Abstract

Starting from the paraxial wave equation, an analytic expression for filament threshold in fiber lasers is derived. The occurrence of filamentation is determined by the larger of two thresholds, one of perturbative gain and one of spatial confinement. The threshold value is approximately a few megawatts, depending on the parameters of the fiber.

© 2007 Optical Society of America

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References

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  1. B. Ortac, A. Hideur, T. Chartier, M. Brunel, C. Özkul, and F. Sanchez, "90-fs stretched-pulse ytterbium-doped double-clad fiber laser," Opt. Lett. 28, 1305-1307 (2003).
    [CrossRef] [PubMed]
  2. J. R. Buckley, F. W. Wise, F. Ö. Ilday, and T. Sosnowski, "Femtosecond fiber lasers with pulse energies above 10nJ," Opt. Lett. 30, 1888-1890 (2005).
    [CrossRef] [PubMed]
  3. Y.-X. Fan, F.-Y. Lu, S.-L. Hu, K.-C. Lu, H.-J. Wang, X.-Y. Dong, and G.-Y. Zhang, "105-kW peak-power double-clad fiber laser," IEEE Photon. Technol. Lett. 15, 652-654 (2003).
    [CrossRef]
  4. F. Röser, J. Rothhard, B. Ortac, A. Liem, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, "131W 220 fs fiber laser system," Opt. Lett. 30, 2754-2756 (2005).
    [CrossRef] [PubMed]
  5. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  6. K. Shiraki, M. Ohashi, and M. Tateda, "Performance of strain-free SBS suppression fiber," J. Lightwave Technol. 14, 549-554 (1996).
    [CrossRef]
  7. J. Toulouse, "Optical nonlinearities in fibers: review, recent, examples and systems applications," J. Lightwave Technol. 23, 3625-3641 (2005).
    [CrossRef]
  8. V. I. Kovalev and R. G. Harrison, "Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers," Opt. Lett. 31, 161-163 (2006).
    [CrossRef] [PubMed]
  9. C.-H. Liu, B. Ehlers, F. Doerfel, S. Heinemann, A. Carter, K. Tankala, J. Farroni, and A. Galvanauskas, "810W continuous-wave and single transverse-mode fibre laser using 20μm core Yb-doped double-clad fibre," Electron. Lett. 40, 1471-1472 (2004).
    [CrossRef]
  10. P. L. Baldeck, F. Raccah, and R. R. Alfano, "Observation of self-focusing in optical fibers with picosecond pulses," Opt. Lett. 12, 588-589 (1987).
    [CrossRef] [PubMed]
  11. Y. M. Hua, Q. Li, Y. L. Chen, and Y. X. Chen, "The ring structure of self-focusing light in optical fibres," Opt. Commun. 79, 459-461 (1990).
    [CrossRef]
  12. H. Adachihara, O. Hess, E. Abraham, P. Ru, and J. V. Moloney, "Spatiotemporal chaos in broad-area semiconductor lasers," J. Opt. Soc. Am. B 10, 658-665 (1993).
    [CrossRef]
  13. D. J. Bossert, J. R. Marciante, and M. W. Wright, "Feedback effects in tapered broad area semiconductor lasers and amplifiers," IEEE Photon. Technol. Lett. 7, 470-472 (1995).
    [CrossRef]
  14. J. R. Marciante and G. P. Agrawal, "Spatio-temporal characteristics of filamentation in broad-area semiconductor lasers," IEEE J. Quantum Electron. 33, 1174-1179 (1997).
    [CrossRef]
  15. G. Tempea and T. Brabec, "Theory of self-focusing in a hollow waveguide," Opt. Lett. 23, 762-764 (1998).
    [CrossRef]
  16. G. Fibich and A. L. Gaeta, "Critical power for self-focusing in bulk media and in hollow waveguides," Opt. Lett. 25, 335-337 (2000).
    [CrossRef]
  17. R. L. Farrow, D. A. V. Kliner, G. R. Hadley, and A. V. Smith, "Peak-power limits on fiber amplifiers imposed by self-focusing," Opt. Lett. 31, 3423-3425 (2006).
    [CrossRef] [PubMed]
  18. P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology, 1st ed. (Acadmic, 1999).
  19. G. P. Agrawal, Lightwave Technology: Telecommunication Systems, 1st ed. (Wiley, 2004).
  20. C. J. Chen, P. K. A. Wai, and C. R. Menyuk, "Self-starting of passively mode-locked lasers with fast saturable absorbers," Opt. Lett. 20, 350-352 (1995).
    [CrossRef] [PubMed]
  21. R. W. Boyd, Nonlinear Optics (Academic, 1992), Sec. 6.2.
  22. C. D. Brooks and F. Di Teodoro, "Multimegawatt peak-power, single-transverse-mode operation of a 100μm core diameter, Yb-doped rodlike photonic crystal fiber amplifier," Appl. Phys. Lett. 89, 111119 (2006).
    [CrossRef]

2006

2005

2004

C.-H. Liu, B. Ehlers, F. Doerfel, S. Heinemann, A. Carter, K. Tankala, J. Farroni, and A. Galvanauskas, "810W continuous-wave and single transverse-mode fibre laser using 20μm core Yb-doped double-clad fibre," Electron. Lett. 40, 1471-1472 (2004).
[CrossRef]

2003

Y.-X. Fan, F.-Y. Lu, S.-L. Hu, K.-C. Lu, H.-J. Wang, X.-Y. Dong, and G.-Y. Zhang, "105-kW peak-power double-clad fiber laser," IEEE Photon. Technol. Lett. 15, 652-654 (2003).
[CrossRef]

B. Ortac, A. Hideur, T. Chartier, M. Brunel, C. Özkul, and F. Sanchez, "90-fs stretched-pulse ytterbium-doped double-clad fiber laser," Opt. Lett. 28, 1305-1307 (2003).
[CrossRef] [PubMed]

2000

1998

1997

J. R. Marciante and G. P. Agrawal, "Spatio-temporal characteristics of filamentation in broad-area semiconductor lasers," IEEE J. Quantum Electron. 33, 1174-1179 (1997).
[CrossRef]

1996

K. Shiraki, M. Ohashi, and M. Tateda, "Performance of strain-free SBS suppression fiber," J. Lightwave Technol. 14, 549-554 (1996).
[CrossRef]

1995

D. J. Bossert, J. R. Marciante, and M. W. Wright, "Feedback effects in tapered broad area semiconductor lasers and amplifiers," IEEE Photon. Technol. Lett. 7, 470-472 (1995).
[CrossRef]

C. J. Chen, P. K. A. Wai, and C. R. Menyuk, "Self-starting of passively mode-locked lasers with fast saturable absorbers," Opt. Lett. 20, 350-352 (1995).
[CrossRef] [PubMed]

1993

1990

Y. M. Hua, Q. Li, Y. L. Chen, and Y. X. Chen, "The ring structure of self-focusing light in optical fibres," Opt. Commun. 79, 459-461 (1990).
[CrossRef]

1987

Appl. Phys. Lett.

C. D. Brooks and F. Di Teodoro, "Multimegawatt peak-power, single-transverse-mode operation of a 100μm core diameter, Yb-doped rodlike photonic crystal fiber amplifier," Appl. Phys. Lett. 89, 111119 (2006).
[CrossRef]

Electron. Lett.

C.-H. Liu, B. Ehlers, F. Doerfel, S. Heinemann, A. Carter, K. Tankala, J. Farroni, and A. Galvanauskas, "810W continuous-wave and single transverse-mode fibre laser using 20μm core Yb-doped double-clad fibre," Electron. Lett. 40, 1471-1472 (2004).
[CrossRef]

IEEE J. Quantum Electron.

J. R. Marciante and G. P. Agrawal, "Spatio-temporal characteristics of filamentation in broad-area semiconductor lasers," IEEE J. Quantum Electron. 33, 1174-1179 (1997).
[CrossRef]

IEEE Photon. Technol. Lett.

Y.-X. Fan, F.-Y. Lu, S.-L. Hu, K.-C. Lu, H.-J. Wang, X.-Y. Dong, and G.-Y. Zhang, "105-kW peak-power double-clad fiber laser," IEEE Photon. Technol. Lett. 15, 652-654 (2003).
[CrossRef]

D. J. Bossert, J. R. Marciante, and M. W. Wright, "Feedback effects in tapered broad area semiconductor lasers and amplifiers," IEEE Photon. Technol. Lett. 7, 470-472 (1995).
[CrossRef]

J. Lightwave Technol.

K. Shiraki, M. Ohashi, and M. Tateda, "Performance of strain-free SBS suppression fiber," J. Lightwave Technol. 14, 549-554 (1996).
[CrossRef]

J. Toulouse, "Optical nonlinearities in fibers: review, recent, examples and systems applications," J. Lightwave Technol. 23, 3625-3641 (2005).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

Y. M. Hua, Q. Li, Y. L. Chen, and Y. X. Chen, "The ring structure of self-focusing light in optical fibres," Opt. Commun. 79, 459-461 (1990).
[CrossRef]

Opt. Lett.

P. L. Baldeck, F. Raccah, and R. R. Alfano, "Observation of self-focusing in optical fibers with picosecond pulses," Opt. Lett. 12, 588-589 (1987).
[CrossRef] [PubMed]

C. J. Chen, P. K. A. Wai, and C. R. Menyuk, "Self-starting of passively mode-locked lasers with fast saturable absorbers," Opt. Lett. 20, 350-352 (1995).
[CrossRef] [PubMed]

G. Tempea and T. Brabec, "Theory of self-focusing in a hollow waveguide," Opt. Lett. 23, 762-764 (1998).
[CrossRef]

B. Ortac, A. Hideur, T. Chartier, M. Brunel, C. Özkul, and F. Sanchez, "90-fs stretched-pulse ytterbium-doped double-clad fiber laser," Opt. Lett. 28, 1305-1307 (2003).
[CrossRef] [PubMed]

J. R. Buckley, F. W. Wise, F. Ö. Ilday, and T. Sosnowski, "Femtosecond fiber lasers with pulse energies above 10nJ," Opt. Lett. 30, 1888-1890 (2005).
[CrossRef] [PubMed]

F. Röser, J. Rothhard, B. Ortac, A. Liem, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, "131W 220 fs fiber laser system," Opt. Lett. 30, 2754-2756 (2005).
[CrossRef] [PubMed]

V. I. Kovalev and R. G. Harrison, "Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers," Opt. Lett. 31, 161-163 (2006).
[CrossRef] [PubMed]

R. L. Farrow, D. A. V. Kliner, G. R. Hadley, and A. V. Smith, "Peak-power limits on fiber amplifiers imposed by self-focusing," Opt. Lett. 31, 3423-3425 (2006).
[CrossRef] [PubMed]

G. Fibich and A. L. Gaeta, "Critical power for self-focusing in bulk media and in hollow waveguides," Opt. Lett. 25, 335-337 (2000).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology, 1st ed. (Acadmic, 1999).

G. P. Agrawal, Lightwave Technology: Telecommunication Systems, 1st ed. (Wiley, 2004).

R. W. Boyd, Nonlinear Optics (Academic, 1992), Sec. 6.2.

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Figures (5)

Fig. 1
Fig. 1

Normalized filament gain versus normalized filament spacing and frequency for d core = 100 μ m , P s = 10 kW .

Fig. 2
Fig. 2

Normalized filament gain versus normalized filament spacing and frequency for d core = 100 μ m , P s = 10 MW .

Fig. 3
Fig. 3

(a) Normalized filament spacing and (b) normalized gain as a function of the signal peak power for various core diameters: 20 μ m (dotted), 50 μ m (dashed–dotted), 100 μ m (dashed), 200 μ m (solid) ( f = 10 GHz ) .

Fig. 4
Fig. 4

Gain threshold power [ NA = 0.2 (dashed), NA = 0.1 (dashed–dotted), NA = 0.05 (solid with “+” symbol)] and spatial threshold power [ NA = 0.2 (solid), NA = 0.1 (dotted), NA = 0.05 (dotted with + symbol)] as functions of core diameter for three NAs ( f = 10 GHz ) .

Fig. 5
Fig. 5

(a) Nonnormalized, (b) normalized filament gain, and (c) normalized filament spacing as a function of the signal peak power for three different cavity lengths: 0.5 m (solid), 2 m (dotted), 4 m (dashed-dotted) ( d core = 100 μ m , f = 10 GHz ).

Tables (1)

Tables Icon

Table 1 Parameters for Yb-Doped Fiber Laser Calculations

Equations (19)

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A s z + 1 v g A s t = i 2 k s T 2 A s + [ 1 2 g s + i ( 2 γ p P p + γ s P s ) ] A s ,
T 2 = 1 r r + 2 r 2 + 1 r 2 2 ϕ 2
N 2 t = N 2 τ ( N 2 σ s e N 1 σ s a ) ϕ s ( N 2 σ p e N 1 σ p a ) ϕ p ,
A ̃ = A s exp [ i ( k s z ω s t ) ] = A s 0 J m ( p s r ) exp ( i m ϕ ) exp [ i ( k s z ω s t ) ] ,
A s z = [ 1 2 g s i p s 2 2 k s + i ( 2 γ p P p + γ s P s ) ] A s .
Δ k s = 1 2 g s i p s 2 2 k s + i ( 2 γ p P p + γ s P s ) .
N 2 = N t ( σ s a σ s e + σ s a P s P s sat + σ p a σ p e + σ p a P p P p sat ) 1 + P s P s sat + P p P p sat ,
P s = [ ( N t σ s a + α cav Γ s ) ( N t ( σ p e σ s a σ p a σ s e ) σ p e + σ p a + α cav Γ s ) P p P p sat ] α cav P s sat Γ s .
a z + 1 v g a t = i 2 k s ( p s 2 a + T 2 a ) + 1 2 g s a + 1 2 Γ s n P s ( σ s e + σ s a ) + i γ s P s ( a + a * ) ,
τ n t = n ( 1 + P s P s sat + P p P p sat ) + ( N 2 N t σ s a σ s e + σ s a ) P s P s sat ( a + a * ) .
a = a 1 J k ϕ ( q r ) exp [ i ( k ϕ ϕ + k z z Ω t ) ] + a 2 J k ϕ ( q r ) exp [ i ( k ϕ ϕ + k z z Ω t ) ] ,
n = n 0 J k ϕ ( q r ) exp [ i ( k ϕ ϕ + k z z Ω t ) ] + n 0 * J k ϕ ( q r ) exp [ i ( k ϕ ϕ + k z z Ω t ) ] ,
k z = Ω v g + i 1 2 [ G ( 1 + i ξ ) g s ] ± p 2 2 k s ( p 2 2 k s 2 γ s P s ) 1 4 [ G 2 ( 1 + i ξ ) 2 + g s 2 ] ,
ξ = Ω τ 1 + P s P s sat + P p P p sat ,
G = Γ s g s P s P s sa t ( 1 + P s P s sat + P p P p sat ) ( 1 + P s P s sat + P p P p sat ) 2 + ( Ω τ ) 2 .
g = Re 2 p 2 k s ( 2 γ s P s p 2 2 k s ) + [ G 2 ( 1 + i ξ ) 2 + g s 2 ] ( G g s ) .
P t h spatial = π 2 a core 2 p s 2 2 γ s k s .
P t h gain = α cav 2 γ s .
P c r = α λ 2 4 π n 0 n ¯ 2 ,

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