Abstract

We achieved displacement metrology with a high-amplitude signal by using a rectangular phase grating as the pupil in a grating imaging system. The imaging phenomenon with a pupil transmission grating that has a bilevel profile with a 50% duty ratio is discussed on the basis of the optical transfer function. By optimizing the imaging condition, we obtained high-contrast images with high light power under a magnified or demagnified imaging system. The amplitude of the signal in the displacement measurement was four times higher than that of the conventional grating imaging system with amplitude gratings.

© 2006 Optical Society of America

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References

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  1. R. M. Pettigrew, "Analysis of grating imaging and its application to displacement metrology," Proc. SPIE 136, 325-332 (1977).
  2. J. Jahns and A. W. Lohman, "The Lau effect: a diffraction experiment with incoherent illumination," Opt. Commun. 28, 263-267 (1979).
    [CrossRef]
  3. F. Gori, "Lau effect and coherence theory," Opt. Commun. 31, 4-8 (1979).
    [CrossRef]
  4. R. Sudol and B. J. Thompson, "Lau effect: theory and experiment," Appl. Opt. 20, 1107-1116 (1981).
  5. K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "Lau effect: OTF theory," Opt. Commun. 46, 14-17 (1983).
    [CrossRef]
  6. D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
    [CrossRef]
  7. G. J. Swanson and E. N. Leith, "Lau effect and grating imaging," J. Opt. Soc. Am. 72, 552-555 (1982).
  8. G. J. Swanson and E. N. Leith, "Analysis of the Lau effect and generalized grating imaging," J. Opt. Soc. Am. A 2, 789-793 (1985).
  9. D. Crespo, J. Alonso, and E. Bernabeu, "Generalized grating imaging using an extended monochromatic light source," J. Opt. Soc. Am. A 17, 1231-1240 (2000).
  10. D. Crespo, J. Alonso, and E. Bernabeu, "Experimental measurement of generalized grating images," Appl. Opt. 41, 1223-1228 (2002).
  11. K. Hane and C. P. Grover, "Imaging with rectangular transmission gratings," J. Opt. Soc. Am. A 4, 706-711 (1987).
  12. K. Hane and C. P. Grover, "Magnified grating images used in displacement sensing," Appl. Opt. 26, 2355-2359 (1987).
  13. Y. Ohmura, T. Oka, T. Nakashima, and K. Hane, "Imaging by a phase grating used in displacement measurement," in Proceedings of American Society for Precision Engineering 2004 Annual Meeting (American Society for Precision Engineering, 2004), Vol. 34, pp. 518-521.

2002

2000

D. Crespo, J. Alonso, and E. Bernabeu, "Generalized grating imaging using an extended monochromatic light source," J. Opt. Soc. Am. A 17, 1231-1240 (2000).

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

1987

1985

1983

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "Lau effect: OTF theory," Opt. Commun. 46, 14-17 (1983).
[CrossRef]

1982

1981

1979

J. Jahns and A. W. Lohman, "The Lau effect: a diffraction experiment with incoherent illumination," Opt. Commun. 28, 263-267 (1979).
[CrossRef]

F. Gori, "Lau effect and coherence theory," Opt. Commun. 31, 4-8 (1979).
[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.

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "Lau effect: OTF theory," Opt. Commun. 46, 14-17 (1983).
[CrossRef]

Crespo, D.

Gori, F.

F. Gori, "Lau effect and coherence theory," Opt. Commun. 31, 4-8 (1979).
[CrossRef]

Grover, C. P.

Hane, K.

K. Hane and C. P. Grover, "Magnified grating images used in displacement sensing," Appl. Opt. 26, 2355-2359 (1987).

K. Hane and C. P. Grover, "Imaging with rectangular transmission gratings," J. Opt. Soc. Am. A 4, 706-711 (1987).

Y. Ohmura, T. Oka, T. Nakashima, and K. Hane, "Imaging by a phase grating used in displacement measurement," in Proceedings of American Society for Precision Engineering 2004 Annual Meeting (American Society for Precision Engineering, 2004), Vol. 34, pp. 518-521.

Jahns, J.

J. Jahns and A. W. Lohman, "The Lau effect: a diffraction experiment with incoherent illumination," Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Leith, E. N.

Lohman, A. W.

J. Jahns and A. W. Lohman, "The Lau effect: a diffraction experiment with incoherent illumination," Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Lohmann, A. W.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "Lau effect: OTF theory," Opt. Commun. 46, 14-17 (1983).
[CrossRef]

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]

Nakashima, T.

Y. Ohmura, T. Oka, T. Nakashima, and K. Hane, "Imaging by a phase grating used in displacement measurement," in Proceedings of American Society for Precision Engineering 2004 Annual Meeting (American Society for Precision Engineering, 2004), Vol. 34, pp. 518-521.

Ohmura, Y.

Y. Ohmura, T. Oka, T. Nakashima, and K. Hane, "Imaging by a phase grating used in displacement measurement," in Proceedings of American Society for Precision Engineering 2004 Annual Meeting (American Society for Precision Engineering, 2004), Vol. 34, pp. 518-521.

Ojeda-Castaneda, J.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "Lau effect: OTF theory," Opt. Commun. 46, 14-17 (1983).
[CrossRef]

Oka, T.

Y. Ohmura, T. Oka, T. Nakashima, and K. Hane, "Imaging by a phase grating used in displacement measurement," in Proceedings of American Society for Precision Engineering 2004 Annual Meeting (American Society for Precision Engineering, 2004), Vol. 34, pp. 518-521.

Pettigrew, R. M.

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

Sudol, R.

Swanson, G. J.

Thompson, B. J.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

K. H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, "Lau effect: OTF theory," Opt. Commun. 46, 14-17 (1983).
[CrossRef]

J. Jahns and A. W. Lohman, "The Lau effect: a diffraction experiment with incoherent illumination," Opt. Commun. 28, 263-267 (1979).
[CrossRef]

F. Gori, "Lau effect and coherence theory," Opt. Commun. 31, 4-8 (1979).
[CrossRef]

Opt. Eng.

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

Proc. SPIE

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

Other

Y. Ohmura, T. Oka, T. Nakashima, and K. Hane, "Imaging by a phase grating used in displacement measurement," in Proceedings of American Society for Precision Engineering 2004 Annual Meeting (American Society for Precision Engineering, 2004), Vol. 34, pp. 518-521.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the optical system.

Fig. 2
Fig. 2

OTF as a function of the normalized spatial frequency γ. The object distance Z 1 and the image distance Z 2 are equal to 3T∕4: (a) θ = λ∕4, (b) θ = λ∕2, (c) θ = 3λ∕4.

Fig. 3
Fig. 3

OTF calculated as a function of Z for N = 1, 2, 3, and 4, when the ratio of the object distance Z 1 to the image distance Z 2 is equal to one: (a) θ = λ∕4, (b) θ = λ∕2, (c) θ = 3λ∕4.

Fig. 4
Fig. 4

OTF calculated as a function of the image distance Z 2 when the object distance Z 1 is not equal to the image distance Z 2: (a) N = 1 with θ = λ∕4 and (b) N = 2 with θ = λ∕2.

Fig. 5
Fig. 5

Experimental setup of the optical system. PD, photodiode.

Fig. 6
Fig. 6

Experimental results of the OTF as a function of Z: (a) N = 1 (T = 4.8 mm), (b) N = 2 (T = 5.25 mm).

Fig. 7
Fig. 7

Normalized amplitude of the fundamental frequency as a function of Z at N = 1 with θ = λ∕4. The open circles correspond to the experimental results, and the solid curve shows the calculated results.

Fig. 8
Fig. 8

Output of the PD as a function of the pupil grating displacement: (a) N = 1, θ = λ∕4, p = 65 μm, and Z 1 = Z 2 = T (= 4.8 mm); (b) N = 2, θ = λ∕2, p = 68 μm, and Z 1 = Z 2 = 3T∕4 (= 3.94 mm).

Equations (14)

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F ( σ 2 ) = exp [ i k a ( λ 2 Z 2     2 σ 2     2 ) ] P ( x λ Z 2 σ 2 ) P * ( x ) × exp ( 2 i k a λ Z 2 σ 2 x ) d x ,
P ( x ) = 1       [ =   exp ( i 0 ) ] ,       | x n p | ε ,
P ( x ) = exp ( i θ ) ,       | x ( n + 1 2 ) p | < p 2 ε ,
{ m p λ Z 2 σ 2 < m p + 2 ε m p 2 ε λ Z 2 σ 2 < mp ,
F ( σ 2 ) = n = exp ( i k a β 2 ) ε + n p β m p + ε + n p [ exp ( i θ ) exp ( 2 i k a β x ) ] d x + n = exp ( i k a β 2 ) β m p ε + n p ε + n p [ exp ( 2 i k a β x ) ] d x + n = exp ( i k a β 2 ) ε + n p β m p ε + n p [ exp ( i θ ) exp ( 2 i k a β x ) ] d x + n = exp ( i k a β 2 ) β m p + ε + n p ε + ( n + 1 ) p { exp [ i ( θ θ ) ] exp ( 2 i k a β x ) } d x ,
F ( σ 2 ) = n = exp ( i k a β 2 ) β m p ε + ( n + 1 ) p ε + ( n + 1 ) p [ exp ( - i θ ) exp ( 2 i k a β x ) ] d x + n = exp ( i k a β 2 ) ε + n p β m p + ε + n p [ exp ( 2 i k a β x ) ] d x + n = exp ( i k a β 2 ) β m p + ε + n p ε + n p [ exp ( i θ ) exp ( 2 i k a β x ) ] d x + n = exp ( i k a β 2 ) ε + n p β m p ε + ( n + 1 ) p { exp [ i ( θ θ ) ] exp ( 2 i k a β x ) } d x ,
β = λ Z 2 σ 2 .
F ( σ 2 ) = 1 + 2 n = 1 L cos [ 2 π σ 2 ( 1 + Z 2 Z 1 ) n p ] 1 + 2 L × [ exp ( i θ ) exp [ i π ( 1 + Z 2 Z 1 ) σ 2 ( 2 ε + m p ) ] [ ( σ 2 m p λ Z 2 ) λ Z 2 4 ε ] ×   sinc [ 2 ε π ( 1 + Z 2 Z 1 ) σ 2 ( σ 2 + m p λ Z 2 ) λ Z 2 2 ε ] + exp [ i π ( 1 + Z 2 Z 1 ) σ 2 ( m p ) ] [ 1 ( σ 2 m p λ Z 2 ) λ Z 2 4 ε ] ×   sinc { 2 ε π ( 1 + Z 2 Z 1 ) σ 2 [ 1 - ( σ 2 m p λ Z 2 ) λ Z 2 2 ε ] } + exp ( i θ ) exp [ i π ( 1 + Z 2 Z 1 ) σ 2 ( 2 ε + m p ) ] [ ( σ 2 m p λ Z 2 ) λ Z 2 4 ε ] ×   sinc [ 2 επ ( 1 + Z 2 Z 1 ) σ 2 ( σ 2 m p λ Z 2 ) λ Z 2 2 ε ] +   exp [ i π ( 1 + Z 2 Z 1 ) σ 2 ( m 1 ) p ] { 1 [ σ 2 ( m + 1 ) p λ Z 2 ] λ Z 2 4 ε } ×   sinc ( 2 ε π ( 1 + Z 2 Z 1 ) σ 2 { 1 + [ σ 2 ( m + 1 ) p λ Z 2 ] λ Z 2 2 ε } ) ] ,
F ( σ 2 ) = 1 + 2 n = 1 L cos [ 2 π σ 2 ( 1 + Z 2 Z 1 ) n p ] 1 + 2 L × [ exp ( i θ ) exp { i π ( 1 + Z 2 Z 1 ) σ 2 [ 2 ε ( m 2 ) p ] } [ ( σ 2 + m p λ Z 2 ) λ Z 2 4 ε ] × sinc [ 2 ε π ( 1 + Z 2 Z 1 ) σ 2 ( σ 2 m p λ Z 2 ) λ Z 2 2 ε ] + exp [ i π ( 1 + Z 2 Z 1 ) σ 2 ( m p ) ] [ 1 + ( σ 2 m p λ Z 2 ) λ Z 2 2 ε ] × sinc { 2 ε π ( 1 + Z 2 Z 1 ) σ 2 [ 1 + ( σ 2 m p λ Z 2 ) λ Z 2 2 ε ] } + exp ( i θ ) exp [ i π ( 1 + Z 2 Z 1 ) σ 2 ( - 2 ε + m p ) ] [ ( σ 2 + m p λ Z 2 ) λ Z 2 4 ε ] × sinc [ 2 ε π ( 1 + Z 2 Z 1 ) σ 2 ( σ 2 m p λ Z 2 ) λ Z 2 2 ε ] + exp [ i π ( 1 + Z 2 Z 1 ) σ 2 ( m 1 ) p ] { 1 + [ σ 2 ( m 1 ) p λ Z 2 ] λ Z 2 4 ε } × sinc ( 2 ε π ( 1 + Z 2 Z 1 ) σ 2 { 1 + [ σ 2 ( m 1 ) p λ Z 2 ] λ Z 2 2 ε } ) ] ,
( 1 + Z 2 Z 1 ) σ 2 p = N ,
Z 1 σ 1 = Z 2 σ 2 ,
σ 1 = Z 2 Z 1 + Z 2 N p .
γ = λ Z 2 σ 2 2 ε .
T = p 2 λ .

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