Abstract

A technique is presented for measuring small displacements by observing the frequency of spectral modulation of white light in a Michelson interferometer. An experiment is described in which the step size of a stepper-motor-driven translation stage was measured by recording the spectrum of light output from an interferometer and performing a cross-correlation calculation with theoretical spectra. Measurements made using standard laboratory-quality optical equipment were accurate to within ~10 nm for a range of over 100 μm.

© 1989 Optical Society of America

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References

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  1. C. H. Palmer, Optics Experiments and Demonstrations (Johns Hopkins, Baltimore, 1962), pp. 147–157.
  2. H. A. Deferrari, F. A. Andrews, “Laser Interferometric Technique for Measuring Small-Order Vibration Displacements,” J. Acoust. Soc. Am. 39, 979–980 (1966).
    [CrossRef]
  3. H. A. Deferrari, R-A. Darby, F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
    [CrossRef]
  4. R. Keller, R. Salathé, T. Tschudi, C. Vourmard, “Michelson Interferometer for Detection of Fast Displacements of Less than a Quarter-Wave over Small Areas,” Appl. Opt. 14, 1616–1620 (1975).
    [CrossRef] [PubMed]
  5. P. A. Flourney, R. W. McClure, G. Wyntjes, “White-Light Interferometric Thickness Gauge,” Appl. Opt. 11, 1907–1915 (1972).
    [CrossRef]
  6. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 191–192.
  7. W. P. Alford, A. Gold, “Laboratory Measurements of the Velocity of Light,” Am. J. Phys. 56, 481–484 (1958).
    [CrossRef]
  8. J-C. Vienot, J-P. Goedgebuer, A. Lacourt, “Space and Time Variables in Optics and Holography: Recent Experimental Aspects,” Appl. Opt. 16, 454–461 (1977).
    [CrossRef] [PubMed]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 300–301.

1977 (1)

1975 (1)

1972 (1)

1967 (1)

H. A. Deferrari, R-A. Darby, F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[CrossRef]

1966 (1)

H. A. Deferrari, F. A. Andrews, “Laser Interferometric Technique for Measuring Small-Order Vibration Displacements,” J. Acoust. Soc. Am. 39, 979–980 (1966).
[CrossRef]

1958 (1)

W. P. Alford, A. Gold, “Laboratory Measurements of the Velocity of Light,” Am. J. Phys. 56, 481–484 (1958).
[CrossRef]

Alford, W. P.

W. P. Alford, A. Gold, “Laboratory Measurements of the Velocity of Light,” Am. J. Phys. 56, 481–484 (1958).
[CrossRef]

Andrews, F. A.

H. A. Deferrari, R-A. Darby, F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[CrossRef]

H. A. Deferrari, F. A. Andrews, “Laser Interferometric Technique for Measuring Small-Order Vibration Displacements,” J. Acoust. Soc. Am. 39, 979–980 (1966).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 300–301.

Darby, R-A.

H. A. Deferrari, R-A. Darby, F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[CrossRef]

Deferrari, H. A.

H. A. Deferrari, R-A. Darby, F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[CrossRef]

H. A. Deferrari, F. A. Andrews, “Laser Interferometric Technique for Measuring Small-Order Vibration Displacements,” J. Acoust. Soc. Am. 39, 979–980 (1966).
[CrossRef]

Flourney, P. A.

Goedgebuer, J-P.

Gold, A.

W. P. Alford, A. Gold, “Laboratory Measurements of the Velocity of Light,” Am. J. Phys. 56, 481–484 (1958).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 191–192.

Keller, R.

Lacourt, A.

McClure, R. W.

Palmer, C. H.

C. H. Palmer, Optics Experiments and Demonstrations (Johns Hopkins, Baltimore, 1962), pp. 147–157.

Salathé, R.

Tschudi, T.

Vienot, J-C.

Vourmard, C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 300–301.

Wyntjes, G.

Am. J. Phys. (1)

W. P. Alford, A. Gold, “Laboratory Measurements of the Velocity of Light,” Am. J. Phys. 56, 481–484 (1958).
[CrossRef]

Appl. Opt. (3)

J. Acoust. Soc. Am. (2)

H. A. Deferrari, F. A. Andrews, “Laser Interferometric Technique for Measuring Small-Order Vibration Displacements,” J. Acoust. Soc. Am. 39, 979–980 (1966).
[CrossRef]

H. A. Deferrari, R-A. Darby, F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[CrossRef]

Other (3)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 191–192.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 300–301.

C. H. Palmer, Optics Experiments and Demonstrations (Johns Hopkins, Baltimore, 1962), pp. 147–157.

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Figures (4)

Fig. 1
Fig. 1

Schematic of configuration of experimental apparatus.

Fig. 2
Fig. 2

Example of unmodulated white light spectrum.

Fig. 3
Fig. 3

Example of modulated white light spectrum.

Fig. 4
Fig. 4

Normalized cross-correlation function between theoretical spectrum and experimental spectrum shown in Fig. 3. Peak value is 0.9966 at a displacement of 1977 nm.

Tables (1)

Tables Icon

Table I Example Set of Displacement Measurements

Equations (14)

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u D ( t ) = a u ( t ) b u ( t 2 x / c ) ,
U D ( ν ) = U ( ν ) [ a b exp ( i 4 π x ν / c ) ] .
G D ( ν ) | U D ( ν ) | 2 = | U ( ν ) | 2 [ a 2 + b 2 2 a b cos ( 4 π x ν / c ) ] = ( a 2 + b 2 ) G ( ν ) [ 1 υ cos ( 4 π x ν / c ) ] ,
υ 2 a b a 2 + b 2 .
I D ( λ ) = I ( λ ) [ 1 υ cos ( 4 π x / λ ) ] .
I ˆ D ( λ n ) = ( a 2 + b 2 ) I ( λ n ) [ 1 υ cos ( 4 π x ˆ / λ n ) ] ; n = 1 , 2 , , N ,
Γ ( x ˆ ) = n = 1 N I ˆ D ( λ n ) I D ( λ n ) n = 1 N I ˆ D 2 ( λ n ) n = 1 N I D 2 ( λ n ) ,
I D ( λ n ) = I D ( true ) ( λ n ) + n ,
I ˆ D ( λ n ) = I D ( true ) ( λ n ) + δ n ,
I ˆ D ( λ n ) I D ( true ) ( λ n ) + 4 π υ Δ x λ n I ( λ n ) sin ( 4 π x ˆ / λ n ) .
n = 1 N n 2 ( N 1 ) σ 2 ,
n = 1 N 4 π υ Δ x λ n I ( λ n ) sin ( 4 π x ˆ / λ n ) n 0 ,
( N 1 ) σ 2 + ( Δ x ) 2 n = 1 N ( 4 π υ λ n ) 2 I 2 ( λ n ) sin 2 ( 4 π x ˆ / λ n ) n = 1 N [ I D ( λ n ) I ˆ D ( λ n ) ] 2 ,
( Δ x ) 2 < n = 1 N [ I D ( λ n ) I ˆ D ( λ n ) ] 2 n = 1 N ( 4 π υ / λ n ) 2 I 2 ( λ n ) sin 2 ( 4 π x ˆ / λ n ) .

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