Abstract

In two preceding works (Appl. Opt. 47, 2201–2206, 2008; Appl. Opt. 48, 414–424, 2009) we introduced the design of an optical encoder based on a nondiffractive beam and studied the dependence of its performance on its parameters (e.g., grating pitch, photodetector size). In those works we proposed different optimization criteria and concluded that the proposed design provides an output sinusoidal signal with high contrast and very low harmonic distortion, while having remarkable tolerance to variations in its parameters and to mechanical perturbations. In this work we (1) study how to improve the system performance by means of selecting appropriate photodetector geometry, (2) study the system performance for different nondiffractive beam geometries, and (3) quantify the output signal tolerance to vertical and lateral misalignment between the centers of the nondiffractive beam and the photodetector. As a consequence, we obtain new sets of optimal parameters that significantly improve the system performance and enhance its tolerance to mechanical perturbations and fabrication errors.

© 2009 Optical Society of America

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References

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  1. A. Lutenberg, F. Perez-Quintian, and M. A. Rebollo, “Optical encoder based on a nondiffractive beam,” Appl. Opt. 47, 2201-2206 (2008).
    [CrossRef] [PubMed]
  2. A. Lutenberg and F. Perez-Quintian, “Optical encoder based on a nondiffractive beam II,” Appl. Opt. 48, 414-424 (2009).
    [CrossRef] [PubMed]
  3. L. Wronkowski, “Diffraction model of an optoelectronic displacement measuring transducer,” Opt. Lasers Technol. 27, 81-88 (1995).
    [CrossRef]
  4. D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
    [CrossRef]
  5. G. Swanson and E. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. A 2, 789-793(1985).
    [CrossRef]
  6. W. Huber and M. Allgauer, “Interferential linear and angular displacement apparatus having scanning and scale grating respectively greater than and less than the source wavelength,” U.S. patent 5,424,833 (13 June 1995).
  7. F. Perez-Quintian, A. Lutenberg, and M. A. Rebollo, “Linear displacement measurement with a grating and speckle pattern illumination,” Appl. Opt. 45, 4821-4825(2006).
    [CrossRef] [PubMed]
  8. C.-C. Wu, W.-J. Wu, Z.-S. Pan, and C.-K. Lee, “Laser linear encoder with both high fabrication and head-to-scale tolerances,”. Appl. Opt. 46, 3169-3176 (2007).
    [CrossRef] [PubMed]
  9. C.-F. Kao, S.-H. Lu, H.-M. Shen, and K.-C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47 , 1833-1837 (2008).
    [CrossRef]
  10. www.heidenhain.com
  11. R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039-3044(1998).
    [CrossRef]

2009 (1)

2008 (2)

C.-F. Kao, S.-H. Lu, H.-M. Shen, and K.-C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47 , 1833-1837 (2008).
[CrossRef]

A. Lutenberg, F. Perez-Quintian, and M. A. Rebollo, “Optical encoder based on a nondiffractive beam,” Appl. Opt. 47, 2201-2206 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

2000 (1)

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

1998 (1)

1995 (1)

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

1985 (1)

Allgauer, M.

W. Huber and M. Allgauer, “Interferential linear and angular displacement apparatus having scanning and scale grating respectively greater than and less than the source wavelength,” U.S. patent 5,424,833 (13 June 1995).

Alonso, J.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

Bernabeu, E.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

Crespo, D.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

Fan, K.-C.

C.-F. Kao, S.-H. Lu, H.-M. Shen, and K.-C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47 , 1833-1837 (2008).
[CrossRef]

Huber, W.

W. Huber and M. Allgauer, “Interferential linear and angular displacement apparatus having scanning and scale grating respectively greater than and less than the source wavelength,” U.S. patent 5,424,833 (13 June 1995).

Kao, C.-F.

C.-F. Kao, S.-H. Lu, H.-M. Shen, and K.-C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47 , 1833-1837 (2008).
[CrossRef]

Lee, C.-K.

Leith, E.

Lu, S.-H.

C.-F. Kao, S.-H. Lu, H.-M. Shen, and K.-C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47 , 1833-1837 (2008).
[CrossRef]

Lutenberg, A.

Morlanes, T.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

Pan, Z.-S.

Perez-Quintian, F.

Piestun, R.

Rebollo, M. A.

Shamir, J.

Shen, H.-M.

C.-F. Kao, S.-H. Lu, H.-M. Shen, and K.-C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47 , 1833-1837 (2008).
[CrossRef]

Swanson, G.

Wronkowski, L.

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

Wu, C.-C.

Wu, W.-J.

Appl. Opt. (4)

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

Jpn. J. Appl. Phys. (1)

C.-F. Kao, S.-H. Lu, H.-M. Shen, and K.-C. Fan, “Diffractive laser encoder with a grating in Littrow configuration,” Jpn. J. Appl. Phys. 47 , 1833-1837 (2008).
[CrossRef]

Opt. Eng. (1)

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, “Optical encoder based on the Lau effect,” Opt. Eng. 39, 817-824(2000).
[CrossRef]

Opt. Lasers Technol. (1)

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

Other (2)

W. Huber and M. Allgauer, “Interferential linear and angular displacement apparatus having scanning and scale grating respectively greater than and less than the source wavelength,” U.S. patent 5,424,833 (13 June 1995).

www.heidenhain.com

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Figures (22)

Fig. 1
Fig. 1

Illustration of TF [ d ( x , y ) ] corresponding to a square photodetector of side a = 7 r o , with r o = 66.5 μm .

Fig. 2
Fig. 2

Illustration of TF [ d ( x , y ) ] corresponding to a circular photodetector of diameter a = 7 r o , with r o = 66.5 μm .

Fig. 3
Fig. 3

f x profiles of the photodetectors of Figs. 1, 2.

Fig. 4
Fig. 4

Diagram of NDB encoder for the circular photodetector implementation.

Fig. 5
Fig. 5

Contrast map numerically obtained for a 96 μm grating, a zero-order Bessel NDB with r o = 66.5 μm , and a circular photodetector.

Fig. 6
Fig. 6

Amplitude map numerically obtained for a 96 μm grating, a zero-order Bessel NDB with r o = 66.5 μm , and a circular photodetector.

Fig. 7
Fig. 7

Mean value map numerically obtained for a 96 μm grating, a zero-order Bessel NDB with r o = 66.5 μm , and a circular photodetector.

Fig. 8
Fig. 8

Contrast map numerically obtained for a 93.6 μm grating, a zero-order Bessel NDB with r o = 66.5 μm , and a circular photodetector.

Fig. 9
Fig. 9

FOM map numerically obtained for a 96 μm grating, a zero-order Bessel NDB with r o = 66.5 μm , and a circular photodetector.

Fig. 10
Fig. 10

Output signal contrast as a function of the normalized z distance for a = 4.37 p and different r o values.

Fig. 11
Fig. 11

Output signal contrast as a function of the normalized z distance for a = 6.89 p and different r o values.

Fig. 12
Fig. 12

Diagram illustrating the zero- and first-order circles. The Greek letters refer to the circles’ phases at the points of interest, as described by Eq. (8).

Fig. 13
Fig. 13

Illustration of the integrand of Eq. (1) for the parameters λ = 650 nm , p = 96 μm , z = 10 mm , r o = 66.5 μm , a = 6.68 r o , a circular photodetector, and a zero-order Bessel NDB, corresponding to Δ x = 24 μm , the maximum of the output signal.

Fig. 14
Fig. 14

Illustration of the integrand of Eq. (1) for the parameters λ = 650 nm , p = 96 μm , z = 10 mm , r o = 66.5 μm , a = 6.68 r o , a circular photodetector, and a zero-order Bessel NDB, corresponding to Δ x = 74 μm , the minimum of the output signal.

Fig. 15
Fig. 15

Illustration of the integrand of Eq. (1) for the parameters λ = 650 nm , p = 96 μm , z = 10 mm , r o = 66.5 μm , a = 6.68 r o , a circular photodetector, and a second-order Bessel NDB, corresponding to Δ x = 24 μm , the maximum of the output signal.

Fig. 16
Fig. 16

Illustration of the integrand of Eq. (1) for the parameters λ = 650 nm , p = 96 μm , z = 10 mm , r o = 66.5 μm , a = 6.68 r o , a circular photodetector, and a second-order Bessel NDB, corresponding to Δ x = 72 μm , the minimum of the output signal.

Fig. 17
Fig. 17

Contrast map numerically obtained for a 96 μm grating, a first-order Bessel NDB with r o = 66.5 μm , and a circular photodetector.

Fig. 18
Fig. 18

Contrast map numerically obtained for a 96 μm grating, a second-order Bessel NDB with r o = 66.5 μm , and a circular photodetector.

Fig. 19
Fig. 19

Output signal contrast as a function of the normalized z distance, for the case of the maximal FOM discussed in Subsection 2C, as the photodetector is misaligned in the x direction.

Fig. 20
Fig. 20

Output signal contrast as a function of the normalized z distance, for the case of maximal FOM discussed in Subsection 2C, as the photodetector is misaligned in the y direction.

Fig. 21
Fig. 21

Output signal contrast as a function of the normalized z distance, for the case of maximal z invariance discussed in Subsection 2D, as the photodetector is misaligned in the x direction.

Fig. 22
Fig. 22

Output signal contrast as a function of the normalized z distance, for the case of maximal z invariance discussed in Subsection 2D, as the photodetector is misaligned in the y direction.

Tables (1)

Tables Icon

Table 1 Comparison of the Predictions of Eq. (2) and the Angular Spectrum Approximation Results for p = 96 μm , r o = 66.5 μm , and λ = 650 nm

Equations (10)

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s ( Δ x ) + + { { TF [ t G ( x Δ x ) ] * TF [ J 0 ( x , y ) ] } H ( f x , f y ) } * TF [ d ( x , y ) ] 2 d f x d f y
min .  amplitude if     1 p = 1.63 a ; 3.70 a ; 5.72 a ; etc., min .  amplitude if     1 p = 2.68 a ; 4.71 a ; 6.72 a ; etc .
r o = 0.71 p , z = 2.05 p 2 λ , a = 4.37 p .
r o = 0.698 p , a = 6.89 p .
E 0 ( r , ϕ , z ) J 0 ( γ r ) ,
E 0 ( r , ϕ , z ) J m ( γ r ) exp [ i m ϕ ] ,
TF [ E 0 ( r , ϕ , z ) ] δ ( f r γ 2 π ) · exp [ i m ϕ ] .
α 1 = m α , β 2 = m ( π α ) + φ , α 2 = m ( π α ) , β 3 = m ( π + α ) + φ , α 3 = m ( π + α ) , χ 1 m α φ , α 4 = m α , χ 1 = m α φ ,
cos α = π γ p .
θ 1 = α 1 + β 2 = φ , θ 3 = α 3 + χ 4 = φ , θ 2 = α 2 + χ 1 = φ , θ 4 = α 4 + χ 3 = φ .

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