Abstract

Phase detection of Young’s fringes is applied to a highly precise retardation measurement. A simple common-path polarizing interferometer is used with a birefringent wedge and a polarizer. The birefringent wedge introduces a spatially linear phase difference between orthogonally polarized light and Young’s fringes are formed on an image sensor. The phase difference between the orthogonally polarized components of light is proportional to the phase of Young’s fringes. Thus, the retardation is equal to the Young’s fringes’ phase change before and after insertion of the retarder into the common-path interferometer. The phase of Young’s fringes is calculated from the Fourier cosine and sine integrals of the fringe profile. The experimental results for wave plates, a Soleil-Babinet compensator, and a Pockels cell are presented with error estimates. The accuracy of the retardation measurement is experimentally estimated to be greater than λ/2100.

© 1990 Optical Society of America

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References

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  1. M. Françon, S. Mallick, Polarization Interferometers: Application in Microscopy and Macroscopy (Wiley, London, 1971).
  2. I. P. Kaminow, E. H. Furner, “Electrooptic Light Modulators,” Proc. IEEE 54, 1374–1390 (1966).
    [CrossRef]
  3. A. J. Fox, T. M. Bruton, “Electro-Optic Effects in the Optically Active Compounds Bi12TiO20 and Bi40Ga2O63,” Appl. Phys. Lett. 27, 360–362 (1975).
    [CrossRef]
  4. J. P. Huignard, H. Rajbenbach, Ph. Refrégier, L. Solyman, “Wave Mixing in Photorefractive Bismuth Silicon Oxide Crystals and Its Applications,” Opt. Eng. 24, 586–592 (1985).
  5. H. G. Jerrard, “Optical Compensators for Measurement of Elliptical Polarization,” J. Opt. Soc. Am. 38, 35–59 (1948).
    [CrossRef]
  6. G. Bruhat, Optique (Cours de Physique Générale) (Masson, Paris, 1959).
  7. L. Yao, Z. Zhiyao, W. Runwen, “Optical Heterodyne Measurement of the Phase Retardation of a Quarter-Wave Plate,” Opt. Lett. 13, 553–555 (1988).
    [CrossRef] [PubMed]
  8. H. Takasaki, “Photoelectric Measurement of Polarized Light by Means of an ADP Polarization Modulator. I. Photoelectric Polarimeter,” J. Opt. Soc. Am. 51, 462–463 (1961).
    [CrossRef]
  9. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1980).
  10. P. S. Hauge, F. H. Dill, “Design and Operation of ETA, an Automated Ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
    [CrossRef]
  11. S. Nakadate, “Phase Detection of Equidistant Fringes for Highly Sensitive Optical Sensing. I. Principle and Error Analyses,” J. Opt. Soc. Am. A 5, 1258–1264 (1988).
    [CrossRef]
  12. S. Nakadate, “Phase Detection of Equidistant Fringes for Highly Sensitive Optical Sensing. II. Experiments,” J. Opt. Soc. Am. A 5, 1265–1269 (1988).
    [CrossRef]

1988 (3)

1985 (1)

J. P. Huignard, H. Rajbenbach, Ph. Refrégier, L. Solyman, “Wave Mixing in Photorefractive Bismuth Silicon Oxide Crystals and Its Applications,” Opt. Eng. 24, 586–592 (1985).

1975 (1)

A. J. Fox, T. M. Bruton, “Electro-Optic Effects in the Optically Active Compounds Bi12TiO20 and Bi40Ga2O63,” Appl. Phys. Lett. 27, 360–362 (1975).
[CrossRef]

1973 (1)

P. S. Hauge, F. H. Dill, “Design and Operation of ETA, an Automated Ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

1966 (1)

I. P. Kaminow, E. H. Furner, “Electrooptic Light Modulators,” Proc. IEEE 54, 1374–1390 (1966).
[CrossRef]

1961 (1)

1948 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1980).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1980).

Bruhat, G.

G. Bruhat, Optique (Cours de Physique Générale) (Masson, Paris, 1959).

Bruton, T. M.

A. J. Fox, T. M. Bruton, “Electro-Optic Effects in the Optically Active Compounds Bi12TiO20 and Bi40Ga2O63,” Appl. Phys. Lett. 27, 360–362 (1975).
[CrossRef]

Dill, F. H.

P. S. Hauge, F. H. Dill, “Design and Operation of ETA, an Automated Ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

Fox, A. J.

A. J. Fox, T. M. Bruton, “Electro-Optic Effects in the Optically Active Compounds Bi12TiO20 and Bi40Ga2O63,” Appl. Phys. Lett. 27, 360–362 (1975).
[CrossRef]

Françon, M.

M. Françon, S. Mallick, Polarization Interferometers: Application in Microscopy and Macroscopy (Wiley, London, 1971).

Furner, E. H.

I. P. Kaminow, E. H. Furner, “Electrooptic Light Modulators,” Proc. IEEE 54, 1374–1390 (1966).
[CrossRef]

Hauge, P. S.

P. S. Hauge, F. H. Dill, “Design and Operation of ETA, an Automated Ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

Huignard, J. P.

J. P. Huignard, H. Rajbenbach, Ph. Refrégier, L. Solyman, “Wave Mixing in Photorefractive Bismuth Silicon Oxide Crystals and Its Applications,” Opt. Eng. 24, 586–592 (1985).

Jerrard, H. G.

Kaminow, I. P.

I. P. Kaminow, E. H. Furner, “Electrooptic Light Modulators,” Proc. IEEE 54, 1374–1390 (1966).
[CrossRef]

Mallick, S.

M. Françon, S. Mallick, Polarization Interferometers: Application in Microscopy and Macroscopy (Wiley, London, 1971).

Nakadate, S.

Rajbenbach, H.

J. P. Huignard, H. Rajbenbach, Ph. Refrégier, L. Solyman, “Wave Mixing in Photorefractive Bismuth Silicon Oxide Crystals and Its Applications,” Opt. Eng. 24, 586–592 (1985).

Refrégier, Ph.

J. P. Huignard, H. Rajbenbach, Ph. Refrégier, L. Solyman, “Wave Mixing in Photorefractive Bismuth Silicon Oxide Crystals and Its Applications,” Opt. Eng. 24, 586–592 (1985).

Runwen, W.

Solyman, L.

J. P. Huignard, H. Rajbenbach, Ph. Refrégier, L. Solyman, “Wave Mixing in Photorefractive Bismuth Silicon Oxide Crystals and Its Applications,” Opt. Eng. 24, 586–592 (1985).

Takasaki, H.

Yao, L.

Zhiyao, Z.

Appl. Phys. Lett. (1)

A. J. Fox, T. M. Bruton, “Electro-Optic Effects in the Optically Active Compounds Bi12TiO20 and Bi40Ga2O63,” Appl. Phys. Lett. 27, 360–362 (1975).
[CrossRef]

IBM J. Res. Dev. (1)

P. S. Hauge, F. H. Dill, “Design and Operation of ETA, an Automated Ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Eng. (1)

J. P. Huignard, H. Rajbenbach, Ph. Refrégier, L. Solyman, “Wave Mixing in Photorefractive Bismuth Silicon Oxide Crystals and Its Applications,” Opt. Eng. 24, 586–592 (1985).

Opt. Lett. (1)

Proc. IEEE (1)

I. P. Kaminow, E. H. Furner, “Electrooptic Light Modulators,” Proc. IEEE 54, 1374–1390 (1966).
[CrossRef]

Other (3)

M. Françon, S. Mallick, Polarization Interferometers: Application in Microscopy and Macroscopy (Wiley, London, 1971).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1980).

G. Bruhat, Optique (Cours de Physique Générale) (Masson, Paris, 1959).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of a common-path interferometer for a retardation measurement.

Fig. 2
Fig. 2

(a) Output voltage from the image sensor and (b) the difference between two successively digitized voltages from the image sensor.

Fig. 3
Fig. 3

(a) Phase fluctuation for 50 s in the stable state of the interferometer shown in Fig. 1, where the standard deviation is 0.026°. (b) Phase difference between the phase averaged over five sample points and the original phase (a).

Fig. 4
Fig. 4

Calculation error caused by data truncation when the frequency of Young’s fringes is 10.0, the frequency error is 0.1, and the fringe visibility is 0.74. The abscissa represents the true phase and the maximum error is 0.015°.

Fig. 5
Fig. 5

Phase measurement for a Soleil-Babinet compensator (SBC), which was moved by 10 μm at sample numbers 32 and 63.

Fig. 6
Fig. 6

Output voltages from the image sensor for (a) a halfwave plate and (b) a quarterwave plate.

Fig. 7
Fig. 7

Experimental results for a Pockels cell. (a) Retardation as a function of the voltage. The halfwave voltage is 296 V. (b) Hysteresis of the retardation is in the range of ±10 V.

Equations (5)

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I 1 ( x ) = W ( x ) [ 1 + cos ( 2 π f x - φ 1 ) ] ,
I 2 ( x ) = W ( x ) [ 1 + cos ( 2 π f x - φ 1 - δ φ ) ] ,
C i = - 0.5 0.5 I i ( x ) cos ( 2 π f x ) d x ,
S i = - 0.5 0.5 I i ( x ) sin ( 2 π f x ) d x ,             ( i = 1 , 2 ) ,
φ i = tan - 1 ( S i / C i ) ,             ( i = 1 , 2 ) ,

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