Abstract

Statistical cross correlations between mode intensities in individual pulses from a multimode dye laser have been studied by using a Fizeau interferometer and a high-resolution linear photodiode array. We find that positive intensity cross correlations develop between modes separated by certain characteristic frequencies. This appears to be a result of spatial hole burning in the standing-wave cavity. The gain competition between certain modes is minimized because of the spatial inhomogeneity of the mode intensity distributions in the gain medium.

© 1984 Optical Society of America

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  1. See, for example, N. C. Wong and J. H. Eberly, "Multiphoton absorption in the presence of two finite-bandwidth lasers," Opt. Lett. 1, 211–213 (1977); B. R. Marx, J. Simons, and L. Allen, "The effect of laser linewidth on two-photon absorption rates," J. Phys. B 11, L273–L277 (1978); A. V. Masalov and L. Allen, "Fluctuation spectroscopy and multiphoton absorption," J. Phys. B 15, 2375–2383 (1982).
    [CrossRef] [PubMed]
  2. See, for example, J. L. F. de Meijere and J. H. Eberly, "Rate of resonant two-photon ionization in the presence of a partially coherent field," Phys. Rev. A 17, 1416–1430 (1978); M. Kus and M. Lewenstein, "The influence of a non-Lorentzian line shape on ionization by the chaotic field," J. Phys. B 14, 4933–4940 (1981).
    [CrossRef]
  3. See, for example, G. S. Agarwal, "Exact solution of laser temporal fluctuations on resonance fluorescence," Phys. Rev. Lett. 37, 1383–1386 (1976); P. Zoller, "Atomic relaxation and resonance fluorescence in intensity on phase-fluctuating laser light," J. Phys. B 11, 2825–2832 (1978).
    [CrossRef]
  4. See, for example, R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, "Theory of Stokes pulse shapes in transient stimulated Raman scattering," Phys. Rev. A 2, 60–72 (1970); S. A. Akhmanov, Yu. E. D'Yakav, and L. I. Pavlov, "Statistical phenomena in Raman scattering by a broad-band pump," Sov. Phys. JETP 39, 249–256 (1974).
    [CrossRef]
  5. See, for example, W. R. Trutna, Jr., Y. K. Park, and R. L. Byer, "The dependence of Raman gain on pump laser bandwidth," IEEE J. Quantum Electron. QE-15, 648–655 (1979); J. Eggleston and R. L. Byer, "Steady-state stimulated Raman scattering by a multimode laser," IEEE J. Quantum Electron. QE-16, 850–853 (1980); G. P. Dzhotyan, Yu. E. D'yakov, I. G. Zubarev, A. B. Mironov, and W. I. Mikhailov, Sov. Phys. JETP 46, 431–435 (1977).
    [CrossRef]
  6. S. M. Curry, R. Cubeddu, and T. W. Hänsch, "Intensity stabilization of dye laser radiation by saturated amplification," Appl. Phys. 1, 153–159 (1973).
    [CrossRef]
  7. H. Statz and C. L. Tang, "Phase locking of modes in lasers," J. Appl. Phys. 36, 3923–3927 (1965).
    [CrossRef]
  8. M. V. R. K. Murty, "The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source," Appl. Opt. 3, 531–535 (1964).
    [CrossRef]
  9. J. Brossel, "Multi-beam localized fringes: intensity distribution and localization," Proc. Phys. Soc. London 59, 224–234 (1947).
    [CrossRef]
  10. M. G. Littman, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08540 (personal communication).
  11. L. A. Rahn, Sandia National Laboratory, Livermore, California 95550 (personal communication).
  12. J. R. Rogers, "Fringe shifts in multiple-beam Fizeau interferometry," J. Opt. Soc. Am. 72, 638–643 (1982); Y. H. Meyer, "Fringe shape with an interferential wedge," J. Opt. Soc. Am. 71, 1255–1263 (1981).
    [CrossRef]
  13. Our circuit, available on request, was designed to trigger Quantel Laser power-supply electronics.
  14. W. Brunner and H. Paul, "Spectral properties of dye lasers," Opt. Quantum Electron. 12, 393–411 (1980).
    [CrossRef]
  15. T. W. Hansch, A. L. Schawlow, and P. E. Toschek, "Ultrasensitive response of a cw dye laser to selective extinction," IEEE J. Quantum Electron. QE-8, 802–804 (1972).
    [CrossRef]
  16. V. M. Baev, T. P. Belikova, E. A. Sviridenkov, and A. F. Suchkov, "Intracavity laser spectroscopy with continuously and quasicontinuously operating lasers," Sov. Phys. JETP 47, 21–29 (1978).
  17. L. A. Westling, M. G. Raymer, M. G. Sceats, and D. F. Coker, "Observation of intensity fluctuations and mode correlations in a broadband cw dye laser," Opt. Commun. 47, 212–217 (1983).
    [CrossRef]

1983 (1)

L. A. Westling, M. G. Raymer, M. G. Sceats, and D. F. Coker, "Observation of intensity fluctuations and mode correlations in a broadband cw dye laser," Opt. Commun. 47, 212–217 (1983).
[CrossRef]

1982 (1)

1980 (1)

W. Brunner and H. Paul, "Spectral properties of dye lasers," Opt. Quantum Electron. 12, 393–411 (1980).
[CrossRef]

1979 (1)

See, for example, W. R. Trutna, Jr., Y. K. Park, and R. L. Byer, "The dependence of Raman gain on pump laser bandwidth," IEEE J. Quantum Electron. QE-15, 648–655 (1979); J. Eggleston and R. L. Byer, "Steady-state stimulated Raman scattering by a multimode laser," IEEE J. Quantum Electron. QE-16, 850–853 (1980); G. P. Dzhotyan, Yu. E. D'yakov, I. G. Zubarev, A. B. Mironov, and W. I. Mikhailov, Sov. Phys. JETP 46, 431–435 (1977).
[CrossRef]

1978 (2)

See, for example, J. L. F. de Meijere and J. H. Eberly, "Rate of resonant two-photon ionization in the presence of a partially coherent field," Phys. Rev. A 17, 1416–1430 (1978); M. Kus and M. Lewenstein, "The influence of a non-Lorentzian line shape on ionization by the chaotic field," J. Phys. B 14, 4933–4940 (1981).
[CrossRef]

V. M. Baev, T. P. Belikova, E. A. Sviridenkov, and A. F. Suchkov, "Intracavity laser spectroscopy with continuously and quasicontinuously operating lasers," Sov. Phys. JETP 47, 21–29 (1978).

1977 (1)

See, for example, N. C. Wong and J. H. Eberly, "Multiphoton absorption in the presence of two finite-bandwidth lasers," Opt. Lett. 1, 211–213 (1977); B. R. Marx, J. Simons, and L. Allen, "The effect of laser linewidth on two-photon absorption rates," J. Phys. B 11, L273–L277 (1978); A. V. Masalov and L. Allen, "Fluctuation spectroscopy and multiphoton absorption," J. Phys. B 15, 2375–2383 (1982).
[CrossRef] [PubMed]

1976 (1)

See, for example, G. S. Agarwal, "Exact solution of laser temporal fluctuations on resonance fluorescence," Phys. Rev. Lett. 37, 1383–1386 (1976); P. Zoller, "Atomic relaxation and resonance fluorescence in intensity on phase-fluctuating laser light," J. Phys. B 11, 2825–2832 (1978).
[CrossRef]

1973 (1)

S. M. Curry, R. Cubeddu, and T. W. Hänsch, "Intensity stabilization of dye laser radiation by saturated amplification," Appl. Phys. 1, 153–159 (1973).
[CrossRef]

1972 (1)

T. W. Hansch, A. L. Schawlow, and P. E. Toschek, "Ultrasensitive response of a cw dye laser to selective extinction," IEEE J. Quantum Electron. QE-8, 802–804 (1972).
[CrossRef]

1970 (1)

See, for example, R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, "Theory of Stokes pulse shapes in transient stimulated Raman scattering," Phys. Rev. A 2, 60–72 (1970); S. A. Akhmanov, Yu. E. D'Yakav, and L. I. Pavlov, "Statistical phenomena in Raman scattering by a broad-band pump," Sov. Phys. JETP 39, 249–256 (1974).
[CrossRef]

1965 (1)

H. Statz and C. L. Tang, "Phase locking of modes in lasers," J. Appl. Phys. 36, 3923–3927 (1965).
[CrossRef]

1964 (1)

1947 (1)

J. Brossel, "Multi-beam localized fringes: intensity distribution and localization," Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

Agarwal, G. S.

See, for example, G. S. Agarwal, "Exact solution of laser temporal fluctuations on resonance fluorescence," Phys. Rev. Lett. 37, 1383–1386 (1976); P. Zoller, "Atomic relaxation and resonance fluorescence in intensity on phase-fluctuating laser light," J. Phys. B 11, 2825–2832 (1978).
[CrossRef]

Baev, V. M.

V. M. Baev, T. P. Belikova, E. A. Sviridenkov, and A. F. Suchkov, "Intracavity laser spectroscopy with continuously and quasicontinuously operating lasers," Sov. Phys. JETP 47, 21–29 (1978).

Belikova, T. P.

V. M. Baev, T. P. Belikova, E. A. Sviridenkov, and A. F. Suchkov, "Intracavity laser spectroscopy with continuously and quasicontinuously operating lasers," Sov. Phys. JETP 47, 21–29 (1978).

Bloembergen, N.

See, for example, R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, "Theory of Stokes pulse shapes in transient stimulated Raman scattering," Phys. Rev. A 2, 60–72 (1970); S. A. Akhmanov, Yu. E. D'Yakav, and L. I. Pavlov, "Statistical phenomena in Raman scattering by a broad-band pump," Sov. Phys. JETP 39, 249–256 (1974).
[CrossRef]

Brossel, J.

J. Brossel, "Multi-beam localized fringes: intensity distribution and localization," Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

Brunner, W.

W. Brunner and H. Paul, "Spectral properties of dye lasers," Opt. Quantum Electron. 12, 393–411 (1980).
[CrossRef]

Byer, R. L.

See, for example, W. R. Trutna, Jr., Y. K. Park, and R. L. Byer, "The dependence of Raman gain on pump laser bandwidth," IEEE J. Quantum Electron. QE-15, 648–655 (1979); J. Eggleston and R. L. Byer, "Steady-state stimulated Raman scattering by a multimode laser," IEEE J. Quantum Electron. QE-16, 850–853 (1980); G. P. Dzhotyan, Yu. E. D'yakov, I. G. Zubarev, A. B. Mironov, and W. I. Mikhailov, Sov. Phys. JETP 46, 431–435 (1977).
[CrossRef]

Carman, R. L.

See, for example, R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, "Theory of Stokes pulse shapes in transient stimulated Raman scattering," Phys. Rev. A 2, 60–72 (1970); S. A. Akhmanov, Yu. E. D'Yakav, and L. I. Pavlov, "Statistical phenomena in Raman scattering by a broad-band pump," Sov. Phys. JETP 39, 249–256 (1974).
[CrossRef]

Coker, D. F.

L. A. Westling, M. G. Raymer, M. G. Sceats, and D. F. Coker, "Observation of intensity fluctuations and mode correlations in a broadband cw dye laser," Opt. Commun. 47, 212–217 (1983).
[CrossRef]

Cubeddu, R.

S. M. Curry, R. Cubeddu, and T. W. Hänsch, "Intensity stabilization of dye laser radiation by saturated amplification," Appl. Phys. 1, 153–159 (1973).
[CrossRef]

Curry, S. M.

S. M. Curry, R. Cubeddu, and T. W. Hänsch, "Intensity stabilization of dye laser radiation by saturated amplification," Appl. Phys. 1, 153–159 (1973).
[CrossRef]

Eberly, J. H.

See, for example, J. L. F. de Meijere and J. H. Eberly, "Rate of resonant two-photon ionization in the presence of a partially coherent field," Phys. Rev. A 17, 1416–1430 (1978); M. Kus and M. Lewenstein, "The influence of a non-Lorentzian line shape on ionization by the chaotic field," J. Phys. B 14, 4933–4940 (1981).
[CrossRef]

See, for example, N. C. Wong and J. H. Eberly, "Multiphoton absorption in the presence of two finite-bandwidth lasers," Opt. Lett. 1, 211–213 (1977); B. R. Marx, J. Simons, and L. Allen, "The effect of laser linewidth on two-photon absorption rates," J. Phys. B 11, L273–L277 (1978); A. V. Masalov and L. Allen, "Fluctuation spectroscopy and multiphoton absorption," J. Phys. B 15, 2375–2383 (1982).
[CrossRef] [PubMed]

Hansch, T. W.

T. W. Hansch, A. L. Schawlow, and P. E. Toschek, "Ultrasensitive response of a cw dye laser to selective extinction," IEEE J. Quantum Electron. QE-8, 802–804 (1972).
[CrossRef]

Hänsch, T. W.

S. M. Curry, R. Cubeddu, and T. W. Hänsch, "Intensity stabilization of dye laser radiation by saturated amplification," Appl. Phys. 1, 153–159 (1973).
[CrossRef]

Littman, M. G.

M. G. Littman, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08540 (personal communication).

Meijere, J. L. F. de

See, for example, J. L. F. de Meijere and J. H. Eberly, "Rate of resonant two-photon ionization in the presence of a partially coherent field," Phys. Rev. A 17, 1416–1430 (1978); M. Kus and M. Lewenstein, "The influence of a non-Lorentzian line shape on ionization by the chaotic field," J. Phys. B 14, 4933–4940 (1981).
[CrossRef]

Murty, M. V. R. K.

Park, Y. K.

See, for example, W. R. Trutna, Jr., Y. K. Park, and R. L. Byer, "The dependence of Raman gain on pump laser bandwidth," IEEE J. Quantum Electron. QE-15, 648–655 (1979); J. Eggleston and R. L. Byer, "Steady-state stimulated Raman scattering by a multimode laser," IEEE J. Quantum Electron. QE-16, 850–853 (1980); G. P. Dzhotyan, Yu. E. D'yakov, I. G. Zubarev, A. B. Mironov, and W. I. Mikhailov, Sov. Phys. JETP 46, 431–435 (1977).
[CrossRef]

Paul, H.

W. Brunner and H. Paul, "Spectral properties of dye lasers," Opt. Quantum Electron. 12, 393–411 (1980).
[CrossRef]

Rahn, L. A.

L. A. Rahn, Sandia National Laboratory, Livermore, California 95550 (personal communication).

Raymer, M. G.

L. A. Westling, M. G. Raymer, M. G. Sceats, and D. F. Coker, "Observation of intensity fluctuations and mode correlations in a broadband cw dye laser," Opt. Commun. 47, 212–217 (1983).
[CrossRef]

Rogers, J. R.

Sceats, M. G.

L. A. Westling, M. G. Raymer, M. G. Sceats, and D. F. Coker, "Observation of intensity fluctuations and mode correlations in a broadband cw dye laser," Opt. Commun. 47, 212–217 (1983).
[CrossRef]

Schawlow, A. L.

T. W. Hansch, A. L. Schawlow, and P. E. Toschek, "Ultrasensitive response of a cw dye laser to selective extinction," IEEE J. Quantum Electron. QE-8, 802–804 (1972).
[CrossRef]

Shimizu, F.

See, for example, R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, "Theory of Stokes pulse shapes in transient stimulated Raman scattering," Phys. Rev. A 2, 60–72 (1970); S. A. Akhmanov, Yu. E. D'Yakav, and L. I. Pavlov, "Statistical phenomena in Raman scattering by a broad-band pump," Sov. Phys. JETP 39, 249–256 (1974).
[CrossRef]

Statz, H.

H. Statz and C. L. Tang, "Phase locking of modes in lasers," J. Appl. Phys. 36, 3923–3927 (1965).
[CrossRef]

Suchkov, A. F.

V. M. Baev, T. P. Belikova, E. A. Sviridenkov, and A. F. Suchkov, "Intracavity laser spectroscopy with continuously and quasicontinuously operating lasers," Sov. Phys. JETP 47, 21–29 (1978).

Sviridenkov, E. A.

V. M. Baev, T. P. Belikova, E. A. Sviridenkov, and A. F. Suchkov, "Intracavity laser spectroscopy with continuously and quasicontinuously operating lasers," Sov. Phys. JETP 47, 21–29 (1978).

Tang, C. L.

H. Statz and C. L. Tang, "Phase locking of modes in lasers," J. Appl. Phys. 36, 3923–3927 (1965).
[CrossRef]

Toschek, P. E.

T. W. Hansch, A. L. Schawlow, and P. E. Toschek, "Ultrasensitive response of a cw dye laser to selective extinction," IEEE J. Quantum Electron. QE-8, 802–804 (1972).
[CrossRef]

Trutna,, W. R.

See, for example, W. R. Trutna, Jr., Y. K. Park, and R. L. Byer, "The dependence of Raman gain on pump laser bandwidth," IEEE J. Quantum Electron. QE-15, 648–655 (1979); J. Eggleston and R. L. Byer, "Steady-state stimulated Raman scattering by a multimode laser," IEEE J. Quantum Electron. QE-16, 850–853 (1980); G. P. Dzhotyan, Yu. E. D'yakov, I. G. Zubarev, A. B. Mironov, and W. I. Mikhailov, Sov. Phys. JETP 46, 431–435 (1977).
[CrossRef]

Wang, C. S.

See, for example, R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, "Theory of Stokes pulse shapes in transient stimulated Raman scattering," Phys. Rev. A 2, 60–72 (1970); S. A. Akhmanov, Yu. E. D'Yakav, and L. I. Pavlov, "Statistical phenomena in Raman scattering by a broad-band pump," Sov. Phys. JETP 39, 249–256 (1974).
[CrossRef]

Westling, L. A.

L. A. Westling, M. G. Raymer, M. G. Sceats, and D. F. Coker, "Observation of intensity fluctuations and mode correlations in a broadband cw dye laser," Opt. Commun. 47, 212–217 (1983).
[CrossRef]

Wong, N. C.

See, for example, N. C. Wong and J. H. Eberly, "Multiphoton absorption in the presence of two finite-bandwidth lasers," Opt. Lett. 1, 211–213 (1977); B. R. Marx, J. Simons, and L. Allen, "The effect of laser linewidth on two-photon absorption rates," J. Phys. B 11, L273–L277 (1978); A. V. Masalov and L. Allen, "Fluctuation spectroscopy and multiphoton absorption," J. Phys. B 15, 2375–2383 (1982).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. (1)

S. M. Curry, R. Cubeddu, and T. W. Hänsch, "Intensity stabilization of dye laser radiation by saturated amplification," Appl. Phys. 1, 153–159 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. W. Hansch, A. L. Schawlow, and P. E. Toschek, "Ultrasensitive response of a cw dye laser to selective extinction," IEEE J. Quantum Electron. QE-8, 802–804 (1972).
[CrossRef]

J. Appl. Phys. (1)

H. Statz and C. L. Tang, "Phase locking of modes in lasers," J. Appl. Phys. 36, 3923–3927 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

L. A. Westling, M. G. Raymer, M. G. Sceats, and D. F. Coker, "Observation of intensity fluctuations and mode correlations in a broadband cw dye laser," Opt. Commun. 47, 212–217 (1983).
[CrossRef]

Opt. Quantum Electron. (1)

W. Brunner and H. Paul, "Spectral properties of dye lasers," Opt. Quantum Electron. 12, 393–411 (1980).
[CrossRef]

Proc. Phys. Soc. London (1)

J. Brossel, "Multi-beam localized fringes: intensity distribution and localization," Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

Sov. Phys. JETP (1)

V. M. Baev, T. P. Belikova, E. A. Sviridenkov, and A. F. Suchkov, "Intracavity laser spectroscopy with continuously and quasicontinuously operating lasers," Sov. Phys. JETP 47, 21–29 (1978).

Other (8)

Our circuit, available on request, was designed to trigger Quantel Laser power-supply electronics.

M. G. Littman, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08540 (personal communication).

L. A. Rahn, Sandia National Laboratory, Livermore, California 95550 (personal communication).

See, for example, N. C. Wong and J. H. Eberly, "Multiphoton absorption in the presence of two finite-bandwidth lasers," Opt. Lett. 1, 211–213 (1977); B. R. Marx, J. Simons, and L. Allen, "The effect of laser linewidth on two-photon absorption rates," J. Phys. B 11, L273–L277 (1978); A. V. Masalov and L. Allen, "Fluctuation spectroscopy and multiphoton absorption," J. Phys. B 15, 2375–2383 (1982).
[CrossRef] [PubMed]

See, for example, J. L. F. de Meijere and J. H. Eberly, "Rate of resonant two-photon ionization in the presence of a partially coherent field," Phys. Rev. A 17, 1416–1430 (1978); M. Kus and M. Lewenstein, "The influence of a non-Lorentzian line shape on ionization by the chaotic field," J. Phys. B 14, 4933–4940 (1981).
[CrossRef]

See, for example, G. S. Agarwal, "Exact solution of laser temporal fluctuations on resonance fluorescence," Phys. Rev. Lett. 37, 1383–1386 (1976); P. Zoller, "Atomic relaxation and resonance fluorescence in intensity on phase-fluctuating laser light," J. Phys. B 11, 2825–2832 (1978).
[CrossRef]

See, for example, R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, "Theory of Stokes pulse shapes in transient stimulated Raman scattering," Phys. Rev. A 2, 60–72 (1970); S. A. Akhmanov, Yu. E. D'Yakav, and L. I. Pavlov, "Statistical phenomena in Raman scattering by a broad-band pump," Sov. Phys. JETP 39, 249–256 (1974).
[CrossRef]

See, for example, W. R. Trutna, Jr., Y. K. Park, and R. L. Byer, "The dependence of Raman gain on pump laser bandwidth," IEEE J. Quantum Electron. QE-15, 648–655 (1979); J. Eggleston and R. L. Byer, "Steady-state stimulated Raman scattering by a multimode laser," IEEE J. Quantum Electron. QE-16, 850–853 (1980); G. P. Dzhotyan, Yu. E. D'yakov, I. G. Zubarev, A. B. Mironov, and W. I. Mikhailov, Sov. Phys. JETP 46, 431–435 (1977).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Pulsed dye laser. A rhodamine 6G flowing dye cell is side pumped by a frequency-doubled Nd:YAG laser. The 4% reflective mirror M1, the diffraction grating DG, and the 100% reflective mirror M2 define the laser cavity. The cavity length is typically about 15 cm.

Fig. 2
Fig. 2

Fizeau spectrum analyzer. The input beam passes through aperture AP and is expanded to a 3-cm-diameter beam. The collimation of the expanded beam is tested by observing the fringes from shearing interferometer SI. The collimated beam passes through a Fizeau interferometer (see text). The interferometer is mounted on a tilting stage to allow for vertical tilting required for localizing the fringes in the observation plane. The linear fringes are detected by a 1024-element photodiode array PDA and recorded with a transient-waveform recorder for output to an oscilloscope, strip-chart recorder, or computer.

Fig. 3
Fig. 3

Six examples of dye-laser spectra taken with the Fizeau spectrum analyzer. Each shows two orders of the interferometer. Three different values of cavity mode spacing are illustrated: (a), (b) Δνc = 0.035 cm−1; (c), (d) Δνc′ = 0.042 cm−1; (e), (f) Δνc″ = 0.022 cm−1

Fig. 4
Fig. 4

Mode-intensity cross-correlation coefficients B(1), B(2), and B(3) for modes separated by one, two, and three mode spacings, respectively, as a function of cavity mode spacing. The points are determined experimentally by using Eq. (2). The solid curves are hand drawn through the data. The arrows indicate the points at which maximum positive correlations are expected on the basis of the theoretical curves in Fig. 6.

Fig. 5
Fig. 5

Theoretical mode-competition coefficients Cn,n+1, Cn,n+2, and Cn,n+3 for modes separated by one, two, and three mode spacings, respectively, from Eq. (4).

Equations (4)

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

B ( j ) = I n I n + j I n I n + j ,
B ( j ) measured = n = 1 N - j I n I n + j n = 1 N - j I n I n + j ,
d d t q n = - 1 τ n q n + G n q n ( 1 - 1 q s m C n m q m ) ,
C n m = 4 x l l + x sin 2 k n z sin 2 k m z d z ,

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