Abstract

The Fourier transform technique, originally developed for spatial fringe pattern analysis, has been applied to the analysis of a temporal fringe signal obtained by a wavelength-shift interferometer used for absolute distance measurements. It has been shown that the error caused by the nonlinear and time-varying current-wavelength characteristic of the laser diode can be removed by combining the Fourier transform technique with the reference technique. A novel technique for distance measurement based on multiple-beam interferometry has been proposed, and an experimental demonstration is given for a three-beam interferometer that includes a reference reflector as an integral part of the system. Error sources and the limitation of the technique are discussed.

© 1991 Optical Society of America

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References

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  1. H. Kikuta, K. Iwata, R. Nagata, “Distance measurement by the wavelength shift of laser diode light,” Appl. Opt. 25, 2976–2980 (1986).
    [CrossRef] [PubMed]
  2. H. Kikuta, K. Iwata, R. Nagata, “Absolute distance measurement by wavelength shift interferometry with a laser diode: some systematic error sources,” Appl. Opt. 26, 1654– 1660 (1987).
    [CrossRef] [PubMed]
  3. K. Hotate, O. Kamatani, “Reflectometry by means of optical-coherence modulation,” Electron. Lett. 25, 1503–1504 (1989).
    [CrossRef]
  4. T. Kubota, M. Nara, T. Yoshino, “Interferometer for measuring displacement and distance,” Opt. Lett. 12, 310–312 (1987).
    [CrossRef] [PubMed]
  5. T. Kobayashi, “Interferometry for measuring distance and displacement using semiconductor lasers,” Kogaku (Jpn. J. Opt.) 17, 279–284 (1988).
  6. S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
    [CrossRef]
  7. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  8. M. Takeda, “Spatial fringe-pattern analysis and its application to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
    [CrossRef]
  9. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  10. M. Takeda, Q. S. Ru, “Computer-based highly sensitive electron-wave interferometry,” Appl. Opt. 24, 3068–3071 (1985).
    [CrossRef] [PubMed]
  11. See, for example, K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]

1990 (1)

M. Takeda, “Spatial fringe-pattern analysis and its application to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

1989 (1)

K. Hotate, O. Kamatani, “Reflectometry by means of optical-coherence modulation,” Electron. Lett. 25, 1503–1504 (1989).
[CrossRef]

1988 (1)

T. Kobayashi, “Interferometry for measuring distance and displacement using semiconductor lasers,” Kogaku (Jpn. J. Opt.) 17, 279–284 (1988).

1987 (2)

1986 (1)

1985 (2)

M. Takeda, Q. S. Ru, “Computer-based highly sensitive electron-wave interferometry,” Appl. Opt. 24, 3068–3071 (1985).
[CrossRef] [PubMed]

S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
[CrossRef]

1983 (1)

1982 (2)

Davies, D. E. N.

S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
[CrossRef]

Hotate, K.

K. Hotate, O. Kamatani, “Reflectometry by means of optical-coherence modulation,” Electron. Lett. 25, 1503–1504 (1989).
[CrossRef]

Ina, H.

Itoh, K.

Iwata, K.

Kamatani, O.

K. Hotate, O. Kamatani, “Reflectometry by means of optical-coherence modulation,” Electron. Lett. 25, 1503–1504 (1989).
[CrossRef]

Kikuta, H.

Kingsley, S. A.

S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
[CrossRef]

Kobayashi, S.

Kobayashi, T.

T. Kobayashi, “Interferometry for measuring distance and displacement using semiconductor lasers,” Kogaku (Jpn. J. Opt.) 17, 279–284 (1988).

Kubota, T.

Mutoh, K.

Nagata, R.

Nara, M.

Ru, Q. S.

Takeda, M.

Yoshino, T.

Appl. Opt. (5)

Electron. Lett. (2)

K. Hotate, O. Kamatani, “Reflectometry by means of optical-coherence modulation,” Electron. Lett. 25, 1503–1504 (1989).
[CrossRef]

S. A. Kingsley, D. E. N. Davies, “OFDR diagnostics for fiber and integrated-optic systems,” Electron. Lett. 21, 434– 435 (1985).
[CrossRef]

Ind. Metrol. (1)

M. Takeda, “Spatial fringe-pattern analysis and its application to precision interferometry and profilometry: an overview,” Ind. Metrol. 1, 79–99 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

Kogaku (Jpn. J. Opt.) (1)

T. Kobayashi, “Interferometry for measuring distance and displacement using semiconductor lasers,” Kogaku (Jpn. J. Opt.) 17, 279–284 (1988).

Opt. Lett. (1)

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Figures (14)

Fig. 1
Fig. 1

Fourier spectra of the interferometric fringe signal obtained by two-beam interferometry. The three spectra are separated by the carrier frequency fS.

Fig. 2
Fig. 2

Multiple-beam Fizeau interferometer with multiple reflectors.

Fig. 3
Fig. 3

Fourier spectra of the interferometric fringe signal obtained by multiple-beam interferometry. The multiple spectra are separated from each other by the carrier frequencies fmn.

Fig. 4
Fig. 4

Pair of two-beam interferometers sharing a common light source: LD, laser diode; CL, collimator lens; BS, beam splitter; CC, corner cube; AMP, amplifier; PD, photodetector; and LPF, low-pass filter.

Fig. 5
Fig. 5

Part of the interferometric fringe signals obtained from the object interferometer (a) and from the reference interferometer (b).

Fig. 6
Fig. 6

Fourier spectra of the fringe signals obtained from the object interferometer (a) and from the reference interferometer (b). Only positive frequency components are shown.

Fig. 7
Fig. 7

(a) Instantaneous angular frequencies ωS(t) and ωR(t); (b) ratio of ωS(t) to ωR(t).

Fig. 8
Fig. 8

Three-beam interferometer: LD, laser diode; CL, collimator lens; BS, beam splitter; CC, corner cube; AMP, amplifier; PD, photodetector; and LPF, low-pass filter.

Fig. 9
Fig. 9

Part of the fringe signal obtained by the three-beam interferometer. The waveform is so different from a pure sinusoid that a conventional fringe-counting technique cannot be used.

Fig. 10
Fig. 10

Fourier spectra of the fringe signal obtained by a three-beam interferometer. Only positive-frequency components are shown.

Fig. 11
Fig. 11

Instantaneous angular frequencies ω12(t), ω13(t), and ω23(t).

Fig. 12
Fig. 12

Optical path differences obtained by the reference technique: (a) ω13(t)/ω12(t) which gives L13/L12; (b) ω23(t)/ω12(t); (c) the ratio of the optical path differences L13/L12 obtained from Eq. (29).

Fig. 13
Fig. 13

Current ramps (upper trace) and fringe signals (lower trace) obtained from (a) fLD = 14.3 Hz and (b) fLD = 143 Hz.

Fig. 14
Fig. 14

Instantaneous frequencies ω(t) obtained for four different repetition frequencies. The influence of the nonlinearity increases with the repetition frequency.

Equations (30)

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Δ ϕ = 2 π L / Λ = 2 π L Δ λ / λ 2 .
ϕ ( t ) = 2 π L λ 2 π α L t λ 2 = ϕ 0 + 2 π f S t ,
f S = α L / λ 2 ,
f R = α L R / λ 2 .
L = ( f S f R ) L R .
α ( t ) = α 0 + Δ α ( t ) ,
ϕ ( t ) = ϕ 0 + 2 π f S t + θ ( t ) ,
θ ( t ) = 2 π L Δ α ( t ) t / λ 2 ,
f S = α 0 L / λ 2 .
g ( t ) = a ( t ) + b ( t ) cos [ 2 π f S t + θ ( t ) + ϕ 0 ] ,
g ( t ) = a ( t ) + c ( t ) exp ( 2 π j f S t ) + c * ( t ) exp ( 2 π j f S t ) ,
c ( t ) = ½ b ( t ) exp j [ θ ( t ) + ϕ 0 ] ,
G ( f ) = g ( t ) exp ( 2 π j f t ) d t = A ( f ) + C ( f f S ) + C * [ ( f + f S ) ] ,
c ( t ) exp ( 2 π j f S t ) = ½ b ( t ) exp j [ 2 π f S t + θ ( t ) + ϕ 0 ] .
log [ c ( t ) exp ( 2 π j f S t ) ] = log [ ½ b ( t ) ] + j [ 2 π f S t + θ ( t ) + ϕ 0 ] .
ϕ R ( t ) = ϕ R 0 + 2 π f R t + θ R ( t ) ,
θ R ( t ) = 2 π L R Δ α ( t ) t / λ 2 ,
f R = α 0 L R / λ 2 .
ω S ( t ) = d ϕ ( t ) d t = 2 π f S + d θ ( t ) d t = 2 π L λ 2 [ α 0 + d Δ α ( t ) t d t ] ,
ω R ( t ) = d ϕ R ( t ) d t = 2 π f R + d θ R ( t ) d t = 2 π L R λ 2 [ α 0 + d Δ α ( t ) t d t ] .
L = [ ω S ( t ) ω R ( t ) ] L R .
g ( t ) = a ( t ) + m < n b m n ( t ) cos [ 2 π f m n t + θ m n ( t ) + ϕ m n ] ,
θ m n ( t ) = 2 π L m n Δ α ( t ) t / λ 2 ,
f m n = α 0 L m n / λ 2 ,
G ( f ) = A ( f ) + m < n { C m n ( f f m n ) + C m n * [ ( f + f m n ) ] } ,
c m n ( t ) = ½ b m n ( t ) exp j [ θ m n ( t ) + ϕ m n ] .
ϕ m n ( t ) = 2 π f m n t + θ m n ( t ) + ϕ m n .
L m n = [ ω m n ( t ) ω 12 ( t ) ] L 12 ,
L 13 = [ 1 + ω 23 ( t ) ω 12 ( t ) ] L 12 .
G ( f ) A ( f ) + n = 2 N { C 1 n ( f f 1 n ) + C 1 n * [ f + f 1 n ] } ,

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