Abstract

An analytical method for phase detection in retardation measurements is proposed. The experimental setup is based on a simple linear polariscope with a λ plate. The intensity modulation at the output of the polariscope is measured when the wavelength is changed and a grid of phase-shifted intensity values is recorded. The phase difference between the components of the light propagating along the principal axes of the birefringent sample is determined with Greivenkamp’s algorithm employed in phase-stepping interferometry. Error analysis for the new method is performed. Simplified algorithms for faster data analysis are proposed. The accuracy attained is comparable with the accuracy of known phase-detection methods used in retardation measurements.

© 1994 Optical Society of America

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References

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  1. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 351–398.
  2. J. Badoz, M. P. Silverman, J. C. Canit, “Wave propagation through a medium with static and dynamic birefringence: theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).
    [CrossRef]
  3. Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 24, 240–244 (1983).
  4. S. Nakadate, “High precision retardation measurement using phase detection of Young’s fringes,” Appl. Opt. 29, 242–246 (1990).
    [CrossRef] [PubMed]
  5. K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Error analysis on the phase calculation from superimposed interferograms generated by a wavelength scanning interferometer,” Opt. Commun. 77, 343–348 (1990).
    [CrossRef]
  6. T. S. Narasimhamurty, M. Ziauddin, “Birefringent compensator for studying very small changes in double refraction,” J. Opt. Soc. Am. 51, 574–581 (1961).
    [CrossRef]
  7. G. Nechev, L. Ivanova, “Digital birefringence measurement using the cross-correlation method,” Appl. Opt. 31, 6716–6719 (1992).
    [CrossRef] [PubMed]
  8. J. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  9. N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference phase shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
    [CrossRef]

1992 (1)

1990 (3)

1988 (1)

1984 (1)

J. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1983 (1)

Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 24, 240–244 (1983).

1961 (1)

Badoz, J.

Canit, J. C.

Cornejo-Rodriguez, A.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 351–398.

Greivenkamp, J.

J. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Hanabusa, H.

Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 24, 240–244 (1983).

Honda, T.

Ivanova, L.

Kinoshita, S.

Nakadate, S.

Narasimhamurty, T. S.

Nechev, G.

Ohyama, N.

Okada, K.

K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Error analysis on the phase calculation from superimposed interferograms generated by a wavelength scanning interferometer,” Opt. Commun. 77, 343–348 (1990).
[CrossRef]

Ose, T.

K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Error analysis on the phase calculation from superimposed interferograms generated by a wavelength scanning interferometer,” Opt. Commun. 77, 343–348 (1990).
[CrossRef]

Sakuta, H.

K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Error analysis on the phase calculation from superimposed interferograms generated by a wavelength scanning interferometer,” Opt. Commun. 77, 343–348 (1990).
[CrossRef]

Shindo, Y.

Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 24, 240–244 (1983).

Silverman, M. P.

Tsujiuchi, J.

K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Error analysis on the phase calculation from superimposed interferograms generated by a wavelength scanning interferometer,” Opt. Commun. 77, 343–348 (1990).
[CrossRef]

N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference phase shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
[CrossRef]

Ziauddin, M.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

K. Okada, H. Sakuta, T. Ose, J. Tsujiuchi, “Error analysis on the phase calculation from superimposed interferograms generated by a wavelength scanning interferometer,” Opt. Commun. 77, 343–348 (1990).
[CrossRef]

Opt. Eng. (1)

J. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Polym. Commun. (1)

Y. Shindo, H. Hanabusa, “Highly sensitive instrument for measuring optical birefringence,” Polym. Commun. 24, 240–244 (1983).

Other (1)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 351–398.

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Figures (3)

Fig. 1
Fig. 1

Setup used to measure the intensity I0 employed in the normalization procedure.

Fig. 2
Fig. 2

Intensity recorded at the polariscope output (a) before the normalization procedure, (b) after the normalization procedure.

Fig. 3
Fig. 3

Intensity recorded at the polariscope output before and after samples of known birefringence have been introduced. Curve 1, the intensity before the sample has been introduced; curve 2, the intensity recorded after a sample with 35.6-nm retardation has been introduced; curve 3, sample with 60.6-nm retardation; curve 4, sample with 81.4-nm retardation.

Equations (12)

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I ( λ ) = 16 ( n o + 1 ) 4 [ 1 2 n o ( n o 1 ) n o + 1 ] I p sin 2 ( δ / 2 ) ,
δ = 2 π λ 0 ( n e n o ) t ,
α ( Δλ ) = 2 π λ 0 Γ Δλ λ 0 ,
I i = I 0 i [ 1 cos ( δ δ Δλ i λ 0 ) ]
I 0 ( λ ) = 16 ( n o + 1 ) 4 [ 1 2 n o ( n o 1 ) n o + 1 ] I p .
I i = 0 . 5 I 0 i { 1 cos [ 2 π λ 0 Γ + 2 π λ 0 ( Q Q Δ λ i λ 0 ) ] } + I f i ,
I i = a 0 + a 1 cos δ i + a 2 sin δ i ,
a 0 = ( I 0 i + I f i ) / I 0 i , a 1 = cos ( 2 π λ 0 Γ ) . a 2 = sin ( 2 π λ 0 Γ ) , δ i = 2 π λ 0 ( Q Q Δλ i λ 0 ) .
n e n 0 = λ 0 2 π 1 t tan 1 ( a 2 a 1 ) .
ϕ = tan 1 ( Σ A n I n / Σ B n I n )
α i = 2 π ( i 1 ) N Q λ 0 , i = 1 , ... , N .
C 12 ( n ) = 1 N 0 N I 1 ( φ ) I 2 ( φ δφ ) d φ ,

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