Abstract

Spontaneous pulsations similar to the type discovered in 3.51-μm Xe laser oscillators were reported several years ago in 3.39-μm He–Ne lasers, but no numerical interpretation of the He–Ne data has been given. The model for the Xe laser instability is adapted here to the He–Ne system, and the results include direct comparisons between theoretical calculations and published experimental data. Good agreement is obtained for the instability threshold, the pulsation frequencies, and many other features; and unmeasured but potentially useful pulsation characteristics are also readily obtained from the model.

© 2003 Optical Society of America

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References

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  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]
  2. L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756–761 (1978).
    [CrossRef]
  3. L. W. Casperson, “Spontaneous pulsations in lasers,” inThird 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, and references therein.
    [CrossRef]
  4. J. R. Ackerhalt, P. W. Milonni, and M.-L. Shih, “Chaos in quantum optics,” Phys. Rep. 128, 205–300 (1985).
    [CrossRef]
  5. R. G. Harrison and D. J. Biswas, “Pulsating instabilities and chaos in lasers,” Prog. Quantum Electron. 10, 147–228 (1985).
    [CrossRef]
  6. N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 25, pp. 1–187.
  7. L. A. Lugiato, L. M. Narducci, J. R. Tredicce, and D. K. Bandy, “25 years of laser instabilities,” in Instabilities and Chaos in Quantum Optics II, N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds. (Plenum, New York, 1988), pp. 1–25.
  8. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (Weinheim, New York, 1991).
  9. G. H. M. van Tartwijk and G. P. Agrawal, “Laser instabilities: a modern perspective,” Prog. Quantum Electron. 22, 43–122 (1998).
    [CrossRef]
  10. F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
    [CrossRef]
  11. F. Sanchez and A. Kellou, “Laser dynamics with excited-state absorption,” J. Opt. Soc. Am. B 14, 209–213 (1997).
    [CrossRef]
  12. W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in phonon-assisted lasers,” J. Opt. Soc. Am. B 15, 2681–2688 (1998).
    [CrossRef]
  13. R. S. Gioggia and N. B. Abraham, “Single-mode self-pulsing instabilities at the Lamb dip of a He–Ne 3.39 μm laser,” Opt. Commun. 47, 278–282 (1983).
    [CrossRef]
  14. R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities, and chaos in a single mode, standing-wave 3.39 μm He–Ne laser,” Phys. Rev. A 29, 1304–1309 (1984).
    [CrossRef]
  15. W. E. Lamb, Jr., “Theory of an optical maser,” Phys. Rev. 134, A1429–A1450 (1964).
    [CrossRef]
  16. L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators,” J. Opt. Soc. Am. B 2, 62–72 (1985).
    [CrossRef]
  17. M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148–3158 (1986).
    [CrossRef] [PubMed]
  18. L. W. Casperson, “Spontaneous coherent pulsations in standing-wave laser oscillators,” J. Opt. Soc. Am. B 5, 958–969 (1988).
    [CrossRef]
  19. 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]
  20. W. R. Bennett, Jr., “Gaseous optical masers,” Appl. Opt. Suppl. 1, 24–61 (1962).
    [CrossRef]
  21. B. Decomps and M. Dumont, “T-9 Polarization of the fluorescent light of Ne atoms interacting with a laser beam measurement of several relaxation times,” IEEE J. Quantum Electron. QE-4, 916–922 (1968).
    [CrossRef]
  22. G. M. Lawrence and H. S. Liszt, “Radiative lifetimes in the resonance series of Ne,” Phys. Rev. 178, 122–125 (1969).
    [CrossRef]
  23. J. Z. Klose, “Transition probabilites and mean lives of the 3s2 laser level in neon I,” Phys. Rev. 188, 45–49 (1969).
    [CrossRef]
  24. I. P. Konovalov and E. D. Protsenko, “Determination of the relaxation constants of levels by the method of resonant interaction of two travelling waves with a two-level system,” Sov. J. Quantum Electron. 6, 1084–1091 (1976).
    [CrossRef]
  25. P. Chenkosol, “Spontaneous pulsations in laser oscillators: effects of spatial field distributions; self-mode-locking dynamics; pulsations in 3.39 μm He–Ne standing-wave lasers,” Ph.D. dissertation (Portland State University, Portland, Ore., 2003).
  26. J. T. Verdeyen, in Lasers, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 328.
  27. M. Ohi, “Measurements of frequency shifts in the 3.39 μm Ne transition with the aid of CH4 Lamb dip,” Jpn. J. Appl. Phys. 12, 1377–1381 (1973).
    [CrossRef]
  28. P. W. Smith, “The effect of cross relaxation on the behavior of gas laser oscillators,” IEEE J. Quantum Electron. QE-8, 704–710 (1972).
    [CrossRef]

1998 (2)

G. H. M. van Tartwijk and G. P. Agrawal, “Laser instabilities: a modern perspective,” Prog. Quantum Electron. 22, 43–122 (1998).
[CrossRef]

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in phonon-assisted lasers,” J. Opt. Soc. Am. B 15, 2681–2688 (1998).
[CrossRef]

1997 (1)

1996 (1)

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

1991 (1)

1988 (1)

1986 (1)

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148–3158 (1986).
[CrossRef] [PubMed]

1985 (3)

L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators,” J. Opt. Soc. Am. B 2, 62–72 (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]

1984 (1)

R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities, and chaos in a single mode, standing-wave 3.39 μm He–Ne laser,” Phys. Rev. A 29, 1304–1309 (1984).
[CrossRef]

1983 (1)

R. S. Gioggia and N. B. Abraham, “Single-mode self-pulsing instabilities at the Lamb dip of a He–Ne 3.39 μm laser,” Opt. Commun. 47, 278–282 (1983).
[CrossRef]

1978 (1)

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

1976 (1)

I. P. Konovalov and E. D. Protsenko, “Determination of the relaxation constants of levels by the method of resonant interaction of two travelling waves with a two-level system,” Sov. J. Quantum Electron. 6, 1084–1091 (1976).
[CrossRef]

1973 (1)

M. Ohi, “Measurements of frequency shifts in the 3.39 μm Ne transition with the aid of CH4 Lamb dip,” Jpn. J. Appl. Phys. 12, 1377–1381 (1973).
[CrossRef]

1972 (2)

P. W. Smith, “The effect of cross relaxation on the behavior of gas laser oscillators,” IEEE J. Quantum Electron. QE-8, 704–710 (1972).
[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]

1969 (2)

G. M. Lawrence and H. S. Liszt, “Radiative lifetimes in the resonance series of Ne,” Phys. Rev. 178, 122–125 (1969).
[CrossRef]

J. Z. Klose, “Transition probabilites and mean lives of the 3s2 laser level in neon I,” Phys. Rev. 188, 45–49 (1969).
[CrossRef]

1968 (1)

B. Decomps and M. Dumont, “T-9 Polarization of the fluorescent light of Ne atoms interacting with a laser beam measurement of several relaxation times,” IEEE J. Quantum Electron. QE-4, 916–922 (1968).
[CrossRef]

1964 (1)

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

1962 (1)

W. R. Bennett, Jr., “Gaseous optical masers,” Appl. Opt. Suppl. 1, 24–61 (1962).
[CrossRef]

Abraham, N. B.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148–3158 (1986).
[CrossRef] [PubMed]

R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities, and chaos in a single mode, standing-wave 3.39 μm He–Ne laser,” Phys. Rev. A 29, 1304–1309 (1984).
[CrossRef]

R. S. Gioggia and N. B. Abraham, “Single-mode self-pulsing instabilities at the Lamb dip of a He–Ne 3.39 μm laser,” Opt. Commun. 47, 278–282 (1983).
[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]

Agrawal, G. P.

G. H. M. van Tartwijk and G. P. Agrawal, “Laser instabilities: a modern perspective,” Prog. Quantum Electron. 22, 43–122 (1998).
[CrossRef]

Bandy, D. K.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148–3158 (1986).
[CrossRef] [PubMed]

Bennett Jr., W. R.

W. R. Bennett, Jr., “Gaseous optical masers,” Appl. Opt. Suppl. 1, 24–61 (1962).
[CrossRef]

Biswas, D. J.

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

Casperson, L. W.

Decomps, B.

B. Decomps and M. Dumont, “T-9 Polarization of the fluorescent light of Ne atoms interacting with a laser beam measurement of several relaxation times,” IEEE J. Quantum Electron. QE-4, 916–922 (1968).
[CrossRef]

Dumont, M.

B. Decomps and M. Dumont, “T-9 Polarization of the fluorescent light of Ne atoms interacting with a laser beam measurement of several relaxation times,” IEEE J. Quantum Electron. QE-4, 916–922 (1968).
[CrossRef]

Gadomski, W.

Gioggia, R. S.

R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities, and chaos in a single mode, standing-wave 3.39 μm He–Ne laser,” Phys. Rev. A 29, 1304–1309 (1984).
[CrossRef]

R. S. Gioggia and N. B. Abraham, “Single-mode self-pulsing instabilities at the Lamb dip of a He–Ne 3.39 μm laser,” Opt. Commun. 47, 278–282 (1983).
[CrossRef]

Harrison, R. G.

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

Kellou, A.

Klose, J. Z.

J. Z. Klose, “Transition probabilites and mean lives of the 3s2 laser level in neon I,” Phys. Rev. 188, 45–49 (1969).
[CrossRef]

Konovalov, I. P.

I. P. Konovalov and E. D. Protsenko, “Determination of the relaxation constants of levels by the method of resonant interaction of two travelling waves with a two-level system,” Sov. J. Quantum Electron. 6, 1084–1091 (1976).
[CrossRef]

Lamb Jr., W. E.

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

Lawrence, G. M.

G. M. Lawrence and H. S. Liszt, “Radiative lifetimes in the resonance series of Ne,” Phys. Rev. 178, 122–125 (1969).
[CrossRef]

Liszt, H. S.

G. M. Lawrence and H. S. Liszt, “Radiative lifetimes in the resonance series of Ne,” Phys. Rev. 178, 122–125 (1969).
[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]

Narducci, L. M.

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148–3158 (1986).
[CrossRef] [PubMed]

Ohi, M.

M. Ohi, “Measurements of frequency shifts in the 3.39 μm Ne transition with the aid of CH4 Lamb dip,” Jpn. J. Appl. Phys. 12, 1377–1381 (1973).
[CrossRef]

Protsenko, E. D.

I. P. Konovalov and E. D. Protsenko, “Determination of the relaxation constants of levels by the method of resonant interaction of two travelling waves with a two-level system,” Sov. J. Quantum Electron. 6, 1084–1091 (1976).
[CrossRef]

Ratajska-Gadomska, B.

Sanchez, F.

F. Sanchez and A. Kellou, “Laser dynamics with excited-state absorption,” J. Opt. Soc. Am. B 14, 209–213 (1997).
[CrossRef]

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[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]

Smith, P. W.

P. W. Smith, “The effect of cross relaxation on the behavior of gas laser oscillators,” IEEE J. Quantum Electron. QE-8, 704–710 (1972).
[CrossRef]

Stephan, G.

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

Tarroja, M. F. H.

van Tartwijk, G. H. M.

G. H. M. van Tartwijk and G. P. Agrawal, “Laser instabilities: a modern perspective,” Prog. Quantum Electron. 22, 43–122 (1998).
[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]

Appl. Opt. Suppl. (1)

W. R. Bennett, Jr., “Gaseous optical masers,” Appl. Opt. Suppl. 1, 24–61 (1962).
[CrossRef]

IEEE J. Quantum Electron. (4)

B. Decomps and M. Dumont, “T-9 Polarization of the fluorescent light of Ne atoms interacting with a laser beam measurement of several relaxation times,” IEEE J. Quantum Electron. QE-4, 916–922 (1968).
[CrossRef]

P. W. Smith, “The effect of cross relaxation on the behavior of gas laser oscillators,” IEEE J. Quantum Electron. QE-8, 704–710 (1972).
[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 coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756–761 (1978).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

M. Ohi, “Measurements of frequency shifts in the 3.39 μm Ne transition with the aid of CH4 Lamb dip,” Jpn. J. Appl. Phys. 12, 1377–1381 (1973).
[CrossRef]

Opt. Commun. (1)

R. S. Gioggia and N. B. Abraham, “Single-mode self-pulsing instabilities at the Lamb dip of a He–Ne 3.39 μm laser,” Opt. Commun. 47, 278–282 (1983).
[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. (3)

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

G. M. Lawrence and H. S. Liszt, “Radiative lifetimes in the resonance series of Ne,” Phys. Rev. 178, 122–125 (1969).
[CrossRef]

J. Z. Klose, “Transition probabilites and mean lives of the 3s2 laser level in neon I,” Phys. Rev. 188, 45–49 (1969).
[CrossRef]

Phys. Rev. A (2)

R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities, and chaos in a single mode, standing-wave 3.39 μm He–Ne laser,” Phys. Rev. A 29, 1304–1309 (1984).
[CrossRef]

M. F. H. Tarroja, N. B. Abraham, D. K. Bandy, and L. M. Narducci, Phys. Rev. A 34, 3148–3158 (1986).
[CrossRef] [PubMed]

Phys. Rev. E (1)

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

Prog. Quantum Electron. (2)

G. H. M. van Tartwijk and G. P. Agrawal, “Laser instabilities: a modern perspective,” Prog. Quantum Electron. 22, 43–122 (1998).
[CrossRef]

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

Sov. J. Quantum Electron. (1)

I. P. Konovalov and E. D. Protsenko, “Determination of the relaxation constants of levels by the method of resonant interaction of two travelling waves with a two-level system,” Sov. J. Quantum Electron. 6, 1084–1091 (1976).
[CrossRef]

Other (6)

P. Chenkosol, “Spontaneous pulsations in laser oscillators: effects of spatial field distributions; self-mode-locking dynamics; pulsations in 3.39 μm He–Ne standing-wave lasers,” Ph.D. dissertation (Portland State University, Portland, Ore., 2003).

J. T. Verdeyen, in Lasers, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 328.

N. B. Abraham, P. Mandel, and L. M. Narducci, “Dynamical instabilities and pulsations in lasers,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 25, pp. 1–187.

L. A. Lugiato, L. M. Narducci, J. R. Tredicce, and D. K. Bandy, “25 years of laser instabilities,” in Instabilities and Chaos in Quantum Optics II, N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds. (Plenum, New York, 1988), pp. 1–25.

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (Weinheim, New York, 1991).

L. W. Casperson, “Spontaneous pulsations in lasers,” inThird 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, and references therein.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Time-series signal shows the laser output intensity close to the threshold of instability (rth=1.61).

Fig. 2
Fig. 2

Homodyne spectrum close to the threshold of instability (rth=1.61).

Fig. 3
Fig. 3

Time-series signals show laser output intensities (left). The corresponding homodyne and heterodyne spectra of the observed time-series signals show peak pulsation frequencies and laser operating frequencies (middle and right, respectively). The cavity is tuned to line center, and the values of the threshold parameter are (a) rth=1.5, (b) rth=1.66, (c) rth=1.7, (d) rth=2.05, and (e) rth=2.6.

Fig. 4
Fig. 4

Average laser intensities (dotted curves) and frequencies of the principal peaks in the homodyne spectra (solid curves) versus cavity detunings (left). Laser operating frequencies versus cavity detunings (right). The threshold parameters are (a) rth=1.5, (b) rth=1.7, (c) rth=2.05, and (d) rth=2.6.

Fig. 5
Fig. 5

Expanded scale of Fig. 4(d) showing the details of the laser operating frequency. Note the existence of sidebands that are not displayed in Fig. 4(d). The threshold parameter is rth=2.6.

Fig. 6
Fig. 6

Laser output intensities (left) and the corresponding homodyne spectra (right) versus cavity detunings. The threshold parameter is rth=2.05. The cavity detunings are (a) line center, (b) 4000 Hz, (c) 0.1 MHz, (d) 0.9 MHz, and (e) 10 MHz.

Tables (1)

Tables Icon

Table 1 Parameters of the 3.39-μm He–Ne Laser Transition

Equations (22)

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

t Pr,k(V, t)=-γ[1+(2k-1)iV]Pr,k(V, t)+yPi,k(V, t)+iAi(t)× exp(-2V2)π1/2 {Ek(t)-Ek+1(t)}+Dk(V, t)-Dk+1(V, t),
t Pi,k(V, t)=-γ[1+(2k-1)iV]Pi,k(V, t)-yPr,k(V, t)-iAr(t)× exp(-2V2)π1/2 {Ek(t)-Ek+1(t)}+Dk(V, t)-Dk+1(V, t),
t Dk(V, t)=-[h1+(2k-2)iγV]Dk(V, t)-h2Mk(V, t)-(2k-2)iγVEk(t) exp(-2V2)π1/2-iγ1[Ar(t)Pi,k-1(V, t)-Ai(t)Pr,k-1(V, t)-Ar(t)Pi,k(V, t)+Ai(t)Pr,k(V, t)],
t Mk(V, t)=-[h3+(2k-2)iγV]Mk(V, t)-h4Dk(V, t)-(2k-2)iγVFk(t) exp(-2V2)π1/2,
ddt Ek(t)=[La(t)-Lb(t)]δk1-h5Ek(t)-h6(t)Fk(t)+Γa2-[Mk(V, t)+Dk(V, t)]dV-Γb2-[Mk(V, t)-Dk(V, t)]dV,
ddt Fk(t)=[La(t)+Lb(t)]δk1-h7Fk(t)-h8(t)Ek(t)+Γa2-[Mk(V, t)+Dk(V, t)]dV+Γb2-[Mk(V, t)-Dk(V, t)]dV,
ddt Ar(t)=-12tcAr(t)+δ(y-y0)Ai(t)--Pi,1i(V, t)dV,
ddt Ai(t)=-12tcAi(t)-δ(y-y0)Ar(t)+-Pr,1i(V, t)dV.
Dk(V, t)=Ek(t)π1/2exp(-2V2)+Dk(V, t),
Mk(V, t)=Fk(t)π1/2exp(-2V2)+Mk(V, t),
h1=(γa+γab+γb)/2,
h2=(γa+γab-γb)/2,
h3=(γa-γab+γb)/2,
h4=(γa-γab-γb)/2,
h5=h1-Γa/2-Γb/2=(γa+γab+γb)/2,
h6=h2-Γa/2+Γb/2=(γa+γab-γb)/2,
h7=h3-Γa/2-Γb/2=(γa-γab+γb)/2,
h8=h4-Γa/2+Γb/2=(γa-γab-γb)/2,
γ1=2γaγb/(γa-γab+γb).
t D1(V, t)=-h1D1(V, t)-h2M1(V, t)-2γ1[Ar(t)Pi,1i(V, t)-Ai(t)Pr,1i(V, t)],
Γa=Γa+xκA(3s2),
i(t)=[E0(t)+Er(t)]2¯12 [|E0(t)|2+|Er|2]+12 {E0(t)Er*exp[-i(ωc-ωr)t]+E0*(t)Erexp[i(ωc-ωr)t]},

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