Abstract

When modulated through the harmonic motion of one mirror, the counterpropagating waves in a ring laser oscillate out of phase. A solution to the wave equation is presented that satisfies both the time-dependent boundary condition and the resonance condition. This theoretical prediction is confirmed experimentally to leading order in terms that are inversely proportional to the speed of light. The method of solution is applicable to arbitrary phase modulation at more than one spatial location in the cavity. Potential uses include the reduction of the locking problem in ring lasers and the testing of higher-order kinematic effects in the theory of relativity.

© 2002 Optical Society of America

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References

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  1. T. Baer, F. V. Kowalski, J. L. Hall, “Frequency stabilization of 0.633-µm He-Ne longitudinal Zeeman laser,” Appl. Opt. 19, 3173–3177 (1980).
    [CrossRef] [PubMed]
  2. T. Midavaine, D. Dangoisse, P. Glorieux, “Observation of chaos in a frequency modulated CO2,” Phys. Rev. Lett. 55, 1989–1992 (1985).
    [CrossRef] [PubMed]
  3. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), chap. 25, sect. 3.
  4. H. A. Haus, H. Statz, I. W. Smith, “Frequency locking of modes in a ring laser,” IEEE J. Quantum Electron. QE-21, 78–85 (1985).
    [CrossRef]
  5. T. J. Hutchings, “Laser gyro with phase dithered mirrors,” U.S. patent4,281,930 (4Aug.1981).
  6. B. H. G. Lijung, J. C. Stiles, “Ring laser gyroscope with Doppler mirrors,” U.S. patent4,410,276 (18Oct.1983).
  7. F. Bretenaker, J. P. Tache, A. Le Floch, “Reverse Sagnac effect in ring lasers,” Europhys. Lett. 21, 291–297 (1993).
    [CrossRef]
  8. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
    [CrossRef]
  9. F. V. Kowalski, J. Murray, A. C. Head, “Phase measurement of light propagating in a linearly accelerating rigid-mirror system,” Phys. Lett. A 174, 190–195 (1993).
    [CrossRef]
  10. F. V. Kowalski, J. Murray, A. C. Head, “Interaction of light with an accelerating dielectric,” Phys. Rev. A 48, 1082–1088 (1993).
    [CrossRef] [PubMed]

1993 (3)

F. Bretenaker, J. P. Tache, A. Le Floch, “Reverse Sagnac effect in ring lasers,” Europhys. Lett. 21, 291–297 (1993).
[CrossRef]

F. V. Kowalski, J. Murray, A. C. Head, “Phase measurement of light propagating in a linearly accelerating rigid-mirror system,” Phys. Lett. A 174, 190–195 (1993).
[CrossRef]

F. V. Kowalski, J. Murray, A. C. Head, “Interaction of light with an accelerating dielectric,” Phys. Rev. A 48, 1082–1088 (1993).
[CrossRef] [PubMed]

1985 (3)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

T. Midavaine, D. Dangoisse, P. Glorieux, “Observation of chaos in a frequency modulated CO2,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

H. A. Haus, H. Statz, I. W. Smith, “Frequency locking of modes in a ring laser,” IEEE J. Quantum Electron. QE-21, 78–85 (1985).
[CrossRef]

1980 (1)

Baer, T.

Bretenaker, F.

F. Bretenaker, J. P. Tache, A. Le Floch, “Reverse Sagnac effect in ring lasers,” Europhys. Lett. 21, 291–297 (1993).
[CrossRef]

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Dangoisse, D.

T. Midavaine, D. Dangoisse, P. Glorieux, “Observation of chaos in a frequency modulated CO2,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Glorieux, P.

T. Midavaine, D. Dangoisse, P. Glorieux, “Observation of chaos in a frequency modulated CO2,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

Hall, J. L.

Haus, H. A.

H. A. Haus, H. Statz, I. W. Smith, “Frequency locking of modes in a ring laser,” IEEE J. Quantum Electron. QE-21, 78–85 (1985).
[CrossRef]

Head, A. C.

F. V. Kowalski, J. Murray, A. C. Head, “Interaction of light with an accelerating dielectric,” Phys. Rev. A 48, 1082–1088 (1993).
[CrossRef] [PubMed]

F. V. Kowalski, J. Murray, A. C. Head, “Phase measurement of light propagating in a linearly accelerating rigid-mirror system,” Phys. Lett. A 174, 190–195 (1993).
[CrossRef]

Hutchings, T. J.

T. J. Hutchings, “Laser gyro with phase dithered mirrors,” U.S. patent4,281,930 (4Aug.1981).

Kowalski, F. V.

F. V. Kowalski, J. Murray, A. C. Head, “Phase measurement of light propagating in a linearly accelerating rigid-mirror system,” Phys. Lett. A 174, 190–195 (1993).
[CrossRef]

F. V. Kowalski, J. Murray, A. C. Head, “Interaction of light with an accelerating dielectric,” Phys. Rev. A 48, 1082–1088 (1993).
[CrossRef] [PubMed]

T. Baer, F. V. Kowalski, J. L. Hall, “Frequency stabilization of 0.633-µm He-Ne longitudinal Zeeman laser,” Appl. Opt. 19, 3173–3177 (1980).
[CrossRef] [PubMed]

Le Floch, A.

F. Bretenaker, J. P. Tache, A. Le Floch, “Reverse Sagnac effect in ring lasers,” Europhys. Lett. 21, 291–297 (1993).
[CrossRef]

Lijung, B. H. G.

B. H. G. Lijung, J. C. Stiles, “Ring laser gyroscope with Doppler mirrors,” U.S. patent4,410,276 (18Oct.1983).

Midavaine, T.

T. Midavaine, D. Dangoisse, P. Glorieux, “Observation of chaos in a frequency modulated CO2,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

Murray, J.

F. V. Kowalski, J. Murray, A. C. Head, “Phase measurement of light propagating in a linearly accelerating rigid-mirror system,” Phys. Lett. A 174, 190–195 (1993).
[CrossRef]

F. V. Kowalski, J. Murray, A. C. Head, “Interaction of light with an accelerating dielectric,” Phys. Rev. A 48, 1082–1088 (1993).
[CrossRef] [PubMed]

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Siegman, A.

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), chap. 25, sect. 3.

Smith, I. W.

H. A. Haus, H. Statz, I. W. Smith, “Frequency locking of modes in a ring laser,” IEEE J. Quantum Electron. QE-21, 78–85 (1985).
[CrossRef]

Statz, H.

H. A. Haus, H. Statz, I. W. Smith, “Frequency locking of modes in a ring laser,” IEEE J. Quantum Electron. QE-21, 78–85 (1985).
[CrossRef]

Stiles, J. C.

B. H. G. Lijung, J. C. Stiles, “Ring laser gyroscope with Doppler mirrors,” U.S. patent4,410,276 (18Oct.1983).

Tache, J. P.

F. Bretenaker, J. P. Tache, A. Le Floch, “Reverse Sagnac effect in ring lasers,” Europhys. Lett. 21, 291–297 (1993).
[CrossRef]

Appl. Opt. (1)

Europhys. Lett. (1)

F. Bretenaker, J. P. Tache, A. Le Floch, “Reverse Sagnac effect in ring lasers,” Europhys. Lett. 21, 291–297 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, H. Statz, I. W. Smith, “Frequency locking of modes in a ring laser,” IEEE J. Quantum Electron. QE-21, 78–85 (1985).
[CrossRef]

Phys. Lett. A (1)

F. V. Kowalski, J. Murray, A. C. Head, “Phase measurement of light propagating in a linearly accelerating rigid-mirror system,” Phys. Lett. A 174, 190–195 (1993).
[CrossRef]

Phys. Rev. A (1)

F. V. Kowalski, J. Murray, A. C. Head, “Interaction of light with an accelerating dielectric,” Phys. Rev. A 48, 1082–1088 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

T. Midavaine, D. Dangoisse, P. Glorieux, “Observation of chaos in a frequency modulated CO2,” Phys. Rev. Lett. 55, 1989–1992 (1985).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[CrossRef]

Other (3)

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), chap. 25, sect. 3.

T. J. Hutchings, “Laser gyro with phase dithered mirrors,” U.S. patent4,281,930 (4Aug.1981).

B. H. G. Lijung, J. C. Stiles, “Ring laser gyroscope with Doppler mirrors,” U.S. patent4,410,276 (18Oct.1983).

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Figures (5)

Fig. 1
Fig. 1

Laser schematic. Mirror B is displaced by way of the piezoelectric crystal, PZT.

Fig. 2
Fig. 2

Schematic of the clockwise (left) and counterclockwise (right) traveling waves in the ring. The frequency shift is greatly exaggerated.

Fig. 3
Fig. 3

Solid lines are the intensity data from an interferometer at mirror C (upper trace) and mirror A (lower trace) for an oscillation frequency of mirror B of 292 Hz. The dotted line is the prediction of Eq. (2).

Fig. 4
Fig. 4

Similar to Fig. 3 but with a 714-Hz oscillation frequency of mirror B.

Fig. 5
Fig. 5

Sagnac interferometer. Mirror B is displaced by way of the piezoelectric crystal, PZT.

Equations (4)

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

ωAcw-ωAccwω0αΩcP0ACB¯-AB¯sin(Ωt0).
IA  1+cosω0cP0-αL+ACB¯-AB¯×cosΩt+ΠΛt
IC  1+cosω0cP0-αL+CAB¯-CB¯×cosΩt+ΠΛt
Δϕt=ϕccwt-ϕcwtω0αΩc2ACB¯-AB¯sinΩt0

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