Abstract

An analytical theory is developed for a four-level cw fiber laser which accounts for saturation of the output due to bleaching of the pump transition. A simple closed form expression is obtained which relates the pump power and output power. Output saturation is most important for systems having at least one long lived excited state other than the upper laser level. The model is used to optimize the fiber length, core radius, and mirror reflectivities for an Er3+ doped fluorozirconate fiber laser operating at 2.7 μm. The optimized efficiency is found to be independent of pump power. Maximum efficiencies of 9.8 and 12.9% are calculated for fibers with attenuation coefficients of 100 and 10 dB/km, respectively.

© 1990 Optical Society of America

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References

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  1. L. M. Frantz, J. S. Nodvik, “Theory of Pulse Propagation in a Laser Amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
    [CrossRef]
  2. W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
    [CrossRef]
  3. M. J. F. Digonnet, C. J. Gaeta, “Theoretical Analysis of Optical Fiber Laser Amplifiers and Oscillators,” Appl. Opt. 24, 333–342 (1985).
    [CrossRef] [PubMed]
  4. T. Y. Fan, R. L. Byer, “Modeling and cw operation of a Quasi-Three-Level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron QE-23, 605–612 (1987).
  5. W. P. Risk, “Modeling of Longitudinally Pumped Solid-State Lasers Exhibiting Reabsorption Losses,” J. Opt. Soc. Am. B 5, 1412–1423 (1988).
    [CrossRef]
  6. J. R. Armitage, “Three-Level Fiber Laser Amplifier: a Theoretical Model,” Appl. Opt. 27, 4831–4836 (1988).
    [CrossRef] [PubMed]
  7. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), pp. 154–156.
  8. R. I. Laming, S. B. Poole, E. J. Tarbox, “Pump Excited-State Absorption in Erbium-Doped Fibers,” Opt. Lett. 13, 1084–1086 (1988).
    [CrossRef] [PubMed]
  9. R. S. Quimby, W. J. Miniscalco, “Continuous-Wave Lasing on a Self-Terminating Transition,” Appl. Opt. 28, 14–16 (1989).
    [CrossRef] [PubMed]

1989 (1)

1988 (3)

1987 (1)

T. Y. Fan, R. L. Byer, “Modeling and cw operation of a Quasi-Three-Level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron QE-23, 605–612 (1987).

1985 (1)

1965 (1)

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

1963 (1)

L. M. Frantz, J. S. Nodvik, “Theory of Pulse Propagation in a Laser Amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

Armitage, J. R.

Byer, R. L.

T. Y. Fan, R. L. Byer, “Modeling and cw operation of a Quasi-Three-Level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron QE-23, 605–612 (1987).

Digonnet, M. J. F.

Fan, T. Y.

T. Y. Fan, R. L. Byer, “Modeling and cw operation of a Quasi-Three-Level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron QE-23, 605–612 (1987).

Frantz, L. M.

L. M. Frantz, J. S. Nodvik, “Theory of Pulse Propagation in a Laser Amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

Gaeta, C. J.

Laming, R. I.

Miniscalco, W. J.

Nodvik, J. S.

L. M. Frantz, J. S. Nodvik, “Theory of Pulse Propagation in a Laser Amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

Poole, S. B.

Quimby, R. S.

Rigrod, W. W.

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

Risk, W. P.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), pp. 154–156.

Tarbox, E. J.

Appl. Opt. (3)

IEEE J. Quantum Electron (1)

T. Y. Fan, R. L. Byer, “Modeling and cw operation of a Quasi-Three-Level 946 nm Nd:YAG laser,” IEEE J. Quantum Electron QE-23, 605–612 (1987).

J. Appl. Phys. (2)

L. M. Frantz, J. S. Nodvik, “Theory of Pulse Propagation in a Laser Amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

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

Opt. Lett. (1)

Other (1)

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), pp. 154–156.

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Figures (11)

Fig. 1
Fig. 1

Energy level diagram for lasing on the 5 → 3 transition, pumping (a) level 7 and (b) level 5.

Fig. 2
Fig. 2

Simplified lasing scheme, pumping (a) level 4 and (b) level 3. The corresponding levels for Er3+ are shown on the right.

Fig. 3
Fig. 3

Variation of the two countercirculating intensities with position along the fiber. The sum of the two is much more constant than either component alone.

Fig. 4
Fig. 4

Fiber laser output power vs incident pump power, for different values of the lifetime τ2. R c = 10 μm and L = 10 cm. The bottom four curves assume degenerate pumping of level 3 while the top curve assumes nondegenerate pumping (or equivalently, χ p = 0).

Fig. 5
Fig. 5

Fiber laser output power vs incident pump power, for different values of W32. τ2 = 9.5 ms, R c = 10 μm and L = 10 cm. Low values of W32 are needed for efficient cw lasing.

Fig. 6
Fig. 6

Fiber laser output power vs incident pump power, for different fiber lengths. R c = 10 μm, W32 = 20 s−1, and τ2 = 9.5 ms.

Fig. 7
Fig. 7

Fiber laser output power vs incident pump power, for different fiber lengths. For higher powers, the optimum fiber length depends on pump power level.

Fig. 8
Fig. 8

Fiber laser output power vs incident pump power, for different fiber core radii. L = 10 cm. Optimum core radius depends on pump power.

Fig. 9
Fig. 9

Same as Fig. 8, for higher pump powers.

Fig. 10
Fig. 10

Optimum fiber length vs fiber core radius, for different output powers. T = 2%, α = 100 dB/km.

Fig. 11
Fig. 11

Minimum input power needed to produce a given output power, vs fiber core radius. Low point of each curve optimizes both L and R c . T = 2%, α = 100 dB/km.

Tables (1)

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Table I Summary of Defined Parameters for Use In Eq. (43)

Equations (47)

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N 1 W p = N 7 ( χ p W p + W 7 ) ,
N 7 W 76 = N 6 W 6 ,
N 7 W 75 + N 6 W 65 + N 3 W 35 ind = N 5 ( W 5 + W 53 ind ) ,
N 7 W 74 + N 6 W 64 + N 5 W 54 = N 4 W 4 ,
N 7 W 73 + N 6 W 63 + N 5 ( W 53 + W 53 ind ) + N 4 W 43 = N 3 ( W 3 + W 35 ind ) ,
N 7 W 72 + N 6 W 62 + N 5 W 52 + N 4 W 42 + N 3 W 32 = N 2 W 2 ,
N o = N 1 + N ex ,
N ex = i = 2 7 N i ,
W p = I p σ p h ν p ,
χ W 35 ind W 53 ind = g 5 g 3 .
b i j = k = j i - 1 β i k b k j ,
b 76 = β 76 ,
b 75 = β 75 + β 76 β 65 ,
b 74 = β 74 + β 75 β 54 + β 76 b 64 ,
b 64 = β 64 + β 65 β 54 .
N 7 = R p τ 7 ,
N 6 = R p b 76 τ 6 ,
N 5 = R p b 75 τ 5 - τ 5 W ind Δ N ,
N 4 = R p b 74 τ 4 - β 54 τ 4 W ind Δ N ,
N 3 = R p b 73 τ 3 + ( 1 - b 53 ) τ 3 W ind Δ N ,
N 2 = R p b 72 τ 2 + ( β 32 - b 52 ) τ 2 W ind Δ N ,
R p N 1 W p 1 + χ p W p τ 7 .
Δ N = R p τ a 1 + W ind τ b ,
τ a = b 75 τ 5 - χ b 73 τ 3 ,
τ b = τ 5 + χ τ 3 ( 1 - b 53 ) .
N 1 = N o 1 + χ p W p τ 7 1 + W p ( τ ex * + χ p τ 7 ) ,
τ ex * τ ex [ 1 + a 2 s 1 + s ] ,
a 2 1 - τ a τ c τ ex τ b ,
τ c τ e - τ 3 - β 32 τ 2 ,
τ e i = 2 5 b 5 i τ i ,
τ ex i = 2 7 b 7 i τ i ,
s W ind τ b = I I sat ,
I sat h ν σ se τ b .
d I p = - I p ( N 1 - χ p N 7 ) σ p d x ,
d I p = - I p N o σ p 1 + W p ( τ ex * + χ p τ 7 ) d x .
l n [ I p I p o ] + σ p ( τ ex * + χ p τ 7 ) h ν p ( I p - I p o ) = - α p L ,
d I + ( x ) = I + ( x ) γ ( x ) d x ,
I + ( L ) = I + ( 0 ) e G ,
G 0 L γ ( x ) d x .
R 1 R 2 e 2 ( G th - α L ) = 1
Δ N ( x ) = - τ a h ν p ( 1 + s ) d I p d x .
G = τ a σ se h ν p ( 1 + s ) ( I p o - I p ) .
P in = G th h ν p ( 1 + s ) A c τ a σ s e 1 1 - exp [ - α p L + σ p σ s e a 3 G th ( 1 + a 4 s ) ] ,
a 3 τ e x + χ p τ 7 τ a ,
a 4 a 2 τ ex + χ p τ 7 τ ex + χ p τ 7 .
G th = α L + l n ( R 1 R 2 ) - 1 / 2 .
P out = ½ T s I sat A c ,

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