Abstract

A parameter compensation method is developed to improve the linearity of the Michelson interferometric angular measurement. The formulas for the exact calculation of the parameters are presented; different cases are discussed. By introducing a weight function in the calculation, we can change the distribution of the linearity over the whole range of measurement. The theoretical analysis and the experimental results show that this method greatly improves the linearity of the measurement even for a large range of measurement.

© 1993 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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  14. P. Shi, E. Stijns, “Highly sensitive angular measurement with Michelson interferometer,” Ind. Metrol. 1, 69–74 (1990).
    [CrossRef]
  15. P. Shi, E. Stijns, “New optical method for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
    [CrossRef] [PubMed]
  16. P. Shi, E. Stijns, “A new optical method for measuring small angle rotations,” in Proceedings of the International Conference on Lasers ’87 (STS, McLean, Va., 1987) pp. 1609–1072.

1990

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

1988

1977

E. Debler, “Winkelinterferometer mit einem Messbereich von 95°,” Feinwerktech. & Messtech. 85, 166–171 (1977).

1974

1971

H. M. Bird, “A computer controlled interferometer system for precision relative angle measurements,” Rev. Sci. Instrum. 42, 1513–1520 (1971).
[CrossRef]

1970

1964

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

1963

1962

R. A. Woodson, “Extended-range Gonimet angle interferometer,” J. Opt. Soc. Am. 52, 1315(A) (1962).

1961

R. A. Woodson, “Differential Woodson interferometer,” J. Opt. Soc. Am. 51, 1467(A) (1961).

1960

1957

1948

Bird, H. M.

H. M. Bird, “A computer controlled interferometer system for precision relative angle measurements,” Rev. Sci. Instrum. 42, 1513–1520 (1971).
[CrossRef]

Chapman, G. D.

Debler, E.

E. Debler, “Winkelinterferometer mit einem Messbereich von 95°,” Feinwerktech. & Messtech. 85, 166–171 (1977).

Harris, O.

Malacara, D.

Marzolf, J. G.

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

Murty, M. V. R. K.

Peck, E. R.

Rohlin, J.

Shi, P.

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

P. Shi, E. Stijns, “New optical method for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
[CrossRef] [PubMed]

P. Shi, E. Stijns, “A new optical method for measuring small angle rotations,” in Proceedings of the International Conference on Lasers ’87 (STS, McLean, Va., 1987) pp. 1609–1072.

Stijns, E.

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

P. Shi, E. Stijns, “New optical method for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
[CrossRef] [PubMed]

P. Shi, E. Stijns, “A new optical method for measuring small angle rotations,” in Proceedings of the International Conference on Lasers ’87 (STS, McLean, Va., 1987) pp. 1609–1072.

Woodson, R. A.

R. A. Woodson, “Extended-range Gonimet angle interferometer,” J. Opt. Soc. Am. 52, 1315(A) (1962).

R. A. Woodson, “Differential Woodson interferometer,” J. Opt. Soc. Am. 51, 1467(A) (1961).

Appl. Opt.

Feinwerktech. & Messtech.

E. Debler, “Winkelinterferometer mit einem Messbereich von 95°,” Feinwerktech. & Messtech. 85, 166–171 (1977).

Ind. Metrol.

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

J. Opt. Soc. Am.

Rev. Sci. Instrum.

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

H. M. Bird, “A computer controlled interferometer system for precision relative angle measurements,” Rev. Sci. Instrum. 42, 1513–1520 (1971).
[CrossRef]

Other

P. Shi, E. Stijns, “A new optical method for measuring small angle rotations,” in Proceedings of the International Conference on Lasers ’87 (STS, McLean, Va., 1987) pp. 1609–1072.

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Figures (10)

Fig. 1
Fig. 1

Michelson interferometric angular measurement setup with one prism: M1, M2, mirrors; D, photodetector; BS, beam splitter.

Fig. 2
Fig. 2

Balanced Michelson interferometric angular measurement setup with two prisms: M1–M3, mirrors; D, photodetector; BS, beam splitter.

Fig. 3
Fig. 3

Principle of the PCM

Fig. 4
Fig. 4

Comparison of the error between the PCM and the sine function for a measuring range of 5°.

Fig. 5
Fig. 5

Comparison of the error between the PCM and the sine function for a measuring range of 10°.

Fig. 6
Fig. 6

Comparison of the error between the PCM and the sine function for a measuring range of 20°.

Fig. 7
Fig. 7

Center of rotation in four different positions of O1, O2, O3, and O4.

Fig. 8
Fig. 8

Rotation in two different directions of +θ and −θ.

Fig. 9
Fig. 9

Experimental setup that was used to check the linearity of the PCM: M1, M2, mirrors; D, photodetector; BS, beam splitter; L, lens.

Fig. 10
Fig. 10

Comparison of the experimental results and the theoretical line of Nideal = θe.

Tables (4)

Tables Icon

Table 1 Comparison of Fringes, Error, and Nonlinearity between the PCM and the Sine Function for a Measuring Range of 5°

Tables Icon

Table 2 Comparison of Fringes, Error, and Nonlinearity between the PCM and the Sine Function for a Measuring Range of 10°

Tables Icon

Table 3 Comparison of Fringes, Error, and Nonlinearity between the PCM and the Sine Function for a Measuring Range of 20°

Tables Icon

Table 4 Comparison of Fringes, Error, and Nonlinearity between the Two PCM’s Using wi = 1 and wi = (θ − i + 1)/∑i

Equations (45)

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Δ P = ( 2 a - 2 x + 2 y tan θ 2 ) sin θ - 2 a n ( 1 - cos α ) ,
Δ P = Δ P 0 + Δ P 1 - Δ P 2 ,
Δ P 0 = ( 2 a - 2 x ) sin θ , Δ P 1 = 2 y tan θ 2 sin θ , Δ P 2 = 2 a n ( 1 - cos α ) .
Δ P = 2 ( 2 a + 2 d ) sin θ ,
Δ P = 4 r sin θ ,
L n = N θ - θ e θ e ,
N = 2 Δ P λ .
N ideal = θ e
Δ P ideal = θ e λ 2 .
error = N - N ideal = 2 Δ P λ - θ e .
N 1 - N 2 = N ideal - N 0 ,
N = N 0 + N 1 - N 2 = N ideal .
Δ P i = ( 2 a + 2 x + 2 y tan i 2 ) sin i - 2 a [ n - ( n 2 - sin 2 i ) 1 / 2 ] ,             i = 1 , 2 , θ ° ,
Error i = N i - N ideal = 2 Δ P i λ - i e ,             i = 1 , 2 , θ ° .
i = 1 θ error i 2 minimum ,
min i = 1 θ ( 2 Δ P i / λ - i e ) 2 = min i = 1 θ 4 λ 2 { 2 x sin i + 2 y tan i 2 sin i + 2 a sin i - 2 a [ n - ( n 2 - sin 2 i ) 1 / 2 ] - λ i e 2 } 2 = min i = 1 θ 4 λ 2 ( A i x + B i y + C i ) 2 ,
A i = 2 sin i ,
B i = 2 tan i 2 sin i ,
C i = 2 a sin i - 2 a [ n - ( n 2 - sin 2 i ) 1 / 2 ] - λ i e 2 ,             i = 1 , 2 , θ ° .
i = 1 θ error i 2 minimum ,
x i = 1 θ error i 2 = 0 ,
y i = 1 θ error i 2 = 0 ,
x i = 1 θ ( A i x + B i y + C i ) 2 = 0 ,
y i = 1 θ ( A i x + B i y + C i ) 2 = 0.
x i = 1 θ A i 2 + y i = 1 θ A i B i = - i = 1 θ A i C i ,
x i = 1 θ A i B i + y i = 1 θ B i 2 = - i = 1 θ B i C i .
x = i = 1 θ A i B i i = 1 θ B i C i - i = 1 θ A i C i i = 1 θ B i 2 i = 1 θ A i 2 i = 1 θ B i 2 - ( i = 1 θ A i B i ) 2 ,
y = i = 1 θ A i B i i = 1 θ A i C i - i = 1 θ A i 2 i = 1 θ B i C i i = 1 θ A i 2 i = 1 θ B i 2 - ( i = 1 θ A i B i ) 2 .
error i = w i ( N i - N ideal ) = w i ( 2 Δ P i λ - i e ) ,             i = 1 , 2 , θ ° .
x = i = 1 θ w i 2 A i B i i = 1 θ w i 2 B i C i - i = 1 θ w i 2 A i C i i = 1 θ w i 2 B i 2 i = 1 θ w i 2 A i 2 i = 1 θ w i 2 B i 2 - ( i = 1 θ w i 2 A i B i ) 2 ,
y = i = 1 θ w i 2 A i B i i = q θ w i 2 A i C i - i = 1 θ w i 2 A i 2 i = 1 θ w i 2 B i C i i = 1 θ w i 2 A i 2 i = 1 θ w i 2 B i 2 - ( i = 1 θ w i 2 A i B i ) 2 .
w i = ( θ - i + 1 ) / i = 1 θ i
w i = ( θ - i + 1 ) / i = 1 θ i ,
Δ p = ( 2 a ± 2 x + 2 y tan θ 2 ) sin θ - 2 a [ n - ( n 2 - sin 2 θ ) 1 / 2 ] ,
Δ p = Δ p 0 + Δ p 1 - Δ p 2 ,
Δ p 0 = ( 2 a ± 2 x ) sin θ ,
Δ p 1 = 2 y tan θ 2 sin θ ,
Δ p 2 = 2 a [ n - ( n 2 - sin 2 θ ) 1 / 2 ] .
Δ p = Δ p 0 - Δ p 1 + Δ p 2 .
Δ p = Δ p 0 - Δ p 1 - Δ p 2 .
Δ p = Δ p 0 + Δ p 1 + Δ p 2 .
Δ p = ( 2 a + 2 x + 2 y tan θ 2 ) sin θ - 2 a [ n - ( n 2 - sin 2 θ ) 1 / 2 ] .
Δ p = Δ p 0 + Δ p 1 - Δ p 2 ,
Δ p = Δ p 0 - Δ p 1 + Δ p 2 ,
Δ p = Δ p 0 + Δ p 1 + Δ p 2 ,

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