Abstract

In interferometric gravitational-wave detectors, most of the optical components are suspended by wires so that they are isolated from all kinds of forces except gravity. The requirement for the alignment of optical components to the laser beam is crucial. We have demonstrated a servo system developed for a Fabry–Perot cavity whose mirrors are suspended independently. We use mechanical modulation and a lock-in detection method to detect any misalignment. This system directly detects the relation between the axis of the laser beam and the axis of the cavity and automatically aligns the cavity to the laser beam. We confirmed that the intensity of the reflected light from the suspended Fabry–Perot cavity can be minimized with this system. Automated control of the alignment of the large-scale detectors is also discussed.

© 1994 Optical Society of America

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References

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  1. R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Q. Prog. Rep. Res. Lab. Electron. M.I.T. 105, 54–76 (1972).
  2. R. L. Forward, “Wideband laser-interferometer gravitational-radiation experiment,” Phys. Rev. D 17, 379–390 (1978).
    [CrossRef]
  3. A. Giazotto, “Interferometric detection of gravitational waves,” Phys. Rep. 182, 365–424 (1989).
    [CrossRef]
  4. B. J. Meers, K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
    [CrossRef]
  5. S. Kawamura, “Test mass orientation noise in the LIGO 40 m prototype,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity (World Scientific, Singapore, 1992), pp. 1486–1488.
  6. D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
    [CrossRef]
  7. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23, 2944–2949 (1984).
    [CrossRef] [PubMed]
  8. N. M. Sampas, D. Z. Anderson, “Stabilization of laser beam alignment to an optical resonator by heterodyne detection of off-axis modes,” Appl. Opt. 29, 394–403 (1990).
    [CrossRef] [PubMed]
  9. F. Barone, L. di Fiore, L. Milano, S. Restaino, “A computer aided system for the automatic alignment of a Michelson interferometer for detecting gravitational waves,” in Proceedings of the Eighth Italian Conference on General Relativity and Gravitational Physics (World Scientific, Singapore, 1989), pp. 551–559.
  10. N. Mio, K. Tsubono, “Short- and long-term frequency stabilization of a He–Ne laser using a Fabry–Perot cavity locked to the Lamb dip,” Appl. Phys. B 54, 202–204 (1992).
    [CrossRef]
  11. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]

1992 (1)

N. Mio, K. Tsubono, “Short- and long-term frequency stabilization of a He–Ne laser using a Fabry–Perot cavity locked to the Lamb dip,” Appl. Phys. B 54, 202–204 (1992).
[CrossRef]

1991 (1)

B. J. Meers, K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
[CrossRef]

1990 (1)

1989 (1)

A. Giazotto, “Interferometric detection of gravitational waves,” Phys. Rep. 182, 365–424 (1989).
[CrossRef]

1988 (1)

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
[CrossRef]

1984 (1)

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1978 (1)

R. L. Forward, “Wideband laser-interferometer gravitational-radiation experiment,” Phys. Rev. D 17, 379–390 (1978).
[CrossRef]

1972 (1)

R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Q. Prog. Rep. Res. Lab. Electron. M.I.T. 105, 54–76 (1972).

Anderson, D. Z.

Barone, F.

F. Barone, L. di Fiore, L. Milano, S. Restaino, “A computer aided system for the automatic alignment of a Michelson interferometer for detecting gravitational waves,” in Proceedings of the Eighth Italian Conference on General Relativity and Gravitational Physics (World Scientific, Singapore, 1989), pp. 551–559.

di Fiore, L.

F. Barone, L. di Fiore, L. Milano, S. Restaino, “A computer aided system for the automatic alignment of a Michelson interferometer for detecting gravitational waves,” in Proceedings of the Eighth Italian Conference on General Relativity and Gravitational Physics (World Scientific, Singapore, 1989), pp. 551–559.

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Forward, R. L.

R. L. Forward, “Wideband laser-interferometer gravitational-radiation experiment,” Phys. Rev. D 17, 379–390 (1978).
[CrossRef]

Giazotto, A.

A. Giazotto, “Interferometric detection of gravitational waves,” Phys. Rep. 182, 365–424 (1989).
[CrossRef]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Kawamura, S.

S. Kawamura, “Test mass orientation noise in the LIGO 40 m prototype,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity (World Scientific, Singapore, 1992), pp. 1486–1488.

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Maischberger, K.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
[CrossRef]

Meers, B. J.

B. J. Meers, K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
[CrossRef]

Milano, L.

F. Barone, L. di Fiore, L. Milano, S. Restaino, “A computer aided system for the automatic alignment of a Michelson interferometer for detecting gravitational waves,” in Proceedings of the Eighth Italian Conference on General Relativity and Gravitational Physics (World Scientific, Singapore, 1989), pp. 551–559.

Mio, N.

N. Mio, K. Tsubono, “Short- and long-term frequency stabilization of a He–Ne laser using a Fabry–Perot cavity locked to the Lamb dip,” Appl. Phys. B 54, 202–204 (1992).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Restaino, S.

F. Barone, L. di Fiore, L. Milano, S. Restaino, “A computer aided system for the automatic alignment of a Michelson interferometer for detecting gravitational waves,” in Proceedings of the Eighth Italian Conference on General Relativity and Gravitational Physics (World Scientific, Singapore, 1989), pp. 551–559.

Rüdiger, A.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
[CrossRef]

Sampas, N. M.

Schilling, R.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
[CrossRef]

Schnupp, L.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
[CrossRef]

Shoemaker, D.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
[CrossRef]

Strain, K. A.

B. J. Meers, K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
[CrossRef]

Tsubono, K.

N. Mio, K. Tsubono, “Short- and long-term frequency stabilization of a He–Ne laser using a Fabry–Perot cavity locked to the Lamb dip,” Appl. Phys. B 54, 202–204 (1992).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Weiss, R.

R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Q. Prog. Rep. Res. Lab. Electron. M.I.T. 105, 54–76 (1972).

Winkler, W.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (2)

N. Mio, K. Tsubono, “Short- and long-term frequency stabilization of a He–Ne laser using a Fabry–Perot cavity locked to the Lamb dip,” Appl. Phys. B 54, 202–204 (1992).
[CrossRef]

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Phys. Rep. (1)

A. Giazotto, “Interferometric detection of gravitational waves,” Phys. Rep. 182, 365–424 (1989).
[CrossRef]

Phys. Rev. D (3)

B. J. Meers, K. A. Strain, “Wave-front distortion in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 43, 3117–3130 (1991).
[CrossRef]

R. L. Forward, “Wideband laser-interferometer gravitational-radiation experiment,” Phys. Rev. D 17, 379–390 (1978).
[CrossRef]

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational-wave detector,” Phys. Rev. D 38, 423–432 (1988).
[CrossRef]

Q. Prog. Rep. Res. Lab. Electron. M.I.T. (1)

R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Q. Prog. Rep. Res. Lab. Electron. M.I.T. 105, 54–76 (1972).

Other (2)

S. Kawamura, “Test mass orientation noise in the LIGO 40 m prototype,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity (World Scientific, Singapore, 1992), pp. 1486–1488.

F. Barone, L. di Fiore, L. Milano, S. Restaino, “A computer aided system for the automatic alignment of a Michelson interferometer for detecting gravitational waves,” in Proceedings of the Eighth Italian Conference on General Relativity and Gravitational Physics (World Scientific, Singapore, 1989), pp. 551–559.

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Figures (4)

Fig. 1
Fig. 1

Relation between the axis of the beam and the axis of the cavity, expressed in terms of two parameters: the displacement (a x ) and the angular tilt (α x ). These parameters are the linear combinations of the transverse displacement of the centers of curvature of the mirrors, represented by x1 and x2.

Fig. 2
Fig. 2

Suspended Fabry–Perot cavity. The mirrors were attached to cylindrical masses independently suspended by two loops of wire. The suspensions were designed so as to restrict the angular and the vertical motions of the masses. The permanent magnets of the actuators were attached to the masses.

Fig. 3
Fig. 3

Experimental apparatus. FP, Fabry–Perot; FI, Faraday isolator; PBS, polarizing beam splitter; λ/4, quarter-wave plate; PD, photodetector.

Fig. 4
Fig. 4

Intensity of the reflected light and the error signals (a) aligned manually while the servo was off, (b) misaligned while the servo was off, (c) while the servo loop was closed.

Equations (37)

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U i j U k l * d x d y = δ i k δ j l ,
U 00 ( 1 - a x 2 2 w 0 2 - k 2 w 0 2 α x 2 8 ) U 00 ( x , y , z ) + ( a x w 0 + i k w 0 α x 2 ) U 10 ( x , y , z ) ,
E = E 0 [ ( 1 - a x 2 2 w 0 2 - k 2 w 0 2 α x 2 8 ) U 00 a ( ν ) + ( a x w 0 + i k w 0 α x 2 ) U 10 a ( ν - ν 10 ) ] ,
I = - - E 2 d x d y = I 0 { a ( ν ) 2 + γ [ a ( ν - ν 10 ) 2 - a ( ν ) 2 ] } ,
γ = ( a x w 0 ) 2 + ( k w 0 α x 2 ) 2
a x = 1 2 ( x 1 + x 2 ) ,
α x = 1 2 R - L ( x 2 - x 1 ) ,
x 1 = x 10 + Δ x 1 sin ω 1 t ,
x 2 = x 20 + Δ x 2 sin ω 2 t ,
I I 0 { a ( ν ) 2 + [ a ( ν - ν 10 ) 2 - a ( ν ) 2 ] × [ γ ( x 10 , x 20 ) + γ x 1 | x 1 = x 10 Δ x 1 sin ω 1 t + γ x 2 | x 2 = x 20 Δ x 2 sin ω 2 t ] }
γ x 1 | x 1 = x 10 = 1 2 w 0 2 [ ( x 10 + x 20 ) + L 2 k 2 w 0 4 ( x 10 - x 20 ) ] ,
γ x 2 | x 2 = x 20 = 1 2 w 0 2 [ ( x 10 + x 20 ) - L 2 k 2 w 0 4 ( x 10 - x 20 ) ] .
e ( 2 P ¯ η h ν ) 1 / 2 ,
i shot = e ( P ¯ η h ν ) 1 / 2 .
P ¯ = P ref ( 1 + γ ¯ u ) = P ref [ 1 + 1 8 w 0 2 ( 1 + L 2 k 2 w 0 4 ) ( Δ x 1 2 + Δ x 2 2 ) u ] ,
P ref a ( ν ) 2 P 0 ,
U a ( ν - ν 10 ) 2 - a ( ν ) 2 a ( ν ) 2 .
i shot = e { η h ν P ref [ 1 + 1 8 w 0 2 ( 1 + L 2 k 2 w 0 4 ) × ( Δ x 1 2 + Δ x 2 2 ) u ] } 1 / 2 .
i error 1 = P ref u Δ x 1 η e 2 h ν γ x 1 | x 1 = x 10
= P ref u Δ x 1 η e 4 h ν ω 0 2 [ ( x 10 + x 20 ) + L 2 k 2 w 0 4 ( x 10 - x 20 ) ] ,
i error 2 = P ref u Δ x 2 η e 2 h ν γ x 2 | x 2 = x 20
= P ref u Δ x 2 η e 4 h ν w 0 2 [ ( x 10 + x 20 ) - L 2 k 2 w 0 4 ( x 10 - x 20 ) ] .
a shot = 1 2 ( x 10 + x 20 ) shot = h ν e η P ref u ( Δ x 1 - 2 + Δ x 2 - 2 ) 1 / 2 w 0 2 i shot ,
α shot = ( x 20 - x 10 ) shot 2 R - L = h c e η P ref u π ( Δ x 1 - 2 + Δ x 2 - 2 ) 1 / 2 | 2 R L - 1 | 1 / 2 i shot ,
w 0 = ( λ L 2 π ) 1 / 2 ~ 2.2 cm .
a x ~ x 2 + β L 2 ,
a x ~ x L + β .
x seism ~ 10 - 7 ( 1 Hz f ) 2 m / Hz ,
a n = w 0 2 u - 1 δ i ref i ref ( Δ x 1 - 2 + Δ x 2 - 2 ) 1 / 2 ,
α n = 2 u - 1 k - 1 δ i ref i ref | 2 R L - 1 | 1 / 2 ( Δ x 1 - 2 + Δ x 2 - 2 ) 1 / 2 ,
i ref = e η P ref h ν .
a n = 1.2 × 10 - 6 1.4 × 10 - 7 m Δ x ( w 0 148 μ m ) 2 m / Hz ,
α n = 4.7 × 10 - 5 1.4 × 10 - 7 m Δ x λ 633 nm × | 2 R L - 1 20 - 1 | 1 / 2 rad / Hz .
γ mod = 1 2 w 0 2 ( Δ x 1 2 sin 2 ω 1 t + Δ x 2 2 sin 2 ω 2 t ) ,
Δ x < 10 - 5 w 0 = 7.0 × 10 - 5 m .
a n = 1.2 × 10 - 6 1.4 × 10 - 7 m 7.0 × 10 - 5 m ( 2.2 cm 148 μ m ) 2 = 5.3 × 10 - 5 m / Hz ,
α n = 4.7 × 10 - 5 1.4 × 10 - 7 m 7.0 × 10 - 5 m 1.06 μ m 633 nm 1 19 = 3.6 × 10 - 8 rad / Hz .

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