Abstract

We report the influence of the nonlinearities in the wavelength-sweeping speed on the resulting interferometric signals in an absolute distance interferometer. The sweeping signal is launched in the reference and target interferometers from an external cavity laser source. The experimental results demonstrate a good resolution in spite of the presence of nonlinearities in the wavelength sweep. These nonlinearities can be modeled by a sum of sinusoids. A simulation is then implemented to analyze the influence of their parameters. It shows that a sinusoidal nonlinearity is robust enough to give a good final measurement uncertainty through a Fourier transform technique. It can be concluded that an optimal value of frequency and amplitude exists in the case of a sinusoidal nonlinearity.

© 2007 Optical Society of America

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References

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  1. S.-W. Kim, K.-N. Joo, J. Jin, and Y. S. Kim, "Absolute distance measurement using femtosecond laser," Proc. SPIE 5858, 187-195 (2005).
  2. K. Minoshima and H. Matsumoto, "High-precision distance measurement using the frequency comb of an ultrashort pulse laser," in Conference on Lasers and Electro-Optics Europe Munich, Germany, 12-17 June 2005.
  3. N. Schuhler, Y. Salvadé, S. Lévêque, R. Dändliker, and R. Holzwarth, "Frequency-comb-referenced two-wavelength source for absolute distance measurement," Opt. Lett. 31, 3101-3103 (2006).
    [Crossref] [PubMed]
  4. J. Thiel, T. Pfeifer, and M. Hartmann, "Interferometric measurement of absolute distances of up to 40 m," Measurement 16, 1-6 (1995).
    [Crossref]
  5. R. Mokdad, B. Pécheux, P. Pfeiffer, and P. Meyrueis, "Fringe pattern analysis with a parametric method for measurement of absolute distance by a frequency-modulated continuous optical wave technique," Appl. Opt. 42, 1008-1012 (2003).
    [Crossref] [PubMed]
  6. P. A. Coe, D. F. Howell, and R. B. Nickerson, "Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous, and precise distance measurements in a hostile environment," Meas. Sci. Technol. 15, 2175-2187 (2004).
    [Crossref]
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    [Crossref]
  8. T. Kinder and K.-D. Salewski, "Absolute distance interferometer with grating-stabilized tunable diode laser at 633 nm," J. Opt. A 4, S364-S368 (2002).
    [Crossref]
  9. P. Pfeiffer, L. Perret, R. Mokdad, and B. Pécheux, "Fringe analysis in scanning frequency interferometry for absolute distance measurement," in Fringe 2005, the Fifth International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2006), pp. 388-395.
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. J. G. Proakis and G. Dimitris Manolakis, Digital Signal Processing; Principles, Algorithms, and Applications (Prentice-Hall Engineering, 1995).
    [PubMed]

2006 (1)

2005 (1)

S.-W. Kim, K.-N. Joo, J. Jin, and Y. S. Kim, "Absolute distance measurement using femtosecond laser," Proc. SPIE 5858, 187-195 (2005).

2004 (1)

P. A. Coe, D. F. Howell, and R. B. Nickerson, "Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous, and precise distance measurements in a hostile environment," Meas. Sci. Technol. 15, 2175-2187 (2004).
[Crossref]

2003 (1)

2002 (1)

T. Kinder and K.-D. Salewski, "Absolute distance interferometer with grating-stabilized tunable diode laser at 633 nm," J. Opt. A 4, S364-S368 (2002).
[Crossref]

1999 (1)

1996 (1)

1995 (1)

J. Thiel, T. Pfeifer, and M. Hartmann, "Interferometric measurement of absolute distances of up to 40 m," Measurement 16, 1-6 (1995).
[Crossref]

1991 (1)

Appl. Opt. (4)

J. Opt. A (1)

T. Kinder and K.-D. Salewski, "Absolute distance interferometer with grating-stabilized tunable diode laser at 633 nm," J. Opt. A 4, S364-S368 (2002).
[Crossref]

Meas. Sci. Technol. (1)

P. A. Coe, D. F. Howell, and R. B. Nickerson, "Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous, and precise distance measurements in a hostile environment," Meas. Sci. Technol. 15, 2175-2187 (2004).
[Crossref]

Measurement (1)

J. Thiel, T. Pfeifer, and M. Hartmann, "Interferometric measurement of absolute distances of up to 40 m," Measurement 16, 1-6 (1995).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (1)

S.-W. Kim, K.-N. Joo, J. Jin, and Y. S. Kim, "Absolute distance measurement using femtosecond laser," Proc. SPIE 5858, 187-195 (2005).

Other (3)

K. Minoshima and H. Matsumoto, "High-precision distance measurement using the frequency comb of an ultrashort pulse laser," in Conference on Lasers and Electro-Optics Europe Munich, Germany, 12-17 June 2005.

P. Pfeiffer, L. Perret, R. Mokdad, and B. Pécheux, "Fringe analysis in scanning frequency interferometry for absolute distance measurement," in Fringe 2005, the Fifth International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2006), pp. 388-395.

J. G. Proakis and G. Dimitris Manolakis, Digital Signal Processing; Principles, Algorithms, and Applications (Prentice-Hall Engineering, 1995).
[PubMed]

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Figures (7)

Fig. 1
Fig. 1

Configuration of our setup. PD obj , PD ref , object and reference photodiodes; ECLD, external cavity laser diode; reference interferometer, a fibered Mach–Zehnder of OPD of 11   m ; RR, retroreflector; BS, beam splitter. Target RR is micropositioned.

Fig. 2
Fig. 2

Frequency ratios for target A at 2.2   m and a 20 nm / s tuning speed; the line is the regression of the mean ratios (circles) over 12 positions and the stars are the computed ratios of each one of the six acquisitions of 2 17 samples at each position.

Fig. 3
Fig. 3

Experimental beat frequency for the 10 nm / s sweeping speed is the continuous curve, and its simulation using a nonlinear sweep term of five sinusoids is the dashed curve.

Fig. 4
Fig. 4

Difference between simulated and ideal frequency ratios for different SNR levels, with a manually fixed filter width (circles, linear sweep; crosses, sinusoidal nonlinearity with amplitude A = 2.2 × 10 3   nm and modulation frequency f m = 94.5   Hz ); 10 computer-generated scans per SNR level with 2 16 samples.

Fig. 5
Fig. 5

(Color online) (a) Experimental instantaneous ratio of the last measurement of Fig. 2. Upper, full graph; lower, zoom. (b) Histogram, distribution function of the instantaneous ratios with steps of 3.7 × 10 6 circle, extracted mean ratio.

Fig. 6
Fig. 6

(Color online) Difference between simulated and ideal frequency ratios for several sweep models (circles, linear; crosses, sinusoidal nonlinearity with A = 2.2 × 10 3   nm and f m = 94.5   Hz ; squares, sinusoidal nonlinearity with A / 2 and f m / 2 ; triangles, sinusoidal nonlinearity with 2A and 2 f m ; diamonds, data-fitting 10 nm / s simulation with five sinusoids). No noise, automated filter, and 2 16 samples per acquisition.

Fig. 7
Fig. 7

(Color online) Difference between simulated and ideal frequency ratios for several sweep models (crosses, sinusoidal nonlinearity with A = 2.2 × 10 3   nm and f m = 94.5   Hz ; squares, sinusoidal nonlinearity with A / 2 and f m / 2 ; triangles, sinusoidal nonlinearity with 2A and 2fm).

Equations (8)

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I i ( r , t ) = ( I 1 i ( r , t ) + I 2 i ( r , t ) ) ( 1 + V i ( t ) cos ( 2 π L i / λ ( t ) + Φ 0 i ) ) .
f b i = α 0 × L i λ 0 2 .
λ ( t ) = λ 0 + α 0 t + α nl λ ( t ) .
Δ Φ = 2 π L i λ 2 π L i λ + Δ λ .
f b i ( t ) = Δ Φ 2 π Δ t = 1 Δ t ( L i Δ λ λ ( λ + Δ λ ) ) .
f b i ( t ) = L i α 0 λ 2 + L i λ 2 α nl λ ( t ) .
λ ( t ) = λ 0 + α 0 t + A k   sin ( 2 π f m k t + φ k ) .
f b i ( t ) = L i α 0 λ 2 + L i λ 2 k = 1 A k   sin ( 2 π f m k t + φ k ) .

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