Abstract

Erbium-doped-fiber lasers with the laser transition and pump absorption sharing common energy bands are studied theoretically by using rate equations based on the interaction between a two-level atom and a two-wavelength field. Explicit and analytical expressions are established for the threshold pump power, output power, and slope efficiency of the laser oscillator under steady-state conditions. Constraints on the choice of the fiber length, the pump wavelength, and the laser wavelength are quantified, and the effects of the residual reflectivity at the pump wavelength on the oscillator characteristics are investigated.

© 1992 Optical Society of America

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References

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  1. Y. Kimura, K. Suzuki, and M. Nakazawa, “Laser-diode-pumped mirror-free Er3+-diode laser,” Opt. Lett. 14, 999–1002 (1989).
    [Crossref] [PubMed]
  2. D. Burns and W. Sibbett, “Controlled amplifier mode locked Er3+fiber ring laser,” Electron. Lett. 26, 505–506 (1990).
    [Crossref]
  3. W. P. Urquhart, “Review of rare earth doped fiber lasers and amplifiers,” Proc. Inst. Electr. Eng. Part J 135, 385–407 (1988).
  4. M. Shimizu, M. Yamada, M. Horiguchi, T. Takeshita, and M. Okayasu, “Erbium-doped fiber amplifiers with an extremely high gain coefficient of 11.0 dB/km;”Electron. Lett. 26, 1641–1643 (1990).
    [Crossref]
  5. E. Desurvire and J. R. Simpson, “Amplification of spontaneous emission in erbium-doped single-mode fibers,” IEEE J. Lightwave Technol. 7, 835–845 (1989).
    [Crossref]
  6. E. Desurvire, C. R. Giles, and J. R. Simpson, “Gain saturation effects in high-speed, multichannel erbium-doped fiber amplifier at λ= 1.53 μ m,” IEEE J. Lightwave Technol. 7, 2095–2104 (1989).
    [Crossref]
  7. M. Peroni and M. Tamburrini, “Gain in erbium-doped fiber amplifiers: a simple analytical solution for the rate equations,” Opt. Lett. 15, 842–844 (1990).
    [Crossref] [PubMed]
  8. P. R. Morkel and R. I. Laming, “Theoretical modeling of erbium-doped fiber amplifiers with excited-state absorption,” Opt. Lett. 14, 1062–1064 (1989).
    [Crossref] [PubMed]
  9. M. Montecchi, A. Mecozzi, M. Settembre, M. Tamburrini, and L. DiGaspare, “Gain and noise in rare-earth-doped optical fibers,” J. Opt. Soc. Am. B 8, 134–141 (1991).
    [Crossref]
  10. E. Desurvire, J. L. Zyskind, and C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 8, 1730–1741 (1990).
    [Crossref]
  11. M. Suyama, R. I. Laming, and D. N. Payne, “Temperature dependent gain and noise characteristics of a 1480 nm-pumped erbium-doped fiber amplifier,” Electron. Lett. 26, 1756–1757 (1990).
    [Crossref]

1991 (1)

1990 (5)

E. Desurvire, J. L. Zyskind, and C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 8, 1730–1741 (1990).
[Crossref]

M. Suyama, R. I. Laming, and D. N. Payne, “Temperature dependent gain and noise characteristics of a 1480 nm-pumped erbium-doped fiber amplifier,” Electron. Lett. 26, 1756–1757 (1990).
[Crossref]

M. Peroni and M. Tamburrini, “Gain in erbium-doped fiber amplifiers: a simple analytical solution for the rate equations,” Opt. Lett. 15, 842–844 (1990).
[Crossref] [PubMed]

D. Burns and W. Sibbett, “Controlled amplifier mode locked Er3+fiber ring laser,” Electron. Lett. 26, 505–506 (1990).
[Crossref]

M. Shimizu, M. Yamada, M. Horiguchi, T. Takeshita, and M. Okayasu, “Erbium-doped fiber amplifiers with an extremely high gain coefficient of 11.0 dB/km;”Electron. Lett. 26, 1641–1643 (1990).
[Crossref]

1989 (4)

E. Desurvire and J. R. Simpson, “Amplification of spontaneous emission in erbium-doped single-mode fibers,” IEEE J. Lightwave Technol. 7, 835–845 (1989).
[Crossref]

E. Desurvire, C. R. Giles, and J. R. Simpson, “Gain saturation effects in high-speed, multichannel erbium-doped fiber amplifier at λ= 1.53 μ m,” IEEE J. Lightwave Technol. 7, 2095–2104 (1989).
[Crossref]

P. R. Morkel and R. I. Laming, “Theoretical modeling of erbium-doped fiber amplifiers with excited-state absorption,” Opt. Lett. 14, 1062–1064 (1989).
[Crossref] [PubMed]

Y. Kimura, K. Suzuki, and M. Nakazawa, “Laser-diode-pumped mirror-free Er3+-diode laser,” Opt. Lett. 14, 999–1002 (1989).
[Crossref] [PubMed]

1988 (1)

W. P. Urquhart, “Review of rare earth doped fiber lasers and amplifiers,” Proc. Inst. Electr. Eng. Part J 135, 385–407 (1988).

Burns, D.

D. Burns and W. Sibbett, “Controlled amplifier mode locked Er3+fiber ring laser,” Electron. Lett. 26, 505–506 (1990).
[Crossref]

Desurvire, E.

E. Desurvire, J. L. Zyskind, and C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 8, 1730–1741 (1990).
[Crossref]

E. Desurvire, C. R. Giles, and J. R. Simpson, “Gain saturation effects in high-speed, multichannel erbium-doped fiber amplifier at λ= 1.53 μ m,” IEEE J. Lightwave Technol. 7, 2095–2104 (1989).
[Crossref]

E. Desurvire and J. R. Simpson, “Amplification of spontaneous emission in erbium-doped single-mode fibers,” IEEE J. Lightwave Technol. 7, 835–845 (1989).
[Crossref]

DiGaspare, L.

Giles, C. R.

E. Desurvire, J. L. Zyskind, and C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 8, 1730–1741 (1990).
[Crossref]

E. Desurvire, C. R. Giles, and J. R. Simpson, “Gain saturation effects in high-speed, multichannel erbium-doped fiber amplifier at λ= 1.53 μ m,” IEEE J. Lightwave Technol. 7, 2095–2104 (1989).
[Crossref]

Horiguchi, M.

M. Shimizu, M. Yamada, M. Horiguchi, T. Takeshita, and M. Okayasu, “Erbium-doped fiber amplifiers with an extremely high gain coefficient of 11.0 dB/km;”Electron. Lett. 26, 1641–1643 (1990).
[Crossref]

Kimura, Y.

Laming, R. I.

M. Suyama, R. I. Laming, and D. N. Payne, “Temperature dependent gain and noise characteristics of a 1480 nm-pumped erbium-doped fiber amplifier,” Electron. Lett. 26, 1756–1757 (1990).
[Crossref]

P. R. Morkel and R. I. Laming, “Theoretical modeling of erbium-doped fiber amplifiers with excited-state absorption,” Opt. Lett. 14, 1062–1064 (1989).
[Crossref] [PubMed]

Mecozzi, A.

Montecchi, M.

Morkel, P. R.

Nakazawa, M.

Okayasu, M.

M. Shimizu, M. Yamada, M. Horiguchi, T. Takeshita, and M. Okayasu, “Erbium-doped fiber amplifiers with an extremely high gain coefficient of 11.0 dB/km;”Electron. Lett. 26, 1641–1643 (1990).
[Crossref]

Payne, D. N.

M. Suyama, R. I. Laming, and D. N. Payne, “Temperature dependent gain and noise characteristics of a 1480 nm-pumped erbium-doped fiber amplifier,” Electron. Lett. 26, 1756–1757 (1990).
[Crossref]

Peroni, M.

Settembre, M.

Shimizu, M.

M. Shimizu, M. Yamada, M. Horiguchi, T. Takeshita, and M. Okayasu, “Erbium-doped fiber amplifiers with an extremely high gain coefficient of 11.0 dB/km;”Electron. Lett. 26, 1641–1643 (1990).
[Crossref]

Sibbett, W.

D. Burns and W. Sibbett, “Controlled amplifier mode locked Er3+fiber ring laser,” Electron. Lett. 26, 505–506 (1990).
[Crossref]

Simpson, J. R.

E. Desurvire and J. R. Simpson, “Amplification of spontaneous emission in erbium-doped single-mode fibers,” IEEE J. Lightwave Technol. 7, 835–845 (1989).
[Crossref]

E. Desurvire, C. R. Giles, and J. R. Simpson, “Gain saturation effects in high-speed, multichannel erbium-doped fiber amplifier at λ= 1.53 μ m,” IEEE J. Lightwave Technol. 7, 2095–2104 (1989).
[Crossref]

Suyama, M.

M. Suyama, R. I. Laming, and D. N. Payne, “Temperature dependent gain and noise characteristics of a 1480 nm-pumped erbium-doped fiber amplifier,” Electron. Lett. 26, 1756–1757 (1990).
[Crossref]

Suzuki, K.

Takeshita, T.

M. Shimizu, M. Yamada, M. Horiguchi, T. Takeshita, and M. Okayasu, “Erbium-doped fiber amplifiers with an extremely high gain coefficient of 11.0 dB/km;”Electron. Lett. 26, 1641–1643 (1990).
[Crossref]

Tamburrini, M.

Urquhart, W. P.

W. P. Urquhart, “Review of rare earth doped fiber lasers and amplifiers,” Proc. Inst. Electr. Eng. Part J 135, 385–407 (1988).

Yamada, M.

M. Shimizu, M. Yamada, M. Horiguchi, T. Takeshita, and M. Okayasu, “Erbium-doped fiber amplifiers with an extremely high gain coefficient of 11.0 dB/km;”Electron. Lett. 26, 1641–1643 (1990).
[Crossref]

Zyskind, J. L.

E. Desurvire, J. L. Zyskind, and C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 8, 1730–1741 (1990).
[Crossref]

Electron. Lett. (3)

M. Shimizu, M. Yamada, M. Horiguchi, T. Takeshita, and M. Okayasu, “Erbium-doped fiber amplifiers with an extremely high gain coefficient of 11.0 dB/km;”Electron. Lett. 26, 1641–1643 (1990).
[Crossref]

D. Burns and W. Sibbett, “Controlled amplifier mode locked Er3+fiber ring laser,” Electron. Lett. 26, 505–506 (1990).
[Crossref]

M. Suyama, R. I. Laming, and D. N. Payne, “Temperature dependent gain and noise characteristics of a 1480 nm-pumped erbium-doped fiber amplifier,” Electron. Lett. 26, 1756–1757 (1990).
[Crossref]

IEEE J. Lightwave Technol. (3)

E. Desurvire, J. L. Zyskind, and C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 8, 1730–1741 (1990).
[Crossref]

E. Desurvire and J. R. Simpson, “Amplification of spontaneous emission in erbium-doped single-mode fibers,” IEEE J. Lightwave Technol. 7, 835–845 (1989).
[Crossref]

E. Desurvire, C. R. Giles, and J. R. Simpson, “Gain saturation effects in high-speed, multichannel erbium-doped fiber amplifier at λ= 1.53 μ m,” IEEE J. Lightwave Technol. 7, 2095–2104 (1989).
[Crossref]

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

Opt. Lett. (3)

Proc. Inst. Electr. Eng. Part J (1)

W. P. Urquhart, “Review of rare earth doped fiber lasers and amplifiers,” Proc. Inst. Electr. Eng. Part J 135, 385–407 (1988).

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Figures (10)

Fig. 1
Fig. 1

Dependence of threshold pump power on (a) the wavelength of the pump radiation when it is oscillating at 1550 nm and (b) the wavelength of the oscilllating radiation when it is pumped at 1480 nm.

Fig. 2
Fig. 2

Variation of threshold pump power with fiber length.

Fig. 3
Fig. 3

Minimum fiber length required to initiate laser action for different cavity losses.

Fig. 4
Fig. 4

Pump power attenuation along the fiber at the threshold condition.

Fig. 5
Fig. 5

Distribution of the population density on the upper laser level inside the cavity for the threshold pump condition.

Fig. 6
Fig. 6

Output power from a frequency-tuned EDFL pumped at 1480 nm with a power of 200 mW at 1480 nm.

Fig. 7
Fig. 7

Dependence of the slope efficiency Γ1 on fiber length.

Fig. 8
Fig. 8

Suggested fiber length lc for achieving high slope efficiency.

Fig. 9
Fig. 9

Fiber length lh for maximized output at the desired wavelengths for various available pump powers.

Fig. 10
Fig. 10

Variation of the population densities N(0)/No (dotted curve) and N(l)/No (solid curve) with pump power.

Equations (42)

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d S ( x ) d x = [ ϕ ( p ) N ( x ) σ ¯ a ( p ) N o ] S ,
d F + ( x ) d x = [ ϕ ( s ) N ( x ) σ ¯ a ( s ) N o ] F + ,
d F ( x ) d x = [ ϕ ( s ) N ( x ) σ ¯ a ( s ) N o ] F ,
N ( x ) u τ = [ σ ¯ a ( p ) N o ϕ ( p ) N ] S + [ σ ¯ a ( s ) N o ϕ ( s ) N ] ( F + + F ) ,
S ( x ) x = 0 = S ( 0 ) ,
F + ( 0 ) = R 1 F ( 0 ) , F ( l ) = R 2 F + ( l ) ,
ϕ ( p ) = σ ¯ a ( p ) + σ ¯ e ( p ) ,
ϕ ( s ) = σ ¯ a ( s ) + σ ¯ e ( s ) .
P p ( x ) = h v p A u S ( x ) ,
P s ( x ) = h v s A u F ( x ) ,
N ( x ) = u τ d S ( x ) d x .
ln [ S ( x ) / S ( 0 ) ] = u τ ϕ ( p ) [ S ( 0 ) S ( x ) ] σ ¯ a ( p ) N o x .
ln ( R 1 R 2 ) 2 + 0 l [ ϕ ( s ) N ( x ) σ ¯ a ( s ) N o ] d x = 0 .
S ( l ) = S ( th ) σ ¯ a ( s ) N o l ln ( R 1 R 2 ) / 2 u τ ϕ ( s ) ,
2 γ = ϕ ( p ) / ϕ ( s ) ,
P p ( th ) = h v p A σ ¯ a ( s ) N o l ln ( R 1 R 2 ) / 2 τ ϕ ( s ) [ 1 ( R 1 R 2 ) γ Q ] ,
Q = exp { [ 2 γ σ ¯ a ( s ) σ ¯ a ( p ) ] N o l } .
N ( x ) N o = u τ σ ¯ a ( p ) S ( x ) 1 + u τ ϕ ( p ) S ( x ) .
P ex ( th ) / P p ( th ) = ( R 1 R 2 ) γ Q .
( R 1 R 2 ) γ = exp { [ 2 γ σ ¯ a ( s ) σ ¯ a ( p ) ] N o l m } .
2 γ σ ¯ a ( s ) σ ¯ a ( p ) = [ σ ¯ e ( s ) σ ¯ a ( p ) σ ¯ e ( p ) σ ¯ a ( s ) ] / ϕ ( s ) .
d S d x + d F + d x d F d x + N ( u τ ) = 0 .
N ( x ) = ( d S / d x ) / S + σ ¯ a ( p ) N o ϕ ( p ) .
ln [ F + ( x ) / F + ( 0 ) ] + σ ¯ a ( s ) N o x ϕ ( s ) = ln [ S ( x ) / S ( 0 ) ] + σ ¯ a ( p ) N o x ϕ ( p ) .
F + ( x ) F ( x ) = D 2 ,
F + ( 0 ) = D R 1 1 / 2 , F + ( l ) = D / R 2 1 / 2 ,
F ( 0 ) = D / R 1 1 / 2 , F ( l ) = D R 2 1 / 2 .
K 1 D = S ( 0 ) S ( l ) + ln [ S ( 0 ) / S ( l ) ] + σ ¯ a ( p ) N o l u τ ϕ ( p ) ,
K 1 = [ 1 ( R 1 R 2 ) 1 / 2 ] [ R 1 1 / 2 + R 2 1 / 2 ] / ( R 1 R 2 ) 1 / 2 .
S ( l ) / S ( 0 ) = ( R 1 R 2 ) γ Q .
P 1 = h v p A u ( 1 R 1 ) D / R 1 1 / 2 ,
P 1 = K o { P p [ 1 ( R 1 R 2 ) γ Q ] E [ σ ¯ a ( s ) N o l ln ( R 1 R 2 ) / 2 ] } ,
K o = ( 1 R 1 ) R 2 1 / 2 v s / v p ( R 1 1 / 2 + R 2 1 / 2 ) [ 1 ( R 1 R 2 ) 1 / 2 ] ,
E = h v p A / τ ϕ ( s ) .
Γ 1 = K o [ 1 ( R 1 R 2 ) γ Q ] .
l c = ln [ θ ( R 1 R 2 ) γ ] [ 2 γ σ ¯ a ( s ) σ ¯ a ( p ) ] N o .
l h = ln V [ 2 γ σ ¯ a ( s ) σ ¯ a ( p ) ] N o ,
V = E σ ¯ a ( s ) ( R 1 R 2 ) γ P p [ σ ¯ a ( p ) 2 γ σ ¯ a ( s ) ] .
S ( 0 ) = S ( 0 ) r 1 + S ( in ) , r 2 S ( l ) = S ( l ) .
S ( 0 ) = { S ( in ) + [ S 2 ( in ) + 4 r 1 B 2 ] 1 / 2 } / 2 .
B = r 1 1 / 2 ( R 1 R 2 ) γ Q S ( in ) .
P 1 = K o { P p [ 1 ( R 1 R 2 ) γ Q ] ( 1 r 2 ) E [ σ ¯ a ( s ) N o l ln ( R 1 R 2 ) / 2 ] } ,

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