Abstract

We describe an efficient and robust method for the extraction of the longitudinal error signal for the automatic control of optical interferometers, which can also be applied when the uncontrolled optical system spans hundreds of fringes. The method is based on classic modulation techniques (phase modulation, mechanical modulation, etc.), but extends their performances by the use of the information available only at the output photodiode. We digitally implemented such a method by following modular hardware and software architectures. We then tested the whole procedure in the automatic control of a suspended Michelson interferometer, showing its feasibility and the good performances.

© 1995 Optical Society of America

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References

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  1. For a review of GW interferometric detectors, see, for example, The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, 1991).
    [CrossRef]
  2. P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).
  3. The Virgo Project: final conceptual design of the Italian–French large-base interferometric antenna Virgo for GW Detection, June 1989, of which the authors are proponents and in whose construction the authors are collaborating.
  4. R. T. Denton, “Modulation techniques,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 703–724.
  5. F. M. Gardiner, Phaselock Techniques (Wiley, New York, 1981).
  6. M. Girard, Boucles a Verrouillage de Phase (McGraw-Hill, Paris, 1988).
  7. C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
    [CrossRef]
  8. F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.
  9. F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to automatic control of the interferometric antenna Virgo for gravitational wave detection,” Meas. Sci. Technol. 5, 1187–1196 (1994).
    [CrossRef]
  10. F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Fringe-counting technique used to lock a suspended interferometer,” Appl. Opt. 33, 1194–1197 (1994).
    [CrossRef] [PubMed]
  11. F. Barone, R. De Rosa, L. Di Fiore, F. Fusco, A. Grado, L. Milano, G. Russo, “Real-time digital control of optical interferometers by means of the mechanical modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
    [CrossRef] [PubMed]
  12. F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
    [CrossRef]
  13. A. V. Oppenheim, G. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  14. F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “The 3m prototype Michelson interferometer in Napoli,” Virgo Note PJT94-020 (Universitá di Napoli, Naples, Italy, 1994).

1995 (1)

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
[CrossRef]

1994 (3)

1989 (1)

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Barone, F.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to automatic control of the interferometric antenna Virgo for gravitational wave detection,” Meas. Sci. Technol. 5, 1187–1196 (1994).
[CrossRef]

F. Barone, R. De Rosa, L. Di Fiore, F. Fusco, A. Grado, L. Milano, G. Russo, “Real-time digital control of optical interferometers by means of the mechanical modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Fringe-counting technique used to lock a suspended interferometer,” Appl. Opt. 33, 1194–1197 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “The 3m prototype Michelson interferometer in Napoli,” Virgo Note PJT94-020 (Universitá di Napoli, Naples, Italy, 1994).

Bradaschia, C.

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Calloni, E.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
[CrossRef]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Fringe-counting technique used to lock a suspended interferometer,” Appl. Opt. 33, 1194–1197 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “The 3m prototype Michelson interferometer in Napoli,” Virgo Note PJT94-020 (Universitá di Napoli, Naples, Italy, 1994).

De Rosa, R.

F. Barone, R. De Rosa, L. Di Fiore, F. Fusco, A. Grado, L. Milano, G. Russo, “Real-time digital control of optical interferometers by means of the mechanical modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Fringe-counting technique used to lock a suspended interferometer,” Appl. Opt. 33, 1194–1197 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.

Del Fabbro, R.

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Denton, R. T.

R. T. Denton, “Modulation techniques,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 703–724.

Di Fiore, L.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to automatic control of the interferometric antenna Virgo for gravitational wave detection,” Meas. Sci. Technol. 5, 1187–1196 (1994).
[CrossRef]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Fringe-counting technique used to lock a suspended interferometer,” Appl. Opt. 33, 1194–1197 (1994).
[CrossRef] [PubMed]

F. Barone, R. De Rosa, L. Di Fiore, F. Fusco, A. Grado, L. Milano, G. Russo, “Real-time digital control of optical interferometers by means of the mechanical modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
[CrossRef] [PubMed]

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “The 3m prototype Michelson interferometer in Napoli,” Virgo Note PJT94-020 (Universitá di Napoli, Naples, Italy, 1994).

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.

Di Virgilio, A.

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Fusco, F.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Fringe-counting technique used to lock a suspended interferometer,” Appl. Opt. 33, 1194–1197 (1994).
[CrossRef] [PubMed]

F. Barone, R. De Rosa, L. Di Fiore, F. Fusco, A. Grado, L. Milano, G. Russo, “Real-time digital control of optical interferometers by means of the mechanical modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.

Gardiner, F. M.

F. M. Gardiner, Phaselock Techniques (Wiley, New York, 1981).

Giazotto, A.

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Girard, M.

M. Girard, Boucles a Verrouillage de Phase (McGraw-Hill, Paris, 1988).

Grado, A.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
[CrossRef]

F. Barone, R. De Rosa, L. Di Fiore, F. Fusco, A. Grado, L. Milano, G. Russo, “Real-time digital control of optical interferometers by means of the mechanical modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “The 3m prototype Michelson interferometer in Napoli,” Virgo Note PJT94-020 (Universitá di Napoli, Naples, Italy, 1994).

Kautzky, H.

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Milano, L.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to automatic control of the interferometric antenna Virgo for gravitational wave detection,” Meas. Sci. Technol. 5, 1187–1196 (1994).
[CrossRef]

F. Barone, R. De Rosa, L. Di Fiore, F. Fusco, A. Grado, L. Milano, G. Russo, “Real-time digital control of optical interferometers by means of the mechanical modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Fringe-counting technique used to lock a suspended interferometer,” Appl. Opt. 33, 1194–1197 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “The 3m prototype Michelson interferometer in Napoli,” Virgo Note PJT94-020 (Universitá di Napoli, Naples, Italy, 1994).

Montelatici, V.

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, G. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Passuello, D.

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Russo, G.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to automatic control of the interferometric antenna Virgo for gravitational wave detection,” Meas. Sci. Technol. 5, 1187–1196 (1994).
[CrossRef]

F. Barone, R. De Rosa, L. Di Fiore, F. Fusco, A. Grado, L. Milano, G. Russo, “Real-time digital control of optical interferometers by means of the mechanical modulation technique,” Appl. Opt. 33, 7846–7856 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Fringe-counting technique used to lock a suspended interferometer,” Appl. Opt. 33, 1194–1197 (1994).
[CrossRef] [PubMed]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “The 3m prototype Michelson interferometer in Napoli,” Virgo Note PJT94-020 (Universitá di Napoli, Naples, Italy, 1994).

Saulson, P. R.

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).

Schafer, G. W.

A. V. Oppenheim, G. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Appl. Opt. (2)

Meas. Sci. Technol. (1)

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to automatic control of the interferometric antenna Virgo for gravitational wave detection,” Meas. Sci. Technol. 5, 1187–1196 (1994).
[CrossRef]

Phys. Lett. A (1)

C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, “First results on the electronic cooling of the Pisa seismic noise super-attenuator for gravitational wave detection,” Phys. Lett. A 137, 329–333 (1989).
[CrossRef]

Rev. Sci. Instrum. (1)

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “High-performance modular digital lock-in amplifier,” Rev. Sci. Instrum. 66, 3697–3702 (1995).
[CrossRef]

Other (9)

A. V. Oppenheim, G. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

F. Barone, E. Calloni, L. Di Fiore, A. Grado, L. Milano, G. Russo, “The 3m prototype Michelson interferometer in Napoli,” Virgo Note PJT94-020 (Universitá di Napoli, Naples, Italy, 1994).

For a review of GW interferometric detectors, see, for example, The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).

The Virgo Project: final conceptual design of the Italian–French large-base interferometric antenna Virgo for GW Detection, June 1989, of which the authors are proponents and in whose construction the authors are collaborating.

R. T. Denton, “Modulation techniques,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 703–724.

F. M. Gardiner, Phaselock Techniques (Wiley, New York, 1981).

M. Girard, Boucles a Verrouillage de Phase (McGraw-Hill, Paris, 1988).

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital systems for automatic control of optical resonators used as gravitational waves detectors,” in Proceedings of the First European Conference on Smart Structures and Materials, B. Culshaw, P. T. Gardiner, A. McDonach, eds. (IOP, Bristol, UK, 1992), pp. 49–53.

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Figures (5)

Fig. 1
Fig. 1

Basic scheme of a Michelson interferometer: BS, beam splitter; M's, mirrors; PD, photodiode.

Fig. 2
Fig. 2

Experimental setup: BS, beam splitter; PD, photodiode; PSD, position-sensing photodiode; PZT, piezoelectric transducer; M's, mirrors; VME, bus interface system.

Fig. 3
Fig. 3

Structure of the digital control system used for the automatic control of the suspended Michelson interferometer.

Fig. 4
Fig. 4

Error-signal extraction of the longitudinal displacement of the suspended Michelson interferometer: top, output signal photodiode; bottom, longitudinal error signal.

Fig. 5
Fig. 5

Longitudinal spectral density noise of the suspended Michelson interferometer: top spectrum, uncontrolled system; bottom spectrum, controlled system.

Equations (9)

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

I ( t ) = I min + I max I min 2 [ 1 + cos ϕ ( t ) ] ,
ϕ ( t ) = 4 π n 0 λ { [ l 2 ( t ) l 1 ( t ) ] + i = 1 2 [ l i ( t ) + d ] j = 1 2 θ i , j 2 } = 4 π n 0 λ l eq ,
l eq = λ 4 π n 0 arccos ( 2 I I min I max I min 1 )
l eq = λ 4 π n 0 [ π arccos ( 2 I I min I max I min 1 ) ]
f res = 1 2 π ( g / l ) 1 / 2 ,
v max = A ( g / l ) 1 / 2 ,
t min = λ 2 v max = λ 2 A ( l / g ) 1 / 2 .
f s min = 2 A n λ ( g / l ) 1 / 2 .
e r = I ( t ) I min I max I min = ( 1 cos θ ) 2 2 π 2 n fringe 2 ,

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