or rotations of any optical elements. Application of a liquid crystal (LC) modulator instead of a quarter-wave plate gives an opportunity to introduce the required phase shift. The space periodic modulation of the polarization of light is achieved by the use of a Wollaston prism placed inside the path of the light beam. Then a fast Fourier transform is used for further calculations. The number of measurements of the light intensity at the output of the system is minimized to two. These assumptions make the proposed method very fast, which is especially important in measurements of the objects with optical anisotropy that is changing in time.

© 2005 Optical Society of America

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References

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  1. J. Jaronski, H. Kasprzak, “Generalized algorithm for photoelastic measurements based on phase-stepping imaging polarimetry,” Appl. Opt. 38, 7018–7025 (1999).
    [CrossRef]
  2. C. Quan, P. Bryanston-Cross, T. Judge, “Photoelasticity stress analysis using carrier fringe and fast Fourier transform techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
    [CrossRef]
  3. J. Jaronski, H. Kasprzak, D. Haszcz, J. Zagorski, “Investigation of the corneal structure by use of phase stepping imaging polarimetry,” in Proceedings of the European Optical Society Topical Meeting in Physiological Optics (European Optical Society, Warsaw, Poland, 23–25 September 1999), pp. 21–22.
  4. J. Bueno, F. Vargas-Martin, “Measurements of the corneal birefringence with a liquid-crystal imaging polariscope,” Appl. Opt. 41, 116–124 (2002).
    [CrossRef] [PubMed]
  5. J. Bueno, “Measurement of parameters of polarization in the living human eye using imaging polarimetry,” Vis. Res. 40, 3791–3799 (2002).
    [CrossRef]
  6. T. Shirai, “Liquid-crystal adaptive optics based on feedback interferometry for high-resolution retinal imaging,” Appl. Opt. 41, 4013–4023 (2002).
    [CrossRef] [PubMed]
  7. B. Laude-Boulesteix, A. De Martino, B. Drevillon, L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824–2832 (2004).
    [CrossRef] [PubMed]
  8. Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
    [CrossRef]
  9. S. Berezhna, I. Berezhnyy, M. Takashi, “High-resolution birefringence imaging in three-dimensional stressed models by Fourier polarimetry,” Appl. Opt. 40, 4940–4946 (2001).
    [CrossRef]
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    [CrossRef]
  11. K. Oka, K. Toshiaki, “Compact complete imaging polarimeter using birefringent wedge prism,” Opt. Express 11, 1510–1519 (2003).
    [CrossRef] [PubMed]
  12. P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), pp. 56–64.
  13. D. Ghiglia, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998), pp. 20–22, 44–46.
  14. S. Drobczynski, H. Kasprzak, “Modeling of influence of liquid crystal modulator adjustment on reconstruction of birefringence and azimuth angle in imaging polarimetry with carrier frequency,” in Proceedings of Optical Security and Safety (International Conference on Systems of Optical Security, Warsaw, Poland, 11–12 December 2003), Vol. 5566, pp. 273–277.

2004 (1)

2003 (1)

2002 (3)

2001 (1)

1999 (1)

1994 (1)

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

1993 (1)

C. Quan, P. Bryanston-Cross, T. Judge, “Photoelasticity stress analysis using carrier fringe and fast Fourier transform techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

1982 (1)

Berezhna, S.

Berezhnyy, I.

Bryanston-Cross, P.

C. Quan, P. Bryanston-Cross, T. Judge, “Photoelasticity stress analysis using carrier fringe and fast Fourier transform techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Bueno, J.

J. Bueno, “Measurement of parameters of polarization in the living human eye using imaging polarimetry,” Vis. Res. 40, 3791–3799 (2002).
[CrossRef]

J. Bueno, F. Vargas-Martin, “Measurements of the corneal birefringence with a liquid-crystal imaging polariscope,” Appl. Opt. 41, 116–124 (2002).
[CrossRef] [PubMed]

De Martino, A.

Drevillon, B.

Drobczynski, S.

S. Drobczynski, H. Kasprzak, “Modeling of influence of liquid crystal modulator adjustment on reconstruction of birefringence and azimuth angle in imaging polarimetry with carrier frequency,” in Proceedings of Optical Security and Safety (International Conference on Systems of Optical Security, Warsaw, Poland, 11–12 December 2003), Vol. 5566, pp. 273–277.

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), pp. 56–64.

Ghiglia, D.

D. Ghiglia, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998), pp. 20–22, 44–46.

Haszcz, D.

J. Jaronski, H. Kasprzak, D. Haszcz, J. Zagorski, “Investigation of the corneal structure by use of phase stepping imaging polarimetry,” in Proceedings of the European Optical Society Topical Meeting in Physiological Optics (European Optical Society, Warsaw, Poland, 23–25 September 1999), pp. 21–22.

Ina, H.

Jaronski, J.

J. Jaronski, H. Kasprzak, “Generalized algorithm for photoelastic measurements based on phase-stepping imaging polarimetry,” Appl. Opt. 38, 7018–7025 (1999).
[CrossRef]

J. Jaronski, H. Kasprzak, D. Haszcz, J. Zagorski, “Investigation of the corneal structure by use of phase stepping imaging polarimetry,” in Proceedings of the European Optical Society Topical Meeting in Physiological Optics (European Optical Society, Warsaw, Poland, 23–25 September 1999), pp. 21–22.

Judge, T.

C. Quan, P. Bryanston-Cross, T. Judge, “Photoelasticity stress analysis using carrier fringe and fast Fourier transform techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Kasprzak, H.

J. Jaronski, H. Kasprzak, “Generalized algorithm for photoelastic measurements based on phase-stepping imaging polarimetry,” Appl. Opt. 38, 7018–7025 (1999).
[CrossRef]

J. Jaronski, H. Kasprzak, D. Haszcz, J. Zagorski, “Investigation of the corneal structure by use of phase stepping imaging polarimetry,” in Proceedings of the European Optical Society Topical Meeting in Physiological Optics (European Optical Society, Warsaw, Poland, 23–25 September 1999), pp. 21–22.

S. Drobczynski, H. Kasprzak, “Modeling of influence of liquid crystal modulator adjustment on reconstruction of birefringence and azimuth angle in imaging polarimetry with carrier frequency,” in Proceedings of Optical Security and Safety (International Conference on Systems of Optical Security, Warsaw, Poland, 11–12 December 2003), Vol. 5566, pp. 273–277.

Kobayashi, S.

Laude-Boulesteix, B.

Oka, K.

Otani, Y.

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Quan, C.

C. Quan, P. Bryanston-Cross, T. Judge, “Photoelasticity stress analysis using carrier fringe and fast Fourier transform techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Schwartz, L.

Shimada, T.

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Shirai, T.

Takashi, M.

Takeda, M.

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), pp. 56–64.

Toshiaki, K.

Umeda, N.

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Vargas-Martin, F.

Yoshizawa, T.

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Zagorski, J.

J. Jaronski, H. Kasprzak, D. Haszcz, J. Zagorski, “Investigation of the corneal structure by use of phase stepping imaging polarimetry,” in Proceedings of the European Optical Society Topical Meeting in Physiological Optics (European Optical Society, Warsaw, Poland, 23–25 September 1999), pp. 21–22.

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

Y. Otani, T. Shimada, T. Yoshizawa, N. Umeda, “Two-dimensional birefringence measurement using the phase shifting technique,” Opt. Eng. 33, 1604–1609 (1994).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

C. Quan, P. Bryanston-Cross, T. Judge, “Photoelasticity stress analysis using carrier fringe and fast Fourier transform techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Vis. Res. (1)

J. Bueno, “Measurement of parameters of polarization in the living human eye using imaging polarimetry,” Vis. Res. 40, 3791–3799 (2002).
[CrossRef]

Other (4)

J. Jaronski, H. Kasprzak, D. Haszcz, J. Zagorski, “Investigation of the corneal structure by use of phase stepping imaging polarimetry,” in Proceedings of the European Optical Society Topical Meeting in Physiological Optics (European Optical Society, Warsaw, Poland, 23–25 September 1999), pp. 21–22.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), pp. 56–64.

D. Ghiglia, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998), pp. 20–22, 44–46.

S. Drobczynski, H. Kasprzak, “Modeling of influence of liquid crystal modulator adjustment on reconstruction of birefringence and azimuth angle in imaging polarimetry with carrier frequency,” in Proceedings of Optical Security and Safety (International Conference on Systems of Optical Security, Warsaw, Poland, 11–12 December 2003), Vol. 5566, pp. 273–277.

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Figures (8)

Fig. 1
Fig. 1

Scheme of the optical system. [P], polarizer; [QW], quarter-wave plate; [OB], examined object; [LC], LC modulator; [W], Wollaston prism; [A], analyzer.

Fig. 2
Fig. 2

Example of the distribution of the total intensity Ic(x, y), with a Gaussian distribution of the illuminating beam intensity.

Fig. 3
Fig. 3

Assumed distribution of (top) the azimuth angle of the object αOB(x, y) and (bottom) the phase retardation γOB(x, y).

Fig. 4
Fig. 4

Recorded light intensity for two states of the LC modulator: top, γLC1 = 90°; bottom, γLC2 = 0°.

Fig. 5
Fig. 5

Fourier spectrum for (top) γLC1 = 90° and (bottom) γLC2 = 0°.

Fig. 6
Fig. 6

Calculated distribution of (top) azimuth angle α OB ( x , y ) and (bottom) phase retardation γ OB ( x , y ).

Fig. 7
Fig. 7

Calculated distribution of (top) azimuth angle α OB ( x , y ) along the line shown in Fig. 6a (top) and (bottom) difference between assumed values αOB(x, y) and the calculated distribution α OB ( x , y ).

Fig. 8
Fig. 8

Calculated distribution of (top) phase retardation γ OB ( x , y ) along the line shown in Fig. 6 (bottom) and (bottom) the difference between assumed function γOB(x, y) and the calculated distribution γ OB ( x , y ).

Equations (22)

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[ S 0 ] = [ I 0 0 0 0 ]
[ P ] = 1 2 [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ]
[ QW ] = T QW 2 [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ] ,
[ OB ] = T OB 2 [ 1 0 0 0 0 cos 2 2 α OB ( 1 cos γ OB ) + cos γ OB sin 2 α OB cos 2 α OB ( 1 cos γ OB ) sin γ OB sin 2 α OB 0 sin 2 α OB cos 2 α OB ( 1 cos γ OB ) sin 2 2 α OB ( 1 cos γ OB ) + cos γ OB sin γ OB cos 2 α OB 0 sin γ OB sin 2 α OB sin γ OB cos 2 α OB cos γ OB ] ,
[ LC ] = T LC 2 [ 1 0 0 0 0 cos γ LC 0 sin γ LC 0 0 1 0 0 sin γ LC 0 cos γ LC ] ,
[ W ] = T W 2 [ 1 0 0 0 0 1 0 0 0 0 cos ( 2 π f 0 x ) sin ( 2 π f 0 x ) 0 0 sin ( 2 π f 0 x ) cos ( 2 π f 0 x ) ] ,
[ A ] = 1 2 [ 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ] ,
γ LC = f ( U ) .
[ S ] = [ A ] [ W ] [ LC ] [ OB ] [ QW ] [ P ] [ S 0 ] .
I c ( x , y ) = 1 4 T QW 2 ( x , y ) T OB 2 ( x , y ) T LC 2 ( x , y ) × T W 2 ( x , y ) I 0 ( x , y ) ,
I n ( x , y ) = I c ( x , y ) { 1 + A n ( x , y ) cos [ 2 π f 0 x + φ n ( x , y ) ] } , n = 1 , 2 .
I 1 ( x , y ) = I c ( x , y ) { 1 + A 1 ( x , y ) cos [ 2 π f 0 x + φ 1 ( x , y ) ] } ,
A 1 2 ( x , y ) = sin 2 [ γ OB ( x , y ) ] ,
tan [ φ 1 ( x , y ) ] = sin [ 2 α OB ( x , y ) ] / cos [ 2 α OB ( x , y ) ] ,
I 2 ( x , y ) = I c ( x , y ) { 1 + A 2 ( x , y ) cos [ 2 π f 0 x + φ 2 ( x , y ) ] } ,
A 2 2 ( x , y ) = { 1 sin 2 [ γ OB ( x , y ) ] sin 2 [ 2 α OB ( x , y ) ] } 2 ,
tan [ φ 2 ( x , y ) ] = cos [ γ OB ( x , y ) ] / { sin [ γ OB ( x , y ) ] cos [ 2 α OB ( x , y ) ] } .
I n ( x , y ) = I c ( x , y ) + c n ( x , y ) exp ( 2 π i f 0 x ) + c n * ( x , y ) × exp ( 2 π i f 0 x ) , n = 1 , 2 ,
c n ( x , y ) = 1 2 I c ( x , y ) A n ( x , y ) exp ( i φ n ( x , y ) ] , n = 1 , 2 .
φ n ( x , y ) = arctan { Im [ c n ( x , y ) ] / Re [ c n ( x , y ) ] } , n = 1 , 2 .
α OB ( x , y ) = 1 2 arctan { Im [ c 1 ( x , y ) ] Re [ c 1 ( x , y ) ] } ,
γ OB ( x , y ) = arctan { Re [ c 2 ( x , y ) ] Im [ c 2 ( x , y ) ] cos [ 2 α OB ( x , y ) ] } .

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