Abstract

The development of stabilized multifrequency lasers makes fractional fringes an increasingly attractive technique for length measurement. Determination of an unknown length from the measured fractional fringes is aided by the development of analytical equations for the length and its uncertainty, and criteria are given for selecting the wavelengths.

© 1977 Optical Society of America

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References

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  1. M. R. Benoit, J. Phys. 7, 57 (1898).
  2. C. Freed, A. Javan, Appl. Phys. Lett. 17, 53 (1970).
    [CrossRef]
  3. K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, Appl. Phys. Lett. 22, 192 (1973).
    [CrossRef]
  4. P. L. M. Heydemann, Rev. Sci. Instrum. 42, 983 (1971).
    [CrossRef]
  5. C. R. Tilford, Rev. Sci. Instrum. 44, 180 (1973).
    [CrossRef]
  6. F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Laser Spectroscopy, R. C. Brewer, A. Moordian, Eds. (Plenum, New York, 1974), p. 555.
    [CrossRef]

1973 (2)

K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, Appl. Phys. Lett. 22, 192 (1973).
[CrossRef]

C. R. Tilford, Rev. Sci. Instrum. 44, 180 (1973).
[CrossRef]

1971 (1)

P. L. M. Heydemann, Rev. Sci. Instrum. 42, 983 (1971).
[CrossRef]

1970 (1)

C. Freed, A. Javan, Appl. Phys. Lett. 17, 53 (1970).
[CrossRef]

1898 (1)

M. R. Benoit, J. Phys. 7, 57 (1898).

Benoit, M. R.

M. R. Benoit, J. Phys. 7, 57 (1898).

Cupp, J. D.

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Laser Spectroscopy, R. C. Brewer, A. Moordian, Eds. (Plenum, New York, 1974), p. 555.
[CrossRef]

Danielson, B. L.

K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, Appl. Phys. Lett. 22, 192 (1973).
[CrossRef]

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Laser Spectroscopy, R. C. Brewer, A. Moordian, Eds. (Plenum, New York, 1974), p. 555.
[CrossRef]

Day, G. W.

K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, Appl. Phys. Lett. 22, 192 (1973).
[CrossRef]

Evenson, K. M.

K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, Appl. Phys. Lett. 22, 192 (1973).
[CrossRef]

Freed, C.

C. Freed, A. Javan, Appl. Phys. Lett. 17, 53 (1970).
[CrossRef]

Heydemann, P. L. M.

P. L. M. Heydemann, Rev. Sci. Instrum. 42, 983 (1971).
[CrossRef]

Javan, A.

C. Freed, A. Javan, Appl. Phys. Lett. 17, 53 (1970).
[CrossRef]

McDonald, D. G.

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Laser Spectroscopy, R. C. Brewer, A. Moordian, Eds. (Plenum, New York, 1974), p. 555.
[CrossRef]

Petersen, F. R.

K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, Appl. Phys. Lett. 22, 192 (1973).
[CrossRef]

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Laser Spectroscopy, R. C. Brewer, A. Moordian, Eds. (Plenum, New York, 1974), p. 555.
[CrossRef]

Tilford, C. R.

C. R. Tilford, Rev. Sci. Instrum. 44, 180 (1973).
[CrossRef]

Wells, J. S.

K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, Appl. Phys. Lett. 22, 192 (1973).
[CrossRef]

Appl. Phys. Lett. (2)

C. Freed, A. Javan, Appl. Phys. Lett. 17, 53 (1970).
[CrossRef]

K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, Appl. Phys. Lett. 22, 192 (1973).
[CrossRef]

J. Phys. (1)

M. R. Benoit, J. Phys. 7, 57 (1898).

Rev. Sci. Instrum. (2)

P. L. M. Heydemann, Rev. Sci. Instrum. 42, 983 (1971).
[CrossRef]

C. R. Tilford, Rev. Sci. Instrum. 44, 180 (1973).
[CrossRef]

Other (1)

F. R. Petersen, D. G. McDonald, J. D. Cupp, B. L. Danielson, in Laser Spectroscopy, R. C. Brewer, A. Moordian, Eds. (Plenum, New York, 1974), p. 555.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Illustration of the use of fractional fringes f1 and f2 to determine a length L. The values of f1, f2, and the estimated length L′ can be used to calculate an improved value for L.

Fig. 2
Fig. 2

Illustration of possible error of one synthetic wavelength due to uncertainty in the measured fringes. This error can be avoided by restricting the usable portion of the synthetic wavelength to (1 − 2γs.

Equations (31)

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L = ( N i + f i ) λ i
L k i = N i + f i ,
L k i = G i + e i ,
δ L k i = N i - G i + f i - e i .
Σ A i δ L k i = Σ A i ( N i - G i + f i - e i ) ,
δ L = Σ A i ( N i - G i ) + Σ A i ( f i - e i ) Σ A i k i
δ L = I + Σ A i ( f i - e i ) Σ A i k i = [ I + Σ A i ( f i - e i ) ] λ s ,
- ½ < I + Σ A i ( f i - e i ) < ½ .
I + Σ A i ( f i - e i ) = F [ Σ A i ( f i - e i ) ]
= F [ Σ A i ( f i - e i ) ] - 1 ,
F [ Σ A i ( f i - e i ) ] = F ( Σ A i f i ) - F ( Σ A i e i )
= F ( Σ A i f i ) - F ( Σ A i e i ) + 1.
δ L = [ I + F ( Σ A i f i ) - F ( Σ A i e i ) ] λ s .
Σ A i e i = Σ A i L k i - Σ A i G i .
F ( Σ A i e i ) = F ( L Σ A i k i ) = F ( L / λ s ) .
δ L = [ I + F ( Σ A i f i ) - F ( L / λ s ) ] λ s ,
- ½ < I + F ( Σ A i f i ) - F ( L / λ s ) < ½ .
L = L + δ L .
d L = L L d L + δ L L d L + Σ δ L f i d f i ,
δ L = [ Σ A i ( N i + f i ) - Σ A i ( G i + e i ) ] λ s = [ Σ A i N i + Σ A i f i - L Σ A i k i ] λ s
( δ L ) / ( L ) = - ( Σ A i k i ) λ s = - 1
( δ L ) / ( f i ) = A i λ s .
d L = d L - d L + ( Σ A i d f i ) λ s = ( Σ A i d f i ) λ s .
- ( 1 - 2 γ ) λ s 2 < δ L < ( 1 - 2 γ ) λ s 2 .
Σ A i Σ A i k i < 1 - 2 Σ A i Σ A i k i ,
γ λ s < L < ( 1 - γ ) λ s             ( L > 0 )
- ( 1 - γ ) λ s < L < - γ λ s             ( L < 0 ) ,
L = F ( Σ A i f i ) λ s             ( L > 0 )
L = [ F ( Σ A i f i ) - 1 ] λ s             ( L < 0 ) .
λ 1 = R ( 28 ) = 9.22953010 μ m ; λ 2 = R ( 24 ) = 9.24994570 μ m ; λ 3 = P ( 32 ) = 9.65741651 μ m .
A 1 = 1 , A 2 = - 1 , A 3 = 0             λ s 1 = 4.1817 mm ; A 1 = 1 , A 2 = 0 , A 3 = - 1             λ s 2 = 0.2083109 m ; A 1 = - 1 , A 2 = 0 , A 3 = 2             λ s 3 = 0.0101269060 mm .

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