Abstract

A theoretical model is considered for an unsteady three-level (single-mode) laser. By use of the rotating wave approximation, the usual semiclassical equations are transformed into a system of first-order space–time (ST) equations. The space oscillations of the population difference (hole-burning effects) and the motion of the atoms are included in the treatment. After the introduction of several assumptions, the ST equations are transformed to a set of time-dependent space-averaged (SA) equations. The numerical calculations were performed for a pulsed two-mirror laser with a finite phase memory decay constant and for resonance and nonresonance frequencies. The calculations show that the SA equations give almost the same results as the much more complicated ST equations.

© 1983 Optical Society of America

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  1. M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).
  2. V. M. Fain, Sov. Phys. JETP 6, 726 (1958); V. M. Fain, Ya. I. Khanin, Sov. Phys. JETP 14, 1069 (1962).
  3. W. W. Rigrod, J. Appl. Phys. 34, 2602 (1963).
    [CrossRef]
  4. L. A. Ostrovski, E. I. Yakubovich, Sov. Phys. JETP 19, 656 (1964).
  5. W. E. Lamb, Phys. Rev. 134, 1429 (1964).
    [CrossRef]
  6. S. Stenholm, W. E. Lamb, Phys. Rev. 181, 618 (1969).
    [CrossRef]
  7. V. DeGiorgio, W. M. Scully, Phys. Rev. A 2, 1170 (1970).
    [CrossRef]
  8. G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
    [CrossRef]
  9. R. Lang, M. O. Scully, W. E. Lamb, Phys. Rev. A 7, 1788 (1973).
    [CrossRef]
  10. F. Najmabadi, M. Sargent, F. A. Hopf, Phys. Rev. A 12, 1553 (1975).
    [CrossRef]
  11. M. Sargent, Appl. Phys. 9, 127 (1976).
    [CrossRef]
  12. R. Bonifacio, L. A. Lugiato, Phys. Rev. A 18, 1129 (1978).
    [CrossRef]
  13. I. Last, M. Baer, Appl. Phys. Lett. 38, 838 (1981).
    [CrossRef]
  14. I. Last, Y. Reuven, M. Baer, Opt. Commun. 39, 83 (1981); I. Last, M. Baer, Appl. Phys. B 28, 2/3, 102 (1982).
    [CrossRef]
  15. N. G. Basov, R. V. Ambartsumjan, V. S. Zuev, P. G. Krynkov, V. S. Letokhov, Sov. Phys. JETP 23, 16 (1966).
  16. A. Icsevgi, W. E. Lamb, Phys. Rev. 185, 519 (1969).
    [CrossRef]
  17. F. A. Hopf, P. Meystre, Phys. Rev. A 12, 2534 (1975).
    [CrossRef]
  18. R. Bonifacio, F. A. Hopf, P. Meystre, M. O. Scully, Phys. Rev. A 12, 2568 (1975).
    [CrossRef]
  19. F. A. Hopf, E. A. Overman, Phys. Rev. A 19, 1180 (1979).
    [CrossRef]
  20. F. A. Hopf, Phys. Rev. A 20, 2064 (1979).
    [CrossRef]
  21. A. T. Rosenberger, T. A. DeTemple, Phys. Rev. A 24, 868 (1981).
    [CrossRef]
  22. Y. Reuven, M. Baer, Opt. Commun. 32, 320 (1980).
    [CrossRef]
  23. Y. Reuven, M. Baer, IEEE J. Quantum Electron. QE-16, 1117 (1980).
    [CrossRef]
  24. Y. Reuven, M. Baer, J. Appl. Phys. 53, 5439 (1982).
    [CrossRef]
  25. W. W. Rigrod, IEEE J. Quantum Electron. QE-14, 377 (1978).
    [CrossRef]

1982

Y. Reuven, M. Baer, J. Appl. Phys. 53, 5439 (1982).
[CrossRef]

1981

A. T. Rosenberger, T. A. DeTemple, Phys. Rev. A 24, 868 (1981).
[CrossRef]

I. Last, M. Baer, Appl. Phys. Lett. 38, 838 (1981).
[CrossRef]

I. Last, Y. Reuven, M. Baer, Opt. Commun. 39, 83 (1981); I. Last, M. Baer, Appl. Phys. B 28, 2/3, 102 (1982).
[CrossRef]

1980

Y. Reuven, M. Baer, Opt. Commun. 32, 320 (1980).
[CrossRef]

Y. Reuven, M. Baer, IEEE J. Quantum Electron. QE-16, 1117 (1980).
[CrossRef]

1979

F. A. Hopf, E. A. Overman, Phys. Rev. A 19, 1180 (1979).
[CrossRef]

F. A. Hopf, Phys. Rev. A 20, 2064 (1979).
[CrossRef]

1978

W. W. Rigrod, IEEE J. Quantum Electron. QE-14, 377 (1978).
[CrossRef]

R. Bonifacio, L. A. Lugiato, Phys. Rev. A 18, 1129 (1978).
[CrossRef]

1976

M. Sargent, Appl. Phys. 9, 127 (1976).
[CrossRef]

1975

F. Najmabadi, M. Sargent, F. A. Hopf, Phys. Rev. A 12, 1553 (1975).
[CrossRef]

F. A. Hopf, P. Meystre, Phys. Rev. A 12, 2534 (1975).
[CrossRef]

R. Bonifacio, F. A. Hopf, P. Meystre, M. O. Scully, Phys. Rev. A 12, 2568 (1975).
[CrossRef]

1973

R. Lang, M. O. Scully, W. E. Lamb, Phys. Rev. A 7, 1788 (1973).
[CrossRef]

1971

G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
[CrossRef]

1970

V. DeGiorgio, W. M. Scully, Phys. Rev. A 2, 1170 (1970).
[CrossRef]

1969

S. Stenholm, W. E. Lamb, Phys. Rev. 181, 618 (1969).
[CrossRef]

A. Icsevgi, W. E. Lamb, Phys. Rev. 185, 519 (1969).
[CrossRef]

1966

N. G. Basov, R. V. Ambartsumjan, V. S. Zuev, P. G. Krynkov, V. S. Letokhov, Sov. Phys. JETP 23, 16 (1966).

1964

L. A. Ostrovski, E. I. Yakubovich, Sov. Phys. JETP 19, 656 (1964).

W. E. Lamb, Phys. Rev. 134, 1429 (1964).
[CrossRef]

1963

W. W. Rigrod, J. Appl. Phys. 34, 2602 (1963).
[CrossRef]

1958

V. M. Fain, Sov. Phys. JETP 6, 726 (1958); V. M. Fain, Ya. I. Khanin, Sov. Phys. JETP 14, 1069 (1962).

Ambartsumjan, R. V.

N. G. Basov, R. V. Ambartsumjan, V. S. Zuev, P. G. Krynkov, V. S. Letokhov, Sov. Phys. JETP 23, 16 (1966).

Baer, M.

Y. Reuven, M. Baer, J. Appl. Phys. 53, 5439 (1982).
[CrossRef]

I. Last, M. Baer, Appl. Phys. Lett. 38, 838 (1981).
[CrossRef]

I. Last, Y. Reuven, M. Baer, Opt. Commun. 39, 83 (1981); I. Last, M. Baer, Appl. Phys. B 28, 2/3, 102 (1982).
[CrossRef]

Y. Reuven, M. Baer, IEEE J. Quantum Electron. QE-16, 1117 (1980).
[CrossRef]

Y. Reuven, M. Baer, Opt. Commun. 32, 320 (1980).
[CrossRef]

Basov, N. G.

N. G. Basov, R. V. Ambartsumjan, V. S. Zuev, P. G. Krynkov, V. S. Letokhov, Sov. Phys. JETP 23, 16 (1966).

Bonifacio, R.

R. Bonifacio, L. A. Lugiato, Phys. Rev. A 18, 1129 (1978).
[CrossRef]

R. Bonifacio, F. A. Hopf, P. Meystre, M. O. Scully, Phys. Rev. A 12, 2568 (1975).
[CrossRef]

DeGiorgio, V.

V. DeGiorgio, W. M. Scully, Phys. Rev. A 2, 1170 (1970).
[CrossRef]

DeTemple, T. A.

A. T. Rosenberger, T. A. DeTemple, Phys. Rev. A 24, 868 (1981).
[CrossRef]

Fain, V. M.

V. M. Fain, Sov. Phys. JETP 6, 726 (1958); V. M. Fain, Ya. I. Khanin, Sov. Phys. JETP 14, 1069 (1962).

Hopf, F. A.

F. A. Hopf, E. A. Overman, Phys. Rev. A 19, 1180 (1979).
[CrossRef]

F. A. Hopf, Phys. Rev. A 20, 2064 (1979).
[CrossRef]

R. Bonifacio, F. A. Hopf, P. Meystre, M. O. Scully, Phys. Rev. A 12, 2568 (1975).
[CrossRef]

F. Najmabadi, M. Sargent, F. A. Hopf, Phys. Rev. A 12, 1553 (1975).
[CrossRef]

F. A. Hopf, P. Meystre, Phys. Rev. A 12, 2534 (1975).
[CrossRef]

Icsevgi, A.

A. Icsevgi, W. E. Lamb, Phys. Rev. 185, 519 (1969).
[CrossRef]

Krynkov, P. G.

N. G. Basov, R. V. Ambartsumjan, V. S. Zuev, P. G. Krynkov, V. S. Letokhov, Sov. Phys. JETP 23, 16 (1966).

Lamb, G. L.

G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
[CrossRef]

Lamb, W. E.

R. Lang, M. O. Scully, W. E. Lamb, Phys. Rev. A 7, 1788 (1973).
[CrossRef]

S. Stenholm, W. E. Lamb, Phys. Rev. 181, 618 (1969).
[CrossRef]

A. Icsevgi, W. E. Lamb, Phys. Rev. 185, 519 (1969).
[CrossRef]

W. E. Lamb, Phys. Rev. 134, 1429 (1964).
[CrossRef]

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Lang, R.

R. Lang, M. O. Scully, W. E. Lamb, Phys. Rev. A 7, 1788 (1973).
[CrossRef]

Last, I.

I. Last, Y. Reuven, M. Baer, Opt. Commun. 39, 83 (1981); I. Last, M. Baer, Appl. Phys. B 28, 2/3, 102 (1982).
[CrossRef]

I. Last, M. Baer, Appl. Phys. Lett. 38, 838 (1981).
[CrossRef]

Letokhov, V. S.

N. G. Basov, R. V. Ambartsumjan, V. S. Zuev, P. G. Krynkov, V. S. Letokhov, Sov. Phys. JETP 23, 16 (1966).

Lugiato, L. A.

R. Bonifacio, L. A. Lugiato, Phys. Rev. A 18, 1129 (1978).
[CrossRef]

Meystre, P.

F. A. Hopf, P. Meystre, Phys. Rev. A 12, 2534 (1975).
[CrossRef]

R. Bonifacio, F. A. Hopf, P. Meystre, M. O. Scully, Phys. Rev. A 12, 2568 (1975).
[CrossRef]

Najmabadi, F.

F. Najmabadi, M. Sargent, F. A. Hopf, Phys. Rev. A 12, 1553 (1975).
[CrossRef]

Ostrovski, L. A.

L. A. Ostrovski, E. I. Yakubovich, Sov. Phys. JETP 19, 656 (1964).

Overman, E. A.

F. A. Hopf, E. A. Overman, Phys. Rev. A 19, 1180 (1979).
[CrossRef]

Reuven, Y.

Y. Reuven, M. Baer, J. Appl. Phys. 53, 5439 (1982).
[CrossRef]

I. Last, Y. Reuven, M. Baer, Opt. Commun. 39, 83 (1981); I. Last, M. Baer, Appl. Phys. B 28, 2/3, 102 (1982).
[CrossRef]

Y. Reuven, M. Baer, Opt. Commun. 32, 320 (1980).
[CrossRef]

Y. Reuven, M. Baer, IEEE J. Quantum Electron. QE-16, 1117 (1980).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, IEEE J. Quantum Electron. QE-14, 377 (1978).
[CrossRef]

W. W. Rigrod, J. Appl. Phys. 34, 2602 (1963).
[CrossRef]

Rosenberger, A. T.

A. T. Rosenberger, T. A. DeTemple, Phys. Rev. A 24, 868 (1981).
[CrossRef]

Sargent, M.

M. Sargent, Appl. Phys. 9, 127 (1976).
[CrossRef]

F. Najmabadi, M. Sargent, F. A. Hopf, Phys. Rev. A 12, 1553 (1975).
[CrossRef]

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Scully, M. O.

R. Bonifacio, F. A. Hopf, P. Meystre, M. O. Scully, Phys. Rev. A 12, 2568 (1975).
[CrossRef]

R. Lang, M. O. Scully, W. E. Lamb, Phys. Rev. A 7, 1788 (1973).
[CrossRef]

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Scully, W. M.

V. DeGiorgio, W. M. Scully, Phys. Rev. A 2, 1170 (1970).
[CrossRef]

Stenholm, S.

S. Stenholm, W. E. Lamb, Phys. Rev. 181, 618 (1969).
[CrossRef]

Yakubovich, E. I.

L. A. Ostrovski, E. I. Yakubovich, Sov. Phys. JETP 19, 656 (1964).

Zuev, V. S.

N. G. Basov, R. V. Ambartsumjan, V. S. Zuev, P. G. Krynkov, V. S. Letokhov, Sov. Phys. JETP 23, 16 (1966).

Appl. Phys.

M. Sargent, Appl. Phys. 9, 127 (1976).
[CrossRef]

Appl. Phys. Lett.

I. Last, M. Baer, Appl. Phys. Lett. 38, 838 (1981).
[CrossRef]

IEEE J. Quantum Electron.

Y. Reuven, M. Baer, IEEE J. Quantum Electron. QE-16, 1117 (1980).
[CrossRef]

W. W. Rigrod, IEEE J. Quantum Electron. QE-14, 377 (1978).
[CrossRef]

J. Appl. Phys.

Y. Reuven, M. Baer, J. Appl. Phys. 53, 5439 (1982).
[CrossRef]

W. W. Rigrod, J. Appl. Phys. 34, 2602 (1963).
[CrossRef]

Opt. Commun.

I. Last, Y. Reuven, M. Baer, Opt. Commun. 39, 83 (1981); I. Last, M. Baer, Appl. Phys. B 28, 2/3, 102 (1982).
[CrossRef]

Y. Reuven, M. Baer, Opt. Commun. 32, 320 (1980).
[CrossRef]

Phys. Rev.

W. E. Lamb, Phys. Rev. 134, 1429 (1964).
[CrossRef]

S. Stenholm, W. E. Lamb, Phys. Rev. 181, 618 (1969).
[CrossRef]

A. Icsevgi, W. E. Lamb, Phys. Rev. 185, 519 (1969).
[CrossRef]

Phys. Rev. A

F. A. Hopf, P. Meystre, Phys. Rev. A 12, 2534 (1975).
[CrossRef]

R. Bonifacio, F. A. Hopf, P. Meystre, M. O. Scully, Phys. Rev. A 12, 2568 (1975).
[CrossRef]

F. A. Hopf, E. A. Overman, Phys. Rev. A 19, 1180 (1979).
[CrossRef]

F. A. Hopf, Phys. Rev. A 20, 2064 (1979).
[CrossRef]

A. T. Rosenberger, T. A. DeTemple, Phys. Rev. A 24, 868 (1981).
[CrossRef]

R. Bonifacio, L. A. Lugiato, Phys. Rev. A 18, 1129 (1978).
[CrossRef]

R. Lang, M. O. Scully, W. E. Lamb, Phys. Rev. A 7, 1788 (1973).
[CrossRef]

F. Najmabadi, M. Sargent, F. A. Hopf, Phys. Rev. A 12, 1553 (1975).
[CrossRef]

V. DeGiorgio, W. M. Scully, Phys. Rev. A 2, 1170 (1970).
[CrossRef]

Rev. Mod. Phys.

G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971).
[CrossRef]

Sov. Phys. JETP

L. A. Ostrovski, E. I. Yakubovich, Sov. Phys. JETP 19, 656 (1964).

V. M. Fain, Sov. Phys. JETP 6, 726 (1958); V. M. Fain, Ya. I. Khanin, Sov. Phys. JETP 14, 1069 (1962).

N. G. Basov, R. V. Ambartsumjan, V. S. Zuev, P. G. Krynkov, V. S. Letokhov, Sov. Phys. JETP 23, 16 (1966).

Other

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of the level system and the processes considered.

Fig. 2
Fig. 2

Hole-burning effect in a two-mirror laser cavity with reflectivities Ro = RL = 1: (a) the standing wave pattern of the electric field; (b) the resulting spatial inhomogeneity of the population difference.

Fig. 3
Fig. 3

Comparison of the solutions of the space–time (ST) equations (37)(41) obtained by neglecting the hole-burning effect (Γ = 0) (—) and by taking this effect into account (- -). The parameters are L = 50 cm, RL = 0.4, Go = 4.8 × 1026 sec−3, τo = 10−8 sec, γ = 10−8 sec−1, νω = 5γ, v = 0.

Fig. 4
Fig. 4

Correlation and normalization functions q(t), η(t), ξ(t), and q ˜ (t) obtained by the numerical solution of Eqs. (37)(41) for Γ = 0 (solid lines), and the interpolation functions q = η = 1 and ξ and q ˜ expressed by Eqs. (96) and (97), respectively (dotted lines). The parameters are L = 50 cm, RL = 0.4, Go = 4.8 × 1026 sec−3, τo = 108 sec, γ = 109 sec−1, ν = ω.

Fig. 5
Fig. 5

Time evolution of the output Q(t) for a resonance case ν = ω. The parameters are L = 50 cm, RL = 0.8, Go = 4.8 × 1025 sec−3, τo = 10−8 sec: —, solution of space–time (ST) Eqs. (37)(41) for Γ = 0 and v = 0; …, solution of space-averaged (SA) Eqs. (77)(81); - - -, solution for the infinite phase memory decay constant γ = ∞ [Eqs. (84) and (85)]; - -, pumping function (Eq. 86).

Fig. 6
Fig. 6

Time evolution of the output Q(t) for a resonance case ν = ω. The parameters are the same as in Fig. 4 except for the pumping Go = 4.8 × 1026 sec−3. For the designation of curves see Fig. 5.

Fig. 7
Fig. 7

Time evolution of the output Q(t) for the resonance and nonresonance cases. The parameters are (a) L = 50 cm, RL = 0.8, Go = 4.8 × 1025 sec−3, τo = 10−8 sec, γ = 108 sec−1, νω = 0,5γ; (b) L = 50 cm, RL = 0.4, Go = 48 × 1026 sec−3, τo = 10−8 sec, γ = 108 sec−1, νω = 0,5γ; (c) L = 5 cm, RL = 0.8, Go = 4.8 × 1026 sec−3τo = 10−8 sec γ = 109 sec−1, νω = 0,3γ. For the designation of the curves see Fig. 5.

Equations (196)

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ω = E a - E b h ,
Ω m = π m c / L ,
c 2 2 E ˜ z 2 - σ ɛ 0 E ˜ t - 2 E ˜ t 2 = 1 ɛ 0 2 P ˜ t 2 ,
2 P ˜ x 2 + 2 γ P ˜ x + ω 2 P ˜ = 2 ω p 2 h E ˜ N ˜ ,
N ˜ x = G ˜ + 2 h ω E ˜ ( P ˜ x + γ P ˜ ) .
x = t + v z ,
v c ,             γ ω .
B = ν p 2 γ h ɛ 0 ,             N = B N ˜ ,             G = B G ˜ , E = ( ɛ 0 B 2 h ν ) 1 / 2 E ˜ ,             P = - ( ν B 2 h ɛ 0 ) 1 / 2 P ˜ ,
2 E t 2 - c 2 2 E z 2 = 1 ν 2 P t 2 ,
2 P x 2 + 2 γ P x + ω 2 P = 2 ω γ E N ,
N x = G - 4 ω E ( P x + γ P ) ,
E = E + + E - .
P = P + + P - .
E ± ( z , t ) = ɛ ± ( z , t ) cos Φ ± ( z , t ) ,
P ± ( z , t ) = C ± ( z , t ) cos Φ ± ( z , t ) + S ± ( z , t ) sin Φ ± ( z , t ) ,
Φ ± ( z , t ) = ν t k z + ϕ ± ( z , t ) ,
t ν ,             z k .
E = ɛ 1 cos τ cos y + ɛ 2 sin τ sin y .
P = C 1 cos γ cos y + C 2 sin τ sin y + S 1 sin τ cos y - S 2 cos τ sin y .
ɛ 1 = ɛ + + ɛ - ,             ɛ 2 = ɛ + - ɛ - , C 1 = C + + C - ,             C 2 = C + - C - ,             S 1 = S + + S - ,             S 2 = S + - S - , ϕ 1 = ½ ( ϕ + + ϕ - ) ,             ϕ 2 = ½ ( ϕ + - ϕ - ) , τ = ν t + ϕ 1 ,             y = k z + ϕ 2 .
N ( z , t ) = g ( z , t ) + U ( z , t ) cos 2 y + V ( z , t ) sin 2 y ,
ɛ 1 t + c ɛ 2 z = S 1 , ɛ 2 t + c ɛ 1 z = S 1 , ɛ 1 ( ϕ 2 t + c ϕ 1 z ) + ɛ 2 ( ϕ 1 t + c ϕ 2 z ) = C 2 , ɛ 1 ( ϕ 1 t + c ϕ 2 z ) + ɛ 2 ( ϕ 2 t + c ϕ 1 z ) = C 1 .
E N = ɛ 1 cos τ [ g cos y - U ( cos y - 2 cos 3 y ) + 2 V ( sin y - sin 3 y ) ] + ɛ 2 sin τ [ g sin y + U ( sin y - 2 sin 3 y ) + 2 V ( cos y - cos 3 y ) ] .
cos 3 y a cos y ,             sin 3 y b sin y ,
d d a [ 0 π ( cos 3 y - a cos y ) 2 d y ] = 0 , d d b [ 0 π ( sin 3 y - b sin y ) 2 d y ] = 0.
E N = ɛ 1 cos τ ( g cos y + 1 2 U cos y + 1 2 V sin y ) + ɛ 2 sin τ ( g sin y - 1 2 U sin y + 1 2 V cos y ) .
- γ C 1 - ( ν - ω + ϕ 1 t ) S 1 + ( k v - ϕ 2 t ) S 2 - γ 2 ɛ 2 V = C 1 t , - γ C 2 - ( ν - ω + ϕ 1 t ) S 2 + ( k v - ϕ 2 t ) S 1 + γ 2 ɛ 1 V = C 2 t , - γ S 1 + ( ν - ω + ϕ 1 t ) C 1 - ( k v - ϕ 2 t ) C 2 + γ ɛ 1 ( g + 1 2 U ) = S 1 t , - γ S 2 + ( ν - ω + ϕ 1 t ) C 2 - ( k v - ϕ 2 t ) C 1 + γ ɛ 2 ( g - 1 2 U ) = S 2 t .
4 ω E ( P t + γ P ) ¯ = ɛ 1 S 1 + ɛ 2 S 2 + ( ɛ 1 S 1 - ɛ 2 S 2 ) cos 2 y + ( ɛ 1 C 2 - ɛ 2 C 1 ) sin 2 y ,
G - ( ɛ 1 S 2 + ɛ 2 S 2 ) = g t , 2 V ϕ 2 t - ( ɛ 1 S 1 - ɛ 2 S 2 ) = U t , - 2 U ϕ 2 t - ( ɛ 1 C 2 - ɛ 2 C 1 ) = V t .
( a )             ɛ ± t ± c ɛ ± z = S ± , ( b )             ɛ ± ( ϕ ± t ± c ϕ ± z ) = C ± , ( c )             - γ S ± + ( ν - ω k v + ϕ ± ϕ t ) C ± + γ ( ɛ ± g + 1 2 ɛ U ) = S ± t , ( d )             - γ C ± - ( ν - ω k v - ϕ ± t ) S ± ± γ 2 ɛ V = C ± t , ( e )             G - 2 ( ɛ + S + + ɛ - S - ) = g t , ( f )             2 V ( ϕ + - ϕ - ) t - 2 ( ɛ + S - + ɛ - S + ) = U t , ( g )             - 2 U ( ϕ + - ϕ - ) t + 2 ( ɛ + C - - ɛ - C + ) = V t .
ɛ + ( z = O , t ) = n o ɛ - ( z = O , t ) ,
Φ + ( z = O , t ) = Φ - ( z = O , t ) .
n L ɛ + ( z = L , T ) = ɛ - ( z = L , t ) ,
Φ + ( z = L , t ) = Φ - ( z = L , t ) - 2 m π .
ϕ + ( z = O , t ) = ϕ - ( z = O , t ) ,
ϕ + ( z = L , t ) = ϕ - ( z = L , t ) - 2 m π + 2 k L .
k L = m π             ( m = integer ) ,
D ± = ɛ ± exp ( i ϕ ± ) ,
M ± = ( S ± + i C ± ) exp ( i ϕ ± ) ,
Γ = ( U + i V ) exp [ i ( ϕ + - ϕ - ) ,
D ± t ± c D ± z = 1 2 M ± + Σ ± ,
M ± t = - [ γ - i ( ω - ν ) i k v ] M ± + γ D ± g + { γ D - Γ , γ D + Γ * ,
Γ t = - [ D + ( M - ) * + ( D - ) * M + ] ,
g t = G - [ D + ( M + ) * + ( D + ) * M + + D - ( M - ) * + ( D - ) * M - ] ,
D + ( O , t ) = n 0 D - ( O , t ) ,             n L D + ( L , t ) = D - ( L , t ) .
D ± t ± c D z = 1 2 M ( v ) d v + Σ ± ( v ) d v .
M ± = γ + i ( ω - ν ) γ 2 + ( ω - ν ) 2 γ D ± g .
D ± t ± c D ± z = 1 2 γ + i ( ω - ν ) γ 2 + ( ω - ν ) 2 γ D ± g + Σ ± ,
g t = G - 2 γ 2 γ 2 + ( ω - ν ) 2 I g ,
I = D + 2 + D - 2 .
I t - c J z = I g + σ ( 2 ) ,
J t + c I z = J g + σ ( 1 ) ,
σ ( α ) = [ Σ + ( D + ) * + ( Σ + ) * D + ] + ( - 1 ) α [ Σ - ( D - ) * + ( Σ - ) * D - ] ;             α = 1 , 2 ,
J = D + 2 - D - 2 .
σ ( 2 ) = A g a ,             σ ( 1 ) = 0 ,
I t + c J z = I g + A g a , J t + c I t = J g , g t = G - 2 γ 2 γ 2 + ( ω - ν ) 2 I g .
( 1 - R o ) I ( z = O , t ) = ( 1 + R o ) J ( O , t ) , ( 1 - R L ) I ( z = L , t ) = ( 1 + R L ) J ( z = L , t ) .
I t + c J z = F ( 2 ) + σ ( 2 ) ,
F ( α ) = ½ [ ( D + ( M + ) * + ( - 1 ) α ( D + ) * M + + ( D - ( M - ) * + ( - 1 ) α ( D - ) M - ]             α = 1 , 2.
H t = - 2 γ [ H - g F ( 2 ) ] ,
H = M + 2 + M - 2 .
F ( 2 ) t + c W ( 2 ) = 1 2 H - γ [ F ( 2 ) - I g ] - i ( ω - ν ) F ( 1 ) + σ M ( 2 ) .
F ( 1 ) t + c W ( 1 ) = - γ F ( 1 ) - i ( ω - ν ) F ( 2 ) ,
W ( α ) = 1 2 { [ ( M + ) * D + z + ( - 1 ) α M + ( D + ) * z ] - [ ( M - ) * D - z + ( - 1 ) α M - ( D - ) * z ] } ,             α = 1 , 2 ,
σ M ( α ) = 1 2 { [ Σ + ( M + ) * + ( - 1 ) α ( Σ + ) * M + ] + [ Σ - ( M - ) * + ( - 1 ) α ( Σ - ) * M - ] } ,             α = 1 , 2
g t = G - 2 F ( 2 ) .
X ( t ) ¯ = 1 L 0 L X ( z , t ) d z ,
d I ¯ d t = F ¯ ( 2 ) - Q + A g ¯ a ,
d H ¯ d t = - 2 γ [ H - η g ¯ F ¯ ( 2 ) ] ,
d F ¯ ( 2 ) d t = - c W ¯ ( 2 ) + 1 2 H ¯ - γ [ F ( 2 ) - q I ¯ g ¯ ] - i ( ν - ω ) F ¯ ( 1 ) + σ ¯ M ( 2 ) ,
d F ¯ ( 1 ) d t = - c W ¯ ( 1 ) - γ F ¯ ( 1 ) - i ( ν - ω ) F ¯ ( 2 ) + σ ¯ M ( 1 ) ,
d g ¯ d t = G ¯ - 2 F ¯ ( 2 ) .
Q = c L [ J ( L ) - J ( O ) ] ,
Q = c β L q ˜ I ¯ ,
β = - 1 2 ln ( R o R L ) .
q ˜ ( t ) = 2 - R L - R 0 2 β [ 1 - ( 2 - R L - R o ) c 2 L 0 t ( t - t ) g ( t ) d t 0 t g ( t ) d t ] .
W ¯ ( 2 ) = 1 2 c ξ Q g ¯ .
ξ = γ 2 0 t g ( t ) d t g ( t ) .
q = I g ¯ I g ¯ ,
η = F ( 2 ) ¯ F ¯ ( 2 ) g ¯ .
d I d t = F ( 2 ) - c β L q ˜ I + A g a ,
d H d t = - 2 γ [ H - g F ( 2 ) ] ,
d F ( 2 ) d t = - c 2 ξ q ˜ β L g I + 1 2 H - γ [ F ( 2 ) - I g ] + ( ν - ω ) F ( 1 ) ,
d F ( 1 ) d t = - γ F ( 1 ) + ( ν - ω ) F ( 2 ) ,
d g d t = G - 2 F ( 2 ) ,
γ d d t .
F ( 2 ) = I g ,
d I d t = ( g - c L β q ˜ ) I + A g a ,
d g d t = G - 2 I g .
G ( t ) = G o t exp ( - t / τ o ) ,
10 23 sec - 3 G o 10 28 sec 3 ,
f = ν - ω γ .
g ¯ ( t ) = 0 t G ¯ ( t ) d t .
g ¯ ( t ) = G o τ o 2 [ 1 - ( t / τ o + 1 ) exp ( - t / τ o ) ] ,
ξ ( t ) = τ o γ 2 τ [ 1 + exp ( - τ ) ] - 2 [ 1 - exp ( - τ ) ] 1 - ( τ + 1 ) exp ( - τ ) ,
ξ ( t ) = γ 6 t .
q ˜ ( t ) = 1 - α 0 ( 1 - α 1 t ) ,
α 0 = 2 ( β - 1 ) + R o + R L 2 β , α 1 = ( 2 - R o - R L ) 2 8 [ 2 ( β - 1 ) + R o + R l ] L ,
α 0 = 1 - R L 2             α 1 = c 4 L .
ξ ( t ) = 1 - exp ( - γ 6 t ) ,
q ˜ ( t ) = 1 - α 0 exp ( - α 1 t ) 1 - 1 2 ( 1 - R L ) exp ( - c t 4 L ) .
I + t + c I + z = I + g + σ ( + ) , I - t - I - z = I - g + σ ( - ) .
σ ± = σ ( 2 ) ± σ ( 1 ) ,
σ ( 1 ) = σ + - σ - ,             σ ( 2 ) = σ + + σ - .
σ ± = 1 2 A g a ,
σ ( 2 ) = A g a ,             σ ( 1 ) = 0.
σ ( α ) = [ Σ + ( D + ) * + ( Σ + ) * D + ] + ( - 1 ) α [ Σ - ( D - ) * + ( Σ - ) * D - ] ,             α = 1 , 2.
Σ ± = Σ ± exp ( i ϕ ± )
D ± = ɛ ± exp ( i ϕ ± ) ,
σ ( α ) = 2 [ | Σ + | ɛ + + ( - 1 ) α Σ - ɛ - ] ,             α = 1 , 2
Σ + ɛ + = Σ - ɛ - ,
σ ( 2 ) = 4 Σ + ɛ + = A g a ,
Σ ± = 1 4 A g a ( D + ) * .
σ M ( 2 ) = 1 4 A g a ( | S + D + | + | S - D - | ) ,
σ M ( 1 ) = - 1 4 A g a ( | C + D + | + | C - D - | ) ,
D ± z = ± 1 2 c M ± ,
M ± = γ γ - i ( ω - ν ) g D ± .
d I ± d t = 1 c γ 2 γ 2 + ( ω - ν ) 2 g I ± ,
I ± = D ± 2 .
I = I + + I - ,             J = I + - I - ,
d I d z = 1 c γ 2 γ 2 + ( ω - ν ) 2 g J ,
d J d z = 1 c γ 2 γ 2 + ( ω - ν ) 2 g I .
Q = c L [ J ( L ) - J ( O ) ] .
Q = γ 2 γ 2 + ( ω - ν ) 2 g I ¯ ,
q ˜ = L c β Q I ¯ ,
q ˜ = L c γ 2 γ 2 + ( ω - ν ) 2 I g ¯ I ¯ ,
β = - 1 2 ln ( R o R L ) .
W ( 2 ) ( z ) = 1 2 c γ 2 γ 2 + ( ω + ν ) 2 I ( z ) g 2 ( z ) ,
W ( 1 ) ( z ) 0.
W ( 2 ) ( z ) = 1 2 J z g .
ξ = 0 L J z g ( z ) d z g ¯ [ J ( L ) - J ( O ) ] ,
ξ = c L 0 L J z g ( z ) d z g ¯ Q = I g 2 ¯ I g ¯ g ¯ ,
W ( 2 ) ¯ = 1 2 c ξ g ¯ Q .
q = I g ¯ I ¯ g ¯ ,
η = F ( 2 ) ¯ F ( 2 ) ¯ g ¯ .
X o = I o / I s ,
I ¯ = I o 1 + X o ( λ / λ s ) λ s β + X o ,
g ¯ = c β L γ 2 + ( ν - ω ) 2 γ 2 ,
F ( 2 ) ¯ = I g ¯ = c L I o λ s γ 2 + ( ν - ω ) 2 γ 2 ,
F ( 2 ) ¯ g = I g 2 ¯ = c 2 L 2 ( β + X o λ s ) [ β + ln ( ψ 0 ψ 2 ) 1 - y o 2 X o 2 ] ,
λ = ( R o + R L ) ( 1 - R o R L ) + 4 R o R L β 2 ( R o + R L ) 2 ( 1 - R o R L ) 2 ,
λ s = ( 1 - R o ) R L ( R 0 + R L ) ( 1 - R o R L ) ,
ψ 0 = 1 y o ( 1 + y o 2 X o - 1 - R o 1 - R o 1 - y o 2 X o 2 ) / ( 1 + X o ) ,
ψ L = 1 y L ( 1 + y o y L X o - 1 - R L 1 + R L 1 - y o 2 X o 2 ) / ( 1 + y o y L X o ) ,
y o = 2 R o / ( 1 + R o ) ;             y L = 2 R L / ( 1 + R L ) .
q = q ˜ = 1 λ β λ s β + X o λ s / λ + X o .
η = ξ = ( 1 + λ s β X o ) [ 1 + ln ( ψ o ψ L ) 1 - y o 2 X o 2 ] .
σ M ( 2 ) = 1 2 γ 2 γ 2 + ( ω - ν ) 2 g σ ( 2 ) , σ M ( 1 ) = i 2 γ ( ω - ν ) γ 2 + ( ω - ν ) 2 σ ( 2 ) ,
D ± t ± c D ± z = 1 2 + Σ ± ,
M ± t = γ D ± g ,
g t = G .
Σ ± = 1 4 A g a ( D ± ) * .
Σ ± = 1 4 A g ( D ± ) * .
q = I g ¯ / I ¯ g ¯ ,
η = I g 2 ¯ / I ¯ g ¯ 2 ,
W ( 2 ) = 1 2 { [ ( M + ) * D + z + M + ( D + ) * z ] - [ ( M - ) * D - z + M - ( D - ) * z ] } .
M ± ( z , t ) = γ 0 t D ± ( z , t ) g ( t ) d t .
W ( 2 ) ( z , t ) = γ 2 0 t g ( t ) d t ( { [ D + ( z , t ) ] * D + ( z , t ) z + D + ( z , t ) [ D + ( z , t ) ] * z } - { [ D - ( z , t ) ] * D - ( z , t ) z + D - ( z , t ) [ D - ( z , t ) ] * z } ) .
W ( 2 ) ( z , t ) = γ [ I + ( z , t ) z - I - ( z , t ) z ] 0 t g ( t ) d t ,
W ( 2 ) ( z , t ) = γ J z 0 t g ( t ) d t .
W ( 2 ) ¯ ( t ) = γ c Q ( t ) 0 t g ( t ) d t ,
Q ( t ) = c L 0 L J z d z = c L [ J ( L , t ) - J ( O , t ) ] .
W ( 2 ) ¯ ( t ) = 1 2 c ξ Q g ¯ .
ξ = γ 2 0 t g ( t ) d t g ( t ) .
J ( z , t ) = I + ( z , t ) - I - ( z , t ) .
I ¯ ( t ) = 1 L [ 0 L I + ( z , t ) d z + 0 L I - ( z , t ) d z ] ,
Q = c L β q ¯ I ¯ ( t ) ,
β = - 1 2 ln ( R o R L ) ,
I + ( z , t ) = 1 2 A [ Y ( t ) - ( 1 - R o ) Y ( t - z c ) ] ,
I - ( z , t ) = 1 2 A [ Y ( t ) - ( 1 - R L ) Y ( t - L - z c ) ] .
Q ( t ) = c A 2 L ( 2 - R L - R o ) Y ( t ) .
I ( t ) = A [ Y ( t ) - ( 2 - R L - R o ) c 2 L 0 t Y ( t ) d t ] .
q ˜ ( t ) = 2 - R L - R o 2 β [ 1 - ( 2 - R L - R o ) c 2 L 0 t Y ( t ) d t / Y ( t ) ] - 1 ,
q ˜ ( t ) = 2 - R L - R o 2 β [ 1 + ( 2 - R L - R o ) c 2 L 0 t Y ( t ) d t Y ( t ) ] .
I ± z ± I ± z = 1 2 A g ,
τ = t z / c , t = τ ± u / c , u = z , z = u ,
I ± ( τ ± u / c , u ) u = 1 2 A g ( τ ± u / c ) .
I + ( τ + u / c , u ) - I + ( τ , O ) = 1 2 A 0 u g ( τ + u / c ) d u ,
I - ( τ - u / c , u ) - I - ( τ - L / c , L ) = 1 2 A u L g ( τ - u / c ) d u .
I ± ( z , O ) 0 ,
I + ( z , t ) = 1 2 A 0 z g [ t - ( z - z ) / c ] d z ,
I - ( z , t ) = 1 2 A z L g [ t - ( z - z ) / c ] d z .
I + ( z , t ) = 1 2 A 0 z / c g ( t - t ) d t ,
I - ( z , t ) = 1 2 A 0 ( L - z ) / c g ( t - t ) d t .
0 t z / c .
z / c t ( z + L ) / c ,
I + ( z , t ) = 1 2 A { 0 z / c g ( t - t ) d t + R o 0 L / c g ( t - z / c - t ) d t } ,
( z + L ) / c t ( z + 2 L ) / c
I + ( z , t ) = 1 q A { 0 z / c g ( t - t ) d t + R o 0 L / c g ( t - z / c - t ) d t + R o 2 R L 0 L / c g [ t - ( z + L ) / c - t ] d t } .
I + ( z , t ) = 1 q A ( 0 z / c g ( t - t ) d t + l = 1 R o l R L l - 1 { 0 L / c g [ t - z + 2 ( l + 1 ) L L - t ] d t + R L 0 L / c g [ t - z + ( 2 l - 1 ) L c - t ] d t } ) ,
u = t + z + 2 ( l - 1 ) L c ,
u = t + z + ( 2 l - 1 ) L c ,
I + ( z , t ) = 1 2 A { ( 1 - R o ) l = 0 R o l R L l 0 ( z + 2 l L ) / c g ( t - u ) d u + ( 1 - R L ) R o l = 0 R o l R L l 0 [ z + ( 2 l + 1 ) L ] / c g ( t - u ) d u } .
I + ( z , t ) = 1 2 { Y ( t ) - ( 1 - R o ) l = 0 R o l R L l Y ( t - z + 2 l L c ) - ( 1 - R L ) R o l = 0 R o l R L l Y [ t - z + ( 2 l + 1 ) L c ] } ,
Y ( x ) = 0 x g ( t ) d t .
I - ( z , t ) = 1 2 A { Y ( t ) - ( 1 - R L ) l = 0 R o l R L l Y ( t - ( 2 l + 1 ) L - z c ) - ( 1 - R o ) l = 0 R o l R L l Y [ t - 2 ( l + 1 ) L - z c ] }

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