Abstract

A new instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry is presented. With this instrument we can achieve a small rotation angle only by measuring the variation in phase difference between s- and p-polarization states. To improve its sensitivity we increase the number of total internal reflections by using two parallelogram prisms instead of two right-angle prisms. The angular resolution of the new instrument is better than 2.2 × 10-6 rad over the measurement range -2.12° ≤ θ ≤ 2.12° for 20 total-internal reflections. The experimental results and the theoretical curve are in good agreement.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1997 (1)

1996 (2)

1995 (1)

1993 (1)

1992 (1)

1988 (1)

1984 (1)

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

1983 (1)

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,” Precis. Eng. 5, 5–8 (1982).
[CrossRef]

1975 (1)

1974 (1)

1970 (1)

1963 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 48–50.

Chapman, G. D.

Chen, C. D.

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

Chickvary, J. L.

Chiu, M. H.

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

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

Deslattes, R. D.

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

Ennos, A. E.

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

Harris, O.

Huang, P. S.

Kamada, O.

Kiyono, S.

Luther, G. G.

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

Malacara, D.

Ni, J.

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, P.

Stijns, E.

Su, D. C.

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

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

Virdee, M. S.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 48–50.

Yoder, P. R.

Appl. Opt. (10)

Precis. Eng. (2)

D. C. Su, M. H. Chiu, C. D. Chen, “Simple two frequency laser,” Precis. Eng. 18, 161–163 (1996).
[CrossRef]

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

Rev. Sci. Instrum. (2)

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

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

Other (1)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 48–50.

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Figures (8)

Fig. 1
Fig. 1

Basic structure of the new instrument for measuring small angles.

Fig. 2
Fig. 2

Two beams on the stage in 20 total internal reflections.

Fig. 3
Fig. 3

Results of simulating the conditions of the two beams undergoing MTIR.

Fig. 4
Fig. 4

Light undergoing MTIR in the parallelogram prisms.

Fig. 5
Fig. 5

Experimental configuration used for measuring small angles by MTIR in heterodyne interferometry.

Fig. 6
Fig. 6

Experimental and theoretical curves of Δϕ versus θ when two beams undergo MTIR in the parallelogram prisms.

Fig. 7
Fig. 7

Δθ versus θ.

Fig. 8
Fig. 8

S versus θ.

Equations (20)

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α1=45°+sin-1sin θnp,
α2=45°-sin-1sin θnp.
ϕ1=2 tan-1sin2 α1-1/np2tan α1 sin α11/2,
ϕ2=2 tan-1sin2 α2-1/np2tan α2 sin α21/2,
ϕBS=tan-1ad-bcac+bd,
a=2v cos β,
b=u2+v2-cos2 β,
c=2v cos βn2-k2-2u2,
d=u2+v2-n2+k22 cos2 β,
β=45°+θ,
2u2=n2-k2-sin2 β+n2-k2-sin2 β2+4n2k21/2,
2v2=-n2-k2-sin2 β+n2-k2-sin2 β2+4n2k21/2.
ϕ=ϕ1-ϕ2+ϕBS,
θmax=sin-1np sin45°-sin-11/np.
ϕ=mϕ1-ϕ2+ϕBS,
1 sin θmax=np sin θ,
l=h tan45°+θ,
Lml,
Δθ=1mdϕ1/dθ-dϕ2/dθ Δϕ.
S=Δϕ/Δθ,

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