Abstract

We propose a new type of tilt sensor. It consists of a grating and an image sensor. It detects the tilt of the collimated wavefront reflected from a plane mirror. Its principle is described and analyzed based on wave optics. Experimental results show its validity. Simulations of the ordinary autocollimator and the proposed tilt sensor show that the effect of noise on the measured angle is smaller for the latter. These results show a possibility of making a smaller and simpler tilt sensor.

© 2009 Optical Society of America

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References

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  1. J. Yuan and X. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74, 1362-1365 (2003).
    [CrossRef]
  2. “Agilent 10770A Angular Interferometer with Agilent 10771A Angular Reflector,” Agilent Technologies user's manual, Chap. 7V.
  3. M.-H. Chiu and D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
    [CrossRef]
  4. T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
    [CrossRef]
  5. Z. Ge and M. Takeda, “High-resolution two-dimensional angle measurement technique based on fringe analysis,” Appl. Opt. 42, 6859-6868 (2003).
    [CrossRef] [PubMed]
  6. S. Prakash, S. Singh, and S. Rana, “Automated small tilt-angle measurement using Lau interferometry,” Appl. Opt. 44, 5905-5909 (2005).
    [CrossRef] [PubMed]
  7. K. Iwata, H. Fukuda, and K. Moriwaki, “Autocollimator detecting wave-front tilt using a grating,” in Proceedings of International Commission for Optics (ICO) Topical Meeting on Optoinformatics /Information Photonics 2006, M.Calvo, A.Pavlov, and J.Jahns, eds. (ICO, 2006), pp.517-518.
  8. M. Takeda and K. Mutoh, “Fourier-transform profilometry for the automatic measurement of 3D object shapes,” Appl. Opt. 22, 3977-3982 (1983).
    [CrossRef] [PubMed]
  9. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 3-108.
    [CrossRef]

2006 (1)

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

2005 (1)

2003 (2)

Z. Ge and M. Takeda, “High-resolution two-dimensional angle measurement technique based on fringe analysis,” Appl. Opt. 42, 6859-6868 (2003).
[CrossRef] [PubMed]

J. Yuan and X. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74, 1362-1365 (2003).
[CrossRef]

1997 (1)

M.-H. Chiu and D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
[CrossRef]

1983 (1)

Chiu, M.-H.

M.-H. Chiu and D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
[CrossRef]

Endo, T.

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

Fukuda, H.

K. Iwata, H. Fukuda, and K. Moriwaki, “Autocollimator detecting wave-front tilt using a grating,” in Proceedings of International Commission for Optics (ICO) Topical Meeting on Optoinformatics /Information Photonics 2006, M.Calvo, A.Pavlov, and J.Jahns, eds. (ICO, 2006), pp.517-518.

Ge, Z.

Greivenkamp, J. E.

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

Iwata, K.

K. Iwata, H. Fukuda, and K. Moriwaki, “Autocollimator detecting wave-front tilt using a grating,” in Proceedings of International Commission for Optics (ICO) Topical Meeting on Optoinformatics /Information Photonics 2006, M.Calvo, A.Pavlov, and J.Jahns, eds. (ICO, 2006), pp.517-518.

Long, X.

J. Yuan and X. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74, 1362-1365 (2003).
[CrossRef]

Moriwaki, K.

K. Iwata, H. Fukuda, and K. Moriwaki, “Autocollimator detecting wave-front tilt using a grating,” in Proceedings of International Commission for Optics (ICO) Topical Meeting on Optoinformatics /Information Photonics 2006, M.Calvo, A.Pavlov, and J.Jahns, eds. (ICO, 2006), pp.517-518.

Mutoh, K.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 3-108.
[CrossRef]

Prakash, S.

Rana, S.

Sasaki, O.

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

Singh, S.

Su, D.-C.

M.-H. Chiu and D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
[CrossRef]

Suzuki, T.

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

Takeda, M.

Yuan, J.

J. Yuan and X. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74, 1362-1365 (2003).
[CrossRef]

Appl. Opt. (3)

Opt. Eng. (2)

M.-H. Chiu and D.-C. Su, “Angle measurement using total-internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
[CrossRef]

T. Suzuki, T. Endo, O. Sasaki, and J. E. Greivenkamp, “Two-dimensional small-rotation-angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006).
[CrossRef]

Rev. Sci. Instrum. (1)

J. Yuan and X. Long, “CCD-area-based autocollimator for precision small-angle measurement,” Rev. Sci. Instrum. 74, 1362-1365 (2003).
[CrossRef]

Other (3)

“Agilent 10770A Angular Interferometer with Agilent 10771A Angular Reflector,” Agilent Technologies user's manual, Chap. 7V.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 3-108.
[CrossRef]

K. Iwata, H. Fukuda, and K. Moriwaki, “Autocollimator detecting wave-front tilt using a grating,” in Proceedings of International Commission for Optics (ICO) Topical Meeting on Optoinformatics /Information Photonics 2006, M.Calvo, A.Pavlov, and J.Jahns, eds. (ICO, 2006), pp.517-518.

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Figures (8)

Fig. 1
Fig. 1

Ordinary autocollimator.

Fig. 2
Fig. 2

Optical system of the proposed tilt sensor.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Grating image on the image sensor. Grating lines are vertical. The results in Fig. 5 are obtained from the data along the horizontal lines (a, b, c, d, and e). Each line corresponds to ten lines of pixels with line numbers shown on the left side.

Fig. 5
Fig. 5

Calculated angle with one-dimensional data along the horizontal lines a, b, c, d, and e in Fig. 4. There are five data points for each setting angle but all five data points are located almost at the same position.

Fig. 6
Fig. 6

Calculated angle with two-dimensional data as shown in Fig. 4. There are two data points for each setting angle but they are located almost at the same position.

Fig. 7
Fig. 7

Simulated standard deviation of the tilting angle measured by the proposed autocollimator (pixel size 7.8 μm ).

Fig. 8
Fig. 8

Simulated standard deviation of the tilting angle measured by the ordinary autocollimator.

Equations (28)

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u = θ z
I ( x , y ) = a + 2 b cos ( 2 π ( x + u ) / p + Φ ) = a + b exp ( i 2 π u / p ) exp ( i 2 π x / p + Φ ) + b exp ( i 2 π u / p ) exp ( i 2 π x / p Φ ) ,
F ( f x , f y ) = a δ ( f x , f y ) + B ( f x + 1 / p , f y ) + B * ( f x 1 / p , f y ) ,
θ = u z = p 2 π z [ tan - 1 { Im { B ( f x , f y ) } Re { B ( f x , f y ) } } Φ ] .
θ = u z = 1 2 π z w ( f x , f y ) 1 f x [ tan - 1 { Im { B ( f x , f y ) } Re { B ( f x , f y ) } } d f x d f y Φ ] , where     w ( f x , f y ) = | B ( f x , f y ) | 2 | B ( f x , f y ) | 2 d f x d f y
T ( X , Y ) = a n exp [ i ( 2 π n / p + ϕ n ) ] .
U ( X ) = exp ( i 2 π X θ / λ ) .
u ( x , y ) = T ( X , Y ) U ( X ) exp [ i 2 π ( x X ) 2 + ( y Y ) 2 + z 2 ) / λ ] d X d Y .
u ( x , y ) = exp [ i 2 π / λ ) { z + ( y Y ) 2 / ( 2 z ) } ] Σ n = a n exp ( i ϕ n ) exp [ i ( 2 π / λ ) ( x 2 + X 2 ) / ( 2 z ) } ] exp [ i ( 2 π { ( n λ / p + θ x / z ) X / λ ] d X d Y .
exp [ i α x 2 ] exp ( i ν x ) d x = π / ( 2 α i ) exp ( i ν 2 / 4 α ) ,
u ( x , y ) = i z λ / 2 exp [ i ( 2 π / λ ) { z + ( y Y ) / ( 2 z ) } ] Σ n = a n exp ( i ϕ n ) exp [ i ( 2 π / λ ) 2 { n λ / p + θ x / z ) 2 } z λ / ( 4 π ) ] .
I ( x , y ) = | u ( x , y ) | 2
I ( x , y ) = Σ n = Σ m = a n a m exp [ i π ( n 2 m 2 ) λ z / p 2 ] exp [ i 2 π ( n m ) ( z θ x } / p } ] exp [ i ( ϕ n ϕ m ) ] .
a n = a n , ϕ n = ϕ n .
I ( x , y ) = Σ n = Σ m = a n a m exp [ i π ( n 2 m 2 ) λ z / p 2 ] cos [ 2 π ( n m ) ( z θ x } / p ( ϕ n ϕ m ) ] .
I ( x , y ) = Σ n = Σ m = a n a m cos [ π ( n 2 m 2 ) λ z / p 2 ] cos [ 2 π ( n m ) ( z θ x } / p ( ϕ n ϕ m ) ] .
I ( x , y ) = Σ n = Σ N = a n a n + N cos [ { π ( 2 N n N 2 ) λ z / p 2 ] cos [ 2 π N ( z θ x } / p ( ϕ n ϕ n + N ) ] .
I 0 ( x , y ) = Σ n = a n 2
I 1 ( x , y ) = 2 Σ n = a n a n + 1 cos [ π ( 2 n 1 ) λ z / p 2 ] cos [ 2 π ( z θ x ) / p ( ϕ n ϕ n + 1 ) ] .
a 0 = 1 , ϕ 0 = 0 , a n = sin ( π n / 2 ) / ( π n / 2 ) , ϕ n = 0 , n = ± 1 , ± 2 , .
a 0 = cos ( κ / 2 ) , ϕ 0 = π / 2 , a n = ( sin κ / 2 ) sin ( π n / 2 ) , ϕ n = 0 , n = ± 1 , ± 2 .
I 1 ( x , y ) = 2 a 1 a 0 cos [ 3 π λ z / p 2 ] cos [ 2 π ( z θ x ) / p ( ϕ 1 ϕ 0 ) ] + 2 a 0 a 1 cos [ π λ z / p 2 ] cos [ 2 π ( z θ x ) / p + ( ϕ 1 ϕ 0 ) ] .
I 1 ( x , y ) = 2 b cos [ 2 π ( z θ x ) / p ] , b = a 1 a 0 cos [ 2 π λ z / p 2 ] cos [ π λ z / p 2 ] .
I 1 ( x , y ) = 2 b sin [ 2 π ( z θ x ) / p ] , b = a 1 a 0 sin [ 2 π λ z / p 2 ] sin [ π λ z / p 2 ] .
u = f θ .
I o ( x , y ) = [ 2 J 1 ( 2 π a γ / λ ) / ( 2 π a γ / λ ) ] 2 where     γ = ( x + u ) 2 + y 2 / f ,
u x = x w ( x , y ) d x d y , u y = y w ( x , y ) d x d y , where     w ( x , y ) = | I ( x , y ) | 2 | I ( x , y ) | 2 d x d y .
θ x = u x / f and θ y = u y / f

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