Abstract

Interferometry with single-point detection is widely used for the precise measurement of short-range displacements, when its range is less than a quarter of the wavelength. The moiré technique is also used for the same purpose when the displacement is smaller than half the period of the superimposed gratings. In both interferometry and the moiré technique, the response function of the system takes a periodic form by increasing the displacement value. For larger values of the displacements, the moment the detected signal experiences one of the extremum values, the signal trend does not reflect the direction of the motion, and it leads to an ambiguity in the motion reconstruction. Since an interference pattern has a sinusoidal intensity profile, by using three-point detection and the aid of conventional spatial phase shifting, we have recently proposed a new method for chasing moving interference fringes and were able to remove the disability of the interferometry in discriminating of the direction of motion for long-range displacements (see Opt. Laser Technol. 103, 387 (2018)). But the transmission function of a moiré pattern is a triangular or trapezoidal function. Therefore, the conventional phase shifting algorithms are not applicable for moiré fringe chasing. In this work, first we introduce a new method for data acquisition in moiré-based displacement and vibration sensors. We use a three-point intensity detection method for chasing moving moiré fringes and we introduce a new algorithm based on the data of three point detectors to remove the disability of the moiré technique to discriminate the direction of motion for long-range displacements. With the aid of three-point intensity detection, a high speed, high accuracy, and long-range displacement sensor based on the moiré technique is built. This sensor can be used in the calibration of mechanical positioning sensors. Also, using a data acquisition procedure we have introduced, in a moiré-based vibration sensor, the vibration parameters can be determined in a simple and reliable manner.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. K. Madanipour and M. T. Tavassoly, “Submicron displacements measurement by measuring autocorrelation of the transmission function of a grating,” Proc. SPIE 8082, 80823O (2011).
    [Crossref]
  2. S. Esmaeili, S. Rasouli, F. Sobouti, and S. Esmaeili, “A moiré micro strain gauge,” Opt. Commun. 285(9), 2243–2246 (2012).
    [Crossref]
  3. S. Rasouli, S. Esmaeili, and F. Sobouti, “Design and construction of a seismometer based on the moiré technique: detailed theoretical analysis, experimental apparatus, and primary results,” Int. J. Opt. Photon. 10, 3–10 (2016).
  4. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993), chap. 5.
  5. A. T‎. Shepherd and G. S. Walker, British Patent 810478 (1995).
  6. L. Wronkowski, “Signal transducing in optoelectronic measurement systems based on the moiré phenomenon,” Opt. Eng. 31(3), 505–516 (1992).
    [Crossref]
  7. S. Rasouli, Z. Eskandari, and Z. Y. Abedini, “Landslide monitoring using moiré technique,” Geosciences 22, 237–240 (2012).
  8. A. H. McIlraith, “A moiré fringe interpolator of high resolution,” J. Sci. Instrum. 41(1), 34–37 (1964).
    [Crossref]
  9. J. Zhu, S. Hu, J. Yu, S. Zhou, Y. Tang, M. Zhong, L. Zhao, M. Chen, L. Li, Y. He, and W. Jiang, “Four-quadrant gratings moiré fringe alignment measurement in proximity lithography,” Opt. Express 21(3), 3463–3473 (2013).
    [Crossref] [PubMed]
  10. M. H. Daemi and S. Rasouli, “Fringe chasing by three-point spatial phase shifting for discrimination of the motion direction in the long-range homodyne laser Doppler vibrometry,” Opt. Laser Technol. 103, 387–395 (2018).
    [Crossref]
  11. B. Košťák, “A new device for in-situ movement detection and measurement,” Exp. Mech. 9(8), 374–379 (1969).
    [Crossref]
  12. M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
    [Crossref] [PubMed]
  13. J.-L. Piro and M. Grediac, “Producing and transferring low-spatial-frequency grids for measuring displacement fields with moiré and grid methods,” Exp. Tech. 28(4), 23–26 (2004).
    [Crossref]
  14. X. Marti, M. D. Rowberry, and J. Blahůt, “A MATLAB code for counting the moiré interference fringes recorded by the optical-mechanical crack gauge TM-71,” Comput. Geosci. 52, 164–167 (2013).
    [Crossref]
  15. G. T. Reid, R. C. Rixon, and H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16(6), 315–319 (1984).
    [Crossref]
  16. I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
    [Crossref]
  17. S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
    [Crossref]
  18. S. Rasouli, M. Dashti, and A. N. Ramaprakash, “An adjustable, high sensitivity, wide dynamic range two channel wave-front sensor based on moiré deflectometry,” Opt. Express 18(23), 23906–23913 (2010).
    [Crossref] [PubMed]
  19. S. Rasouli, “Use of a moiré deflectometer on a telescope for atmospheric turbulence measurements,” Opt. Lett. 35(9), 1470–1472 (2010).
    [Crossref] [PubMed]

2018 (1)

M. H. Daemi and S. Rasouli, “Fringe chasing by three-point spatial phase shifting for discrimination of the motion direction in the long-range homodyne laser Doppler vibrometry,” Opt. Laser Technol. 103, 387–395 (2018).
[Crossref]

2016 (3)

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

S. Rasouli, S. Esmaeili, and F. Sobouti, “Design and construction of a seismometer based on the moiré technique: detailed theoretical analysis, experimental apparatus, and primary results,” Int. J. Opt. Photon. 10, 3–10 (2016).

2013 (2)

X. Marti, M. D. Rowberry, and J. Blahůt, “A MATLAB code for counting the moiré interference fringes recorded by the optical-mechanical crack gauge TM-71,” Comput. Geosci. 52, 164–167 (2013).
[Crossref]

J. Zhu, S. Hu, J. Yu, S. Zhou, Y. Tang, M. Zhong, L. Zhao, M. Chen, L. Li, Y. He, and W. Jiang, “Four-quadrant gratings moiré fringe alignment measurement in proximity lithography,” Opt. Express 21(3), 3463–3473 (2013).
[Crossref] [PubMed]

2012 (2)

S. Esmaeili, S. Rasouli, F. Sobouti, and S. Esmaeili, “A moiré micro strain gauge,” Opt. Commun. 285(9), 2243–2246 (2012).
[Crossref]

S. Rasouli, Z. Eskandari, and Z. Y. Abedini, “Landslide monitoring using moiré technique,” Geosciences 22, 237–240 (2012).

2011 (1)

K. Madanipour and M. T. Tavassoly, “Submicron displacements measurement by measuring autocorrelation of the transmission function of a grating,” Proc. SPIE 8082, 80823O (2011).
[Crossref]

2010 (3)

2004 (1)

J.-L. Piro and M. Grediac, “Producing and transferring low-spatial-frequency grids for measuring displacement fields with moiré and grid methods,” Exp. Tech. 28(4), 23–26 (2004).
[Crossref]

1992 (1)

L. Wronkowski, “Signal transducing in optoelectronic measurement systems based on the moiré phenomenon,” Opt. Eng. 31(3), 505–516 (1992).
[Crossref]

1984 (1)

G. T. Reid, R. C. Rixon, and H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16(6), 315–319 (1984).
[Crossref]

1969 (1)

B. Košťák, “A new device for in-situ movement detection and measurement,” Exp. Mech. 9(8), 374–379 (1969).
[Crossref]

1964 (1)

A. H. McIlraith, “A moiré fringe interpolator of high resolution,” J. Sci. Instrum. 41(1), 34–37 (1964).
[Crossref]

Abedini, Z. Y.

S. Rasouli, Z. Eskandari, and Z. Y. Abedini, “Landslide monitoring using moiré technique,” Geosciences 22, 237–240 (2012).

Baron, I.

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

Blahut, J.

X. Marti, M. D. Rowberry, and J. Blahůt, “A MATLAB code for counting the moiré interference fringes recorded by the optical-mechanical crack gauge TM-71,” Comput. Geosci. 52, 164–167 (2013).
[Crossref]

Chen, M.

Daemi, M. H.

M. H. Daemi and S. Rasouli, “Fringe chasing by three-point spatial phase shifting for discrimination of the motion direction in the long-range homodyne laser Doppler vibrometry,” Opt. Laser Technol. 103, 387–395 (2018).
[Crossref]

Dashti, M.

Eskandari, Z.

S. Rasouli, Z. Eskandari, and Z. Y. Abedini, “Landslide monitoring using moiré technique,” Geosciences 22, 237–240 (2012).

Esmaeili, S.

S. Rasouli, S. Esmaeili, and F. Sobouti, “Design and construction of a seismometer based on the moiré technique: detailed theoretical analysis, experimental apparatus, and primary results,” Int. J. Opt. Photon. 10, 3–10 (2016).

S. Esmaeili, S. Rasouli, F. Sobouti, and S. Esmaeili, “A moiré micro strain gauge,” Opt. Commun. 285(9), 2243–2246 (2012).
[Crossref]

S. Esmaeili, S. Rasouli, F. Sobouti, and S. Esmaeili, “A moiré micro strain gauge,” Opt. Commun. 285(9), 2243–2246 (2012).
[Crossref]

Frontera, C.

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

Fujigaki, M.

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

Grasemann, B.

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

Grediac, M.

J.-L. Piro and M. Grediac, “Producing and transferring low-spatial-frequency grids for measuring displacement fields with moiré and grid methods,” Exp. Tech. 28(4), 23–26 (2004).
[Crossref]

Hausmann, H.

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

He, Y.

Holy, V.

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

Hu, S.

Jiang, W.

Košták, B.

B. Košťák, “A new device for in-situ movement detection and measurement,” Exp. Mech. 9(8), 374–379 (1969).
[Crossref]

Kriegner, D.

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

Lenhardt, W.

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

Li, L.

Llull, M.

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

Madanipour, K.

K. Madanipour and M. T. Tavassoly, “Submicron displacements measurement by measuring autocorrelation of the transmission function of a grating,” Proc. SPIE 8082, 80823O (2011).
[Crossref]

Marti, X.

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

X. Marti, M. D. Rowberry, and J. Blahůt, “A MATLAB code for counting the moiré interference fringes recorded by the optical-mechanical crack gauge TM-71,” Comput. Geosci. 52, 164–167 (2013).
[Crossref]

McIlraith, A. H.

A. H. McIlraith, “A moiré fringe interpolator of high resolution,” J. Sci. Instrum. 41(1), 34–37 (1964).
[Crossref]

Messer, H. I.

G. T. Reid, R. C. Rixon, and H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16(6), 315–319 (1984).
[Crossref]

Mitrovic, I.

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

Morimoto, Y.

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

Olejnik, K.

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

Piro, J.-L.

J.-L. Piro and M. Grediac, “Producing and transferring low-spatial-frequency grids for measuring displacement fields with moiré and grid methods,” Exp. Tech. 28(4), 23–26 (2004).
[Crossref]

Plan, L.

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

Ramaprakash, A. N.

Rasouli, S.

M. H. Daemi and S. Rasouli, “Fringe chasing by three-point spatial phase shifting for discrimination of the motion direction in the long-range homodyne laser Doppler vibrometry,” Opt. Laser Technol. 103, 387–395 (2018).
[Crossref]

S. Rasouli, S. Esmaeili, and F. Sobouti, “Design and construction of a seismometer based on the moiré technique: detailed theoretical analysis, experimental apparatus, and primary results,” Int. J. Opt. Photon. 10, 3–10 (2016).

S. Esmaeili, S. Rasouli, F. Sobouti, and S. Esmaeili, “A moiré micro strain gauge,” Opt. Commun. 285(9), 2243–2246 (2012).
[Crossref]

S. Rasouli, Z. Eskandari, and Z. Y. Abedini, “Landslide monitoring using moiré technique,” Geosciences 22, 237–240 (2012).

S. Rasouli, “Use of a moiré deflectometer on a telescope for atmospheric turbulence measurements,” Opt. Lett. 35(9), 1470–1472 (2010).
[Crossref] [PubMed]

S. Rasouli, M. Dashti, and A. N. Ramaprakash, “An adjustable, high sensitivity, wide dynamic range two channel wave-front sensor based on moiré deflectometry,” Opt. Express 18(23), 23906–23913 (2010).
[Crossref] [PubMed]

Reid, G. T.

G. T. Reid, R. C. Rixon, and H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16(6), 315–319 (1984).
[Crossref]

Ri, S.

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

Rixon, R. C.

G. T. Reid, R. C. Rixon, and H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16(6), 315–319 (1984).
[Crossref]

Rowberry, M. D.

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

X. Marti, M. D. Rowberry, and J. Blahůt, “A MATLAB code for counting the moiré interference fringes recorded by the optical-mechanical crack gauge TM-71,” Comput. Geosci. 52, 164–167 (2013).
[Crossref]

Sobouti, F.

S. Rasouli, S. Esmaeili, and F. Sobouti, “Design and construction of a seismometer based on the moiré technique: detailed theoretical analysis, experimental apparatus, and primary results,” Int. J. Opt. Photon. 10, 3–10 (2016).

S. Esmaeili, S. Rasouli, F. Sobouti, and S. Esmaeili, “A moiré micro strain gauge,” Opt. Commun. 285(9), 2243–2246 (2012).
[Crossref]

Stemberk, J.

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

Tang, Y.

Tavassoly, M. T.

K. Madanipour and M. T. Tavassoly, “Submicron displacements measurement by measuring autocorrelation of the transmission function of a grating,” Proc. SPIE 8082, 80823O (2011).
[Crossref]

Wronkowski, L.

L. Wronkowski, “Signal transducing in optoelectronic measurement systems based on the moiré phenomenon,” Opt. Eng. 31(3), 505–516 (1992).
[Crossref]

Yu, J.

Zhao, L.

Zhong, M.

Zhou, S.

Zhu, J.

Comput. Geosci. (1)

X. Marti, M. D. Rowberry, and J. Blahůt, “A MATLAB code for counting the moiré interference fringes recorded by the optical-mechanical crack gauge TM-71,” Comput. Geosci. 52, 164–167 (2013).
[Crossref]

Exp. Mech. (2)

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50(4), 501–508 (2010).
[Crossref]

B. Košťák, “A new device for in-situ movement detection and measurement,” Exp. Mech. 9(8), 374–379 (1969).
[Crossref]

Exp. Tech. (1)

J.-L. Piro and M. Grediac, “Producing and transferring low-spatial-frequency grids for measuring displacement fields with moiré and grid methods,” Exp. Tech. 28(4), 23–26 (2004).
[Crossref]

Geomorphology (1)

I. Baroň, L. Plan, B. Grasemann, I. Mitrovic, W. Lenhardt, H. Hausmann, and J. Stemberk, “Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps,” Geomorphology 259, 81–89 (2016).
[Crossref]

Geosciences (1)

S. Rasouli, Z. Eskandari, and Z. Y. Abedini, “Landslide monitoring using moiré technique,” Geosciences 22, 237–240 (2012).

Int. J. Opt. Photon. (1)

S. Rasouli, S. Esmaeili, and F. Sobouti, “Design and construction of a seismometer based on the moiré technique: detailed theoretical analysis, experimental apparatus, and primary results,” Int. J. Opt. Photon. 10, 3–10 (2016).

J. Sci. Instrum. (1)

A. H. McIlraith, “A moiré fringe interpolator of high resolution,” J. Sci. Instrum. 41(1), 34–37 (1964).
[Crossref]

Measurement (Lond) (1)

M. D. Rowberry, D. Kriegner, V. Holy, C. Frontera, M. Llull, K. Olejnik, and X. Marti, “The instrumental resolution of a moire extensometer in light of its recent automatisation,” Measurement (Lond) 91, 258–265 (2016).
[Crossref] [PubMed]

Opt. Commun. (1)

S. Esmaeili, S. Rasouli, F. Sobouti, and S. Esmaeili, “A moiré micro strain gauge,” Opt. Commun. 285(9), 2243–2246 (2012).
[Crossref]

Opt. Eng. (1)

L. Wronkowski, “Signal transducing in optoelectronic measurement systems based on the moiré phenomenon,” Opt. Eng. 31(3), 505–516 (1992).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (2)

M. H. Daemi and S. Rasouli, “Fringe chasing by three-point spatial phase shifting for discrimination of the motion direction in the long-range homodyne laser Doppler vibrometry,” Opt. Laser Technol. 103, 387–395 (2018).
[Crossref]

G. T. Reid, R. C. Rixon, and H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring moiré topography,” Opt. Laser Technol. 16(6), 315–319 (1984).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (1)

K. Madanipour and M. T. Tavassoly, “Submicron displacements measurement by measuring autocorrelation of the transmission function of a grating,” Proc. SPIE 8082, 80823O (2011).
[Crossref]

Other (2)

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier Science, 1993), chap. 5.

A. T‎. Shepherd and G. S. Walker, British Patent 810478 (1995).

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Figures (8)

Fig. 1
Fig. 1 Moiré pattern created by superimposing two similar Ronchi gratings and illustration of the averaging slit S used for scanning of the transmittance intensity.
Fig. 2
Fig. 2 Simulated transmittance function of moiré fringes of two similar Ronchi gratings along the direction perpendicular to the fringes, (a) with different length values l, and (b) with different width values eof the scanning silt S.
Fig. 3
Fig. 3 Illustration of a moiré-based displacement sensor with a single-point detection system.
Fig. 4
Fig. 4 (a) The reference moiré pattern. (b) and (c) two displaced moiré patterns with the same displacement values of Δ d m along two opposite directions, respectively. (d), (e), and (f) the corresponding intensity profiles and intensity values on the detector.
Fig. 5
Fig. 5 The same patterns and plots of Fig. 4 and the intensity values recorded by three point detectors.
Fig. 6
Fig. 6 (a) Top view schematic diagram of the moiré unit, and (b)-(d) three views of the displacement sensor. (b) shows all parts of the sensor. (c) shows G1 and G2, and the array detector from a different view. In (d) different parts are indicated by numbers: 1 chassis, 2 rail, 3 movable grating holder, 4 light source, 5 G1, and 6 shows array detector. G2 is fixed on the array detector close to G1.
Fig. 7
Fig. 7 Verification of the sensor operation. (a) Normalized simulated and experimentally recorded intensity profiles of D 2 detector. (b) Rescaled difference of the auxiliary detection values, I 3 I 1 in a range of (−1.0, 1.0). In (b) and (c) we have e=63.5 μmand d m =410 μm.
Fig. 8
Fig. 8 Verification of the sensor operation under a couple of displacements and returns. The displacement values were measured by the sensor and a fine micrometer. In the measurements e=63.5 μmand d m =410 μm.

Equations (17)

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

t(x,y)= k= + a k exp( j2kπy d ),
T(x,y)= k= + n= + a k a n exp{ j2π d [(k+n)ycos( θ 2 )+(nk)xsin( θ 2 )]}.
I(x)= I 0 l 2 + l 2 d y x e 2 x+ e 2 T (x,y)dx,
I(x)= I 0 k= + n= + a k a n lsinc[ (k+n)lcos( θ 2 ) d ]× x e 2 x+ e 2 e xp[ j2π d (nk)xsin( θ 2 )]dx
= I 0 el k= + n= + a k a n sinc[ (k+n)lcos( θ 2 ) d ]×sinc[ (nk)esin( θ 2 ) d ]exp[j 2π d (nk)xsin( θ 2 )].
I(x)= I 0 el k= + n= + a k a n sinc[ (k+n)lcos( θ 2 ) d ]×sinc[ (nk)e 2 d m ]exp[j π(nk)x d m ].
I 0 el=2( I max + I min ).
I a (x)= k= + n= + a k a n sinc[ (k+n)lcos( θ 2 ) d ]×sinc[ (nk)e 2 d m ]cos[ π(nk)x d m ].
a k = 1 2 sinc( k 2 ).
I a (x)= 1 4 k= + n= + sinc( k 2 ) sinc( n 2 )sinc[ (k+n)lcos( θ 2 ) d ]×sinc[ (nk)e 2 d m ]cos[ π(nk)x d m ].
I a (x)= 1 4 n= + sinc 2 ( n 2 )sinc( en d m )cos( 2nπx d m ) .
I a (x)= 1 4 + 1 2 n=1 + sinc 2 ( n 2 )sinc( en d m )cos( 2nπx d m ) .
I a,normalized (x)= I a (x) I a,min I a,max I a,min ,
I 2,normalized ( x )= I 2 I 2,min I 2,max I 2,min .
I a,normalized (x)= I 2,normalized (x).
Displacement=md+x( I 2,normalized ),
x{ [md,md+d/2 ], ( I 1 I 3 )<0, [md+d/2 ,(m+1)d], ( I 3 I 1 )<0.

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