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," Proc. SPIE 6102, 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," Proc. SPIE 6102, 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. Agrawal, Nonlinear 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. Fève, P. Schrader, R. Farrow and D. Kliner, "Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers," Proc. SPIE 6453, (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]

2007

J. P. Fève, P. Schrader, R. Farrow and D. Kliner, "Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers," Proc. SPIE 6453, (2007).

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," Proc. SPIE 6102, 425-432 (2006).

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]

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley and R. Schmitt, "High-peak-power (>1.2MW) pulsed fiber amplifier," Proc. SPIE 6102, 138 -148 (2006).

2005

2004

2000

1997

Broeng, J.

Brooks, C.

Clarkson, W.

Cooper, L.

Di Teodoro, F.

Farrow, R.

J. P. Fève, P. Schrader, R. Farrow and D. Kliner, "Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers," Proc. SPIE 6453, (2007).

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley and R. Schmitt, "High-peak-power (>1.2MW) pulsed fiber amplifier," Proc. SPIE 6102, 138 -148 (2006).

Fève, J. P.

J. P. Fève, P. Schrader, R. Farrow and D. Kliner, "Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers," Proc. SPIE 6453, (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," Proc. SPIE 6102, 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," Proc. SPIE 6102, 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," Proc. SPIE 6102, 425-432 (2006).

Jakobsen, C.

Kliner, D.

J. P. Fève, P. Schrader, R. Farrow and D. Kliner, "Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers," Proc. SPIE 6453, (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," Proc. SPIE 6102, 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," Proc. SPIE 6102, 138 -148 (2006).

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

Koplow, J.

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," Proc. SPIE 6102, 425-432 (2006).

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

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," Proc. SPIE 6102, 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," Proc. SPIE 6102, 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," Proc. SPIE 6102, 138 -148 (2006).

Schrader, P.

J. P. Fève, P. Schrader, R. Farrow and D. Kliner, "Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers," Proc. SPIE 6453, (2007).

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley and R. Schmitt, "High-peak-power (>1.2MW) pulsed fiber amplifier," Proc. SPIE 6102, 138 -148 (2006).

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," Proc. SPIE 6102, 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," Proc. SPIE 6102, 425-432 (2006).

Thomson, M.

Tünnermann, A.

Wang, P.

Zellmer, H.

Opt. Express

Opt. Lett.

Proc. SPIE

J. P. Fève, P. Schrader, R. Farrow and D. Kliner, "Limiting effects of four-wave mixing in high-power pulsed fiber amplifiers," Proc. SPIE 6453, (2007).

R. Farrow, D. Kliner, P. Schrader, A. Hoops, S. Moore, G. Hadley and R. Schmitt, "High-peak-power (>1.2MW) pulsed fiber amplifier," Proc. SPIE 6102, 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," Proc. SPIE 6102, 425-432 (2006).

Other

G. Agrawal, Nonlinear 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).

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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}.

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.

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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