Abstract

We discuss the application of digital systems to the automatic control of dual-wave optical interferometers. We show that, if the mechanical-modulation technique is used for error-signal extraction, digital techniques can be used both for error-signal extraction and for control-signal generation. Therefore, apart from two front/end amplifiers that are necessary to match the dynamics of the detectors and actuators to the dynamics of the analog-to-digital converters and digital-to-analog converters, no other analog devices are required. In particular, the mechanical-modulation technique requires the synchronous demodulation of the photodiode output signal. Hence we need to implement a digital lock-in amplifier whose algorithm is described here. Finally, we describe one of the possible applications of this digital control procedure, such as the control of a classic Mach–Zehnder interferometer in air.

© 1994 Optical Society of America

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  1. F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” IEEE Trans. Nucl. Sci. 39, 232–237 (1992).
    [CrossRef]
  2. F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” Measurement Sci. Technol. (to be published).
  3. R. T. Denton, “Modulation techniques,” in Laser Handbook,F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), pp. 703–724.
  4. H. Melchior, “Demodulation and photodetection techniques,” in Laser Handbook,F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), pp. 725–835.
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), Chap. 7, p. 259.
  6. The Virgo Project: Proposal for the Italian–French Very Long Baseline Interferometric Antenna for Gravitational Waves Detection (Instituto Nazionale di Fisica Nucleare, Rome, and Centre National de la Recherche Scientifique, Paris, 1989).
  7. D. Shoemaker, “Contributions à l'étude de la détection inter-férométrique des ondes de gravitation,” Pd.D dissertation (Centre National de la Recherche Scientifique, Paris, 1987).
  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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.
  9. F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).
  10. M. Girard, Boucles a Verrouillage de Phase (McGraw-Hill, Paris, 1988), Chap. 7, p. 208.
  11. F. M. Gardiner, Phaselock Techniques (Wiley, New York, 1981), Chap. 3.
  12. I. Horowitz, Synthesis of Feedback Systems (Academic, New York, 1963).
  13. T. Kailath, Linear System (Prentice-Hall, Englewood Cliffs, N.J., 1980).
  14. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  15. M. Bellanger, Digital Processing of Signals (Wiley, New York, 1984).
  16. J. A. Cadzow, H. R. Martens, Discrete-Time and Computer Control Systems (Prentice-Hall, Englewood Cliffs, N.J., 1970).
  17. F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” in Proceedings of the Seventh Conference on Real Time '91,K. D. Müller, ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 292–297.
  18. F. Barone, L. Di Fiore, L. Milano, G. Russo, “Automatic control of a Michelson interferometer,” in Proceedings of the Ninth Italian Conference on General Relativity and Gravitational Physics,R. Cianci, R. De Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro, eds. (World Scientific, Singapore, 1990), pp. 562–567.
  19. A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.
  20. F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.
  21. F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
    [CrossRef]

1994

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
[CrossRef]

1992

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” IEEE Trans. Nucl. Sci. 39, 232–237 (1992).
[CrossRef]

Augurio, A.

A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.

Barone, F.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” IEEE Trans. Nucl. Sci. 39, 232–237 (1992).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, “Automatic control of a Michelson interferometer,” in Proceedings of the Ninth Italian Conference on General Relativity and Gravitational Physics,R. Cianci, R. De Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro, eds. (World Scientific, Singapore, 1990), pp. 562–567.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” Measurement Sci. Technol. (to be published).

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).

A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” in Proceedings of the Seventh Conference on Real Time '91,K. D. Müller, ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 292–297.

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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.

Bellanger, M.

M. Bellanger, Digital Processing of Signals (Wiley, New York, 1984).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), Chap. 7, p. 259.

Cadzow, J. A.

J. A. Cadzow, H. R. Martens, Discrete-Time and Computer Control Systems (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Calloni, E.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
[CrossRef]

A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.

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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

De Rosa, R.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
[CrossRef]

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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).

Denton, R. T.

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

Di Fiore, L.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” IEEE Trans. Nucl. Sci. 39, 232–237 (1992).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, “Automatic control of a Michelson interferometer,” in Proceedings of the Ninth Italian Conference on General Relativity and Gravitational Physics,R. Cianci, R. De Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro, eds. (World Scientific, Singapore, 1990), pp. 562–567.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” Measurement Sci. Technol. (to be published).

A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” in Proceedings of the Seventh Conference on Real Time '91,K. D. Müller, ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 292–297.

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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.

Fusco, F.

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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).

Gardiner, F. M.

F. M. Gardiner, Phaselock Techniques (Wiley, New York, 1981), Chap. 3.

Girard, M.

M. Girard, Boucles a Verrouillage de Phase (McGraw-Hill, Paris, 1988), Chap. 7, p. 208.

Horowitz, I.

I. Horowitz, Synthesis of Feedback Systems (Academic, New York, 1963).

Kailath, T.

T. Kailath, Linear System (Prentice-Hall, Englewood Cliffs, N.J., 1980).

Martens, H. R.

J. A. Cadzow, H. R. Martens, Discrete-Time and Computer Control Systems (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Melchior, H.

H. Melchior, “Demodulation and photodetection techniques,” in Laser Handbook,F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), pp. 725–835.

Milano, L.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” IEEE Trans. Nucl. Sci. 39, 232–237 (1992).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, “Automatic control of a Michelson interferometer,” in Proceedings of the Ninth Italian Conference on General Relativity and Gravitational Physics,R. Cianci, R. De Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro, eds. (World Scientific, Singapore, 1990), pp. 562–567.

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” Measurement Sci. Technol. (to be published).

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).

A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” in Proceedings of the Seventh Conference on Real Time '91,K. D. Müller, ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 292–297.

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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.

Oppenheim, A. V.

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

Russo, G.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” IEEE Trans. Nucl. Sci. 39, 232–237 (1992).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, “Automatic control of a Michelson interferometer,” in Proceedings of the Ninth Italian Conference on General Relativity and Gravitational Physics,R. Cianci, R. De Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro, eds. (World Scientific, Singapore, 1990), pp. 562–567.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” Measurement Sci. Technol. (to be published).

A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” in Proceedings of the Seventh Conference on Real Time '91,K. D. Müller, ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 292–297.

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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.

Schafer, R. W.

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

Shoemaker, D.

D. Shoemaker, “Contributions à l'étude de la détection inter-férométrique des ondes de gravitation,” Pd.D dissertation (Centre National de la Recherche Scientifique, Paris, 1987).

Solarino, F.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

Solimeno, S.

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” IEEE Trans. Nucl. Sci. 39, 232–237 (1992).
[CrossRef]

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” in Proceedings of the Seventh Conference on Real Time '91,K. D. Müller, ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 292–297.

A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), Chap. 7, p. 259.

IEEE Trans. Nucl. Sci.

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” IEEE Trans. Nucl. Sci. 39, 232–237 (1992).
[CrossRef]

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, L. Milano, G. Russo, “High-speed low-noise digital control system,” IEEE Trans. Nucl. Sci. 41, 190–199 (1994).
[CrossRef]

Other

F. Barone, L. Di Fiore, L. Milano, G. Russo, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” Measurement Sci. Technol. (to be published).

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

H. Melchior, “Demodulation and photodetection techniques,” in Laser Handbook,F. T. Arecchi, E. O. Schulz-DuBois, eds. (North-Holland, Amsterdam, 1972), pp. 725–835.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), Chap. 7, p. 259.

The Virgo Project: Proposal for the Italian–French Very Long Baseline Interferometric Antenna for Gravitational Waves Detection (Instituto Nazionale di Fisica Nucleare, Rome, and Centre National de la Recherche Scientifique, Paris, 1989).

D. Shoemaker, “Contributions à l'étude de la détection inter-férométrique des ondes de gravitation,” Pd.D dissertation (Centre National de la Recherche Scientifique, Paris, 1987).

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 Structure and Materials,B. Culshaw, P. T. Gardiner, A. McDonach, eds. (Institute of Physics Publishing, Bristol, England, 1992), pp. 49–53.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, “Digital alignment control of the Virgo interferometric antenna mirrors,” in Proceedings of the Tenth Italian Conference on General Relativity and Gravitational Physics,M. Cerdonio, R. Cianci, M. Francaviglia, G. Magnano, eds. (World Scientific, Singapore, 1992).

M. Girard, Boucles a Verrouillage de Phase (McGraw-Hill, Paris, 1988), Chap. 7, p. 208.

F. M. Gardiner, Phaselock Techniques (Wiley, New York, 1981), Chap. 3.

I. Horowitz, Synthesis of Feedback Systems (Academic, New York, 1963).

T. Kailath, Linear System (Prentice-Hall, Englewood Cliffs, N.J., 1980).

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

M. Bellanger, Digital Processing of Signals (Wiley, New York, 1984).

J. A. Cadzow, H. R. Martens, Discrete-Time and Computer Control Systems (Prentice-Hall, Englewood Cliffs, N.J., 1970).

F. Barone, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic alignment of a Michelson interferometer,” in Proceedings of the Seventh Conference on Real Time '91,K. D. Müller, ed. (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 292–297.

F. Barone, L. Di Fiore, L. Milano, G. Russo, “Automatic control of a Michelson interferometer,” in Proceedings of the Ninth Italian Conference on General Relativity and Gravitational Physics,R. Cianci, R. De Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro, eds. (World Scientific, Singapore, 1990), pp. 562–567.

A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G. Russo, S. Solimeno, “Automatic control system for mirrors alignment of the interferometric antenna Virgo,” in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity,H. Sato, T. Nakamura, eds. (World Scientific, Singapore, 1992), p. 1453.

F. Barone, E. Calloni, R. De Rosa, L. Di Fiore, F. Fusco, L. Milano, G. Russo, F. Solarino, “A digital approach to the automatic control of the interferometric antenna Virgo for gravitational wave detection,” in Proceedings of the 13th International Conference on General Relativity and Gravitation,P. W. Lamberti, O. E. Ortiz, eds. (Universidad National de Cordoba, Cordoba, Spain, 1992), p. 204.

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Figures (7)

Fig. 1
Fig. 1

Control system scheme for one degree of freedom.

Fig. 2
Fig. 2

Synchronous demodulation of the photodiode output signal for the mechanical-modulation technique.

Fig. 3
Fig. 3

Digital-lock-in-amplifier scheme.

Fig. 4
Fig. 4

Digital-phase-locked-loop equivalent scheme.

Fig. 5
Fig. 5

Digital-automatic-control scheme for the Mach–Zehnder interferometer.

Fig. 6
Fig. 6

Residual spectral density noise in the Mach–Zehnder interferometer digital control (0–100 Hz).

Fig. 7
Fig. 7

Residual spectral density noise in the Mach–Zehnder interferometer digital control (0–1 kHz).

Equations (54)

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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 ) ] ,
ϕ ( t ) = 4 π n 0 λ [ l ̅ 2 + n l 2 ( t ) l ̅ 1 n l 1 ( t ) ] = 4 π n 0 λ [ Δ l ̅ + n l ( t ) ] ,
l m = l m c cos ω c t ,
I ( t ) = I min + ( I max I min 4 ) ( 4 π n 0 λ ) 2 { [ n l 2 ( t ) + l m c 2 2 ] + 2 l m c n l ( t ) cos ω c t + l m c 2 2 cos 2 ω c t } .
Δ l ̅ = ( 2 m + 1 ) ( λ / 4 ) ,
n l ( t ) λ .
l m c λ .
e ( t ) = k 0 ( I max I min 4 ) ( 4 π n 0 λ ) 2 l m c n l ( t ) = k 1 n l ( t ) ,
y 1 ( n ) = k 2 e ( n ) cos ψ err ( n ) ,
y 2 ( n ) = k 2 e ( n ) sin ψ err ( n ) ,
ψ err ( n ) = g ( n ) * ψ in ( n ) = m = + g ( m ) ψ in ( n m )
y 3 ( n ) = y 1 2 ( n ) + y 2 2 ( n ) = k 2 e 2 ( n ) 4 ,
y 4 ( n ) = y 1 ( n ) y 2 ( n ) = k 2 e 2 ( n ) 8 sin 2 ψ err ( n ) .
y 4 ( n ) y 3 ( n ) = 1 2 sin 2 ψ err ( n ) ψ err ( n ) .
ψ car ( n ) = ψ 0 u ( n ) ,
u ( n ) = { 0 n < 0 1 n 0 ,
ψ err ( n ) = ψ 0 m = + w ( m ) u ( n m ) ,
ψ car ( n ) = ψ 0 r ( n ) = ψ 0 n T s u ( n ) .
ψ car ( n ) = ( ω c ω r ) n T s ,
Δ ϕ = 2 π f f s ,
B f s 10 ,
l m = l m c cos ω c t ,
I ( t ) = I min + ( I max I min 2 ) × ( 1 + cos { 4 π n 0 λ [ e ( t ) + l m c cos ω c t ] } ) ,
I ( t ) = I min + ( I max I min 2 ) × { 1 + cos [ 4 π n 0 λ e ( t ) ] [ J 0 ( 4 π n 0 λ l m c ) + 2 k = 1 ( 1 ) k J 2 k ( 4 π n 0 λ l m c ) cos ( 2 k ω c t ) ] 2 sin [ 4 π n 0 λ e ( t ) ] k = 0 ( 1 ) k J 2 k + 1 × ( 4 π n 0 λ l m c ) cos [ ( 2 k + 1 ) ω c t ] } ,
cos ( z cos θ ) = J 0 ( z ) + 2 k = 1 ( 1 ) k J 2 k ( z ) cos ( 2 k θ ) ,
sin ( z cos θ ) = 2 k = 1 ( 1 ) k J 2 k + 1 ( z ) cos [ ( 2 k + 1 ) θ ] .
I ( t ) = I min + ( I max I min 2 ) { 1 + [ 1 1 2 ( 4 π n 0 λ ) 2 n l 2 ( t ) ] × [ J 0 ( 4 π n 0 λ l m c ) + 2 k = 1 ( 1 ) k J 2 k ( 4 π n 0 λ l m c ) × cos ( 2 k ω c t ) ] 2 ( 4 π n 0 λ ) n l ( t ) k = 0 ( 1 ) k × J 2 k + 1 ( 4 π n 0 λ l m c ) cos [ ( 2 k + 1 ) ω c t ] } ,
J ν ( z ) = ( 1 2 z ) ν k = 0 ( 1 4 z 2 ) k k ! Γ ( ν + k + 1 ) ,
I ( t ) = I min + ( I max I min 2 ) ( 4 π n 0 λ ) 2 { [ n l 2 ( t ) + l m c 2 2 ] + 2 l m c n l ( t ) cos ω c t + l m c 2 2 cos 2 ω c t } .
y ( n ) = i = 0 M b i x ( n i ) j = 1 N a j y ( n j ) .
Z [ f ( k ) ] = F ( z ) = k = 0 f ( k ) z k ,
H ( z ) = Y ( z ) X ( z ) = i = 0 M b i z i 1 + j = 1 N a j z i ,
ψ ref ( n ) = ψ ref ( n 1 ) + ψ h ( n ) ,
M ( z ) = Ψ con ( z ) Ψ h ( z ) = z z 1 .
Ψ err ( z ) = ( z 1 ) G ( z ) ( z 1 ) + z G ( z ) H ( z ) Ψ car ( z ) .
lim n Ψ err ( n ) = lim z 1 ( z 1 ) Ψ err ( z ) .
ψ car ( n ) = ψ 0 u ( n ) Ψ car ( z ) = ( z z 1 ) ψ 0 ,
lim n ψ err ( n ) = lim z 1 ( z 1 ) [ ( z 1 ) G ( z ) ( z 1 ) + z G ( z ) H ( z ) ] ( z z 1 ) ψ 0 .
ψ car ( n ) = ψ 0 r ( n ) Ψ car ( z ) = z T s ( z 1 ) 2 ψ 0 ,
H ( s ) = k 1 b 0 + b 1 s + + s m s h ( a h + a h + 1 s + + s n ) ,
H ( s ) = H 0 s h Π i = 1 m 1 ( s + 1 τ z , i ) Π i = 1 m 2 ( s 2 + 2 δ z i ω n z , i s + ω n z , i 2 ) Π i = 1 n 1 ( s + 1 τ p , i ) Π i = 1 n 2 ( s 2 + 2 δ p i ω n p , i s + ω n p , i 2 ) .
H int ( s ) = 1 s
H ( s ) = 1 ( s + 1 τ ) .
y ( t ) = 0 t x ( t ) u ( t t ) d t = x ( t ) * u ( t ) ,
y ( t 2 ) y ( t 1 ) = 0 t 2 x ( t ) u ( t 2 t ) d t 0 t 1 x ( t ) u ( t 1 t ) d t ,
y ( t 2 ) y ( t 1 ) = 0 t 2 x ( t ) d t .
y ( t 2 ) y ( t 1 ) = t 2 t 1 2 [ x ( t 2 + x ( t 1 ) ] ,
y ( n ) y ( n 1 ) = T s 2 [ x ( n ) + x ( n 1 ) ] .
Y ( z ) Y ( z ) z 1 = T s 2 [ X ( z ) + X ( z ) z 1 ] ,
H ( z ) = T s 2 1 + z 1 1 z 1 .
2 T s 1 z 1 1 + z 1 s .
H ( z ) = T s τ ( 1 + z 1 ) T s + 2 τ + ( T 2 τ ) z 1 ,
π f A i T s = tan ( π f D i T s ) .

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