Abstract

The influence of quadrature phase shift on the measured displacement error was experimentally investigated using a two-detector polarizing homodyne laser interferometer with a quadrature detection system. Common nonlinearities, including the phase-shift error, were determined and effectively corrected by a robust data-processing algorithm. The measured phase-shift error perfectly agrees with the theoretically determined phase-shift error region. This error is systematic, periodic and severely asymmetrical around the nominal displacement value. The main results presented in this paper can also be used to assess and correct the detector errors of other interferometric and non-interferometric displacement-measuring devices based on phase-quadrature detection.

© 2009 OSA

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References

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  1. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
    [CrossRef]
  2. R. Reibold and W. Molkenstruck, “Laser interferometric measurement and computerized evaluation of ultrasonic displacements,” Acustica 49, 205–211 (1981).
  3. V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66(7), 3729–3734 (1995).
    [CrossRef]
  4. T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43(12), 2443–2448 (2004).
    [CrossRef] [PubMed]
  5. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
    [CrossRef] [PubMed]
  6. L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. 19(11), 115104 (2008).
    [CrossRef]
  7. T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. 92(23), 234101–234103 (2008).
    [CrossRef]
  8. G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
    [CrossRef]
  9. N.-I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vazquez-Castillo, “Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry,” Opt. Express 16(23), 19330–19341 (2008).
    [CrossRef]
  10. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20(19), 3382–3384 (1981).
    [CrossRef] [PubMed]
  11. C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
    [CrossRef]
  12. T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia 36(4), 375–383 (1999).
    [CrossRef]
  13. K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12(4), 195–198 (1990).
    [CrossRef]
  14. C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
    [CrossRef]
  15. T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
    [CrossRef]
  16. J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009).
    [CrossRef]
  17. C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications, (Adam Hilger, Bristol, 1990).
  18. I. Dániel, “Advanced successive phase unwrapping algorithm for quadrature output Michelson interferometers,” Measurement 37(2), 95–102 (2005).
    [CrossRef]
  19. M. Pisani, “Multiple reflection Michelson interferometer with picometer resolution,” Opt. Express 16(26), 21558–21563 (2008).
    [CrossRef] [PubMed]

2009 (1)

J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009).
[CrossRef]

2008 (4)

L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. 19(11), 115104 (2008).
[CrossRef]

T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. 92(23), 234101–234103 (2008).
[CrossRef]

N.-I. Toto-Arellano, G. Rodriguez-Zurita, C. Meneses-Fabian, and J. F. Vazquez-Castillo, “Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry,” Opt. Express 16(23), 19330–19341 (2008).
[CrossRef]

M. Pisani, “Multiple reflection Michelson interferometer with picometer resolution,” Opt. Express 16(26), 21558–21563 (2008).
[CrossRef] [PubMed]

2005 (2)

2004 (2)

G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef]

T. Keem, S. Gonda, I. Misumi, Q. Huang, and T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43(12), 2443–2448 (2004).
[CrossRef] [PubMed]

2001 (1)

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

1999 (1)

T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia 36(4), 375–383 (1999).
[CrossRef]

1996 (2)

C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[CrossRef]

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

1995 (1)

V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66(7), 3729–3734 (1995).
[CrossRef]

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[CrossRef]

1990 (1)

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12(4), 195–198 (1990).
[CrossRef]

1981 (2)

R. Reibold and W. Molkenstruck, “Laser interferometric measurement and computerized evaluation of ultrasonic displacements,” Acustica 49, 205–211 (1981).

P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20(19), 3382–3384 (1981).
[CrossRef] [PubMed]

Ahn, J.

J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009).
[CrossRef]

Birch, K. P.

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12(4), 195–198 (1990).
[CrossRef]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[CrossRef]

Brock, N.

Dai, G. L.

G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef]

Dániel, I.

I. Dániel, “Advanced successive phase unwrapping algorithm for quadrature output Michelson interferometers,” Measurement 37(2), 95–102 (2005).
[CrossRef]

Danzebrink, H. U.

G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef]

Eom, T.

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

Eom, T. B.

J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009).
[CrossRef]

Gonda, S.

Greco, V.

V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66(7), 3729–3734 (1995).
[CrossRef]

Hasche, K.

G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef]

Hayes, J.

Heydemann, P. L. M.

Huang, Q.

Jeong, K.

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

Kang, C.-S.

J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009).
[CrossRef]

Keem, T.

Kim, J.

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

Kim, J. W.

J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009).
[CrossRef]

Kim, J.-A.

J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009).
[CrossRef]

Kurosawa, T.

Meneses-Fabian, C.

Millerd, J.

Misumi, I.

Molesini, G.

V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66(7), 3729–3734 (1995).
[CrossRef]

Molkenstruck, W.

R. Reibold and W. Molkenstruck, “Laser interferometric measurement and computerized evaluation of ultrasonic displacements,” Acustica 49, 205–211 (1981).

Morlanes, T.

L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. 19(11), 115104 (2008).
[CrossRef]

Možina, J.

T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. 92(23), 234101–234103 (2008).
[CrossRef]

North-Morris, M.

Novak, M.

Peng, G.-S.

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

Petkovšek, R.

T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. 92(23), 234101–234103 (2008).
[CrossRef]

Pisani, M.

Pohlenz, F.

G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef]

Požar, T.

T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. 92(23), 234101–234103 (2008).
[CrossRef]

Quercioli, F.

V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66(7), 3729–3734 (1995).
[CrossRef]

Reibold, R.

R. Reibold and W. Molkenstruck, “Laser interferometric measurement and computerized evaluation of ultrasonic displacements,” Acustica 49, 205–211 (1981).

Rodriguez-Zurita, G.

Sanchez-Brea, L. M.

L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. 19(11), 115104 (2008).
[CrossRef]

Su, C.-S.

C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[CrossRef]

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

Toto-Arellano, N.-I.

Usuda, T.

T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia 36(4), 375–383 (1999).
[CrossRef]

Vazquez-Castillo, J. F.

Wilkening, G.

G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef]

Wu, C.-M.

C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[CrossRef]

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

Wyant, J.

Acustica (1)

R. Reibold and W. Molkenstruck, “Laser interferometric measurement and computerized evaluation of ultrasonic displacements,” Acustica 49, 205–211 (1981).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

T. Požar, R. Petkovšek, and J. Možina, “Dispersion of an optodynamic wave during its multiple transitions in a rod,” Appl. Phys. Lett. 92(23), 234101–234103 (2008).
[CrossRef]

Meas. Sci. Technol. (7)

G. L. Dai, F. Pohlenz, H. U. Danzebrink, K. Hasche, and G. Wilkening, “Improving the performance of interferometers in metrological scanning probe microscopes,” Meas. Sci. Technol. 15(2), 444–450 (2004).
[CrossRef]

C.-M. Wu, C.-S. Su, and G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7(4), 520–524 (1996).
[CrossRef]

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12(10), 1734–1738 (2001).
[CrossRef]

J.-A. Kim, J. W. Kim, C.-S. Kang, T. B. Eom, and J. Ahn, “A digital signal processing module for real-time compensation of nonlinearity in a homodyne interferometer using a field-programmable gate array,” Meas. Sci. Technol. 20(1), 017003 (2009).
[CrossRef]

C.-M. Wu and C.-S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7(1), 62–68 (1996).
[CrossRef]

L. M. Sanchez-Brea and T. Morlanes, “Metrological errors in optical encoders,” Meas. Sci. Technol. 19(11), 115104 (2008).
[CrossRef]

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[CrossRef]

Measurement (1)

I. Dániel, “Advanced successive phase unwrapping algorithm for quadrature output Michelson interferometers,” Measurement 37(2), 95–102 (2005).
[CrossRef]

Metrologia (1)

T. Usuda and T. Kurosawa, “Calibration methods for vibration transducers and their uncertainties,” Metrologia 36(4), 375–383 (1999).
[CrossRef]

Opt. Express (2)

Precis. Eng. (1)

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12(4), 195–198 (1990).
[CrossRef]

Rev. Sci. Instrum. (1)

V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66(7), 3729–3734 (1995).
[CrossRef]

Other (1)

C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications, (Adam Hilger, Bristol, 1990).

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Figures (4)

Fig. 1
Fig. 1

Schematic top view of the HQLI. The exiting light from the stabilized He-Ne laser is linearly polarized at 45° in the x-y plane. The beamsplitter (BS) evenly splits the beam into the reference and measurement arms. The octadic-wave plate (OWP) and high-reflectivity (HR) mirror are placed in the reference arm. The HR mirror in the measurement arm is driven by a piezoelectric transducer (PZT). The polarizing beamsplitter (PBS) transmits the x-polarization and reflects the y-polarization. An optically narrow band-pass filter (BPF) is placed before the photodiodes labeled PDx and PDy.

Fig. 2
Fig. 2

The data processing of the two photodiode signals in the HQLI measurement of the harmonically vibrating mirror (f = 100 Hz, A = 270 nm) mounted on a PZT. (a) Raw signals: Vx (t) and Vy (t). (b) Processed error-corrected signals: sx (t) and sy (t). (c) Displacement as a function of the time obtained after the unwrapping of the phase. (d) Lissajous figure of the processed data: (sx (t),sy (t)).

Fig. 3
Fig. 3

The measured displacement of the harmonically vibrating mirror (f = 100 Hz, A = 270 nm). (a) Displacement as a function of time for the reference measurement ur obtained with an adjusted HQMI (the black line), the inaccurate measurement um with a HQMI lacking the phase quadrature (the red line), and the software-corrected displacement uc of the inaccurate measurement (the blue circles). (b) The Lissajous representation of the signals from which the displacements in (a) were derived. (c) The displacement error uerr between the distorted and the reference measurement um ur (the red line) and the one between the software-corrected and the reference displacement uc ur (the blue line).

Fig. 4
Fig. 4

The influence of the phase shift on the error in the displacement. (a) The reference (α = 90°; the black line) and distorted (α = 43°; the red line) harmonic displacements and the corresponding error marked as “error H”. (b) The reference (α = 90°; the black line) and distorted (α = 140°; the red line) triangular displacements and the corresponding error marked as “error T”. (c) The phase-shift displacement error uerr as a function of the phase shift α for λ = 632.8 nm (left scale) and the arbitrary position period p (right scale). The measured border errors of the harmonic (f = 100 Hz, A = 270 nm) and the triangular (f = 70 Hz, A = 330 nm) displacements for several values of α are marked as circles and squares. The theoretical error borders are calculated from Eqs. (8) (the blue and red lines).

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

E(45°)=12[11], bs=12[1001], pbsx=[1000], pbsy=[0001],opd=eiδ[1001], owp(φ)=[cosπ8isinπ8cos2φisinπ8sin2φisinπ8sin2φcosπ8+isinπ8cos2φ].
Erx,ry=pbsx,ybsowp(φ)owp(φ)bsE(45°),
Emx,my=pbsx,ybsopdbsE(45°).
Ix,y=I0(Erx,ry+Emx,my)(Erx,ry+Emx,my)
Ix(δ,φ)=I016(4+sin4φ+22(cosδsinδ(cos2φ+sin2φ))),
Iy(δ,φ)=I016(4sin4φ+22(cosδ+sinδ(cos2φsin2φ))).
Ix(δ45°,0°)=I04(1+cosδ)   and   Iy(δ45°,0°)=I04(1+sinδ).
u(t)=λ4π(arctanIyI0/4IxI0/4+mπ),   m=0,±1,±2,  .
Vx,y(u(t))=Vx0,y0cos(4πλu(t)δx,y)+Vxoff,yoff
sx(t)=cosδ(t)   and   sy(t)=sinδ(t).
sxe=cosδ=sx   and   sye=cos(δα)=sxcosα+sysinα,
sx=sxe   and   sy=(syesxecosα)/sinα
r(θ)=sinα(1cosαsin2θ)12
(r=sxe2+sye2,θ=arctansyesxe)
α(φ)=arctan(cos2φsin2φ)+arctan(cos2φ+sin2φ).
uerr(δ,α)=umur=λ4π(arctansyesxearctansysx).
uerrB(α)=uerr(δB(α),α);   where   δB(α)=12(αarccos(sinα1cosα)),
uerrb(α)=uerr(δb(α),α);   where   δb(α)=12(α+arccos(sinα1cosα)).
λ4π(90°α)cos2δ

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