Abstract

We propose what we believe to be a novel method for highly accurate three-dimensional (3D) angle measurement based on propagation vector analysis of digital holography. Three-dimensional rotations in space can be achieved by use of a CCD camera and a multifacet object, which reflects an incident wave into different directions. The propagation vectors of the reflected waves from the object can be extracted by analyzing the object spectrum of the recorded hologram. Any small rotation of the object will induce a change in the propagation vectors in space, which can then be used for 3D angle measurement. Experimental results are presented to verify the idea.

© 2007 Optical Society of America

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  1. http://www.warrenknight.com/WKClinometers.html.
  2. P. R. Yoder, Jr., E. R. Schlesinger, and J. L. Chickvary, "Active annular-beam laser autocollimator system," Appl. Opt. 14, 1890-1895 (1975).
    [CrossRef] [PubMed]
  3. G. G. Luther and R. D. Deslattes, "Single axis photoelectronic autocollimator," Rev. Sci. Instrum. 55, 747-750 (1984).
    [CrossRef]
  4. J. Rohlin, "An interferometer for precision angle measurement," Appl. Opt. 2, 762-763 (1963).
    [CrossRef]
  5. D. Malacara and O. Harris, "Interferometric measurement of angles," Appl. Opt. 9, 1630-1633 (1970).
    [CrossRef] [PubMed]
  6. P. Shi and E. Stijns, "New optical methods for measuring small-angle rotations," Appl. Opt. 27, 4342-4344 (1988).
    [CrossRef] [PubMed]
  7. P. S. Huang, S. Kiyono, and O. Kamada "Angle measurement based on the internal reflection effect: a new method," Appl. Opt. 31, 6047-6055 (1992).
    [CrossRef] [PubMed]
  8. P. S. Huang and J. Ni, "Angle measurement based on the internal-reflection effect and the use of right-angle prisms," Appl. Opt. 34, 4976-4981 (1995).
    [CrossRef] [PubMed]
  9. M. H. Chiu, S. F. Wang, and R. S. Chang, "Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry," Appl. Opt. 43, 5438-5442 (2004).
    [CrossRef] [PubMed]
  10. Z. Ge and M. Takeda, "High-resolution two-dimensional angle measurement technique based on fringe analysis," Appl. Opt. 42, 6859-6868 (2003).
    [CrossRef] [PubMed]
  11. B. Rose, H. Imam, and S. G. Hanson, "Non-contact laser speckle sensor for measuring one- and two-dimensional angular displacement," J. Opt. 29, 115-120 (1998).
    [CrossRef]
  12. X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, "Measurement of two-dimensional small rotation angles by using orthogonal parallel interference patterns," Appl. Opt. 35, 5657-5666 (1996).
    [CrossRef] [PubMed]
  13. C. H. Liu, W. Y. Jywe, and S. C. Tzeng, "Simple three-dimensional laser angle sensor for three-dimensional small-angle measurement," Appl. Opt. 43, 2840-2845 (2004).
    [CrossRef] [PubMed]
  14. L. Yu and M. Kim, "Wavelength scanning digital interference holography for variable tomographic scanning," Opt. Express 13, 5621-5627 (2005).
    [CrossRef] [PubMed]
  15. L. Yu and M. Kim, "Variable tomographic scanning with wavelength scanning digital interference holography," Opt. Commun 260, 462-468 (2006).
    [CrossRef]
  16. U. Schnars and W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
    [CrossRef] [PubMed]
  17. E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
    [CrossRef]
  18. L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
    [CrossRef]
  19. T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, "Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation," Appl. Opt. 45, 851-863 (2006).
    [CrossRef] [PubMed]
  20. T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, "Numerical parametric lens for shifting, magnification and complete aberration compensation in digital holographic microscopy," J. Opt. Soc. Am. A 23, 3177-3190 (2006).
    [CrossRef]

2007 (1)

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

2006 (3)

2005 (1)

2004 (2)

2003 (1)

2000 (1)

1998 (1)

B. Rose, H. Imam, and S. G. Hanson, "Non-contact laser speckle sensor for measuring one- and two-dimensional angular displacement," J. Opt. 29, 115-120 (1998).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

1992 (1)

1988 (1)

1984 (1)

G. G. Luther and R. D. Deslattes, "Single axis photoelectronic autocollimator," Rev. Sci. Instrum. 55, 747-750 (1984).
[CrossRef]

1975 (1)

1970 (1)

1963 (1)

Alfieri, D.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Aspert, N.

Bourquin, S.

Chang, R. S.

Charrière, F.

Chickvary, J. L.

Chiu, M. H.

Colomb, T.

Cuche, E.

Dai, X.

De Petrocellis, L.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Depeursinge, C.

Deslattes, R. D.

G. G. Luther and R. D. Deslattes, "Single axis photoelectronic autocollimator," Rev. Sci. Instrum. 55, 747-750 (1984).
[CrossRef]

Ferraro, P.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Finizio, A.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Ge, Z.

Greivenkamp, J. E.

Grilli, S.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Hanson, S. G.

B. Rose, H. Imam, and S. G. Hanson, "Non-contact laser speckle sensor for measuring one- and two-dimensional angular displacement," J. Opt. 29, 115-120 (1998).
[CrossRef]

Harris, O.

Huang, P. S.

Imam, H.

B. Rose, H. Imam, and S. G. Hanson, "Non-contact laser speckle sensor for measuring one- and two-dimensional angular displacement," J. Opt. 29, 115-120 (1998).
[CrossRef]

Jüptner, W.

Jywe, W. Y.

Kamada, O.

Kim, M.

L. Yu and M. Kim, "Variable tomographic scanning with wavelength scanning digital interference holography," Opt. Commun 260, 462-468 (2006).
[CrossRef]

L. Yu and M. Kim, "Wavelength scanning digital interference holography for variable tomographic scanning," Opt. Express 13, 5621-5627 (2005).
[CrossRef] [PubMed]

Kiyono, S.

Kühn, J.

Liu, C. H.

Luther, G. G.

G. G. Luther and R. D. Deslattes, "Single axis photoelectronic autocollimator," Rev. Sci. Instrum. 55, 747-750 (1984).
[CrossRef]

Malacara, D.

Marian, A.

Marquet, P.

Miccio, L.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Montfort, F.

Ni, J.

Nicola, S. D.

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

Rohlin, J.

Rose, B.

B. Rose, H. Imam, and S. G. Hanson, "Non-contact laser speckle sensor for measuring one- and two-dimensional angular displacement," J. Opt. 29, 115-120 (1998).
[CrossRef]

Sasaki, O.

Schlesinger, E. R.

Schnars, U.

Shi, P.

Stijns, E.

Suzuki, T.

Takeda, M.

Tzeng, S. C.

Wang, S. F.

Yoder, P. R.

Yu, L.

L. Yu and M. Kim, "Variable tomographic scanning with wavelength scanning digital interference holography," Opt. Commun 260, 462-468 (2006).
[CrossRef]

L. Yu and M. Kim, "Wavelength scanning digital interference holography for variable tomographic scanning," Opt. Express 13, 5621-5627 (2005).
[CrossRef] [PubMed]

Appl. Opt. (13)

D. Malacara and O. Harris, "Interferometric measurement of angles," Appl. Opt. 9, 1630-1633 (1970).
[CrossRef] [PubMed]

P. R. Yoder, Jr., E. R. Schlesinger, and J. L. Chickvary, "Active annular-beam laser autocollimator system," Appl. Opt. 14, 1890-1895 (1975).
[CrossRef] [PubMed]

P. Shi and E. Stijns, "New optical methods for measuring small-angle rotations," Appl. Opt. 27, 4342-4344 (1988).
[CrossRef] [PubMed]

U. Schnars and W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
[CrossRef] [PubMed]

P. S. Huang and J. Ni, "Angle measurement based on the internal-reflection effect and the use of right-angle prisms," Appl. Opt. 34, 4976-4981 (1995).
[CrossRef] [PubMed]

X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, "Measurement of two-dimensional small rotation angles by using orthogonal parallel interference patterns," Appl. Opt. 35, 5657-5666 (1996).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, "Spatial filtering for zero-order and twin-image elimination in digital off-axis holography," Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

P. S. Huang, S. Kiyono, and O. Kamada "Angle measurement based on the internal reflection effect: a new method," Appl. Opt. 31, 6047-6055 (1992).
[CrossRef] [PubMed]

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

C. H. Liu, W. Y. Jywe, and S. C. Tzeng, "Simple three-dimensional laser angle sensor for three-dimensional small-angle measurement," Appl. Opt. 43, 2840-2845 (2004).
[CrossRef] [PubMed]

M. H. Chiu, S. F. Wang, and R. S. Chang, "Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry," Appl. Opt. 43, 5438-5442 (2004).
[CrossRef] [PubMed]

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, "Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation," Appl. Opt. 45, 851-863 (2006).
[CrossRef] [PubMed]

J. Rohlin, "An interferometer for precision angle measurement," Appl. Opt. 2, 762-763 (1963).
[CrossRef]

Appl. Phys. Lett. (1)

L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, "Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram," Appl. Phys. Lett. 90, 041104 (2007).
[CrossRef]

J. Opt. (1)

B. Rose, H. Imam, and S. G. Hanson, "Non-contact laser speckle sensor for measuring one- and two-dimensional angular displacement," J. Opt. 29, 115-120 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun (1)

L. Yu and M. Kim, "Variable tomographic scanning with wavelength scanning digital interference holography," Opt. Commun 260, 462-468 (2006).
[CrossRef]

Opt. Express (1)

Rev. Sci. Instrum. (1)

G. G. Luther and R. D. Deslattes, "Single axis photoelectronic autocollimator," Rev. Sci. Instrum. 55, 747-750 (1984).
[CrossRef]

Other (1)

http://www.warrenknight.com/WKClinometers.html.

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Figures (6)

Fig. 1
Fig. 1

(Color online) Apparatus for 3D angle measurement.

Fig. 2
Fig. 2

(Color online) (a) Top view of the object in an absolute x–y–z frame; (b) propagation vectors of the two-facet object; (c) local frame {B} attached to the object; (d) shift of the propagation vectors due to 3D rotations in space.

Fig. 3
Fig. 3

(Color online) (a) Hologram of a split resolution target; (b) spectrum of the hologram; (c) filtered spectrum component of the right-down part of the resolution target; (d), (e) reconstructed amplitude and phase images, respectively, from (c); (f) obtained by shifting (c) to near the center of the spectrum domain; (g) reconstructed phase map from (f) containing fringes with lower frequency; (h), (i) reconstructed amplitude and phase images, respectively, of the up-left part of the target.

Fig. 4
Fig. 4

(a) Hologram or interferogram of the two-facet object; (b) spectrum of (a); (c) spectrum after rotation; (d), (e), (f) spectrum, phase map, and unwrapped phase map, respectively, of the right facet; (g), (h), (i) those of the left facet.

Fig. 5
Fig. 5

(Color online) Angle measurement around the x axis: (a) calibration test of the proposed method with an autocollimator; (b) comparison with the fringe analysis method [10].

Fig. 6
Fig. 6

(Color online) Angle measurement around the z axis: calibration test of the proposed method within a measurement range of 10°; the inset shows a calibration error of ± 0 .005 ° by use of an autocollimator.

Equations (11)

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E ( x , y , z ) = S ( ξ , η ; z ) exp [ i 2 π ( ξ x + η y ) ] d ξ d η ,
n = [ cos α , cos β , cos γ ] T = λ [ ξ , η , ζ ] T ,
z A B = ( n 1 + n 2 ) / | n 1 + n 2 | ,
x A B = ( n 1 n 2 ) / | n 1 n 2 | ,
y A B = z A B × x A B ,
R B A = R x A ( θ 3 ) R y A ( θ 2 ) R z A ( θ 1 ) ,
θ 2 = arctan ( t 13 , t 23 2 + t 33 2 ) ,
θ 3 = arctan ( t 23 / cos θ 2 , t 33 / cos θ 2 ) ,
θ 1 = arctan ( t 12 / cos θ 2 , t 11 / cos θ 2 ) ,
ξ 1 = X 1 N s + ξ 1 r , η 1 = Y 1 M s + η 1 r ,
θ x ,max = sin 1 ( λ M 2 Δ x ) ,

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