Abstract

Based on the heterodyne interferometric technique and the discrimination technique of using two light beams with different wavelengths, a novel method for identifying the fast axis of a wave plate and evaluating its phase retardation is presented. Some of the merits of the method, such as, a simple optical setup, high stability, better resolution, and easier operation, are presented, and the validity of the method is demonstrated.

© 1996 Optical Society of America

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References

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  1. H. Takasaki, “Photoelectric measurement of polarized light by means of an ADP polarization modulator,” J. Opt. Soc. Am. 51, 461–462 (1961).
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 5, p. 364.
  3. P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
    [CrossRef]
  4. D. C. Su, L. H. Shyu, “Phase shifting scatter plate interferometer using a polarization technique,” J. Mod. Opt. 38, 951–959 (1991).
    [CrossRef]
  5. P. Hariharan, “Double-pass interferometers,” in OpticalShop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 247–263.
  6. G. Li, J. Li, Y. Li, “Determination of the fast axis with an infrared spectrometer for quartz and mica waveplates,” Appl. Opt. 29, 1870–1871 (1990).
    [CrossRef] [PubMed]
  7. N. O’Flaherty, N. Kiyomoto, I. Shirahama, Y. Mochida, “A system for high sensitivity measurement of birefringence using a photo-elastic modulator, and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
    [CrossRef]
  8. B. R. Grunstra, H. B. Perkins, “A method for measurement of optical retardation angles near 90 degrees,” Appl. Opt. 5, 585–587 (1966).
    [CrossRef] [PubMed]
  9. C. M. McIntyre, S. E. Harris, “Achromatic wave plates for the visible spectrum,” J. Opt. Soc. Am. 58, 1575–1580 (1968).
    [CrossRef]
  10. Y. Lin, Z. Zhou, R. Wang, “Optical heterodyne measurement of the phase retardation of a quarter-wave plate,” Opt. Lett. 13, 553–555 (1988).
    [CrossRef]
  11. E. Collett, “The measurement of the phase shift of a retarder,” in Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 130–138.
  12. S. Nakadate, “High precision retardation measurement using phase detection of Young’s fringes,” Appl. Opt. 29, 242–246 (1990).
    [CrossRef] [PubMed]
  13. M. Sypek, “A new technique for the measurement of phase retardation,” Opt. Laser Technol. 23, 42–44 (1991).
    [CrossRef]
  14. L.-H. Shyu, C.-L. Chen, D.-C. Su, “Method for measuring the retardation of a wave plate,” Appl. Opt. 32, 4228–4230 (1993).
    [CrossRef] [PubMed]
  15. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5, pp. 121–131.

1993

1991

D. C. Su, L. H. Shyu, “Phase shifting scatter plate interferometer using a polarization technique,” J. Mod. Opt. 38, 951–959 (1991).
[CrossRef]

M. Sypek, “A new technique for the measurement of phase retardation,” Opt. Laser Technol. 23, 42–44 (1991).
[CrossRef]

1990

1988

1973

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

1968

1966

1961

H. Takasaki, “Photoelectric measurement of polarized light by means of an ADP polarization modulator,” J. Opt. Soc. Am. 51, 461–462 (1961).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 5, p. 364.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 5, p. 364.

Chen, C.-L.

Collett, E.

E. Collett, “The measurement of the phase shift of a retarder,” in Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 130–138.

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]

Grunstra, B. R.

Hariharan, P.

P. Hariharan, “Double-pass interferometers,” in OpticalShop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 247–263.

Harris, S. E.

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]

Kiyomoto, N.

N. O’Flaherty, N. Kiyomoto, I. Shirahama, Y. Mochida, “A system for high sensitivity measurement of birefringence using a photo-elastic modulator, and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

Li, G.

Li, J.

Li, Y.

Lin, Y.

McIntyre, C. M.

Mochida, Y.

N. O’Flaherty, N. Kiyomoto, I. Shirahama, Y. Mochida, “A system for high sensitivity measurement of birefringence using a photo-elastic modulator, and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

Nakadate, S.

O’Flaherty, N.

N. O’Flaherty, N. Kiyomoto, I. Shirahama, Y. Mochida, “A system for high sensitivity measurement of birefringence using a photo-elastic modulator, and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

Perkins, H. B.

Shirahama, I.

N. O’Flaherty, N. Kiyomoto, I. Shirahama, Y. Mochida, “A system for high sensitivity measurement of birefringence using a photo-elastic modulator, and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

Shyu, L. H.

D. C. Su, L. H. Shyu, “Phase shifting scatter plate interferometer using a polarization technique,” J. Mod. Opt. 38, 951–959 (1991).
[CrossRef]

Shyu, L.-H.

Su, D. C.

D. C. Su, L. H. Shyu, “Phase shifting scatter plate interferometer using a polarization technique,” J. Mod. Opt. 38, 951–959 (1991).
[CrossRef]

Su, D.-C.

Sypek, M.

M. Sypek, “A new technique for the measurement of phase retardation,” Opt. Laser Technol. 23, 42–44 (1991).
[CrossRef]

Takasaki, H.

H. Takasaki, “Photoelectric measurement of polarized light by means of an ADP polarization modulator,” J. Opt. Soc. Am. 51, 461–462 (1961).

Wang, R.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5, pp. 121–131.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5, pp. 121–131.

Zhou, Z.

Appl. Opt.

IBM J. Res. Dev.

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

J. Mod. Opt.

D. C. Su, L. H. Shyu, “Phase shifting scatter plate interferometer using a polarization technique,” J. Mod. Opt. 38, 951–959 (1991).
[CrossRef]

J. Opt. Soc. Am.

H. Takasaki, “Photoelectric measurement of polarized light by means of an ADP polarization modulator,” J. Opt. Soc. Am. 51, 461–462 (1961).

C. M. McIntyre, S. E. Harris, “Achromatic wave plates for the visible spectrum,” J. Opt. Soc. Am. 58, 1575–1580 (1968).
[CrossRef]

Opt. Laser Technol.

M. Sypek, “A new technique for the measurement of phase retardation,” Opt. Laser Technol. 23, 42–44 (1991).
[CrossRef]

Opt. Lett.

Other

E. Collett, “The measurement of the phase shift of a retarder,” in Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 130–138.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Chap. 5, p. 364.

P. Hariharan, “Double-pass interferometers,” in OpticalShop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 247–263.

N. O’Flaherty, N. Kiyomoto, I. Shirahama, Y. Mochida, “A system for high sensitivity measurement of birefringence using a photo-elastic modulator, and its applications,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. SPIE1274, 122–131 (1990).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5, pp. 121–131.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the method: P, polarizer; BS, beam splitter; M, mirror; EO, electro-optic modulator; AN, analyzer; W, wave plate to be measured; D, photodetector.

Fig. 2
Fig. 2

Relation curves of ϕ versus θ for different δ: (a) δ = −180° to 0°, (b) δ = 0° to 180°, at 30° intervals.

Fig. 3
Fig. 3

Theoretical and experimental results for wave plates with 632.8-nm wavelength: (a) a quarter-wave plate, (b) a half-wave plate. Theoretical and experimental results with 441.6-nm wavelength are also included.

Fig. 4
Fig. 4

Comparison of the relation curves of ϕ versus θ for different δ as (a) = 5°, (b) = −5°. Solid and dashed curves represent results without and with azimuth angular error , respectively.

Fig. 5
Fig. 5

Measurement errors for wave plates as || = 2° and 5°.

Fig. 6
Fig. 6

Optical setup for locating the fast axis of the EO modulator at either θ = 0° or θ = 90°: P, polarizer; EO, electro-optic modulator; A, analyzer; D, photodetector.

Fig. 7
Fig. 7

Optical setup for identifying the fast axis of EO modulator: P, polarizer; AOM, acoustic-optic modulator; M, mirror; EO, electro-optic modulator; PBS, polarization beam splitter; A’s, analyzer; D’s, photodetectors.

Equations (21)

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( E r x E r y ) = [ 0 1 0 1 ] [ exp ( i ψ x ) 0 0 exp ( i ψ y ) ] × [ cos Γ 2 i sin Γ 2 i sin Γ 2 cos Γ 2 ] ( 1 0 ) = [ 0 i sin Γ 2 exp ( i ψ y ) ] ,
I r = E r 2 = 1 2 ( 1 - cos Γ ) ,
( E t x E t y ) = [ 0 0 0 1 ] [ cos δ 2 + i sin δ 2 cos 2 θ i sin δ 2 sin 2 θ i sin δ 2 sin 2 θ cos δ 2 - i sin δ 2 cos 2 θ ] [ cos Γ 2 i sin Γ 2 i sin Γ 2 cos Γ 2 ] ( 1 0 ) = [ 0 i sin 2 θ sin δ 2 cos Γ 2 + i cos δ 2 sin Γ 2 + cos 2 θ sin δ 2 sin Γ 2 ] .
I t = 1 2 [ 1 - A 2 + B 2 cos ( Γ + ϕ ) ] ;
A = cos 2 2 θ + sin 2 2 θ cos δ ,
B = sin 2 θ sin δ .
ϕ = tan - 1 ( B / A ) .
A 0 ,             then ϕ = tan - 1 ( B / A ) ;
A < 0 and B 0 ,             then ϕ = π + tan - 1 ( B / A ) ;
A < 0 and B < 0 ,             then ϕ = - π + tan - 1 ( B / A ) .
ϕ 2 λ 1 λ 2 ϕ 1 ,
EO = [ cos Γ 2 - i sin 2 sin Γ 2 i cos 2 sin Γ 2 i cos 2 sin Γ 2 cos Γ 2 + i sin 2 sin Γ 2 ] .
I r = cos 2 2 ( 1 - cos w t ) ,
I t = 1 2 ( γ + β ) - 1 2 [ ( γ - β cos 2 α ) 2 + ( β sin 2 α ) 2 ] 1 / 2 cos ( w t + ϕ ) ,
α = tan - 1 ( cos 2 cos ( δ / 2 ) sin 2 θ sin ( δ / 2 ) ) , β = [ sin 2 θ sin ( δ / 2 ) 2 ] + [ cos 2 cos ( δ / 2 ) ] 2 , γ = cos 2 ( 2 θ - 2 ) sin 2 ( δ / 2 ) ,
ϕ = tan - 1 { cos 2 sin 2 θ sin δ [ cos 2 ( 2 θ - 2 ) - sin 2 2 θ ] sin 2 ( δ / 2 ) + cos 2 2 cos 2 ( δ / 2 ) } .
Δ δ = tan - 1 ( sin δ cos 2 cos δ ) - δ .
I = sin 2 2 η 2 ( 1 - cos w t ) ,
E 0 = { exp [ i 2 π ( f 0 ± f 2 ) t ] exp [ i 2 π ( f 0 f 2 ) t ] } ,
f 1 = f s ± f / 2 ,
f 2 = f s f / 2 ,

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