Abstract

We present the optimization of a Fabry–Perot velocimeter designed to measure speed at a few millimeters per second with a relative uncertainty of 10-8. We focus on the accuracy and the optimization of the Fabry–Perot, with a review of the uncertainties related to the geometry, the beam shape, and the Doppler frequency measurement. These errors are quantified to ensure that the required accuracy is reached. We then describe the practical implementation and show the results.

© 2000 Optical Society of America

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References

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  1. E. L. Canuteson, M. A. Zumberge, “Fiber-optic extrinsic Fabry–Perot vibration-isolated interferometer for use in absolute gravity meters,” Appl. Opt. 35, 3500–3504 (1996).
    [CrossRef] [PubMed]
  2. J. L. Santos, A. P. Leite, D. A. Jackson, “Optical fiber sensing with a low-finesse Fabry–Perot cavity,” Appl. Opt. 31, 7361–7366 (1992).
    [CrossRef] [PubMed]
  3. W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
    [CrossRef]
  4. D. Z. Anderson, “Alignment of optical resonant cavities,” Appl. Opt. 23, 2944–2949 (1984).
    [CrossRef]
  5. E. Morrison, B. J. Meers, D. I. Robertson, H. Wards, “Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
    [CrossRef] [PubMed]
  6. F. Durst, W. H. Stevenson, “Influence of Gaussian beam properties on laser Doppler signals,” Appl. Opt. 18, 516–524 (1979).
    [CrossRef] [PubMed]
  7. P. C. Miles, “Geometry of the fringe field formed in the intersection of two Gaussian beams,” Appl. Opt. 35, 5887–5895 (1996).
    [CrossRef] [PubMed]
  8. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  9. A. K. Ghatak, K. Thyagfarajan, Optical Electronics (Cambridge U. Press, Cambridge, UK, 1989), pp. 46–48.
  10. Hewlett-Packard, Fundamentals of the Electronic Counters, Application Note 200, Electronic Counter Series (Hewlett-Packard, P.O. Box 4026, Englewood, Colo., 1997).

1999 (1)

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

1996 (2)

1994 (1)

1992 (1)

1984 (1)

1979 (1)

1966 (1)

Anderson, D. Z.

Beer, W.

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

Canuteson, E. L.

Courteville, A.

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

Dändliker, R.

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

Durst, F.

Ghatak, A. K.

A. K. Ghatak, K. Thyagfarajan, Optical Electronics (Cambridge U. Press, Cambridge, UK, 1989), pp. 46–48.

Hewlett-Packard,

Hewlett-Packard, Fundamentals of the Electronic Counters, Application Note 200, Electronic Counter Series (Hewlett-Packard, P.O. Box 4026, Englewood, Colo., 1997).

Jackson, D. A.

Jeanneret, B.

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

Jeckelmann, B.

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

Kogelnik, H.

Leite, A. P.

Li, T.

Meers, B. J.

Miles, P. C.

Morrison, E.

Richard, P.

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

Robertson, D. I.

Salvadé, Y.

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

Santos, J. L.

Stevenson, W. H.

Thyagfarajan, K.

A. K. Ghatak, K. Thyagfarajan, Optical Electronics (Cambridge U. Press, Cambridge, UK, 1989), pp. 46–48.

Wards, H.

Zumberge, M. A.

Appl. Opt. (7)

IEEE Trans. Instrum. Meas. (1)

W. Beer, B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, R. Dändliker “A proposal for a new moving-coil experiment,” IEEE Trans. Instrum. Meas. 48, 192–195 (1999).
[CrossRef]

Other (2)

A. K. Ghatak, K. Thyagfarajan, Optical Electronics (Cambridge U. Press, Cambridge, UK, 1989), pp. 46–48.

Hewlett-Packard, Fundamentals of the Electronic Counters, Application Note 200, Electronic Counter Series (Hewlett-Packard, P.O. Box 4026, Englewood, Colo., 1997).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the Fabry–Perot interferometer. Mirror M2 moves at angle γ with respect to the z axis with a speed ν. M1 is tilted by angle θ, and the incident beam E propagates at angle α with respect to the z axis.

Fig. 2
Fig. 2

Variation of the Doppler frequency f G that is due to a Gaussian beam profile as a function of the beam waist w 0, which is computed from Eq. (24) with λ = 633 nm. f p = -2ν/λ is the Doppler frequency for a plane wave.

Fig. 3
Fig. 3

Photodetector output as a function of the normalized distance d/λ between the two mirrors for R 2 = 0.9 and different values of R 1 [relation (26): n 0 = 1 and α = 0)].

Fig. 4
Fig. 4

Maximum slope of the photocurrent, normalized to the amplitude, and visibility as a function of reflectivity R 1 of the first mirror and different values for R 2.

Fig. 5
Fig. 5

Power spectrum of the photocurrent integrated from 0 to f, as a function of the frequency f normalized to the Doppler frequency f D , for R 2 = 0.9 and different values of R 1.

Fig. 6
Fig. 6

Experimental setup for the velocity measurement of the Watt balance. GPIB, general-purpose interface bus; PBS, polarizing beam splitter.

Fig. 7
Fig. 7

Sinusoidal mirror displacement of amplitude λ/2 and corresponding interference signal.

Equations (27)

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fD1=-2νλcos γ cos α,
αr1=2θ+α,
fD2=-2νλcos γ cos2θ+α.
fDN=fD1+fD2++fDn=-2νλcos γ n=0N-1cos2nθ+α-2νλcos γ cos αN-2 n=0N-1nθ2,
ν=-fD1λ2 cos γ cos α.
Δνν=1fD1 ΔfD1-1fl Δfl+tan γΔγ+tan αΔα.
Δνν1fD1 ΔfD1-1fl Δfl+Δγ2+Δα2.
Ex, y, z=2π1wzexp-x2+y2w2z×exp-jkz+jφz-j k2x2+y2Rz=Ax, y, zexpjϕx, y, z,
φz=tan-1z/zR
zR=πw02/λ;
Rz=z+zR2/z
wz=w01+z/zR21/2
it   |Ex, y, z0|2 expjΔϕx, ydxdy+c.c.,
Δϕx, y=ϕx, y, z1-ϕx, y, z0ϕzz=z0z1-z0=-kz1-z0×1-1k φd+12x2+y2Rd,
φd=φzz=z0=zRz02+zR2,
Rd=1/Rzz=z0=zR2-z02zR2-z022.
Δφx, y=-4πλd0+νt1-1k φd+12x2+y2Rd=2πt-τfx, y,
fx, y=-2νλ1-1k φd+12x2+y2Rd
τ=-d0/ν.
f0=-2νλ1-1kzRz02+zR2.
f=-2νλ1-12kzR,
it  0exp-2r2w2z0expj2πfrt-τr dr+c.c. 2w2z0+jkνRdt-τ-1 expj2πf0t-τ+c.c. cos2πf0t-τ-tan-1νRdφdt-τ.
fG=-2νλ1-12kzR-12kRdφd1-1+d2zR2×zR2-z02zR2-z022-1.
fG=-2νλ1-12kzR.
Δννz02-zR22zR2-2kzRz02+zR2.
it  1-1-R2+R1-R221-R2+4 R sin2ϕ/2.
SNR|AC=V22SNR|DC,

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