Abstract

We describe a new method of angle measurement that is based on the internal-reflection effect at an air–glass boundary. The method uses a differential detection scheme to largely reduce the inherent nonlinearity of the reflectance versus the angle of incidence in internal reflection. With nonlinearity reduced, the displacement of the angle of incidence can be determined accurately by measuring the reflectance. The resolution and measurement range are determined by the initial angle of incidence, the polarization state of the light, and the number of reflections. Compared with interferometers and autocollimators, this method has the advantage of a simple sensor design for applications ranging from a wide measurement range to extremely high resolution. Other advantages are compact size, simple structure, and low cost. A theoretical analysis of the method and some experimental results of a prototype sensor are presented. The possible applications of the method are also discussed.

© 1992 Optical Society of America

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References

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  1. J. Rohlin, “An interferometer for precision angle measurement,” Appl. Opt. 2, 762–763 (1963).
    [CrossRef]
  2. D. Malacara, O. Harris, “Interferometric measurement of angles,” Appl. Opt. 9, 1630–1633 (1970).
    [CrossRef] [PubMed]
  3. G. D. Chapman, “Interferometric angular measurement,” Appl. Opt. 13, 1646–1651 (1974).
    [CrossRef] [PubMed]
  4. Pan Shi, Erik Stijns, “New optical method for measuring small-angle rotations,” Appl. Opt. 27, 4342–4344 (1988).
    [CrossRef] [PubMed]
  5. T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
    [CrossRef]
  6. R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. (1983), p. 3.
  7. F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
    [CrossRef]
  8. A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surface by laser autocollimation,” J. Prec. Eng. 4, 5–8 (1982).
    [CrossRef]
  9. J. Ni, P. S. Huang, S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” ASME J. Eng. Ind. (to be published).
  10. P. R. Yoder, E. R. Schlesinger, J. L. Chickvary, “Active annular-beam laser auto-collimator system,” Appl. Opt. 14, 1890–1895 (1975).
    [CrossRef] [PubMed]
  11. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1989), Chap. 4, p. 100.

1990 (1)

T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
[CrossRef]

1988 (1)

1983 (2)

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. (1983), p. 3.

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

1982 (1)

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

1975 (1)

1974 (1)

1970 (1)

1963 (1)

Chapman, G. D.

Chickvary, J. L.

Ennos, A. E.

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

Harris, O.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1989), Chap. 4, p. 100.

Huang, P. S.

J. Ni, P. S. Huang, S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” ASME J. Eng. Ind. (to be published).

Malacara, D.

Ni, J.

J. Ni, P. S. Huang, S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” ASME J. Eng. Ind. (to be published).

Quenelle, R. C.

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. (1983), p. 3.

Rohlin, J.

Schlesinger, E. R.

Schuda, F. J.

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

Shi, Pan

Stijns, Erik

Takano, T.

T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
[CrossRef]

Virdee, M. S.

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

Wu, S. M.

J. Ni, P. S. Huang, S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” ASME J. Eng. Ind. (to be published).

Wuerz, L. J.

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. (1983), p. 3.

Yoder, P. R.

Yonehara, S.

T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
[CrossRef]

Appl. Opt. (5)

Hewlett-Packard J. (1)

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J. (1983), p. 3.

IEEE Trans. Aerosp. Electron. Syst. (1)

T. Takano, S. Yonehara, “Basic investigations on an angle measurement system using a laser,” IEEE Trans. Aerosp. Electron. Syst. 26, 657–662 (1990).
[CrossRef]

J. Prec. Eng. (1)

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

Rev. Sci. Instrum. (1)

F. J. Schuda, “High-precision, wide-range, dual-axis, angle monitoring system,” Rev. Sci. Instrum. 54, 1648–1652 (1983).
[CrossRef]

Other (2)

J. Ni, P. S. Huang, S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” ASME J. Eng. Ind. (to be published).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1989), Chap. 4, p. 100.

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Figures (18)

Fig. 1
Fig. 1

Internal reflection of light beams at an air–glass interface.

Fig. 2
Fig. 2

Reflectance of internal reflection at an air–glass interface. Angle of incidence is in degrees.

Fig. 3
Fig. 3

Reflectance and linearized reflectance. Angular displacement is in degrees.

Fig. 4
Fig. 4

Principle of the differential detection method. Prism 1 and prism 2 are identical.

Fig. 5
Fig. 5

Linearized reflectances of s-polarized light for different initial reflectances; Rs0, initial reflectance of s-polarized light. Angular displacement is in degrees.

Fig. 6
Fig. 6

Linearized reflectances of p-polarized light for different initial reflectances; Rp0, initial reflectance of p-polarized light. Angular displacement is in degrees.

Fig. 7
Fig. 7

Sensitivity of the linearized reflectance. Initial angle of incidence is in degrees.

Fig. 8
Fig. 8

Nonlinearity error of the linearized reflectance of s-polarized light. Range is the percentage of the total measurable range and initial angle of incidence is in degrees.

Fig. 9
Fig. 9

Nonlinearity error of the linearized reflectance of p-polarized light. Range is the percentage of the total measurable range and initial angle of incidence is in degrees.

Fig. 10
Fig. 10

Residual error of the linearized reflectance of s-polarized light fit with a third-order polynomial. Initial angle of incidence is in degrees.

Fig. 11
Fig. 11

Residual error of the linearized reflectance of p-polarized light fit with a third-order polynominal. Initial angle of incidence is in degrees.

Fig. 12
Fig. 12

Linearized reflectances for multiple reflections. Angular displacement is in degrees.

Fig. 13
Fig. 13

Measurement error caused by the angular displacement in the plane perpendicular to the plane of incidence.

Fig. 14
Fig. 14

Optical schematic of a prototype sensor.

Fig. 15
Fig. 15

Characteristic curve of the prototype sensor.

Fig. 16
Fig. 16

Central part of the characteristic curve and its residual error of curve fitting.

Fig. 17
Fig. 17

Noise and drift of the prototype sensor. Time is in minutes.

Fig. 18
Fig. 18

Prototype sensor for angle measurement.

Equations (19)

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R s = [ sin ( θ i - θ t ) sin ( θ i + θ t ) ] 2 ,
R p = [ tan ( θ i - θ t ) tan ( θ i + θ t ) ] 2 ,
n i sin θ i = n t sin θ t .
R s ( Δ θ ) = a 0 + a 1 Δ θ + a 2 Δ θ 2 + a 3 Δ θ 3 + ,
a 0 = R s ( θ 0 ) , a 1 = R s ( θ 0 ) , a 2 = 1 2 ! R s ( θ 0 ) , a 3 = 1 3 ! R s ( θ 0 ) .
R s ( - Δ θ ) = a 0 - a 1 Δ θ + a 2 Δ θ 2 - a 3 Δ θ 3 + .
R s ( Δ θ ) - R s ( - Δ θ ) = 2 a 1 Δ θ + 2 a 3 Δ θ 3 + .
R s l = R s ( Δ θ ) - R s ( - Δ θ ) R s ( Δ θ ) + R s ( - Δ θ ) ,
R s ( Δ θ ) + R s ( - Δ θ ) = 2 a 0 + 2 a 2 Δ θ 2 + .
R p ( Δ θ ) = b 0 + b 1 Δ θ + b 2 Δ θ 2 + b 3 Δ θ 3 + ,
R p l = R p ( Δ θ ) - R p ( - Δ θ ) R p ( Δ θ ) + R p ( - Δ θ ) .
R s l ( Δ θ ) = c 1 Δ θ + c 3 Δ θ 3 + .
c 1 = a 1 a 0 = 4 tan θ t ,
c 3 = a 0 a 3 - a 1 a 2 a 0 2 = 4 tan θ t [ 1 2 ( n 2 - 1 ) sec 4 θ t - 5 sec 2 θ t + 16 3 ] ,
R p l ( Δ θ ) = d 1 Δ θ + d 3 Δ θ 3 + .
d 1 = b 1 b 0 = 4 tan θ t u ,
d 3 = b 0 b 3 - b 1 b 2 b 0 2 = 4 tan θ t [ 1 2 ( n 2 - 1 ) sec 4 θ t + 1 3 sec 2 θ t - 1 3 n ( n 2 + 1 ) ( 2 cos 2 θ 0 sec 2 θ t + cos 2 θ 0 ) + 1 3 u 2 ( n 2 + 1 ) 2 sin 2 2 θ 0 - 16 3 u 2 tan 2 θ t ] ,
θ 0 = sin - 1 { 1 n [ 17 ± ( 321 - 96 n 2 ) 1 / 2 32 ] 1 / 2 } = 17.83 ° ,             37.96 ° ,
cos θ i = cos θ e cos θ i

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