Abstract

The physical phenomena that dominate the power characteristics of a laser depend on the detailed nature of the amplifying medium and the resonator structure. In predicting the power characteristics, numerous approximations are always required. The most important approximations are considered here in detail, and error estimates are presented so that a designer can select the appropriate model for a particular application. Emphasis is placed on analytic solutions and specific phenomena considered include longitudinal and transverse spatial hole burning, large single-pass gain, and mixed line broadening.

© 1980 Optical Society of America

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References

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  1. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 3.
  2. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1976), Chap. 5.
  3. W. E. Lamb, Phys. Rev. A: 134, 1429 (1964).
  4. W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
    [CrossRef]
  5. L. W. Casperson, IEEE J. Quantum Electron. QE-9, 250 (1973).
    [CrossRef]
  6. A similar integration was performed by W. W. Rigrod, J. Appl. Phys. 34, 2602 (1963).
    [CrossRef]
  7. An early discussion of longitudinal spatial hole burning is by C. L. Tang, H. Statz, G. DeMars, J. Appl. Phys. 34, 2289 (1963).
    [CrossRef]
  8. I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (3.615-1).
  9. Ref. 8, Eq. (3.681-1).
  10. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. GPO, Washington, D.C., 1970), Eq. (6.2.2).
  11. Ref. 10, Eq. (15.1.1).
  12. H. G. Danielmeyer, J. Appl. Phys. 42, 3125 (1971).
    [CrossRef]
  13. V. Evtuhov, A. E. Siegman, Appl. Opt. 4, 142 (1965).
    [CrossRef]
  14. O. Ersoy, Opt. Quantum Electron. 7, 247 (1975).
    [CrossRef]
  15. Ref. 8, Eq. (3.466-1).
  16. Ref. 8, Eq. (3.383-8).
  17. B. J. Feldman, M. S. Feld, Phys. Rev. A: 1, 1375 (1970).
    [CrossRef]

1975 (1)

O. Ersoy, Opt. Quantum Electron. 7, 247 (1975).
[CrossRef]

1973 (1)

L. W. Casperson, IEEE J. Quantum Electron. QE-9, 250 (1973).
[CrossRef]

1971 (1)

H. G. Danielmeyer, J. Appl. Phys. 42, 3125 (1971).
[CrossRef]

1970 (1)

B. J. Feldman, M. S. Feld, Phys. Rev. A: 1, 1375 (1970).
[CrossRef]

1965 (2)

1964 (1)

W. E. Lamb, Phys. Rev. A: 134, 1429 (1964).

1963 (2)

A similar integration was performed by W. W. Rigrod, J. Appl. Phys. 34, 2602 (1963).
[CrossRef]

An early discussion of longitudinal spatial hole burning is by C. L. Tang, H. Statz, G. DeMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Casperson, L. W.

L. W. Casperson, IEEE J. Quantum Electron. QE-9, 250 (1973).
[CrossRef]

Danielmeyer, H. G.

H. G. Danielmeyer, J. Appl. Phys. 42, 3125 (1971).
[CrossRef]

DeMars, G.

An early discussion of longitudinal spatial hole burning is by C. L. Tang, H. Statz, G. DeMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Ersoy, O.

O. Ersoy, Opt. Quantum Electron. 7, 247 (1975).
[CrossRef]

Evtuhov, V.

Feld, M. S.

B. J. Feldman, M. S. Feld, Phys. Rev. A: 1, 1375 (1970).
[CrossRef]

Feldman, B. J.

B. J. Feldman, M. S. Feld, Phys. Rev. A: 1, 1375 (1970).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (3.615-1).

Lamb, W. E.

W. E. Lamb, Phys. Rev. A: 134, 1429 (1964).

Rigrod, W. W.

W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

A similar integration was performed by W. W. Rigrod, J. Appl. Phys. 34, 2602 (1963).
[CrossRef]

Ryzhik, I. W.

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (3.615-1).

Siegman, A. E.

V. Evtuhov, A. E. Siegman, Appl. Opt. 4, 142 (1965).
[CrossRef]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 3.

Statz, H.

An early discussion of longitudinal spatial hole burning is by C. L. Tang, H. Statz, G. DeMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Tang, C. L.

An early discussion of longitudinal spatial hole burning is by C. L. Tang, H. Statz, G. DeMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1976), Chap. 5.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

L. W. Casperson, IEEE J. Quantum Electron. QE-9, 250 (1973).
[CrossRef]

J. Appl. Phys. (4)

A similar integration was performed by W. W. Rigrod, J. Appl. Phys. 34, 2602 (1963).
[CrossRef]

An early discussion of longitudinal spatial hole burning is by C. L. Tang, H. Statz, G. DeMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

W. W. Rigrod, J. Appl. Phys. 36, 2487 (1965).
[CrossRef]

H. G. Danielmeyer, J. Appl. Phys. 42, 3125 (1971).
[CrossRef]

Opt. Quantum Electron. (1)

O. Ersoy, Opt. Quantum Electron. 7, 247 (1975).
[CrossRef]

Phys. Rev. A (2)

B. J. Feldman, M. S. Feld, Phys. Rev. A: 1, 1375 (1970).
[CrossRef]

W. E. Lamb, Phys. Rev. A: 134, 1429 (1964).

Other (8)

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 3.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1976), Chap. 5.

I. S. Gradshteyn, I. W. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (3.615-1).

Ref. 8, Eq. (3.681-1).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. GPO, Washington, D.C., 1970), Eq. (6.2.2).

Ref. 10, Eq. (15.1.1).

Ref. 8, Eq. (3.466-1).

Ref. 8, Eq. (3.383-8).

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Figures (7)

Fig. 1
Fig. 1

Normalized intensity sI incident on the coupling mirror of a high-gain homogeneously broadened laser as a function of the threshold parameter r for various values of the single-pass gain. The curve labeled ghol = 0 is the same as the standard low-gain approximation of Eq. (12).

Fig. 2
Fig. 2

Normalized intensity sI incident on the coupling mirror of a high-gain non-Doppler inhomogeneously broadened laser for various values of the single-pass gain. The curve labeled gil = 0 is the same as the standard low-gain approximation of Eq. (14).

Fig. 3
Fig. 3

Normalized intensity sI incident on the coupling mirror of a high-gain Doppler laser tuned away from line center for various values of the single-pass gain. The curve labeled gil = 0 is the same as the standard low-gain approximation r2 − 1.

Fig. 4
Fig. 4

Normalized internal one-way intensity sI vs the threshold parameter r for various types of lasers. The curve labeled ho is the homogeneously broadened laser of Eq. (12) neglecting longitudinal spatial hole burning, while hl denotes the homogeneously broadened laser of Eq. (38) with hole burning included. Similarly, io is the inhomogeneously broadened laser of Eq. (14), and il is the laser of Eq. (45). The curve id is the extreme case of a Doppler laser described by Eq. (50), and most Doppler lasers would be closer to io.

Fig. 5
Fig. 5

Normalized intensity sI vs the threshold parameter r for various lasers. The curve labeled ho is the homogeneously broadened laser of Eq. (12) neglecting transverse spatial hole burning, while ht denotes the laser of Eq. (60). Similarly, io is the inhomogeneously broadened laser of Eq. (14), and it is the laser of Eq. (65).

Fig. 6
Fig. 6

Normalized intensity sI vs the threshold parameter r for various values of the mized broadening parameter ρ. For small values of ρ the curve approaches the relationship of Eq. (14), and for large ρ the curve approaches Eq. (12).

Fig. 7
Fig. 7

Saturation exponent as a function of the mixed broadening parameter ρ.

Equations (128)

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d I d z = g ( ν , I ) I - η I ,
g h ( ν , I ) = g h o 1 + [ 2 ( ν - ν 0 ) / Δ ν h ] 2 + s I ,
g i ( ν , I ) = g h o 0 p ( ν a ) d ν a 1 + [ 2 ( ν - ν a ) / Δ ν h ] 2 + s I ,
g i ( ν , I ) = g h o p ( ν ) 0 d ν a 1 + [ 2 ( ν - ν a ) / Δ ν h ] 2 + s I = g h o p ( ν ) ( π Δ ν h / 2 ) ( 1 + s I ) 1 / 2 .
p ( ν a ) = 2 ( ln 2 ) 1 / 2 π 1 / 2 Δ ν D exp { - [ 2 ( ν a - ν 0 ) Δ ν D ] 2 ln 2 } ,
g i ( ν , I ) = g i o exp { - [ 2 ( ν - ν 0 ) / Δ ν D ] 2 ln 2 } ( 1 + s I ) 1 / 2 ,
g i o = g h o Δ ν h Δ ν D ( π ln 2 ) 1 / 2 .
d I d z = g h o I 1 + [ 2 ( ν - ν 0 ) / Δ ν h ] 2 + s I - η I .
Δ I = 2 g h o l I 1 + [ 2 ( ν - ν 0 ) / Δ ν h ] 2 + 2 s I - 2 η l I ,
Δ I = ( 1 - R l ) I + ( 1 - R r ) I ,
I = 1 2 s { 2 g h o l ( 1 - R l ) + ( 1 - R r ) + 2 η l - 1 - [ 2 ( ν - ν 0 ) Δ ν h ] 2 } .
s I = ( r - 1 ) / 2 ,
d I d z = g i o exp { - [ 2 ( ν - ν 0 ) / Δ ν D ] 2 ln 2 } ( 1 + s I ) 1 / 2 - η I .
s I = ( r 2 - 1 ) / 2.
d I + d z = g h o I + 1 + s ( I + + I - ) - η I + ,
d I - d z = - g h o I - 1 + s ( I + + I - ) + η I - .
I + ( z ) I - ( z ) = const .
2 s I + ( z r ) = [ ( 1 - R r ) + ( R r / R l ) 1 / 2 ( 1 - R l ) ] - 1 [ 2 g h o l + ln ( R l R r ) ] .
2 s I + ( z r ) = ( 1 - R r ) - 1 ( 2 g h o l + ln R r ) .
r = - 2 g h o l / ln R r .
s I + ( z r ) = g h o l ( 1 - r - 1 ) 1 - exp ( - 2 g h o l / r ) .
d I + d z = g i I + [ 1 + s ( I + + I - ) ] 1 / 2 - η I + ,
d I - d z = - g i I - [ 1 + s ( I + + I - ) ] 1 / 2 + η I - .
d I + d z = g i I + ( 1 + s I + ) 1 / 2 - η I + ,
d I - d z = - g i I - ( 1 + s I - ) 1 / 2 + η I - ,
2 g i l = ln [ ( 1 + s I 2 ) 1 / 2 - 1 ( 1 + s I 2 ) 1 / 2 + 1 × ( 1 + s I 1 ) 1 / 2 + 1 ( 1 + s I 1 ) 1 / 2 - 1 ] + 2 ( 1 + s I 2 ) 1 / 2 - 2 ( 1 + s I 1 ) 1 / 2 ,
2 g i l = ln [ ( 1 + s I 2 ) 1 / 2 - 1 ( 1 + s I 2 ) 1 / 2 + 1 × ( 1 + R s I 2 ) 1 / 2 + 1 ( 1 + R s I 2 ) 1 / 2 - 1 ] + 2 ( 1 + s I 2 ) 1 / 2 - 2 ( 1 + R s I 2 ) 1 / 2 .
1 2 t c = ω 0 μ 2 0 L γ 0 0 l sin 2 ( k z ) N ( ω a ) d z d ω a 1 + [ ( ω - ω a ) / γ ] 2 + 4 sin 2 ( k z ) s I ,
ω - Ω = ω 0 μ 2 0 L γ 0 0 l ( ω a - ω γ ) × sin 2 ( k z ) N ( ω a ) d z d ω a 1 + [ ( ω - ω a ) / γ ] 2 + 4 sin 2 ( k z ) s I ,
1 2 t c = ω 0 μ 2 N 0 L γ 0 l sin 2 ( k z ) d z 1 + [ ( ω - ω 0 ) / γ ] 2 + 4 sin 2 ( k z ) s I ,
ω - Ω = ω 0 μ 2 N 0 L γ ( ω 0 - ω γ ) 0 l sin 2 ( k z ) d z 1 + [ ( ω - ω 0 ) / γ ] 2 + 4 sin 2 ( k z ) s I ,
N = 0 N ( ω a ) d ω a
1 = 2 r ( 1 + 4 s I 1 + [ ω - ω 0 ) / γ ] 2 + { 1 + 4 s I 1 + [ ( ω - ω 0 ) / γ ] 2 } 1 / 2 ) - 1 ,
( ω - Ω ) t c = r ( ω 0 - ω γ ) ( 1 + 4 s I 1 + [ ( ω - ω 0 ) / γ ] 2 + { 1 + 4 s I 1 + [ ( ω - ω 0 ) / γ ] 2 } 1 / 2 ) - 1 ,
r = t c ω 0 μ 2 N l 0 γ L [ 1 + ( ω - ω 0 γ ) 2 ] - 1 .
s I 1 + [ ( ω - ω 0 ) / γ ] 2 = 4 r - 1 - ( 8 r + 1 ) 1 / 2 8 .
( ω - Ω ) t c = ω 0 - ω 2 γ .
ω = ( Ω + ω 0 / 2 γ t c ) ( 1 + 1 / 2 γ t c ) - 1 ,
s I = 4 r - 1 - ( 8 r + 1 ) 1 / 2 8 .
s I ( r - 1 ) / 3.
1 2 t c = ω 0 μ 2 N ( ω ) 0 L γ 0 l 0 sin 2 ( k z ) d ω a d z 1 + [ ( ω - ω a ) / γ ] 2 + 4 sin 2 ( k z ) s I ,
ω - Ω = ω 0 μ 2 N ( ω ) 0 L γ 0 1 0 ( ω a - ω γ ) × sin 2 ( k z ) d ω a d z 1 + [ ( ω - ω a ) / γ ] 2 + 4 sin 2 ( k z ) s I .
1 2 t c = π ω 0 μ 2 N ( ω ) 0 L 0 l sin 2 ( k z ) d z [ 1 + 4 s I sin 2 ( k z ) ] 1 / 2 .
0 π / 2 sin 2 μ - 1 x cos 2 ν - 1 x d x ( 1 - a 2 sin 2 x ) ρ = B ( μ , ν ) F ( ρ , μ ; μ + ν ; a 2 ) 2 ,
1 2 t c = ω 0 μ 2 N ( ω ) l 0 L B ( / 2 3 , ½ ) F ( ½ , / 2 3 ; 2 ; - 4 s I ) .
1 - r F ( ½ , / 2 3 ; 2 ; - 4 s I ) ,
r = π ω 0 t c μ 2 N ( ω ) l 0 L .
1 r = 1 - / 2 3 s I + .
r ( 1 + 3 s I ) 1 / 2 .
s I ( r 2 - 1 ) / 3.
1 r = - N ( v ) W r ( v ) d v 1 + 2 W r ( v ) s I / - N ( v ) α o r ( v ) d v ,
2 ( ω - Ω ) t c r = - N ( v ) Re [ ω 0 - ω i k v + γ W ( v ) 1 + 2 W r ( v ) s I ] d v / - N ( v ) α o r ( v ) d v ,
1 r = - N ( v ) α o r ( v ) d v 1 + 2 α o r ( v ) s I / - N ( v ) α o r ( v ) d v = - N ( v ) d ν [ γ 2 / 2 γ 2 + ( k v + ω - ω 0 ) 2 + γ 2 / 2 γ 2 + ( k v - ω + ω 0 ) 2 ] - 1 + 2 s I ÷ - [ γ 2 / 2 γ 2 + ( k v + ω - ω 0 ) 2 + γ 2 / 2 γ 2 + ( k v + ω - ω 0 ) 2 ] N ( v ) d v .
1 r = - N ( v ) d v 1 + ( k v / γ ) 2 + 2 s I / - N ( v ) d v 1 + ( k v / γ ) 2 .
I ( r , ϕ , z ) = P ( z ) f ( r , ϕ , z ) ,
f ( r , ϕ , z ) = [ 2 / π w 2 ( z ) ] exp [ - 2 r 2 / w 2 ( z ) ] ,
d I d z = g h o I 1 + s I - η I .
d P d z = g h o π w 2 2 s ln ( 1 + 2 s P π w 2 ) - η P .
d P d z = g h o π w 2 4 s ln ( 1 + 4 s P π w 2 ) - η P .
π w 2 4 s P ln ( 1 + 4 s P π w 2 ) = 1 r ,
1 4 s I ln ( 1 + 4 s I ) = 1 r .
d P d z = g i π ω 2 s [ ( 1 + 2 s P π w 2 ) 1 / 2 - 1 ] - η P .
d P d z = g i π w 2 2 s [ ( 1 + 4 s P π w 2 ) 1 / 2 - 1 ] - η P .
π w 2 2 s P [ ( 1 + 4 s P π w 2 ) 1 / 2 - 1 ] = 1 r .
s P π w 2 = r ( r - 1 ) .
s I = r ( r - 1 ) .
1 r = - exp ( - v 2 / u 2 ) d v 1 + ( k v / γ ) 2 + 2 s I / - exp ( - v 2 / u 2 ) d v 1 + ( k v / γ ) 2 ,
1 r = - exp ( - ρ y 2 ) d y 1 + y 2 + 2 s I / - exp ( - ρ y 2 ) d y 1 + y 2 .
s I = [ r f ( ρ ) - 1 ] / 2.
f ( ρ ) = 0 exp ( - ρ y 2 ) d y 1 + y 2 / 0 exp ( - ρ y 2 ) d y ( 1 + y 2 ) 2 .
f ( ρ ) 1 + 1 1 + exp [ ζ ( log ρ - log ρ 0 ) ] ,
( t + v z ) ρ a b ( v , ω a , z , t ) = - ( i ω a + γ ) ρ a b ( v , ω a , z , t ) - i μ E ( z , t ) [ ρ a a ( v , ω a , z , t ) - ρ b b ( v , ω a , z , t ) ] ,
( t + v z ) ρ a a ( v , w a , z , t ) = λ a ( v , ω a , z , t ) - γ a ρ a a ( v , ω a , z , t ) + [ i μ E ( z , t ) ρ b a ( v , ω a , z , t ) + c . c . ] ,
( t + v z ) ρ b b ( v , ω a , z , t ) = λ b ( v , ω a , z , t ) - γ b ρ b b ( v , ω a , z , t ) - [ i μ E ( z , t ) ρ b a ( v , ω a , z , t ) + c . c . ] ,
ρ b a ( v , ω a , z , t ) = ρ a b * ( v , ω a , z , t ) ,
2 E ( z , t ) z 2 - μ 0 σ E ( z , t ) t - μ 0 0 2 E ( z , t ) t 2 = μ 0 2 P ( z , t ) t 2 .
P ( z , t ) = 0 - μ ρ a b ( v , ω a , z , t ) d v d ω a + c . c . ,
E ( z , t ) = ½ E sin ( k z ) exp ( - i ω t ) + c . c . ,
ρ a b ( v , ω a , z , t ) = [ C ( v , ω a , z ) + i S ( v , ω a , z ) ] exp ( - i ω t ) / 2 μ .
v z S ( v , ω a , z ) = ( ω - ω a ) C ( v , ω a , z ) - γ S ( v , ω a , z ) - μ 2 E D ( v , ω a , z ) sin ( k z ) ,
v z C ( v , ω a , z ) = - ( ω - ω a ) S ( v , ω a , z ) - γ C ( v , ω a , z ) ,
v z D ( v , ω a , z ) = λ a ( v , ω a ) - λ b ( v , ω a ) - γ a + γ b 2 D ( v , ω a , z ) - γ a - γ b 2 M ( v , ω a , z ) + E S ( v , ω a , z ) sin ( k z ) ,
v z M ( v , ω a , z ) = λ a ( v , ω a ) + λ b ( v , ω a ) - γ a + γ b 2 M ( v , ω a , z ) - γ a - γ B 2 D ( v , w a , z ) ,
σ 2 0 E = - ω 0 0 L - 0 0 l sin ( k z ) S ( v , ω a , z ) d z d ω a d v ,
( ω - Ω ) E = - ω 0 0 L - 0 0 l sin ( k z ) C ( v , ω a , z ) d z d ω a d v ,
0 = ( ω - ω a ) C ( ω a , z ) - γ S ( ω a , z ) - μ 2 E D ( ω a , z ) sin ( k z ) ,
0 = - ( ω - ω a ) S ( ω a , z ) - γ C ( ω a , z ) ,
0 = λ a ( ω a ) - λ b ( ω a ) - γ a + γ b 2 D ( ω a , z ) - γ a - γ b 2 M ( ω a , z ) + E S ( ω a , z ) sin ( k z ) ,
0 = λ a ( ω a ) + λ b ( ω a ) - γ a + γ b 2 M ( ω a , z ) - γ a - γ b 2 D ( ω a , z ) ,
σ 2 0 E = - ω 0 0 L 0 0 l sin ( k z ) S ( ω a , z ) d z d ω a ,
( ω - Ω ) E = - ω 0 0 L 0 0 l sin ( k z ) C ( ω a , z ) d z d ω a ,
S ( ω a , z ) = - S ( v , ω a , z ) d v , C ( ω a , z ) = - C ( v , ω a , z ) d v , D ( ω a , z ) = - D ( v , ω a , z ) d v , M ( ω a , z ) = - M ( v , ω a , z ) d v , λ a ( ω a ) = - λ a ( v , ω a ) d v , λ b ( ω a ) = - λ b ( v , ω a ) d v .
S ( ω a , z ) = - μ 2 E D ( ω a , z ) sin ( k z ) / γ 1 + [ ( ω - ω a ) / γ ] 2 .
D ( ω a , z ) = γ a + γ b 2 γ a γ b E S ( ω a , z ) sin ( k z ) + N ( ω a ) ,
N ( ω a ) = λ a ( ω a ) / γ a - λ b ( ω a ) / γ b .
S ( ω a , z ) = - μ 2 E N ( ω a ) sin ( k z ) / γ 1 + [ ( ω - ω a ) / γ ] 2 + 4 sin 2 ( k z ) s I ,
s I = μ 2 E 2 8 2 γ a + γ b γ γ a γ b .
C ( ω a , z ) = ( ω - ω a γ ) μ 2 E N ( ω a ) sin ( k z ) / γ 1 + [ ( ω - ω a ) / γ ] 2 + 4 sin 2 ( k z ) s I .
1 2 t c = ω 0 μ 2 0 L γ 0 0 l sin 2 ( k z ) N ( ω a ) d z d ω a 1 + [ ( ω - ω a ) / γ ] 2 + 4 sin 2 ( k z ) s I ,
ω - Ω = ω 0 μ 2 0 L γ 0 0 l ( ω a - ω γ ) × sin 2 ( k z ) N ( ω a ) d z d ω a 1 + [ ( ω - ω a ) / γ ] 2 + 4 sin 2 ( k z ) s I .
v z S ( v , z ) = ( ω - ω 0 ) C ( v , z ) - γ S ( v , z ) - μ 2 E D ( v , z ) sin ( k z ) ,
v z C ( v , z ) = - ( ω - ω 0 ) S ( v , z ) - γ C ( v , z ) ,
v z D ( v , z ) = λ a ( v ) - λ b ( v ) - γ a + γ b 2 D ( v , z ) - γ a - γ b 2 M ( v , z ) + E S ( v , z ) sin ( k z ) ,
v z M ( v , z ) = λ a ( v ) + λ b ( v ) - γ a + γ b 2 M ( v , z ) - γ a - γ b 2 D ( v , z ) ,
σ 2 0 E = - ω 0 0 L - 0 l sin ( k z ) S ( v , z ) d z d v ,
( ω - Ω ) E = - ω 0 0 L - 0 l sin ( k z ) C ( v , z ) d z d v ,
S ( v , z ) = - S ( v , ω a , z ) d ω a , C ( v , z ) = - C ( v , ω a , z ) d ω a , D ( v , z ) = - D ( v , ω a , z ) d ω a , M ( v , z ) = - M ( v , ω a , z ) d ω a .
S ( v , z ) = j = - S 2 j + 1 ( v ) exp [ ( 2 j + 1 ) i k z ] ,
C ( v , z ) = j = - C 2 j + 1 ( v ) exp [ ( 2 j + 1 ) i k z ] ,
D ( v , z ) = j = - D 2 j ( v ) exp [ ( 2 j ) i k z ] ,
M ( v , z ) = j = - M 2 j ( v ) exp [ ( 2 j ) i k z ] ,
0 = - [ ( 2 j + 1 ) i k v + γ ] S 2 j + 1 ( v ) + ( ω - ω 0 ) C 2 j + 1 ( v ) + i μ 2 E 2 [ D 2 j ( v ) - D 2 j + 2 ( v ) ] ,
0 = - [ ( 2 j + 1 ) i k v + γ ] C 2 j + 1 ( v ) - ( ω - ω 0 ) S 2 j + 1 ( v ) ,
0 = [ λ a ( v ) - λ b ( v ) ] δ j o - [ ( 2 j ) i k v + γ a + γ b 2 ] D 2 j ( v ) - γ a - γ b 2 M 2 j ( v ) - i E 2 [ S 2 j - 1 ( v ) - S 2 j + 1 ( v ) ] ,
0 = [ λ a ( v ) + λ b ( v ) ] δ j o - [ ( 2 j ) i k v + γ a + γ b 2 ] M 2 j ( v ) - γ a - γ b 2 D 2 j ( v ) ,
E 2 t c = ω 0 l 0 L - S 1 i ( v ) d v ,
( ω - Ω ) E = ω 0 l 0 L - C 1 i ( v ) d v ,
S 2 j + 1 ( v ) = i μ 2 E 2 γ α j ( v ) [ D 2 j ( v ) - D 2 j + 2 ( v ) ] ,
α j ( v ) = γ / 2 ( 2 j + 1 ) i k v + i ( ω - ω 0 ) + γ + γ / 2 ( 2 j + 1 ) i k v - i ( ω - ω 0 ) + γ .
D 2 j ( v ) = - i E 4 γ a + γ b γ a γ b β j ( v ) [ S 2 j - 1 ( v ) - S 2 j + 1 ( v ) ] + [ λ a ( v ) γ a - λ b ( v ) γ b ] δ jo ,
β j ( v ) = γ a γ b γ a + γ b [ 1 ( 2 j ) i k v + γ a + 1 ( 2 j ) i k v + γ b ] .
S 1 ( v ) = i 4 E γ a γ b γ a + γ b D 0 ( v ) W ( v ) s I ,
W ( v ) = α 0 ( v ) _ 1 + α 0 ( v ) β 1 ( v ) s I _ 1 + α 1 ( v ) β 1 ( v ) s I _ 1 + α 1 ( v ) β 2 ( v ) s I _ . 1 +
S 1 i ( v ) = 4 E γ a γ b γ a + γ b N ( v ) s I W r ( v ) 1 + 2 W r ( v ) s I ,
C 1 i ( v ) = 4 E γ a γ b γ a + γ b N ( v ) s I Re [ ω 0 - ω i k v + γ W ( v ) 1 + 2 W r ( v ) s I ] .
g = 2 ω 0 c 0 E - S 1 i ( v ) d v = μ 2 ω 0 c 0 γ - N ( v ) α o r ( v ) d v .
1 t c = g c l L - N ( v ) W r ( v ) d v 1 + 2 W r ( v ) s I / - N ( v ) α o r ( v ) d v .
ω - Ω = g c l 2 L - N ( v ) Re [ ω 0 - ω i k v + γ W ( v ) 1 + 2 W r ( v ) s I ] d v / - N ( v ) α o r ( v ) d v .

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