Abstract

In this paper, a high-sensitivity total-internal-reflection (TIR) heterodyne interferometer is proposed for measuring small angles. In the proposed interferometer, a half-wave plate and two quarter-wave plates that exhibit specific optic-axis azimuths are combined to form a phase shifter. When a rhomboid prism is placed between the phase shifter and an analyzer that exhibits suitable transmission-axis azimuth, it shifts and enhances the phase difference of the s- and p-polarization states at double TIR. The enhanced phase difference is dependent on the incident angle; thus small angles can be easily and accurately measured by estimating the phase difference. The experimental results demonstrate the feasibility of this method. Angular resolution and sensitivity levels superior to 1.2×104deg (2.1×106rad) and 100 (deg/deg), respectively, were attainable in a dynamic range of 0.5 deg.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
    [CrossRef]
  2. G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
    [CrossRef]
  3. P. Shi and E. Stijns, “New optical methods for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
    [CrossRef]
  4. 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]
  5. 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]
  6. P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
    [CrossRef]
  7. A. Y. Zhang and P. S. Huang, “Total internal reflection for precision small-angle measurement,” Appl. Opt. 40, 1617–1622 (2001).
    [CrossRef]
  8. W. Y. Zhang, J. Zhang, and L. Y. Wu, “Small-angle measurement of laser beam steering based on total Internal-reflection effect,” J. Phys. Conf. Ser. 48, 766–770 (2006).
    [CrossRef]
  9. M. H. Chiu and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104–7106 (1997).
    [CrossRef]
  10. W. D. Zhou and L. L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37, 5957–5963 (1998).
    [CrossRef]
  11. W. D. Zhou and L. L. Cai, “Improved angle interferometer based on total internal reflection,” Appl. Opt. 38, 1179–1185 (1999).
    [CrossRef]
  12. 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]
  13. S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48, 2566–2573 (2009).
    [CrossRef]
  14. E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).
  15. S. S. Huard, Polarization of Light (Wiley, 1997).
  16. N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
    [CrossRef]
  17. J. M. De Freitas and M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
    [CrossRef]
  18. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
    [CrossRef]
  19. A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
    [CrossRef]

2009 (1)

2006 (1)

W. Y. Zhang, J. Zhang, and L. Y. Wu, “Small-angle measurement of laser beam steering based on total Internal-reflection effect,” J. Phys. Conf. Ser. 48, 766–770 (2006).
[CrossRef]

2004 (1)

2001 (1)

1999 (1)

1998 (1)

1997 (1)

1996 (1)

1993 (3)

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]

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

J. M. De Freitas and M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
[CrossRef]

1992 (2)

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[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]

1990 (1)

A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

1988 (1)

1984 (1)

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

1982 (1)

A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
[CrossRef]

Bobroff, N.

A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Cai, L. L.

Chang, R. S.

Chen, W. W.

Chiu, M. H.

De Freitas, J. M.

J. M. De Freitas and M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
[CrossRef]

Deslattes, R. D.

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

Ennos, A. E.

A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).

Hetrick, P. S.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Hou, W.

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Huang, P. S.

Huard, S. S.

S. S. Huard, Polarization of Light (Wiley, 1997).

Kamada, O.

Kao, F. H.

Kiyono, S.

Kramar, J. A.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Luther, G. G.

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

Ni, J.

Oldham, N. M.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Player, M. A.

J. M. De Freitas and M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
[CrossRef]

Rosenbluth, A. E.

A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Shi, P.

Stijns, E.

Su, D. C.

Teague, E. C.

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Towler, W. R.

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

Virdee, M. S.

A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
[CrossRef]

Wang, S. F.

Wilkening, G.

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

Wu, L. Y.

W. Y. Zhang, J. Zhang, and L. Y. Wu, “Small-angle measurement of laser beam steering based on total Internal-reflection effect,” J. Phys. Conf. Ser. 48, 766–770 (2006).
[CrossRef]

Zhang, A. Y.

Zhang, J.

W. Y. Zhang, J. Zhang, and L. Y. Wu, “Small-angle measurement of laser beam steering based on total Internal-reflection effect,” J. Phys. Conf. Ser. 48, 766–770 (2006).
[CrossRef]

Zhang, W. Y.

W. Y. Zhang, J. Zhang, and L. Y. Wu, “Small-angle measurement of laser beam steering based on total Internal-reflection effect,” J. Phys. Conf. Ser. 48, 766–770 (2006).
[CrossRef]

Zhou, W. D.

Appl. Opt. (10)

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

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]

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]

P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
[CrossRef]

A. Y. Zhang and P. S. Huang, “Total internal reflection for precision small-angle measurement,” Appl. Opt. 40, 1617–1622 (2001).
[CrossRef]

M. H. Chiu and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104–7106 (1997).
[CrossRef]

W. D. Zhou and L. L. Cai, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37, 5957–5963 (1998).
[CrossRef]

W. D. Zhou and L. L. Cai, “Improved angle interferometer based on total internal reflection,” Appl. Opt. 38, 1179–1185 (1999).
[CrossRef]

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]

S. F. Wang, M. H. Chiu, W. W. Chen, F. H. Kao, and R. S. Chang, “Small-displacement sensing system based on multiple total internal reflections in heterodyne interferometry,” Appl. Opt. 48, 2566–2573 (2009).
[CrossRef]

J. Phys. Conf. Ser. (1)

W. Y. Zhang, J. Zhang, and L. Y. Wu, “Small-angle measurement of laser beam steering based on total Internal-reflection effect,” J. Phys. Conf. Ser. 48, 766–770 (2006).
[CrossRef]

Meas. Sci. Technol. (1)

J. M. De Freitas and M. A. Player, “Importance of rotational beam alignment in the generation of second harmonic errors in laser heterodyne interferometry,” Meas. Sci. Technol. 4, 1173–1176 (1993).
[CrossRef]

Precis. Eng. (4)

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[CrossRef]

A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
[CrossRef]

N. M. Oldham, J. A. Kramar, P. S. Hetrick, and E. C. Teague, “Electronic limitations in phase meters for heterodyne interferometry,” Precis. Eng. 15, 173–179 (1993).
[CrossRef]

Rev. Sci. Instrum. (1)

G. G. Luther, R. D. Deslattes, and W. R. Towler, “Single axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747–750 (1984).
[CrossRef]

Other (2)

E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).

S. S. Huard, Polarization of Light (Wiley, 1997).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Total internal reflection in a small-angle measurement apparatus.

Fig. 2.
Fig. 2.

Phase difference ϕ versus the incident angle θi for 0degβ90deg.

Fig. 3.
Fig. 3.

Schematic diagram of small-angle measurements.

Fig. 4.
Fig. 4.

Measurement results and theoretical curves of ϕ versus θi.

Fig. 5.
Fig. 5.

Angular resolution δθi versus the incident angle θi.

Fig. 6.
Fig. 6.

Sensitivity S versus the incident angle θi.

Fig. 7.
Fig. 7.

Phase difference ϕ versus the incident angle θi without the phase shifter.

Equations (19)

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

Ei=(cosθpsinθp).
θ1=45°+sin1(sinθinp).
Et=(cos2βsinβcosβsinβcosβsin2β)(taexp(iδt/2)00tbexp(iδt/2))(100i)12(1ii1)×(cosΔsinΔsinΔcosΔ)(cosθpsinθp)=(At1cosθpexp(iϕ)+At2sinθpexp(iπ/2))(cosβsinβ),
At1={12[(tptpcosβ)2+(tstssinβ)2+tptptstssin2β·cos(2Δ+δt)]}1/2,
At2={12[(tptpcosβ)2+(tstssinβ)2tptptstssin2β·cos(2Δ+δt)]}1/2,
ϕ=tan1[tan(45°σ)·tan(Δ+δt/2)]tan1[tan(45°+σ)·tan(Δ+δt/2)].
δt=4tan1{[sin2[45°+sin1(sinθi/np)](1/np)2]1/2tan[45°+sin1(sinθi/np)]·sin[45°+sin1(sinθi/np)]},
σ=tan1(tststptptanβ),
tp=2cosθinpcosθi+[1(sinθi/np)2]1/2,
ts=2cosθicosθi+np[1(sinθi/np)2]1/2,
tp=2np[1(sinθi/np)2]1/2npcosθi+[1(sinθi/np)2]1/2,
ts=2np[1(sinθi/np)2]1/2cosθi+np[1(sinθi/np)2]1/2.
δt,max=4tan1(np212np).
Ir=12[2(cosαcosθp)2+2(sinαsinθp)2+sin(2θp)sin(2α)cos(ωtϕBS)],
It=|Et|2=[(At1cosθp)2+(At2sinθp)2+At1At2sin2θp·cos(ωt+ϕπ/2)],
ψ=ϕ+ϕBSπ/2
δθi=|θiϕ|δϕ+|θiβ|δΔ+|θiβ|δβ,
S=|ϕθi|.
ϕ=tan1[tan(45°σ)·tan(δt/2)]tan1[tan(45°+σ)·tan(δt/2)].

Metrics