Abstract

Interferometry with laser diodes is a cost-effective way to perform displacement measurement. The tunability of laser diodes is also of great interest in multiple-wavelength interferometry. However, the additional flicker noise in the frequency-noise spectrum of semiconductor lasers may become a limiting factor. Investigations on the limitations due to the 1/f noise of laser diodes are presented for both classical and multiple-wavelength interferometry. Measurements at the limit of the coherence length of laser diodes with the corresponding phase fluctuations are reported. The theoretical results are verified experimentally.

© 2000 Optical Society of America

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References

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  1. G. P. Barwood, P. Gill, W. R. C. Rowley, “A simple rubidium-stabilized laser diode for interferometric applications,” J. Phys. E 21, 966–971 (1988).
    [Crossref]
  2. A. Abou-Zeid, “Diode lasers for interferometry,” Precis. Eng. 11, 139–144 (1989).
    [Crossref]
  3. S. Manhart, R. Maurer, “Diode laser and fiber optics for dual-wavelength heterodyne interferometry,” in Optics in Complex Systems, F. Lanzl, G. Weigelt, eds., Proc. SPIE1319, 214–216 (1990).
    [Crossref]
  4. P. de Groot, S. Kishner, “Synthetic wavelength stabilization for two-color laser diode interferometry,” Appl. Opt. 30, 4026–4033 (1991).
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  5. E. Zimmermann, Y. Salvadé, R. Dändliker, “Stabilized three-wavelength source calibrated by electronic means for high-accuracy absolute distance measurement,” Opt. Lett. 21, 531–533 (1996).
    [Crossref] [PubMed]
  6. R. Dändliker, Y. Salvadé, E. Zimmermann, “Distance measurement by multiple-wavelength interferometry,” J. Opt. 29, 105–114 (1998).
    [Crossref]
  7. J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10, 2113–2118 (1971).
    [Crossref] [PubMed]
  8. C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12, 2071–2074 (1973).
    [Crossref] [PubMed]
  9. K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1988), Chap. 7.
  10. Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of multiple-wavelength interferometry due to frequency fluctuations of laser diodes,” in Proceedings of International Workshop on Interferometry (The Institute of Physical and Chemical Research, Wako, Japan, 1996), pp. 9–10.
  11. Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of interferometry due to frequency fluctuations of laser diodes,” Proceedings of Topical Meeting on Optoelectronic/Displacement Measurements and Applications (European Optical Society, Orsay, France, 1997).
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).
  13. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 10.3–10.4.
  14. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.1.3.
  15. R. Dändliker, R. Thalmann, D. Prongué, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339–341 (1988).
    [Crossref] [PubMed]
  16. E. Zimmermann, R. Dändliker, “Full-dynamic phase demodulator for heterodyne interferometric vibration measurements,” in Second International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, E. P. Tomasini, ed., Proc. SPIE2868, 488–489 (1996).
    [Crossref]
  17. N. Sagna, C. Mandache, P. Thomann, “Noise measurements in single-mode GaAlAs diode lasers,” in Proceedings of the Sixth European Frequency and Time Forum, (European Space Research and Technology Centre, Noordwijk, The Netherlands, 1992), pp. 521–525.
  18. Y. Shevy, H. Deng, “Frequency-stable and ultranarrow-linewidth semiconductor laser locked directly to an atomic-cesium transition,” Opt. Lett. 23, 472–474 (1998).
    [Crossref]

1998 (2)

R. Dändliker, Y. Salvadé, E. Zimmermann, “Distance measurement by multiple-wavelength interferometry,” J. Opt. 29, 105–114 (1998).
[Crossref]

Y. Shevy, H. Deng, “Frequency-stable and ultranarrow-linewidth semiconductor laser locked directly to an atomic-cesium transition,” Opt. Lett. 23, 472–474 (1998).
[Crossref]

1996 (1)

1991 (1)

1989 (1)

A. Abou-Zeid, “Diode lasers for interferometry,” Precis. Eng. 11, 139–144 (1989).
[Crossref]

1988 (2)

G. P. Barwood, P. Gill, W. R. C. Rowley, “A simple rubidium-stabilized laser diode for interferometric applications,” J. Phys. E 21, 966–971 (1988).
[Crossref]

R. Dändliker, R. Thalmann, D. Prongué, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339–341 (1988).
[Crossref] [PubMed]

1973 (1)

1971 (1)

Abou-Zeid, A.

A. Abou-Zeid, “Diode lasers for interferometry,” Precis. Eng. 11, 139–144 (1989).
[Crossref]

Barwood, G. P.

G. P. Barwood, P. Gill, W. R. C. Rowley, “A simple rubidium-stabilized laser diode for interferometric applications,” J. Phys. E 21, 966–971 (1988).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 10.3–10.4.

Dändliker, R.

R. Dändliker, Y. Salvadé, E. Zimmermann, “Distance measurement by multiple-wavelength interferometry,” J. Opt. 29, 105–114 (1998).
[Crossref]

E. Zimmermann, Y. Salvadé, R. Dändliker, “Stabilized three-wavelength source calibrated by electronic means for high-accuracy absolute distance measurement,” Opt. Lett. 21, 531–533 (1996).
[Crossref] [PubMed]

R. Dändliker, R. Thalmann, D. Prongué, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339–341 (1988).
[Crossref] [PubMed]

E. Zimmermann, R. Dändliker, “Full-dynamic phase demodulator for heterodyne interferometric vibration measurements,” in Second International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, E. P. Tomasini, ed., Proc. SPIE2868, 488–489 (1996).
[Crossref]

Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of interferometry due to frequency fluctuations of laser diodes,” Proceedings of Topical Meeting on Optoelectronic/Displacement Measurements and Applications (European Optical Society, Orsay, France, 1997).

Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of multiple-wavelength interferometry due to frequency fluctuations of laser diodes,” in Proceedings of International Workshop on Interferometry (The Institute of Physical and Chemical Research, Wako, Japan, 1996), pp. 9–10.

de Groot, P.

Deng, H.

Gill, P.

G. P. Barwood, P. Gill, W. R. C. Rowley, “A simple rubidium-stabilized laser diode for interferometric applications,” J. Phys. E 21, 966–971 (1988).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.1.3.

Kishner, S.

Mandache, C.

N. Sagna, C. Mandache, P. Thomann, “Noise measurements in single-mode GaAlAs diode lasers,” in Proceedings of the Sixth European Frequency and Time Forum, (European Space Research and Technology Centre, Noordwijk, The Netherlands, 1992), pp. 521–525.

Manhart, S.

S. Manhart, R. Maurer, “Diode laser and fiber optics for dual-wavelength heterodyne interferometry,” in Optics in Complex Systems, F. Lanzl, G. Weigelt, eds., Proc. SPIE1319, 214–216 (1990).
[Crossref]

Maurer, R.

S. Manhart, R. Maurer, “Diode laser and fiber optics for dual-wavelength heterodyne interferometry,” in Optics in Complex Systems, F. Lanzl, G. Weigelt, eds., Proc. SPIE1319, 214–216 (1990).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1988), Chap. 7.

Polhemus, C.

Prongué, D.

Rowley, W. R. C.

G. P. Barwood, P. Gill, W. R. C. Rowley, “A simple rubidium-stabilized laser diode for interferometric applications,” J. Phys. E 21, 966–971 (1988).
[Crossref]

Sagna, N.

N. Sagna, C. Mandache, P. Thomann, “Noise measurements in single-mode GaAlAs diode lasers,” in Proceedings of the Sixth European Frequency and Time Forum, (European Space Research and Technology Centre, Noordwijk, The Netherlands, 1992), pp. 521–525.

Salvadé, Y.

R. Dändliker, Y. Salvadé, E. Zimmermann, “Distance measurement by multiple-wavelength interferometry,” J. Opt. 29, 105–114 (1998).
[Crossref]

E. Zimmermann, Y. Salvadé, R. Dändliker, “Stabilized three-wavelength source calibrated by electronic means for high-accuracy absolute distance measurement,” Opt. Lett. 21, 531–533 (1996).
[Crossref] [PubMed]

Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of interferometry due to frequency fluctuations of laser diodes,” Proceedings of Topical Meeting on Optoelectronic/Displacement Measurements and Applications (European Optical Society, Orsay, France, 1997).

Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of multiple-wavelength interferometry due to frequency fluctuations of laser diodes,” in Proceedings of International Workshop on Interferometry (The Institute of Physical and Chemical Research, Wako, Japan, 1996), pp. 9–10.

Shevy, Y.

Thalmann, R.

Thomann, P.

N. Sagna, C. Mandache, P. Thomann, “Noise measurements in single-mode GaAlAs diode lasers,” in Proceedings of the Sixth European Frequency and Time Forum, (European Space Research and Technology Centre, Noordwijk, The Netherlands, 1992), pp. 521–525.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 10.3–10.4.

Wyant, J. C.

Zimmermann, E.

R. Dändliker, Y. Salvadé, E. Zimmermann, “Distance measurement by multiple-wavelength interferometry,” J. Opt. 29, 105–114 (1998).
[Crossref]

E. Zimmermann, Y. Salvadé, R. Dändliker, “Stabilized three-wavelength source calibrated by electronic means for high-accuracy absolute distance measurement,” Opt. Lett. 21, 531–533 (1996).
[Crossref] [PubMed]

E. Zimmermann, R. Dändliker, “Full-dynamic phase demodulator for heterodyne interferometric vibration measurements,” in Second International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, E. P. Tomasini, ed., Proc. SPIE2868, 488–489 (1996).
[Crossref]

Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of multiple-wavelength interferometry due to frequency fluctuations of laser diodes,” in Proceedings of International Workshop on Interferometry (The Institute of Physical and Chemical Research, Wako, Japan, 1996), pp. 9–10.

Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of interferometry due to frequency fluctuations of laser diodes,” Proceedings of Topical Meeting on Optoelectronic/Displacement Measurements and Applications (European Optical Society, Orsay, France, 1997).

Appl. Opt. (3)

J. Opt. (1)

R. Dändliker, Y. Salvadé, E. Zimmermann, “Distance measurement by multiple-wavelength interferometry,” J. Opt. 29, 105–114 (1998).
[Crossref]

J. Phys. E (1)

G. P. Barwood, P. Gill, W. R. C. Rowley, “A simple rubidium-stabilized laser diode for interferometric applications,” J. Phys. E 21, 966–971 (1988).
[Crossref]

Opt. Lett. (3)

Precis. Eng. (1)

A. Abou-Zeid, “Diode lasers for interferometry,” Precis. Eng. 11, 139–144 (1989).
[Crossref]

Other (9)

S. Manhart, R. Maurer, “Diode laser and fiber optics for dual-wavelength heterodyne interferometry,” in Optics in Complex Systems, F. Lanzl, G. Weigelt, eds., Proc. SPIE1319, 214–216 (1990).
[Crossref]

E. Zimmermann, R. Dändliker, “Full-dynamic phase demodulator for heterodyne interferometric vibration measurements,” in Second International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, E. P. Tomasini, ed., Proc. SPIE2868, 488–489 (1996).
[Crossref]

N. Sagna, C. Mandache, P. Thomann, “Noise measurements in single-mode GaAlAs diode lasers,” in Proceedings of the Sixth European Frequency and Time Forum, (European Space Research and Technology Centre, Noordwijk, The Netherlands, 1992), pp. 521–525.

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1988), Chap. 7.

Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of multiple-wavelength interferometry due to frequency fluctuations of laser diodes,” in Proceedings of International Workshop on Interferometry (The Institute of Physical and Chemical Research, Wako, Japan, 1996), pp. 9–10.

Y. Salvadé, E. Zimmermann, R. Dändliker, “Limitations of interferometry due to frequency fluctuations of laser diodes,” Proceedings of Topical Meeting on Optoelectronic/Displacement Measurements and Applications (European Optical Society, Orsay, France, 1997).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 10.3–10.4.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.1.3.

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Figures (6)

Fig. 1
Fig. 1

Two-wavelength source stabilized on a common stable Fabry–Perot resonator: LD, laser diodes; PD, photodetector.

Fig. 2
Fig. 2

Measurements of the beat-note spectrum between two stabilized laser diodes (solid curve), fit to a Lorentzian function (dashed curve).

Fig. 3
Fig. 3

Optical setup used for measurements at the limit of the coherence length: LD, laser diodes; PBS, polarizing beam splitters; AOM, acousto-optic modulators; BS, beam splitter; P, polarizers; λ/4, quarter-wave plates.

Fig. 4
Fig. 4

Measured frequency-noise spectrum for free-running and stabilized laser diodes, compared with the frequency-noise power spectral density (PSD) estimated with Eq. (25).

Fig. 5
Fig. 5

Phase fluctuations as a function of the interferometric delay τ for an integration time of T=1 ms, calculated for white-frequency noise only (dashed curve) and for the estimated frequency-noise spectrum with flicker noise (solid curve).

Fig. 6
Fig. 6

Interferometric phase fluctuations for different integration times T for an interferometric delay of τ=τc compared with calculated values for white-frequency noise only (dashed curve) and for the estimated frequency-noise spectrum with flicker noise (solid curve).

Equations (29)

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V(t)=V0 exp[iϕ(t)]exp(i2πνt),
I=|V(t)+V(t-τ)|2=2V02+2V02 Re{exp[iΔϕτ(t)]}cos(2πντ)
Δϕτ(t)=ϕ(t)-ϕ(t+τ).
p(Δϕτ)=12πΔϕτ2exp-12Δϕτ2ϕτ2.
exp[iΔϕτ(t)]=exp-12Δϕτ2.
g(τ)=V(t)V*(t-τ)|V(t)|2=exp-12Δϕτ2exp(i2πντ).
Imax(τ)-Imin(τ)Imax(τ)+Imin(τ)=|g(τ)|=exp-12Δϕτ2.
Δϕτc2=2.
Δϕτ(t)=ϕ(t)-ϕ(t-τ)=t-τtϕ˙(t)dt,
SΔϕτ(f)=Sϕ˙(f)sin πfτπfτ2τ2.
SΔϕτ(f)=4π2τ2Sδν(f)sin πfτπfτ2.
Δϕτ2=0SΔϕτ(f)df.
Δϕτ,T(t)=1Tt-TtΔϕτ(t)dt.
SΔϕτ,T(f)=SΔϕτ(f)sin πfTπfT2.
SΔϕτ,T(f)=4π2τ2Sδν(f)sin πfτπfτ2sin πfTπfT2.
SΔϕτ,T(f)=4π2τ2Sδν(f)sin πfTπfT2.
Δϕτ,T2=0SΔϕτ,T(f)df.
Δϕτ2=2π2C0τ.
C0=1π2τc.
Δϕτ,T2=2 τ2τcT1-τ3T
Δϕτ,T2=2 τ2τcT.
Δϕτ,τc2=2 τ2τc2.
Δϕτ,T2=Δϕτ,τc2 τcT.
Sν(f)|withfeedback=1|1+H(f)|2Sν(f)|free-running,
Sv(f)|withfeedback=11+fmin/fSv(f)|free-running,
I(t)=A0+A1 cos(2πf1t+Δϕ1)+A2 cos(2πf2t+Δϕ2),
Idem(t)=A12 cos[2π(f1-f2)t+(Δϕ1-Δϕ2)].
SΔΦτ,T(f)=4π2τ22Sδv(f)|G(f)|2,
SΔΦτ,T(f)8π2τ2Sδv(f),f<6 kHz.

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