Abstract

A theoretical analysis of a three-level fiber laser amplifier in the small-signal regime is presented in which the effects of the transverse variation of the pump and signal mode intensities, the transverse variation of the density of the lasing species, and excited-state absorption of pump photons are included. Guidelines for the design of an optimum fiber are given.

© 1988 Optical Society of America

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References

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  1. C. J. Koester, E. Snitzer, “Amplification in a Fiber Laser,” Appl. Opt. 3, 1182 (1964).
    [CrossRef]
  2. J. Stone, C. A. Burrus, “Neodymium Doped Silica Lasers in End-Pumped Fiber Geometry,” Appl. Phys. Lett. 23, 388 (1973).
    [CrossRef]
  3. S. B. Poole, D. N. Payne, M. E. Fermann, “Fabrication of Low-Loss Optical Fibres Containing Rare-Earth Ions,” Electron. Lett. 21, 737 (1985).
    [CrossRef]
  4. R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Neodymium Doped Silica Single-Mode Fibre Lasers,” Electron. Lett. 21, 738 (1985).
    [CrossRef]
  5. M. J. F. Digonnet, C. J. Gaeta, “Theoretical Analysis of Optical Fiber Laser Amplifiers and Oscillators,” Appl. Opt. 24, 333 (1985).
    [CrossRef] [PubMed]
  6. R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Low Threshold Tunable cw and Q-Switched Fibre Laser Operating at 1.5 μm,” Electron. Lett. 22, 159 (1986).
    [CrossRef]
  7. I. P. Alcock, A. I. Ferguson, D. C. Hanna, A. C. Tropper, “Continuous Wave Oscillation of a Monomode Neodymium Doped Fibre Laser at 0.9 μm on the 4F3/2 → I9/2 Transition,” Opt. Commun. 58, 405 (1986).
    [CrossRef]
  8. L. Reekie, R. J. Mears, S. B. Poole, D. N. Payne, “A Pr3+ Doped Singlemode Fibre Laser,” presented at IOP Meeting on Solid-State Lasers, Imperial College, London (1986).
  9. J. R. Armitage, C. G. Atkins, R. Wyatt, B. J. Ainslie, S. P. Craig, “Spectroscopic Studies of Er3+ Doped Singlemode Silica Fiber,” in Technical Digest of Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, DC, 1987), paper WD3.
  10. D. Gloge, “Weakly Guiding Fibers,” Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]

1986 (2)

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Low Threshold Tunable cw and Q-Switched Fibre Laser Operating at 1.5 μm,” Electron. Lett. 22, 159 (1986).
[CrossRef]

I. P. Alcock, A. I. Ferguson, D. C. Hanna, A. C. Tropper, “Continuous Wave Oscillation of a Monomode Neodymium Doped Fibre Laser at 0.9 μm on the 4F3/2 → I9/2 Transition,” Opt. Commun. 58, 405 (1986).
[CrossRef]

1985 (3)

S. B. Poole, D. N. Payne, M. E. Fermann, “Fabrication of Low-Loss Optical Fibres Containing Rare-Earth Ions,” Electron. Lett. 21, 737 (1985).
[CrossRef]

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Neodymium Doped Silica Single-Mode Fibre Lasers,” Electron. Lett. 21, 738 (1985).
[CrossRef]

M. J. F. Digonnet, C. J. Gaeta, “Theoretical Analysis of Optical Fiber Laser Amplifiers and Oscillators,” Appl. Opt. 24, 333 (1985).
[CrossRef] [PubMed]

1973 (1)

J. Stone, C. A. Burrus, “Neodymium Doped Silica Lasers in End-Pumped Fiber Geometry,” Appl. Phys. Lett. 23, 388 (1973).
[CrossRef]

1971 (1)

1964 (1)

Ainslie, B. J.

J. R. Armitage, C. G. Atkins, R. Wyatt, B. J. Ainslie, S. P. Craig, “Spectroscopic Studies of Er3+ Doped Singlemode Silica Fiber,” in Technical Digest of Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, DC, 1987), paper WD3.

Alcock, I. P.

I. P. Alcock, A. I. Ferguson, D. C. Hanna, A. C. Tropper, “Continuous Wave Oscillation of a Monomode Neodymium Doped Fibre Laser at 0.9 μm on the 4F3/2 → I9/2 Transition,” Opt. Commun. 58, 405 (1986).
[CrossRef]

Armitage, J. R.

J. R. Armitage, C. G. Atkins, R. Wyatt, B. J. Ainslie, S. P. Craig, “Spectroscopic Studies of Er3+ Doped Singlemode Silica Fiber,” in Technical Digest of Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, DC, 1987), paper WD3.

Atkins, C. G.

J. R. Armitage, C. G. Atkins, R. Wyatt, B. J. Ainslie, S. P. Craig, “Spectroscopic Studies of Er3+ Doped Singlemode Silica Fiber,” in Technical Digest of Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, DC, 1987), paper WD3.

Burrus, C. A.

J. Stone, C. A. Burrus, “Neodymium Doped Silica Lasers in End-Pumped Fiber Geometry,” Appl. Phys. Lett. 23, 388 (1973).
[CrossRef]

Craig, S. P.

J. R. Armitage, C. G. Atkins, R. Wyatt, B. J. Ainslie, S. P. Craig, “Spectroscopic Studies of Er3+ Doped Singlemode Silica Fiber,” in Technical Digest of Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, DC, 1987), paper WD3.

Digonnet, M. J. F.

Ferguson, A. I.

I. P. Alcock, A. I. Ferguson, D. C. Hanna, A. C. Tropper, “Continuous Wave Oscillation of a Monomode Neodymium Doped Fibre Laser at 0.9 μm on the 4F3/2 → I9/2 Transition,” Opt. Commun. 58, 405 (1986).
[CrossRef]

Fermann, M. E.

S. B. Poole, D. N. Payne, M. E. Fermann, “Fabrication of Low-Loss Optical Fibres Containing Rare-Earth Ions,” Electron. Lett. 21, 737 (1985).
[CrossRef]

Gaeta, C. J.

Gloge, D.

Hanna, D. C.

I. P. Alcock, A. I. Ferguson, D. C. Hanna, A. C. Tropper, “Continuous Wave Oscillation of a Monomode Neodymium Doped Fibre Laser at 0.9 μm on the 4F3/2 → I9/2 Transition,” Opt. Commun. 58, 405 (1986).
[CrossRef]

Koester, C. J.

Mears, R. J.

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Low Threshold Tunable cw and Q-Switched Fibre Laser Operating at 1.5 μm,” Electron. Lett. 22, 159 (1986).
[CrossRef]

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Neodymium Doped Silica Single-Mode Fibre Lasers,” Electron. Lett. 21, 738 (1985).
[CrossRef]

L. Reekie, R. J. Mears, S. B. Poole, D. N. Payne, “A Pr3+ Doped Singlemode Fibre Laser,” presented at IOP Meeting on Solid-State Lasers, Imperial College, London (1986).

Payne, D. N.

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Low Threshold Tunable cw and Q-Switched Fibre Laser Operating at 1.5 μm,” Electron. Lett. 22, 159 (1986).
[CrossRef]

S. B. Poole, D. N. Payne, M. E. Fermann, “Fabrication of Low-Loss Optical Fibres Containing Rare-Earth Ions,” Electron. Lett. 21, 737 (1985).
[CrossRef]

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Neodymium Doped Silica Single-Mode Fibre Lasers,” Electron. Lett. 21, 738 (1985).
[CrossRef]

L. Reekie, R. J. Mears, S. B. Poole, D. N. Payne, “A Pr3+ Doped Singlemode Fibre Laser,” presented at IOP Meeting on Solid-State Lasers, Imperial College, London (1986).

Poole, S. B.

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Low Threshold Tunable cw and Q-Switched Fibre Laser Operating at 1.5 μm,” Electron. Lett. 22, 159 (1986).
[CrossRef]

S. B. Poole, D. N. Payne, M. E. Fermann, “Fabrication of Low-Loss Optical Fibres Containing Rare-Earth Ions,” Electron. Lett. 21, 737 (1985).
[CrossRef]

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Neodymium Doped Silica Single-Mode Fibre Lasers,” Electron. Lett. 21, 738 (1985).
[CrossRef]

L. Reekie, R. J. Mears, S. B. Poole, D. N. Payne, “A Pr3+ Doped Singlemode Fibre Laser,” presented at IOP Meeting on Solid-State Lasers, Imperial College, London (1986).

Reekie, L.

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Low Threshold Tunable cw and Q-Switched Fibre Laser Operating at 1.5 μm,” Electron. Lett. 22, 159 (1986).
[CrossRef]

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Neodymium Doped Silica Single-Mode Fibre Lasers,” Electron. Lett. 21, 738 (1985).
[CrossRef]

L. Reekie, R. J. Mears, S. B. Poole, D. N. Payne, “A Pr3+ Doped Singlemode Fibre Laser,” presented at IOP Meeting on Solid-State Lasers, Imperial College, London (1986).

Snitzer, E.

Stone, J.

J. Stone, C. A. Burrus, “Neodymium Doped Silica Lasers in End-Pumped Fiber Geometry,” Appl. Phys. Lett. 23, 388 (1973).
[CrossRef]

Tropper, A. C.

I. P. Alcock, A. I. Ferguson, D. C. Hanna, A. C. Tropper, “Continuous Wave Oscillation of a Monomode Neodymium Doped Fibre Laser at 0.9 μm on the 4F3/2 → I9/2 Transition,” Opt. Commun. 58, 405 (1986).
[CrossRef]

Wyatt, R.

J. R. Armitage, C. G. Atkins, R. Wyatt, B. J. Ainslie, S. P. Craig, “Spectroscopic Studies of Er3+ Doped Singlemode Silica Fiber,” in Technical Digest of Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, DC, 1987), paper WD3.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. Stone, C. A. Burrus, “Neodymium Doped Silica Lasers in End-Pumped Fiber Geometry,” Appl. Phys. Lett. 23, 388 (1973).
[CrossRef]

Electron. Lett. (3)

S. B. Poole, D. N. Payne, M. E. Fermann, “Fabrication of Low-Loss Optical Fibres Containing Rare-Earth Ions,” Electron. Lett. 21, 737 (1985).
[CrossRef]

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Neodymium Doped Silica Single-Mode Fibre Lasers,” Electron. Lett. 21, 738 (1985).
[CrossRef]

R. J. Mears, L. Reekie, S. B. Poole, D. N. Payne, “Low Threshold Tunable cw and Q-Switched Fibre Laser Operating at 1.5 μm,” Electron. Lett. 22, 159 (1986).
[CrossRef]

Opt. Commun. (1)

I. P. Alcock, A. I. Ferguson, D. C. Hanna, A. C. Tropper, “Continuous Wave Oscillation of a Monomode Neodymium Doped Fibre Laser at 0.9 μm on the 4F3/2 → I9/2 Transition,” Opt. Commun. 58, 405 (1986).
[CrossRef]

Other (2)

L. Reekie, R. J. Mears, S. B. Poole, D. N. Payne, “A Pr3+ Doped Singlemode Fibre Laser,” presented at IOP Meeting on Solid-State Lasers, Imperial College, London (1986).

J. R. Armitage, C. G. Atkins, R. Wyatt, B. J. Ainslie, S. P. Craig, “Spectroscopic Studies of Er3+ Doped Singlemode Silica Fiber,” in Technical Digest of Topical Meeting on Tunable Solid State Lasers (Optical Society of America, Washington, DC, 1987), paper WD3.

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Figures (8)

Fig. 1
Fig. 1

Energy level diagram of a three-level laser medium.

Fig. 2
Fig. 2

Variation of R × PS with (PP/PS) for a fiber with a step dopant profile (νP = 4.5).

Fig. 3
Fig. 3

Variation of R × PS with (PP/PS) for a fiber with a Gaussian dopant profile with r1/2 = 0.25a(νP = 4.5).

Fig. 4
Fig. 4

Variation of R × PS with (PP/PS) for the LP01 mode for a series of fibers with Gaussian dopant profiles of varying halfwidths, normalized to the core radius (νP = 4.5).

Fig. 5
Fig. 5

Variation of R × PS with (PP/PS) for the LP02 mode for a series of fibers with Gaussian dopant profiles of varying halfwidths, normalized to the core radius (νP = 4.5).

Fig. 6
Fig. 6

Variation of R with PP for the LP02 mode of a fiber with a Gaussian dopant profile (r1/2 = 0.25a) for various v-values.

Fig. 7
Fig. 7

Variation of R with PP for the LP01 mode of a fiber with a Gaussian dopant profile (r1/2 = 0.25a) for various v-values.

Fig. 8
Fig. 8

Variation of R × PS with (PP/PS) for the LP01 mode of a fiber with a Gaussian dopant profile (r1/2 = 0.25a) for various values of the ratio σESA/σA(νP = 3.0).

Equations (20)

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I pump ( r , θ , z ) = I P ( z ) f ( r , θ ) , I signal ( r , θ , z ) = I 0 ( z ) g ( r ) , N T ( r , θ , z ) = N 0 n ( r ) .
N 1 ( r , θ , z ) = N T ( r , θ , z ) 1 + I pump ( r , θ , z ) / I S ,
N 2 ( r , θ , z ) = N T ( r , θ , z ) 1 + I S / I pump ( r , θ , z ) ,
P P ( z + δ z ) - P P ( z ) = 0 0 2 π [ I pump ( r , θ , z + δ z ) - I pump ( r , θ , z ) ] r d r d θ = ± 0 0 2 π N 0 n ( r ) σ A I P ( z ) f ( r , θ ) 1 + I P ( z ) f ( r , θ ) / I S δ z · r d r d θ ,
P 0 ( z + δ z ) - P 0 ( z ) = 0 0 2 π [ I signal ( r , θ , z + δ z ) - I signal ( r , θ , z ) ] r d r d θ = ± 0 0 2 π N 0 n ( r ) σ G [ I P ( z ) f ( r , θ ) / I S - 1 ] 1 + I P ( z ) f ( r , θ ) / I S × I 0 ( z ) g ( r ) δ z · r d r d θ .
R = gain per cm pump power absorbed per cm = ( α 0 ) eff ( α P ) eff P P ( z ) ,
P P ( z + δ z ) - P P ( z ) = ± ( α P ) eff P P ( z ) δ z ,
P 0 ( z + δ z ) - P 0 ( z ) = + ( α 0 ) eff P 0 ( z ) δ z .
f ( r , θ ) = F ( r ) cos 2 l θ = { J l 2 ( u r / a ) cos 2 l θ r < a , [ J l ( u ) / K l ( w ) ] 2 K l 2 ( w r / a ) cos 2 l θ r > a , g ( r ) = { J l 2 ( u r / a ) r < a , [ J 0 ( u ) / K 0 ( w ) ] 2 K 0 2 ( w r / a ) r > a ,
I P = t P ( ν P ) × P P / π a 2 ,
t P = π a 2 / 0 0 2 π F ( r ) cos 2 l θ d θ r d r .
I 0 = t 0 ( ν 0 ) × P 0 / π a 2 , t 0 = π a 2 / 0 g ( r ) 2 π r d r .
( α P ) eff P P = N 0 σ A I S 0 n ( r ) 0 2 π [ P P t P F ( r ) cos 2 l θ / P S 1 + P P t P F ( r ) cos 2 l θ / P S ] d θ r d r , ( α P ) eff = N 0 σ G t 0 π a 2 0 n ( r ) g ( r ) × 0 2 π { P P t P F ( r ) cos 2 l θ / P S 1 + P P t P F ( r ) cos 2 l θ / P S } d θ r d r ,
( α P ) eff P P = N 0 σ A I S 0 n ( r ) A ( r ) 2 π r d r , ( α 0 ) eff = N 0 σ G t 0 / π a 2 0 n ( r ) g ( r ) { 2 A ( r ) - 1 } 2 π r d r ,
A ( r ) = { 1 - ( 1 + a ) - 1 l = 0 , 1 - ( 1 + a ) - 1 / 2 l 0 , a = ( P P s / P S ) × t P F ( r ) .
R = ( α 0 ) eff ( α A ) eff P P = σ G σ A P S × [ t 0 × 0 n ( r ) g ( r ) { 2 A ( r ) - 1 } r d r 0 n ( r ) A ( r ) d r ] .
P P ( z + δ z ) - P P ( z ) = ± 0 0 2 π N 0 n ( r ) [ σ A + σ ESA I P ( z ) f ( r , θ ) / I S ] I P ( z ) f ( r , θ ) 1 + I P ( z ) f ( r , θ ) / I S × δ z · r d r d θ ,
( α P ) eff P P = N 0 σ A I S 0 n ( r ) 0 2 π × [ [ 1 + ( σ ESA / σ A ) P P t P F ( r ) cos 2 l θ / P S ] P P t P F ( r ) cos 2 l θ / P S 1 + P P t P F ( r ) cos 2 l θ / P S ] × d θ r d r .
d P P d z = ± ( α P ) eff P P , d P 0 d z = + ( α 0 ) eff P 0 .
Gain = z = 0 z = L ( α 0 ) eff d z = P ( z = 0 ) P ( z = L ) ( α 0 ) eff ( d z d P P ) d P P = P ( z = 0 ) P ( z = L ) ( α 0 ) eff ( d z d P P ) d P P = P OUT P IN [ ( α 0 ) eff ( α P ) eff P P ] d P P = P OUT P IN R ( P P ) d P P = area under the R vs P P curves between P IN and P OUT .

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