Abstract

A new technique for the measurement of two-dimensional small angular deviation is presented. A compound prism, which effectively produces a combination of two right-angled prisms in orthogonal directions, and plane reference surfaces have been utilized for the measurement of the orthogonal components of the angular tilt of an incident plane wavefront. Each orthogonal component of the angular tilt is separately measured from the angular rotation of the resultant wedge fringes between two plane wavefronts generated due to splitting of the incident plane wavefront by the corresponding set of right-angled prism and plane reference surface. The technique is shown to have high sensitivity for the measurement of small angle deviation. A monolithic prism interferometer, which is practically insensitive to vibration, is also proposed. Results obtained for the measurement of a known tilt angle are presented.

© 2008 Optical Society of America

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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. J. Yuan, X. Long, and K. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
    [CrossRef]
  3. D. Malacara and O. Harris, “Interferometric measurement of angles,” Appl. Opt. 9, 1630-1633 (1970).
    [CrossRef] [PubMed]
  4. D. Tentori and M. Celaya, “Continuous angle measurement with a Jamin interferometer,” Appl. Opt. 25, 215-220 (1986).
    [CrossRef] [PubMed]
  5. J. Rohlin, “An interferometer for precision angle measurement,” Appl. Opt. 2, 762-763 (1963).
    [CrossRef]
  6. G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646-1651 (1974).
    [CrossRef] [PubMed]
  7. P. Shi and E. Stijns, “New optical method for measuring small angle rotations,” Appl. Opt. 27, 4342-4344 (1988).
    [CrossRef] [PubMed]
  8. P. Shi and E. Stijns, “Improving the linearity of the Michelson interferometric angular measurement by a parameter compensation method,” Appl. Opt. 32, 44-51 (1993).
    [CrossRef] [PubMed]
  9. M. Ikram and G. Hussain, “Michelson interferometer for precision angle measurement,” Appl. Opt. 38, 113-120 (1999).
    [CrossRef]
  10. D. Zheng, X. Wang, and O. Sasaki, “Small dynamic angle measurement using tangent relationship and phase modulating techniques,” Opt. Eng. 46, 093602 (2007).
    [CrossRef]
  11. 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]
  12. 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]
  13. M. H. Chiu and D. C. Su, “Angle measurement using total internal reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
    [CrossRef]
  14. 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]
  15. J. Guo, Z. Zhu, W. Deng, and S. Shen, “Angle measurement using surface plasmon resonance heterodyne interferometry: a new method,” Opt. Eng. 37, 2998-3001 (1998).
    [CrossRef]
  16. S. F. Wang, M. H. Chiu, C. W. Lai, and R. S. Chang, “High sensitivity small angle sensor based on surface plasmon resonance technology and heterodyne interferometry,” Appl. Opt. 45, 6702-6707 (2006).
    [CrossRef] [PubMed]
  17. X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380-6388 (1995).
    [CrossRef] [PubMed]
  18. X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of two dimensional small rotation angles by using orthogonal parallel interference pattern,” Appl. Opt. 35, 5657-5666 (1996).
    [CrossRef] [PubMed]
  19. T. Suzuki, T. Endo, and O. Sasaki, “Two-dimensional small rotation angle measurement using an imaging method,” Opt. Eng. 45, 043604 (2006)
    [CrossRef]
  20. J. Masajada, “Small angle rotation measurement using optical vortex interferometer,” Opt. Commun. 239, 373-381 (2004).
    [CrossRef]
  21. A. P. Massajada, P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Measurement of the small wave tilt using the optical vortex interferometer with the Wollaston compensator,” Appl. Opt. 46, 8039-8044 (2007).
    [CrossRef]
  22. A. P. Massajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave's tilt and orientation using an optical vortex interferometer,” Opt. Eng. 46, 073604 (2007)
    [CrossRef]
  23. O. Sasaki, C. Togashi, and T. Suzuki, “ Two-dimensional rotation angle measurement using a sinusoidal phase modulating laser diode interferometer,” Opt. Eng. 42, 1132-1136 (2003).
    [CrossRef]
  24. Z. Ge and M. Takeda, “High resolution two-dimensional angle measurement technique based on fringe analysis,” Appl. Opt. 42, 6859-6868 (2003).
    [CrossRef] [PubMed]
  25. M. V. R. K. Murty, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 22-23.

2007

D. Zheng, X. Wang, and O. Sasaki, “Small dynamic angle measurement using tangent relationship and phase modulating techniques,” Opt. Eng. 46, 093602 (2007).
[CrossRef]

A. P. Massajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave's tilt and orientation using an optical vortex interferometer,” Opt. Eng. 46, 073604 (2007)
[CrossRef]

A. P. Massajada, P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Measurement of the small wave tilt using the optical vortex interferometer with the Wollaston compensator,” Appl. Opt. 46, 8039-8044 (2007).
[CrossRef]

2006

2005

J. Yuan, X. Long, and K. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

2004

2003

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

O. Sasaki, C. Togashi, and T. Suzuki, “ Two-dimensional rotation angle measurement using a sinusoidal phase modulating laser diode interferometer,” Opt. Eng. 42, 1132-1136 (2003).
[CrossRef]

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

1999

1998

J. Guo, Z. Zhu, W. Deng, and S. Shen, “Angle measurement using surface plasmon resonance heterodyne interferometry: a new method,” Opt. Eng. 37, 2998-3001 (1998).
[CrossRef]

1997

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

1996

1995

1993

1992

1988

1986

1974

1970

1963

Borwinska, M.

A. P. Massajada, P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Measurement of the small wave tilt using the optical vortex interferometer with the Wollaston compensator,” Appl. Opt. 46, 8039-8044 (2007).
[CrossRef]

A. P. Massajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave's tilt and orientation using an optical vortex interferometer,” Opt. Eng. 46, 073604 (2007)
[CrossRef]

Celaya, M.

Chang, R. S.

Chapman, G. D.

Chiu, M. H.

Dai, X.

Deng, W.

J. Guo, Z. Zhu, W. Deng, and S. Shen, “Angle measurement using surface plasmon resonance heterodyne interferometry: a new method,” Opt. Eng. 37, 2998-3001 (1998).
[CrossRef]

Dubik, B.

A. P. Massajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave's tilt and orientation using an optical vortex interferometer,” Opt. Eng. 46, 073604 (2007)
[CrossRef]

Endo, T.

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

Ge, Z.

Greivenkamp, J. E.

Guo, J.

J. Guo, Z. Zhu, W. Deng, and S. Shen, “Angle measurement using surface plasmon resonance heterodyne interferometry: a new method,” Opt. Eng. 37, 2998-3001 (1998).
[CrossRef]

Harris, O.

Huang, P. S.

Hussain, G.

Ikram, M.

Kamada, O.

Kiyono, S.

Kurzynowski, P.

Lai, C. W.

Long, X.

J. Yuan, X. Long, and K. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

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

Malacara, D.

Masajada, J.

J. Masajada, “Small angle rotation measurement using optical vortex interferometer,” Opt. Commun. 239, 373-381 (2004).
[CrossRef]

Massajada, A. P.

A. P. Massajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave's tilt and orientation using an optical vortex interferometer,” Opt. Eng. 46, 073604 (2007)
[CrossRef]

A. P. Massajada, P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Measurement of the small wave tilt using the optical vortex interferometer with the Wollaston compensator,” Appl. Opt. 46, 8039-8044 (2007).
[CrossRef]

Murty, M. V. R. K.

M. V. R. K. Murty, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 22-23.

Ni, J.

Rohlin, J.

Sasaki, O.

D. Zheng, X. Wang, and O. Sasaki, “Small dynamic angle measurement using tangent relationship and phase modulating techniques,” Opt. Eng. 46, 093602 (2007).
[CrossRef]

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

O. Sasaki, C. Togashi, and T. Suzuki, “ Two-dimensional rotation angle measurement using a sinusoidal phase modulating laser diode interferometer,” Opt. Eng. 42, 1132-1136 (2003).
[CrossRef]

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

X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380-6388 (1995).
[CrossRef] [PubMed]

Shen, S.

J. Guo, Z. Zhu, W. Deng, and S. Shen, “Angle measurement using surface plasmon resonance heterodyne interferometry: a new method,” Opt. Eng. 37, 2998-3001 (1998).
[CrossRef]

Shi, P.

Stijns, E.

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

O. Sasaki, C. Togashi, and T. Suzuki, “ Two-dimensional rotation angle measurement using a sinusoidal phase modulating laser diode interferometer,” Opt. Eng. 42, 1132-1136 (2003).
[CrossRef]

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

X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, “Measurement of small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380-6388 (1995).
[CrossRef] [PubMed]

Takeda, M.

Tentori, D.

Togashi, C.

O. Sasaki, C. Togashi, and T. Suzuki, “ Two-dimensional rotation angle measurement using a sinusoidal phase modulating laser diode interferometer,” Opt. Eng. 42, 1132-1136 (2003).
[CrossRef]

Wang, S. F.

Wang, X.

D. Zheng, X. Wang, and O. Sasaki, “Small dynamic angle measurement using tangent relationship and phase modulating techniques,” Opt. Eng. 46, 093602 (2007).
[CrossRef]

Wozniak, W. A.

Yang, K.

J. Yuan, X. Long, and K. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

Yuan, J.

J. Yuan, X. Long, and K. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

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

Zheng, D.

D. Zheng, X. Wang, and O. Sasaki, “Small dynamic angle measurement using tangent relationship and phase modulating techniques,” Opt. Eng. 46, 093602 (2007).
[CrossRef]

Zhu, Z.

J. Guo, Z. Zhu, W. Deng, and S. Shen, “Angle measurement using surface plasmon resonance heterodyne interferometry: a new method,” Opt. Eng. 37, 2998-3001 (1998).
[CrossRef]

Appl. Opt.

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

G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646-1651 (1974).
[CrossRef] [PubMed]

D. Tentori and M. Celaya, “Continuous angle measurement with a Jamin interferometer,” Appl. Opt. 25, 215-220 (1986).
[CrossRef] [PubMed]

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

P. Shi and E. Stijns, “Improving the linearity of the Michelson interferometric angular measurement by a parameter compensation method,” Appl. Opt. 32, 44-51 (1993).
[CrossRef] [PubMed]

M. Ikram and G. Hussain, “Michelson interferometer for precision angle measurement,” Appl. Opt. 38, 113-120 (1999).
[CrossRef]

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 small rotation angles by using a parallel interference pattern,” Appl. Opt. 34, 6380-6388 (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 pattern,” Appl. Opt. 35, 5657-5666 (1996).
[CrossRef] [PubMed]

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]

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]

S. F. Wang, M. H. Chiu, C. W. Lai, and R. S. Chang, “High sensitivity small angle sensor based on surface plasmon resonance technology and heterodyne interferometry,” Appl. Opt. 45, 6702-6707 (2006).
[CrossRef] [PubMed]

A. P. Massajada, P. Kurzynowski, W. A. Wozniak, and M. Borwinska, “Measurement of the small wave tilt using the optical vortex interferometer with the Wollaston compensator,” Appl. Opt. 46, 8039-8044 (2007).
[CrossRef]

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

Opt. Commun.

J. Masajada, “Small angle rotation measurement using optical vortex interferometer,” Opt. Commun. 239, 373-381 (2004).
[CrossRef]

Opt. Eng.

A. P. Massajada, M. Borwinska, and B. Dubik, “Reconstruction of a plane wave's tilt and orientation using an optical vortex interferometer,” Opt. Eng. 46, 073604 (2007)
[CrossRef]

O. Sasaki, C. Togashi, and T. Suzuki, “ Two-dimensional rotation angle measurement using a sinusoidal phase modulating laser diode interferometer,” Opt. Eng. 42, 1132-1136 (2003).
[CrossRef]

D. Zheng, X. Wang, and O. Sasaki, “Small dynamic angle measurement using tangent relationship and phase modulating techniques,” Opt. Eng. 46, 093602 (2007).
[CrossRef]

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

J. Guo, Z. Zhu, W. Deng, and S. Shen, “Angle measurement using surface plasmon resonance heterodyne interferometry: a new method,” Opt. Eng. 37, 2998-3001 (1998).
[CrossRef]

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

Rev. Sci. Instrum.

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

J. Yuan, X. Long, and K. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

Other

M. V. R. K. Murty, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 22-23.

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Figures (15)

Fig. 1
Fig. 1

Optical schematic showing the generation of interfering plane wavefronts with a RP ( 90 ° , 45 ° ) and a plane reference surface.

Fig. 2
Fig. 2

Optical schematic for explaining the principle of tilt measurement.

Fig. 3
Fig. 3

Resultant wedge and fringe direction for combinations of ω x and ω y . OA, OA’, OB, and OB’ represent the resultant wedge directions for the orthogonal pairs ( ω x , ω y ), ( ω x , ω y ), ( ω x , ω y ), and ( ω x , ω y ), respectively. F 1 F 1 indicates the fringe direction for ( ω x , ω y ), and ( ω x , ω y ). F 2 F 2 shows the fringe direction for ( ω x , ω y ), and ( ω x , ω y ).

Fig. 4
Fig. 4

Plan and side views of CP.

Fig. 5
Fig. 5

Ray paths in CP.

Fig. 6
Fig. 6

Optical schematic of the setup for the measurement of tilt/angular errors of the carriage of an air-bearing-type linear translation stage.

Fig. 7
Fig. 7

Plot of θ versus α for d = 25 mm , 20 mm , 15 mm .

Fig. 8
Fig. 8

Plot of S versus α for d = 25 mm , 20 mm , 15 mm , 10 mm , 5 mm , 1 mm .

Fig. 9
Fig. 9

Plot of δ θ versus α for d = 25 mm , 20 mm , 15 mm ( d d = 0.5 mm and d α = 0.5 ° )

Fig. 10
Fig. 10

Grabbed fringes in the initial setting.

Fig. 11
Fig. 11

Rotation of fringes due to wavefront tilt.

Fig. 12
Fig. 12

Typical interferograms in V of the CP. (a) Initial fringes with d 11.25 mm . Angular rotations of the fringes with (b) θ 1 0.43 " , (c) θ 2 0.93 " , (d) θ 3 1.74 " , (e) θ 4 3.77 " , and (f) θ 5 5.27 " , respectively.

Fig. 13
Fig. 13

Plot of the mean values of α versus θ for d 11.25 mm and 6.25 mm with error bars for θ.

Fig. 14
Fig. 14

Typical interferograms in V for the measurement point θ 3 . (a) Initial fringes with d 11.25 mm . (b) Fringe rotation with θ 3 . Angular rotation of the fringes with (c) ( θ 3 + δ θ ) and (d) ( θ 3 δ θ ).

Fig. 15
Fig. 15

Plot of experimentally obtained values of S versus α for d 11.25 mm .

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

ω = ( ω x 2 + ω y 2 ) ,
tan α = ( ω y / ω x ) .
θ = [ ( λ tan α ) / 2 d ] .
tan α = [ ( ω y + ω a sin β ) / ( ω a cos β ) ] .
θ a = ( θ x 2 + θ y 2 ) ,
tan α a = ( θ y / θ x ) .
S = δ α / δ θ = ( 2 d / λ ) cos 2 α .
δ θ α = ( δ θ / δ α ) d α = ( λ / 2 d ) ( 1 / cos 2 α ) d α ,
δ θ d = ( δ θ / δ d ) d d = ( λ / 2 d 2 ) tan α d d ,
δ θ = δ θ α + δ θ d ,

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