Abstract

In a previous work, we introduced the design of an optical encoder based on a nondiffractive beam and demonstrated that it generates a suitable output sinusoidal signal [Appl. Opt. 47, 2201–2206 (2008)]. In this work, we experimentally, analytically, and numerically study the dependence of the system performance on its parameters (grating pitch, photodetector size, etc.) and propose three different optimization criteria for which the tolerance to variations in the system parameters is also analyzed. We conclude that the proposed design generates a suitable output signal, with high contrast and very low harmonic distortion, while having a remarkable tolerance to variations in its parameters and to mechanical perturbations.

© 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. B. Hafizi, A. K. Ganguly, A. Ting, C. I. Moore, and P. Sprangle, “Analysis of Gaussian beam and Bessel beam driven laser accelerators,” Phys. Rev. E 60, 4779-4792 (1999).
    [CrossRef]
  3. A. Papoulis, “Random signals,” in Systems and Transforms with Applications in Optics (McGraw-Hill, 1986), pp. 259-260.
  4. K. Ching-Fen and L. Mao-Hong, “Optical encoder based on the fractional Talbot effect,” Opt. Commun. 250, 16-23 (2005).
    [CrossRef]
  5. R. M. Pettigrew, “Analysis of grating imaging and its application to displacement metrology,” Proc. SPIE 136, 325-332(1977).
  6. L. M. Sanchez-Brea, J. Saez-Landete, J. Alonso, and E. Bernabeu, “Invariant grating pseudoimaging using polychromatic light and a finite extension source,” Appl. Opt. 47, 1470-1477 (2008).
    [CrossRef] [PubMed]

2008

2005

K. Ching-Fen and L. Mao-Hong, “Optical encoder based on the fractional Talbot effect,” Opt. Commun. 250, 16-23 (2005).
[CrossRef]

1999

B. Hafizi, A. K. Ganguly, A. Ting, C. I. Moore, and P. Sprangle, “Analysis of Gaussian beam and Bessel beam driven laser accelerators,” Phys. Rev. E 60, 4779-4792 (1999).
[CrossRef]

1977

R. M. Pettigrew, “Analysis of grating imaging and its application to displacement metrology,” Proc. SPIE 136, 325-332(1977).

Alonso, J.

Bernabeu, E.

Ching-Fen, K.

K. Ching-Fen and L. Mao-Hong, “Optical encoder based on the fractional Talbot effect,” Opt. Commun. 250, 16-23 (2005).
[CrossRef]

Ganguly, A. K.

B. Hafizi, A. K. Ganguly, A. Ting, C. I. Moore, and P. Sprangle, “Analysis of Gaussian beam and Bessel beam driven laser accelerators,” Phys. Rev. E 60, 4779-4792 (1999).
[CrossRef]

Hafizi, B.

B. Hafizi, A. K. Ganguly, A. Ting, C. I. Moore, and P. Sprangle, “Analysis of Gaussian beam and Bessel beam driven laser accelerators,” Phys. Rev. E 60, 4779-4792 (1999).
[CrossRef]

Lutenberg, A.

Mao-Hong, L.

K. Ching-Fen and L. Mao-Hong, “Optical encoder based on the fractional Talbot effect,” Opt. Commun. 250, 16-23 (2005).
[CrossRef]

Moore, C. I.

B. Hafizi, A. K. Ganguly, A. Ting, C. I. Moore, and P. Sprangle, “Analysis of Gaussian beam and Bessel beam driven laser accelerators,” Phys. Rev. E 60, 4779-4792 (1999).
[CrossRef]

Papoulis, A.

A. Papoulis, “Random signals,” in Systems and Transforms with Applications in Optics (McGraw-Hill, 1986), pp. 259-260.

Perez-Quintian, F.

Pettigrew, R. M.

R. M. Pettigrew, “Analysis of grating imaging and its application to displacement metrology,” Proc. SPIE 136, 325-332(1977).

Rebollo, M. A.

Saez-Landete, J.

Sanchez-Brea, L. M.

Sprangle, P.

B. Hafizi, A. K. Ganguly, A. Ting, C. I. Moore, and P. Sprangle, “Analysis of Gaussian beam and Bessel beam driven laser accelerators,” Phys. Rev. E 60, 4779-4792 (1999).
[CrossRef]

Ting, A.

B. Hafizi, A. K. Ganguly, A. Ting, C. I. Moore, and P. Sprangle, “Analysis of Gaussian beam and Bessel beam driven laser accelerators,” Phys. Rev. E 60, 4779-4792 (1999).
[CrossRef]

Appl. Opt.

Opt. Commun.

K. Ching-Fen and L. Mao-Hong, “Optical encoder based on the fractional Talbot effect,” Opt. Commun. 250, 16-23 (2005).
[CrossRef]

Phys. Rev. E

B. Hafizi, A. K. Ganguly, A. Ting, C. I. Moore, and P. Sprangle, “Analysis of Gaussian beam and Bessel beam driven laser accelerators,” Phys. Rev. E 60, 4779-4792 (1999).
[CrossRef]

Proc. SPIE

R. M. Pettigrew, “Analysis of grating imaging and its application to displacement metrology,” Proc. SPIE 136, 325-332(1977).

Other

A. Papoulis, “Random signals,” in Systems and Transforms with Applications in Optics (McGraw-Hill, 1986), pp. 259-260.

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Figures (18)

Fig. 1
Fig. 1

Diagram of the NDB encoder implemented by means of an axicon.

Fig. 2
Fig. 2

Diagram of the experimental setup used to study the dependence of the NDB encoder performance on its design parameters.

Fig. 3
Fig. 3

Contrast map experimentally obtained using a 100 μm grating and a NDB with r o = 66.5 μm .

Fig. 4
Fig. 4

SNR map experimentally obtained using a 100 μm grating and a NDB with r o = 66.5 μm .

Fig. 5
Fig. 5

Mean-value map experimentally obtained using a 100 μm grating and a NDB with r o = 66.5 μm .

Fig. 6
Fig. 6

FOM map experimentally obtained using a 100 μm grating and a NDB with r o = 66.5 μm .

Fig. 7
Fig. 7

Diagram illustrating the way in which the system parameters modify the contrast.

Fig. 8
Fig. 8

Illustration of the integrand of Eq. (11) for the parameters λ = 650 nm , p = 100 μm , z = 10 mm , r o = 66.5 μm , and a = 7 r o , corresponding to Δ x = 25 μm , the maximum of the output signal.

Fig. 9
Fig. 9

Illustration of the integrand of Eq. (11) for the parameters λ = 650 nm , p = 100 μm , z = 10     mm , r o = 66.5 μm , and a = 7 r o , corresponding to Δ x = 75 μm , the minimum of the output signal.

Fig. 10
Fig. 10

Contrast profiles at z = 0 and z = z 1 = 16 mm corresponding to the ASM simulations for p = 100 μm and r o = 66.5 μm .

Fig. 11
Fig. 11

Comparison between the z 1 values predicted by Eq. (17) with the corresponding results obtained by means of the ASM implementation for r o = 66.5 μm , λ = 0.650 μm , and p between 86 and 155 μm .

Fig. 12
Fig. 12

Correlation coefficient between the contrast profile at z = 0 and the contrast profile at z = z 1 for r o = 66.5 μm , λ = 0.650 μm , and p between 86 and 155 μm .

Fig. 13
Fig. 13

Contrast map numerically obtained for a 96 μm grating and a NDB with r o = 66.5 μm .

Fig. 14
Fig. 14

FOM map numerically obtained for a 96 μm grating and a NDB with r o = 66.5 μm .

Fig. 15
Fig. 15

Output signal contrast as a function of the normalized z distance for a = 4.36 p and different r o values.

Fig. 16
Fig. 16

Output signal contrast as a function of the normalized z distance for a = 4.45 p and different r o values.

Fig. 17
Fig. 17

FOM map numerically obtained for a 96 μm grating and a NDB with r o = 69.8 μm .

Fig. 18
Fig. 18

Output signal contrast as a function of the normalized z distance for a = 5 p and different r o values.

Tables (2)

Tables Icon

Table 1 Correlation Coefficient Between the Measurements and the ASM Predictions

Tables Icon

Table 2 Comparison Between the Predictions of Eq. (12) and the ASM Results

Equations (23)

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I ( r , ϕ , z ) J 0 2 ( 2 π ( n 1 ) tan α λ r ) ,
t G ( x Δ x ) = 1 2 + n = 1 , 3 , 5 , .. + 2 n π sen [ 2 π p n ( x Δ x ) ] ,
E ( x , y , Δ x ) TF 1 { TF [ t G ( x Δ x ) J 0 ( x , y ) ] H ( ω x , ω y ) } ,
H ( ω x , ω y ) = { exp [ j 2 π z λ 1 ( λ f x ) 2 ( λ f y ) 2 ] f x 2 + f y 2 < 1 λ 0 otherwise ,
s ( Δ x ) + + E 2 ( x , y , Δ x ) d ( x , y ) d x d y ,
d ( x , y ) { 1 | x | , | y | a 2 0 otherwise ,
SNR = TF [ s ( Δ x ) ] 2 | f = 1 / p f = 0.05 / p 10 / p TF [ s ( Δ x ) ] 2 TF [ s ( Δ x ) ] 2 | f = 1 / p .
FOM | a , z = min ( C ( s ( Δ x ) ) , 0.4 ) 0.4 × min ( s ( Δ x ) , 0.4 ) 0.4 × min ( SNR ( s ( Δ x ) ) , 140 ) 140 × min ( a , 6 r o ) 6 r o ,
s ( Δ x ) + + E ˜ 2 ( x , y , Δ x ) d x d y ,
E ˜ ( x , y , Δ x ) TF 1 { { TF [ t G ( x Δ x ) J 0 ( x , y ) ] H ( f x , f y ) } } d ( x , y ) .
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 .
π r 0 2.405 < p < 2 π r 0 2.405 .
min contrast   if 1 p = 1.43 a ; 3.47 a ; 5.48 a ; etc . max contrast   if 1 p = 2.46 a ; 4.48 a ; 6.48 a ; etc .
p sin α = λ .
δ = z tan α .
δ 1 = r o · 1.5936.
z 1 = 1.5936 r o · p λ ,
ψ = 2 ( 1 p 2.405 2 π r o ) .
ψ × 4.48 p = n ,
r o = 0.692 p .
r o = 0.692 p , z = 1.10 p 2 λ , a = 4.48 p .
r o = 0.727 p , z = 3 p 2 λ , a = 4.48 p ,
r o = 0.692 p , z = 1.10 p 2 λ , a = 5 p ,

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