Abstract

We present a theoretical study of four-wave mixing interactions in fibers in the presence of gain. In contrast to passive fibers, positive gain at the pump wavelength leads to constructive generation of the signal and idler waves, even in the case of large phase-mismatch, so that FWM processes can be very efficient even in isotropic single-mode fibers with normal dispersion. We also propose simple ways to mitigate these parametric interactions by applying a controlled variation of the phase-mismatch along the fiber. These concepts apply to all optical amplifiers.

© 2007 Optical Society of America

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References

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  1. F. Di Teodoro and C. Brooks, “Multistage Yb-doped fiber amplifier generating megawatt peak-power, subnanosecond pulses,” Opt. Lett. 30,3299–3301 (2005).
    [Crossref]
  2. R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).
  3. J. Koplow, D. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25,442–444 (2000).
    [Crossref]
  4. J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,“ Opt. Express 12,1313–1319 (2004).
    [Crossref] [PubMed]
  5. M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).
  6. P. Wang, L. Cooper, J. Sahu, and W. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber amplifier,” Opt. Lett. 31,226–228 (2006).
    [Crossref] [PubMed]
  7. G. AgrawalNonlinear Fiber Optics, 3rd ed., Optics and Photonics Series (Academic, San Diego, Calif., 2001).
  8. C. Brooks and F.Di Teodoro, “1-mJ energy, 1-MW peak-power, 10 W-average power, spectrally-narrow, diffraction-limited pulses from a photonic-crystal fiber amplifier,” Opt. Express 13,8999–9002 (2005).
    [Crossref] [PubMed]
  9. Complete calculation of modal propagation shows waveguide contribution to phase-mismatch is less than 3%, so that the single-mode approximation is justified (R. Farrow, personal communication).
  10. J. P. Féve, P. Schrader, R. Farrow, and D. Kliner, “Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers,” in Fiber Lasers IV: Technology, Systems and Applications, Proc. SPIE6453 (2007).
  11. S. Murdoch, M. Thomson, R. Leonhardt, and J. Harvey, “Quasi-phase-matched modulation instability in birefringent fibers,” Opt. Lett. 22,682 (1997)
    [Crossref] [PubMed]

2006 (1)

2005 (2)

2004 (1)

2000 (1)

1997 (1)

Broeng, J.

Brooks, C.

Clarkson, W.

Cooper, L.

Di Teodoro, F.

Farrow, R.

J. P. Féve, P. Schrader, R. Farrow, and D. Kliner, “Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers,” in Fiber Lasers IV: Technology, Systems and Applications, Proc. SPIE6453 (2007).

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

Féve, J. P.

J. P. Féve, P. Schrader, R. Farrow, and D. Kliner, “Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers,” in Fiber Lasers IV: Technology, Systems and Applications, Proc. SPIE6453 (2007).

Goldberg, L.

Hadley, G.

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

Harvey, J.

Hoops, A.

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

Hotoleanu, M.

M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).

Jakobsen, C.

Kliner, D.

J. Koplow, D. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25,442–444 (2000).
[Crossref]

J. P. Féve, P. Schrader, R. Farrow, and D. Kliner, “Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers,” in Fiber Lasers IV: Technology, Systems and Applications, Proc. SPIE6453 (2007).

M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

Koplow, J.

J. Koplow, D. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25,442–444 (2000).
[Crossref]

M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).

Leonhardt, R.

Liem, A.

Limpert, J.

Moore, S.

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

Murdoch, S.

Nolte, S.

Petersson, A.

Philipov, V.

M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).

Reich, M.

Sahu, J.

Schmitt, R.

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

Schrader, P.

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

J. P. Féve, P. Schrader, R. Farrow, and D. Kliner, “Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers,” in Fiber Lasers IV: Technology, Systems and Applications, Proc. SPIE6453 (2007).

Schreiber, T.

Söderlund, M.

M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).

Tammela, S.

M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).

Teodoro, F.Di

Thomson, M.

Tünnermann, A.

Wang, P.

Zellmer, H.

Opt. Express (2)

Opt. Lett. (4)

Other (5)

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley, and R. Schmitt, “High-peak-power (>1.2MW) pulsed fiber amplifier,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,138–148 (2006).

M. Hotoleanu, M. Söderlund, D. Kliner, J. Koplow, S. Tammela, and V. Philipov, “Higher-order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant,” in Fiber Lasers III: Technology, Systems and Applications, A. Brown, J. Nilsson, D. Harter, and A. Tünnermann, eds., Proc. SPIE6102,425–432 (2006).

G. AgrawalNonlinear Fiber Optics, 3rd ed., Optics and Photonics Series (Academic, San Diego, Calif., 2001).

Complete calculation of modal propagation shows waveguide contribution to phase-mismatch is less than 3%, so that the single-mode approximation is justified (R. Farrow, personal communication).

J. P. Féve, P. Schrader, R. Farrow, and D. Kliner, “Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers,” in Fiber Lasers IV: Technology, Systems and Applications, Proc. SPIE6453 (2007).

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Figures (2)

Fig. 1.
Fig. 1.

Peak powers along propagation in the fiber: (a) Fixed seed energy; (b) Zoom towards fiber entrance. Solid curves are signal (λ3=1094nm), green dashed curve is pump (λ1=1064nm). Different values of seed energy, gain and Raman coefficient: red {E in=5μJ, g=0.5m-1, g R=5.10-14m2/W} ; magenta {E in=5μJ, g=0m-1, g R=0m2/W} ; blue {E in=5μJ, g=0m-1, g R=5.10-14m2/W} ; black {E in=50μJ, g=0m-1, g R=0m2/W}.

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Fig. 2.
Fig. 2.

Signal power along the fiber with longitudinally controlled phase-mismatch. Red: constant Δk=41.5m-1. Magenta: linear increase of Δk, slope a= 4pm-1. Blue: exponential increase of δk , a=b=1.4m-1. Inset: zoom towards fiber exit.

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Equations (7)

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{ ∂ E 1 ∂ z = j γ [ ∣ E 1 ∣ 2 + ( 2 − ρ ) ( ∣ E 3 ∣ 2 + ∣ E 4 ∣ 2 ) ] E 1 − g R 2 A eff ∣ E 3 ∣ 2 E 1 + g 2 E 1 + j γ E 1 * E 3 E 4 e j Δ k z ∂ E 3 ∂ z = j γ [ ∣ E 3 ∣ 2 + ( 2 − ρ ) ( ∣ E 1 ∣ 2 + ∣ E 4 ∣ 2 ) ] E 3 + g R 2 A eff ∣ E 1 ∣ 2 E 3 + j γ E 4 * E 1 E 1 e − j Δ k z ∂ E 4 ∂ z = j γ [ ∣ E 4 ∣ 2 + ( 2 − ρ ) ( ∣ E 1 ∣ 2 + ∣ E 3 ∣ 2 ) ] E 4 + j γ E 3 * E 1 E 1 e − j Δ k z
κ ( z ) = Δ kz + 2 γ P 0 g e gz
d 2 B 3 dz 2 + [ j ( Δ k + 2 γ P 0 e gz ) − g ] dB 3 dz − γ 2 P 0 2 e 2 gz B 3 = 0
P 3 ( z ) = [ A exp ( u + z ) + B exp ( u − z ) ] 2
P 3 ( z ) ∝ sin 2 ( Δ k z 2 )
P 3 ( z ) ∝ exp ( gz )
L PM = 1 g ln ( Δ k 2 γ P 0 ) = 1 g ln ( Δ k λ 1 A eff 4 π n 2 P 0 )

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