Abstract

For the thin-disk laser the increased amplification of spontaneous emission for larger disks limits the scalability. An analytical model of the influence of the amplified spontaneous emission on the effective lifetime of the excited ions is developed and with this model optimized parameters for the minimization of the lifetime reduction are found. It is shown that output powers up to the megawatt level are achievable with a single disk, but with disk dimensions far beyond the actual technical limits. The model is also used to evaluate the limits of achievable energy.

© 2008 Optical Society of America

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  1. A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
    [CrossRef]
  2. A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13, 598-609 (2007).
    [CrossRef]
  3. C. Stolzenburg, A. Voss, T. Graf, M. Larionov, and A. Giesen, “Advanced pulsed thin disk laser sources,” Proc. SPIE 6871, 68710H (2008).
    [CrossRef]
  4. D. Kouznetsov, J.-F. Bisson, J. Dong, and K.-I. Ueda, “Surface loss limit of the power scaling of a thin-disk laser,” J. Opt. Soc. Am. B 23, 1074-1082 (2006).
    [CrossRef]
  5. K. Contag, Modellierung und Numerische Auslegung des Yb:YAG-Scheibenlasers (Utz, 2002).
  6. D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22, 1165-1173 (1986).
    [CrossRef]
  7. A. Sasaki, K.-I. Ueda, H. Takuma, and K. Kasuya, “Amplified spontaneous emission in high power KrF lasers,” J. Appl. Phys. 65, 231-236 (1989).
    [CrossRef]
  8. I. Okuda and M. J. Shaw, “Gain depletion due to amplified spontaneous emission in multi-pass laser amplifiers,” Appl. Phys. B 54, 506-512 (1992).
    [CrossRef]
  9. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, 1972).
  10. D. Kouznetsov and J -F. Bisson, “Role of undoped cap in the scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25, 338-345 (2008).
    [CrossRef]
  11. C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650-657 (2000).
    [CrossRef]
  12. J. Speiser and A. Giesen, “Numerical modeling of high power continuous-wave Yb:YAG thin disk lasers, scaling to 14 kW,” in Advanced Solid-State Photonics, OSA Technical Digest Series (Optical Society of America, 2007), paper WB9.
  13. K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
    [CrossRef]
  14. U. Griebner, V. Petrov, K. Petermann, and V. Peters, “Passively mode-locked Yb:Lu2O3 laser,” Opt. Express 12, 3125-3130 (2004).
    [CrossRef] [PubMed]
  15. R. Peters, C. Kränkel, K. Petermann, and G. Huber, “Broadly tunable high-power Yb:Lu2O3 thin disk laser with 80% slope efficiency,” Opt. Express 15, 7075-7082 (2007).
    [CrossRef] [PubMed]

2008 (3)

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

C. Stolzenburg, A. Voss, T. Graf, M. Larionov, and A. Giesen, “Advanced pulsed thin disk laser sources,” Proc. SPIE 6871, 68710H (2008).
[CrossRef]

D. Kouznetsov and J -F. Bisson, “Role of undoped cap in the scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25, 338-345 (2008).
[CrossRef]

2007 (2)

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13, 598-609 (2007).
[CrossRef]

R. Peters, C. Kränkel, K. Petermann, and G. Huber, “Broadly tunable high-power Yb:Lu2O3 thin disk laser with 80% slope efficiency,” Opt. Express 15, 7075-7082 (2007).
[CrossRef] [PubMed]

2006 (1)

2004 (1)

2000 (2)

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650-657 (2000).
[CrossRef]

K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
[CrossRef]

1992 (1)

I. Okuda and M. J. Shaw, “Gain depletion due to amplified spontaneous emission in multi-pass laser amplifiers,” Appl. Phys. B 54, 506-512 (1992).
[CrossRef]

1989 (1)

A. Sasaki, K.-I. Ueda, H. Takuma, and K. Kasuya, “Amplified spontaneous emission in high power KrF lasers,” J. Appl. Phys. 65, 231-236 (1989).
[CrossRef]

1986 (1)

D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22, 1165-1173 (1986).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, 1972).

Basun, S. A.

K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
[CrossRef]

Bisson, J -F.

Bisson, J.-F.

Contag, K.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650-657 (2000).
[CrossRef]

K. Contag, Modellierung und Numerische Auslegung des Yb:YAG-Scheibenlasers (Utz, 2002).

Dong, J.

Eggleston, J. M.

D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22, 1165-1173 (1986).
[CrossRef]

Fornasiero, L.

K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
[CrossRef]

Giesen, A.

C. Stolzenburg, A. Voss, T. Graf, M. Larionov, and A. Giesen, “Advanced pulsed thin disk laser sources,” Proc. SPIE 6871, 68710H (2008).
[CrossRef]

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13, 598-609 (2007).
[CrossRef]

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650-657 (2000).
[CrossRef]

J. Speiser and A. Giesen, “Numerical modeling of high power continuous-wave Yb:YAG thin disk lasers, scaling to 14 kW,” in Advanced Solid-State Photonics, OSA Technical Digest Series (Optical Society of America, 2007), paper WB9.

Graf, T.

C. Stolzenburg, A. Voss, T. Graf, M. Larionov, and A. Giesen, “Advanced pulsed thin disk laser sources,” Proc. SPIE 6871, 68710H (2008).
[CrossRef]

Griebner, U.

Huber, G.

R. Peters, C. Kränkel, K. Petermann, and G. Huber, “Broadly tunable high-power Yb:Lu2O3 thin disk laser with 80% slope efficiency,” Opt. Express 15, 7075-7082 (2007).
[CrossRef] [PubMed]

K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
[CrossRef]

Hügel, H.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650-657 (2000).
[CrossRef]

Kasuya, K.

A. Sasaki, K.-I. Ueda, H. Takuma, and K. Kasuya, “Amplified spontaneous emission in high power KrF lasers,” J. Appl. Phys. 65, 231-236 (1989).
[CrossRef]

Killi, A.

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

Kleinbauer, J.

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

Kouznetsov, D.

Kränkel, C.

Kuch, S.

K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
[CrossRef]

Larionov, M.

C. Stolzenburg, A. Voss, T. Graf, M. Larionov, and A. Giesen, “Advanced pulsed thin disk laser sources,” Proc. SPIE 6871, 68710H (2008).
[CrossRef]

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650-657 (2000).
[CrossRef]

Lowenthal, D. D.

D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22, 1165-1173 (1986).
[CrossRef]

Mix, E.

K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
[CrossRef]

Neuhaus, J.

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

Okuda, I.

I. Okuda and M. J. Shaw, “Gain depletion due to amplified spontaneous emission in multi-pass laser amplifiers,” Appl. Phys. B 54, 506-512 (1992).
[CrossRef]

Petermann, K.

Peters, R.

Peters, V.

U. Griebner, V. Petrov, K. Petermann, and V. Peters, “Passively mode-locked Yb:Lu2O3 laser,” Opt. Express 12, 3125-3130 (2004).
[CrossRef] [PubMed]

K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
[CrossRef]

Petrov, V.

Sasaki, A.

A. Sasaki, K.-I. Ueda, H. Takuma, and K. Kasuya, “Amplified spontaneous emission in high power KrF lasers,” J. Appl. Phys. 65, 231-236 (1989).
[CrossRef]

Schad, S.

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

Schmitz, C.

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

Shaw, M. J.

I. Okuda and M. J. Shaw, “Gain depletion due to amplified spontaneous emission in multi-pass laser amplifiers,” Appl. Phys. B 54, 506-512 (1992).
[CrossRef]

Speiser, J.

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13, 598-609 (2007).
[CrossRef]

J. Speiser and A. Giesen, “Numerical modeling of high power continuous-wave Yb:YAG thin disk lasers, scaling to 14 kW,” in Advanced Solid-State Photonics, OSA Technical Digest Series (Optical Society of America, 2007), paper WB9.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, 1972).

Stewen, C.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650-657 (2000).
[CrossRef]

Stolzenburg, C.

C. Stolzenburg, A. Voss, T. Graf, M. Larionov, and A. Giesen, “Advanced pulsed thin disk laser sources,” Proc. SPIE 6871, 68710H (2008).
[CrossRef]

Sutter, D.

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

Takuma, H.

A. Sasaki, K.-I. Ueda, H. Takuma, and K. Kasuya, “Amplified spontaneous emission in high power KrF lasers,” J. Appl. Phys. 65, 231-236 (1989).
[CrossRef]

Ueda, K.-I.

D. Kouznetsov, J.-F. Bisson, J. Dong, and K.-I. Ueda, “Surface loss limit of the power scaling of a thin-disk laser,” J. Opt. Soc. Am. B 23, 1074-1082 (2006).
[CrossRef]

A. Sasaki, K.-I. Ueda, H. Takuma, and K. Kasuya, “Amplified spontaneous emission in high power KrF lasers,” J. Appl. Phys. 65, 231-236 (1989).
[CrossRef]

Voss, A.

C. Stolzenburg, A. Voss, T. Graf, M. Larionov, and A. Giesen, “Advanced pulsed thin disk laser sources,” Proc. SPIE 6871, 68710H (2008).
[CrossRef]

Zawischa, I.

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

Appl. Phys. B (1)

I. Okuda and M. J. Shaw, “Gain depletion due to amplified spontaneous emission in multi-pass laser amplifiers,” Appl. Phys. B 54, 506-512 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. D. Lowenthal and J. M. Eggleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22, 1165-1173 (1986).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13, 598-609 (2007).
[CrossRef]

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW thin disc laser,” IEEE J. Sel. Top. Quantum Electron. 6, 650-657 (2000).
[CrossRef]

J. Appl. Phys. (1)

A. Sasaki, K.-I. Ueda, H. Takuma, and K. Kasuya, “Amplified spontaneous emission in high power KrF lasers,” J. Appl. Phys. 65, 231-236 (1989).
[CrossRef]

J. Lumin. (1)

K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87, 913-975 (2000).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (2)

Proc. SPIE (2)

A. Killi, I. Zawischa, D. Sutter, J. Kleinbauer, S. Schad, J. Neuhaus, and C. Schmitz, “Current status and development trends of disk laser technology,” Proc. SPIE 6871, 68710L (2008).
[CrossRef]

C. Stolzenburg, A. Voss, T. Graf, M. Larionov, and A. Giesen, “Advanced pulsed thin disk laser sources,” Proc. SPIE 6871, 68710H (2008).
[CrossRef]

Other (3)

K. Contag, Modellierung und Numerische Auslegung des Yb:YAG-Scheibenlasers (Utz, 2002).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, 1972).

J. Speiser and A. Giesen, “Numerical modeling of high power continuous-wave Yb:YAG thin disk lasers, scaling to 14 kW,” in Advanced Solid-State Photonics, OSA Technical Digest Series (Optical Society of America, 2007), paper WB9.

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Figures (11)

Fig. 1
Fig. 1

Integration of ASE flux density without reflection at the boundaries. Hatched area represents approximation by segment of a sphere.

Fig. 2
Fig. 2

ASE flux density with reflection at the HR or AR face.

Fig. 3
Fig. 3

ASE flux density with either one reflection at the HR or AR face.

Fig. 4
Fig. 4

ASE flux density at the HR face.

Fig. 5
Fig. 5

Optimized ξ and τ ASE for cw operation.

Fig. 6
Fig. 6

Laser power as function of gain, round-trip loss β = 0.25 % , 80% pump absorption.

Fig. 7
Fig. 7

Laser power, efficiency, and pump spot radius, round-trip loss β = 0.25 % .

Fig. 8
Fig. 8

Laser power for different round-trip losses, P abs = 0.01 P abs , opt , 80% pump absorption.

Fig. 9
Fig. 9

Optimized ξ and τ ASE for fluoresence operation.

Fig. 10
Fig. 10

Extractable energy as a function of gain, round-trip loss β = 0.25 % , 80% pump absorption.

Fig. 11
Fig. 11

Maximum extractable energy for different round-trip losses.

Tables (4)

Tables Icon

Table 1 Material Parameters

Tables Icon

Table 2 Results of Power Optimization

Tables Icon

Table 3 Results of Efficiency Optimization

Tables Icon

Table 4 Results of Energy Optimization

Equations (57)

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γ = g h .
d Φ = N 2 τ 1 4 π s 2 exp ( γ s ) d V .
d Φ = N 2 τ 1 4 π r 2 exp ( γ r ) r 2 d r sin ϑ d ϑ d ϕ .
Φ = N 2 τ 1 4 π r 2 exp ( γ r ) r 2 d r sin ( ϑ ) d ϑ d ϕ = N 2 4 π τ exp ( γ r ) d r sin ϑ d ϑ d ϕ .
Φ 0 = N 2 4 π τ 0 2 π 0 Θ 0 h ( 2 cos ϑ ) exp ( γ r ) d r sin ϑ d ϑ d ϕ + N 2 4 π τ 0 2 π π Θ π 0 h ( 2 cos ϑ ) exp ( γ r ) d r sin ϑ d ϑ d ϕ + N 2 4 π τ 0 2 π Θ π Θ 0 R 2 + h 2 4 exp ( γ r ) d r sin ϑ d ϑ d ϕ = N 2 τ 0 Θ 0 h ( 2 cos ϑ ) exp ( γ r ) d r sin ϑ d ϑ + N 2 τ Θ π 2 0 R 2 + h 2 4 e γ r d r sin ϑ d ϑ = N 2 γ τ [ exp ( h γ 2 ) + h γ 2 Ei ( γ 2 4 R 2 + h 2 ) h γ 2 Ei ( h γ 2 ) 1 ] ,
Φ r = N 2 2 τ 0 α tr 0 h ( 2 cos ϑ ) exp ( γ r ) d r sin ϑ d ϑ + N 2 2 τ α tr Θ 1 0 3 h ( 2 cos ϑ ) exp ( γ r ) d r sin ϑ d ϑ + N 2 2 τ Θ 1 π 2 0 R 2 + 9 h 2 4 exp ( γ r ) d r sin ϑ d ϑ + N 2 2 τ π 2 π Θ 0 R 2 + h 2 4 exp ( γ r ) d r sin ϑ d ϑ + N 2 2 τ π Θ π α tr 0 h ( 2 cos ϑ ) exp ( γ r ) d r sin ϑ d ϑ + N 2 2 τ π α tr π 0 3 h ( 2 cos ϑ ) exp ( γ r ) d r sin ϑ d ϑ ,
Φ HR = N 2 2 τ 0 α tr 0 h cos ϑ e γ r d r sin ϑ d ϑ + N 2 2 τ α tr Θ 2 0 2 h cos ϑ e γ r d r sin ϑ d ϑ + N 2 2 τ Θ 2 π 2 0 R 2 + 4 h 2 e γ r d r sin ϑ d ϑ ,
Φ ̃ = N 2 τ 0 Θ 3 0 2 h cos ϑ e γ r d r sin ϑ d ϑ + N 2 τ Θ 3 π 2 0 4 R 2 + 4 h 2 e γ r d r sin ϑ d ϑ = N 2 γ τ [ exp ( 2 h γ ) + 2 h γ Ei ( 2 γ R 2 + h 2 ) 2 h γ Ei ( 2 h γ ) 1 ] ,
ξ = R 2 + h 2 h , R 2 = h 2 ( ξ 2 1 ) ,
Φ ̃ = N 2 γ τ [ exp ( 2 g ) + 2 g Ei ( 2 g ξ ) 2 g Ei ( 2 g ) 1 ] .
N ̇ 2 = Q N 2 τ γ Φ ̃ = Q N 2 τ N 2 τ [ exp ( 2 g ) + 2 g Ei ( 2 g ξ ) 2 g Ei ( 2 g ) 1 ] = Q N 2 τ [ exp ( 2 g ) + 2 g Ei ( 2 g ξ ) 2 g Ei ( 2 g ) ] .
τ ASE = τ exp ( 2 g ) + 2 g Ei ( 2 g ξ ) 2 g Ei ( 2 g ) ,
N 2 = Q τ ASE .
Ei ( x ) exp ( x ) x [ 1 + 1 x + 2 x 2 + 6 x 3 + ] ,
Ei ( x ) = γ Euler + ln ( x ) + n = 1 x n n n ! ,
τ τ ASE exp ( 2 g ) 2 g ( γ Euler + ln ( 2 g ) ) + exp ( 2 g ξ ) ξ 1 + 2 g ( 1 γ Euler ln ( 2 g ) ) + exp ( 2 g ξ ) ξ 1 + χ ( g ) + exp ( 2 g ξ ) ξ ,
0 < χ ( g ) < exp ( γ Euler ) 0.56
0 < g < exp ( 1 γ Euler ) 2 0.76 .
τ τ ̃ ASE = 1 + exp ( 2 g ξ ) 2 ξ ,
τ ASE τ ξ exp ( 2 g ξ ) .
g = h ( σ em , l N 2 σ abs , l N 1 ) = h ( ( σ em , l + σ abs , l ) N 2 σ abs , l N 0 ) ,
Q = P abs π R 2 h q p ,
g = P abs τ ASE π R 2 q p ( σ em , l + σ abs , l ) h σ abs , l N 0 ,
ln t = h ( σ abs , p N 0 ( σ abs , p + σ em , p ) N 2 ) .
f a , l = σ abs , l σ em , l ,
f e , p = σ em , p σ abs , p ,
g = h σ em , l ( ( 1 + f a , l ) N 2 f a , l N 0 ) ,
ln t = h σ abs , p ( N 0 ( 1 + f e , p ) N 2 ) .
N 0 = ln t h σ abs , p 1 + f a , l 1 f a , l f e , p + g h σ em , l 1 + f e , p 1 f a , l f e , p ,
g = h σ em , l N 2 ( 1 f a , l f e , p ) + ln t σ em , l σ abs , p f a , l = σ em , l P abs τ ASE π R 2 q p ( 1 f a , l f e , p ) + ln t σ em , l σ abs , p f a , l = σ em , l P abs τ ASE π R 2 q p f A ,
A = ln t σ em , l σ abs , p f a , l ,
f = 1 f a , l f e , p .
Δ T = 1 2 q p q l q p P abs π R 2 h λ th ,
P abs = π R 2 C h ,
T AR = 1 2 q p q l q p C λ th + T HR ,
T ¯ = 1 3 q p q l q p C λ th + T HR ,
P abs = π h ( ξ 2 1 ) C ,
h = P abs π C ( ξ 2 1 ) ,
π R 2 = P abs 2 π C 2 ( ξ 2 1 ) ,
R = P abs π C ξ 2 1 .
P out = q l q p T o c T o c + β ( P abs P th ) = q l q p 2 g β 2 g ( P abs P th ) ,
P th = π R 2 q p g + A τ ASE σ em , l f ,
P out = q l q p 2 g β 2 g ( P abs π R 2 q p g + A τ ASE σ em , l f ) .
P out = q l q p 2 g β 2 g ( P abs q p P abs 2 π C 2 ( ξ 2 1 ) g + A τ ASE σ em , l f ) .
P abs , opt = π C 2 ( ξ 2 1 ) τ ASE σ em , l f 2 q p ( g + A ) ,
P out , opt = q l q p 2 g β 2 g π C 2 ( ξ 2 1 ) τ ASE σ em , l f 4 q p ( g + A ) .
ξ [ ( ξ 2 1 ) τ ASE ] = 0 .
ξ 2 e 2 g + 2 ξ 2 g Ei ( 2 ξ g ) 2 ξ 2 g Ei ( 2 g ) ξ 2 g e 2 ξ g + g e 2 ξ g = 0 .
η rel = q l q p 2 g β 2 g ( 1 q p P abs π C 2 ( ξ 2 1 ) g + A τ ASE σ em , l f ) .
Q int = q l π R 2 h N 2 .
2 h σ em , l ( ( 1 + f a , l ) N 2 , th f a , l N 0 ) = β ,
Q e x = q l π R 2 h ( N 2 N 2 , th ) = q l π R 2 h ( N 2 β + 2 h σ em , l N 0 f a , l 2 h σ em , l ( 1 + f a , l ) ) = q l π R 2 2 h σ em , l ( N 2 ( 1 + f a , l ) N 0 f a , l ) β 2 σ em , l ( 1 + f a , l ) = q l π R 2 2 g β 2 σ em , l ( 1 + f a , l ) = q l π h 2 ( ξ 2 1 ) 2 g β 2 σ em , l ( 1 + f a , l ) .
g = σ em , l P abs τ ASE π R 2 q p ( 1 f a , l f e , p ) + ln t σ em , l σ abs , p f a , l = σ em , l P abs τ ASE π R 2 q p f A = σ em , l C τ ASE h q p f A .
h = σ em , l C τ ASE f q p ( g + A ) ,
Q e x = q l q p 2 π C 2 2 σ em , l f 2 1 + f a , l 2 g β ( g + A ) 2 ( ξ 2 1 ) τ ASE 2 .
ξ [ ( ξ 2 1 ) τ ASE 2 ] = 0 .
ξ 2 e 2 g + 2 ξ 2 g Ei ( 2 ξ g ) 2 ξ 2 g Ei ( 2 g ) 2 ξ 2 g e 2 ξ g + 2 g e 2 ξ g = 0 ,

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