Abstract

Based on measuring one-dimensional small rotation angles by using a parallel interference pattern (PIP), a method for measuring two-dimensional (2D) small rotation angles by using two different PIP’s that are orthogonal to each other is proposed. We simultaneously measure the 2D small rotation angles Δθ and Δϕ by detecting the phases of the orthogonal PIP’s reflected by an object at two detection points. A sensitivity of 4.9 mrad/arcsec and a spatial resolution of 1.5 × 1.5 mm2 are achieved in the measurement. Theoretical analysis and experimental results show that error ɛ1 in the measurement of Δϕ is almost equal to −0.01Δθ and error ɛ2 in the measurement of Δθ is almost equal to −0.01Δϕ. For small rotation angles of less than a few tens of arcseconds, the random errors whose standard deviations are 0.6 arcsec are dominant.

© 1996 Optical Society of America

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References

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  1. X. Dai, O. Sasaki, J. E. Greivenkamp, T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380–6388 (1995).
    [CrossRef] [PubMed]
  2. J. G. Marzolf, “Angle measuring interferometer,” Rev. Sci. Instrum. 35, 1212–1215 (1964).
    [CrossRef]
  3. G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646–1651 (1974).
    [CrossRef] [PubMed]
  4. P. Shi, E. Stijns, “Highly sensitive angular measurement with Michelson interferometer,” Ind. Metrol. 1, 69–74 (1990).
    [CrossRef]
  5. P. R. Yoder, E. R. Schlesinger, J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890–1895 (1975).
    [CrossRef] [PubMed]
  6. F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
    [CrossRef]
  7. O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
    [CrossRef]
  8. O. Sasaki, H. Sasazaki, T. Suzuki, “Two-wavelength sinusoidal phase-modulating laser-diode interferometer insensitive to external disturbances,” Appl. Opt. 30, 4040–4045 (1991).
    [CrossRef] [PubMed]

1995 (1)

1991 (1)

1990 (2)

P. Shi, E. Stijns, “Highly sensitive angular measurement with Michelson interferometer,” Ind. Metrol. 1, 69–74 (1990).
[CrossRef]

O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
[CrossRef]

1983 (1)

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

1975 (1)

1974 (1)

1964 (1)

J. G. Marzolf, “Angle measuring interferometer,” Rev. Sci. Instrum. 35, 1212–1215 (1964).
[CrossRef]

Chapman, G. D.

Chickvary, J. L.

Dai, X.

Greivenkamp, J. E.

Marzolf, J. G.

J. G. Marzolf, “Angle measuring interferometer,” Rev. Sci. Instrum. 35, 1212–1215 (1964).
[CrossRef]

Sasaki, O.

Sasazaki, H.

Schlesinger, E. R.

Schuda, F. J.

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

Shi, P.

P. Shi, E. Stijns, “Highly sensitive angular measurement with Michelson interferometer,” Ind. Metrol. 1, 69–74 (1990).
[CrossRef]

Stijns, E.

P. Shi, E. Stijns, “Highly sensitive angular measurement with Michelson interferometer,” Ind. Metrol. 1, 69–74 (1990).
[CrossRef]

Suzuki, T.

Takahashi, K.

O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
[CrossRef]

Yoder, P. R.

Appl. Opt. (4)

Ind. Metrol. (1)

P. Shi, E. Stijns, “Highly sensitive angular measurement with Michelson interferometer,” Ind. Metrol. 1, 69–74 (1990).
[CrossRef]

Opt. Eng. (1)

O. Sasaki, K. Takahashi, T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29, 1511–1515 (1990).
[CrossRef]

Rev. Sci. Instrum. (2)

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

J. G. Marzolf, “Angle measuring interferometer,” Rev. Sci. Instrum. 35, 1212–1215 (1964).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Reflection of a parallel interference pattern.

Fig. 2
Fig. 2

Change in the unit normal vector n ^ of the reflected equiphase plane. (a) Unit normal vector n ^ 0 at θ = ϕ = 0. (b) Rotation of vector n ^ 0 by angle θ of the object. (c) Rotation of vector n ^ 2 by angle ϕ of the object.

Fig. 3
Fig. 3

Ideal configuration for measuring small rotation angle Δϕ.

Fig. 4
Fig. 4

Ideal configuration for measuring small rotation angle Δθ.

Fig. 5
Fig. 5

Relationships of ratios R11, R12, R21, R22 and angles δ1, δ2, θ, ϕ. (a) Ratio R11 versus angles δ1, θ, ϕ. (b) Ratio R12 versus angles δ1, θ, ϕ. (c) Ratio R21 versus angles δ2, θ, ϕ. (d) Ratio R22 versus angles δ2, θ, ϕ.

Fig. 6
Fig. 6

Experimental setup.

Fig. 7
Fig. 7

Measurement results of errors EΔϕ and EΔθ at Δθc = 10 and 60 arcsec, respectively.

Fig. 8
Fig. 8

Measurement results of errors EΔθ and EΔϕ at Δϕc = 10 and 60 arcsec, respectively.

Fig. 9
Fig. 9

Measurement results of 2D small rotation angles Δϕ, Δθ within ±10 arcsec.

Fig. 10
Fig. 10

Measurement results of 2D small rotation angles from displacements of a stage.

Equations (61)

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n ^ 0 = { n 0 x , n 0 y , n 0 z } ,
n 0 x = cos β ,             n 0 y = cos γ ,             n 0 z = cos 90 ° = 0.
n ^ 1 = { n 1 x , n 1 y , n 1 z } ,
n 1 x = cos β ,             n 1 y = cos γ ,             n 1 z = 0.
n ^ 2 = { n 2 x , n 2 y , n 2 z } ,
n 2 x = n 1 x cos 2 θ ,             n 2 y = n 1 y ,             n 2 z = n 1 x sin 2 θ .
n ^ 3 = { n 3 x , n 3 y , n 3 z } ,
n 3 x = n 2 x ,             n 3 y = n 1 y cos 2 ϕ + n 2 z sin 2 ϕ , n 3 z = - n 1 y sin 2 ϕ + n 2 z cos 2 ϕ .
n 3 x = cos β cos 2 θ , n 3 y = cos γ cos 2 ϕ + cos β sin 2 θ sin 2 ϕ , n 3 z = - cos γ sin 2 ϕ + cos β cos 2 ϕ sin 2 θ .
n ^ = { n x , n y , n z } ,
n x = n 3 x cos 2 θ + n 3 z sin 2 θ ,             n y = n 3 y , n z = - n 3 x sin 2 θ + n 3 z cos 2 θ .
d 0 ( β , γ , θ , ϕ ) = n · AB = n x x 0 + n y y 0 - n z z 0 .
d 0 ( β , γ , θ , ϕ ) = α 0 2 π S ,
S = λ 2 sin ( η / 2 ) ,
d ( β , γ , θ , ϕ , Δ θ , Δ ϕ ) = n · AB .
d ( β , γ , θ , ϕ , Δ θ , Δ ϕ ) = α 2 π S .
d ( β , γ , θ , ϕ , Δ θ , Δ ϕ ) - d 0 ( β , γ , θ , ϕ ) = Δ α 2 π S ,
Δ α = α - α 0 .
d 0 = y 0 ,
d = y 0 + 2 Δ ϕ ( z 0 cos 2 θ - x 0 sin 2 θ ) .
Δ ϕ = d - d 0 2 z 0 ,
z 0 = ( z 0 cos 2 θ - x 0 sin 2 θ ) .
Δ ϕ = Δ α 4 π z 0 S .
d 0 = x 0 ,
d = x 0 - 2 z 0 Δ θ + 2 y 0 Δ ϕ sin 2 θ .
Δ θ = - d - d 0 2 z 0 + φ ,
φ = ( y 0 Δ ϕ sin 2 θ ) / z 0 .
Δ θ = - Δ α 4 π z 0 S .
β = 90 ° - δ 1 ,             γ = δ 1 ,
β = δ 2 ,             γ = 90 ° - δ 2 ,
Δ ϕ m = - R 11 Δ θ + R 12 Δ ϕ ,
Δ θ m = R 21 Δ θ - R 22 Δ ϕ ,
R 11 = sin δ 1 [ z 0 z 0 - 2 cos 2 θ sin ϕ ( sin ϕ + y 0 z 0 cos ϕ ) ] ,
R 12 = sin δ 1 sin 2 θ ( sin 2 ϕ + y 0 z 0 cos 2 ϕ ) + cos δ 1 ( cos 2 ϕ - y 0 z 0 sin 2 ϕ ) ,
R 21 = cos δ 2 [ 1 - 2 cos 2 θ sin ϕ ( z 0 z 0 sin ϕ + y 0 z 0 cos ϕ ) ] ,
R 22 = [ cos δ 2 sin 2 θ ( z 0 z 0 sin 2 ϕ + y 0 z 0 cos 2 ϕ ) + sin δ 2 ( z 0 z 0 cos 2 ϕ - y 0 z 0 sin 2 ϕ ) ] .
Δ ϕ m = Δ ϕ + ɛ 1 ,
Δ θ m = Δ θ + ɛ 2 ,
ɛ 1 = ( R 12 - 1 ) Δ ϕ - R 11 Δ θ ,
ɛ 2 = ( R 21 - 1 ) Δ θ - R 22 Δ ϕ .
S e 1 = 4 π z 0 / S 1 ,
S e 2 = 4 π z 0 / S 2 ,
Δ ϕ 1 4 z 0 S 1 ,
Δ θ 1 4 z 0 S 2 .
S R = S 1 × S 2 .
σ α 0 ( Δ ϕ ) = σ α ( Δ ϕ ) = σ α 0 ( Δ θ ) = σ α ( Δ θ ) = σ .
σ Δ ϕ = 2 σ 4 π z 0 S ,
σ Δ θ = 2 σ 4 π z 0 S .
E Δ ϕ = ɛ 1 + ɛ Δ ϕ ,
E Δ θ = ɛ 2 + ɛ Δ θ .
S A = S 1 A [ cos ( z 1 cos ω 1 t + α 1 A ) ] + S 2 A [ cos ( z 2 cos ω 2 t + α 2 A ) ] ,
S B = S 1 B [ cos ( z 1 cos ω 1 t + α 1 B ) ] + S 2 B [ cos ( z 2 cos ω 2 t + α 2 B ) ] .
n x = cos β cos 2 ( θ + Δ θ ) cos 2 θ - [ cos γ sin 2 ( ϕ + Δ ϕ ) - cos β cos 2 ( ϕ + Δ ϕ ) sin 2 ( θ + Δ θ ) ] sin 2 θ , n y = cos γ cos 2 ( ϕ + Δ ϕ ) + cos β sin 2 ( ϕ + Δ ϕ ) × sin 2 ( θ + Δ θ ) , n z = - cos β cos 2 ( θ + Δ θ ) sin 2 θ - [ cos γ sin 2 × ( ϕ + Δ ϕ ) - cos β cos 2 ( ϕ + Δ ϕ ) sin 2 × ( θ + Δ θ ) ] cos 2 θ .
n x = sin δ 1 cos 2 ( θ + Δ θ ) cos 2 θ - [ cos δ 1 sin 2 ( ϕ + Δ ϕ ) - sin δ 1 cos 2 ( ϕ + Δ ϕ ) × sin 2 ( θ + Δ θ ) ] sin 2 θ , n y = cos δ 1 cos 2 ( ϕ + Δ ϕ ) + sin δ 1 sin 2 ( ϕ + Δ ϕ ) × sin 2 ( θ + Δ θ ) , n z = - sin δ 1 cos 2 ( θ + Δ θ ) sin 2 θ - [ cos δ 1 sin 2 ( ϕ + Δ ϕ ) - sin δ 1 × cos 2 ( ϕ + Δ ϕ ) sin 2 ( θ + Δ θ ) ] cos 2 θ .
n x = sin δ 1 cos 2 2 θ - 2 Δ θ sin δ 1 sin 2 θ cos 2 θ - cos δ 1 sin 2 ϕ sin 2 θ - 2 Δ ϕ cos δ 1 cos 2 ϕ sin 2 θ + sin δ 1 sin 2 θ ( sin 2 θ cos 2 ϕ - 2 Δ ϕ sin 2 ϕ sin 2 θ + 2 Δ θ cos 2 θ cos 2 ϕ ) ,
n y = cos δ 1 ( cos 2 ϕ - 2 Δ ϕ sin 2 ϕ ) + sin δ 1 ( sin 2 ϕ sin 2 θ + 2 Δ ϕ cos 2 ϕ sin 2 θ + 2 Δ θ cos 2 θ sin 2 ϕ ) , n z = - sin δ 1 cos 2 θ sin 2 θ + 2 Δ θ sin δ 1 sin 2 2 θ - cos 2 θ cos δ 1 ( sin 2 ϕ + 2 Δ ϕ cos 2 ϕ ) + cos 2 θ sin δ 1 ( cos 2 ϕ sin 2 θ - 2 Δ ϕ sin 2 ϕ sin 2 θ + 2 Δ θ cos 2 θ cos 2 ϕ ) .
d 0 ( β , γ , θ , ϕ ) = x 0 [ sin δ 1 ( 1 - 2 sin 2 2 θ sin 2 ϕ ) - cos δ 1 sin 2 ϕ sin 2 θ ] + y 0 ( cos δ 1 cos 2 ϕ + sin δ 1 sin 2 ϕ sin 2 θ ) + 2 z 0 cos 2 θ sin ϕ ( sin δ 1 sin 2 θ sin ϕ + cos δ 1 cos ϕ ) .
d ( β , γ , θ , ϕ , Δ θ , Δ ϕ ) - d 0 ( β , γ , θ , ϕ ) = 2 Δ θ sin δ 1 [ 2 cos 2 θ sin ϕ ( z 0 sin ϕ + y 0 cos ϕ ) - z 0 ] + 2 Δ ϕ [ sin δ 1 sin 2 θ ( z 0 sin 2 ϕ + y 0 cos 2 ϕ ) + cos δ 1 ( z 0 cos 2 ϕ - y 0 sin 2 ϕ ) ] ,
Δ ϕ m = - Δ θ sin δ 1 [ z 0 z 0 - 2 cos 2 θ sin ϕ × ( sin ϕ + y 0 z 0 cos ϕ ) ] + Δ ϕ [ sin δ 1 sin 2 θ ( sin 2 ϕ + y 0 z 0 cos 2 ϕ ) + cos δ 1 ( cos 2 ϕ - y 0 z 0 sin 2 ϕ ) ] .
d ( β , γ , θ , ϕ , Δ θ , Δ ϕ ) - d 0 ( β , γ , θ , ϕ ) = 2 Δ θ cos δ 2 [ 2 cos 2 θ sin ϕ ( z 0 sin ϕ + y 0 cos ϕ ) - z 0 ] + 2 Δ ϕ [ cos δ 2 sin 2 θ ( z 0 sin 2 ϕ + y 0 cos 2 ϕ ) + sin δ 2 ( z 0 cos 2 ϕ - y 0 sin 2 ϕ ) ] .
Δ θ m = Δ θ cos δ 2 [ 1 - 2 cos 2 θ sin ϕ × ( z 0 z 0 sin ϕ + y 0 z 0 cos ϕ ) ] - Δ ϕ [ cos δ 2 sin 2 θ ( z 0 z 0 sin 2 ϕ + y 0 z 0 cos 2 ϕ ) + sin δ 2 ( z 0 z 0 cos 2 ϕ - y 0 z 0 sin 2 ϕ ) ] .

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