Abstract

Semiclassical models for the dynamical behavior of single-longitudinal-mode, homogeneously broadened, unidirectional ring-laser oscillators have long been available. However, in most of these models the beam’s cross section is assumed to be independent of longitudinal position in the laser, while in practical laser systems it is common for the diameter to vary with position. A model for lasers with focusing-beam field variations is developed here. Linear stability analysis shows that higher values of excitation are required to reach the type 1 instability threshold for lasers with longitudinal variations of the beam diameter than for the uniform plane-wave laser model.

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  1. M. E. Globus, Yu. V. Naboikin, A. M. Ratner, I. A. Rom-Krichevskaya, and Yu. A. Tiunov, "Stationary generation in an optical resonator with lenses," Sov. Phys. JETP 25, 562-567 (1967).
  2. R. Polloni and O. Svelto, "Static and dynamic behavior of a single-mode Nd-YAG laser," IEEE J. Quantum Electron. QE-4, 481-485 (1968).
    [CrossRef]
  3. L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, "Instabilities in passive and active optical systems with a Gaussian transverse intensity profile," Phys. Rev. A 30, 1366-1376 (1984).
    [CrossRef]
  4. L. A. Lugiato and M. Milani, "Disappearance of laser instabilities in a Gaussian cavity mode," Opt. Commun. 46, 57-60 (1983).
    [CrossRef]
  5. L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
    [CrossRef] [PubMed]
  6. R. J. Horowicz, L. A. Lugiato, and G. Strini, "Steady-state and stability analysis for a laser with a saturable absorber with a Gaussian transverse profile," Z. Phys. B: Condens. Matter 58, 71-78 (1984).
    [CrossRef]
  7. L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
    [CrossRef]
  8. C. P. Smith and R. Dykstra, "Lorenz like chaos in a Gaussian mode laser with a radially dependent gain," Opt. Commun. 117, 107-110 (1995).
    [CrossRef]
  9. P. Chenkosol and L. W. Casperson, "Spontaneous coherent pulsations in standing-wave laser oscillators: stability criteria for homogeneous broadening," J. Opt. Soc. Am. B 10, 817-826 (1993).
    [CrossRef]
  10. L. W. Casperson, "Spontaneous coherent pulsations in ring-laser oscillators: stability criteria," J. Opt. Soc. Am. B 2, 993-997 (1985).
    [CrossRef]
  11. See, for example, C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, "Instabilities and chaos of a single-mode NH3 ring laser," Opt. Commun. 52, 405-408 (1985).
    [CrossRef]
  12. L. W. Casperson, "Spontaneous coherent pulsations in ring-laser oscillators," J. Opt. Soc. Am. B 2, 62-72 (1985).
    [CrossRef]
  13. L. W. Casperson, "Spontaneous coherent pulsations in ring-laser oscillators: simplified models," J. Opt. Soc. Am. B 2, 73-80 (1985).
    [CrossRef]

1995

C. P. Smith and R. Dykstra, "Lorenz like chaos in a Gaussian mode laser with a radially dependent gain," Opt. Commun. 117, 107-110 (1995).
[CrossRef]

1993

1988

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
[CrossRef] [PubMed]

1987

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
[CrossRef]

1985

1984

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, "Instabilities in passive and active optical systems with a Gaussian transverse intensity profile," Phys. Rev. A 30, 1366-1376 (1984).
[CrossRef]

R. J. Horowicz, L. A. Lugiato, and G. Strini, "Steady-state and stability analysis for a laser with a saturable absorber with a Gaussian transverse profile," Z. Phys. B: Condens. Matter 58, 71-78 (1984).
[CrossRef]

1983

L. A. Lugiato and M. Milani, "Disappearance of laser instabilities in a Gaussian cavity mode," Opt. Commun. 46, 57-60 (1983).
[CrossRef]

1968

R. Polloni and O. Svelto, "Static and dynamic behavior of a single-mode Nd-YAG laser," IEEE J. Quantum Electron. QE-4, 481-485 (1968).
[CrossRef]

1967

M. E. Globus, Yu. V. Naboikin, A. M. Ratner, I. A. Rom-Krichevskaya, and Yu. A. Tiunov, "Stationary generation in an optical resonator with lenses," Sov. Phys. JETP 25, 562-567 (1967).

Bandy, D. K.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
[CrossRef]

Casperson, L. W.

Chenkosol, P.

Cooper, M.

See, for example, C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, "Instabilities and chaos of a single-mode NH3 ring laser," Opt. Commun. 52, 405-408 (1985).
[CrossRef]

Dykstra, R.

C. P. Smith and R. Dykstra, "Lorenz like chaos in a Gaussian mode laser with a radially dependent gain," Opt. Commun. 117, 107-110 (1995).
[CrossRef]

Ering, P. S.

See, for example, C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, "Instabilities and chaos of a single-mode NH3 ring laser," Opt. Commun. 52, 405-408 (1985).
[CrossRef]

Globus, M. E.

M. E. Globus, Yu. V. Naboikin, A. M. Ratner, I. A. Rom-Krichevskaya, and Yu. A. Tiunov, "Stationary generation in an optical resonator with lenses," Sov. Phys. JETP 25, 562-567 (1967).

Horowicz, R. J.

R. J. Horowicz, L. A. Lugiato, and G. Strini, "Steady-state and stability analysis for a laser with a saturable absorber with a Gaussian transverse profile," Z. Phys. B: Condens. Matter 58, 71-78 (1984).
[CrossRef]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, "Instabilities in passive and active optical systems with a Gaussian transverse intensity profile," Phys. Rev. A 30, 1366-1376 (1984).
[CrossRef]

Klische, W.

See, for example, C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, "Instabilities and chaos of a single-mode NH3 ring laser," Opt. Commun. 52, 405-408 (1985).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
[CrossRef]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, "Instabilities in passive and active optical systems with a Gaussian transverse intensity profile," Phys. Rev. A 30, 1366-1376 (1984).
[CrossRef]

R. J. Horowicz, L. A. Lugiato, and G. Strini, "Steady-state and stability analysis for a laser with a saturable absorber with a Gaussian transverse profile," Z. Phys. B: Condens. Matter 58, 71-78 (1984).
[CrossRef]

L. A. Lugiato and M. Milani, "Disappearance of laser instabilities in a Gaussian cavity mode," Opt. Commun. 46, 57-60 (1983).
[CrossRef]

Milani, M.

L. A. Lugiato and M. Milani, "Disappearance of laser instabilities in a Gaussian cavity mode," Opt. Commun. 46, 57-60 (1983).
[CrossRef]

Naboikin, Yu. V.

M. E. Globus, Yu. V. Naboikin, A. M. Ratner, I. A. Rom-Krichevskaya, and Yu. A. Tiunov, "Stationary generation in an optical resonator with lenses," Sov. Phys. JETP 25, 562-567 (1967).

Narducci, L. M.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
[CrossRef]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, "Instabilities in passive and active optical systems with a Gaussian transverse intensity profile," Phys. Rev. A 30, 1366-1376 (1984).
[CrossRef]

Polloni, R.

R. Polloni and O. Svelto, "Static and dynamic behavior of a single-mode Nd-YAG laser," IEEE J. Quantum Electron. QE-4, 481-485 (1968).
[CrossRef]

Prati, F.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
[CrossRef]

Ratner, A. M.

M. E. Globus, Yu. V. Naboikin, A. M. Ratner, I. A. Rom-Krichevskaya, and Yu. A. Tiunov, "Stationary generation in an optical resonator with lenses," Sov. Phys. JETP 25, 562-567 (1967).

Rom-Krichevskaya, I. A.

M. E. Globus, Yu. V. Naboikin, A. M. Ratner, I. A. Rom-Krichevskaya, and Yu. A. Tiunov, "Stationary generation in an optical resonator with lenses," Sov. Phys. JETP 25, 562-567 (1967).

Ru, P.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
[CrossRef]

Smith, C. P.

C. P. Smith and R. Dykstra, "Lorenz like chaos in a Gaussian mode laser with a radially dependent gain," Opt. Commun. 117, 107-110 (1995).
[CrossRef]

Strini, G.

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, "Instabilities in passive and active optical systems with a Gaussian transverse intensity profile," Phys. Rev. A 30, 1366-1376 (1984).
[CrossRef]

R. J. Horowicz, L. A. Lugiato, and G. Strini, "Steady-state and stability analysis for a laser with a saturable absorber with a Gaussian transverse profile," Z. Phys. B: Condens. Matter 58, 71-78 (1984).
[CrossRef]

Svelto, O.

R. Polloni and O. Svelto, "Static and dynamic behavior of a single-mode Nd-YAG laser," IEEE J. Quantum Electron. QE-4, 481-485 (1968).
[CrossRef]

Tiunov, Yu. A.

M. E. Globus, Yu. V. Naboikin, A. M. Ratner, I. A. Rom-Krichevskaya, and Yu. A. Tiunov, "Stationary generation in an optical resonator with lenses," Sov. Phys. JETP 25, 562-567 (1967).

Tredicce, J. R.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
[CrossRef]

Weiss, C. O.

See, for example, C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, "Instabilities and chaos of a single-mode NH3 ring laser," Opt. Commun. 52, 405-408 (1985).
[CrossRef]

IEEE J. Quantum Electron.

R. Polloni and O. Svelto, "Static and dynamic behavior of a single-mode Nd-YAG laser," IEEE J. Quantum Electron. QE-4, 481-485 (1968).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

L. A. Lugiato and M. Milani, "Disappearance of laser instabilities in a Gaussian cavity mode," Opt. Commun. 46, 57-60 (1983).
[CrossRef]

L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci, P. Ru, and J. R. Tredicce, "Low threshold instabilities in unidirectional ring lasers," Opt. Commun. 64, 167-171 (1987).
[CrossRef]

C. P. Smith and R. Dykstra, "Lorenz like chaos in a Gaussian mode laser with a radially dependent gain," Opt. Commun. 117, 107-110 (1995).
[CrossRef]

See, for example, C. O. Weiss, W. Klische, P. S. Ering, and M. Cooper, "Instabilities and chaos of a single-mode NH3 ring laser," Opt. Commun. 52, 405-408 (1985).
[CrossRef]

Phys. Rev. A

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, "Role of transverse effects in laser instabilities," Phys. Rev. A 37, 3847-3866 (1988).
[CrossRef] [PubMed]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, "Instabilities in passive and active optical systems with a Gaussian transverse intensity profile," Phys. Rev. A 30, 1366-1376 (1984).
[CrossRef]

Sov. Phys. JETP

M. E. Globus, Yu. V. Naboikin, A. M. Ratner, I. A. Rom-Krichevskaya, and Yu. A. Tiunov, "Stationary generation in an optical resonator with lenses," Sov. Phys. JETP 25, 562-567 (1967).

Z. Phys. B: Condens. Matter

R. J. Horowicz, L. A. Lugiato, and G. Strini, "Steady-state and stability analysis for a laser with a saturable absorber with a Gaussian transverse profile," Z. Phys. B: Condens. Matter 58, 71-78 (1984).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Unidirectional ring cavity. The lengths of the laser medium and the cavity are l and L, respectively. Mirrors A, B, and C have 100% reflectivity, whereas the output mirror O has reflectivity R.

Fig. 2
Fig. 2

Focused-beam function A ( z ) and several notation definitions employed in the analysis.

Fig. 3
Fig. 3

Plots of the real part of the complex rate constant for lasers with focusing beam or uniform plane-wave fields as functions of the threshold parameter R and decay ratio δ. These are obtained by numerically solving Eq. (78), where the minimum spotsize of the field in the uniform plane-wave case is set to a very large number. The values of the other laser parameters are w 0 = 0.004 m , l = 1.5 m , ρ = 0.01 , and λ = 81.5 μ m . The lower curve in each pair is for the focusing-beam field.

Fig. 4
Fig. 4

Plots of the imaginary part of the complex rate constant for lasers with focusing-beam or uniform plane-wave fields as functions of the threshold parameter R and decay ratio δ. These are obtained by numerically solving Eq. (78), where the minimum spotsize of the field in the uniform plane-wave case is set to a very large number. The values of the other laser parameters are w 0 = 0.004 m , l = 1.5 m , ρ = 0.01 , and λ = 81.5 μ m .

Fig. 5
Fig. 5

Plots of the type 1 stability boundaries for lasers with focusing-beam and uniform plane-wave fields with l = 1.5 m , ρ = 0.01 , and λ = 81.5 μ m , and values of the minimum spotsize w 0 as labeled. The lowest curve is the result from the uniform plane-wave model, i.e., w 0 is infinitely large.[9]

Fig. 6
Fig. 6

Plots of the type 1 stability boundaries for lasers with focusing beams as functions of normalized length of the laser gain medium and the decay ratio δ. The length of the gain medium is normalized to the Rayleigh length z 0 . The values of the other laser parameters are w 0 = 0.004 m , l = 1.5 m , ρ = 0.01 , and λ = 81.5 μ m .

Equations (80)

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ρ a b ( r , z , t ) t = ( i ω 0 + γ ) ρ a b ( r , z , t ) i μ E ( r , z , t ) [ ρ a a ( r , z , t ) ρ b b ( r , z , t ) ] ,
ρ a a ( r , z , t ) t = λ a ( r , z , t ) λ a ρ a a ( r , z , t ) + [ i μ E ( r , z , t ) ρ b a ( r , z , t ) + c.c. ] ,
ρ b b ( r , z , t ) t = λ b ( r , z , t ) γ b ρ b b ( r , z , t ) + γ a b ρ a a ( r , z , t ) [ i μ E ( r , z , t ) ρ b a ( r , z , t ) + c.c. ] ,
ρ b a ( r , z , t ) = ρ a b * ( r , z , t ) ,
t 2 E ( r , z , t ) + 2 E ( r , z , t ) z 2 μ 1 ϵ 1 2 E ( r , z , t ) t 2 μ 1 σ E ( r , z , t ) t = μ 1 2 P ( r , z , t ) t 2 ,
t 2 2 r 2 + 1 r r
P ( r , z , t ) = 1 2 P ( r , z , t ) exp [ i ( k z ω t ) ] + c.c. = μ ρ a b ( r , z , t ) + c.c. ,
ρ a b ( r , z , t ) = P ( r , z , t ) 2 μ exp [ i ( k z ω t ) ] .
E ( r , z , t ) = 1 2 E 0 A ( r , z ) exp [ i ( k 0 z ω t ) ] + c.c. ,
t 2 A ( r , z ) + i ( 2 k 0 ) A ( r , z ) z = 0 .
E ( r , z , t ) = 1 2 E ( r , z , t ) exp [ i ( k z ω t ) ] + c.c. = 1 2 E ( z , t ) A ( r , z ) exp [ i ( k z ω t ) ] + c.c.
exp [ i ( k z ω t ) ] { E ( z , t ) [ t 2 A ( r , z ) + i ( 2 k ) A ( r , z ) z ] + E ( z , t ) A ( r , z ) ( μ 1 ϵ 1 ω 2 k 2 + i μ 1 σ ω ) + i 2 A ( r , z ) [ k E ( z , t ) z + μ 1 ϵ 1 ω E ( z , t ) t ] } + c.c. = exp [ i ( k z ω t ) ] [ μ 1 ω 2 P ( r , z , t ) ] + c.c.
exp [ i ( k z ω t ) ] { E ( z , t ) A ( r , z ) ( μ 1 ϵ 1 ω 2 k 2 + i μ 1 σ ω ) + i 2 A ( r , z ) [ k E ( z , t ) z + μ 1 ϵ 1 ω E ( z , t ) t ] } + c.c. = exp [ i ( k z ω t ) ] [ μ 1 ω 2 P ( r , z , t ) ] + c.c.
L 2 L 2 { E ( z , t ) A ( r , z ) ( μ 1 ϵ 1 ω 2 k 2 + i μ 1 σ ω ) + i 2 A ( r , z ) [ k E ( z , t ) z + μ 1 ϵ 1 ω E ( z , t ) t ] } d z = L 2 L 2 [ μ 1 ω 2 P ( r , z , t ) ] d z ,
E ( t ) ( μ 1 ϵ 1 ω 2 k 2 + i μ 1 σ ω ) L 2 L 2 A ( z ) d z + i 2 μ 1 ϵ 1 ω d E ( t ) d t L 2 L 2 A ( z ) d z = μ 1 ω 2 L 2 L 2 P ( z , t ) d z ,
E ( z , t ) = 1 2 E ( t ) A ( z ) exp [ i ( k z ω t ) ] + c.c. ,
P ( z , t ) = 1 2 P ( z , t ) exp [ i ( k z ω t ) ] + c.c.
w ( z ) = w 0 [ 1 + ( z z 0 ) 2 ] 1 2 ,
d E ( t ) d t + ( μ 1 ϵ 1 ω 2 k 2 + i μ 1 σ ω ) i 2 μ 1 ϵ 1 ω E ( t ) = i ω 2 ϵ 1 l 2 l 2 P ( z , t ) d z L 2 L 2 A ( z ) d z ,
d E ( t ) d t + ( σ 2 ϵ 1 i ω 2 Ω 2 2 ω ) E ( t ) = i ω 2 ϵ 1 l 2 l 2 P ( z , t ) d z L 2 L 2 A ( z ) d z ,
d E ( t ) d t + [ σ 2 ϵ 1 i γ ( y y 0 ) ] E ( t ) = i ω 0 2 ϵ 1 l 2 l 2 P ( z , t ) d z L 2 L 2 A ( z ) d z ,
y = ( ω ω 0 ) γ ,
y 0 = ( Ω ω 0 ) γ .
P ( z , t ) = 1 2 P ( z , t ) exp [ i ( k z ω t ) ] + c.c. = μ ρ a b ( z , t ) + c.c. ,
ρ a b ( z , t ) = P ( z , t ) 2 μ exp [ i ( k z ω t ) ] .
P ( z , t ) t = i ( ω ω 0 ) P ( z , t ) γ P ( z , t ) i μ 2 E ( t ) A ( z ) D ( z , t ) ,
D ( z , t ) t = λ d ( z , t ) ( γ a + γ a b + γ b 2 ) D ( z , t ) ( γ a + γ a b γ b 2 ) M ( z , t ) + i 2 [ E ( t ) A ( z ) P * ( z , t ) E * ( t ) A ( z ) P ( z , t ) ] ,
M ( z , t ) t = λ m ( z , t ) ( γ a γ a b γ b 2 ) D ( z , t ) ( γ a γ a b + γ b 2 ) M ( z , t ) ,
P n ( z , t ) t = γ [ ( 1 i y ) P n ( z , t ) + i E n ( t ) A ( z ) D n ( z , t ) ] ,
D n ( z , t ) t = λ d n ( z , t ) ( γ a + γ a b + γ b 2 ) D n ( z , t ) ( γ a + γ a b γ b 2 ) M n ( z , t ) + i γ n A ( z ) [ E n ( t ) P n * ( z , t ) E n * ( t ) P n ( z , t ) ] ,
M n ( z , t ) t = λ m n ( z , t ) ( γ a γ a b γ b 2 ) D n ( z , t ) ( γ a γ a b + γ b 2 ) M n ( z , t ) ,
d E n ( t ) d t = γ c [ 1 + i δ ( y y 0 ) ] E n + i γ c l 2 l 2 P n ( z , t ) d z L 2 L 2 A ( z ) d z ,
E n ( t ) = ( γ a γ a b + γ b 2 γ γ a γ b ) 1 2 ( μ ) E ( t ) ,
P n ( z , t ) = ( ω 0 σ ) ( γ a γ a b + γ b 2 γ γ a γ b ) 1 2 ( μ ) P ( z , t ) ,
D n ( z , t ) = ( ω 0 σ ) ( μ 2 γ ) D ( z , t ) ,
M n ( z , t ) = ( ω 0 σ ) ( μ 2 γ ) M ( z , t ) ,
λ d n ( z , t ) = ( ω 0 σ ) ( μ 2 γ ) λ d ( z , t ) ,
λ m n ( z , t ) = ( ω 0 σ ) ( μ 2 γ ) λ m ( z , t ) ,
γ n = γ a γ b γ a γ a b + γ b ,
γ c = 1 2 t c = σ 2 ϵ 1 ,
δ = γ γ c .
y = y 0 = 0 .
γ b = γ a + γ a b = γ d .
P n ( z , t ) t = γ [ P n ( z , t ) + i E n ( t ) A ( z ) D n ( z , t ) ] ,
D n ( z , t ) t = γ d { D n ( z , t ) λ d n γ d i 2 A ( z ) [ E n ( t ) P n * ( z , t ) E n * ( t ) P n ( z , t ) ] } ,
d E n ( t ) d t = γ c [ E n ( t ) i l 2 l 2 P n ( z , t ) d z L 2 L 2 A ( z ) d z ] ;
P ̃ ( z , t ) t = γ [ P ̃ ( z , t ) + i E ̃ ( t ) A ( z ) D ̃ ( z , t ) ] ,
D ̃ ( z , t ) t = γ d { D ̃ ( z , t ) R i 2 A ( z ) [ E ̃ ( t ) P ̃ * ( z , t ) E ̃ * ( t ) P ̃ ( z , t ) ] } ,
d E ̃ ( t ) d t = γ c [ E ̃ ( t ) i l 2 l 2 P ̃ ( z , t ) d z l 2 l 2 A ( z ) d z ] ,
P ̃ ( z , t ) = l 2 l 2 A ( z ) d z L 2 L 2 A ( z ) d z P n ( z , t ) ,
D ̃ ( z , t ) = l 2 l 2 A ( z ) d z L 2 L 2 A ( z ) d z D n ( z , t ) ,
E ̃ ( t ) = E n ( t ) ,
λ d n = R λ th = R [ L 2 L 2 A ( z ) d z l 2 l 2 A ( z ) d z ] γ d .
P ̃ i ( z , t ) t = γ [ P ̃ i ( z , t ) + E ̃ r ( t ) A ( z ) D ̃ ( z , t ) ] ,
D ̃ ( z , t ) d t = γ d [ D ̃ ( z , t ) R E ̃ r ( t ) A ( z ) P ̃ i ( z , t ) ] ,
d E ̃ r ( t ) d t = γ c [ E ̃ r ( t ) + l 2 l 2 P ̃ i ( z , t ) d z l 2 l 2 A ( z ) d z ] .
P ̃ i s ( z ) = E ̃ r s A ( z ) D ̃ s ( z ) ,
D ̃ s ( z ) = R + E ̃ r s A ( z ) P ̃ i s ( z ) ,
E ̃ r s = l 2 l 2 P ̃ i s ( z ) d z l 2 l 2 A ( z ) d z γ d ,
D ̃ s ( z ) = [ R 1 + E ̃ r s 2 A 2 ( z ) ] ,
P ̃ i s ( z ) = [ R E ̃ r s A ( z ) 1 + E ̃ r s 2 A 2 ( z ) ] .
E ̃ r s = l 2 l 2 [ R E ̃ r s A ( z ) 1 + E ̃ r s 2 A 2 ( z ) ] d z l 2 l 2 A ( z ) d z ,
1 = R l 2 l 2 [ A ( z ) 1 + E ̃ r s 2 A 2 ( z ) ] d z l 2 l 2 A ( z ) d z ,
1 = R 0 l 2 { [ 1 + ( z z 0 ) 2 ] 1 2 E ̃ r s 2 + 1 + ( z z 0 ) 2 } d z 0 l 2 [ 1 + ( z z 0 ) 2 ] 1 2 d z ,
P ̃ i ( z , t ) t = δ [ P ̃ i ( z , t ) + E ̃ r ( t ) A ( z ) D ̃ ( z , t ) ] ,
D ̃ ( z , t ) t = δ ρ [ D ̃ ( z , t ) R E ̃ r ( t ) A ( z ) P ̃ i ( z , t ) ] ,
d E ̃ r ( t ) d t = [ E ̃ r ( t ) + l 2 l 2 P ̃ i ( z , t ) d z l 2 l 2 A ( z ) d z ] .
P ̃ i ( z , t ) = P ̃ i s ( z ) + P ̃ i ( z , t ) ,
D ̃ ( z , t ) = D ̃ s ( z ) + D ̃ ( z , t ) ,
E ̃ r ( t ) = E ̃ r s + E ̃ r ( t ) ,
P ̃ i ( z , t ) t = δ { P ̃ i ( z , t ) + A ( z ) [ E ̃ r s D ̃ ( z , t ) + D ̃ s ( z ) E ̃ r ( t ) ] } ,
D ̃ ( z , t ) t = δ ρ { D ̃ ( z , t ) A ( z ) [ E ̃ r s P ̃ i ( z , t ) + P ̃ i s ( z ) E ̃ r ( t ) ] } ,
d E ̃ r ( t ) d t = [ E ̃ r ( t ) + l 2 l 2 P ̃ i ( z , t ) d z l 2 l 2 A ( z ) d z ] ,
P ̃ i ( z , t ) = P ̃ i ( z ) exp ( λ c t ) ,
D ̃ ( z , t ) = D ̃ ( z ) exp ( λ c t ) ,
E ̃ r ( t ) = E ̃ r exp ( λ c t ) ,
( λ c + δ ) P ̃ i ( z ) = δ A ( z ) [ E ̃ r s D ̃ ( z ) + D ̃ s ( z ) E ̃ r ] ,
( λ c + δ ρ ) D ̃ ( z ) = δ ρ A ( z ) [ E ̃ r s P ̃ i ( z ) + P ̃ i s ( z ) E ̃ r ] ,
( λ c + 1 ) E ̃ r = l 2 l 2 P ̃ i ( z , t ) d z l 2 l 2 A ( z ) d z .
λ c = 1 + δ R 0 l 2 A ( z ) d z × 0 l 2 A ( z ) { ( λ c + δ ρ ) δ ρ [ E ̃ r s A ( z ) ] 2 } d z { 1 + [ E ̃ r s A ( z ) ] 2 } { ( λ c + δ ρ ) ( λ c + δ ) + δ 2 ρ [ E ̃ r s A ( z ) ] 2 } ,

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