Abstract

A new, to our knowledge, heterodyne interferometer for differential displacement measurements is presented. It is, in principle, free of periodic nonlinearity. A pair of spatially separated light beams with different frequencies is produced by two acousto-optic modulators, avoiding the main source of periodic nonlinearity in traditional heterodyne interferometers that are based on a Zeeman split laser. In addition, laser beams of the same frequency are used in the measurement and the reference arms, giving the interferometer theoretically perfect immunity from common-mode displacement. We experimentally demonstrated a residual level of periodic nonlinearity of less than 20 pm in amplitude. The remaining periodic error is attributed to unbalanced ghost reflections that drift slowly with time.

© 1999 Optical Society of America

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References

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  1. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
    [CrossRef]
  2. C. M. Wu, R. D. Deslattes, “Analytical modeling of the periodic nonlinearity in heterodyne interferometry,” Appl. Opt. 37, 6696–6700 (1998).
    [CrossRef]
  3. C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
    [CrossRef]
  4. W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
    [CrossRef]
  5. C. M. Wu, C. S. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
    [CrossRef]
  6. M. Tanaka, T. Yamagami, K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels,” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
    [CrossRef]
  7. This relation has also been discussed in A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
    [CrossRef]

1998 (1)

1996 (2)

C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

C. M. Wu, C. S. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

1992 (1)

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

1990 (1)

This relation has also been discussed in A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

1989 (1)

M. Tanaka, T. Yamagami, K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels,” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[CrossRef]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

This relation has also been discussed in A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Deslattes, R. D.

Hou, W.

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Nakayama, K.

M. Tanaka, T. Yamagami, K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels,” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[CrossRef]

Peng, G. S.

C. M. Wu, C. S. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Rosenbluth, A. E.

This relation has also been discussed in A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Su, C. S.

C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

C. M. Wu, C. S. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Tanaka, M.

M. Tanaka, T. Yamagami, K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels,” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[CrossRef]

Wilkening, G.

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Wu, C. M.

C. M. Wu, R. D. Deslattes, “Analytical modeling of the periodic nonlinearity in heterodyne interferometry,” Appl. Opt. 37, 6696–6700 (1998).
[CrossRef]

C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

C. M. Wu, C. S. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Yamagami, T.

M. Tanaka, T. Yamagami, K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels,” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Instrum. Meas. (1)

M. Tanaka, T. Yamagami, K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels,” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[CrossRef]

Meas. Sci. Technol. (3)

C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

C. M. Wu, C. S. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Precis. Eng. (2)

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

This relation has also been discussed in A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Typical setup for heterodyne interferometry. BS, beam splitter; E s , reference beam at frequency f 1; E p , measurement beam at frequency f 2; E p ′, Doppler-shifted measurement beam at frequency f 2′; I m , measurement signal; I r , reference signal.

Fig. 2
Fig. 2

(a) Configuration of the differential heterodyne interferometer tested in this study. A synthetic displacement was generated by means of a pressure cell. PM SM-fiber, polarization-maintaining single-mode fiber; BS, beam splitter; RAP, right-angle prism; BC, beam combiner. (b) Equivalent topology designed to measure an actual displacement.

Fig. 3
Fig. 3

Periodic nonlinearity measured at two different times: (a) periodic error at the fringe frequency and (b) periodic error at twice the fringe frequency.

Fig. 4
Fig. 4

Investigation of the periodic errors associated with the digital lock-in amplifier. A purely electronic signal at 80.001 kHz is demodulated at 80.000 kHz, resulting in a difference signal at 1 Hz: (a) components C (solid curve) and S (dotted curve) of demodulated signal and (b) residuals dϕ (solid curve) and dR/R (dotted curve).

Equations (22)

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Im AB cosΔωt-ϕt,
Ir AB cosΔωt,
Δω=2π|f1-f2|,
ϕt=2 2πλ  dtvt,  =2 2πλ st,
CAB cosΔωt-ϕtcosΔωt=12AB cosϕt,  SAB sinΔωt-ϕtsinΔωt=12AB sinϕt.
Im  AB cosΔωt-ϕ+Aβ+BαcosΔωt+αβ+βpfαpfcosΔωt+ϕ+Aβpf-BαpfsinΔωt+αβpf-βαpfsinΔωt+ϕ,
Ct=Im cosΔωt,  St=Im sinΔωt,
Ct=Im cosΔωt,St=Im sinΔωt,
Rt=C2t+S2t1/2,
Φt=tan-1St/Ct,
Im=cosΔωt-ϕ+ cosΔωt-δ,
C=12cos ϕ+ cos δ,  S=12ϕ+ sin δ,
Φt=tan-1sin ϕ+ sin δcos ϕ+ cos δ.
dΦ=Φ-ϕ= sinδ-ϕ.
R=12cos ϕ+ cos δ2+sin ϕ+ sin δ21/2,
dRR= cosδ-ϕ.
=AB+Bα,  δ=0,
dΦ=-Aβ+Bαsinϕ,  dRR=Aβ+Bαcosϕ,
=αβ+βpfαpf,  δ=-ϕ,
dΦ=-αβ+βpfαpfsin2ϕ,  dRR=αβ+βpfαpfcos2ϕ,
|dϕ|=dRR,
|dx|=λ4πdRR.

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