Abstract

Spontaneous pulsation phenomena in xenon lasers are now well known, and related effects have been observed with other laser types. Most theoretical analyses of such pulsations are formulated for unidirectional ring lasers, whereas most experiments involve standing-wave resonator geometries. Described here are rigorous semiclassical models governing the spontaneous pulsation instability in standing-wave laser oscillators. The standing-wave results are more complex and closer to experimental observations than the ring results for operation in the Lambdip region near line center.

© 1988 Optical Society of America

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  1. A. G. Gurtovnik, Izv. Vyssh. Uchebn. Zavedenii Radiofiz. 1, 83 (1958) [English translation TG 230-T382 in Applied Physics Laboratory Library Bulletin (Johns Hopkins University, Baltimore, Md., 1963)].
  2. G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399 (1958).
    [CrossRef]
  3. C. Kikuchi, J. Lambe, G. Makhov, and R. W. Terhune, J. Appl. Phys. 30, 1061 (1959).
    [CrossRef]
  4. L. W. Casperson, 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), p. 88.
  5. L. W. Casperson, J. Opt. Soc. Am. B 2, 62 (1985).
    [CrossRef]
  6. J. C. Englund, R. R. Snapp, and W. C. Schieve, Prog. Opt. 21, 355 (1984).
    [CrossRef]
  7. N. B. Abraham, L. A. Lugiato, and L. M. Narducci, J. Opt. Soc. Am. B 2, 7 (1985).
    [CrossRef]
  8. J. R. Ackerhalt, P. W. Milonni, and M. L. Shih, Phys. Rep. 128, 205 (1985).
    [CrossRef]
  9. R. G. Harrison and D. J. Biswas, Prog. Quantum Electron. 10, 147 (1985).
    [CrossRef]
  10. N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” in Progress in Optics, Vol. 25, E. Wolf, ed. (North-Holland, Amsterdam, to be published).
  11. J. Bentley and N. B. Abraham, Opt. Commun. 41, 52 (1982).
    [CrossRef]
  12. M. Maeda and N. B. Abraham, Phys. Rev. A 26, 3395 (1982).
    [CrossRef]
  13. N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.
  14. R. S. Gioggia and N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
    [CrossRef]
  15. R. S. Gioggia and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 563.
  16. L. E. Urbach, S. N. Liu, and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 593.
  17. L. M. Hoffer, T. H. Chyba, and N. B. Abraham, J. Opt. Soc. Am. B 2, 102 (1985).
    [CrossRef]
  18. M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.
  19. M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148 (1986).
    [CrossRef] [PubMed]
  20. L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
    [CrossRef]
  21. L. W. Casperson, J. Opt. Soc. Am B 5, 970 (1988).
    [CrossRef]
  22. L. W. Casperson, Phys. Rev. A 21, 911 (1980), Eq. (45).
    [CrossRef]
  23. L. W. Casperson, Phys. Rev. A 23, 248 (1981), Eq. (87).
    [CrossRef]
  24. L. W. Casperson, J. Opt. Soc. Am. B 2, 73 (1985).
    [CrossRef]
  25. L. W. Casperson, J. Opt. Soc. Am. B 2, 993 (1985).
    [CrossRef]
  26. L. W. Casperson, Opt. Quantum Electron. 19, 29 (1987).
    [CrossRef]
  27. L. W. Casperson and A. Yariv, Appl. Opt. 11, 462 (1972).
    [CrossRef] [PubMed]

1988 (1)

L. W. Casperson, J. Opt. Soc. Am B 5, 970 (1988).
[CrossRef]

1987 (1)

L. W. Casperson, Opt. Quantum Electron. 19, 29 (1987).
[CrossRef]

1986 (1)

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148 (1986).
[CrossRef] [PubMed]

1985 (7)

1984 (1)

J. C. Englund, R. R. Snapp, and W. C. Schieve, Prog. Opt. 21, 355 (1984).
[CrossRef]

1983 (1)

R. S. Gioggia and N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
[CrossRef]

1982 (2)

J. Bentley and N. B. Abraham, Opt. Commun. 41, 52 (1982).
[CrossRef]

M. Maeda and N. B. Abraham, Phys. Rev. A 26, 3395 (1982).
[CrossRef]

1981 (1)

L. W. Casperson, Phys. Rev. A 23, 248 (1981), Eq. (87).
[CrossRef]

1980 (1)

L. W. Casperson, Phys. Rev. A 21, 911 (1980), Eq. (45).
[CrossRef]

1978 (1)

L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

1972 (1)

1959 (1)

C. Kikuchi, J. Lambe, G. Makhov, and R. W. Terhune, J. Appl. Phys. 30, 1061 (1959).
[CrossRef]

1958 (2)

A. G. Gurtovnik, Izv. Vyssh. Uchebn. Zavedenii Radiofiz. 1, 83 (1958) [English translation TG 230-T382 in Applied Physics Laboratory Library Bulletin (Johns Hopkins University, Baltimore, Md., 1963)].

G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399 (1958).
[CrossRef]

Abraham, N. B.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148 (1986).
[CrossRef] [PubMed]

L. M. Hoffer, T. H. Chyba, and N. B. Abraham, J. Opt. Soc. Am. B 2, 102 (1985).
[CrossRef]

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, J. Opt. Soc. Am. B 2, 7 (1985).
[CrossRef]

R. S. Gioggia and N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
[CrossRef]

J. Bentley and N. B. Abraham, Opt. Commun. 41, 52 (1982).
[CrossRef]

M. Maeda and N. B. Abraham, Phys. Rev. A 26, 3395 (1982).
[CrossRef]

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” in Progress in Optics, Vol. 25, E. Wolf, ed. (North-Holland, Amsterdam, to be published).

R. S. Gioggia and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 563.

L. E. Urbach, S. N. Liu, and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 593.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

Ackerhalt, J. R.

J. R. Ackerhalt, P. W. Milonni, and M. L. Shih, Phys. Rep. 128, 205 (1985).
[CrossRef]

Adams, S. P.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

Bandy, D. K.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148 (1986).
[CrossRef] [PubMed]

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

Bentley, J.

J. Bentley and N. B. Abraham, Opt. Commun. 41, 52 (1982).
[CrossRef]

Biswas, D. J.

R. G. Harrison and D. J. Biswas, Prog. Quantum Electron. 10, 147 (1985).
[CrossRef]

Casperson, L. W.

L. W. Casperson, J. Opt. Soc. Am B 5, 970 (1988).
[CrossRef]

L. W. Casperson, Opt. Quantum Electron. 19, 29 (1987).
[CrossRef]

L. W. Casperson, J. Opt. Soc. Am. B 2, 62 (1985).
[CrossRef]

L. W. Casperson, J. Opt. Soc. Am. B 2, 73 (1985).
[CrossRef]

L. W. Casperson, J. Opt. Soc. Am. B 2, 993 (1985).
[CrossRef]

L. W. Casperson, Phys. Rev. A 23, 248 (1981), Eq. (87).
[CrossRef]

L. W. Casperson, Phys. Rev. A 21, 911 (1980), Eq. (45).
[CrossRef]

L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

L. W. Casperson and A. Yariv, Appl. Opt. 11, 462 (1972).
[CrossRef] [PubMed]

L. W. Casperson, 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), p. 88.

Chyba, T.

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

Chyba, T. H.

Coleman, M.

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

Englund, J. C.

J. C. Englund, R. R. Snapp, and W. C. Schieve, Prog. Opt. 21, 355 (1984).
[CrossRef]

Gioggia, R. S.

R. S. Gioggia and N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
[CrossRef]

R. S. Gioggia and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 563.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

Gurtovnik, A. G.

A. G. Gurtovnik, Izv. Vyssh. Uchebn. Zavedenii Radiofiz. 1, 83 (1958) [English translation TG 230-T382 in Applied Physics Laboratory Library Bulletin (Johns Hopkins University, Baltimore, Md., 1963)].

Halas, N. J.

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

Harrison, R. G.

R. G. Harrison and D. J. Biswas, Prog. Quantum Electron. 10, 147 (1985).
[CrossRef]

Hoffer, L. M.

L. M. Hoffer, T. H. Chyba, and N. B. Abraham, J. Opt. Soc. Am. B 2, 102 (1985).
[CrossRef]

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

Isaacs, T.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

Kikuchi, C.

C. Kikuchi, J. Lambe, G. Makhov, and R. W. Terhune, J. Appl. Phys. 30, 1061 (1959).
[CrossRef]

G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399 (1958).
[CrossRef]

Lambe, J.

C. Kikuchi, J. Lambe, G. Makhov, and R. W. Terhune, J. Appl. Phys. 30, 1061 (1959).
[CrossRef]

G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399 (1958).
[CrossRef]

Liu, S. N.

L. E. Urbach, S. N. Liu, and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 593.

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

Lugiato, L. A.

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, J. Opt. Soc. Am. B 2, 7 (1985).
[CrossRef]

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

Maeda, M.

M. Maeda and N. B. Abraham, Phys. Rev. A 26, 3395 (1982).
[CrossRef]

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

Makhov, G.

C. Kikuchi, J. Lambe, G. Makhov, and R. W. Terhune, J. Appl. Phys. 30, 1061 (1959).
[CrossRef]

G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399 (1958).
[CrossRef]

Mandel, P.

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” in Progress in Optics, Vol. 25, E. Wolf, ed. (North-Holland, Amsterdam, to be published).

Milonni, P. W.

J. R. Ackerhalt, P. W. Milonni, and M. L. Shih, Phys. Rep. 128, 205 (1985).
[CrossRef]

Narducci, L. M.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148 (1986).
[CrossRef] [PubMed]

N. B. Abraham, L. A. Lugiato, and L. M. Narducci, J. Opt. Soc. Am. B 2, 7 (1985).
[CrossRef]

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” in Progress in Optics, Vol. 25, E. Wolf, ed. (North-Holland, Amsterdam, to be published).

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

Schieve, W. C.

J. C. Englund, R. R. Snapp, and W. C. Schieve, Prog. Opt. 21, 355 (1984).
[CrossRef]

Shih, M. L.

J. R. Ackerhalt, P. W. Milonni, and M. L. Shih, Phys. Rep. 128, 205 (1985).
[CrossRef]

Snapp, R. R.

J. C. Englund, R. R. Snapp, and W. C. Schieve, Prog. Opt. 21, 355 (1984).
[CrossRef]

Tarroja, M. F. H.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148 (1986).
[CrossRef] [PubMed]

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

Terhune, R. W.

C. Kikuchi, J. Lambe, G. Makhov, and R. W. Terhune, J. Appl. Phys. 30, 1061 (1959).
[CrossRef]

G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399 (1958).
[CrossRef]

Urbach, L. E.

L. E. Urbach, S. N. Liu, and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 593.

Wesson, J. C.

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

Yariv, A.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

Izv. Vyssh. Uchebn. Zavedenii Radiofiz. (1)

A. G. Gurtovnik, Izv. Vyssh. Uchebn. Zavedenii Radiofiz. 1, 83 (1958) [English translation TG 230-T382 in Applied Physics Laboratory Library Bulletin (Johns Hopkins University, Baltimore, Md., 1963)].

J. Appl. Phys. (1)

C. Kikuchi, J. Lambe, G. Makhov, and R. W. Terhune, J. Appl. Phys. 30, 1061 (1959).
[CrossRef]

J. Opt. Soc. Am B (1)

L. W. Casperson, J. Opt. Soc. Am B 5, 970 (1988).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

J. Bentley and N. B. Abraham, Opt. Commun. 41, 52 (1982).
[CrossRef]

Opt. Quantum Electron. (1)

L. W. Casperson, Opt. Quantum Electron. 19, 29 (1987).
[CrossRef]

Phys. Rep. (1)

J. R. Ackerhalt, P. W. Milonni, and M. L. Shih, Phys. Rep. 128, 205 (1985).
[CrossRef]

Phys. Rev. (1)

G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399 (1958).
[CrossRef]

Phys. Rev. A (4)

M. Maeda and N. B. Abraham, Phys. Rev. A 26, 3395 (1982).
[CrossRef]

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148 (1986).
[CrossRef] [PubMed]

L. W. Casperson, Phys. Rev. A 21, 911 (1980), Eq. (45).
[CrossRef]

L. W. Casperson, Phys. Rev. A 23, 248 (1981), Eq. (87).
[CrossRef]

Phys. Rev. Lett. (1)

R. S. Gioggia and N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
[CrossRef]

Prog. Opt. (1)

J. C. Englund, R. R. Snapp, and W. C. Schieve, Prog. Opt. 21, 355 (1984).
[CrossRef]

Prog. Quantum Electron. (1)

R. G. Harrison and D. J. Biswas, Prog. Quantum Electron. 10, 147 (1985).
[CrossRef]

Other (6)

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” in Progress in Optics, Vol. 25, E. Wolf, ed. (North-Holland, Amsterdam, to be published).

L. W. Casperson, 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), p. 88.

N. B. Abraham, T. Chyba, M. Coleman, R. S. Gioggia, N. J. Halas, L. M. Hoffer, S. N. Liu, M. Maeda, and J. C. Wesson, 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), p. 107.

R. S. Gioggia and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 563.

L. E. Urbach, S. N. Liu, and N. B. Abraham, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 593.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, T. Isaacs, R. S. Gioggia, S. P. Adams, L. M. Narducci, and L. A. Lugiato, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 246.

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Figures (6)

Fig. 1
Fig. 1

Theoretical spontaneous pulsation intensity waveforms for a single-mode standing-wave xenon laser with line-center tuning and threshold parameter values of (a) r = 2.0, (b) r = 1.8, (c) r = 1.6, (d) r = 1.4, (e) r = 1.2, and (f) r = 1.1. The amplitude, frequency, and waveform complexity of the pulsations all tend to decrease with decreasing values of r.

Fig. 2
Fig. 2

Experimental time-domain plot of the output intensity from a low-pressure xenon laser at a discharge current of 40 mA and a time scale of 0.2 μsec/division.

Fig. 3
Fig. 3

Theoretical spontaneous pulsation waveforms for a xenon laser with a threshold parameter of r = 1.8 and detunings of (a) Δν = 0Δνh, (b) Δν = 2.5Δνh, (c) Δν = 5Δνh, and (d) Δν = 7.5Δνh. In contrast to the ring-laser results, the pulsation intensity and periodicity tend initially to increase with detuning from line center.

Fig. 4
Fig. 4

Theoretical Lamb-dip curves of average intensity versus frequency tuning for a spontaneously pulsing low-pressure xenon laser. The frequency detuning is measured in units of the homogeneous line width Δνh.

Fig. 5
Fig. 5

Experimental Lamb-dip power curve for a low-pressure xenon laser with decreasing cavity length (increasing frequency) and a discharge current of 18 mA (after Ref. 27).

Fig. 6
Fig. 6

Theoretical Lamb-dip power curves from Fig. 4 after multiplication by the square of the gain and dispersion focusing correction given in Eq. (95).

Equations (98)

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( t + υ z ) ρ a b ( υ , ω α , z , t ) = ( i ω α + γ ) ρ a b ( υ , ω α , z , t ) i μ E ( z , t ) [ ρ a a ( υ , ω α , z , t ) ρ b b ( υ , ω α , z , t ) ] ,
( t + υ z ) ρ a b ( υ , ω α , z , t ) = λ a ( υ , ω α , z , t ) γ a ρ a a ( υ , ω α , z , t ) + [ i μ E ( z , t ) ρ b a ( υ , ω α , z , t ) + c . c . ] Γ a ( υ , υ ) ρ a a ( υ , ω α , z , t ) d υ + Γ a ( υ , υ ) ρ a a ( υ , ω α , z , t ) d υ ,
( t + υ z ) ρ b b ( υ , ω α , z , t ) = λ b ( υ , ω α , z , t ) γ b ρ b b ( υ , ω α , z , t ) + γ a b ρ a a ( υ , ω α , z , t ) [ i μ E ( z , t ) ρ b a ( υ , ω α , z , t ) + c . c . ] Γ b ( υ , υ ) ρ b b ( υ , ω α , z , t ) d υ + Γ b ( υ , υ ) ρ b b ( υ , ω α , z , t ) d υ ,
ρ b a ( υ , ω α , z , t ) = ρ a b * ( υ , ω α , z , t ) ,
2 E ( z , t ) z 2 μ 1 σ E ( z , t ) t μ 1 1 2 E ( z , t ) t 2 = μ 1 2 P ( z , t ) t 2 .
P ( z , t ) = 0 μ ρ a b ( υ , ω α , z , t ) d υ d ω α + c . c .
E ( z , t ) = ½ sin ( k z ) E ( t ) exp ( i ω t ) + c . c . ,
ρ a b ( υ , ω α , z , t ) = P ( υ , ω α , z , t ) exp ( i ω t ) / 2 μ .
( t + υ z ) P ( υ , ω α , z , t ) = i ( ω ω α ) P ( υ , ω α , z , t ) γ P ( υ , ω α , z , t ) i μ 2 sin ( k z ) E ( t ) × [ ρ a a ( υ , ω α , z , t ) ρ b b ( υ , ω α , z , t ) ] ,
( t + υ z ) ρ a a ( υ , ω α , z , t ) = λ a ( υ , ω α , z , t ) γ a ρ a a ( υ , ω α , z , t ) + i 4 sin ( k z ) [ E ( t ) P * ( υ , ω α , z , t ) E * ( t ) P ( υ , ω α , z , t ) ] Γ a ( υ , υ ) ρ a a ( υ , ω α , z , t ) d υ + Γ a ( υ , υ ) ρ a a ( υ , ω α , z , t ) d υ ,
( t + υ z ) ρ b b ( υ , ω α , z , t ) = λ b ( υ , ω α , z , t ) γ b ρ b b ( υ , ω α , z , t ) + γ a b ρ a a ( υ , ω α , z , t ) i 4 sin ( k z ) [ E ( t ) P * ( υ , ω α , z , t ) E * ( t ) P ( υ , ω α , z , t ) ] Γ b ( υ , υ ) ρ b b ( υ , ω α , z , t ) d υ + Γ b ( υ , υ ) ρ b b ( υ , ω α , z , t ) d υ .
sin ( k z ) [ i μ 1 1 ω d E ( t ) d t + i μ 1 σ ω 2 E ( t ) + ( ω 2 μ 1 1 2 k 2 2 ) E ( t ) ] = μ 1 ω 2 2 0 P ( υ , ω α , z , t ) d υ d ω α ,
sin ( k z ) [ i d E ( t ) d t + i σ 2 1 E ( t ) + ω 2 Ω 2 2 ω E ( t ) ] = ω 2 1 0 P ( υ , ω α , z , t ) d υ d ω α .
d E ( t ) d t = σ 2 1 E ( t ) + i ( ω Ω ) E ( t ) + i ω 0 2 1 0 0 L sin ( k z ) P ( υ , ω α , z , t ) d z d υ d ω α .
P ( υ , ω α , z , t ) = P r ( υ , ω α , z , t ) + i P i ( υ , ω α , z , t )
E ( t ) = E r ( t ) + i E i ( t ) .
( t + υ z ) P r ( υ , ω α , z , t ) = ( ω ω α ) P i ( υ , ω α , z , t ) γ P r ( υ , ω α , z , t ) + μ 2 sin ( k z ) E i ( t ) D ( υ , ω α , z , t ) ,
( t + υ z ) P i ( υ , ω α , z , t ) = ( ω ω α ) P r ( υ , ω α , z , t ) γ P i ( υ , ω α , z , t ) μ 2 sin ( k z ) E r ( t ) D ( υ , ω α , z , t ) ,
( t + υ z ) D ( υ , ω α , z , t ) = λ a ( υ , ω α , z , t ) λ b ( υ , ω α , z , t ) γ a + γ a b + λ b 2 D ( υ , ω α , z , t ) γ a + γ a b γ b 2 M ( υ , ω α , z , t ) + sin ( k z ) [ E r ( t ) P i ( υ , ω α , z , t ) E i ( t ) P r ( υ , ω α , z , t ) ] 1 2 Γ a ( υ , υ ) [ M ( υ , ω α , z , t ) + D ( υ , ω α , z , t ) ] d υ + 1 2 Γ a ( υ , υ ) [ M ( υ , ω α , z , t ) + D ( υ , ω α , z , t ) ] d υ + 1 2 Γ b ( υ , υ ) [ M ( υ , ω α , z , t ) D ( υ , ω α , z , t ) ] d υ 1 2 Γ b ( υ , υ ) [ M ( υ , ω α , z , t ) D ( υ , ω α , z , t ) ] d υ
( t + υ z ) M ( υ , ω α , z , t ) = λ a ( υ , ω α , z , t ) + λ b ( υ , ω α , z , t ) γ a γ a b γ b 2 D ( υ , ω α , z , t ) γ a γ a b + γ b 2 M ( υ , ω α , z , t ) 1 2 Γ a ( υ , υ ) [ M ( υ , ω α , z , t ) + D ( υ , ω α , z , t ) ] d υ + 1 2 Γ a ( υ , υ ) [ M ( υ , ω α , z , t ) + D ( υ , ω α , z , t ) ] d υ 1 2 Γ b ( υ , υ ) [ M ( υ , ω α , z , t ) D ( υ , ω α , z , t ) ] d υ + 1 2 Γ b ( υ , υ ) [ M ( υ , ω α , z , t ) D ( υ , ω α , z , t ) ] d υ ,
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 ( υ , ω α , z , t ) d z d υ 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 ( υ , ω α , z , t ) d z d υ d ω α ,
Γ a ( υ , υ ) = Γ a u π 1 / 2 exp ( υ 2 / u 2 ) ,
Γ b ( υ , υ ) = Γ b u π 1 / 2 exp ( υ 2 / u 2 ) ,
( t + υ z ) D ( υ , ω α , z , t ) = λ a ( υ , ω α , z , t ) λ b ( υ , ω α , z , t ) γ a + γ a b + γ b 2 D ( υ , ω α , z , t ) γ a + γ a b γ b 2 M ( υ , ω α , z , t ) + sin ( k z ) [ E r ( t ) P i ( υ , ω α , z , t ) E i ( t ) P r ( υ , ω α , z , t ) ] + Γ a 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M ( υ , ω α , z , t ) + D ( υ , ω α , z , t ) ] d υ Γ b 2 u π 1 / 2 exp ( υ 2 / u 2 ) × [ M ( υ , ω α , z , t ) D ( υ , ω α , z , t ) ] d υ ,
( t + υ z ) M ( υ , ω α , z , t ) = λ a ( υ , ω α , z , t ) + λ b ( υ , ω α , z , t ) γ a γ a b γ b 2 D ( υ , ω α , z , t ) γ a γ a b + γ b 2 M ( υ , ω α , z , t ) + Γ a 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M ( υ , ω α , z , t ) + D ( υ , ω α , z , t ) ] d υ + Γ b 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M ( υ , ω α , z , t ) D ( υ , ω α , z , t ) ] d υ ,
P r ( υ , ω α , z , t ) = j = P r , 2 j + 1 ( υ , ω α , t ) exp [ ( 2 j + 1 ) ikz ] ,
P i ( υ , ω α , z , t ) = j = P i , 2 j + 1 ( υ , ω α , t ) exp [ ( 2 j + 1 ) ikz ] ,
D ( υ , ω α , z , t ) = j = D 2 j ( υ , ω α , t ) exp [ ( 2 j ) ikz ] ,
M ( υ , ω α , z , t ) = j = M 2 j ( υ , ω α , t ) exp [ ( 2 j ) ikz ] ,
P r , j ( υ , ω α , t ) = P r , j * ( υ , ω α , t ) ,
P i , j ( υ , ω α , t ) = P i , j * ( υ , ω α , t ) ,
D j ( υ , ω α , t ) = D j * ( υ , ω α , t ) ,
M j ( υ , ω α , t ) = M j * ( υ , ω α , t ) .
P r , 2 j + 1 ( υ , ω α , t ) t = ( ω ω α ) P i , 2 j + 1 ( υ , ω α , t ) [ ( 2 j + 1 ) i k υ + γ ] P r , 2 j + 1 ( υ , ω α , t ) i μ 2 E i ( t ) [ D 2 j ( υ , ω α , t ) D 2 j + 2 ( υ , ω α , t ) ] ,
P i , 2 j + 1 ( υ , ω α , t ) t = ( ω ω α ) P i , 2 j + 1 ( υ , ω α , t ) [ ( 2 j + 1 ) i k υ + γ ] P i , 2 j + 1 ( υ , ω α , t ) + i μ 2 2 E r ( t ) [ D 2 j ( υ , ω α , t ) D 2 j + 2 ( υ , ω α , t ) ] ,
D 2 j ( υ , ω α , t ) t = [ λ a ( υ , ω α , t ) λ b ( υ , ω α , t ) ] δ j 0 [ ( 2 j ) i k υ + h 1 ] D 2 j ( υ , ω α , t ) h 2 M 2 j ( υ , ω α , t ) i 2 { [ E r ( t ) P i , 2 j 1 ( υ , ω α , t ) E i ( t ) P r , 2 j 1 ( υ , ω α , t ) ] [ E r ( t ) P i , 2 j + 1 ( υ , ω α , t ) E i ( t ) P i , 2 j + 1 ( υ , ω α , t ) ] } + Γ a 2 u π 1 / 2 exp ( υ 2 / υ 2 ) [ M 2 j ( υ , ω α , t ) + D 2 j ( υ , ω α , t ) ] d υ Γ b 2 u π 1 / 2 exp ( υ 2 / υ 2 ) [ M 2 j ( υ , ω α , t ) D 2 j ( υ , ω α , t ) ] d υ ,
M 2 j ( υ , ω α , t ) t = [ λ a ( υ , ω α , t ) + λ b ( υ , ω α , t ) ] δ j 0 [ ( 2 j ) i k υ + h 3 ] M 2 j ( υ , ω α , t ) h 4 D 2 j ( υ , ω α , t ) + Γ a 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M 2 j ( υ , ω α , t ) + D 2 j ( υ , ω α , t ) ] d υ + Γ b 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M 2 j ( υ , ω α , t ) D 2 j ( υ , ω α , t ) ] d υ ,
d E r ( t ) d t = E r ( t ) 2 t c ( ω Ω ) E i ( t ) ω 0 l 1 L 0 P i , 1 ( υ , ω α , t ) P i , 1 ( υ , ω α , t ) 2 i d υ d ω α ,
d E i ( t ) d t = E i ( t ) 2 t c + ( ω Ω ) E r ( t ) + ω 0 l 1 L 0 P r , 1 ( υ , ω α , t ) P r , 1 ( υ , ω α , t ) 2 i d υ d ω α ,
h 1 = ( γ a + γ a b + γ b ) / 2 ,
h 2 = ( γ a + γ a b γ b ) / 2 ,
h 3 = ( γ a γ a b + γ b ) / 2 ,
h 4 = ( γ a γ a b γ b ) / 2 .
0 = ( ω ω α ) P i , 2 j + 1 ( υ , ω α ) [ ( 2 j + 1 ) i k υ + γ ] P r , 2 j + 1 ( υ , ω α ) i μ 2 2 E i [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] ,
0 = ( ω ω α ) P r , 2 j + 1 ( υ , ω α ) [ ( 2 j + 1 ) i k υ + γ ] P i , 2 j + 1 ( υ , ω α ) + i μ 2 2 E r [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] ,
0 = [ λ a ( υ , ω α ) λ b ( υ , ω α ) ] δ j 0 [ ( 2 j ) i k υ + h 1 ] D 2 j ( υ , ω α ) h 2 M 2 j ( υ , ω α ) i 2 { [ E r , P i , 2 j 1 ( υ , ω α ) E i P r , 2 j 1 ( υ , ω α ) ] [ E r P i , 2 j + 1 ( υ , ω α ) E i P r , 2 j + 1 ( υ , ω α ) ] } + Γ a 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M 2 j ( υ , ω α ) + D 2 j ( υ , ω α ) ] d υ , Γ b 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M 2 j ( υ , ω α ) D 2 j ( υ , ω α ) ] d υ ,
0 = [ λ a ( υ , ω α ) + λ b ( υ , ω α ) ] δ j 0 [ ( 2 j ) i k υ + h 3 ] M 2 j ( υ , ω α ) h 4 D 2 j ( υ , ω α ) + Γ a 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M 2 j ( υ , ω α ) + D 2 j ( υ , ω α ) ] d υ + Γ b 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M 2 j ( υ , ω α ) D 2 j ( υ , ω α ) ] d υ ,
0 = E r 2 t c ( ω Ω ) E i + ω 0 l 1 L 0 P i , 1 i ( υ , ω α ) d υ d ω α ,
0 = E i 2 t c + ( ω Ω ) E r ω 0 l 1 L 0 P r , 1 i ( υ , ω α ) d υ d ω α ,
P i , 2 j + 1 ( υ , ω α ) = x j P r , 2 j + 1 ( υ , ω α ) i μ 2 E i 2 ( ω ω α ) [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] ,
P r , 2 j + 1 ( υ , ω α ) = x j P i , 2 j + 1 ( υ , ω α ) i μ 2 E r 2 ( ω ω α ) [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] ,
x j = ( 2 j + 1 ) i k υ + γ ω ω α .
P i , 2 j + 1 ( υ , ω α ) = i μ 2 2 ( ω ω α ) ( 1 + x j 2 ) ( E r x j E i ) × [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] ,
P r , 2 j + 1 ( υ , ω α ) = i μ 2 2 ( ω ω α ) ( 1 + x j 2 ) ( E i x j E r ) × [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] .
0 = [ λ a ( υ , ω α ) λ b ( υ , ω α ) ] δ j 0 [ ( 2 j ) i k υ + h 1 ] D 2 j ( υ , ω α ) h 2 M 2 j ( υ , ω α ) + μ 2 ( E r 2 + E i 2 ) 4 2 ( ω ω α ) × { x j 1 1 + x j 1 2 [ D 2 j 2 ( υ , ω α ) D 2 j ( υ , ω α ) ] x j 1 + x j 2 [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] } + Γ a 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M 2 j ( υ , ω α ) + D 2 j ( υ , ω α ) ] d υ Γ b 2 u π 1 / 2 exp ( υ 2 / u 2 ) [ M 2 j ( υ , ω α ) D 2 j ( υ , ω α ) ] d υ .
λ a , j ( υ , ω α ) = { L a ( ω α ) δ j 0 + Γ a 2 [ M 2 j ( υ , ω α ) + D 2 j ( υ , ω α ) ] d υ } × exp ( υ 2 / u 2 ) u π 1 / 2 ,
λ b , j ( υ , ω α ) = { L b ( ω α ) δ j 0 + Γ b 2 [ M 2 j ( υ , ω α ) D 2 j ( υ , ω α ) ] d υ } × exp ( υ 2 / u 2 ) u π 1 / 2 ,
λ a ( υ , ω α ) = L a ( ω α ) exp ( υ 2 / u 2 ) / u π 1 / 2 ,
λ b ( υ , ω α ) = L b ( ω ) exp ( υ 2 / u 2 ) / u π 1 / 2 .
0 = λ a , j ( υ , ω α ) λ b , j ( υ , ω α ) [ ( 2 j ) i k υ + h 1 ] D 2 j ( υ , ω α ) h 2 M 2 j ( υ , ω α ) + μ 2 ( E r 2 + E i 2 ) 4 2 ( ω ω α ) { x j 1 1 + x j 1 2 [ D 2 j 2 ( υ , ω α ) D 2 j ( υ , ω α ) ] x j 1 + x j 2 [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] } ,
0 = λ a , j ( υ , ω α ) + λ b , j ( υ , ω α ) [ ( 2 j ) i k υ + h 3 ] M 2 j ( υ , ω α ) h 4 D 2 j ( υ , ω α ) .
M 2 j ( υ , ω α ) = λ a , j ( υ , ω α ) + λ b , j ( υ , ω α ) h 4 D 2 j ( υ , ω α ) ( 2 j ) i k υ + h 3 .
D 2 j ( υ , ω α ) = { [ ( 2 j ) i k υ + h 3 h 2 ] λ a , j ( υ , ω α ) [ ( 2 j ) i k υ + h 3 + h 2 ] λ b , j ( υ , ω α ) } ÷ { [ ( 2 j ) i k υ + h 1 ] [ ( 2 j ) i k υ + h 3 ] h 2 h 4 } + [ μ 2 4 2 ( ω ω α ) ] × [ ( 2 j ) i k υ + h 3 ] ( E r 2 + E i 2 ) [ ( 2 j ) i k υ + h 1 ] [ ( 2 j ) i k υ + h 3 ] h 2 h 4 × { x j 1 1 + x j 1 2 [ D 2 j 2 ( υ , ω α ) D 2 j ( υ , ω α ) ] x j 1 + x j 2 [ D 2 j ( υ , ω α ) D 2 j + 2 ( υ , ω α ) ] } ,
D 0 ( υ , ω α ) = ( h 3 h 2 ) λ a , 0 ( υ , ω α ) ( h 3 + h 2 ) λ b , 0 ( υ , ω α ) h 1 h 3 h 2 h 4 + [ μ 2 4 2 ( ω ω α ) ] h 3 ( E r 2 + E i 2 ) h 1 h 3 h 2 h 4 × { x 1 1 + x 1 2 [ D 2 ( υ , ω α ) D 0 ( υ , ω α ) ] x 0 1 + x 0 2 [ D 0 ( υ , ω α ) D 2 ( υ , ω α ) ] } .
{ 1 + μ 2 4 2 h 3 ( E r 2 + E i 2 ) ( h 1 h 3 h 2 h 4 ) γ [ 1 i k υ / γ ( ω ω α ) 2 / γ 2 + ( 1 i k υ / γ ) 2 + 1 + i k υ / γ ( ω ω α ) 2 / γ 2 + ( 1 + i k υ / γ ) 2 ] } D 0 ( υ , ω α ) = ( h 3 h 2 ) λ a , 0 ( υ , ω α ) ( h 3 + h 2 ) λ b , 0 ( υ , ω α ) h 1 h 3 h 2 h 4 + μ 2 4 2 h 3 ( E r 2 + E i 2 ) ( h 1 h 3 h 2 h 4 ) γ × [ 1 i k υ / γ ( ω ω α ) 2 / γ 2 + ( 1 i k υ / γ ) 2 D 2 ( υ , ω α ) + 1 + i k υ / γ ( ω ω α ) 2 / γ 2 + ( 1 + i k υ / γ ) 2 D 2 ( υ , ω α ) ] .
A r = μ 2 [ h 3 ( h 1 h 3 h 2 h 4 ) γ ] 1 / 2 E r = μ 2 [ γ a γ a b + γ b 2 γ γ a γ b ] 1 / 2 E r ,
A i = μ 2 [ h 3 ( h 1 h 3 h 2 h 4 ) γ ] 1 / 2 E i = μ 2 [ γ a γ a b + γ b 2 γ γ a γ b ] 1 / 2 E i .
{ 1 + 2 ( A r 2 + A i 2 ) } D 0 ( υ , ω α ) = ( h 3 h 2 ) λ a , 0 ( υ , ω α ) ( h 3 + h 2 ) λ b , 0 ( υ , ω α ) h 1 h 3 h 2 h 4 + ( A r 2 + A i 2 ) [ D 2 ( υ , ω α ) + D 2 ( υ , ω α ) ] .
P i , j ( V , U ) = 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 ( υ , ω α ) ,
P r , j ( V , U ) = 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 ( υ , ω α ) ,
D 2 j ( V , U ) = u t c ω 0 l μ 2 1 L D 2 j ( υ , ω α ) ,
M 2 j ( V , U ) = u t c ω 0 l μ 2 1 L M 2 j ( υ , ω α ) ,
λ a ( V , U ) = u t c ω 0 l μ 2 1 L λ a ( υ , ω α ) ,
λ b ( V , U ) = u t c ω 0 l μ 2 1 L λ b ( υ , ω α ) ,
V = υ u = k υ 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 [ h 1 + ( 2 j ) i γ V ] D 2 j ( V , U , t ) h 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 ) ] } + Γ a 2 π 1 / 2 exp ( 2 V 2 ) [ M 2 j ( V , U , t ) + D 2 j ( V , U , t ) ] d V Γ b 2 π 1 / 2 exp ( 2 V 2 ) [ M 2 j ( V , U , t ) D 2 j ( V , U , t ) ] d V ,
M 2 j ( V , U , t ) t = [ λ a ( V , U , t ) + λ b ( V , U , t ) ] δ j 0 [ h 3 + ( 2 j ) i γ V ] M 2 j ( V , U , t ) h 4 D 2 j ( V , U , t ) + Γ a 2 π 1 / 2 exp ( 2 V 2 ) [ M 2 j ( V , U , t ) + D 2 j ( V , U , t ) ] d V + Γ b 2 π 1 / 2 exp ( 2 V 2 ) [ M 2 j ( V , U , t ) D 2 j ( V , U , t ) ] d V ,
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 ( V , t ) = P r , 2 j + 1 ( V , U , t ) d U ,
P i , 2 j + 1 ( V , t ) = P i , 2 j + 1 ( V , U , t ) d U ,
D 2 j ( V , t ) = D 2 j ( V , U , t ) d U ,
M 2 j ( V , t ) = M 2 j ( V , U , t ) d U ,
λ a ( V , t ) = λ a ( V , U , t ) d U ,
λ b ( V , t ) = λ b ( V , U , t ) d U .
P r , 2 j + 1 ( V , t ) t = γ { [ 1 + ( 2 j + 1 ) i V ] P r , 2 j + 1 ( V , t ) + y P i , 2 j + 1 ( V , t ) + i A i ( t ) [ D 2 j ( V , t ) D 2 j + 2 ( V , t ) ] } ,
P i , 2 j + 1 ( V , t ) t = γ { [ 1 + ( 2 j + 1 ) i V ] P i , 2 j + 1 ( V , t ) y P r , 2 j + 1 ( V , t ) i A r ( t ) [ D 2 j ( V , t ) D 2 j + 2 ( V , t ) ] } ,
D 2 j ( V , t ) t = [ λ a ( V , t ) λ b ( V , t ) ] δ j 0 [ h 1 + ( 2 j ) i γ V ] D 2 j ( V , t ) h 2 M 2 j ( V , t ) i γ 1 { [ A r ( t ) P i , 2 j 1 ( V , t ) A i ( t ) P r , 2 j 1 ( V , t ) ] [ A r ( t ) P i , 2 j + 1 ( V , t ) A i ( t ) P r , 2 j + 1 ( V , t ) ] } + Γ a 2 π 1 / 2 exp ( 2 V 2 ) [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V Γ b 2 π 1 / 2 exp ( 2 V 2 ) [ M 2 j ( V , t ) D 2 j ( V , t ) ] d V ,
M 2 j ( V , t ) t = [ λ a ( V , t ) + γ b ( V , t ) ] δ j 0 [ h 3 + ( 2 j ) i γ V ] M 2 j ( V , t ) h 4 D 2 j ( V , t ) + Γ a 2 π 1 / 2 exp ( 2 V 2 ) [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V + Γ a 2 π 1 / 2 exp ( 2 V 2 ) [ M 2 j ( V , t ) D 2 j ( V , t ) ] d V ,
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 , t ) d V ] ,
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 , t ) d V ] .
I = ( γ a γ a b + γ b ) γ a γ b ( γ a γ a b + γ b ) γ a γ b ( A r 2 + A i 2 ) .
w * ( x ) = exp ( x 2 4 ) { [ 1 + ( 2 F ( x ) exp x 2 π 1 / 2 ) 2 ] 1 / 2 + 2 F ( x ) exp x 2 π 1 / 2 } 1 / 4 ,
F ( x ) = exp ( x 2 ) 0 x exp ( t 2 ) d t ,

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