Abstract

Multilongitudinal mode instability in ring Nd:YAG lasers is theoretically analyzed. After we review the way in which the standard two-level laser theory applies to this laser we extend the theoretical treatment to include transverse effects. We do this by taking into account the finite transverse section of the active medium and by assuming a Gaussian transverse distribution for the intracavity field. Finally we demonstrate that multimode emission develops whenever the intracavity field waist diameter is almost equal to the active rod diameter. We conclude that continuous-wave diode-pumped Nd:YAG lasers with low cavity losses are good candidates for the observation of the Risken–Nummedal–Graham–Haken instability.

© 2003 Optical Society of America

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References

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  1. H. Risken and K. Nummedal, “Instability of off resonance mode in lasers,” Phys. Lett. A 26, 275–276 (1968).
    [CrossRef]
  2. H. Risken and K. Nummedal, “Self-pulsing in laser,” J. Appl. Phys. 39, 4662–4672 (1968).
    [CrossRef]
  3. R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
    [CrossRef]
  4. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Weinheim, Germany, 1991).
  5. E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
    [CrossRef]
  6. E. Roldán and G. J. de Valcárcel, “On the observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped fiber lasers,” Europhys. Lett. 43, 255–260 (1998).
    [CrossRef]
  7. E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fibre lasers,” Phys. Rev. A 60, 2517–2528 (1999).
    [CrossRef]
  8. T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” in International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, S. N. Bagaev, V. N. Zadkov, and S. M. Arakelian, eds., Proc. SPIE 4429, 112–115 (2001).
    [CrossRef]
  9. F. Mitschke, Universität Rostock, Rostock, Germany (personal communication, 2002).
  10. E. Roldán, G. J. de Valcárcel, F. Silva, and F. Prati, “Multimode emission in inhomogeneously-brodened ring lasers,” J. Opt. Soc. Am. B 18, 1601–1611 (2001).
    [CrossRef]
  11. E. Roldán and G. J. de Valcárcel, “Multimode instability in inhomogeneously broadened class-B ring lasers: beyond the uniform field limit,” Phys. Rev. A 64, 053805 (2001).
    [CrossRef]
  12. E. Roldán, G. J. de Valcárcel, and F. Mitschke, “Localized and distributed intracavity losses and the Risken–Nummedal–Graham–Haken instability,” Appl. Phys. B (to be published).
  13. L. A. Lugiato and M. Milani, “Effects of Gaussian-beam averaging on laser instabilities,” J. Opt. Soc. Am. B 2, 15–17 (1985).
    [CrossRef]
  14. C. P. Smith and R. Dykstra, “Lorenz-like chaos in a Gaussian mode laser with radially dependent gain,” Opt. Commun. 117, 107–110 (1995).
    [CrossRef]
  15. J. F. Urchueguía, G. J. de Valcárcel, and E. Roldán, “Laser instabilities in a Gaussian cavity mode with Gaussian pump profile,” J. Opt. Soc. Am. B 15, 1512–1520 (1998).
    [CrossRef]
  16. J. F. Urchueguía, G. J. de Valcárcel, E. Roldán, and F. Prati, “Transverse effects in ring fibre laser multimode instabilities,” Phys. Rev. A 62, 041801 (2000).
    [CrossRef]
  17. In principle there could be a small inhomogeneous contribution to the gain line caused by the proximity of some Nd ions to the sparing defects in the YAG crystal. But such inhomogeneous broadening has not been reported as far as we know, and thus the assumption of pure homogeneous broadening is a good approximation.
  18. A slightly different model was derived in Ref. 7 because equal relaxation rates for the two lasing levels were assumed. This is not a realistic approximation for Nd:YAG lasers, and thus the model given here corrects that of Ref. 7. The present derivation follows Ref. 19.
  19. Ya. I. Khanin, Principles of Laser Dynamics (Elsevier, Amsterdam, 1995).
  20. T=1−R, where R is the fraction of light exiting the medium that is reinjected by the cavity into the medium after one round trip. 0≤R≤1 by definition, and the uniform field limit requires that R→1, i.e., that T→0.
  21. W. Koechner, Solid State Laser Engineering, Vol. X of Springer Series in Optical Sciences (Springer, New York, 1976).
  22. In practice this means that the detuning between a longitudinal mode and the gain line, which at most equals half of the cavity’s free spectral range Δω, must be much smaller than the width of the gain line γ; i.e., Δω≡πc/Lc≪γ. With the given parameters for Nd:YAG lasers we have Δω/γ~10−4, so the inequality is safely verified.
  23. P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge U. Press, Cambridge, 1997).
  24. A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
    [CrossRef]
  25. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986).
  26. J. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969).
    [CrossRef] [PubMed]
  27. J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
    [CrossRef] [PubMed]
  28. D. Dangoisse, D. Hennequin, C. Leppers, E. Iouvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A 46, 5955–5958 (1992).
    [CrossRef] [PubMed]
  29. Lamp pumping is not considered because thermal lensing effects make effective pumping more that 1.5 times above threshold impossible, whereas pumping at least 9 times above threshold is required for observing instability.

2001 (3)

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” in International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, S. N. Bagaev, V. N. Zadkov, and S. M. Arakelian, eds., Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

E. Roldán and G. J. de Valcárcel, “Multimode instability in inhomogeneously broadened class-B ring lasers: beyond the uniform field limit,” Phys. Rev. A 64, 053805 (2001).
[CrossRef]

E. Roldán, G. J. de Valcárcel, F. Silva, and F. Prati, “Multimode emission in inhomogeneously-brodened ring lasers,” J. Opt. Soc. Am. B 18, 1601–1611 (2001).
[CrossRef]

2000 (1)

J. F. Urchueguía, G. J. de Valcárcel, E. Roldán, and F. Prati, “Transverse effects in ring fibre laser multimode instabilities,” Phys. Rev. A 62, 041801 (2000).
[CrossRef]

1999 (1)

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fibre lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

1998 (2)

J. F. Urchueguía, G. J. de Valcárcel, and E. Roldán, “Laser instabilities in a Gaussian cavity mode with Gaussian pump profile,” J. Opt. Soc. Am. B 15, 1512–1520 (1998).
[CrossRef]

E. Roldán and G. J. de Valcárcel, “On the observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped fiber lasers,” Europhys. Lett. 43, 255–260 (1998).
[CrossRef]

1997 (1)

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

1995 (1)

C. P. Smith and R. Dykstra, “Lorenz-like chaos in a Gaussian mode laser with radially dependent gain,” Opt. Commun. 117, 107–110 (1995).
[CrossRef]

1992 (1)

D. Dangoisse, D. Hennequin, C. Leppers, E. Iouvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

1989 (1)

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

1985 (1)

1969 (1)

1968 (3)

H. Risken and K. Nummedal, “Instability of off resonance mode in lasers,” Phys. Lett. A 26, 275–276 (1968).
[CrossRef]

H. Risken and K. Nummedal, “Self-pulsing in laser,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
[CrossRef]

1963 (1)

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Arnaud, J.

Bonfrate, G.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

Dangoisse, D.

D. Dangoisse, D. Hennequin, C. Leppers, E. Iouvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

de Valcárcel, G. J.

E. Roldán and G. J. de Valcárcel, “Multimode instability in inhomogeneously broadened class-B ring lasers: beyond the uniform field limit,” Phys. Rev. A 64, 053805 (2001).
[CrossRef]

E. Roldán, G. J. de Valcárcel, F. Silva, and F. Prati, “Multimode emission in inhomogeneously-brodened ring lasers,” J. Opt. Soc. Am. B 18, 1601–1611 (2001).
[CrossRef]

J. F. Urchueguía, G. J. de Valcárcel, E. Roldán, and F. Prati, “Transverse effects in ring fibre laser multimode instabilities,” Phys. Rev. A 62, 041801 (2000).
[CrossRef]

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fibre lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

J. F. Urchueguía, G. J. de Valcárcel, and E. Roldán, “Laser instabilities in a Gaussian cavity mode with Gaussian pump profile,” J. Opt. Soc. Am. B 15, 1512–1520 (1998).
[CrossRef]

E. Roldán and G. J. de Valcárcel, “On the observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped fiber lasers,” Europhys. Lett. 43, 255–260 (1998).
[CrossRef]

Dykstra, R.

C. P. Smith and R. Dykstra, “Lorenz-like chaos in a Gaussian mode laser with radially dependent gain,” Opt. Commun. 117, 107–110 (1995).
[CrossRef]

Fontana, F.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

Fox, A. G.

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Ghazzawi, A. M.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

Glorieux, P.

D. Dangoisse, D. Hennequin, C. Leppers, E. Iouvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Graham, R.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
[CrossRef]

Green, C.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

Haken, H.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
[CrossRef]

Hennequin, D.

D. Dangoisse, D. Hennequin, C. Leppers, E. Iouvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Iouvergneaux, E.

D. Dangoisse, D. Hennequin, C. Leppers, E. Iouvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Lenz, M. O.

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” in International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, S. N. Bagaev, V. N. Zadkov, and S. M. Arakelian, eds., Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

Leppers, C.

D. Dangoisse, D. Hennequin, C. Leppers, E. Iouvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

Li, T.

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Lugiato, L. A.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

L. A. Lugiato and M. Milani, “Effects of Gaussian-beam averaging on laser instabilities,” J. Opt. Soc. Am. B 2, 15–17 (1985).
[CrossRef]

Milani, M.

Mitschke, F.

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” in International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, S. N. Bagaev, V. N. Zadkov, and S. M. Arakelian, eds., Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

Narducci, L. M.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

Nummedal, K.

H. Risken and K. Nummedal, “Self-pulsing in laser,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

H. Risken and K. Nummedal, “Instability of off resonance mode in lasers,” Phys. Lett. A 26, 275–276 (1968).
[CrossRef]

Pernigo, M. A.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

Pessina, E. M.

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fibre lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

Prati, F.

E. Roldán, G. J. de Valcárcel, F. Silva, and F. Prati, “Multimode emission in inhomogeneously-brodened ring lasers,” J. Opt. Soc. Am. B 18, 1601–1611 (2001).
[CrossRef]

J. F. Urchueguía, G. J. de Valcárcel, E. Roldán, and F. Prati, “Transverse effects in ring fibre laser multimode instabilities,” Phys. Rev. A 62, 041801 (2000).
[CrossRef]

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fibre lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

Quel, E. J.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

Redondo, J.

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fibre lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

Risken, H.

H. Risken and K. Nummedal, “Instability of off resonance mode in lasers,” Phys. Lett. A 26, 275–276 (1968).
[CrossRef]

H. Risken and K. Nummedal, “Self-pulsing in laser,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Roldán, E.

E. Roldán, G. J. de Valcárcel, F. Silva, and F. Prati, “Multimode emission in inhomogeneously-brodened ring lasers,” J. Opt. Soc. Am. B 18, 1601–1611 (2001).
[CrossRef]

E. Roldán and G. J. de Valcárcel, “Multimode instability in inhomogeneously broadened class-B ring lasers: beyond the uniform field limit,” Phys. Rev. A 64, 053805 (2001).
[CrossRef]

J. F. Urchueguía, G. J. de Valcárcel, E. Roldán, and F. Prati, “Transverse effects in ring fibre laser multimode instabilities,” Phys. Rev. A 62, 041801 (2000).
[CrossRef]

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fibre lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

J. F. Urchueguía, G. J. de Valcárcel, and E. Roldán, “Laser instabilities in a Gaussian cavity mode with Gaussian pump profile,” J. Opt. Soc. Am. B 15, 1512–1520 (1998).
[CrossRef]

E. Roldán and G. J. de Valcárcel, “On the observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped fiber lasers,” Europhys. Lett. 43, 255–260 (1998).
[CrossRef]

Silva, F.

Smith, C. P.

C. P. Smith and R. Dykstra, “Lorenz-like chaos in a Gaussian mode laser with radially dependent gain,” Opt. Commun. 117, 107–110 (1995).
[CrossRef]

Tredicce, J. R.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

Urchueguía, J. F.

J. F. Urchueguía, G. J. de Valcárcel, E. Roldán, and F. Prati, “Transverse effects in ring fibre laser multimode instabilities,” Phys. Rev. A 62, 041801 (2000).
[CrossRef]

J. F. Urchueguía, G. J. de Valcárcel, and E. Roldán, “Laser instabilities in a Gaussian cavity mode with Gaussian pump profile,” J. Opt. Soc. Am. B 15, 1512–1520 (1998).
[CrossRef]

Voigt, T.

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” in International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, S. N. Bagaev, V. N. Zadkov, and S. M. Arakelian, eds., Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

Appl. Opt. (1)

Europhys. Lett. (1)

E. Roldán and G. J. de Valcárcel, “On the observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped fiber lasers,” Europhys. Lett. 43, 255–260 (1998).
[CrossRef]

J. Appl. Phys. (1)

H. Risken and K. Nummedal, “Self-pulsing in laser,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

C. P. Smith and R. Dykstra, “Lorenz-like chaos in a Gaussian mode laser with radially dependent gain,” Opt. Commun. 117, 107–110 (1995).
[CrossRef]

Phys. Lett. A (1)

H. Risken and K. Nummedal, “Instability of off resonance mode in lasers,” Phys. Lett. A 26, 275–276 (1968).
[CrossRef]

Phys. Rev. A (5)

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fibre lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

J. F. Urchueguía, G. J. de Valcárcel, E. Roldán, and F. Prati, “Transverse effects in ring fibre laser multimode instabilities,” Phys. Rev. A 62, 041801 (2000).
[CrossRef]

D. Dangoisse, D. Hennequin, C. Leppers, E. Iouvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A 46, 5955–5958 (1992).
[CrossRef] [PubMed]

E. Roldán and G. J. de Valcárcel, “Multimode instability in inhomogeneously broadened class-B ring lasers: beyond the uniform field limit,” Phys. Rev. A 64, 053805 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, “Spatial and temporal instabilities in a CO2 laser,” Phys. Rev. Lett. 62, 1274–1277 (1989).
[CrossRef] [PubMed]

Proc. IEEE (1)

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Proc. SPIE (1)

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” in International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, S. N. Bagaev, V. N. Zadkov, and S. M. Arakelian, eds., Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

Z. Phys. (1)

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
[CrossRef]

Other (12)

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Weinheim, Germany, 1991).

F. Mitschke, Universität Rostock, Rostock, Germany (personal communication, 2002).

In principle there could be a small inhomogeneous contribution to the gain line caused by the proximity of some Nd ions to the sparing defects in the YAG crystal. But such inhomogeneous broadening has not been reported as far as we know, and thus the assumption of pure homogeneous broadening is a good approximation.

A slightly different model was derived in Ref. 7 because equal relaxation rates for the two lasing levels were assumed. This is not a realistic approximation for Nd:YAG lasers, and thus the model given here corrects that of Ref. 7. The present derivation follows Ref. 19.

Ya. I. Khanin, Principles of Laser Dynamics (Elsevier, Amsterdam, 1995).

T=1−R, where R is the fraction of light exiting the medium that is reinjected by the cavity into the medium after one round trip. 0≤R≤1 by definition, and the uniform field limit requires that R→1, i.e., that T→0.

W. Koechner, Solid State Laser Engineering, Vol. X of Springer Series in Optical Sciences (Springer, New York, 1976).

In practice this means that the detuning between a longitudinal mode and the gain line, which at most equals half of the cavity’s free spectral range Δω, must be much smaller than the width of the gain line γ; i.e., Δω≡πc/Lc≪γ. With the given parameters for Nd:YAG lasers we have Δω/γ~10−4, so the inequality is safely verified.

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge U. Press, Cambridge, 1997).

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986).

E. Roldán, G. J. de Valcárcel, and F. Mitschke, “Localized and distributed intracavity losses and the Risken–Nummedal–Graham–Haken instability,” Appl. Phys. B (to be published).

Lamp pumping is not considered because thermal lensing effects make effective pumping more that 1.5 times above threshold impossible, whereas pumping at least 9 times above threshold is required for observing instability.

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Figures (4)

Fig. 1
Fig. 1

Switch-on threshold R on as a function of the size of the beam in units of active rod radius x for three values of G.

Fig. 2
Fig. 2

(a) Minimum instability threshold R c , (b) instability to lasing threshold ratio R c / R on , and (c) critical cavity length L c c as functions of x for three values of G.

Fig. 3
Fig. 3

Instability to the lasing threshold (upper figures) and instability threshold (lower figures) as functions of x for G = 10 4 and (a) L c = 9   m and (b) L c = 12.5   m .

Fig. 4
Fig. 4

Instability threshold as a function of L c for G = 10 4 and x = 1 (top) and x = 1.24 (bottom). The lasing threshold is R on = 0.688   s - 1 for x = 1 and R on = 0.910   s - 1 for x = 1.24 .

Equations (62)

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v z E + t E = g   Im   ρ 12 - κ E ,
t Im   ρ 12 = E ρ ρ 22 - γ Im   ρ 12 ,
t ρ 22 = - 2 E   Im   ρ 12 - ( γ 02 + γ int ) ρ 22 + R ( 1 - ρ 22 ) ,
t = t + L c - L m c z L m ,
E ( z = 0 ,   t ) = E ( z = L m ,   t ) .
R = n 0 B 03 W .
v = c   L m L c , κ = c T 2 L c , g = 1 2   v N σ γ ,
L c = L c + ( n - 1 ) L m
g = 5.05 × 10 21 s - 2 .
κ = c T 2 L c 8 × 10 5 s - 1 ,
G g κ γ = N σ L m T 1.5 × 10 4 ,
γ = γ 02 + γ int ,
β = 1 + R / γ
F = 2 [ γ ( R + γ ) ] 1 / 2   E ,
P = [ 2 γ ( R + γ ) ] 1 / 2 R Im   ρ 21 ,
D = R + γ R   ρ 22 ,
v z F + t F = - κ ( F - AP ) ,
t P = - γ ( P - FD ) ,
t D = - γ β ( D - 1 + FP ) ,
A = G   R R + γ
R = γ A ( G - A ) .
τ = γ γ t , ζ = z / L m ,
η ζ F + τ F = - σ ( F - AP ) ,
τ P = - γ - 1 ( P - FD ) ,
τ D = - γ β ( D - 1 + FP ) ,
η = c γ γ L c , σ = κ / γ γ , γ = γ / γ .
F ( ζ = 0 ,   τ ) = F ( ζ = 1 ,   τ ) .
R on = γ / ( G - 1 ) .
0 = λ 3 + c 2 λ 2 + c 1 λ + c 0 ,
c 2 = γ - 1 + γ β + σ + i α ,
c 1 = β ( A + γ σ ) + i α ( 1 + γ 2 β ) ,
c 0 = β [ 2 σ ( A - 1 ) + i α A ] ,
α = k α ˜ = k   2 π c γ γ L c , k = 0 ,   ± 1 ,   ± 2 , ,
α ± 2 = ( β / 2 ) { 3 ( A - 1 ) ± [ ( A - 1 ) 2 - 8 ( A - 1 ) ] 1 / 2 } .
R c = 9 γ / ( G - 9 ) .
R c / R on = 9 ( G - 1 ) / ( G - 9 ) 9 ,
α c 2 = 12 ( 1 + R c / γ ) = 12 G / ( G - 9 ) 12 .
L c c = 2 π c / 12 γ γ 13 m ,
η ζ F ( ζ ,   τ ) + τ F ( ζ ,   τ )
= - σ F - A 0 u m du   exp ( - u / 2 P ) ,
τ P ( u ,   ζ ,   τ ) = - γ - 1 [ P - exp ( - u / 2 ) FD ] ,
τ D ( u ,   ζ ,   τ ) = - γ β [ D - 1 + exp ( - u / 2 ) FP ] ,
u m = ( ρ m / w 0 ) 2
D ( u ) = 1 1 + exp ( - u ) F 2 , P ( u ) = exp ( - u / 2 ) F 1 + exp ( - u ) F 2 ,
I = A   ln 1 + I 1 + exp ( - u m ) I .
A on = 1 1 - exp ( - u m ) .
R on = γ G [ 1 - exp ( - u m ) ] - 1 .
δ F ( ζ ,   τ ) δ P ( u ,   ζ ,   τ ) δ D ( u ,   ζ ,   τ ) = f p ( u ) d ( u ) exp ( λ τ ) exp ( i η - 1 α ζ ) ,
λ = - i α - σ + σ A 0 u m d u exp ( - u ) 1 + exp ( - u ) I × λ + γ β [ 1 - exp ( - u ) I ] ( 1 + γ λ ) ( λ + γ β ) + γ β   exp ( - u ) I ,
λ = - i α - σ + σ   λ + 2 γ β λ ( 1 + γ λ + γ 2 β )
+ σ A I ( 2 + γ λ ) ( λ + γ β ) λ ( 1 + γ λ + γ 2 β ) × ln 1 - γ β [ 1 - exp ( - u m ) ] I ( 1 + γ λ ) ( λ + γ β ) + γ β I .
λ = λ 0 + γ λ 1 + γ 2 λ 2 +   .
Re   λ 2 = - σ α 2   ( c 0 - c 2 α 2 + α 4 ) ,
c 0 = β 2 AI [ 1 - exp ( - 2 u m ) ] ,
c 2 = 3 β ( A / A on - 1 ) ,
α ± 2 = c 2 / 2 ± [ ( c 2 / 2 ) 2 - c 0 ] 1 / 2 ,
9 ( A / A on - 1 ) 2 4 AI [ 1 - exp ( - 2 u m ) ] .
α c = [ 3 β ( A c / A on - 1 ) / 2 ] 1 / 2 .
η ζ F + τ F = - σ ( F - u m AP ) ,
τ P = - γ - 1 ( P - FD ) ,
τ D = - γ β ( D - 1 + FP )
x = w 0 / ρ m = u m - 1 / 2 ,

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