Abstract

The theory of homogeneously broadened four level fiber lasers, which use fiber loops as distributed reflective elements, is examined. Such cavities can be made entirely from rare earth doped fiber. The amplifying characteristics of doped fiber loops are examined. The threshold pump power and the loop reflectivity necessary to optimize the lasing output power from an oscillator formed from two loops in series are predicted.

© 1989 Optical Society of America

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References

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  1. M. Shimitzu, H. Suda, M. Horiguchi, “High Efficiency Nd-Doped Fibre Lasers Using Direct-Coated Dielectric Mirrors,” Electron. Lett. 23, 768–769 (1987).
    [CrossRef]
  2. R. Wyatt, B. J. Ainslie, S. P. Craig, “Efficient Operation of an Array-Pumped Er3+ Doped Silica Fibre Laser at 1.5 μm,”Electron. Lett. 22, 1362–1363 (1988).
    [CrossRef]
  3. M. C. Brierley, P. W. France, “Neodymium-Doped Fluoro-Zirconate Fibre Laser,” Electron. Lett. 23, 815–817 (1987).
    [CrossRef]
  4. E. Desurvire, J. R. Simpson, P. C. Becker, “High-Gain Erbium-Doped Traveling-wave Amplifier,” Opt. Lett. 12, 888–890 (1987).
    [CrossRef] [PubMed]
  5. T. J. Whitley, “Laser-Diode Pumped Operation of Er3+-Doped Fibre Amplifier,” Electron. Lett. 24, 1537–1539 (1988).
    [CrossRef]
  6. P. Urquhart, “Review of Rare Earth Doped Fibre Lasers and Amplifiers,” Proc. IEE (Part J Optoelectronics) 134, 385–407 (1988).
    [CrossRef]
  7. P. Barnsley, P. Urquhart, C. Millar, M. Brierley, “Fiber Fox-Smith Resonators: Application to Single Longitudinal Mode Operation of Fiber Lasers,” J. Opt. Soc. Am. A 5, 1339–1345 (1988),
    [CrossRef]
  8. I. M. Jauncey, L. Reekie, J. E. Townsend, D. Payne, C. J. Rowe, “Single Longitudinal Mode Operation of Nd3+-Doped Fibre Laser,” Electron. Lett. 24, 24–26 (1988).
    [CrossRef]
  9. I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-Doped cw Fiber Laser Using All-Fiber Reflectors,” Appl. Opt. 26, 2197–2201 (1987).
    [CrossRef] [PubMed]
  10. C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, P. Urquhart, “Fibre Laser with Adjustable Fibre Output for Wavelength Tuning and Variable Output Coupling,” Proc. IEE (Part J Optoelectronics) 135, 303–309 (1988).
    [CrossRef]
  11. W. W. Rigrod, “Gain Saturation and Output Power of Optical Masers,” J. Appl. Phys. 34, 2602–2609 (1963).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA1986), see especially chapters 8 and 12.
  13. D. B. Mortimore, “Fiber Loop Reflectors,” IEEE/OSA J. Lightwave Technol. LT-6, 1217–1224 (1988).
    [CrossRef]
  14. M. J. F. Digonnet, “Theory of Superfluorescent Fibre Lasers,” IEEE/OSA J. Lightwave Technol. LT-4, 1631–1639 (1986).
    [CrossRef]
  15. J. R. Armitage, “Three-Level Fiber Laser Amplifier: A Theoretical Model,” Appl. Opt. 27, 4831–4836 (1988).
    [CrossRef] [PubMed]
  16. D. Gloge, “Weakly Guiding Fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  17. W. W. Rigord, “Homogeneously Broadened CW Lasers with Uniform Distributed Loss,” IEEE J. Quantum Electron. QE-14, 377–381 (1978).
    [CrossRef]

1988 (8)

T. J. Whitley, “Laser-Diode Pumped Operation of Er3+-Doped Fibre Amplifier,” Electron. Lett. 24, 1537–1539 (1988).
[CrossRef]

P. Urquhart, “Review of Rare Earth Doped Fibre Lasers and Amplifiers,” Proc. IEE (Part J Optoelectronics) 134, 385–407 (1988).
[CrossRef]

I. M. Jauncey, L. Reekie, J. E. Townsend, D. Payne, C. J. Rowe, “Single Longitudinal Mode Operation of Nd3+-Doped Fibre Laser,” Electron. Lett. 24, 24–26 (1988).
[CrossRef]

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, P. Urquhart, “Fibre Laser with Adjustable Fibre Output for Wavelength Tuning and Variable Output Coupling,” Proc. IEE (Part J Optoelectronics) 135, 303–309 (1988).
[CrossRef]

R. Wyatt, B. J. Ainslie, S. P. Craig, “Efficient Operation of an Array-Pumped Er3+ Doped Silica Fibre Laser at 1.5 μm,”Electron. Lett. 22, 1362–1363 (1988).
[CrossRef]

D. B. Mortimore, “Fiber Loop Reflectors,” IEEE/OSA J. Lightwave Technol. LT-6, 1217–1224 (1988).
[CrossRef]

P. Barnsley, P. Urquhart, C. Millar, M. Brierley, “Fiber Fox-Smith Resonators: Application to Single Longitudinal Mode Operation of Fiber Lasers,” J. Opt. Soc. Am. A 5, 1339–1345 (1988),
[CrossRef]

J. R. Armitage, “Three-Level Fiber Laser Amplifier: A Theoretical Model,” Appl. Opt. 27, 4831–4836 (1988).
[CrossRef] [PubMed]

1987 (4)

E. Desurvire, J. R. Simpson, P. C. Becker, “High-Gain Erbium-Doped Traveling-wave Amplifier,” Opt. Lett. 12, 888–890 (1987).
[CrossRef] [PubMed]

I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-Doped cw Fiber Laser Using All-Fiber Reflectors,” Appl. Opt. 26, 2197–2201 (1987).
[CrossRef] [PubMed]

M. Shimitzu, H. Suda, M. Horiguchi, “High Efficiency Nd-Doped Fibre Lasers Using Direct-Coated Dielectric Mirrors,” Electron. Lett. 23, 768–769 (1987).
[CrossRef]

M. C. Brierley, P. W. France, “Neodymium-Doped Fluoro-Zirconate Fibre Laser,” Electron. Lett. 23, 815–817 (1987).
[CrossRef]

1986 (1)

M. J. F. Digonnet, “Theory of Superfluorescent Fibre Lasers,” IEEE/OSA J. Lightwave Technol. LT-4, 1631–1639 (1986).
[CrossRef]

1978 (1)

W. W. Rigord, “Homogeneously Broadened CW Lasers with Uniform Distributed Loss,” IEEE J. Quantum Electron. QE-14, 377–381 (1978).
[CrossRef]

1971 (1)

1963 (1)

W. W. Rigrod, “Gain Saturation and Output Power of Optical Masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[CrossRef]

Ainslie, B. J.

R. Wyatt, B. J. Ainslie, S. P. Craig, “Efficient Operation of an Array-Pumped Er3+ Doped Silica Fibre Laser at 1.5 μm,”Electron. Lett. 22, 1362–1363 (1988).
[CrossRef]

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, P. Urquhart, “Fibre Laser with Adjustable Fibre Output for Wavelength Tuning and Variable Output Coupling,” Proc. IEE (Part J Optoelectronics) 135, 303–309 (1988).
[CrossRef]

I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-Doped cw Fiber Laser Using All-Fiber Reflectors,” Appl. Opt. 26, 2197–2201 (1987).
[CrossRef] [PubMed]

Armitage, J. R.

Barnsley, P.

Becker, P. C.

Brierley, M.

Brierley, M. C.

M. C. Brierley, P. W. France, “Neodymium-Doped Fluoro-Zirconate Fibre Laser,” Electron. Lett. 23, 815–817 (1987).
[CrossRef]

Craig, S. P.

R. Wyatt, B. J. Ainslie, S. P. Craig, “Efficient Operation of an Array-Pumped Er3+ Doped Silica Fibre Laser at 1.5 μm,”Electron. Lett. 22, 1362–1363 (1988).
[CrossRef]

I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-Doped cw Fiber Laser Using All-Fiber Reflectors,” Appl. Opt. 26, 2197–2201 (1987).
[CrossRef] [PubMed]

Desurvire, E.

Digonnet, M. J. F.

M. J. F. Digonnet, “Theory of Superfluorescent Fibre Lasers,” IEEE/OSA J. Lightwave Technol. LT-4, 1631–1639 (1986).
[CrossRef]

France, P. W.

M. C. Brierley, P. W. France, “Neodymium-Doped Fluoro-Zirconate Fibre Laser,” Electron. Lett. 23, 815–817 (1987).
[CrossRef]

Gloge, D.

Horiguchi, M.

M. Shimitzu, H. Suda, M. Horiguchi, “High Efficiency Nd-Doped Fibre Lasers Using Direct-Coated Dielectric Mirrors,” Electron. Lett. 23, 768–769 (1987).
[CrossRef]

Jauncey, I. M.

I. M. Jauncey, L. Reekie, J. E. Townsend, D. Payne, C. J. Rowe, “Single Longitudinal Mode Operation of Nd3+-Doped Fibre Laser,” Electron. Lett. 24, 24–26 (1988).
[CrossRef]

Millar, C.

Millar, C. A.

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, P. Urquhart, “Fibre Laser with Adjustable Fibre Output for Wavelength Tuning and Variable Output Coupling,” Proc. IEE (Part J Optoelectronics) 135, 303–309 (1988).
[CrossRef]

I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-Doped cw Fiber Laser Using All-Fiber Reflectors,” Appl. Opt. 26, 2197–2201 (1987).
[CrossRef] [PubMed]

Miller, I. D.

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, P. Urquhart, “Fibre Laser with Adjustable Fibre Output for Wavelength Tuning and Variable Output Coupling,” Proc. IEE (Part J Optoelectronics) 135, 303–309 (1988).
[CrossRef]

I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-Doped cw Fiber Laser Using All-Fiber Reflectors,” Appl. Opt. 26, 2197–2201 (1987).
[CrossRef] [PubMed]

Mortimore, D. B.

D. B. Mortimore, “Fiber Loop Reflectors,” IEEE/OSA J. Lightwave Technol. LT-6, 1217–1224 (1988).
[CrossRef]

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, P. Urquhart, “Fibre Laser with Adjustable Fibre Output for Wavelength Tuning and Variable Output Coupling,” Proc. IEE (Part J Optoelectronics) 135, 303–309 (1988).
[CrossRef]

I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-Doped cw Fiber Laser Using All-Fiber Reflectors,” Appl. Opt. 26, 2197–2201 (1987).
[CrossRef] [PubMed]

Payne, D.

I. M. Jauncey, L. Reekie, J. E. Townsend, D. Payne, C. J. Rowe, “Single Longitudinal Mode Operation of Nd3+-Doped Fibre Laser,” Electron. Lett. 24, 24–26 (1988).
[CrossRef]

Payne, D. B.

Reekie, L.

I. M. Jauncey, L. Reekie, J. E. Townsend, D. Payne, C. J. Rowe, “Single Longitudinal Mode Operation of Nd3+-Doped Fibre Laser,” Electron. Lett. 24, 24–26 (1988).
[CrossRef]

Rigord, W. W.

W. W. Rigord, “Homogeneously Broadened CW Lasers with Uniform Distributed Loss,” IEEE J. Quantum Electron. QE-14, 377–381 (1978).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, “Gain Saturation and Output Power of Optical Masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[CrossRef]

Rowe, C. J.

I. M. Jauncey, L. Reekie, J. E. Townsend, D. Payne, C. J. Rowe, “Single Longitudinal Mode Operation of Nd3+-Doped Fibre Laser,” Electron. Lett. 24, 24–26 (1988).
[CrossRef]

Shimitzu, M.

M. Shimitzu, H. Suda, M. Horiguchi, “High Efficiency Nd-Doped Fibre Lasers Using Direct-Coated Dielectric Mirrors,” Electron. Lett. 23, 768–769 (1987).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA1986), see especially chapters 8 and 12.

Simpson, J. R.

Suda, H.

M. Shimitzu, H. Suda, M. Horiguchi, “High Efficiency Nd-Doped Fibre Lasers Using Direct-Coated Dielectric Mirrors,” Electron. Lett. 23, 768–769 (1987).
[CrossRef]

Townsend, J. E.

I. M. Jauncey, L. Reekie, J. E. Townsend, D. Payne, C. J. Rowe, “Single Longitudinal Mode Operation of Nd3+-Doped Fibre Laser,” Electron. Lett. 24, 24–26 (1988).
[CrossRef]

Urquhart, P.

P. Urquhart, “Review of Rare Earth Doped Fibre Lasers and Amplifiers,” Proc. IEE (Part J Optoelectronics) 134, 385–407 (1988).
[CrossRef]

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, P. Urquhart, “Fibre Laser with Adjustable Fibre Output for Wavelength Tuning and Variable Output Coupling,” Proc. IEE (Part J Optoelectronics) 135, 303–309 (1988).
[CrossRef]

P. Barnsley, P. Urquhart, C. Millar, M. Brierley, “Fiber Fox-Smith Resonators: Application to Single Longitudinal Mode Operation of Fiber Lasers,” J. Opt. Soc. Am. A 5, 1339–1345 (1988),
[CrossRef]

I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-Doped cw Fiber Laser Using All-Fiber Reflectors,” Appl. Opt. 26, 2197–2201 (1987).
[CrossRef] [PubMed]

Whitley, T. J.

T. J. Whitley, “Laser-Diode Pumped Operation of Er3+-Doped Fibre Amplifier,” Electron. Lett. 24, 1537–1539 (1988).
[CrossRef]

Wyatt, R.

R. Wyatt, B. J. Ainslie, S. P. Craig, “Efficient Operation of an Array-Pumped Er3+ Doped Silica Fibre Laser at 1.5 μm,”Electron. Lett. 22, 1362–1363 (1988).
[CrossRef]

Appl. Opt. (3)

Electron. Lett. (5)

M. Shimitzu, H. Suda, M. Horiguchi, “High Efficiency Nd-Doped Fibre Lasers Using Direct-Coated Dielectric Mirrors,” Electron. Lett. 23, 768–769 (1987).
[CrossRef]

R. Wyatt, B. J. Ainslie, S. P. Craig, “Efficient Operation of an Array-Pumped Er3+ Doped Silica Fibre Laser at 1.5 μm,”Electron. Lett. 22, 1362–1363 (1988).
[CrossRef]

M. C. Brierley, P. W. France, “Neodymium-Doped Fluoro-Zirconate Fibre Laser,” Electron. Lett. 23, 815–817 (1987).
[CrossRef]

T. J. Whitley, “Laser-Diode Pumped Operation of Er3+-Doped Fibre Amplifier,” Electron. Lett. 24, 1537–1539 (1988).
[CrossRef]

I. M. Jauncey, L. Reekie, J. E. Townsend, D. Payne, C. J. Rowe, “Single Longitudinal Mode Operation of Nd3+-Doped Fibre Laser,” Electron. Lett. 24, 24–26 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. W. Rigord, “Homogeneously Broadened CW Lasers with Uniform Distributed Loss,” IEEE J. Quantum Electron. QE-14, 377–381 (1978).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (2)

D. B. Mortimore, “Fiber Loop Reflectors,” IEEE/OSA J. Lightwave Technol. LT-6, 1217–1224 (1988).
[CrossRef]

M. J. F. Digonnet, “Theory of Superfluorescent Fibre Lasers,” IEEE/OSA J. Lightwave Technol. LT-4, 1631–1639 (1986).
[CrossRef]

J. Appl. Phys. (1)

W. W. Rigrod, “Gain Saturation and Output Power of Optical Masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Proc. IEE (Part J Optoelectronics) (2)

C. A. Millar, I. D. Miller, D. B. Mortimore, B. J. Ainslie, P. Urquhart, “Fibre Laser with Adjustable Fibre Output for Wavelength Tuning and Variable Output Coupling,” Proc. IEE (Part J Optoelectronics) 135, 303–309 (1988).
[CrossRef]

P. Urquhart, “Review of Rare Earth Doped Fibre Lasers and Amplifiers,” Proc. IEE (Part J Optoelectronics) 134, 385–407 (1988).
[CrossRef]

Other (1)

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA1986), see especially chapters 8 and 12.

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Figures (7)

Fig. 1
Fig. 1

Fiber loop reflector showing the input signal and the transmitted and reflected amplified output signal.

Fig. 2
Fig. 2

Variation of signal output intensity with respect to signal input intensity. The loop reflectivity is held at R = ½. The gain values are G = (a) 0, (b) 1, (c) 2, (d) 3, (e) 4, (f) 5.

Fig. 3
Fig. 3

Variation of signal output intensity with respect to the splitting ratio of the coupler. The input intensity is held at half the saturation level. The gain values are G = (a) 0, (b) 1, (c) 2, (d) 3, (e) 4, (f) 5.

Fig. 4
Fig. 4

Fiber laser resonator formed from two loop reflectors in series. The points marked 1–8 form the subscripts on the intensity terms used to calculate the lasing performance.

Fig. 5
Fig. 5

Variation of extraction efficiency with respect to reflectivity of the output loop, R l 2. The input loop has a reflectivity, R1 = 95%. The gain values, G tot are: (a) 0.2, (b) 0.5, (c) 1.0, (d) 2.0 and (e) 5.0. In all cases the values of G i for the individual cavity segments are G1 = ½G tot , G2 = ⅓G tot and G3 = ⅙G tot . The coupler losses, γ l 1 = γ l 2 = 0.01.

Fig. 6
Fig. 6

Variation of the output intensity I 8 a with respect to the gain G tot when the input loop has reflectivity R l 1 = 95% and the output loop has reflectivity, R l 2 = (a) 80.7% and (b) 54.1%. The coupler losses are γ l 1 = γ l 2 = 0.01. The effective gain, G eff = ⅔G tot . The markers are the output intensity values indicated from the curves of Fig. 5 when R l 2 = (a) 80.7% and (b) 54.1%, respectively.

Fig. 7
Fig. 7

Variation of signal output power with respect to splitting ratio of the coupler. The input power is held at half the saturation level. The fibre fields are taken to satisfy the weakly guiding analysis. The values of the parameters used in computation are given in the text of the Appendix.

Equations (45)

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E 2 a = ( 1 K l ) 1 / 2 ( 1 γ l ) 1 / 2 E 1 t ,
E 3 a = j K l 1 / 2 ( 1 γ l ) 1 / 2 E 1 t .
E 1 a = ( 1 K l ) 1 / 2 ( 1 γ l ) 1 / 2 E 2 t + j K l 1 / 2 ( 1 γ l ) 1 / 2 E 3 t ,
E 4 a = ( 1 K l ) 1 / 2 ( 1 γ l ) 1 / 2 E 3 t + j K l 1 / 2 ( 1 γ l ) 1 / 2 E 2 t .
d E ± d z = ± [ α l ( z ) 2 + j β l ( z ) ] E ± .
E 2 t E 2 a = E 3 t E 3 a = C .
T l = ( 1 2 K l ) 2 ( 1 γ l ) 2 exp ( α l L ) ,
R l = 4 K l ( 1 K l ) ( 1 γ l ) 2 exp ( α l L ) .
I 4 a = ( 1 2 K l ) 2 K l ( 1 γ l ) I 2 t = ( 1 2 K l ) 2 ( 1 K l ) ( 1 γ l ) I 3 t .
I 1 a = 4 ( 1 K l ) ( 1 γ l ) I 2 t = 4 K l ( 1 γ l ) I 3 t .
I 2 a = ( 1 K l ) ( 1 γ l ) I 1 t .
I 3 a = K l ( 1 γ l ) I 1 t .
d I ± d z = ± g l I ± [ 1 + y 2 + ( I + + I ) ] α l I ± .
I ± = I ± / I s ,
I s = ( h ν l σ l τ ) ;
y = ( ν ν l ) δ ν / 2 .
g l ( z ) = ( σ l τ h ν p ) α p I p 1 exp [ ( α i + α p ) z ] .
I a b s = 0 L α p I p 1 exp [ ( α i + α p ) z ] d z , = I p 1 ( α p α p + α i ) { 1 exp [ ( α p + α i ) L ] } .
I t r a n s a b s = T p I a b s ,
I r e f a b s = R p I a b s ;
I t o t a l a b s = ( 1 γ p ) 2 I a b s .
| C | 2 = I + I = I 2 a I 2 t = I 3 a I 3 t ;
( 1 1 γ l ) ln [ I o u t X l I 1 t ] + ( I o u t X l ) I 1 t = G 0 ( 1 γ l ) ;
G 0 = ( σ l τ h ν p ) I t o t a l a b s .
R l I 1 a = T l I 4 a ,
K l = [ 2 ± ( 4 2 ( 1 γ l ) 2 ) ] / 4 0 . 143 , 0 . 857
4 I 2 t I 2 a = 4 I 3 t I 3 a = I 4 t I 4 a = I 5 t I 5 a = 4 I 6 t I 6 a = 4 I 7 t I 7 a
( 1 1 γ l 1 ) ln [ I 4 a R l 1 I 4 t ] + I 4 a R l 1 I 4 t = G 1 ( 1 γ l 1 ) ,
ln [ I 4 t I 5 a ] + ( I 5 t I 5 a ) + ( I 4 t I 4 a ) = G 2 ,
( 1 1 γ l 2 ) ln [ I 5 a R l 2 I 5 t ] + I 5 a R l 2 I 5 t = G 3 ( 1 γ l 2 ) ,
I 4 t I 4 a = I 5 t I 5 a .
I 1 a = ( T l 1 R l 1 ) I 4 a
I 8 a = ( T l 2 R l 2 ) I 5 a
G 1 = I p 1 ( 1 γ p 1 ) 2 ( σ l τ h ν p ) ( α p α p + α i ) { 1 exp [ ( α p + α i ) L 1 ] } ,
G 2 = I p 1 ( 1 γ p 1 ) 2 ( σ l τ h ν p ) ( α p α p + α i ) { exp [ ( α p + α i ) L 1 ] exp [ ( α p + α i ) ( L 1 + L 2 ) ] } ,
G 3 = I p 1 ( 1 γ p 1 ) 2 ( 1 γ p 2 ) 2 ( σ l τ h ν p ) ( α p α p + α i ) × { exp [ ( α p + α i ) ( L 1 + L 2 ) ] exp [ ( α p + α i ) ( L 1 + L 2 + L 3 ) ] } .
2 G e f f = ( G 1 + 2 G 2 + G 3 ) = ln [ 1 R l 1 R l 2 ] + ( 1 + γ l 1 ) I 4 t + ( 1 + γ l 2 ) I 5 t + [ ( 1 γ l 1 ) R l 1 2 ] I 4 a + [ ( 1 γ l 2 ) R l 2 2 ] I 5 a
G e f f + 1 2 ln [ R l 1 R l 2 ] = 0 .
( L 1 + 2 L 2 + L 3 ) g + 1 2 ln [ R l 1 R l 2 ] 0
G t o t = G 1 + G 2 + G 3 I p 1 ( σ l τ h ν p ) ( 1 γ p 1 ) 2 ( α p α p + α i ) × { 1 ( 1 γ p 2 ) 2 exp [ ( α p + α i ) ( L 1 + L 2 + L 3 ) ] }
I a v a i l = G t o t
I p ± ( r , z ) = ψ p , l ( r ) P p , ± ( z ) P p , ± ( z ) J 0 2 ( u p , l r a ) 0 < r < a
d P ± d z = ± 2 π 0 a Γ l ( z ) P ± ( z ) ψ p ( r ) ψ l ( r ) r d r 1 + m [ P + ( z ) + P ( z ) ] ψ l ( r ) .
P ± = P ± / P S .
Γ l ( z ) = 2 π 0 a g l ( r , z ) r d r = ( σ l τ h ν p ) α p P p 1 exp [ ( α i + α p ) z ] ,

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