Abstract

The influence of mode nonorthogonality on the correlation function of the intensity fluctuation and of the amplitude that determines laser linewidth as well as the coherence of light is investigated. The semiclassical approach based on a time-dependent solution of the Fokker–Planck equation is used. Numerical results obtained for a distributed-feedback laser with nonvanishing end reflectivity and a complex coupling coefficient reveal the difference between the standard approach (for orthogonal modes) and the more realistic model (mode nonorthogonality included).

© 1996 Optical Society of America

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  1. K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. 15, 566 (1979).
    [CrossRef]
  2. H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser amplifiers,” IEEE J. Quantum Electron. 21, 63 (1985).
    [CrossRef]
  3. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253 (1989).
    [CrossRef] [PubMed]
  4. A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264 (1989).
    [CrossRef] [PubMed]
  5. W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785 (1989).
    [CrossRef] [PubMed]
  6. S. J. M. Kuppens, M. P. van Exter, M. van Duin, and J. P. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Electron. 31, 1237 (1995).
    [CrossRef]
  7. S. J. M. Kuppens, M. P. van Exter, and J. P. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815 (1994).
    [CrossRef] [PubMed]
  8. P. Szczepański and A. Kujawski, “Non-orthogonality of the longitudinal eigenmodes of a distributed feedback laser,” Opt. Commun. 87, 259 (1992).
    [CrossRef]
  9. A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Influence of end reflectivity on the excess-noise factor in distributed feedback lasers,” IEEE J. Quantum Electron. 29, 2873 (1993).
    [CrossRef]
  10. A. Tyszka-Zawadzka and P. Szczepański, “Excess-noise factor in partly gain coupled DFB lasers,” Opt. Commun. 111, 502 (1994).
    [CrossRef]
  11. Y. Nakano, Y. Deguchi, Y. Luo, and K. Tada, “Reduction of excess noise induced by external reflection in a gain-coupled distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 27, 1732 (1991).
    [CrossRef]
  12. M. M-Tehrani and L. Mandel, “Coherence theory of the ring laser,” Phys. Rev. A 17, 677 (1978).
    [CrossRef]
  13. H. Risken, “Correlation function of the amplitude and of the intensity fluctuation for a laser model near threshold,” Z. Phys. 191, 302 (1966).
    [CrossRef]
  14. H. Risken, “Distribution- and correlation-functions for a laser amplitude,” Z. Phys. 186, 85 (1965).
    [CrossRef]
  15. A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Effect of mode nonorthogonality in distributed feedback lasers,” Opt. Lett. 19, 1222 (1994).
    [CrossRef] [PubMed]
  16. H. Risken, The Fokker-Planck Equation, Methods of Solution and Applications, 2nd ed. (Springer-Verlag, Berlin, 1989).
    [CrossRef]
  17. P. Szczepański, “Semiclassical theory of multimode operation of a distributed feedback laser,” IEEE J. Quantum Electron. 24, 1248 (1988).
    [CrossRef]
  18. S. R. Chinn, “Effect of mirror reflectivity in a distributed-feedback laser,” IEEE J. Quantum Electron. 19, 574 (1973).
    [CrossRef]

1995 (1)

S. J. M. Kuppens, M. P. van Exter, M. van Duin, and J. P. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Electron. 31, 1237 (1995).
[CrossRef]

1994 (3)

S. J. M. Kuppens, M. P. van Exter, and J. P. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815 (1994).
[CrossRef] [PubMed]

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Effect of mode nonorthogonality in distributed feedback lasers,” Opt. Lett. 19, 1222 (1994).
[CrossRef] [PubMed]

A. Tyszka-Zawadzka and P. Szczepański, “Excess-noise factor in partly gain coupled DFB lasers,” Opt. Commun. 111, 502 (1994).
[CrossRef]

1993 (1)

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Influence of end reflectivity on the excess-noise factor in distributed feedback lasers,” IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

1992 (1)

P. Szczepański and A. Kujawski, “Non-orthogonality of the longitudinal eigenmodes of a distributed feedback laser,” Opt. Commun. 87, 259 (1992).
[CrossRef]

1991 (1)

Y. Nakano, Y. Deguchi, Y. Luo, and K. Tada, “Reduction of excess noise induced by external reflection in a gain-coupled distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 27, 1732 (1991).
[CrossRef]

1989 (3)

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253 (1989).
[CrossRef] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785 (1989).
[CrossRef] [PubMed]

1988 (1)

P. Szczepański, “Semiclassical theory of multimode operation of a distributed feedback laser,” IEEE J. Quantum Electron. 24, 1248 (1988).
[CrossRef]

1985 (1)

H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser amplifiers,” IEEE J. Quantum Electron. 21, 63 (1985).
[CrossRef]

1979 (1)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. 15, 566 (1979).
[CrossRef]

1978 (1)

M. M-Tehrani and L. Mandel, “Coherence theory of the ring laser,” Phys. Rev. A 17, 677 (1978).
[CrossRef]

1973 (1)

S. R. Chinn, “Effect of mirror reflectivity in a distributed-feedback laser,” IEEE J. Quantum Electron. 19, 574 (1973).
[CrossRef]

1966 (1)

H. Risken, “Correlation function of the amplitude and of the intensity fluctuation for a laser model near threshold,” Z. Phys. 191, 302 (1966).
[CrossRef]

1965 (1)

H. Risken, “Distribution- and correlation-functions for a laser amplitude,” Z. Phys. 186, 85 (1965).
[CrossRef]

Chinn, S. R.

S. R. Chinn, “Effect of mirror reflectivity in a distributed-feedback laser,” IEEE J. Quantum Electron. 19, 574 (1973).
[CrossRef]

Deguchi, Y.

Y. Nakano, Y. Deguchi, Y. Luo, and K. Tada, “Reduction of excess noise induced by external reflection in a gain-coupled distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 27, 1732 (1991).
[CrossRef]

Hamel, W. A.

W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785 (1989).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser amplifiers,” IEEE J. Quantum Electron. 21, 63 (1985).
[CrossRef]

Kawakami, S.

H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser amplifiers,” IEEE J. Quantum Electron. 21, 63 (1985).
[CrossRef]

Kujawski, A.

P. Szczepański and A. Kujawski, “Non-orthogonality of the longitudinal eigenmodes of a distributed feedback laser,” Opt. Commun. 87, 259 (1992).
[CrossRef]

Kuppens, S. J. M.

S. J. M. Kuppens, M. P. van Exter, M. van Duin, and J. P. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Electron. 31, 1237 (1995).
[CrossRef]

S. J. M. Kuppens, M. P. van Exter, and J. P. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815 (1994).
[CrossRef] [PubMed]

Luo, Y.

Y. Nakano, Y. Deguchi, Y. Luo, and K. Tada, “Reduction of excess noise induced by external reflection in a gain-coupled distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 27, 1732 (1991).
[CrossRef]

Mandel, L.

M. M-Tehrani and L. Mandel, “Coherence theory of the ring laser,” Phys. Rev. A 17, 677 (1978).
[CrossRef]

M-Tehrani, M.

M. M-Tehrani and L. Mandel, “Coherence theory of the ring laser,” Phys. Rev. A 17, 677 (1978).
[CrossRef]

Nakano, Y.

Y. Nakano, Y. Deguchi, Y. Luo, and K. Tada, “Reduction of excess noise induced by external reflection in a gain-coupled distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 27, 1732 (1991).
[CrossRef]

Petermann, K.

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. 15, 566 (1979).
[CrossRef]

Risken, H.

H. Risken, “Correlation function of the amplitude and of the intensity fluctuation for a laser model near threshold,” Z. Phys. 191, 302 (1966).
[CrossRef]

H. Risken, “Distribution- and correlation-functions for a laser amplitude,” Z. Phys. 186, 85 (1965).
[CrossRef]

H. Risken, The Fokker-Planck Equation, Methods of Solution and Applications, 2nd ed. (Springer-Verlag, Berlin, 1989).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253 (1989).
[CrossRef] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

Szczepanski, P.

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Effect of mode nonorthogonality in distributed feedback lasers,” Opt. Lett. 19, 1222 (1994).
[CrossRef] [PubMed]

A. Tyszka-Zawadzka and P. Szczepański, “Excess-noise factor in partly gain coupled DFB lasers,” Opt. Commun. 111, 502 (1994).
[CrossRef]

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Influence of end reflectivity on the excess-noise factor in distributed feedback lasers,” IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

P. Szczepański and A. Kujawski, “Non-orthogonality of the longitudinal eigenmodes of a distributed feedback laser,” Opt. Commun. 87, 259 (1992).
[CrossRef]

P. Szczepański, “Semiclassical theory of multimode operation of a distributed feedback laser,” IEEE J. Quantum Electron. 24, 1248 (1988).
[CrossRef]

Tada, K.

Y. Nakano, Y. Deguchi, Y. Luo, and K. Tada, “Reduction of excess noise induced by external reflection in a gain-coupled distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 27, 1732 (1991).
[CrossRef]

Tyszka-Zawadzka, A.

A. Tyszka-Zawadzka and P. Szczepański, “Excess-noise factor in partly gain coupled DFB lasers,” Opt. Commun. 111, 502 (1994).
[CrossRef]

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Effect of mode nonorthogonality in distributed feedback lasers,” Opt. Lett. 19, 1222 (1994).
[CrossRef] [PubMed]

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Influence of end reflectivity on the excess-noise factor in distributed feedback lasers,” IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

van Duin, M.

S. J. M. Kuppens, M. P. van Exter, M. van Duin, and J. P. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Electron. 31, 1237 (1995).
[CrossRef]

van Exter, M. P.

S. J. M. Kuppens, M. P. van Exter, M. van Duin, and J. P. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Electron. 31, 1237 (1995).
[CrossRef]

S. J. M. Kuppens, M. P. van Exter, and J. P. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815 (1994).
[CrossRef] [PubMed]

Woerdman, J. P.

S. J. M. Kuppens, M. P. van Exter, M. van Duin, and J. P. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Electron. 31, 1237 (1995).
[CrossRef]

S. J. M. Kuppens, M. P. van Exter, and J. P. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815 (1994).
[CrossRef] [PubMed]

W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785 (1989).
[CrossRef] [PubMed]

Wolinski, W.

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Effect of mode nonorthogonality in distributed feedback lasers,” Opt. Lett. 19, 1222 (1994).
[CrossRef] [PubMed]

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Influence of end reflectivity on the excess-noise factor in distributed feedback lasers,” IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

IEEE J. Quantum Electron. (7)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. 15, 566 (1979).
[CrossRef]

H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain-guided laser amplifiers,” IEEE J. Quantum Electron. 21, 63 (1985).
[CrossRef]

S. J. M. Kuppens, M. P. van Exter, M. van Duin, and J. P. Woerdman, “Evidence of nonuniform phase-diffusion in a bad-cavity laser,” IEEE J. Quantum Electron. 31, 1237 (1995).
[CrossRef]

A. Tyszka-Zawadzka, P. Szczepański, and W. Woliński, “Influence of end reflectivity on the excess-noise factor in distributed feedback lasers,” IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

Y. Nakano, Y. Deguchi, Y. Luo, and K. Tada, “Reduction of excess noise induced by external reflection in a gain-coupled distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 27, 1732 (1991).
[CrossRef]

P. Szczepański, “Semiclassical theory of multimode operation of a distributed feedback laser,” IEEE J. Quantum Electron. 24, 1248 (1988).
[CrossRef]

S. R. Chinn, “Effect of mirror reflectivity in a distributed-feedback laser,” IEEE J. Quantum Electron. 19, 574 (1973).
[CrossRef]

Opt. Commun. (2)

A. Tyszka-Zawadzka and P. Szczepański, “Excess-noise factor in partly gain coupled DFB lasers,” Opt. Commun. 111, 502 (1994).
[CrossRef]

P. Szczepański and A. Kujawski, “Non-orthogonality of the longitudinal eigenmodes of a distributed feedback laser,” Opt. Commun. 87, 259 (1992).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (4)

M. M-Tehrani and L. Mandel, “Coherence theory of the ring laser,” Phys. Rev. A 17, 677 (1978).
[CrossRef]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Phys. Rev. A 39, 1253 (1989).
[CrossRef] [PubMed]

A. E. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators,” Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

W. A. Hamel and J. P. Woerdman, “Nonorthogonality of the longitudinal eigenmodes of a laser,” Phys. Rev. A 40, 2785 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

S. J. M. Kuppens, M. P. van Exter, and J. P. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72, 3815 (1994).
[CrossRef] [PubMed]

Z. Phys. (2)

H. Risken, “Correlation function of the amplitude and of the intensity fluctuation for a laser model near threshold,” Z. Phys. 191, 302 (1966).
[CrossRef]

H. Risken, “Distribution- and correlation-functions for a laser amplitude,” Z. Phys. 186, 85 (1965).
[CrossRef]

Other (1)

H. Risken, The Fokker-Planck Equation, Methods of Solution and Applications, 2nd ed. (Springer-Verlag, Berlin, 1989).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Eigenvalue λ10 versus the normalized pump parameter a ¯ for the amplitudes of the end reflectivity r = 0.1 and r = 0.5, with the phase of the end reflectivity φr as a parameter. The coupling coefficient is κL = 0.1. The dashed curve is obtained for orthogonal laser modes.

Fig. 2
Fig. 2

Eigenvalue λ10 as a function of the normalized pump parameter a ¯ for the coupling coefficients κL = 0.1 and κL = 5, with the phase of the end reflectivity φr as a parameter. The amplitude of the end reflectivity r = 0.1. The dashed curve is obtained for orthogonal modes.

Fig. 3
Fig. 3

Variation of the eigenvalue λ01 with the normalized pump parameter a ¯ for the amplitudes of the end reflectivity r = 0.1 and r = 0.5, with the phase of the end reflectivity φe as a parameter. The coupling coefficient is κL = 0.1 The dashed curve is obtained for orthogonal modes.

Fig. 4
Fig. 4

Eigenvalue λ01 versus the normalized pump parameter a ¯ for the coupling coefficients κL = 0.1 and κL = 5, with the phase of the end reflectivity φr as a parameter. The amplitude of the end reflectivity is r = 0.1. The dashed curve is obtained for orthogonal laser modes.

Fig. 5
Fig. 5

Eigenvalue λ10 as a function of the normalized pump parameter a ¯ for the coupling strengths |κL| = 1 and |κL| = 5, with the phase of the complex coupling coefficient φ as a parameter. The dashed curve is obtained for orthogonal modes.

Fig. 6
Fig. 6

Eigenvalue λ01 versus the normalized pump parameter a ¯ for the coupling strengths |κL| = 1 and |κL| = 5, with the phase of the complex coupling coefficient φ as a parameter. The dashed curve is obtained for orthogonal laser modes.

Equations (31)

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d E d T = ( a β | E | 2 ) E + Γ ( T ) , d E * d T = ( a β | E | 2 ) E * + Γ * ( T ) ,
K = 0 L d z Φ N ( z ) Φ N * ( z ) ,
U N ( z ) = C [ R N ( z ) exp ( ik N z ) S N ( z ) exp ( ik N z ) Z ]
Φ N ( z ) = C + [ S N ( z ) exp ( ik N z ) R N ( z ) exp ( ik N z ) ] ,
0 L d z U N ( z ) Φ M ( z ) = δ NM ,
R N = sinh γ N ( z + L / 2 ) r exp ( i ϕ r ) sinh γ N ( z L / 2 ) , S N = sinh γ N ( z L / 2 ) + r exp ( i ϕ r ) sinh γ N ( z + L / 2 ) ,
γ N = i κ sinh ( γ N L ) × [ 1 r 2 exp ( 2 i ϕ r ] [ 1 r exp ( i φ r ) exp ( γ N L ) ] [ 1 r exp ( i φ r ) exp ( γ N L ) ] ,
K = | γ N [ 1 + r 2 exp ( 2 i φ r ) ] A + r exp ( i φ r ) B | 2 × | { ( 1 + r 2 ) [ γ b sinh ( γ a L ) γ a sinh ( γ b L ) ] + 4 r cos ( φ r ) [ γ a cosh ( γ a L / 2 ) sinh ( γ b L / 2 ) γ b cosh ( γ b L / 2 ) sinh ( γ a L / 2 ) ] } ( γ a γ b ) 1 | 2
K = | γ N γ N L cosh ( γ N L ) sinh ( γ N L ) × γ b sinh ( γ a L ) γ a sinh ( γ b L ) γ a γ b | 2 ,
γ N = i κ sinh ( γ N L ) .
W ( ρ , ϕ , T ) T = 1 ρ ρ [ ( a β ρ 2 ) ρ 2 W ] + KP [ 1 ρ ρ ( ρ W ρ ) ] + 1 ρ 2 2 W ϕ 2 .
ρ ¯ = β P 4 ρ , t = β P T , a ¯ = a β P .
W ( ρ ¯ , ϕ , t ) t = 1 ρ ¯ ρ ¯ [ ( a ¯ ρ ¯ 2 ) ρ ¯ 2 W ] + K [ 1 ρ ¯ ρ ¯ ( ρ ¯ W ρ ¯ ) ] + 1 ρ ¯ 2 2 W ϕ 2 .
W ( ρ ¯ , ϕ , t ) = m = 0 n = A nm f nm ( ρ ¯ ) exp ( in ϕ ) exp ( λ nm t ) ,
ρ ¯ ( p f nm ρ ¯ ) q n f nm + λ ¯ nm p f nm = 0 ,
p ( ρ ¯ ) = ρ ¯ exp [ 1 K ( ρ ¯ 4 4 a ¯ ρ ¯ 2 2 ) ] , q n ( ρ ¯ ) = [ 1 K ( 2 a ¯ 4 ρ ¯ 2 ) + n 2 ρ ¯ 2 ] p ( ρ ¯ ) .
g ( τ ) = ρ ¯ ( t + τ ) exp [ i ϕ ( t + τ ) ] ρ ¯ ( t ) exp [ i ϕ ( t ) ]
( τ ) = ( ρ ¯ 2 ( t + τ ) ρ ¯ 2 ) ( ρ ¯ 2 ( t ) ρ ¯ 2 )
g ( τ ) = N m = 0 [ 0 ρ ¯ 2 f 1 m ( ρ ¯ ) d ρ ¯ ] 2 exp ( λ 1 m τ ) ,
( τ ) = N m = 1 [ 0 ρ ¯ 3 f 0 m ( ρ ¯ ) d ρ ¯ ] 2 exp ( λ 0 m τ ) ,
λ ¯ nm = 0 ( pf nm 2 + q n f nm 2 ) d ρ ¯ 0 pf nm 2 d ρ ¯ ,
f 10 = ρ ¯ exp [ ρ ¯ 4 4 K + ( a ¯ K + ) ρ ¯ 2 2 ] ,
f 01 = ( 1 A ρ ¯ 2 ) exp ( ρ ¯ 4 4 K + a ¯ ρ ¯ 2 2 K ) ,
I n ( y ) = 0 v n exp ( v 2 4 K + y v 2 K ) d v
I n ( y ) = 2 K ( n 1 ) I n 2 ( y ) + yI n 1 ( y ) , I 1 ( y ) = 2 K + yI 0 ( y ) , I 0 ( y ) = π K exp ( y 2 / 4 K ) [ 1 + Φ ( y / 2 K ) ] ,
λ ¯ 10 Min [ h ( ) ] = Min [ 2 K + a ¯ 2 K 2 + 2 3 K 2 + 2 ( 1 + 2 K ) I 0 ( a ¯ + 2 ) I 1 ( a ¯ + 2 ) ] ,
λ ¯ 01 ( 4 K 2 8 K + 8 ) I 0 ( a ¯ ) I 1 ( a ¯ ) [ I 2 ( a ¯ ) I 0 ( a ¯ ) I 1 2 ( a ¯ ) ] .
g ( τ ) g ( 0 ) exp ( λ 10 τ ) ,
( τ ) ( 0 ) exp ( λ 01 τ ) ,
g ( T ) = g ( 0 ) exp ( Δ ν T ) ,
Δ ν = λ 10 β P .

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