Abstract

A method to amplify the rotation angle of a mirror, based on multiple reflections between two quasi-parallel mirrors, is presented. The method allows rotations of fractions of nanoradians to be measured with a simple setup. The working principle, the experimental setup, and the results are presented.

© 2006 Optical Society of America

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References

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  1. A. Bergamin, G. Cavegnero, and G. Mana, "A displacement and angle interferometer with subatomic resolution," Rev. Sci. Instrum. 64, 3076-3081 (1993).
    [CrossRef]
  2. M. A. C. Perryman, "An astrometric and photometric survey of our galaxy," in Astrophysics and Space Science (Springer Science Business Media, 2002).
  3. M. P. Fitzgerald, T. R. Armstrong, R. B. Hurst, and A. C. Corney, "A method to measure Newton's gravitational constant," Metrologia 31, 301-310 (1994).
    [CrossRef]
  4. T. J. Quinn, C. C. Speake, and R. S. Davis, "Novel torsion balance for the measurement of the Newtonian gravitational constant," Metrologia 34, 245-249 (1997).
    [CrossRef]
  5. S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6 µm range," Phys. Rev. Lett. 78, 5-8 (1997).
    [CrossRef]
  6. R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy: Methods and Applications (Cambridge U. Press, 1994).
    [CrossRef]
  7. D. H. Rank, "High precision achromatic autocollimator," Rev. Sci. Instrum. 17, 243-244 (1946).
    [CrossRef] [PubMed]
  8. J. Rohlin, "An interferometer for precision angle measurement," Appl. Opt. 2, 762-763 (1963).
    [CrossRef]
  9. 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]
  10. M.-H. Chiu and D.-C. Su, "Angle measurement using total internal-reflection heterodyne interferometry," Opt. Eng. 36, 1750-1753 (1997).
    [CrossRef]

1997 (3)

T. J. Quinn, C. C. Speake, and R. S. Davis, "Novel torsion balance for the measurement of the Newtonian gravitational constant," Metrologia 34, 245-249 (1997).
[CrossRef]

S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6 µm range," Phys. Rev. Lett. 78, 5-8 (1997).
[CrossRef]

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

1994 (1)

M. P. Fitzgerald, T. R. Armstrong, R. B. Hurst, and A. C. Corney, "A method to measure Newton's gravitational constant," Metrologia 31, 301-310 (1994).
[CrossRef]

1993 (1)

A. Bergamin, G. Cavegnero, and G. Mana, "A displacement and angle interferometer with subatomic resolution," Rev. Sci. Instrum. 64, 3076-3081 (1993).
[CrossRef]

1992 (1)

1963 (1)

1946 (1)

D. H. Rank, "High precision achromatic autocollimator," Rev. Sci. Instrum. 17, 243-244 (1946).
[CrossRef] [PubMed]

Armstrong, T. R.

M. P. Fitzgerald, T. R. Armstrong, R. B. Hurst, and A. C. Corney, "A method to measure Newton's gravitational constant," Metrologia 31, 301-310 (1994).
[CrossRef]

Bergamin, A.

A. Bergamin, G. Cavegnero, and G. Mana, "A displacement and angle interferometer with subatomic resolution," Rev. Sci. Instrum. 64, 3076-3081 (1993).
[CrossRef]

Cavegnero, G.

A. Bergamin, G. Cavegnero, and G. Mana, "A displacement and angle interferometer with subatomic resolution," Rev. Sci. Instrum. 64, 3076-3081 (1993).
[CrossRef]

Chiu, M.-H.

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

Corney, A. C.

M. P. Fitzgerald, T. R. Armstrong, R. B. Hurst, and A. C. Corney, "A method to measure Newton's gravitational constant," Metrologia 31, 301-310 (1994).
[CrossRef]

Davis, R. S.

T. J. Quinn, C. C. Speake, and R. S. Davis, "Novel torsion balance for the measurement of the Newtonian gravitational constant," Metrologia 34, 245-249 (1997).
[CrossRef]

Fitzgerald, M. P.

M. P. Fitzgerald, T. R. Armstrong, R. B. Hurst, and A. C. Corney, "A method to measure Newton's gravitational constant," Metrologia 31, 301-310 (1994).
[CrossRef]

Huang, P. S.

Hurst, R. B.

M. P. Fitzgerald, T. R. Armstrong, R. B. Hurst, and A. C. Corney, "A method to measure Newton's gravitational constant," Metrologia 31, 301-310 (1994).
[CrossRef]

Kamada, O.

Kiyono, S.

Lamoreaux, S. K.

S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6 µm range," Phys. Rev. Lett. 78, 5-8 (1997).
[CrossRef]

Mana, G.

A. Bergamin, G. Cavegnero, and G. Mana, "A displacement and angle interferometer with subatomic resolution," Rev. Sci. Instrum. 64, 3076-3081 (1993).
[CrossRef]

Perryman, M. A. C.

M. A. C. Perryman, "An astrometric and photometric survey of our galaxy," in Astrophysics and Space Science (Springer Science Business Media, 2002).

Quinn, T. J.

T. J. Quinn, C. C. Speake, and R. S. Davis, "Novel torsion balance for the measurement of the Newtonian gravitational constant," Metrologia 34, 245-249 (1997).
[CrossRef]

Rank, D. H.

D. H. Rank, "High precision achromatic autocollimator," Rev. Sci. Instrum. 17, 243-244 (1946).
[CrossRef] [PubMed]

Rohlin, J.

Speake, C. C.

T. J. Quinn, C. C. Speake, and R. S. Davis, "Novel torsion balance for the measurement of the Newtonian gravitational constant," Metrologia 34, 245-249 (1997).
[CrossRef]

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]

Wiesendanger, R.

R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy: Methods and Applications (Cambridge U. Press, 1994).
[CrossRef]

Appl. Opt. (2)

Metrologia (2)

M. P. Fitzgerald, T. R. Armstrong, R. B. Hurst, and A. C. Corney, "A method to measure Newton's gravitational constant," Metrologia 31, 301-310 (1994).
[CrossRef]

T. J. Quinn, C. C. Speake, and R. S. Davis, "Novel torsion balance for the measurement of the Newtonian gravitational constant," Metrologia 34, 245-249 (1997).
[CrossRef]

Opt. Eng. (1)

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

Phys. Rev. Lett. (1)

S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6 µm range," Phys. Rev. Lett. 78, 5-8 (1997).
[CrossRef]

Rev. Sci. Instrum. (2)

D. H. Rank, "High precision achromatic autocollimator," Rev. Sci. Instrum. 17, 243-244 (1946).
[CrossRef] [PubMed]

A. Bergamin, G. Cavegnero, and G. Mana, "A displacement and angle interferometer with subatomic resolution," Rev. Sci. Instrum. 64, 3076-3081 (1993).
[CrossRef]

Other (2)

M. A. C. Perryman, "An astrometric and photometric survey of our galaxy," in Astrophysics and Space Science (Springer Science Business Media, 2002).

R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy: Methods and Applications (Cambridge U. Press, 1994).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Angle-amplification principle: (a) effect of a rotating mirror, (b) parallel-mirror amplifier, mirror A rotated, (c) parallel-mirror amplifier, mirror B rotated.

Fig. 2
Fig. 2

Autocollimation angle amplifier.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Multiple reflections on the mirrors. The mirrors are slightly divergent in the vertical axis; thus the input and output beams are not superimposed and show a parabolic shape. The arrows show the first (dashed) and the last (dotted) reflections. The ghost image visible behind the reflection patterns is due to a small fraction of the beam that passes through the reflecting coating and hits the back of the mirror.

Fig. 5
Fig. 5

Relation between the amplifier sensitivity and the number of reflections. The amplification principle is demonstrated.

Fig. 6
Fig. 6

Equivalent-angle noise spectral density. The two upper curves are the noise of the amplifier, the lower curve is the noise of the electronics alone (see text for details).

Fig. 7
Fig. 7

Response of the amplifier to an angular square modulation of 20 nrad p.p. with a frequency of 1 Hz. The signal is filtered with a 10 Hz low-pass filter.

Equations (2)

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G A ( gain  for  rotations  of  mirror  A ) = 2 N A = 2 ( 1 + α / β ) ,
G B ( gain  for  rotations  of  mirror  B ) = 2 N B = 2 α / β .

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