Abstract

The spatial variation of gain and saturation is analyzed in weakly and heavily saturated cases for a fast axial flow (FAF) CO2 laser amplifier with laser gas in laminar and turbulent flow regions. Expressiong the laser parameters by fourth-order even functions of the radial distance, the development of the amplified radiation is predicted and compared successfully to the experimental results for the 10P(18) line. The distinctive feature of a FAF system is enhancement of the gain and saturation irradiance as well as the mode volume. The condition of spatial soliton is also discussed in a heavily saturated case.

© 1989 Optical Society of America

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References

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  1. T. F. Deutsch, F. A. Horrigan, R. I. Rudko, “cw Operation of High-Pressure Flowing CO2 Lasers,” Appl. Phys. Lett. 15, 88–91 (1969).
    [CrossRef]
  2. W. B. Tiffany, R. Targ, J. D. Foster, “Kilowatt CO2 Gas-Transport Laser,” Appl. Phys. Lett. 15, 91–93 (1969).
    [CrossRef]
  3. J. E. Harry, D. R. Evans, “Multikilowatt Compact Axial Flow CO2 Laser,” Appl. Phys. Lett. 50, 313–315 (1987).
    [CrossRef]
  4. H. Sugawara, K. Kuwabara, S. Takemori, A. Wada, K. Sasaki, “20-kW Fast-Axial-Flow CO2 Laser with High-Frequency Turboblower,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1984), paper TUC3.
  5. M. Kasamatsu, S. Shiratori, T. Sato, “Characteristics of Two Cathode Systems for CW Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-22, 2026–2031 (1986).
    [CrossRef]
  6. E. Tsuchida, H. Sato, “Theoretical Analysis of Transient Gain Phenomena in a Fast-Axial Flow Type CO2 Laser Amplifier,” IEEE J. Quantum Electron. 25, 121–131 (1989).
    [CrossRef]
  7. E. Tsuchida, H. Sato, “Effect of Gas-Flow Velocity on Transient Behavior of Gain Constant in a Fast-Axial Flow Type CO2 Laser Amplifier,” Jpn. J. Appl. Phys. 27, 1445–1453 (1988).
    [CrossRef]
  8. H. Sato, E. Tsuchida, “Theoretical Analysis of the Dynamic Behavior of a Fast Axial Flow CO2 Laser Amplifier,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1988), paper FD5.
  9. H. Gamo, J. S. Ostrem, “Determination of Laser Parameters from Saturated-Gain Measurement for a Partially Homogeneously Broadened Gas-Laser Amplifier,” J. Opt. Soc. Am. 65, 29–32 (1975).
    [CrossRef]
  10. E. Tsuchida, H. Sato, “Dependence of Spatial Gain Distribution on Gas-Flow Velocity and Discharge Current in a Fast-Axial Flow CO2 laser Amplifier,” Jpn. J. Appl. Phys. 28, 396–405 (1989).
    [CrossRef]
  11. I. H. Shames, Mechanics of Fluids (MacGraw-Hill, New York, 1982), Chap. 8.
  12. A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 10.
  13. R. K. Brimacombe, J. Reid, “Accurate Measurements of Pressure-Broadened Linewidths in a Transversely Excited CO2 Discharge,” IEEE J. Quantum Electron. QE-19, 1668–1673 (1983).
    [CrossRef]
  14. R. K. Brimacombe, J. Reid, “Measurements of Anomalous Gain Coefficients in Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-19, 1674–1679 (1983).
    [CrossRef]
  15. S. Singer, “Observations of Anomalous Gain Coefficients in TEA Double-Discharge CO2 Lasers,” IEEE J. Quantum Electron. QE-10, 829–831 (1974).
    [CrossRef]
  16. S. Maneuf, F. Reynaud, “First Observation of Higher Order Planar Soliton Beams,” in Proceedings, Fourteenth Congress of the International Commission for Optics (International Commission for Optics, Quebec, 1987), paper A11.2.

1989 (2)

E. Tsuchida, H. Sato, “Dependence of Spatial Gain Distribution on Gas-Flow Velocity and Discharge Current in a Fast-Axial Flow CO2 laser Amplifier,” Jpn. J. Appl. Phys. 28, 396–405 (1989).
[CrossRef]

E. Tsuchida, H. Sato, “Theoretical Analysis of Transient Gain Phenomena in a Fast-Axial Flow Type CO2 Laser Amplifier,” IEEE J. Quantum Electron. 25, 121–131 (1989).
[CrossRef]

1988 (1)

E. Tsuchida, H. Sato, “Effect of Gas-Flow Velocity on Transient Behavior of Gain Constant in a Fast-Axial Flow Type CO2 Laser Amplifier,” Jpn. J. Appl. Phys. 27, 1445–1453 (1988).
[CrossRef]

1987 (1)

J. E. Harry, D. R. Evans, “Multikilowatt Compact Axial Flow CO2 Laser,” Appl. Phys. Lett. 50, 313–315 (1987).
[CrossRef]

1986 (1)

M. Kasamatsu, S. Shiratori, T. Sato, “Characteristics of Two Cathode Systems for CW Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-22, 2026–2031 (1986).
[CrossRef]

1983 (2)

R. K. Brimacombe, J. Reid, “Accurate Measurements of Pressure-Broadened Linewidths in a Transversely Excited CO2 Discharge,” IEEE J. Quantum Electron. QE-19, 1668–1673 (1983).
[CrossRef]

R. K. Brimacombe, J. Reid, “Measurements of Anomalous Gain Coefficients in Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-19, 1674–1679 (1983).
[CrossRef]

1975 (1)

1974 (1)

S. Singer, “Observations of Anomalous Gain Coefficients in TEA Double-Discharge CO2 Lasers,” IEEE J. Quantum Electron. QE-10, 829–831 (1974).
[CrossRef]

1969 (2)

T. F. Deutsch, F. A. Horrigan, R. I. Rudko, “cw Operation of High-Pressure Flowing CO2 Lasers,” Appl. Phys. Lett. 15, 88–91 (1969).
[CrossRef]

W. B. Tiffany, R. Targ, J. D. Foster, “Kilowatt CO2 Gas-Transport Laser,” Appl. Phys. Lett. 15, 91–93 (1969).
[CrossRef]

Brimacombe, R. K.

R. K. Brimacombe, J. Reid, “Accurate Measurements of Pressure-Broadened Linewidths in a Transversely Excited CO2 Discharge,” IEEE J. Quantum Electron. QE-19, 1668–1673 (1983).
[CrossRef]

R. K. Brimacombe, J. Reid, “Measurements of Anomalous Gain Coefficients in Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-19, 1674–1679 (1983).
[CrossRef]

Deutsch, T. F.

T. F. Deutsch, F. A. Horrigan, R. I. Rudko, “cw Operation of High-Pressure Flowing CO2 Lasers,” Appl. Phys. Lett. 15, 88–91 (1969).
[CrossRef]

Evans, D. R.

J. E. Harry, D. R. Evans, “Multikilowatt Compact Axial Flow CO2 Laser,” Appl. Phys. Lett. 50, 313–315 (1987).
[CrossRef]

Foster, J. D.

W. B. Tiffany, R. Targ, J. D. Foster, “Kilowatt CO2 Gas-Transport Laser,” Appl. Phys. Lett. 15, 91–93 (1969).
[CrossRef]

Gamo, H.

Harry, J. E.

J. E. Harry, D. R. Evans, “Multikilowatt Compact Axial Flow CO2 Laser,” Appl. Phys. Lett. 50, 313–315 (1987).
[CrossRef]

Horrigan, F. A.

T. F. Deutsch, F. A. Horrigan, R. I. Rudko, “cw Operation of High-Pressure Flowing CO2 Lasers,” Appl. Phys. Lett. 15, 88–91 (1969).
[CrossRef]

Kasamatsu, M.

M. Kasamatsu, S. Shiratori, T. Sato, “Characteristics of Two Cathode Systems for CW Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-22, 2026–2031 (1986).
[CrossRef]

Kuwabara, K.

H. Sugawara, K. Kuwabara, S. Takemori, A. Wada, K. Sasaki, “20-kW Fast-Axial-Flow CO2 Laser with High-Frequency Turboblower,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1984), paper TUC3.

Maneuf, S.

S. Maneuf, F. Reynaud, “First Observation of Higher Order Planar Soliton Beams,” in Proceedings, Fourteenth Congress of the International Commission for Optics (International Commission for Optics, Quebec, 1987), paper A11.2.

Ostrem, J. S.

Reid, J.

R. K. Brimacombe, J. Reid, “Accurate Measurements of Pressure-Broadened Linewidths in a Transversely Excited CO2 Discharge,” IEEE J. Quantum Electron. QE-19, 1668–1673 (1983).
[CrossRef]

R. K. Brimacombe, J. Reid, “Measurements of Anomalous Gain Coefficients in Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-19, 1674–1679 (1983).
[CrossRef]

Reynaud, F.

S. Maneuf, F. Reynaud, “First Observation of Higher Order Planar Soliton Beams,” in Proceedings, Fourteenth Congress of the International Commission for Optics (International Commission for Optics, Quebec, 1987), paper A11.2.

Rudko, R. I.

T. F. Deutsch, F. A. Horrigan, R. I. Rudko, “cw Operation of High-Pressure Flowing CO2 Lasers,” Appl. Phys. Lett. 15, 88–91 (1969).
[CrossRef]

Sasaki, K.

H. Sugawara, K. Kuwabara, S. Takemori, A. Wada, K. Sasaki, “20-kW Fast-Axial-Flow CO2 Laser with High-Frequency Turboblower,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1984), paper TUC3.

Sato, H.

E. Tsuchida, H. Sato, “Theoretical Analysis of Transient Gain Phenomena in a Fast-Axial Flow Type CO2 Laser Amplifier,” IEEE J. Quantum Electron. 25, 121–131 (1989).
[CrossRef]

E. Tsuchida, H. Sato, “Dependence of Spatial Gain Distribution on Gas-Flow Velocity and Discharge Current in a Fast-Axial Flow CO2 laser Amplifier,” Jpn. J. Appl. Phys. 28, 396–405 (1989).
[CrossRef]

E. Tsuchida, H. Sato, “Effect of Gas-Flow Velocity on Transient Behavior of Gain Constant in a Fast-Axial Flow Type CO2 Laser Amplifier,” Jpn. J. Appl. Phys. 27, 1445–1453 (1988).
[CrossRef]

H. Sato, E. Tsuchida, “Theoretical Analysis of the Dynamic Behavior of a Fast Axial Flow CO2 Laser Amplifier,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1988), paper FD5.

Sato, T.

M. Kasamatsu, S. Shiratori, T. Sato, “Characteristics of Two Cathode Systems for CW Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-22, 2026–2031 (1986).
[CrossRef]

Shames, I. H.

I. H. Shames, Mechanics of Fluids (MacGraw-Hill, New York, 1982), Chap. 8.

Shiratori, S.

M. Kasamatsu, S. Shiratori, T. Sato, “Characteristics of Two Cathode Systems for CW Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-22, 2026–2031 (1986).
[CrossRef]

Singer, S.

S. Singer, “Observations of Anomalous Gain Coefficients in TEA Double-Discharge CO2 Lasers,” IEEE J. Quantum Electron. QE-10, 829–831 (1974).
[CrossRef]

Sugawara, H.

H. Sugawara, K. Kuwabara, S. Takemori, A. Wada, K. Sasaki, “20-kW Fast-Axial-Flow CO2 Laser with High-Frequency Turboblower,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1984), paper TUC3.

Takemori, S.

H. Sugawara, K. Kuwabara, S. Takemori, A. Wada, K. Sasaki, “20-kW Fast-Axial-Flow CO2 Laser with High-Frequency Turboblower,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1984), paper TUC3.

Targ, R.

W. B. Tiffany, R. Targ, J. D. Foster, “Kilowatt CO2 Gas-Transport Laser,” Appl. Phys. Lett. 15, 91–93 (1969).
[CrossRef]

Tiffany, W. B.

W. B. Tiffany, R. Targ, J. D. Foster, “Kilowatt CO2 Gas-Transport Laser,” Appl. Phys. Lett. 15, 91–93 (1969).
[CrossRef]

Tsuchida, E.

E. Tsuchida, H. Sato, “Theoretical Analysis of Transient Gain Phenomena in a Fast-Axial Flow Type CO2 Laser Amplifier,” IEEE J. Quantum Electron. 25, 121–131 (1989).
[CrossRef]

E. Tsuchida, H. Sato, “Dependence of Spatial Gain Distribution on Gas-Flow Velocity and Discharge Current in a Fast-Axial Flow CO2 laser Amplifier,” Jpn. J. Appl. Phys. 28, 396–405 (1989).
[CrossRef]

E. Tsuchida, H. Sato, “Effect of Gas-Flow Velocity on Transient Behavior of Gain Constant in a Fast-Axial Flow Type CO2 Laser Amplifier,” Jpn. J. Appl. Phys. 27, 1445–1453 (1988).
[CrossRef]

H. Sato, E. Tsuchida, “Theoretical Analysis of the Dynamic Behavior of a Fast Axial Flow CO2 Laser Amplifier,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1988), paper FD5.

Wada, A.

H. Sugawara, K. Kuwabara, S. Takemori, A. Wada, K. Sasaki, “20-kW Fast-Axial-Flow CO2 Laser with High-Frequency Turboblower,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1984), paper TUC3.

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 10.

Appl. Phys. Lett. (3)

T. F. Deutsch, F. A. Horrigan, R. I. Rudko, “cw Operation of High-Pressure Flowing CO2 Lasers,” Appl. Phys. Lett. 15, 88–91 (1969).
[CrossRef]

W. B. Tiffany, R. Targ, J. D. Foster, “Kilowatt CO2 Gas-Transport Laser,” Appl. Phys. Lett. 15, 91–93 (1969).
[CrossRef]

J. E. Harry, D. R. Evans, “Multikilowatt Compact Axial Flow CO2 Laser,” Appl. Phys. Lett. 50, 313–315 (1987).
[CrossRef]

IEEE J. Quantum Electron. (5)

M. Kasamatsu, S. Shiratori, T. Sato, “Characteristics of Two Cathode Systems for CW Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-22, 2026–2031 (1986).
[CrossRef]

E. Tsuchida, H. Sato, “Theoretical Analysis of Transient Gain Phenomena in a Fast-Axial Flow Type CO2 Laser Amplifier,” IEEE J. Quantum Electron. 25, 121–131 (1989).
[CrossRef]

R. K. Brimacombe, J. Reid, “Accurate Measurements of Pressure-Broadened Linewidths in a Transversely Excited CO2 Discharge,” IEEE J. Quantum Electron. QE-19, 1668–1673 (1983).
[CrossRef]

R. K. Brimacombe, J. Reid, “Measurements of Anomalous Gain Coefficients in Transversely Excited CO2 Lasers,” IEEE J. Quantum Electron. QE-19, 1674–1679 (1983).
[CrossRef]

S. Singer, “Observations of Anomalous Gain Coefficients in TEA Double-Discharge CO2 Lasers,” IEEE J. Quantum Electron. QE-10, 829–831 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (2)

E. Tsuchida, H. Sato, “Dependence of Spatial Gain Distribution on Gas-Flow Velocity and Discharge Current in a Fast-Axial Flow CO2 laser Amplifier,” Jpn. J. Appl. Phys. 28, 396–405 (1989).
[CrossRef]

E. Tsuchida, H. Sato, “Effect of Gas-Flow Velocity on Transient Behavior of Gain Constant in a Fast-Axial Flow Type CO2 Laser Amplifier,” Jpn. J. Appl. Phys. 27, 1445–1453 (1988).
[CrossRef]

Other (5)

H. Sato, E. Tsuchida, “Theoretical Analysis of the Dynamic Behavior of a Fast Axial Flow CO2 Laser Amplifier,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1988), paper FD5.

H. Sugawara, K. Kuwabara, S. Takemori, A. Wada, K. Sasaki, “20-kW Fast-Axial-Flow CO2 Laser with High-Frequency Turboblower,” in Technical Digest, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1984), paper TUC3.

I. H. Shames, Mechanics of Fluids (MacGraw-Hill, New York, 1982), Chap. 8.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 10.

S. Maneuf, F. Reynaud, “First Observation of Higher Order Planar Soliton Beams,” in Proceedings, Fourteenth Congress of the International Commission for Optics (International Commission for Optics, Quebec, 1987), paper A11.2.

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Figures (12)

Fig. 1
Fig. 1

Experimental setups: (a) for a weakly saturated case and (b) modification for a heavily saturated case.

Fig. 2
Fig. 2

Spatial distribution of amplified radiation in a weakly saturated case: (a) the laminar flow region (υ = 160 m/s) and (b) the turbulent flow region (υ = 180 m/s).

Fig. 3
Fig. 3

Spatial distribution of amplified radiation in a heavily saturated case: (a) laminar and (b) turbulent flow regions.

Fig. 4
Fig. 4

Spatial distribution of the gain parameter in the laminar and turbulent flow regions.

Fig. 5
Fig. 5

Spatial distribution of the saturation irradiance in the laminar and turbulent flow regions.

Fig. 6
Fig. 6

Spatial distribution of the first-order coefficient |m| and y parameter in the laminar and turbulent flow regions.

Fig. 7
Fig. 7

Spatial distribution of a small signal gain in the laminar and turbulent flow regions.

Fig. 8
Fig. 8

Theoretical calculation of amplified radiation distribution: (a) laminar and (b) turbulent flow regions.

Fig. 9
Fig. 9

Comparison of the amplification factor between measurements and theory for a weakly saturated case: (a) laminar and (b) turbulent flow regions.

Fig. 10
Fig. 10

Comparison of the amplification factor between measurements and theory for a heavily saturated case: (a) laminar and (b) turbulent flow regions.

Fig. 11
Fig. 11

Comparison of the beamwidth (FWHM) variation between the theory and experiment along the plasma length for a weakly saturated case: (a) laminar and (b) turbulent flow regions.

Fig. 12
Fig. 12

Beam width prediction along the plasma length for the spatial soliton in a heavily saturated case, where lines a and c, respectively, correspond to beamwidth narrowing and broadening and b to the steady beamwidth, i.e., a spatial soliton.

Tables (1)

Tables Icon

Table 1 Various Parameters Determined by the Experiments for the Laminar and Turbulent Flow Regions

Equations (37)

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γ ( x , y ; I in / I s ) = 2 G ψ ( x , y 1 + I in / I s ) / 1 + I in / I s ,
ψ ( x , ξ ) = 1 π ξ exp ( t 2 ) ( x t ) 2 + ξ 2 d t .
ξ = y 1 + I in / I s , x = 2 ln 2 ( ν ν 0 ) / Δ ν D ,
y = ln 2 Δ ν L / Δ ν D .
υ ( r ) = υ [ ( d | r | ) / d ] 1 / 7 ,
G ( r ) = G ( 0 ) ( 1 g 2 r 2 g 4 r 4 ) ,
I s = 4 π n 2 Δ ν L h ν ( t 2 / t s ) λ 2 ,
I s ( r ) = I s ( 0 ) ( 1 s 2 r 2 s 4 r 4 ) ,
I in ( r ) = I in ( 0 ) exp ( r 2 / w 2 i r 4 / w 4 i ) ;
I out ( r ) = I out ( 0 ) exp ( r 2 / w 2 r 4 / w 4 ) ,
I out ( r ) = I in ( r ) exp { γ [ x , y ; I in ( r ) / I s ( r ) ] δ z } .
γ [ r , I in ( r ) ] = γ 0 ( 1 g 2 r 2 g 4 r 4 ) [ 1 | m ( 0 , y ) | I in ( r ) / I s ( r ) ] ,
| m ( 0 , y ) | = 1 2 [ 1 y ψ ( 0 , y ) y / ψ ( 0 , y ) ] ,
m ( r ) = | m ( 0 , y ) | = m ( 0 ) ( 1 m 2 r 2 m 4 r 4 ) ,
y ( r ) = y ( 0 ) ( 1 y 2 r 2 y 4 r 4 ) ,
I out ( 0 ) = I in ( 0 ) exp { γ 0 [ 1 m ( 0 ) I in ( 0 ) / I s ( 0 ) ] δ z } ;
1 / w 2 = 1 / w 2 i + A w γ 0 δ z ;
1 / w 4 = 1 / w 4 i + B w γ 0 δ z ;
A w = g 2 ( g 2 + m 2 + 1 / w 2 i s 2 ) m ( 0 ) I in ( 0 ) / I s ( 0 ) ,
B w = g 4 { g 4 + m 4 + 1 / w 4 i s 4 g 2 m 2 + g 2 s 2 + s 2 m 2 ( g 2 + m 2 s 2 ) / w 2 i s 2 2 } m ( 0 ) I in ( 0 ) / I s ( 0 ) .
I out ( w 1 / 2 / 2 ) = 1 2 ,
w 1 / 2 = 2 2 ln 2 [ 1 / w 2 + ( 1 / w 2 2 + 4 ln 2 / w 4 ) 1 / 2 ] 1 / 2 .
w 1 / 2 = 2 2 ln 2 { 1 / w 2 i + A w γ 0 δ z + [ ( 1 / w 2 i + A w γ 0 δ z ) 2 + 4 ln 2 ( 1 / w 4 i + B w γ 0 δ z ) ] 1 / 2 } 1 / 2 .
w 1 / 2 = 2 2 ln 2 { 1 w 2 i + 1 w 2 i 2 + 4 ln 2 w 4 i + [ A w + A w w 2 i + ( 2 ln 2 ) B w 1 w 2 i 2 + 4 ln 2 w 4 i ] γ 0 δ z } 1 / 2 .
γ [ 0 , y ; I in ( r ) / I s ( r ) ] = 2 G π y exp ( t 2 ) t 2 + y 2 [ I in ( r ) / I s ( r ) ] d t .
γ [ 0 , y ; I in ( r ) / I s ( r ) ] = 2 G π I s ( r ) I in ( r ) exp [ y 2 I in ( r ) I s ( r ) ] × erfc [ y I in ( r ) I s ( r ) ] ,
erfc ( z ) = 1 erf ( z ) .
erfc ( z ) = exp ( z 2 ) { 1 2 z 1 2 2 z 3 + 1 2 3 z 5 } .
γ [ 0 , y ; I in ( r ) / I s ( r ) ] = G ( r ) π y ( r ) I s ( r ) I in ( r ) .
I out ( 0 ) = I in ( 0 ) exp [ γ h I s ( 0 ) / I in ( 0 ) δ z ] ,
1 / w 2 = 1 / w 2 i + A h γ h δ z ,
1 / w 4 = 1 / w 4 i + B h γ h δ z ,
γ h = 2 G ( 0 ) / π y ( 0 ) , A h = ( g 2 + s 2 y 2 1 / w 2 i ) I s ( 0 ) / I in ( 0 ) ,
B h = { g 4 + s 4 y 4 1 / w 4 i g 2 s 2 + y 2 g 2 + y 2 s 2 + ( g 2 + s 2 y 2 / w 2 i y 2 2 1 / w 2 i 2 } I s ( 0 ) / I in ( 0 ) .
w 1 / 2 = 2 2 ln 2 { 1 / w 2 i + A h γ h δ z + [ ( 1 / w 2 i + A h γ h δ z ) 2 + 4 ln 2 ( 1 / w 4 i + B h γ h δ z ) ] 1 / 2 } 1 / 2 .
w 1 / 2 = 2 2 ln 2 { 1 w 2 i + 1 w 2 i 2 + 4 ln 2 w 4 i + [ A h + A h w 2 i + ( 2 ln 2 ) B h 1 w 2 i 2 + 4 ln 2 w 4 i ] γ h δ z } 1 / 2 .
1 w 2 i s = 1 2 { ( g 2 + s 2 y 2 ) + [ ( g 2 + s 2 y 2 ) 2 4 g 4 + s 4 y 4 g 2 s 2 + y 2 g 2 + y 2 s 2 y 2 2 1 + 1 / ln 2 ] 1 / 2 } .

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