Abstract

Low-threshold spontaneous pulsations are known to occur in the output beams of certain high-gain gas lasers, and good agreement between the experimental observations and numerical models has been achieved. There have also been several analytical studies of threshold criteria for spontaneously pulsing lasers. However, the analytical studies are mostly not applicable to the lasers in which the pulsations have been observed. Stability criteria for high-gain gas lasers are derived in this study, and these criteria are compared with a previously modeled gas laser instability.

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  1. L. W. Casperson, in Laser Physics, J.D.Harvey and D.F.Walls, eds., Vol. 182 of Springer Lecture Notes in Physics (Springer-Verlag, 1983), pp. 88-106.
    [CrossRef]
  2. L. W. Casperson, “Spontaneous coherent pulsations in laser oscillators,” IEEE J. Quantum Electron. QE-14, 756-761 (1978).
    [CrossRef]
  3. J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in high-gain lasers,” J. Opt. Soc. Am. 70, 1622 (1980).
  4. J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in a high gain He-Xe laser,” Opt. Commun. 41, 52-56 (1982).
    [CrossRef]
  5. M. Maeda and N. B. Abraham, “Measurements of mode-splitting self-pulsing in a single-mode, Fabry-Perot laser,” Phys. Rev. A 26, 3395-3403 (1982).
    [CrossRef]
  6. 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]
  7. 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]
  8. P. Chenkosol and L. W. Casperson, “Spontaneous coherent pulsations in 3.39 μm He-Ne Standing-Wave Laser Oscillators,” J. Opt. Soc. Am. B 20, 2539-2547 (2003).
    [CrossRef]
  9. L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators,” J. Opt. Soc. Am. B 2, 62-72 (1985).
  10. D. K. Bandy, L. M. Narducci, L. A. Lugiato, and N. B. Abraham, “Time-dependent behavior of a unidirectional ring laser with inhomogeneous broadening,” J. Opt. Soc. Am. B 2, 56-61 (1985).
    [CrossRef]
  11. L. W. Casperson, “Spontaneous coherent pulsations in standing-wave laser oscillators,” J. Opt. Soc. Am. B 5, 958-969 (1988).
  12. P. Mandel, “Influence of Lorentz broadening on the stability of monomode ring lasers,” Opt. Commun. 45, 269-272 (1983).
    [CrossRef]
  13. L. A. Lugiato, L. M. Narducci, D. K. Bandy, and N. B. Abraham, “Instabilities in inhomogeneously broadened single-mode lasers,” Opt. Commun. 46, 115-120 (1983).
    [CrossRef]
  14. P. Mandel, “Influence of Doppler broadening on the stability of monomode ring lasers,” Opt. Commun. 44, 400-404 (1983).
    [CrossRef]
  15. L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators: stability criteria,” J. Opt. Soc. Am. 2, 993-997 (1985), and references.
  16. L. W. Casperson, “Stability criteria for lasers with mixed line broadening,” Opt. Quantum Electron. 19, 29-36 (1987).
    [CrossRef]
  17. M. L. Minden and L. W. Casperson, “Dispersion-induced instability in cw laser oscillators,” IEEE J. Quantum Electron. QE-18, 1952-1957 (1982).
  18. C. S. Willett, “Neutral gas lasers,” in Handbook of Lasers with Selected Data on Optical Technology, R.J.Pressley, ed. (CRC Press, 1971) pp. 183-241.
  19. K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65-68 (1993).
    [CrossRef] [PubMed]

2003 (1)

1993 (1)

K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65-68 (1993).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

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

1985 (3)

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 (4)

P. Mandel, “Influence of Lorentz broadening on the stability of monomode ring lasers,” Opt. Commun. 45, 269-272 (1983).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and N. B. Abraham, “Instabilities in inhomogeneously broadened single-mode lasers,” Opt. Commun. 46, 115-120 (1983).
[CrossRef]

P. Mandel, “Influence of Doppler broadening on the stability of monomode ring lasers,” Opt. Commun. 44, 400-404 (1983).
[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]

1982 (3)

M. L. Minden and L. W. Casperson, “Dispersion-induced instability in cw laser oscillators,” IEEE J. Quantum Electron. QE-18, 1952-1957 (1982).

J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in a high gain He-Xe laser,” Opt. Commun. 41, 52-56 (1982).
[CrossRef]

M. Maeda and N. B. Abraham, “Measurements of mode-splitting self-pulsing in a single-mode, Fabry-Perot laser,” Phys. Rev. A 26, 3395-3403 (1982).
[CrossRef]

1980 (1)

J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in high-gain lasers,” J. Opt. Soc. Am. 70, 1622 (1980).

1978 (1)

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

Abraham, N. B.

D. K. Bandy, L. M. Narducci, L. A. Lugiato, and N. B. Abraham, “Time-dependent behavior of a unidirectional ring laser with inhomogeneous broadening,” J. Opt. Soc. Am. B 2, 56-61 (1985).
[CrossRef]

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]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and N. B. Abraham, “Instabilities in inhomogeneously broadened single-mode lasers,” Opt. Commun. 46, 115-120 (1983).
[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]

M. Maeda and N. B. Abraham, “Measurements of mode-splitting self-pulsing in a single-mode, Fabry-Perot laser,” Phys. Rev. A 26, 3395-3403 (1982).
[CrossRef]

J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in a high gain He-Xe laser,” Opt. Commun. 41, 52-56 (1982).
[CrossRef]

J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in high-gain lasers,” J. Opt. Soc. Am. 70, 1622 (1980).

Bandy, D. K.

D. K. Bandy, L. M. Narducci, L. A. Lugiato, and N. B. Abraham, “Time-dependent behavior of a unidirectional ring laser with inhomogeneous broadening,” J. Opt. Soc. Am. B 2, 56-61 (1985).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and N. B. Abraham, “Instabilities in inhomogeneously broadened single-mode lasers,” Opt. Commun. 46, 115-120 (1983).
[CrossRef]

Bentley, J.

J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in a high gain He-Xe laser,” Opt. Commun. 41, 52-56 (1982).
[CrossRef]

J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in high-gain lasers,” J. Opt. Soc. Am. 70, 1622 (1980).

Casperson, L. W.

P. Chenkosol and L. W. Casperson, “Spontaneous coherent pulsations in 3.39 μm He-Ne Standing-Wave Laser Oscillators,” J. Opt. Soc. Am. B 20, 2539-2547 (2003).
[CrossRef]

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

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

L. W. Casperson, “Spontaneous coherent pulsations in ring-laser oscillators,” J. Opt. Soc. Am. B 2, 62-72 (1985).

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

M. L. Minden and L. W. Casperson, “Dispersion-induced instability in cw laser oscillators,” IEEE J. Quantum Electron. QE-18, 1952-1957 (1982).

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

L. W. Casperson, in Laser Physics, J.D.Harvey and D.F.Walls, eds., Vol. 182 of Springer Lecture Notes in Physics (Springer-Verlag, 1983), pp. 88-106.
[CrossRef]

Chenkosol, P.

Cuomo, K. M.

K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65-68 (1993).
[CrossRef] [PubMed]

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]

Lugiato, L. A.

D. K. Bandy, L. M. Narducci, L. A. Lugiato, and N. B. Abraham, “Time-dependent behavior of a unidirectional ring laser with inhomogeneous broadening,” J. Opt. Soc. Am. B 2, 56-61 (1985).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and N. B. Abraham, “Instabilities in inhomogeneously broadened single-mode lasers,” Opt. Commun. 46, 115-120 (1983).
[CrossRef]

Maeda, M.

M. Maeda and N. B. Abraham, “Measurements of mode-splitting self-pulsing in a single-mode, Fabry-Perot laser,” Phys. Rev. A 26, 3395-3403 (1982).
[CrossRef]

Mandel, P.

P. Mandel, “Influence of Lorentz broadening on the stability of monomode ring lasers,” Opt. Commun. 45, 269-272 (1983).
[CrossRef]

P. Mandel, “Influence of Doppler broadening on the stability of monomode ring lasers,” Opt. Commun. 44, 400-404 (1983).
[CrossRef]

Minden, M. L.

M. L. Minden and L. W. Casperson, “Dispersion-induced instability in cw laser oscillators,” IEEE J. Quantum Electron. QE-18, 1952-1957 (1982).

Narducci, L. M.

D. K. Bandy, L. M. Narducci, L. A. Lugiato, and N. B. Abraham, “Time-dependent behavior of a unidirectional ring laser with inhomogeneous broadening,” J. Opt. Soc. Am. B 2, 56-61 (1985).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and N. B. Abraham, “Instabilities in inhomogeneously broadened single-mode lasers,” Opt. Commun. 46, 115-120 (1983).
[CrossRef]

Oppenheim, A. V.

K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65-68 (1993).
[CrossRef] [PubMed]

Willett, C. S.

C. S. Willett, “Neutral gas lasers,” in Handbook of Lasers with Selected Data on Optical Technology, R.J.Pressley, ed. (CRC Press, 1971) pp. 183-241.

IEEE J. Quantum Electron. (2)

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

M. L. Minden and L. W. Casperson, “Dispersion-induced instability in cw laser oscillators,” IEEE J. Quantum Electron. QE-18, 1952-1957 (1982).

J. Opt. Soc. Am. (2)

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

J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in high-gain lasers,” J. Opt. Soc. Am. 70, 1622 (1980).

J. Opt. Soc. Am. B (4)

Opt. Commun. (5)

J. Bentley and N. B. Abraham, “Mode-pulling, mode-splitting and pulsing in a high gain He-Xe laser,” Opt. Commun. 41, 52-56 (1982).
[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]

P. Mandel, “Influence of Lorentz broadening on the stability of monomode ring lasers,” Opt. Commun. 45, 269-272 (1983).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and N. B. Abraham, “Instabilities in inhomogeneously broadened single-mode lasers,” Opt. Commun. 46, 115-120 (1983).
[CrossRef]

P. Mandel, “Influence of Doppler broadening on the stability of monomode ring lasers,” Opt. Commun. 44, 400-404 (1983).
[CrossRef]

Opt. Quantum Electron. (1)

L. W. Casperson, “Stability criteria for lasers with mixed line broadening,” Opt. Quantum Electron. 19, 29-36 (1987).
[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. Maeda and N. B. Abraham, “Measurements of mode-splitting self-pulsing in a single-mode, Fabry-Perot laser,” Phys. Rev. A 26, 3395-3403 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65-68 (1993).
[CrossRef] [PubMed]

Other (2)

L. W. Casperson, in Laser Physics, J.D.Harvey and D.F.Walls, eds., Vol. 182 of Springer Lecture Notes in Physics (Springer-Verlag, 1983), pp. 88-106.
[CrossRef]

C. S. Willett, “Neutral gas lasers,” in Handbook of Lasers with Selected Data on Optical Technology, R.J.Pressley, ed. (CRC Press, 1971) pp. 183-241.

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Figures (1)

Fig. 1
Fig. 1

Type-1 stability boundaries for inhomogeneously broadened unidirectional ring laser oscillators with uniform plane-wave electric fields.

Equations (59)

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P r ( V , t ) t = γ [ P r ( V , t ) V P i ( V , t ) ] ,
P i t ( V , t ) = γ { P i ( V , t ) + V P r ( V , t ) + A ( t ) [ D 0 + D ( V , t ) ] } ,
D ( V , t ) t = γ d [ D ( V , t ) A ( t ) P i ( V , t ) ] ,
d A ( t ) d t = γ c [ A ( t ) + P i ( V , t ) d V ] .
P r s ( V ) = V P i s ( V ) ,
P i s ( V ) = V P r s ( V ) A s ( D 0 + D s ( V ) ) ,
D s ( V ) = A s P i s ( V ) ,
A s = P i s ( V ) d V .
P i s ( V ) = A s D 0 ( 1 + V 2 + A s 2 ) .
1 = D 0 d V ( 1 + V 2 + A s 2 ) = 2 D 0 0 d V ( 1 + V 2 + A s 2 ) = D 0 π 1 + A s 2 .
R = π D 0 = 1 + A s 2 .
A s 2 = R 2 1 .
P i ( V , t ) t = γ [ P i ( V , t ) + V P r ( V , t ) + A ( t ) ( R π + D ( V , t ) ) ] .
P r ( V , t ) = P r s ( V ) + P r ( V , t ) ,
P i ( V , t ) = P i s ( V ) + P i ( V , t ) ,
D ( V , t ) = D s ( V ) + D ( V , t ) ,
A ( t ) = A s + A ( t ) ,
P r ( V , t ) t = γ [ ( P r s ( V ) + P r ( V , t ) ) V ( P i s ( V ) + P i ( V , t ) ) ] = γ [ P r ( V , t ) V P i ( V , t ) ] ,
P i ( V , t ) t = γ [ ( P i s ( V ) + P i ( V , t ) ) + V ( P r s ( V ) + P r ( V , t ) ) + ( A s + A ( t ) ) ( R π + D s ( V ) + D ( V , t ) ) ] γ [ P i ( V , t ) + V P r ( V , t ) + A s D ( V , t ) + A ( t ) ( R π + D s ( V ) ) ] ,
D ( V , t ) t = γ d [ ( D s ( V ) + D ( V , t ) ) ( A s + A ( t ) ) ( P i s ( V ) + P i ( V , t ) ) ] γ d [ D ( V , t ) ( A s P i ( V , t ) + A ( t ) P i s ( V ) ) ] ,
d A ( t ) d t = γ c [ ( A s + A ( t ) ) + ( P i s ( V ) + P i ( V , t ) ) d V ] = γ c [ A ( t ) + P i ( V , t ) d V ] ,
P r ( V , t ) t = δ [ P r ( V , t ) V P i ( V , t ) ] ,
P i ( V , t ) t = δ [ P i ( V , t ) + V P r ( V , t ) + A s D ( V , t ) + A ( t ) ( R π + D s ( V ) ) ] ,
D ( V , t ) t = δ ρ [ D ( V , t ) ( A s P i ( V , t ) + A ( t ) P i s ( V ) ) ] ,
d A ( t ) d t = [ A ( t ) + P i ( V , t ) d V ] .
P r ( V , t ) = P r ( V ) e λ t ,
P i ( V , t ) = P i ( V ) e λ t ,
D ( V , t ) = D ( V ) e λ t ,
A ( t ) = A e λ t ,
λ P r ( V ) = δ [ P r ( V ) V P i ( V ) ] ,
( λ + δ ) P r ( V ) = δ V P i ( V ) ,
λ P i ( V ) = δ [ P i ( V ) + V P r ( V ) + A s D ( V ) + A ( R π + D s ( V ) ) ] ,
( λ + δ ) P i ( V ) = δ [ V P r ( V ) + A s D ( V ) + A ( R π + D s ( V ) ) ] ,
λ D ( V ) = δ ρ [ D ( V ) ( A s P i ( V ) + A P i s ( V ) ) ] ,
( λ + δ ρ ) D ( V ) = δ ρ [ A s P i ( V ) + A P i s ( V ) ] ,
λ A = [ A + P i ( V ) d V ] ,
( λ + 1 ) A = P i ( V ) d V .
( λ + δ ) P i ( V ) = δ [ V ( δ V λ + δ ) P i ( V ) + A s D ( V ) + A ( R π + D s ( V ) ) ] ,
( λ + δ + δ 2 V 2 λ + δ ) P i ( V ) = δ [ A s D ( V ) + A ( R π + D s ( V ) ) ] .
( λ + δ + δ 2 V 2 λ + δ ) P i ( V ) = δ [ A s { δ ρ λ + δ ρ ( A s P i ( V ) + A P i s ( V ) ) } + A ( R π + D s ( V ) ) ] ,
( λ + δ + δ 2 V 2 λ + δ + δ 2 ρ A s 2 λ + δ ρ ) P i ( V ) = δ 2 ρ A s ( λ + δ ρ ) [ A s ( R π ) 1 + V 2 + A s 2 ] A δ [ ( R π ) A s 2 ( R π ) 1 + V 2 + A s 2 ] A = δ 2 ρ ( R π ) A s 2 A ( λ + δ ρ ) ( 1 + V 2 + A s 2 ) δ ( R π ) ( 1 + V 2 ) A ( 1 + V 2 + A s 2 ) = [ δ ( R π ) A 1 + V 2 + A s 2 ] [ 1 + V 2 δ ρ A s 2 λ + δ ρ ] ,
P i ( V ) = [ δ ( R π ) A 1 + V 2 + A s 2 ] [ 1 + V 2 δ ρ A s 2 λ + δ ρ ] ( λ + δ + δ 2 V 2 λ + δ + δ 2 ρ A s 2 λ + δ ρ ) ,
P i s ( V ) = A s D 0 1 + V 2 + A s 2 = A s ( R π ) 1 + V 2 + A s 2 ,
D s ( V ) = A s P i s ( V ) = A s 2 ( R π ) 1 + V 2 + A s 2 ,
( λ + 1 ) = [ δ ( R π ) 1 + V 2 + A s 2 ] [ 1 + V 2 δ ρ A s 2 λ + δ ρ ] ( λ + δ + δ 2 V 2 λ + δ + δ 2 ρ A s 2 λ + δ ρ ) d V
= δ ( R π ) ( λ + δ ) [ 1 + V 2 + A s 2 ] × [ ( λ + δ ρ ) ( 1 + V 2 ) δ ρ A s 2 ] d V [ ( λ + δ ρ ) ( ( λ + δ ) 2 + δ 2 V 2 ) + ( λ + δ ) δ 2 ρ A s 2 ] .
( λ + 1 ) = δ ( R π ) ( λ + δ ) [ V 2 + R 2 ] × [ ( λ + δ ρ ) ( 1 + V 2 ) δ ρ ( R 2 1 ) ] d V [ ( λ + δ ρ ) ( ( λ + δ ) 2 + δ 2 V 2 ) + ( λ + δ ) δ 2 ρ ( R 2 1 ) ] = 2 δ ( R π ) ( λ + δ ) 1 ( V 2 + R 2 ) × [ ( λ + δ ρ ) ( 1 + V 2 ) δ ρ ( R 2 1 ) ] d V [ ( λ + δ ρ ) ( ( λ + δ ) 2 + δ 2 V 2 ) + ( λ + δ ) δ 2 ρ ( R 2 1 ) ] .
A 2 = ( λ + δ ρ ) ,
B 2 = δ ρ ( R 2 1 ) ,
C 2 = ( λ + δ ) 2 ,
D 2 = δ 2 ,
E 2 = ( λ + δ ) δ 2 ρ ( R 2 1 ) ,
F 2 = A 2 + B 2 A 2 ,
G 2 = A 2 C 2 + E 2 A 2 D 2 .
0 1 ( V 2 + R 2 ) [ A 2 ( 1 + V 2 ) + B 2 ] [ A 2 ( C 2 + D 2 V 2 ) + E 2 ] d V = 0 1 ( V 2 + R 2 ) [ ( A 2 + B 2 ) + A 2 V 2 ] [ ( A 2 C 2 + E 2 ) + A 2 D 2 V 2 ] d V = 1 D 2 0 1 ( V 2 + R 2 ) [ ( A 2 + B 2 A 2 ) + V 2 ] [ ( A 2 C 2 + E 2 A 2 D 2 ) + V 2 ] d V = 1 D 2 0 ( F 2 + V 2 ) ( V 2 + R 2 ) ( G 2 + V 2 ) d V = 1 D 2 ( R 2 G 2 ) [ ( R 2 F 2 ) 0 d V V 2 + R 2 + ( F 2 G 2 ) 0 d V G 2 + V 2 ] = 1 D 2 ( R 2 G 2 ) [ ( R 2 F 2 ) ( π 2 R ) + ( F 2 G 2 ) ( π 2 G 2 ) ] = ( π 2 ) D 2 ( R 2 G 2 ) [ R 2 F 2 R + F 2 G 2 G 2 ] .
( λ + 1 ) 2 δ ( R π ) ( λ + δ ) ( π 2 ) D 2 ( R 2 G 2 ) [ R 2 F 2 R + F 2 G 2 G 2 ] = 0 ,
( λ + 1 ) δ R ( λ + δ ) D 2 ( R 2 G 2 ) [ R 2 F 2 R + F 2 G 2 G 2 ] = 0 .
0 = ( λ + 1 ) δ R ( λ + δ ) δ 2 ( R 2 A 2 C 2 + E 2 A 2 D 2 ) × [ R 2 A 2 + B 2 A 2 R + A 2 + B 2 A 2 A 2 C 2 + E 2 A 2 D 2 A 2 C 2 + E 2 A 2 D 2 ] = ( λ + 1 ) δ R ( λ + δ ) δ 2 [ R 2 ( λ + δ ρ ) ( λ + δ ) 2 + ( λ + δ ) δ 2 ρ ( R 2 1 ) ( λ + δ ρ ) δ 2 ] [ R 2 ( λ + δ ρ ) δ ρ ( R 2 1 ) ( λ + δ ρ ) R + ( λ + δ ρ ) δ ρ ( R 2 1 ) ( λ + δ ρ ) ( λ + δ ρ ) ( λ + δ ) 2 + ( λ + δ ) δ 2 ρ ( R 2 1 ) ( λ + δ ρ ) δ 2 ( λ + δ ρ ) ( λ + δ ) 2 + ( λ + δ ) δ 2 ρ ( R 2 1 ) ( λ + δ ρ ) δ 2 ] .
0 = ( i v + 1 ) δ R ( i v + δ ) δ 2 [ R 2 ( i v + δ ρ ) ( i v + δ ) 2 + ( i v + δ ) δ 2 ρ ( R 2 1 ) ( i v + δ ρ ) δ 2 ] [ R 2 i v + δ ρ δ ρ ( R 2 1 ) ( i v + δ ρ ) R + ( i v + δ ρ ) δ ρ ( R 2 1 ) ( i v + δ ρ ) ( i v + δ ρ ) ( i v + δ ) 2 + ( i v + δ ) δ 2 ρ ( R 2 1 ) ( i v + δ ρ ) δ 2 ( i v + δ ρ ) ( i v + δ ) 2 + ( i v + δ ) δ 2 ρ ( R 2 1 ) ( i v + δ ρ ) δ 2 ] .

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