Abstract

The dynamics of a laser with excited-state absorption at the lasing wavelength is theoretically studied. The model is based on the rate equations for a four-level system. The stationary state is analytically calculated, permitting both the investigation of the laser characteristics and linear stability analysis. The latter shows that, in some conditions, the steady state is not stable in a particular range of pumping rates. However, a stable solution is restored for sufficiently high pumping rates. Stable self-pulsing solutions are obtained by numerical integration of the coupled equations. Also, the transient regimes are numerically analyzed.

© 1997 Optical Society of America

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References

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  1. See, for example, the features on nonlinear laser instabilities, J. Opt. Soc. Am. B 2, 7–264 (1985); J. Opt. Soc. Am. B 5, 879–1215 (1988).
    [CrossRef]
  2. H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
    [CrossRef]
  3. L. W. Casperson, “Spontaneous coherent pulsations in ring laser oscillators,” J. Opt. Soc. Am. B 2, 62–72 (1985).
    [CrossRef]
  4. J. L. A. Chilla and O. E. Martinez, “Spatiotemporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638–643 (1993).
    [CrossRef]
  5. F. Sanchez, P. Le Boudec, P. L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
    [CrossRef] [PubMed]
  6. S. A. Pollack, D. B. Chang, and N. L. Moise, “Upconversion-pumped infrared erbium laser,” J. Appl. Phys. 60, 4077–4086 (1986).
    [CrossRef]
  7. S. A. Pollack and D. B. Chang, “Ion-pair upconversion pumped laser emission in Er3+ ions in YAG, YLF, SrF2 and CaF2 crystals,” J. Appl. Phys. 64, 2885–2893 (1988).
    [CrossRef]
  8. A. C. Tropper, J. N. Carter, R. D. T. Lauder, D. C. Hanna, S. T. Davey, and D. Szebesta, “Analysis of blue and red laser performance of the infrared-pumped praseodymium-doped fluoride fiber laser,” J. Opt. Soc. Am. B 11, 886–893 (1994).
    [CrossRef]
  9. P. Xie and S. C. Rand, “Nonlinear dynamics of cooperative upconversion,” J. Opt. Soc. Am. B 11, 901–912 (1994).
    [CrossRef]
  10. R. A. McFarlane, “High-power visible upconversion laser,” Opt. Lett. 16, 1397–1399 (1991).
    [CrossRef] [PubMed]
  11. R. A. McFarlane, “Dual wavelength visible upconversion laser,” Appl. Phys. Lett. 54, 2301–2303 (1989).
    [CrossRef]
  12. I. P. Alcock, A. I. Ferguson, D. C. Hanna, and A. C. Tropper, “Tunable, continuous-wave neodymium-doped monomode-fiber laser operating at 0.900–0.945 and 1.070–1.135 µm,” Opt. Lett. 11, 709–711 (1986).
    [CrossRef] [PubMed]
  13. F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
    [CrossRef]
  14. J. D. Crawford, “Introduction to bifurcation theory,” Rev. Mod. Phys. 63, 991–1037 (1991).
    [CrossRef]
  15. P. LeBoudec, M. LeFlohic, P. L. François, F. Sanchez, and G. Stephan, “Self-pulsing in Er3+-doped fiber laser,” Opt. Quantum Electron. 25, 359–367 (1993).
    [CrossRef]

1996 (1)

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

1994 (2)

1993 (3)

J. L. A. Chilla and O. E. Martinez, “Spatiotemporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638–643 (1993).
[CrossRef]

P. LeBoudec, M. LeFlohic, P. L. François, F. Sanchez, and G. Stephan, “Self-pulsing in Er3+-doped fiber laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

1991 (2)

J. D. Crawford, “Introduction to bifurcation theory,” Rev. Mod. Phys. 63, 991–1037 (1991).
[CrossRef]

R. A. McFarlane, “High-power visible upconversion laser,” Opt. Lett. 16, 1397–1399 (1991).
[CrossRef] [PubMed]

1989 (1)

R. A. McFarlane, “Dual wavelength visible upconversion laser,” Appl. Phys. Lett. 54, 2301–2303 (1989).
[CrossRef]

1988 (1)

S. A. Pollack and D. B. Chang, “Ion-pair upconversion pumped laser emission in Er3+ ions in YAG, YLF, SrF2 and CaF2 crystals,” J. Appl. Phys. 64, 2885–2893 (1988).
[CrossRef]

1986 (2)

1985 (2)

1968 (1)

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Alcock, I. P.

Carter, J. N.

Casperson, L. W.

Chang, D. B.

S. A. Pollack and D. B. Chang, “Ion-pair upconversion pumped laser emission in Er3+ ions in YAG, YLF, SrF2 and CaF2 crystals,” J. Appl. Phys. 64, 2885–2893 (1988).
[CrossRef]

S. A. Pollack, D. B. Chang, and N. L. Moise, “Upconversion-pumped infrared erbium laser,” J. Appl. Phys. 60, 4077–4086 (1986).
[CrossRef]

Chilla, J. L. A.

Crawford, J. D.

J. D. Crawford, “Introduction to bifurcation theory,” Rev. Mod. Phys. 63, 991–1037 (1991).
[CrossRef]

Davey, S. T.

Ferguson, A. I.

François, P. L.

P. LeBoudec, M. LeFlohic, P. L. François, F. Sanchez, and G. Stephan, “Self-pulsing in Er3+-doped fiber laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

Hanna, D. C.

Lauder, R. D. T.

Le Boudec, P.

F. Sanchez, P. Le Boudec, P. L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

LeBoudec, P.

P. LeBoudec, M. LeFlohic, P. L. François, F. Sanchez, and G. Stephan, “Self-pulsing in Er3+-doped fiber laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

LeFlohic, M.

P. LeBoudec, M. LeFlohic, P. L. François, F. Sanchez, and G. Stephan, “Self-pulsing in Er3+-doped fiber laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

Martinez, O. E.

McFarlane, R. A.

R. A. McFarlane, “High-power visible upconversion laser,” Opt. Lett. 16, 1397–1399 (1991).
[CrossRef] [PubMed]

R. A. McFarlane, “Dual wavelength visible upconversion laser,” Appl. Phys. Lett. 54, 2301–2303 (1989).
[CrossRef]

Moise, N. L.

S. A. Pollack, D. B. Chang, and N. L. Moise, “Upconversion-pumped infrared erbium laser,” J. Appl. Phys. 60, 4077–4086 (1986).
[CrossRef]

Nummedal, K.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Pollack, S. A.

S. A. Pollack and D. B. Chang, “Ion-pair upconversion pumped laser emission in Er3+ ions in YAG, YLF, SrF2 and CaF2 crystals,” J. Appl. Phys. 64, 2885–2893 (1988).
[CrossRef]

S. A. Pollack, D. B. Chang, and N. L. Moise, “Upconversion-pumped infrared erbium laser,” J. Appl. Phys. 60, 4077–4086 (1986).
[CrossRef]

Rand, S. C.

Risken, H.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Sanchez, F.

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

P. LeBoudec, M. LeFlohic, P. L. François, F. Sanchez, and G. Stephan, “Self-pulsing in Er3+-doped fiber laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

Stephan, G.

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

P. LeBoudec, M. LeFlohic, P. L. François, F. Sanchez, and G. Stephan, “Self-pulsing in Er3+-doped fiber laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P. L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

Szebesta, D.

Tropper, A. C.

Xie, P.

Appl. Phys. Lett. (1)

R. A. McFarlane, “Dual wavelength visible upconversion laser,” Appl. Phys. Lett. 54, 2301–2303 (1989).
[CrossRef]

J. Appl. Phys. (3)

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

S. A. Pollack, D. B. Chang, and N. L. Moise, “Upconversion-pumped infrared erbium laser,” J. Appl. Phys. 60, 4077–4086 (1986).
[CrossRef]

S. A. Pollack and D. B. Chang, “Ion-pair upconversion pumped laser emission in Er3+ ions in YAG, YLF, SrF2 and CaF2 crystals,” J. Appl. Phys. 64, 2885–2893 (1988).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

P. LeBoudec, M. LeFlohic, P. L. François, F. Sanchez, and G. Stephan, “Self-pulsing in Er3+-doped fiber laser,” Opt. Quantum Electron. 25, 359–367 (1993).
[CrossRef]

Phys. Rev. A (1)

F. Sanchez, P. Le Boudec, P. L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (1)

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

Rev. Mod. Phys. (1)

J. D. Crawford, “Introduction to bifurcation theory,” Rev. Mod. Phys. 63, 991–1037 (1991).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic representation of the energy levels. The single arrows indicate relaxations, the double arrows show the lasing and ESA transitions, ni represents the population of level |i〉, and P is the pump parameter.

Fig. 2
Fig. 2

Evolution of the laser intensity versus the pumping ratio for different values of ESA: (a) a2=10-5, a3=10-1, a4=10-2; (b) a2=10-5, a3=5×10-4, a4=10-2; (c) a2=10-6, a3=5×10-5, a4=10-3.

Fig. 3
Fig. 3

Evolution of I versus r for different values of a3: (a) x=0 and (b) x=0.3. The remaining parameters are a2=10-5 and a4=10-2.

Fig. 4
Fig. 4

Evolution of α and ω versus r: (a) x=0, a2=10-5, a4=10-2; (b) x=0.3, a2=10-5, a4=10-2; (c) x=0.3, a2=10-6, a4=10-3.

Fig. 5
Fig. 5

Stability diagram in the plane (r, a3) showing the stable steady-state regions and the unstable regions (self-pulsing behavior): (a) a2=10-5, a4=10-2; (b) a2=10-6, a4=10-3.

Fig. 6
Fig. 6

Envelope of the transient regimes obtained by numerical resolution of the system of Eqs. (1): (a) self-pulsing operation for x=0.3 and r=1.5, (b) cw operation for x=0.3 and r=3.8. The other parameters are a2=10-5, a3=10-3, and a4=10-2.

Equations (19)

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n1t=-Pn1+a2n2-(n1-n2)I,
n2t=-a2n2+a3n3+(n1-n2)I,
n3t=Pn1-a3n3-x(n3-n4)I+a4n4,
n4t=-a4n4+x(n3-n4)I,
It=-I+A(n2-n1)I+xA(n4-n3)I,
n¯1=a3(a2+I¯)(a4+xI¯)D(I¯),
n¯2=a3(I¯+P)(a4+xI¯)D(I¯),
n¯3=P(a2+I¯)(a4+xI¯)D(I¯),
n¯4=xPI¯(a2+I¯)D(I¯),
D(I¯)=a2a3a4+a4P(a2+a3)+I¯[2a3a4+xa2a3+P(a4+2xa2+xa3)]+2xI¯2(a3+P).
-2I¯2x(a3+P)+I¯[-2a3a4-xa2a3(1+A)
-a4P-xP(2a2+a3)+AxP(a3-a4)]+[-a2a3a4(1+A)-a4P(a2+a3)+Aa4P(a3-xa2)]=0.
Pth=a2a3(1+A)A(a3-xa2)-a2-a3.
tδ=Lδ,
δ=n1-n¯1n2-n¯2n3-n¯3n4-n¯4I-I¯,
L=-P-I¯I¯P0-AI¯a2+I¯-a2-I¯00AI¯0a3-a3-xI¯xI¯-AxI¯00a4+xI¯-a4-xI¯xAI¯n¯2-n¯1n¯1-n¯2x(n¯4-n¯3)x(n¯3-n¯4)0.
Det(L-λI)=0.
λ(λ-λ1)(λ-λ2)(λ-λ3)(λ-λ3*)=0.
ωω0r-1,

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