Abstract

Reflection optical encoders are studied as three-grating moiré systems. An analysis is made of the differences that may appear between it and the standard case in which an optical encoder is regarded as a two-grating system.

© 2000 Optical Society of America

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References

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  1. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, The Netherlands, 1993), Chap. 5, pp. 99–139.
  2. F. Talbot, “Facts relating to optical science: Number IV,” Philos. Mag. 9, 401–407 (1836).
  3. A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
    [CrossRef]
  4. E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
    [CrossRef]
  5. K. Patorski, Moiré Metrology (Pergamon, New York, 1998).
  6. G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part I,” Opt. Spectrosc. 22, 73–78 (1967); G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part II,” Opt. Spectrosc. 22, 255–258 (1967); G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part III,” Opt. Spectrosc. 22, 335–340 (1967).
  7. L. Liu, X. Liu, L. Ye, “Joint Talbot effect and logic-operated moiré patterns,” J. Opt. Soc. Am. A 7, 970–976 (1990).
    [CrossRef]
  8. L. Wronkowski, “Diffraction model of an optoelectronic displacement measuring transducer,” Opt. Laser Technol. 27, 81–88 (1995).
    [CrossRef]
  9. L. Wronkowski, “Opto-electronic analog-impulse transducer accuracy from the point of view of diffraction phenomena,” in New Measurement Technology to Serve Mankind: Acta IMEKO 1985, Volume III. Measurement in Mechanics and Laser Metrology, G. Striker, T. Boromisza, T. Kemeny, eds. (Omikk-Technoinform, Budapest, Hungary, 1985), pp. 445–462.

1997 (1)

A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

1995 (1)

L. Wronkowski, “Diffraction model of an optoelectronic displacement measuring transducer,” Opt. Laser Technol. 27, 81–88 (1995).
[CrossRef]

1990 (1)

1985 (1)

1967 (1)

G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part I,” Opt. Spectrosc. 22, 73–78 (1967); G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part II,” Opt. Spectrosc. 22, 255–258 (1967); G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part III,” Opt. Spectrosc. 22, 335–340 (1967).

1836 (1)

F. Talbot, “Facts relating to optical science: Number IV,” Philos. Mag. 9, 401–407 (1836).

Kafri, O.

Keren, E.

Liu, L.

Liu, X.

Olszak, A.

A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

Patorski, K.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, The Netherlands, 1993), Chap. 5, pp. 99–139.

K. Patorski, Moiré Metrology (Pergamon, New York, 1998).

Rassudova, G. N.

G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part I,” Opt. Spectrosc. 22, 73–78 (1967); G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part II,” Opt. Spectrosc. 22, 255–258 (1967); G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part III,” Opt. Spectrosc. 22, 335–340 (1967).

Talbot, F.

F. Talbot, “Facts relating to optical science: Number IV,” Philos. Mag. 9, 401–407 (1836).

Wronkowski, L.

A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

L. Wronkowski, “Diffraction model of an optoelectronic displacement measuring transducer,” Opt. Laser Technol. 27, 81–88 (1995).
[CrossRef]

L. Wronkowski, “Opto-electronic analog-impulse transducer accuracy from the point of view of diffraction phenomena,” in New Measurement Technology to Serve Mankind: Acta IMEKO 1985, Volume III. Measurement in Mechanics and Laser Metrology, G. Striker, T. Boromisza, T. Kemeny, eds. (Omikk-Technoinform, Budapest, Hungary, 1985), pp. 445–462.

Ye, L.

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

Opt. Laser Technol. (1)

L. Wronkowski, “Diffraction model of an optoelectronic displacement measuring transducer,” Opt. Laser Technol. 27, 81–88 (1995).
[CrossRef]

Opt. Spectrosc. (1)

G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part I,” Opt. Spectrosc. 22, 73–78 (1967); G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part II,” Opt. Spectrosc. 22, 255–258 (1967); G. N. Rassudova, “Moiré interference fringes in a system consisting of a transmission and a reflection diffraction grating. Part III,” Opt. Spectrosc. 22, 335–340 (1967).

Philos. Mag. (1)

F. Talbot, “Facts relating to optical science: Number IV,” Philos. Mag. 9, 401–407 (1836).

Other (3)

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, The Netherlands, 1993), Chap. 5, pp. 99–139.

K. Patorski, Moiré Metrology (Pergamon, New York, 1998).

L. Wronkowski, “Opto-electronic analog-impulse transducer accuracy from the point of view of diffraction phenomena,” in New Measurement Technology to Serve Mankind: Acta IMEKO 1985, Volume III. Measurement in Mechanics and Laser Metrology, G. Striker, T. Boromisza, T. Kemeny, eds. (Omikk-Technoinform, Budapest, Hungary, 1985), pp. 445–462.

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Figures (9)

Fig. 1
Fig. 1

Typical scheme of a moiré reflection encoder.

Fig. 2
Fig. 2

Optical setup of a three-grating moiré system.

Fig. 3
Fig. 3

Variation of the modulation of the signal with the distance z between gratings for the three-grating (reflection) and the two-grating (transmission) cases. The distance z is normalized to the Talbot distance.

Fig. 4
Fig. 4

Under a geometric approximation, when the shadow of the first grating for a given angle of incidence is projected onto the third grating it completely darkens the observation plane.

Fig. 5
Fig. 5

Variation of the signal given by the encoder plotted as a function of the inclination of the incident beam. The solid curve represents q 0λz/ p = 0; the dashed curve represents q 0λz/ p = π/4; the dashed-dotted line represents q 0λz/ p = π/4.

Fig. 6
Fig. 6

Schematic representation of the incident field on each observation window when the light source is defocused.

Fig. 7
Fig. 7

Modulation curves for different sizes of the light source. The solid curve represents the results from the diode laser; the dotted curve represents the results from the 200-µm emitting IRED laser; the long-dashed curve represents the results from the 350-µm emitting IRED laser.

Fig. 8
Fig. 8

Modulation curves for different angles θ of the incident beam. The solid curve represents normal incidence; the dotted curve represents an angle of incidence of θ = 2°; the long-dashed curve represents an angle of incidence of θ = 4°.

Fig. 9
Fig. 9

Modulation curves for different defocusing positions for (a) the central observation window and (b) the lateral observation window. The solid curve represents the results for no defocus, Δ = 0; the dotted curve represents results for a defocus value of Δ = 1.0 mm; the long-dashed curve represents results for a defocus value of Δ = 2.0 mm; the dashed–dotted curve represents results for a defocus value of Δ = 2.5 mm.

Equations (32)

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φ1Bx=expiq0x,
t1x=n an expiqnx,
a0=12,  a2n+1=-1n2n+1,  a2n=0.
φ1Ax=t1xφ1Bx=n an expixqn+q0.
Φ1Akx=TFφ1Ax=n anδkx-qn-q0.
φ2Bx=- expikxxexp-ikx2z12kn anδkx-qn-q0=n an expixqn+q0exp-iqn+q02z12k,
t2x=m am expiϕmexpiqmx,
φ3Bx=nm anam expiϕmexpixqn+m+q0exp-iqn+q02z12kexp-iqn+m+q02z22k.
I3Bx=|φ3Bx|2=nnmm ananamam expiϕm-m×expixqn-n+m-m×exp-iq2n2-n2z1+z22k×exp-iq2m2-m2z22k×exp-iq2nm-nmz2k×exp-iqq0n-nz1+z2k×exp-iqq0m-mz2k.
I3Bx=lh expiϕlexpixqh+l×exp-iq2h2z1+z22kexp-iq2l2z22k×exp-iq2lh z2kexp-iqq0h z1+z2k×exp-iqq0l z2kAh, lBh, l,
Ah, l=n an+han exp-iq2nh z1+z2k×exp-iq2nl z2k,
Bh, l=m am+lam exp-iq2ml+hz2k.
I3Ax=nlh an expiϕlexpixqh+l+n×exp-iq2h2z1+z22kexp-iq2l2z22k×exp-iq2lh z2kexp-iqq0h z1+z2k×exp-iqq0l z2kAh, lBh, l.
Iϕ=I3Ax=-p/2p/2 I3Axdx.
Iϕ=lh ah+l expiϕlexp-iq2h2z1+z22k×exp-iq2l2z22kexp-iq2lh z2k×exp-iqq0h z1+z2k×exp-iqq0l z2kAh, lBh, l.
Iϕ=lh ah+l expiϕlexp-iq2h2zk×exp-iq2l2z2kexp-iq2lh zk×exp-iqq02h+lzkAh, lBh, l,
Iϕ=lh ah+l expiϕlexp-iq2h2zk×exp-iq2l2z2kexp-iq2lh zkAh, lBh, l.
MR=Imax-Imin=l0 |al|2 cosz2k q2l2×12+n|an|2 exp-i zk q2nl.
MD=l0 |al|2 cosz2k q2l2.
ISϕ=-πS/fλπS/fλ Iq0dq0.
IS=lh ah+l expiϕlexp-iq2h2zk×exp-iq2l2z2kexp-iq2lh zkAh, lBh, l×sincπ Sfzp2h+l,
MST=ISmax-ISmin=l0 |al|2 cosz2k q2l2sincπ Sfzp l×12+n|an|2 exp-i zk q2nl.
MSD=l0 |al|2 cosz2k q2l2sincπ Sfzp l.
ITϕ=18+h0 ah2cosϕhcosqq0h zk+12 cos2qq0h zk.
MT=2 h0 ah2 cosqq0h zk.
Df<p8z.
1/RΔ/f2,
θw=w/RwΔ/f2.
IO=lh ah+l expiϕlexp-iq2h2zk×exp-iq2l2z2kexp-iq2lh zkAh, lBh, l
Iw=lh ah+l expiϕ+π2l×exp-iq2h2zkexp-iq2l2z2k×exp-iq2lh zkexp-iqθw2h+lz Ah, lBh, l.
IOϕ=14+h0 |ah|2 cosϕh,
IWϕ=18+h0 |ah|2cosϕ+π2hcoshqθwz+12 cos2hqθwz,

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