Abstract

We show theoretically and numerically that in an instantaneous-response Kerr medium, a coherent signal light can be generated spontaneously from an incoherent pump light in the degenerate four-wave mixing process. Our analysis shows that this phenomenon requires the same group velocities of pump wave and idler wave as well as the convection (walk-off) between the signal wave and pump/idler waves.

© 2009 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008).
  2. A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010-2013 (2001).
    [CrossRef] [PubMed]
  3. V. N. Tsytovich, Nonlinear Effects in Plasma (Plenum, 1970).
  4. D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. 51, 275-310 (1979).
    [CrossRef]
  5. S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409-1412 (1998).
    [CrossRef]
  6. A. Picozzi and M. Haelterman, “Dispersion-induced dynamical transition in parametric solitary waves,” Phys. Rev. Lett. 84, 5760-5763 (2000).
    [CrossRef] [PubMed]
  7. A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E 66, 56605-56605 (2002).
    [CrossRef]
  8. C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator,” Opt. Commun. 237, 437-449 (2004).
    [CrossRef]
  9. C. Montes, W. Grundkötter, H. Suche, and W. Sohler, “Coherent signal from incoherently cw-pumped singly resonant Ti: LiNbO̱3 integrated optical parametric oscillators,” J. Opt. Soc. Am. B 24, 2796-2806 (2007).
    [CrossRef]
  10. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
    [CrossRef] [PubMed]
  11. Y. Yan and Y. Changxi, “Four-wave mixing between coherent signal and incoherent pump light in nonlinear fiber,” J. Lightwave Technol. 27 (2009). doi: 10.1109/JLT.2009.2027213
  12. K. Hammani, C. Finot, and G. Millot, “Emergence of extreme events in fiber-based parametric processes driven by a partially incoherent pump wave,” Opt. Lett. 34, 1138-1140 (2009).
    [CrossRef] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
  14. F. S. Yang, M. E. Marhic, and L. G. Kazovsky, “cw fiber optical parametric amplifier with net gain and wavelengthconversion efficiency,” Electron. Lett. 32, 2336-2338 (1996).
    [CrossRef]
  15. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
    [CrossRef]
  16. K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Phase-conjugate pump dithering for high-quality idler generation in a fiber optical parametric amplifier,” IEEE Photonics Technol. Lett. 15, 33-35 (2003).
    [CrossRef]
  17. J. W. Goodman, Statistical Optics (Wiley, 2000).
  18. H. Feshfach and P. M. Morse, Methods of Theoretical Physics (McGraw-Hill, 1953).
  19. R. L. Stratonovich, Topics in the Theory of Random Noise (Science Publishers, 1963).
  20. W. Astar, A. S. Lenihan, and G. M. Carter, “Polarization-insensitive wavelength conversion by FWM in a highly nonlinear PCF of polarization-scrambled 10-Gb/s RZ-OOK and RZ-DPSK signals,” IEEE Photonics Technol. Lett. 19, 1676-1678 (2007).
    [CrossRef]
  21. C. J. McKinstrie and C. Xie, “Polarization-independent amplification and frequency conversion in strongly-birefringent fibers,” Opt. Express 16, 16774-16797 (2008).
    [CrossRef] [PubMed]
  22. A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre, “Tailoring cw supercontinuum generation in microstructured fibers with two-zero dispersion wavelengths,” Opt. Express 15, 11553-11563 (2007).
    [CrossRef] [PubMed]

2009 (2)

Y. Yan and Y. Changxi, “Four-wave mixing between coherent signal and incoherent pump light in nonlinear fiber,” J. Lightwave Technol. 27 (2009). doi: 10.1109/JLT.2009.2027213

K. Hammani, C. Finot, and G. Millot, “Emergence of extreme events in fiber-based parametric processes driven by a partially incoherent pump wave,” Opt. Lett. 34, 1138-1140 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (3)

2004 (2)

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator,” Opt. Commun. 237, 437-449 (2004).
[CrossRef]

2003 (2)

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Phase-conjugate pump dithering for high-quality idler generation in a fiber optical parametric amplifier,” IEEE Photonics Technol. Lett. 15, 33-35 (2003).
[CrossRef]

2002 (1)

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E 66, 56605-56605 (2002).
[CrossRef]

2001 (1)

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

2000 (1)

A. Picozzi and M. Haelterman, “Dispersion-induced dynamical transition in parametric solitary waves,” Phys. Rev. Lett. 84, 5760-5763 (2000).
[CrossRef] [PubMed]

1998 (1)

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409-1412 (1998).
[CrossRef]

1996 (1)

F. S. Yang, M. E. Marhic, and L. G. Kazovsky, “cw fiber optical parametric amplifier with net gain and wavelengthconversion efficiency,” Electron. Lett. 32, 2336-2338 (1996).
[CrossRef]

1979 (1)

D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. 51, 275-310 (1979).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Astar, W.

W. Astar, A. S. Lenihan, and G. M. Carter, “Polarization-insensitive wavelength conversion by FWM in a highly nonlinear PCF of polarization-scrambled 10-Gb/s RZ-OOK and RZ-DPSK signals,” IEEE Photonics Technol. Lett. 19, 1676-1678 (2007).
[CrossRef]

Beaugeois, M.

Bers, A.

D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. 51, 275-310 (1979).
[CrossRef]

Bouazaoui, M.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008).

Brar, K.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

Carter, G. M.

W. Astar, A. S. Lenihan, and G. M. Carter, “Polarization-insensitive wavelength conversion by FWM in a highly nonlinear PCF of polarization-scrambled 10-Gb/s RZ-OOK and RZ-DPSK signals,” IEEE Photonics Technol. Lett. 19, 1676-1678 (2007).
[CrossRef]

Centanni, J. C.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

Changxi, Y.

Y. Yan and Y. Changxi, “Four-wave mixing between coherent signal and incoherent pump light in nonlinear fiber,” J. Lightwave Technol. 27 (2009). doi: 10.1109/JLT.2009.2027213

Chraplyvy, A. R.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

Feshfach, H.

H. Feshfach and P. M. Morse, Methods of Theoretical Physics (McGraw-Hill, 1953).

Finot, C.

Gallo, K.

C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator,” Opt. Commun. 237, 437-449 (2004).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Grundkötter, W.

Haelterman, M.

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E 66, 56605-56605 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

A. Picozzi and M. Haelterman, “Dispersion-induced dynamical transition in parametric solitary waves,” Phys. Rev. Lett. 84, 5760-5763 (2000).
[CrossRef] [PubMed]

Hammani, K.

Headley, C.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

Jopson, R. M.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

Jorgensen, C. G.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

Kaup, D. J.

D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. 51, 275-310 (1979).
[CrossRef]

Kazovsky, L. G.

K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Phase-conjugate pump dithering for high-quality idler generation in a fiber optical parametric amplifier,” IEEE Photonics Technol. Lett. 15, 33-35 (2003).
[CrossRef]

F. S. Yang, M. E. Marhic, and L. G. Kazovsky, “cw fiber optical parametric amplifier with net gain and wavelengthconversion efficiency,” Electron. Lett. 32, 2336-2338 (1996).
[CrossRef]

Lenihan, A. S.

W. Astar, A. S. Lenihan, and G. M. Carter, “Polarization-insensitive wavelength conversion by FWM in a highly nonlinear PCF of polarization-scrambled 10-Gb/s RZ-OOK and RZ-DPSK signals,” IEEE Photonics Technol. Lett. 19, 1676-1678 (2007).
[CrossRef]

Marhic, M. E.

K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Phase-conjugate pump dithering for high-quality idler generation in a fiber optical parametric amplifier,” IEEE Photonics Technol. Lett. 15, 33-35 (2003).
[CrossRef]

F. S. Yang, M. E. Marhic, and L. G. Kazovsky, “cw fiber optical parametric amplifier with net gain and wavelengthconversion efficiency,” Electron. Lett. 32, 2336-2338 (1996).
[CrossRef]

McKinstrie, C. J.

C. J. McKinstrie and C. Xie, “Polarization-independent amplification and frequency conversion in strongly-birefringent fibers,” Opt. Express 16, 16774-16797 (2008).
[CrossRef] [PubMed]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

Millot, G.

K. Hammani, C. Finot, and G. Millot, “Emergence of extreme events in fiber-based parametric processes driven by a partially incoherent pump wave,” Opt. Lett. 34, 1138-1140 (2009).
[CrossRef] [PubMed]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409-1412 (1998).
[CrossRef]

Montes, C.

C. Montes, W. Grundkötter, H. Suche, and W. Sohler, “Coherent signal from incoherently cw-pumped singly resonant Ti: LiNbO̱3 integrated optical parametric oscillators,” J. Opt. Soc. Am. B 24, 2796-2806 (2007).
[CrossRef]

C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator,” Opt. Commun. 237, 437-449 (2004).
[CrossRef]

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E 66, 56605-56605 (2002).
[CrossRef]

Morse, P. M.

H. Feshfach and P. M. Morse, Methods of Theoretical Physics (McGraw-Hill, 1953).

Mussot, A.

Picozzi, A.

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator,” Opt. Commun. 237, 437-449 (2004).
[CrossRef]

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E 66, 56605-56605 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

A. Picozzi and M. Haelterman, “Dispersion-induced dynamical transition in parametric solitary waves,” Phys. Rev. Lett. 84, 5760-5763 (2000).
[CrossRef] [PubMed]

Pitois, S.

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409-1412 (1998).
[CrossRef]

Radic, S.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

Reiman, A.

D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. 51, 275-310 (1979).
[CrossRef]

Sohler, W.

Stratonovich, R. L.

R. L. Stratonovich, Topics in the Theory of Random Noise (Science Publishers, 1963).

Suche, H.

Sylvestre, T.

Tsytovich, V. N.

V. N. Tsytovich, Nonlinear Effects in Plasma (Plenum, 1970).

Wabnitz, S.

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409-1412 (1998).
[CrossRef]

Wong, K. K. Y.

K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Phase-conjugate pump dithering for high-quality idler generation in a fiber optical parametric amplifier,” IEEE Photonics Technol. Lett. 15, 33-35 (2003).
[CrossRef]

Xie, C.

Yan, Y.

Y. Yan and Y. Changxi, “Four-wave mixing between coherent signal and incoherent pump light in nonlinear fiber,” J. Lightwave Technol. 27 (2009). doi: 10.1109/JLT.2009.2027213

Yang, F. S.

F. S. Yang, M. E. Marhic, and L. G. Kazovsky, “cw fiber optical parametric amplifier with net gain and wavelengthconversion efficiency,” Electron. Lett. 32, 2336-2338 (1996).
[CrossRef]

Electron. Lett. (1)

F. S. Yang, M. E. Marhic, and L. G. Kazovsky, “cw fiber optical parametric amplifier with net gain and wavelengthconversion efficiency,” Electron. Lett. 32, 2336-2338 (1996).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol. Lett. 15, 673-675 (2003).
[CrossRef]

K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Phase-conjugate pump dithering for high-quality idler generation in a fiber optical parametric amplifier,” IEEE Photonics Technol. Lett. 15, 33-35 (2003).
[CrossRef]

W. Astar, A. S. Lenihan, and G. M. Carter, “Polarization-insensitive wavelength conversion by FWM in a highly nonlinear PCF of polarization-scrambled 10-Gb/s RZ-OOK and RZ-DPSK signals,” IEEE Photonics Technol. Lett. 19, 1676-1678 (2007).
[CrossRef]

J. Lightwave Technol. (1)

Y. Yan and Y. Changxi, “Four-wave mixing between coherent signal and incoherent pump light in nonlinear fiber,” J. Lightwave Technol. 27 (2009). doi: 10.1109/JLT.2009.2027213

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

Opt. Commun. (1)

C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator,” Opt. Commun. 237, 437-449 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. E (1)

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E 66, 56605-56605 (2002).
[CrossRef]

Phys. Rev. Lett. (4)

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409-1412 (1998).
[CrossRef]

A. Picozzi and M. Haelterman, “Dispersion-induced dynamical transition in parametric solitary waves,” Phys. Rev. Lett. 84, 5760-5763 (2000).
[CrossRef] [PubMed]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010-2013 (2001).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. 51, 275-310 (1979).
[CrossRef]

Other (6)

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008).

V. N. Tsytovich, Nonlinear Effects in Plasma (Plenum, 1970).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

J. W. Goodman, Statistical Optics (Wiley, 2000).

H. Feshfach and P. M. Morse, Methods of Theoretical Physics (McGraw-Hill, 1953).

R. L. Stratonovich, Topics in the Theory of Random Noise (Science Publishers, 1963).

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Figures (4)

Fig. 1
Fig. 1

Amplitude and phase of the waves with phase-locking mechanism ( Δ β 1 i = Δ β 1 s = 0 ) . (a) Amplitude of the input signal wave; (b) amplitude of output signal and idler waves; (c) phase of pump f p ( z , T ) ; (d) phase of signal wave f s ( z , T ) ; (e) phase of idler wave f i ( z , T ) ; (f) 2 f p ( z , T ) f i ( z , T ) . Other parameters used in simulation: length of fiber Z = 200 m , nonlinear parameter γ = 5.8 W 1 km 1 Pump power is 1 W .

Fig. 2
Fig. 2

Illustration of the relative β 1 of the PCF versus frequency. The circles mark the β 1 of the three waves.

Fig. 3
Fig. 3

Numerical results of high-coherence signal generated from incoherent pump. PCF length z = 1000 m ; pump average power 0.1 W ; nonlinear parameter γ = 23 W 1 km 1 . (a) Amplitudes of the input signal wave and three output waves. The input signal amplitude is given in an arbitrary unit. (b) Numerical results of the normalized autocorrelation functions R j ( t ) obtained from temporal profiles of the input signal wave and the three output waves.

Fig. 4
Fig. 4

Numerical results of normalized amplitude and correlation function for output signal waves with and without the second-order and third-order dispersion in simulation. PCF length z = 2000 m ; pump average power 0.1 W ; nonlinear parameter γ = 23 W 1 km 1 . (a) With second-order and third-order dispersion; (b) without second and third-order dispersion.

Tables (1)

Tables Icon

Table 1 Frequencies, Dispersion Parameters, and the Relative Value of β 1 j of the Three Waves

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

D s A s = i γ ( 2 | A p | 2 + | A s | 2 + 2 | A i | 2 ) A s + 2 i γ A p A p A i * exp ( i Δ β z ) ,
D i A i = i γ ( 2 | A p | 2 + 2 | A s | 2 + | A i | 2 ) A i + 2 i γ A p A p A s * exp ( i Δ β z ) ,
D p A p = i γ ( | A p | 2 + 2 | A s | 2 + 2 | A i | 2 ) A p + 2 i γ A i A s A p * exp ( i Δ β z ) ,
[ z + Δ β 1 s T 2 i γ | A p ( z , T ) 2 | + α s ] A s = 2 i γ A p A p A i * exp ( i Δ β z ) ( a ) ,
[ z + Δ β 1 i T 2 i γ | A p ( z , T ) | 2 + α i ] A i = 2 i γ A p A p A s * exp ( i Δ β z ) ( b ) ,
A p ( z , T ) = A p ( 0 , T ) exp [ i γ | A p ( 0 , T ) | 2 z ] ( c ) ,
A i ( z , T ) = 2 i γ exp ( 2 i γ z ) 0 z exp ( i Δ β z ) × exp [ α i ( z z ) ] [ A p ( 0 , T ) ] 2 A s * ( z , T ) d z ,
T = T + Δ β 1 i ( z z ) .
[ z + Δ β 1 s T 2 i γ P + α s ] A s ( z , T ) = 4 γ 2 0 z M ( z z ) × [ A p * ( 0 , T ) ] 2 [ A p ( 0 , T ) ] 2 A s ( z , T ) d z ,
( z + Δ β 1 s T 2 i γ | A p | 2 ) A s = 2 i γ A p A p A i * ( a ) ,
( z 2 i γ | A p | 2 ) A i = 2 i γ A p A p A s * ( b ) ,
A p ( z , T ) = A p ( 0 , T ) exp ( i γ | A p ( 0 , T ) | 2 z ) ( c ) .
z [ z + Δ β 1 s T 2 i γ P ( T ) ] A s = 4 γ 2 [ P ( T ) ] 2 A s ,
A s ( z , T ) = 1 2 π + A ̃ ( k , T ) exp ( i k z ) d k .
A s ( T , z ) = 1 2 π + A ̃ ( 0 , k ) exp [ f ( k ) z ] d k ,
f ( k ) = i { 4 γ 2 P 2 [ 1 + m ( T ) ] + k 2 2 γ P 0 [ 1 + n ( T ) ] k } T k Δ β 1 s z + i k ,
m ( T ) = 0 T η ( t ) d t n ( T ) = 0 T ϵ ( t ) d t .
A s ( z , T ) exp [ f ( k 0 ) z ] d k ,
k 0 = 2 γ P [ T + n ( T ) ] ( T Δ β 1 s z ) 0 < T < Δ β 1 s z ,
A s ( z , T ) exp { 4 P γ Δ β 1 s [ T + n ( T ) ] [ ( Δ β 1 s z T ) ] } × exp { i 2 P 0 γ Δ β 1 s [ 1 + m ( T ) ] T } .
A s ( z , T ) exp [ 4 P 0 γ Δ β 1 s T ( Δ β 1 s z T ) ] exp [ i 2 P 0 γ Δ β 1 s T ] .

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