Abstract

The stability criteria for single-mode standing-wave laser oscillators in the homogeneously broadened limit are reported, and two types of criteria are distinguished. The first type (type 1) corresponds to the minimum value of the threshold parameter for which an infinitesimal perturbation away from steady state grows into an oscillatory solution. A second type (type 2) corresponds to the minimum value of threshold parameter for which large-amplitude oscillations do not decay to the steady-state solution. Undamped pulsations in single-mode homogeneously broadened standing-wave laser oscillators are found to occur at a much higher excitation level than in ring-laser oscillators with homogeneous broadening. The effects of detuning on the stability criteria are also investigated.

© 1993 Optical Society of America

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  1. L. W. Casperson, “Spontaneous pulsations in lasers,” in Third New Zealand Symposium on Laser Physics, J. D. Harvey and D. F. Walls, eds., Vol. 182 of Springer Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), pp. 88–106.
  2. J. C. Englund, R. R. Snapp, and W. C. Schieve, “Fluctuations, instabilities and chaos in the laser driven nonlinear ring cavity,” Prog. Opt. 21, 355–428 (1984).
    [CrossRef]
  3. J. R. Ackerhalt, P. W. Milonni, and M.-L. Shih, “Chaos in quantum optics,” Phys. Rep. 128, 205–300 (1985).
    [CrossRef]
  4. R. G. Harrison and D. J. Biswas, “Pulsating instabilities and chaos in lasers,” Prog. Quantum Electron. 10, 147–228 (1985).
    [CrossRef]
  5. W. J. Firth, “Instabilities and chaos in lasers and optical resonators,” in Chaos, A. V. Holden, ed. (Princeton U. Press, Princeton, N.J., 1986), pp. 135–157.
  6. N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1–190 (1988).
    [CrossRef]
  7. C. O. Weiss, “Chaotic laser dynamics,” Opt. Quantum Electron. 20, 1–22 (1988).
    [CrossRef]
  8. P. W. Milonni, M.-L. Shih, and J. R. Ackerhalt, Chaos in Laser Matter Interactions (World Scientific, Singapore, 1987).
  9. E. N. Lorenz, “Deterministic non-periodic flows,” J. Atmos. Sci. 20, 130–141 (1963).
    [CrossRef]
  10. H. Haken, “Analogy between higher instabilities in fluids and lasers,” Phys. Lett.53A, 77–78 (1975).
  11. C. O. Weiss and W. Klische, “On observability of Lorenz instabilities in lasers,” Opt. Commun. 51, 47–48 (1984).
    [CrossRef]
  12. C. O. Weiss, “Observation of instabilities and chaos in optically pumped far-infrared lasers,” J. Opt. Soc. Am. B 2, 137–140 (1985).
    [CrossRef]
  13. C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, “Instabilities and chaos of a single-mode NH3ring laser,” Opt. Commun. 52, 405–408 (1985).
    [CrossRef]
  14. W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser,” Phys. Rev. A 31, 4049–4051 (1985).
    [CrossRef] [PubMed]
  15. C. O. Weiss and J. Brock, “Evidence of Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804–2806 (1986).
    [CrossRef] [PubMed]
  16. E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571–2574 (1985).
    [CrossRef] [PubMed]
  17. N. B. Abraham, D. Dangoisse, P. Glorieux, and P. Mandel, “Observation of undamped pulsations in a low-pressure, far-infrared laser and comparison with a simple theoretical model,” J. Opt. Soc. Am. B 2, 23–24 (1985).
    [CrossRef]
  18. M. Y. Li, T. Win, C. O. Weiss, and N. R. Heckenberg, “Attractor properties of laser dynamics: a comparison of NH3-laser emission with the Lorenz model,” Opt. Commun. 80, 119–126 (1990).
    [CrossRef]
  19. D. Y. Tang, C. O. Weiss, E. Roldan, and G. J. de Valcarcel, “Deviation from Lorenz-type dynamics of an NH3ring laser,” Opt. Commun. 89, 47–53 (1992).
    [CrossRef]
  20. L. W. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
    [CrossRef]
  21. L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756–761 (1978).
    [CrossRef]
  22. M. A. Dupertuis, R. R. E. Salomaa, and M. R. Siegrist, “The conditions for Lorenz chaos in an optically-pumped far-infrared laser,” Opt. Commun. 57, 410–414 (1986).
    [CrossRef]
  23. R. Corbalan, F. Laguarta, J. Pujol, and R. Vilaseca, “Lorenz-like dynamics in Doppler broadened coherently pumped lasers,” Opt. Commun. 71, 290–294 (1989).
    [CrossRef]
  24. U. Hubner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3laser,” Phys. Rev. A 40, 6354–6365 (1989).
    [CrossRef]
  25. C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
    [CrossRef] [PubMed]
  26. N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 68, 437–441 (1988).
    [CrossRef]
  27. C. O. Weiss and N. B. Abraham, “Characterizing chaotic attractors underlying single mode laser emission by quantitative laser field and phase measurement,” in Measures of Complexity and Chaos, B. Abraham, Al. M. Albano, A. Passamante, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 269–280.
    [CrossRef]
  28. E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
    [CrossRef]
  29. C. O. Weiss, W. Klische, N. B. Abraham, and U. Hubner, “Comparison of NH3laser dynamics with the extended Lorenz model,” Appl. Phys. B 49, 211–215 (1989).
    [CrossRef]
  30. D. Y. Tang, M. Y. Li, and C. O. Weiss, “Field dynamics of a single-mode laser,” Phys. Rev. A 44, 7597–7604 (1991).
    [CrossRef] [PubMed]
  31. H. Zeghlache and P. Mandel, “Phase and amplitude dynamics in the laser Lorenz model,” Phys. Rev. A 38, 3128–3131 (1988).
    [CrossRef] [PubMed]
  32. R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1991).
    [CrossRef]
  33. G. J. de Valcarcel, E. Roldan, and R. Vilaseca, “Lorenz character of the Doppler-broadened far-infrared laser,” J. Opt. Soc. Am. B 8, 2420–2428 (1991).
    [CrossRef]
  34. L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators: stability criteria,” J. Opt. Soc. Am. B 2, 993–997 (1985).
    [CrossRef]
  35. L. W. Casperson, “spontaneous coherent pulsations in standing-wave laser oscillators,” J. Opt. Soc. Am. B 5, 958–969 (1988).
    [CrossRef]
  36. L. W. Casperson, “Stability criteria for lasers with mixed line broadening,” Opt. Quantum Electron. 19, 29–36 (1987).
    [CrossRef]
  37. L. W. Casperson, “Recent progress in modeling single-mode laser instabilities,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds., Vol. 4 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, 1986), pp. 58–71.
  38. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, A1429–A1450 (1964).
    [CrossRef]
  39. L. A. Lugiato and L. M. Narducci, “Nonlinear dynamics in a Fabry–Perot resonator,” Z. Phys. B 71, 129–138 (1988).
    [CrossRef]
  40. Ref. 35, Eqs. (15)–(20).
  41. L. W. Casperson and M. F. H. Tarroja, “Spontaneous coherent pulsations in standing-wave laser oscillators: simplified models,” J. Opt. Soc. Am. B 8, 250–261 (1991).
    [CrossRef]
  42. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1979), Eq. (3.165.1), p. 368.
  43. L. W. Casperson, “Laser power calculations: sources of error,” Appl. Opt. 19, 422–434 (1980), Eq. (38).
    [CrossRef] [PubMed]

1992 (1)

D. Y. Tang, C. O. Weiss, E. Roldan, and G. J. de Valcarcel, “Deviation from Lorenz-type dynamics of an NH3ring laser,” Opt. Commun. 89, 47–53 (1992).
[CrossRef]

1991 (4)

D. Y. Tang, M. Y. Li, and C. O. Weiss, “Field dynamics of a single-mode laser,” Phys. Rev. A 44, 7597–7604 (1991).
[CrossRef] [PubMed]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1991).
[CrossRef]

G. J. de Valcarcel, E. Roldan, and R. Vilaseca, “Lorenz character of the Doppler-broadened far-infrared laser,” J. Opt. Soc. Am. B 8, 2420–2428 (1991).
[CrossRef]

L. W. Casperson and M. F. H. Tarroja, “Spontaneous coherent pulsations in standing-wave laser oscillators: simplified models,” J. Opt. Soc. Am. B 8, 250–261 (1991).
[CrossRef]

1990 (1)

M. Y. Li, T. Win, C. O. Weiss, and N. R. Heckenberg, “Attractor properties of laser dynamics: a comparison of NH3-laser emission with the Lorenz model,” Opt. Commun. 80, 119–126 (1990).
[CrossRef]

1989 (4)

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

C. O. Weiss, W. Klische, N. B. Abraham, and U. Hubner, “Comparison of NH3laser dynamics with the extended Lorenz model,” Appl. Phys. B 49, 211–215 (1989).
[CrossRef]

R. Corbalan, F. Laguarta, J. Pujol, and R. Vilaseca, “Lorenz-like dynamics in Doppler broadened coherently pumped lasers,” Opt. Commun. 71, 290–294 (1989).
[CrossRef]

U. Hubner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3laser,” Phys. Rev. A 40, 6354–6365 (1989).
[CrossRef]

1988 (7)

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 68, 437–441 (1988).
[CrossRef]

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1–190 (1988).
[CrossRef]

C. O. Weiss, “Chaotic laser dynamics,” Opt. Quantum Electron. 20, 1–22 (1988).
[CrossRef]

H. Zeghlache and P. Mandel, “Phase and amplitude dynamics in the laser Lorenz model,” Phys. Rev. A 38, 3128–3131 (1988).
[CrossRef] [PubMed]

L. W. Casperson, “spontaneous coherent pulsations in standing-wave laser oscillators,” J. Opt. Soc. Am. B 5, 958–969 (1988).
[CrossRef]

L. A. Lugiato and L. M. Narducci, “Nonlinear dynamics in a Fabry–Perot resonator,” Z. Phys. B 71, 129–138 (1988).
[CrossRef]

1987 (1)

L. W. Casperson, “Stability criteria for lasers with mixed line broadening,” Opt. Quantum Electron. 19, 29–36 (1987).
[CrossRef]

1986 (2)

M. A. Dupertuis, R. R. E. Salomaa, and M. R. Siegrist, “The conditions for Lorenz chaos in an optically-pumped far-infrared laser,” Opt. Commun. 57, 410–414 (1986).
[CrossRef]

C. O. Weiss and J. Brock, “Evidence of Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804–2806 (1986).
[CrossRef] [PubMed]

1985 (8)

E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571–2574 (1985).
[CrossRef] [PubMed]

N. B. Abraham, D. Dangoisse, P. Glorieux, and P. Mandel, “Observation of undamped pulsations in a low-pressure, far-infrared laser and comparison with a simple theoretical model,” J. Opt. Soc. Am. B 2, 23–24 (1985).
[CrossRef]

J. R. Ackerhalt, P. W. Milonni, and M.-L. Shih, “Chaos in quantum optics,” Phys. Rep. 128, 205–300 (1985).
[CrossRef]

R. G. Harrison and D. J. Biswas, “Pulsating instabilities and chaos in lasers,” Prog. Quantum Electron. 10, 147–228 (1985).
[CrossRef]

C. O. Weiss, “Observation of instabilities and chaos in optically pumped far-infrared lasers,” J. Opt. Soc. Am. B 2, 137–140 (1985).
[CrossRef]

C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, “Instabilities and chaos of a single-mode NH3ring laser,” Opt. Commun. 52, 405–408 (1985).
[CrossRef]

W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser,” Phys. Rev. A 31, 4049–4051 (1985).
[CrossRef] [PubMed]

L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators: stability criteria,” J. Opt. Soc. Am. B 2, 993–997 (1985).
[CrossRef]

1984 (2)

J. C. Englund, R. R. Snapp, and W. C. Schieve, “Fluctuations, instabilities and chaos in the laser driven nonlinear ring cavity,” Prog. Opt. 21, 355–428 (1984).
[CrossRef]

C. O. Weiss and W. Klische, “On observability of Lorenz instabilities in lasers,” Opt. Commun. 51, 47–48 (1984).
[CrossRef]

1980 (1)

1978 (1)

L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756–761 (1978).
[CrossRef]

1972 (1)

L. W. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
[CrossRef]

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, A1429–A1450 (1964).
[CrossRef]

1963 (1)

E. N. Lorenz, “Deterministic non-periodic flows,” J. Atmos. Sci. 20, 130–141 (1963).
[CrossRef]

Abraham, N. B.

U. Hubner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3laser,” Phys. Rev. A 40, 6354–6365 (1989).
[CrossRef]

C. O. Weiss, W. Klische, N. B. Abraham, and U. Hubner, “Comparison of NH3laser dynamics with the extended Lorenz model,” Appl. Phys. B 49, 211–215 (1989).
[CrossRef]

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 68, 437–441 (1988).
[CrossRef]

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1–190 (1988).
[CrossRef]

N. B. Abraham, D. Dangoisse, P. Glorieux, and P. Mandel, “Observation of undamped pulsations in a low-pressure, far-infrared laser and comparison with a simple theoretical model,” J. Opt. Soc. Am. B 2, 23–24 (1985).
[CrossRef]

C. O. Weiss and N. B. Abraham, “Characterizing chaotic attractors underlying single mode laser emission by quantitative laser field and phase measurement,” in Measures of Complexity and Chaos, B. Abraham, Al. M. Albano, A. Passamante, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 269–280.
[CrossRef]

Ackerhalt, J. R.

J. R. Ackerhalt, P. W. Milonni, and M.-L. Shih, “Chaos in quantum optics,” Phys. Rep. 128, 205–300 (1985).
[CrossRef]

P. W. Milonni, M.-L. Shih, and J. R. Ackerhalt, Chaos in Laser Matter Interactions (World Scientific, Singapore, 1987).

Biswas, D. J.

R. G. Harrison and D. J. Biswas, “Pulsating instabilities and chaos in lasers,” Prog. Quantum Electron. 10, 147–228 (1985).
[CrossRef]

Brock, J.

C. O. Weiss and J. Brock, “Evidence of Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804–2806 (1986).
[CrossRef] [PubMed]

Casperson, L. W.

L. W. Casperson and M. F. H. Tarroja, “Spontaneous coherent pulsations in standing-wave laser oscillators: simplified models,” J. Opt. Soc. Am. B 8, 250–261 (1991).
[CrossRef]

L. W. Casperson, “spontaneous coherent pulsations in standing-wave laser oscillators,” J. Opt. Soc. Am. B 5, 958–969 (1988).
[CrossRef]

L. W. Casperson, “Stability criteria for lasers with mixed line broadening,” Opt. Quantum Electron. 19, 29–36 (1987).
[CrossRef]

L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators: stability criteria,” J. Opt. Soc. Am. B 2, 993–997 (1985).
[CrossRef]

L. W. Casperson, “Laser power calculations: sources of error,” Appl. Opt. 19, 422–434 (1980), Eq. (38).
[CrossRef] [PubMed]

L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756–761 (1978).
[CrossRef]

L. W. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
[CrossRef]

L. W. Casperson, “Spontaneous pulsations in lasers,” in Third New Zealand Symposium on Laser Physics, J. D. Harvey and D. F. Walls, eds., Vol. 182 of Springer Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), pp. 88–106.

L. W. Casperson, “Recent progress in modeling single-mode laser instabilities,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds., Vol. 4 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, 1986), pp. 58–71.

Cooper, M.

C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, “Instabilities and chaos of a single-mode NH3ring laser,” Opt. Commun. 52, 405–408 (1985).
[CrossRef]

Corbalan, R.

R. Corbalan, F. Laguarta, J. Pujol, and R. Vilaseca, “Lorenz-like dynamics in Doppler broadened coherently pumped lasers,” Opt. Commun. 71, 290–294 (1989).
[CrossRef]

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

Dangoisse, D.

de Valcarcel, G. J.

D. Y. Tang, C. O. Weiss, E. Roldan, and G. J. de Valcarcel, “Deviation from Lorenz-type dynamics of an NH3ring laser,” Opt. Commun. 89, 47–53 (1992).
[CrossRef]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1991).
[CrossRef]

G. J. de Valcarcel, E. Roldan, and R. Vilaseca, “Lorenz character of the Doppler-broadened far-infrared laser,” J. Opt. Soc. Am. B 8, 2420–2428 (1991).
[CrossRef]

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

Dupertuis, M. A.

M. A. Dupertuis, R. R. E. Salomaa, and M. R. Siegrist, “The conditions for Lorenz chaos in an optically-pumped far-infrared laser,” Opt. Commun. 57, 410–414 (1986).
[CrossRef]

Englund, J. C.

J. C. Englund, R. R. Snapp, and W. C. Schieve, “Fluctuations, instabilities and chaos in the laser driven nonlinear ring cavity,” Prog. Opt. 21, 355–428 (1984).
[CrossRef]

Ering, P. S.

C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, “Instabilities and chaos of a single-mode NH3ring laser,” Opt. Commun. 52, 405–408 (1985).
[CrossRef]

Firth, W. J.

W. J. Firth, “Instabilities and chaos in lasers and optical resonators,” in Chaos, A. V. Holden, ed. (Princeton U. Press, Princeton, N.J., 1986), pp. 135–157.

Glorieux, P.

Godone, A.

E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571–2574 (1985).
[CrossRef] [PubMed]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1979), Eq. (3.165.1), p. 368.

Haken, H.

H. Haken, “Analogy between higher instabilities in fluids and lasers,” Phys. Lett.53A, 77–78 (1975).

Harrison, R. G.

R. G. Harrison and D. J. Biswas, “Pulsating instabilities and chaos in lasers,” Prog. Quantum Electron. 10, 147–228 (1985).
[CrossRef]

Heckenberg, N. R.

M. Y. Li, T. Win, C. O. Weiss, and N. R. Heckenberg, “Attractor properties of laser dynamics: a comparison of NH3-laser emission with the Lorenz model,” Opt. Commun. 80, 119–126 (1990).
[CrossRef]

Hogenboom, E. H. M.

E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571–2574 (1985).
[CrossRef] [PubMed]

Hubner, U.

U. Hubner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3laser,” Phys. Rev. A 40, 6354–6365 (1989).
[CrossRef]

C. O. Weiss, W. Klische, N. B. Abraham, and U. Hubner, “Comparison of NH3laser dynamics with the extended Lorenz model,” Appl. Phys. B 49, 211–215 (1989).
[CrossRef]

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

Klische, W.

C. O. Weiss, W. Klische, N. B. Abraham, and U. Hubner, “Comparison of NH3laser dynamics with the extended Lorenz model,” Appl. Phys. B 49, 211–215 (1989).
[CrossRef]

E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571–2574 (1985).
[CrossRef] [PubMed]

C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, “Instabilities and chaos of a single-mode NH3ring laser,” Opt. Commun. 52, 405–408 (1985).
[CrossRef]

W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser,” Phys. Rev. A 31, 4049–4051 (1985).
[CrossRef] [PubMed]

C. O. Weiss and W. Klische, “On observability of Lorenz instabilities in lasers,” Opt. Commun. 51, 47–48 (1984).
[CrossRef]

Laguarta, F.

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

R. Corbalan, F. Laguarta, J. Pujol, and R. Vilaseca, “Lorenz-like dynamics in Doppler broadened coherently pumped lasers,” Opt. Commun. 71, 290–294 (1989).
[CrossRef]

Lamb, W. E.

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, A1429–A1450 (1964).
[CrossRef]

Li, M. Y.

D. Y. Tang, M. Y. Li, and C. O. Weiss, “Field dynamics of a single-mode laser,” Phys. Rev. A 44, 7597–7604 (1991).
[CrossRef] [PubMed]

M. Y. Li, T. Win, C. O. Weiss, and N. R. Heckenberg, “Attractor properties of laser dynamics: a comparison of NH3-laser emission with the Lorenz model,” Opt. Commun. 80, 119–126 (1990).
[CrossRef]

Lorenz, E. N.

E. N. Lorenz, “Deterministic non-periodic flows,” J. Atmos. Sci. 20, 130–141 (1963).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato and L. M. Narducci, “Nonlinear dynamics in a Fabry–Perot resonator,” Z. Phys. B 71, 129–138 (1988).
[CrossRef]

Mandel, P.

H. Zeghlache and P. Mandel, “Phase and amplitude dynamics in the laser Lorenz model,” Phys. Rev. A 38, 3128–3131 (1988).
[CrossRef] [PubMed]

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1–190 (1988).
[CrossRef]

N. B. Abraham, D. Dangoisse, P. Glorieux, and P. Mandel, “Observation of undamped pulsations in a low-pressure, far-infrared laser and comparison with a simple theoretical model,” J. Opt. Soc. Am. B 2, 23–24 (1985).
[CrossRef]

Milonni, P. W.

J. R. Ackerhalt, P. W. Milonni, and M.-L. Shih, “Chaos in quantum optics,” Phys. Rep. 128, 205–300 (1985).
[CrossRef]

P. W. Milonni, M.-L. Shih, and J. R. Ackerhalt, Chaos in Laser Matter Interactions (World Scientific, Singapore, 1987).

Narducci, L. M.

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1–190 (1988).
[CrossRef]

L. A. Lugiato and L. M. Narducci, “Nonlinear dynamics in a Fabry–Perot resonator,” Z. Phys. B 71, 129–138 (1988).
[CrossRef]

Pujol, J.

R. Corbalan, F. Laguarta, J. Pujol, and R. Vilaseca, “Lorenz-like dynamics in Doppler broadened coherently pumped lasers,” Opt. Commun. 71, 290–294 (1989).
[CrossRef]

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

Roldan, E.

D. Y. Tang, C. O. Weiss, E. Roldan, and G. J. de Valcarcel, “Deviation from Lorenz-type dynamics of an NH3ring laser,” Opt. Commun. 89, 47–53 (1992).
[CrossRef]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1991).
[CrossRef]

G. J. de Valcarcel, E. Roldan, and R. Vilaseca, “Lorenz character of the Doppler-broadened far-infrared laser,” J. Opt. Soc. Am. B 8, 2420–2428 (1991).
[CrossRef]

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1979), Eq. (3.165.1), p. 368.

Salomaa, R. R. E.

M. A. Dupertuis, R. R. E. Salomaa, and M. R. Siegrist, “The conditions for Lorenz chaos in an optically-pumped far-infrared laser,” Opt. Commun. 57, 410–414 (1986).
[CrossRef]

Schieve, W. C.

J. C. Englund, R. R. Snapp, and W. C. Schieve, “Fluctuations, instabilities and chaos in the laser driven nonlinear ring cavity,” Prog. Opt. 21, 355–428 (1984).
[CrossRef]

Shih, M.-L.

J. R. Ackerhalt, P. W. Milonni, and M.-L. Shih, “Chaos in quantum optics,” Phys. Rep. 128, 205–300 (1985).
[CrossRef]

P. W. Milonni, M.-L. Shih, and J. R. Ackerhalt, Chaos in Laser Matter Interactions (World Scientific, Singapore, 1987).

Siegrist, M. R.

M. A. Dupertuis, R. R. E. Salomaa, and M. R. Siegrist, “The conditions for Lorenz chaos in an optically-pumped far-infrared laser,” Opt. Commun. 57, 410–414 (1986).
[CrossRef]

Silba, F.

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

Snapp, R. R.

J. C. Englund, R. R. Snapp, and W. C. Schieve, “Fluctuations, instabilities and chaos in the laser driven nonlinear ring cavity,” Prog. Opt. 21, 355–428 (1984).
[CrossRef]

Tang, D. Y.

D. Y. Tang, C. O. Weiss, E. Roldan, and G. J. de Valcarcel, “Deviation from Lorenz-type dynamics of an NH3ring laser,” Opt. Commun. 89, 47–53 (1992).
[CrossRef]

D. Y. Tang, M. Y. Li, and C. O. Weiss, “Field dynamics of a single-mode laser,” Phys. Rev. A 44, 7597–7604 (1991).
[CrossRef] [PubMed]

Tarroja, M. F. H.

Vilaseca, R.

G. J. de Valcarcel, E. Roldan, and R. Vilaseca, “Lorenz character of the Doppler-broadened far-infrared laser,” J. Opt. Soc. Am. B 8, 2420–2428 (1991).
[CrossRef]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1991).
[CrossRef]

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

R. Corbalan, F. Laguarta, J. Pujol, and R. Vilaseca, “Lorenz-like dynamics in Doppler broadened coherently pumped lasers,” Opt. Commun. 71, 290–294 (1989).
[CrossRef]

Weiss, C. O.

D. Y. Tang, C. O. Weiss, E. Roldan, and G. J. de Valcarcel, “Deviation from Lorenz-type dynamics of an NH3ring laser,” Opt. Commun. 89, 47–53 (1992).
[CrossRef]

D. Y. Tang, M. Y. Li, and C. O. Weiss, “Field dynamics of a single-mode laser,” Phys. Rev. A 44, 7597–7604 (1991).
[CrossRef] [PubMed]

M. Y. Li, T. Win, C. O. Weiss, and N. R. Heckenberg, “Attractor properties of laser dynamics: a comparison of NH3-laser emission with the Lorenz model,” Opt. Commun. 80, 119–126 (1990).
[CrossRef]

U. Hubner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3laser,” Phys. Rev. A 40, 6354–6365 (1989).
[CrossRef]

C. O. Weiss, W. Klische, N. B. Abraham, and U. Hubner, “Comparison of NH3laser dynamics with the extended Lorenz model,” Appl. Phys. B 49, 211–215 (1989).
[CrossRef]

N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 68, 437–441 (1988).
[CrossRef]

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

C. O. Weiss, “Chaotic laser dynamics,” Opt. Quantum Electron. 20, 1–22 (1988).
[CrossRef]

C. O. Weiss and J. Brock, “Evidence of Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804–2806 (1986).
[CrossRef] [PubMed]

E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571–2574 (1985).
[CrossRef] [PubMed]

C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, “Instabilities and chaos of a single-mode NH3ring laser,” Opt. Commun. 52, 405–408 (1985).
[CrossRef]

W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser,” Phys. Rev. A 31, 4049–4051 (1985).
[CrossRef] [PubMed]

C. O. Weiss, “Observation of instabilities and chaos in optically pumped far-infrared lasers,” J. Opt. Soc. Am. B 2, 137–140 (1985).
[CrossRef]

C. O. Weiss and W. Klische, “On observability of Lorenz instabilities in lasers,” Opt. Commun. 51, 47–48 (1984).
[CrossRef]

C. O. Weiss and N. B. Abraham, “Characterizing chaotic attractors underlying single mode laser emission by quantitative laser field and phase measurement,” in Measures of Complexity and Chaos, B. Abraham, Al. M. Albano, A. Passamante, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 269–280.
[CrossRef]

Win, T.

M. Y. Li, T. Win, C. O. Weiss, and N. R. Heckenberg, “Attractor properties of laser dynamics: a comparison of NH3-laser emission with the Lorenz model,” Opt. Commun. 80, 119–126 (1990).
[CrossRef]

Yariv, A.

L. W. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
[CrossRef]

Zeghlache, H.

H. Zeghlache and P. Mandel, “Phase and amplitude dynamics in the laser Lorenz model,” Phys. Rev. A 38, 3128–3131 (1988).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (1)

C. O. Weiss, W. Klische, N. B. Abraham, and U. Hubner, “Comparison of NH3laser dynamics with the extended Lorenz model,” Appl. Phys. B 49, 211–215 (1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

L. W. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
[CrossRef]

L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756–761 (1978).
[CrossRef]

J. Atmos. Sci. (1)

E. N. Lorenz, “Deterministic non-periodic flows,” J. Atmos. Sci. 20, 130–141 (1963).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Commun. (8)

E. Roldan, G. J. de Valcarcel, R. Vilaseca, F. Silba, J. Pujol, R. Corbalan, and F. Laguarta, “Phase dynamics in a Doppler broadened optically-pumped laser,” Opt. Commun. 73, 506–510 (1989).
[CrossRef]

N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 68, 437–441 (1988).
[CrossRef]

M. A. Dupertuis, R. R. E. Salomaa, and M. R. Siegrist, “The conditions for Lorenz chaos in an optically-pumped far-infrared laser,” Opt. Commun. 57, 410–414 (1986).
[CrossRef]

R. Corbalan, F. Laguarta, J. Pujol, and R. Vilaseca, “Lorenz-like dynamics in Doppler broadened coherently pumped lasers,” Opt. Commun. 71, 290–294 (1989).
[CrossRef]

M. Y. Li, T. Win, C. O. Weiss, and N. R. Heckenberg, “Attractor properties of laser dynamics: a comparison of NH3-laser emission with the Lorenz model,” Opt. Commun. 80, 119–126 (1990).
[CrossRef]

D. Y. Tang, C. O. Weiss, E. Roldan, and G. J. de Valcarcel, “Deviation from Lorenz-type dynamics of an NH3ring laser,” Opt. Commun. 89, 47–53 (1992).
[CrossRef]

C. O. Weiss and W. Klische, “On observability of Lorenz instabilities in lasers,” Opt. Commun. 51, 47–48 (1984).
[CrossRef]

C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, “Instabilities and chaos of a single-mode NH3ring laser,” Opt. Commun. 52, 405–408 (1985).
[CrossRef]

Opt. Quantum Electron. (2)

C. O. Weiss, “Chaotic laser dynamics,” Opt. Quantum Electron. 20, 1–22 (1988).
[CrossRef]

L. W. Casperson, “Stability criteria for lasers with mixed line broadening,” Opt. Quantum Electron. 19, 29–36 (1987).
[CrossRef]

Phys. Rep. (1)

J. R. Ackerhalt, P. W. Milonni, and M.-L. Shih, “Chaos in quantum optics,” Phys. Rep. 128, 205–300 (1985).
[CrossRef]

Phys. Rev. (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. 134, A1429–A1450 (1964).
[CrossRef]

Phys. Rev. A (5)

U. Hubner, N. B. Abraham, and C. O. Weiss, “Dimensions and entropies of chaotic intensity pulsations in a single-mode far-infrared NH3laser,” Phys. Rev. A 40, 6354–6365 (1989).
[CrossRef]

D. Y. Tang, M. Y. Li, and C. O. Weiss, “Field dynamics of a single-mode laser,” Phys. Rev. A 44, 7597–7604 (1991).
[CrossRef] [PubMed]

H. Zeghlache and P. Mandel, “Phase and amplitude dynamics in the laser Lorenz model,” Phys. Rev. A 38, 3128–3131 (1988).
[CrossRef] [PubMed]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1991).
[CrossRef]

W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser,” Phys. Rev. A 31, 4049–4051 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

C. O. Weiss and J. Brock, “Evidence of Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804–2806 (1986).
[CrossRef] [PubMed]

E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571–2574 (1985).
[CrossRef] [PubMed]

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

Prog. Opt. (2)

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” Prog. Opt. 25, 1–190 (1988).
[CrossRef]

J. C. Englund, R. R. Snapp, and W. C. Schieve, “Fluctuations, instabilities and chaos in the laser driven nonlinear ring cavity,” Prog. Opt. 21, 355–428 (1984).
[CrossRef]

Prog. Quantum Electron. (1)

R. G. Harrison and D. J. Biswas, “Pulsating instabilities and chaos in lasers,” Prog. Quantum Electron. 10, 147–228 (1985).
[CrossRef]

Z. Phys. B (1)

L. A. Lugiato and L. M. Narducci, “Nonlinear dynamics in a Fabry–Perot resonator,” Z. Phys. B 71, 129–138 (1988).
[CrossRef]

Other (8)

Ref. 35, Eqs. (15)–(20).

C. O. Weiss and N. B. Abraham, “Characterizing chaotic attractors underlying single mode laser emission by quantitative laser field and phase measurement,” in Measures of Complexity and Chaos, B. Abraham, Al. M. Albano, A. Passamante, and P. E. Rapp, eds. (Plenum, New York, 1989), pp. 269–280.
[CrossRef]

L. W. Casperson, “Recent progress in modeling single-mode laser instabilities,” in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds., Vol. 4 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, 1986), pp. 58–71.

W. J. Firth, “Instabilities and chaos in lasers and optical resonators,” in Chaos, A. V. Holden, ed. (Princeton U. Press, Princeton, N.J., 1986), pp. 135–157.

L. W. Casperson, “Spontaneous pulsations in lasers,” in Third New Zealand Symposium on Laser Physics, J. D. Harvey and D. F. Walls, eds., Vol. 182 of Springer Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), pp. 88–106.

P. W. Milonni, M.-L. Shih, and J. R. Ackerhalt, Chaos in Laser Matter Interactions (World Scientific, Singapore, 1987).

H. Haken, “Analogy between higher instabilities in fluids and lasers,” Phys. Lett.53A, 77–78 (1975).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1979), Eq. (3.165.1), p. 368.

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Figures (6)

Fig. 1
Fig. 1

Type 1 (perturbation) stability criteria for a line center tuned homogeneously broadened standing-wave oscillator for various values of the decay rate ratios δ = γ/γc and ρ = γd/γ0. Above or inside one of these curves an infinitesimal perturbation will exhibit oscillatory growth with time, while outside or below a curve the perturbation will decay away.

Fig. 2
Fig. 2

Type 1 stability criteria for a line center tuned homogeneously broadened ring-laser oscillator (after Ref. 34).

Fig. 3
Fig. 3

Type 2 (large-amplitude) stability criteria for a line center tuned homogeneously broadened standing-wave laser oscillator. Above or inside one of these curves a large-amplitude oscillation can exist indefinitely, while outside or below a curve the oscillation will decay.

Fig. 4
Fig. 4

Type 2 stability criteria for a line center tuned homogeneously broadened ring laser oscillator (after Ref. 34).

Fig. 5
Fig. 5

Type 1 stability criteria for a standing-wave homogeneously broadened laser oscillator that is detuned by 0.5 Δνh.

Fig. 6
Fig. 6

Type 2 stability criteria for a standing-wave homogeneously broadened laser oscillator that is detuned by 0.5 Δνh.

Equations (95)

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

( t + v z ) P r ( v , w α , z , t ) = - ( ω - ω α ) P i ( v , ω α , z , t ) - γ P r ( v , ω α , z , t ) + μ 2 sin ( k z ) E i ( t ) D ( v , ω α , z , t ) ,
( t + v z ) P i ( v , ω α , z , t ) = ( ω - ω α ) P r ( v , ω α , z , t ) - γ P i ( v , ω α , z , t ) - μ 2 sin ( k z ) E r ( t ) D ( v , ω α , z , t ) ,
( t + v z ) D ( v , ω α , z , t ) = λ a ( v , ω α , z , t ) - λ b ( v , ω α , z , t ) - γ a + γ a b - γ b 2 D ( v , ω α , z , t ) - γ a + γ a b - γ b 2 M ( v , ω α , z , t ) + sin ( k z ) [ E r ( t ) P i ( v , ω α , z , t ) - E i ( t ) P r ( v , ω α , z , t ) ] ,
( t + v z ) M ( v , ω α , z , t ) = λ a ( v , ω α , z , t ) + λ b ( v , ω α , z , t ) - γ a - γ a b - γ b 2 D ( v , ω α , z , t ) - γ a - γ a b + γ b 2 M ( v , ω α , z , t ) ,
d E r ( t ) d t = - E r ( t ) 2 t c - ( ω - Ω ) E i ( t ) - ω 0 1 L 0 - 0 l sin ( k z ) P i ( v , ω α , z , t ) d z d v d ω α ,
d E i ( t ) d t = - E i ( t ) 2 t c + ( ω - Ω ) E r ( t ) + ω 0 1 L 0 - 0 l sin ( k z ) P r ( v , ω α , z , t ) d z d v d ω α ,
ρ a b ( v , ω α , z , t ) = P ( v , ω α , z , t ) exp ( - i ω t ) / 2 μ ,
E ( z , t ) = ½ sin ( k z ) E ( t ) exp ( - i ω t ) + c . c . ,
D ( v , ω α , z , t ) = ρ a a ( v , ω α , z , t ) - ρ b b ( v , ω α , z , t ) ,
M ( v , ω α , z , t ) = ρ a a ( v , ω α , z , t ) + ρ b b ( v , ω α , z , t ) .
P r ( v , ω α , z , t ) = j = - P r , 2 j + 1 ( v , ω α , t ) exp [ ( 2 j + 1 ) i k z ] ,
P i ( v , ω α , z , t ) = j = - P i , 2 j + 1 ( v , ω α , t ) exp [ ( 2 j + 1 ) i k z ] ,
D ( v , ω α , z , t ) = j = - D 2 j ( v , ω α , t ) exp [ ( 2 j ) i k z ] ,
M ( v , ω α , z , t ) = j = - M 2 j ( v , ω α , t ) exp [ ( 2 j ) i k z ] ,
A r ( t ) = μ 2 ( γ a - γ a b + γ b 2 γ γ a γ b ) 1 / 2 E r ( t ) ,
A i ( t ) = μ 2 ( γ a - γ a b + γ b 2 γ γ a γ b ) 1 / 2 E i ( t ) ,
P i , j ( V , U , t ) = u γ t c ω 0 l μ 1 L ( γ a - γ a b + γ b 2 γ γ a γ b ) 1 / 2 P i , j ( v , ω α , t ) ,
P r , j ( V , U , t ) = u γ t c ω 0 l μ 1 L ( γ a - γ a b + γ b 2 γ γ a γ b ) 1 / 2 P r , j ( v , ω α , t ) ,
D 2 j ( V , U , t ) = u t c ω 0 l μ 2 1 L D 2 j ( v , ω α , z , t ) ,
M 2 j ( V , U , t ) = u t c ω 0 l μ 2 1 L M 2 j ( v , ω α , t ) ,
λ a ( V , U , t ) = u t c ω 0 l μ 2 1 L λ a ( v , ω α , t ) ,
λ b ( V , U , t ) = u t c ω 0 l μ 2 1 L λ b ( v , ω α , t ) ,
V = v u = k v γ ,
U = ω α - ω 0 γ .
P r , 2 j + 1 ( V , U , t ) t = - γ { [ 1 + ( 2 j + 1 ) i V ] P r , 2 j + 1 ( V , U , t ) + ( y - U ) P i , 2 j + 1 ( V , U , t ) + i A i ( t ) × [ D 2 j ( V , U , t ) - D 2 j + 2 ( V , U , t ) ] } ,
P i , 2 j + 1 ( V , U , t ) t = - γ { [ 1 + ( 2 j + 1 ) i V ] P i , 2 j + 1 ( V , U , t ) - ( y - U ) P r , 2 j + 1 ( V , U , t ) - i A r ( t ) × [ D 2 j ( V , U , t ) - D 2 j + 2 ( V , U , t ) ] } ,
D 2 j ( V , U , t ) t = [ λ a ( V , U , t ) - λ b ( V , U , t ) ] δ j 0 - [ γ a + γ a b + γ b 2 + ( 2 j ) i γ V ] D 2 j ( V , U , t ) - γ a + γ a b - γ b 2 M 2 j ( V , U , t ) - i γ 1 { [ A r ( t ) P i , 2 j - 1 ( V , U , t ) - A i ( t ) × P r , 2 j - 1 ( V , U , t ) ] - [ A r ( t ) P i , 2 j + 1 ( V , U , t ) - A i ( t ) P r , 2 j + 1 ( V , U , t ) ] } ,
M 2 j ( V , U , t ) t = [ λ a ( V , U , t ) + λ b ( V , U , t ) ] δ j 0 - [ γ a - γ a b + γ b 2 + ( 2 j ) i γ V ] M 2 j ( V , U , t ) - γ a - γ a b - γ b 2 D 2 j ( V , U , t ) ,
d A r ( t ) d t = - 1 2 t c [ A r ( t ) + δ ( y - y 0 ) A i ( t ) - - - P i , 1 i ( V , U , t ) d V d U ] ,
d A i ( t ) d t = - 1 2 t c [ A i ( t ) - δ ( y - y 0 ) A r ( t ) + - - P r , 1 i ( V , U , t ) d v d U ] ,
P r , 2 j + 1 ( t ) = - - P r , 2 j + 1 ( V , U , t ) d V d U ,
P i , 2 j + 1 ( t ) = - - P i , 2 j + 1 ( V , U , t ) d V d U ,
D 2 j ( t ) = - - D 2 j ( V , U , t ) d V d U ,
M 2 j ( t ) = - - M 2 j ( V , U , t ) d V d U ,
λ a ( t ) = - - λ a ( V , U , t ) d V d U ,
λ b ( t ) = - - λ b ( V , U , t ) d V d U .
P r , 2 j + 1 ( t ) t = - γ { P r , 2 j + 1 ( t ) + y P i , 2 j + 1 ( t ) + i A i ( t ) × [ D 2 j ( t ) - D 2 j + 2 ( t ) ] } ,
P i , 2 j + 1 ( t ) t = - γ { P i , 2 j + 1 ( t ) - y P r , 2 j + 1 ( t ) - i A r ( t ) × [ D 2 j ( t ) - D 2 j + 2 ( t ) ] } ,
D 2 j ( t ) t = [ λ a ( t ) - λ b ( t ) ] δ j 0 - γ a + γ a b + γ b 2 D 2 j ( t ) - γ a + γ a b - γ b 2 M 2 j ( t ) - i γ 1 { [ A r ( t ) P i , 2 j - 1 ( t ) - A i ( t ) P r , 2 j - 1 ( t ) ] - [ A r ( t ) P i , 2 j + 1 ( t ) - A i ( t ) P r , 2 j + 1 ( t ) ] } ,
M 2 j ( t ) t = [ λ a ( t ) + λ b ( t ) ] δ j 0 - γ a - γ a b + γ b 2 M 2 j ( t ) - γ a - γ a b - γ b 2 D 2 j ( t ) ,
A r ( t ) t = - 1 2 t c [ A r ( t ) + δ ( y - y 0 ) A i ( t ) - P i , 1 i ( t ) ] ,
A i ( t ) t = - 1 2 t c [ A i ( t ) - δ ( y - y 0 ) A r ( t ) + P r , 1 i ( t ) ] .
P r , 2 j + 1 ( t ) t = - δ { P r , 2 j + 1 ( t ) + y P i , 2 j + 1 ( t ) + i A i ( t ) × [ D 2 j ( t ) - D 2 j + 2 ( t ) ] } ,
P i , 2 j + 1 ( t ) t = - δ { P i , 2 j + 1 ( t ) - y P r , 2 j + 1 ( t ) - i A r ( t ) × [ D 2 j ( t ) - D 2 j + 2 ( t ) ] } ,
D 2 j ( t ) t = λ d ( t ) γ c δ j 0 - ρ δ D 2 j ( t ) - i ρ δ × { [ A r ( t ) P i , 2 j - 1 ( t ) - A i ( t ) P r , 2 j - 1 ( t ) ] - [ A r ( t ) P i , 2 j + 1 ( t ) - A i ( t ) P r , 2 j + 1 ( t ) ] } ,
d A r ( t ) t = - [ A r ( t ) + δ ( y - y 0 ) A i ( t ) - P i , 1 i ( t ) ] ,
d A i ( t ) t = - [ A i ( t ) - δ ( y - y 0 ) A r ( t ) + P r , 1 i ( t ) ] ,
P r ( z , t ) = 0 - P r ( v , ω α , z , t ) d v d ω α ,
P i ( z , t ) = 0 - P i ( v , ω α , z , t ) d v d ω α ,
D ( z , t ) = 0 - D ( v , ω α , z , t ) d v d ω α ,
M ( z , t ) = 0 - M ( v , ω α , z , t ) d v d ω α ,
λ a ( z , t ) = 0 - λ a ( v , ω α , z , t ) d v d ω α ,
λ b ( z , t ) = 0 - λ b ( v , ω α , z , t ) d v d ω α .
P r ( z , t ) t = - γ P r ( z , t ) + μ 2 sin ( k z ) E i ( t ) D ( z , t ) ,
P i ( z , t ) t = - γ P i ( z , t ) - μ 2 sin ( k z ) E r ( t ) D ( z , t ) ,
D ( z , t ) t = λ d ( z , t ) - γ d D ( z , t ) + 1 sin ( k z ) [ E r ( t ) P i ( z , t ) - E i ( t ) P r ( z , t ) ] ,
d E r ( t ) d t = - E r ( t ) 2 t c - ω 0 1 L 0 l sin ( k z ) P i ( z , t ) d z ,
d E i ( t ) d t = - E i ( t ) 2 t c + ω 0 1 L 0 l sin ( k z ) P r ( z , t ) d z ,
P r ( z , t ) t = - γ P r ( z , t ) + μ 2 sin ( k z ) E i ( t ) D ( z , t ) ,
D ( z , t ) t = λ d - γ d D ( z , t ) + 1 sin ( k z ) E r ( t ) P i ( z , t ) ,
d E r ( t ) d t = - E r ( t ) 2 t c - ω 0 1 L 0 l sin ( k z ) P i ( z , t ) d z .
A r ( t ) = - μ 2 ( 1 γ γ d ) 1 / 2 E r ( t ) ,
P i ( ζ , t ) = ω 0 1 L μ 2 ( 1 γ γ d ) 1 / 2 1 γ c l π P i ( z , t ) ,
D ( ζ , t ) = μ 2 γ ω 0 1 L 1 γ c l 2 π D ( z , t ) ,
D 0 = μ 2 γ ω 0 1 L 1 γ c l 2 π λ d γ d .
P i ( z , t ) t = - γ [ P i ( ζ , t ) + 2 sin ( ζ ) A r ( t ) D ( ζ , t ) ] ,
D ( ζ , t ) t = - γ d [ D ( ζ , t ) - D 0 - 2 sin ( ζ ) A r ( t ) P i ( ζ , t ) ] ,
A r ( t ) t = - γ c [ A r ( t ) + 2 0 π / 2 sin ( ζ ) P i ( ζ , t ) d ζ ] .
P i s ( ζ ) = - 2 sin ( ζ ) A r s D s ( ζ ) ,
D s ( ζ ) = D 0 1 + 4 s i n 2 ( ζ ) A rs 2 .
1 = 4 0 π / 2 sin 2 ( ζ ) D 0 1 + 4 sin 2 ( ζ ) A r s 2 d ζ .
1 = 4 0 π / 2 sin 2 ( ζ ) D 0 d ζ = π D 0 .
1 = 4 r π 0 π / 2 sin 2 ( ζ ) 1 + 4 sin 2 ( ζ ) A r s 2 d ζ = 2 r π 0 π / 2 1 - cos ( ζ ) 1 + 4 sin 2 ( ζ ) A r s 2 d ζ = 2 r 1 + 4 A r s 2 + ( 1 + 4 A r s 2 ) 1 / 2 ,
0 π / 2 cos ( 2 n x ) 1 - a 2 sin 2 ( x ) d x = ( - 1 ) n π 2 ( 1 - a 2 ) 1 / 2 ( 1 - ( 1 - a 2 ) 1 / 2 a ) 2 n .
( 1 + 4 A r s 2 ) 1 / 2 = - 1 + ( 1 + 8 r ) 1 / 2 2 .
A r s 2 = 4 r - 1 - ( 1 + 8 r ) 1 / 2 8 .
P i ( ζ , t ) = P i z ( ζ ) + P i ( ζ , t ) ,
D ( ζ , t ) = D s ( ζ ) + D ( ζ , t ) ,
A r ( t ) = A r s + A r ( t ) ,
d P i ( ζ , t ) d t = - γ { P i ( ζ , t ) + 2 sin ( ζ ) × [ A r s D ( ζ , t ) + A r ( t ) D s ( ζ ) ] } ,
d D ( ζ , t ) d t = - γ d { D ( ζ , t ) - 2 sin ( ζ ) × [ A r s P i ( ζ , t ) + A r ( t ) P i s ( ζ ) ] } ,
d A r ( t ) d t = - γ c [ A r ( t ) + 2 0 π / 2 sin ( ζ ) P i ( ζ , t ) d ζ ] .
P i ( ζ , t ) = P i ( ζ ) exp ( s t ) ,
D ( ζ , t ) = D ( ζ ) exp ( s t ) ,
A r ( t ) = A r exp ( s t ) ,
s P i ( ζ ) = - γ { P i ( ζ ) + 2 sin ( ζ ) [ A r s D ( ζ ) + A r D s ( ζ ) ] } ,
s D ( ζ ) = - γ d { D ( ζ ) - 2 sin ( ζ ) [ A r s P i ( ζ ) + A r P i s ( ζ ) ] } ,
s A r = - γ c [ A r + 2 0 π / 2 sin ( ζ ) P i ( ζ ) d ζ ] .
P i ( ζ ) = - 2 γ sin ( ζ ) s + γ [ A r r / π 1 + 4 sin 2 ( ζ ) A r s 2 + A r s D ( ζ ) ] ,
D ( ζ ) = 2 γ d sin ( ζ ) s + γ d [ - 2 sin ( ζ ) A r s A r r / π 1 + 4 sin 2 ( ζ ) A r s 2 + A r s P i ( ζ ) ] ,
A r = - 2 γ c s + γ c 0 π / 2 sin ( ζ ) P i ( ζ ) d ζ ,
P i ( ζ ) = - { 2 γ sin ( ζ ) A r r / π ( s + γ ) [ 1 + 4 sin 2 ( ζ ) A r s 2 ] } [ 1 - 4 γ d sin 2 ( ζ ) A r s 2 s + γ d ] 1 + 4 γ γ d sin 2 ( ζ ) A r s 2 ( s + γ ) ( s + γ d ) .
1 = 4 γ γ c r / π ( s + γ ) ( s + γ c ) × 0 π / 2 sin 2 ( ζ ) [ 1 - 4 γ d sin 2 ( ζ ) A r s 2 s + γ d ] [ 1 + 4 sin 2 ( ζ ) A r s 2 ] [ 1 + 4 γ γ d sin 2 ( ζ ) A r s 2 ( s + γ ) ( s + γ d ) ] d ζ .
1 = 2 γ γ c r / [ ( s + γ ) ( s + γ c ) ] 4 γ γ d A r s 2 / [ ( s + γ ) ( s + γ d ) ] - 4 A r s 2 × ( { 1 ( 1 + 4 A r s 2 ) 1 / 2 - 1 [ 1 + 4 γ γ d A r s 2 ( s + γ ) ( s + γ d ) ] 1 / 2 } - 4 γ d A r s 2 s + γ d { 1 1 + 4 A r s 2 + ( 1 + 4 A r s 2 ) 1 / 2 - 1 1 + 4 γ γ s A r s 2 ( s + γ ) ( s + γ d ) + [ 1 + 4 γ γ s A r s 2 ( s + γ ) ( s + γ d ) ] 1 / 2 } ) .
1 = 2 δ r ( s + 1 ) ( s + δ ) 4 δ 2 ρ A r s 2 ( s + δ ) ( s + δ ρ ) - 4 A r s 2 × ( { 1 ( 1 + 4 A r s 2 ) 1 / 2 - 1 [ 1 + 4 δ 2 ρ A r s 2 ( s + δ ) ( s + δ ρ ) ] 1 / 2 } - 4 δ ρ A r s 2 s + δ ρ { 1 1 + 4 A r s 2 + ( 1 + 4 A r s 2 ) 1 / 2 - 1 1 + 4 δ 2 ρ A r s 2 ( s + δ ) ( s + δ ρ ) + [ 1 + 4 δ 2 ρ A r s 2 ( s + δ ) ( s + δ ρ ) ] 1 / 2 } ) ,

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