Abstract

A series of simplifications and approximations is introduced into a recently described model for spontaneous pulsations in standing-wave laser oscillators. To the extent that these simplifications are valid they can lead to significant reductions in computation time and sometimes also to a better understanding of the relative importance of various physical effects. Of special interest is the number of spatial harmonics required to represent adequately the effects of longitudinal spatial hole burning. Other approximations investigated include neglect of spectral cross relaxation, neglect of multiple-energy-level equations, neglect of electric-field derivatives, and neglect of polarization derivatives. In addition, a detailed discussion is included that concerns the most efficient numerical implementation of the model.

© 1991 Optical Society of America

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References

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  1. L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
    [CrossRef]
  2. 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.
  3. N. B. Abraham, L. A. Lugiato, and L. M. Narducci, J. Opt. Soc. Am. B 2, 7 (1985).
    [CrossRef]
  4. R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds., Optical Instabilities (Cambridge U. Press, Cambridge, 1986).
  5. N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds., Instabilities and Chaos in Quantum Optics II (Plenum, New York, 1988).
    [CrossRef]
  6. L. W. Casperson, J. Opt. Soc. Am. B 5, 958 (1988).
    [CrossRef]
  7. L. W. Casperson, J. Opt. Soc. Am. B 5, 970 (1988).
    [CrossRef]
  8. L. W. Casperson, J. Opt. Soc. Am. B 2, 73 (1985).
    [CrossRef]
  9. L. W. Casperson, Phys. Rev. 21, 911 (1980).
    [CrossRef]
  10. Y. I. Khanin, in Ref. 4, p. 212.
  11. M. G. Raymer, Z. Deng, and M. Beck, J. Opt. Soc. Am. B 5, 1588 (1988).
    [CrossRef]

1988 (3)

1985 (2)

1980 (1)

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

1978 (1)

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

IEEE J. Quantum Electron. (1)

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

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

Phys. Rev. (1)

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

Other (4)

Y. I. Khanin, in Ref. 4, p. 212.

R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds., Optical Instabilities (Cambridge U. Press, Cambridge, 1986).

N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds., Instabilities and Chaos in Quantum Optics II (Plenum, New York, 1988).
[CrossRef]

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.

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Figures (9)

Fig. 1
Fig. 1

Experimental power spectra of the output from a low-pressure xenon laser at discharge currents of (a) 50 mA (r ~ 2.1) and (b) 40 mA (r ~ 1.7) (from Ref. 7).

Fig. 2
Fig. 2

Theoretical power spectra corresponding to threshold parameters and detunings of (a) r = 2.1, y0 = 2.0 and (b) r = 1.7, y0 = 15.0.

Fig. 3
Fig. 3

Theoretical spontaneous pulsation intensity waveforms for a standing-wave laser with threshold parameter r = 2.0, line-center tuning (y0 = 0), and number of spatial harmonics corresponding to (a) kmax = 8, (b) kmax = 6, (c) kmax = 4, (d) kmax = 2, (e) kmax = 1. The time period for which accurate predictions can be made becomes shorter for smaller values of kmax.

Fig. 4
Fig. 4

Theoretical intensity waveforms for a laser with r = 1.1 and y0 = 0 and the number of spatial harmonics corresponding to (a) kmax = 8, (b) kmax = 6, (c) kmax = 4, (d) kmax = 2, (e) kmax = 1. As the threshold parameter is reduced, fewer spatial harmonics are required for accurate results.

Fig. 5
Fig. 5

Theoretical intensity waveforms for a laser with line-center tuning, kmax = 1, and threshold parameter values (a) r = 2.0, (b) r = 1.8, (c) r = 1.6, (d) r = 1.4, (e) r = 1.2, (f) r = 1.1. Generally the pulsation amplitude, frequency, and waveform complexity decrease with decreasing values of r.

Fig. 6
Fig. 6

Theoretical intensity waveforms for a laser with line-center tuning and kmax = 1 but with spectral cross relaxation neglected. The threshold parameter values are (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. Comparison with data such as those shown in Fig. 5 indicates that spectral cross relaxation causes a modest increase in pulsation frequency but no major qualitative change in the general pulsation characteristics.

Fig. 7
Fig. 7

Theoretical intensity waveforms for a laser with line-center tuning but with simplified energy-level structure and with spectral cross relaxation neglected. The threshold parameter values are (a) r = 2.0, (b) r = 1.8, (c) r = 1.6, (d) r = 1.4, (e) r = 1.2, (f) r = 1.1.

Fig. 8
Fig. 8

Theoretical intensity waveforms for a laser with line-center tuning but with field derivatives and spectral cross relaxation neglected and with a simplified energy-level structure. The threshold parameter values are (a) r = 2.0, (b) r = 1.8, (c) r = 1.6, (d) r = 1.4, (e) r = 1.2, (f) r = 1.1. Comparison with Fig. 7 indicates that the neglect of the field derivatives in this example has little effect on the pulsations.

Fig. 9
Fig. 9

Theoretical intensity waveforms for a laser with line-center tuning in the rate-equation approximation. The threshold parameter values are (a) r = 2.0, (b) r = 1.8, (c) r = 1.6, (d) r = 1.4, (e) r = 1.2, (f) r = 1.1. In the rate-equation approximation the relaxation oscillations are always damped.

Equations (92)

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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 ) ] } + ɛ Γ a 2 π 1 / 2 exp ( - ɛ 2 V 2 ) - [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V - ɛ Γ b 2 π 1 / 2 exp ( - ɛ 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 ) + λ 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 ) + ɛ Γ a 2 π 1 / 2 exp ( - ɛ 2 V 2 ) - [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V + ɛ Γ b 2 π 1 / 2 exp ( - ɛ 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 ] .
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 a ) / 2 ,
γ 1 = 2 γ a γ b / ( γ a - γ a b + γ b ) .
D 2 j ( V , t ) = ɛ E 2 j ( t ) π 1 / 2 exp ( - ɛ 2 V 2 ) + D 2 j ( V , t ) ,
M 2 j ( V , t ) = ɛ F 2 j ( t ) π 1 / 2 exp ( - ɛ 2 V 2 ) + M 2 j ( V , t ) ,
λ a ( V , t ) = ɛ L a ( t ) π 1 / 2 exp ( - ɛ 2 V 2 ) ,
λ b ( V , t ) = ɛ L b ( t ) π 1 / 2 exp ( - ɛ 2 V 2 ) ,
E 2 j ( t ) t = [ L a ( t ) - L b ( t ) ] δ j 0 - h 1 E 2 j ( t ) - h 2 F 2 j ( t ) + Γ a 2 { F 2 j ( t ) + E 2 j ( t ) + - [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V } - Γ b 2 { F 2 j ( t ) - E 2 j ( t ) + - [ M 2 j ( V , t ) - D 2 j ( V , t ) ] d V } ,
F 2 j ( t ) t = [ L a ( t ) + L b ( t ) ] δ j 0 - h 3 F 2 j ( t ) - h 4 E 2 j ( t ) + Γ a 2 { F 2 j ( t ) + E 2 j ( t ) + - [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V } + Γ b 2 { F 2 j ( t ) - E 2 j ( t ) + - [ M 2 j ( V , t ) - D 2 j ( V , t ) ] d V } ,
2 D 2 j ( V , t ) t = - [ h 1 + ( 2 j ) i γ V ] D 2 j ( V , t ) - h 2 M 2 j ( V , t ) - ( 2 j ) i γ V E 2 j ( t ) ɛ π 1 / 2 exp ( - ɛ 2 V 2 ) - 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 ) ] } ,
M 2 j ( V , t ) t = - [ h 3 + ( 2 j ) i γ V ] M 2 j ( V , t ) - h 4 D 2 j ( V , t ) - ( 2 j ) i γ V F 2 j ( t ) ɛ π 1 / 2 exp ( - ɛ 2 V 2 ) .
d E 2 j ( t ) d t = [ L a ( t ) - L b ( t ) ] δ j 0 - h 5 E 2 j ( t ) - h 6 F 2 j ( t ) + Γ 1 2 - [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V - Γ b 2 - [ M 2 j ( V , t ) - D 2 j ( V , t ) ] d V ,
d F 2 j ( t ) d t = [ L a ( t ) + L b ( t ) ] δ j 0 - h 7 F 2 j ( t ) - h 8 E 2 j ( t ) + Γ a 2 - [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V + Γ b 2 - [ M 2 j ( V , t ) - D 2 j ( V , t ) ] d V ,
h 5 = h 1 - Γ a / 2 - Γ b / 2 = ( γ a + γ a b + γ b ) / 2 ,
h 6 = h 2 - Γ a / 2 + Γ b / 2 = ( γ a + γ a b - γ b ) / 2 ,
h 7 = h 3 - Γ a / 2 - Γ b / 2 = ( γ a - γ a b + γ b ) / 2 ,
h 8 = h 4 - Γ a / 2 + Γ b / 2 = ( γ a - γ a b - γ b ) / 2.
P r , k ( V , t ) t = - γ { [ 1 + ( 2 k - 1 ) i V ] P r , k ( V , t ) + y P i , k ( V , t ) + i A i ( t ) [ ɛ E k ( t ) exp ( - ɛ 2 V 2 ) π 1 / 2 - ɛ E k + 1 ( t ) exp ( - ɛ 2 V 2 ) π 1 / 2 + D k ( V , t ) - D k + 1 ( V , t ) ] } ,
P i , k ( V , t ) t = - γ { [ 1 + ( 2 k - 1 ) i V ] P i , k ( V , t ) - y P r , k ( V , t ) - i A r ( t ) [ ɛ E k ( t ) exp ( - ɛ 2 V 2 ) π 1 / 2 - ɛ E k + 1 ( t ) exp ( - ɛ 2 V 2 ) π 1 / 2 + D k ( V , t ) - D k + 1 ( V , t ) ] } ,
D k ( V , t ) t = - [ h 1 + ( 2 k - 2 ) i γ V ] D k ( V , t ) - h 2 M k ( V , t ) - ( 2 k - 2 ) i γ V E k ( t ) ɛ exp ( - ɛ 2 V 2 ) π 1 / 2 - i γ 1 [ A r ( t ) P i , k - 1 ( V , t ) - A i ( t ) P r , k - 1 ( V , t ) - A r ( t ) P i , k ( V , t ) + A i ( t ) P r , k ( V , t ) ] ,
M k ( V , t ) t = - [ h 3 + ( 2 k - 2 ) i γ V ] M k ( V , t ) - h 4 D 4 ( V , t ) - ( 2 k - 2 ) i γ V F k ( t ) ɛ exp ( - ɛ 2 V 2 ) π 1 / 2 ,
d E k ( t ) d t = [ L a ( t ) - L b ( t ) ] δ k 1 - h 5 E k ( t ) - h 6 F k ( t ) + Γ a 2 - [ M k ( V , t ) + D k ( V , t ) ] d V - Γ b 2 - [ M k ( V , t ) - D k ( V , t ) ] d V ,
d F k ( t ) d t = [ L a ( t ) + L b ( t ) ] δ k 1 - h 7 F k ( t ) - h 8 E k ( t ) + Γ a 2 - [ M k ( V , t ) + D k ( V , t ) ] d V + Γ b 2 - [ M k ( V , t ) - D k ( 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 ] .
P r , j ( V , t ) = P r , - j * ( V , t ) ,
P i , j ( V , t ) = P i , - j * ( V , t ) ,
D j ( V , t ) = D - j * ( V , t ) ,
M j ( V , t ) = M - j * ( V , t ) .
D 1 ( V , t ) t = - h 1 D 1 ( V , t ) - h 2 M 1 ( V , t ) - 2 γ 1 [ A r ( t ) P i , 1 i ( V , t ) - A i ( t ) P r , 1 i ( V , t ) ] .
D 2 j ( V , t ) t = [ λ a ( V ) - λ b ( V ) ] δ j 0 - [ h 1 - ( 2 j ) i γ V ] D 2 j ( V , t ) - h 2 M 2 j ( V , t ) + ɛ Γ 1 2 π 1 / 2 exp ( - ɛ 2 V 2 ) - [ M 2 j ( V , t ) + D 2 j ( V , t ) ] d V - ɛ Γ b 2 π 1 / 2 exp ( - ɛ 2 V 2 ) - [ M 2 j ( V , t ) - D 2 j ( V , t ) ] d V ,
D 2 j ( V , t ) t = [ λ a ( V ) - λ b ( V ) ] δ j 0 - [ h 5 + ( 2 j ) i γ V ] D 2 j ( V , t ) - h 6 M 2 j ( V , t ) ,
M 2 j ( V , t ) t = [ λ a ( V ) + λ b ( V ) ] δ j 0 - [ h 7 + ( 2 j ) i γ V ] M 2 j ( V , t ) - h 8 D 2 j ( V , t ) ,
0 = ( 1 + i V ) P r , 1 ( V ) + y P i , 1 ( V ) + i A i D 0 ( V ) ,
0 = ( 1 + i V ) P i , 1 ( V ) - y P r , 1 ( V ) - i A r D 0 ( V ) ,
0 = λ a ( V ) - λ b ( V ) - h 5 D 0 ( V ) - h 6 M 0 ( V ) ,
0 = λ a ( V ) + λ b ( V ) - h 7 M 0 ( V ) - h 8 D 0 ( V ) ,
0 = A r + δ ( y - y 0 ) A i - - P i , 1 i ( V ) d V ,
0 = A i - δ ( y - y 0 ) A r + - P r , 1 i ( V ) d V .
P r , 1 ( V ) = - i 1 + i V ( y 1 + i V A r + A i ) × D 0 ( V ) 1 1 + [ y / ( 1 + i V ) ] 2 = - i α 0 ( V , y ) [ β 0 ( V , y ) A r + A i ] D 0 ( V ) ,
P i , 1 ( V ) = - i 1 + i V ( y 1 + i V A i - A r ) × D 0 ( V ) 1 1 + [ y / ( 1 + i V ) ] 2 = - i α 0 ( V , y ) [ β 0 ( V , y ) A i - A r ] D 0 ( V ) ,
α j ( V , y ) = 1 / 2 1 + ( 2 j + 1 ) V + i y + 1 / 2 1 + ( 2 j + 1 ) i V - i y ,
β j ( V , y ) = y 1 + ( 2 j + 1 ) i V .
D 0 ( V ) = ( h 7 - h 6 ) λ a ( V ) - ( h 7 - h 6 ) λ b ( V ) h 5 h 7 - h 6 h 8 = ( 1 - γ a b γ b ) λ a ( V ) γ a - λ b ( V ) γ b ,
0 = A r + δ ( y - y 0 ) A i + A i - Re [ α 0 ( V , y ) β 0 ( V , y ) ] D 0 ( V ) d V - A r - Re [ α 0 ( V , y ) ] D 0 ( V ) d V ,
0 = A i - δ ( y - y 0 ) A r - A r - Re [ α 0 ( V , y ) β 0 ( V , y ) ] D 0 ( V ) d V - A i - Re [ α 0 ( V , y ) ] D 0 ( V ) d V .
1 = - Re [ α 0 ( V , y ) ] D 0 ( V ) d V ,
δ ( y - y 0 ) = - Re [ α 0 ( V , y ) β 0 ( V , y ) ] D 0 ( V ) d V .
1 = - Re ( 1 1 + i V ) [ ( 1 - γ a b γ b ) λ a ( V ) γ a - λ b ( V ) γ b ] d V .
1 = ɛ π 1 / 2 [ ( 1 - γ a b γ b ) L a γ a - L b γ b ] - exp ( - ɛ 2 V 2 ) 1 + V 2 d V = ɛ π 1 / 2 L a [ ( 1 - γ a b γ b ) 1 γ a - q γ b ] - exp ( - ɛ 2 V 2 ) 1 + V 2 d V ,
L a = r L ath = r π 1 / 2 ɛ { [ ( 1 - γ a b γ b ) 1 γ a - q γ b ] - exp ( - ɛ 2 V 2 ) 1 + V 2 d V } - 1 .
0 = L a - L b - γ a + γ a b + γ b 2 E 1 - γ a + γ a b - γ b 2 F 1 ,
0 = L a + L b - γ a - γ a b - γ b 2 E 1 - γ a - γ a b + γ b 2 F 1 .
E 1 = ( 1 - γ a b γ b ) L a γ a - L b γ b ,
F 1 = ( 1 + γ a b γ b ) L a γ a + L b γ b .
P r , 1 ( V , t ) t = - γ [ ( 1 + i V ) P r , 1 ( V , t ) + y P i , 1 ( V , t ) + i A i ( t ) D 0 ( V , t ) ] ,
P i , 1 ( V , t ) t = - γ [ ( 1 + i V ) P i , 1 ( V , t ) - y P r , 1 ( V , t ) - i A r ( t ) D 0 ( V , t ) ] ,
D 0 ( V , t ) t = λ a ( V ) - λ b ( V ) - h 1 D 0 ( V , t ) - h 2 M 0 ( V , t ) - 2 γ 1 [ A r ( t ) P i , 1 i ( V , t ) - A i ( t ) P r , 1 i ( V , t ) ] + ɛ Γ a 2 π 1 / 2 exp ( - ɛ 2 V 2 ) × - [ M 0 ( V , t ) + D 0 ( V , t ) ] d V - ɛ Γ b 2 π 1 / 2 exp ( - ɛ 2 V 2 ) × - [ M 0 ( V , t ) - D 0 ( V , t ) ] d V ,
M 0 ( V , t ) t = λ a ( V ) + λ b ( V ) - h 3 M 0 ( V , t ) - h 4 D 0 ( V , t ) + ɛ Γ a 2 π 1 / 2 exp ( - ɛ 2 V 2 ) × - [ M 0 ( V , t ) + D 0 ( V , t ) ] d V + ɛ Γ b 2 π 1 / 2 exp ( - ɛ 2 V 2 ) × - [ M 0 ( V , t ) - D 0 ( 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 ] ,
P r , 1 ( V , t ) t = - γ [ ( 1 + i V ) P r , 1 ( V , t ) + y P i , 1 ( V , t ) + i A i ( t ) D 0 ( V , t ) ] ,
P i , 1 ( V , t ) t = - γ [ ( 1 + i V ) P i , 1 ( V , t ) - y P r , 1 ( V , t ) - i A r ( t ) D 0 ( V , t ) ] ,
D 0 ( V , t ) t = λ a ( V ) - λ b ( V ) - γ a + γ a b + γ b 2 D 0 ( V , t ) - γ a + γ a b - γ b 2 M 0 ( V , t ) - 4 γ a γ b γ a - γ a b + γ b × [ A r ( t ) P i , 1 i ( V , t ) - A i ( t ) P r , 1 i ( V , t ) ] ,
M 0 ( V , t ) t = λ a ( V ) + λ b ( V ) - γ a - γ a b + γ b 2 M 0 ( V , t ) - γ a - γ a b - γ b 2 D 0 ( V , 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 , 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 ] ,
P r , 1 ( V , t ) t = - γ [ ( 1 + i V ) P r , 1 ( V , t ) + y P i , 1 ( V , t ) + i A i ( t ) D 0 ( V , t ) ] ,
P i , 1 ( V , t ) t = - γ [ ( 1 + i V ) P i , 1 ( V , t ) - y P r , 1 ( V , t ) - i A r ( t ) D 0 ( V , t ) ] ,
D 0 ( V , t ) t = λ a ( V ) - λ b ( V ) - ( γ a + γ a b ) D 0 ( V , t ) - 2 ( γ a + γ a b ) [ A r ( t ) P i , 1 i ( V , t ) - A i ( t ) P r , 1 i ( V , t ) ] ,
d A r ( t ) d t = 1 2 t c [ A r ( t ) + δ ( y - y 0 ) A i ( t ) - - P 1 , 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 ] .
A r ( t ) = δ ( y - y 0 ) - P r , 1 i ( V , t ) d V + - P i , 1 i ( V , t ) d V 1 + δ 2 ( y - y 0 ) 2 ,
A i ( t ) = δ ( y - y 0 ) - P i , 1 i ( V , t ) d V - - P r , 1 i ( V , t ) d V 1 + δ 2 ( y - y 0 ) 2 .
A r ( t ) = + - P i , 1 i ( V , t ) d V ,
A i ( t ) = - - P r , 1 i ( V , t ) d V .
P r , 1 ( V , t ) t = - γ [ ( 1 + i V ) P r , 1 ( V , t ) - i D 0 ( V , t ) - P r , 1 i ( V , t ) d V ] ,
P i , 1 ( V , t ) t = - γ [ ( 1 + i V ) P i , 1 ( V , t ) - i D 0 ( V , t ) - P i , 1 i ( V , t ) d V ] ,
D 0 ( V , t ) t = λ d ( V ) - γ d D 0 ( V , t ) - 2 γ d [ P i , 1 i ( V , t ) - P i , 1 i ( V , t ) d V + P r , 1 i ( V , t ) - P r , 1 i ( V , t ) d V ] ,
P i 1 , r ( V , t ) t = - γ [ P i , 1 r ( V , t ) - V P i , 1 i ( V , t ) ] ,
P i , 1 i ( V , t ) t = - γ [ P i , 1 i ( V , t ) + V P i , 1 r ( V , t ) - D 0 ( V , t ) - P i , 1 i ( V , t ) d V ] ,
D 0 ( V , t ) t = λ d ( V ) - γ d D 0 ( V , t ) - 2 γ d [ P i , 1 i ( V , t ) - P i , 1 i ( V , t ) d V ] .
P r , 2 j + 1 ( V , t ) = - i { [ 1 + ( 2 j + 1 ) i V ] A i ( t ) + y A r ( t ) } [ 1 + ( 2 j + 1 ) i V ] 2 + y 2 × [ D 2 j ( V , t ) - D 2 j + 1 ( V , t ) ] ,
P i , 2 j + 1 ( V , t ) = - i { - [ 1 + ( 2 j + 1 ) i V ] A r ( t ) + y A i ( t ) } [ 1 + ( 2 j + 1 ) i V ] 2 + y 2 × [ D 2 j ( V , t ) - D 2 j + 2 ( V , t ) ] .

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