Abstract

We investigate the propagation of two parallel polarized pump beams in the framework of a single nonlinear Schrödinger equation in order to thoroughly understand the modulation instability process in the regime of normal group-velocity dispersion. A linear stability analysis based on Floquet theory allows us to account for four-wave mixing and to contrast the results with those arising from incoherent nonlinear coupling due to cross-phase modulation only. Based on the nature of the unstable eigenvectors, we explain why, in the normal dispersion regime, modulation instability is not observed in the scalar configuration. Numerical simulations validate the analysis.

© 2014 Optical Society of America

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  1. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
    [CrossRef]
  2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
    [CrossRef]
  3. G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
    [CrossRef]
  4. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).
  5. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [CrossRef]
  6. C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
    [CrossRef]
  7. C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
    [CrossRef]
  8. G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
    [CrossRef]
  9. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
    [CrossRef]
  10. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhart, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
    [CrossRef]
  11. C. De Angelis, M. Santagiustina, and S. Trillo, “Induced modulational instability in high-birefringence fibers: the strong conversion regime,” Opt. Lett. 19, 335–337 (1994).
    [CrossRef]
  12. C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51, 774–791 (1995).
    [CrossRef]
  13. J. S. Y. Chen, G. K. L. Wong, S. G. Murdoch, R. J. Kruhlak, R. Leonhardt, J. D. Harvey, N. Y. Joly, and J. C. Knight, “Cross-phase modulation instability in photonic crystal fibers,” Opt. Lett. 31, 873–875 (2006).
    [CrossRef]
  14. E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
    [CrossRef]
  15. A. Kudlinski, A. Bendahmane, D. Labat, S. Virally, R. T. Murray, E. J. R. Kelleher, and A. Mussot, “Simultaneous scalar and cross-phase modulation instabilities in highly birefringent photonic crystal fiber,” Opt. Express 21, 8437–8443 (2013).
    [CrossRef]
  16. A different type of MI in birefringent fibers involves coherent coupling between the two polarizations; see S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
    [CrossRef]
  17. G. Millot, S. Pitois, and P. Tchofo-Dinda, “Modulational instability processes in optical isotropic fibers under dual frequency circular polarization pumping,” J. Opt. Soc. Am. B 19, 454–460 (2002).
    [CrossRef]
  18. J. R. Thompson and R. Roy, “Nonlinear dynamics of multiple four-wave mixing processes in a single-mode fiber,” Phys. Rev. A 43, 4987–4996 (1991).
    [CrossRef]
  19. D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, “Conservation law for multiple four-wave-mixing processes in a nonlinear optical medium,” Phys. Rev. A 50, 1807–1813 (1994).
    [CrossRef]
  20. S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
    [CrossRef]
  21. D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
    [CrossRef]
  22. X. Liu, X. Zhou, and C. Lou, “Multiple four-wave mixing self-stability in optical fibers,” Phys. Rev. A 72, 013811 (2005).
    [CrossRef]
  23. J. E. Rothenberg, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813 (1990).
    [CrossRef]
  24. G. P. Agrawal, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813 (1990).
    [CrossRef]
  25. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
    [CrossRef]
  26. T. Tanemura and K. Kikuchi, “Unified analysis of modulational instability induced by cross-phase modulation in optical fibers,” J. Opt. Soc. Am. B 20, 2502–2514 (2003).
    [CrossRef]
  27. A. Armaroli and S. Trillo, “Collective modulation instability of multiple four-wave mixing,” Opt. Lett. 36, 1999–2001 (2011).
    [CrossRef]
  28. J. Fatome, C. Finot, A. Armaroli, and S. Trillo, “Observation of modulationally unstable multi-wave mixing,” Opt. Lett. 38, 181–183 (2013).
    [CrossRef]
  29. S. Trillo and S. Wabnitz, “Dynamic spontaneous fluorescence in parametric wave coupling,” Phys. Rev. E 55, R4897–R4900 (1997).
    [CrossRef]
  30. S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E 56, 1048–1058 (1997).
    [CrossRef]
  31. R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
    [CrossRef]
  32. E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13, 142–144 (2001).
    [CrossRef]
  33. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
    [CrossRef]
  34. C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
    [CrossRef]
  35. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge University, 2008).
  36. Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
  37. J. Fatome, S. Pitois, and G. Millot, “20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave-mixing in optical fibers,” IEEE J. Quantum Electron. 42, 1038–1046 (2006).
    [CrossRef]
  38. L. A. Bui, M. D. Pelusi, T. D. Vo, N. Sarkhosh, H. Emami, B. J. Eggleton, and A. Mitchell, “Instantaneous frequency measurement system using optical mixing in highly nonlinear fiber,” Opt. Express 17, 22983–22991 (2009).
    [CrossRef]
  39. V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. G. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett. 35, 1440–1442 (2010).
    [CrossRef]
  40. S. Trillo and A. Valiani, “Hydrodynamic instability of four-wave-mixing,” Opt. Lett. 35, 3967–3969 (2010).
    [CrossRef]

2013

2011

A. Armaroli and S. Trillo, “Collective modulation instability of multiple four-wave mixing,” Opt. Lett. 36, 1999–2001 (2011).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

2010

2009

2007

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
[CrossRef]

2006

J. S. Y. Chen, G. K. L. Wong, S. G. Murdoch, R. J. Kruhlak, R. Leonhardt, J. D. Harvey, N. Y. Joly, and J. C. Knight, “Cross-phase modulation instability in photonic crystal fibers,” Opt. Lett. 31, 873–875 (2006).
[CrossRef]

J. Fatome, S. Pitois, and G. Millot, “20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave-mixing in optical fibers,” IEEE J. Quantum Electron. 42, 1038–1046 (2006).
[CrossRef]

2005

X. Liu, X. Zhou, and C. Lou, “Multiple four-wave mixing self-stability in optical fibers,” Phys. Rev. A 72, 013811 (2005).
[CrossRef]

2003

2002

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[CrossRef]

G. Millot, S. Pitois, and P. Tchofo-Dinda, “Modulational instability processes in optical isotropic fibers under dual frequency circular polarization pumping,” J. Opt. Soc. Am. B 19, 454–460 (2002).
[CrossRef]

2001

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13, 142–144 (2001).
[CrossRef]

2000

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

1998

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

1997

S. Trillo and S. Wabnitz, “Dynamic spontaneous fluorescence in parametric wave coupling,” Phys. Rev. E 55, R4897–R4900 (1997).
[CrossRef]

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E 56, 1048–1058 (1997).
[CrossRef]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

1995

C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51, 774–791 (1995).
[CrossRef]

1994

C. De Angelis, M. Santagiustina, and S. Trillo, “Induced modulational instability in high-birefringence fibers: the strong conversion regime,” Opt. Lett. 19, 335–337 (1994).
[CrossRef]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, “Conservation law for multiple four-wave-mixing processes in a nonlinear optical medium,” Phys. Rev. A 50, 1807–1813 (1994).
[CrossRef]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef]

1993

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

1991

J. R. Thompson and R. Roy, “Nonlinear dynamics of multiple four-wave mixing processes in a single-mode fiber,” Phys. Rev. A 43, 4987–4996 (1991).
[CrossRef]

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
[CrossRef]

1990

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
[CrossRef]

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhart, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

J. E. Rothenberg, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813 (1990).
[CrossRef]

G. P. Agrawal, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813 (1990).
[CrossRef]

1989

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef]

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[CrossRef]

1988

A different type of MI in birefringent fibers involves coherent coupling between the two polarizations; see S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[CrossRef]

1987

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef]

1986

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef]

1982

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[CrossRef]

1970

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Agrawal, G. P.

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

G. P. Agrawal, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813 (1990).
[CrossRef]

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef]

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef]

Alfano, R. R.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef]

Andrekson, P. A.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Armaroli, A.

Baboiu, D. M.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Baldeck, P. L.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef]

Beletic, J. W.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

Bendahmane, A.

Berkhoer, A. L.

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Bingham, R.

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[CrossRef]

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[CrossRef]

Blessing, D. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Bui, L. A.

Cappellini, G.

Cestier, I.

Chen, J. S. Y.

Chraplyvy, A. R.

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[CrossRef]

Ciaramella, E.

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13, 142–144 (2001).
[CrossRef]

Colman, P.

Combrié, S.

Curti, F.

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13, 142–144 (2001).
[CrossRef]

De Angelis, C.

C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51, 774–791 (1995).
[CrossRef]

C. De Angelis, M. Santagiustina, and S. Trillo, “Induced modulational instability in high-birefringence fibers: the strong conversion regime,” Opt. Lett. 19, 335–337 (1994).
[CrossRef]

De Rossi, A.

Drummond, P. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhart, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Dudley, J. M.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhart, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Eckhouse, V.

Eggleton, B. J.

Eisenstein, G.

Emami, H.

Fatome, J.

J. Fatome, C. Finot, A. Armaroli, and S. Trillo, “Observation of modulationally unstable multi-wave mixing,” Opt. Lett. 38, 181–183 (2013).
[CrossRef]

J. Fatome, S. Pitois, and G. Millot, “20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave-mixing in optical fibers,” IEEE J. Quantum Electron. 42, 1038–1046 (2006).
[CrossRef]

Finot, C.

Foster, M. A.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
[CrossRef]

Fuerst, R. A.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Gaeta, A. L.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
[CrossRef]

Geraghty, D. F.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
[CrossRef]

Grüner-Nielsen, L.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Hart, D. L.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, “Conservation law for multiple four-wave-mixing processes in a nonlinear optical medium,” Phys. Rev. A 50, 1807–1813 (1994).
[CrossRef]

Harvey, J. D.

J. S. Y. Chen, G. K. L. Wong, S. G. Murdoch, R. J. Kruhlak, R. Leonhardt, J. D. Harvey, N. Y. Joly, and J. C. Knight, “Cross-phase modulation instability in photonic crystal fibers,” Opt. Lett. 31, 873–875 (2006).
[CrossRef]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhart, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Hasegawa, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef]

Joly, N. Y.

Judy, A. F.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, “Conservation law for multiple four-wave-mixing processes in a nonlinear optical medium,” Phys. Rev. A 50, 1807–1813 (1994).
[CrossRef]

Karlsson, M.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Kelleher, E. J. R.

Kennedy, T. A. B.

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, “Conservation law for multiple four-wave-mixing processes in a nonlinear optical medium,” Phys. Rev. A 50, 1807–1813 (1994).
[CrossRef]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhart, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Kikuchi, K.

Knight, J. C.

Kruhlak, R. J.

Kudlinski, A.

Labat, D.

Lawrence, B.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Leonhardt, R.

Leonhart, R.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhart, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Lipson, M.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
[CrossRef]

Liu, X.

X. Liu, X. Zhou, and C. Lou, “Multiple four-wave mixing self-stability in optical fibers,” Phys. Rev. A 72, 013811 (2005).
[CrossRef]

Lou, C.

X. Liu, X. Zhou, and C. Lou, “Multiple four-wave mixing self-stability in optical fibers,” Phys. Rev. A 72, 013811 (2005).
[CrossRef]

Lundström, C.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Luther, G. G.

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
[CrossRef]

Marhic, M. E.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge University, 2008).

McKinstrie, C. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[CrossRef]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
[CrossRef]

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[CrossRef]

Millot, G.

J. Fatome, S. Pitois, and G. Millot, “20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave-mixing in optical fibers,” IEEE J. Quantum Electron. 42, 1038–1046 (2006).
[CrossRef]

G. Millot, S. Pitois, and P. Tchofo-Dinda, “Modulational instability processes in optical isotropic fibers under dual frequency circular polarization pumping,” J. Opt. Soc. Am. B 19, 454–460 (2002).
[CrossRef]

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

Mitchell, A.

Murdoch, S. G.

Murray, R. T.

Mussot, A.

Pelusi, M. D.

Pitois, S.

J. Fatome, S. Pitois, and G. Millot, “20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave-mixing in optical fibers,” IEEE J. Quantum Electron. 42, 1038–1046 (2006).
[CrossRef]

G. Millot, S. Pitois, and P. Tchofo-Dinda, “Modulational instability processes in optical isotropic fibers under dual frequency circular polarization pumping,” J. Opt. Soc. Am. B 19, 454–460 (2002).
[CrossRef]

Puttnam, B. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Radic, S.

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813 (1990).
[CrossRef]

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef]

Roy, R.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, “Conservation law for multiple four-wave-mixing processes in a nonlinear optical medium,” Phys. Rev. A 50, 1807–1813 (1994).
[CrossRef]

J. R. Thompson and R. Roy, “Nonlinear dynamics of multiple four-wave mixing processes in a single-mode fiber,” Phys. Rev. A 43, 4987–4996 (1991).
[CrossRef]

Salem, R.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
[CrossRef]

Santagiustina, M.

Sarkhosh, N.

Seve, E.

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

Someda, C. G.

Stegeman, G. I.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Stoev, K.

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, “Conservation law for multiple four-wave-mixing processes in a nonlinear optical medium,” Phys. Rev. A 50, 1807–1813 (1994).
[CrossRef]

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[CrossRef]

Tai, K.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef]

Tanemura, T.

Tchofo-Dinda, P.

Thompson, J. R.

J. R. Thompson and R. Roy, “Nonlinear dynamics of multiple four-wave mixing processes in a single-mode fiber,” Phys. Rev. A 43, 4987–4996 (1991).
[CrossRef]

Tipsuwannakul, E.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Toda, H.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Tomita, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef]

Tong, Z.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Torruellas, W. E.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Trillo, S.

J. Fatome, C. Finot, A. Armaroli, and S. Trillo, “Observation of modulationally unstable multi-wave mixing,” Opt. Lett. 38, 181–183 (2013).
[CrossRef]

A. Armaroli and S. Trillo, “Collective modulation instability of multiple four-wave mixing,” Opt. Lett. 36, 1999–2001 (2011).
[CrossRef]

S. Trillo and A. Valiani, “Hydrodynamic instability of four-wave-mixing,” Opt. Lett. 35, 3967–3969 (2010).
[CrossRef]

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13, 142–144 (2001).
[CrossRef]

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

S. Trillo and S. Wabnitz, “Dynamic spontaneous fluorescence in parametric wave coupling,” Phys. Rev. E 55, R4897–R4900 (1997).
[CrossRef]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E 56, 1048–1058 (1997).
[CrossRef]

C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51, 774–791 (1995).
[CrossRef]

C. De Angelis, M. Santagiustina, and S. Trillo, “Induced modulational instability in high-birefringence fibers: the strong conversion regime,” Opt. Lett. 19, 335–337 (1994).
[CrossRef]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef]

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
[CrossRef]

Turner, A. C.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
[CrossRef]

Vadalà, G.

Valiani, A.

Virally, S.

Vo, T. D.

Wabnitz, S.

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E 56, 1048–1058 (1997).
[CrossRef]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

S. Trillo and S. Wabnitz, “Dynamic spontaneous fluorescence in parametric wave coupling,” Phys. Rev. E 55, R4897–R4900 (1997).
[CrossRef]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef]

A different type of MI in birefringent fibers involves coherent coupling between the two polarizations; see S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[CrossRef]

Wong, G. K. L.

Yu, M.

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

Zakharov, V. E.

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Zhou, X.

X. Liu, X. Zhou, and C. Lou, “Multiple four-wave mixing self-stability in optical fibers,” Phys. Rev. A 72, 013811 (2005).
[CrossRef]

IEEE J. Quantum Electron.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[CrossRef]

J. Fatome, S. Pitois, and G. Millot, “20-GHz-to-1-THz repetition rate pulse sources based on multiple four-wave-mixing in optical fibers,” IEEE J. Quantum Electron. 42, 1038–1046 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13, 142–144 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photonics

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nat. Photonics 2, 35–38 (2007).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).

Opt. Commun.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhart, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Fluids B

C. J. McKinstrie and R. Bingham, “The modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
[CrossRef]

Phys. Rev. A

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406–3413 (1989).
[CrossRef]

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef]

C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51, 774–791 (1995).
[CrossRef]

J. R. Thompson and R. Roy, “Nonlinear dynamics of multiple four-wave mixing processes in a single-mode fiber,” Phys. Rev. A 43, 4987–4996 (1991).
[CrossRef]

D. L. Hart, A. F. Judy, T. A. B. Kennedy, R. Roy, and K. Stoev, “Conservation law for multiple four-wave-mixing processes in a nonlinear optical medium,” Phys. Rev. A 50, 1807–1813 (1994).
[CrossRef]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
[CrossRef]

A different type of MI in birefringent fibers involves coherent coupling between the two polarizations; see S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[CrossRef]

X. Liu, X. Zhou, and C. Lou, “Multiple four-wave mixing self-stability in optical fibers,” Phys. Rev. A 72, 013811 (2005).
[CrossRef]

Phys. Rev. E

S. Trillo and S. Wabnitz, “Dynamic spontaneous fluorescence in parametric wave coupling,” Phys. Rev. E 55, R4897–R4900 (1997).
[CrossRef]

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E 56, 1048–1058 (1997).
[CrossRef]

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave mixing processes in an optical fiber,” Phys. Rev. E 57, 4757–4774 (1998).
[CrossRef]

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase modulation induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

Phys. Rev. Lett.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
[CrossRef]

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisoliton-like generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

J. E. Rothenberg, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813 (1990).
[CrossRef]

G. P. Agrawal, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813 (1990).
[CrossRef]

Phys. Scr.

C. J. McKinstrie and G. G. Luther, “The modulational instability of collinear waves,” Phys. Scr. T30, 31–40 (1990).
[CrossRef]

Sov. Phys. JETP

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Other

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge University, 2008).

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Figures (6)

Fig. 1.
Fig. 1.

Scheme of the spectral lines involved in the FWM process and MI analysis along with the complex amplitude notation used throughout the text. The central zero frequency in this scheme corresponds to the carrier frequency ω0, while all symbols denote actually normalized angular frequency offsets.

Fig. 2.
Fig. 2.

(a) Color level plot of MI gain in the frequency plane (δω,Ω), as obtained from the IC-NLS model in Eq. (2). The black solid line δω=2Ω gives the frequency offset of n=3 FWM modes, which are neglected in the IC-NLS model. The white-dashed curve yields the nonlinear phase-matching frequency δω [Eq. (4)]. (b) Gain profile g=g(δω) sampled at Ω=2 (solid line) compared with the analytic expression arising from two-wave ansatz [Eq. (5), dashed curve]. The inset shows the power fractions (red solid stems) of the unstable eigenvector components versus frequency ω, corresponding to the peak gain. The arrows indicate to which pump beam (dashed vertical lines) the sidebands refer.

Fig. 3.
Fig. 3.

(a) Output spectrum (log-scale) at z=8, showing the growth of strongly asymmetric sidebands around the pumps, ruled by the IC-NLS model, in the normal GVD regime. The dashed blue line indicates the optimum detuning yielding maximum gain or nonlinear phase matching [Eq. (4)], while the dashed red lines labeled As1,s2 stand for the frequency of the first-order FWM products. Here Ω=1.5 and the input is symmetric (u1=u2=1/2, |u1|2+|u2|2=1) in the presence of noise.

Fig. 4.
Fig. 4.

Gain curves from Floquet analysis (solid blue line) versus modulation frequency δω, compared with those arising from the IC-NLS model (dashed red line), for pump detunings (a) Ω=1.25, (b) Ω=1.5, and (c) Ω=2. The insets report for each main panel the structure of the unstable eigenvector (blue solid line) at the peak of each of the three main bands labeled 1, 2, and 3. The dashed red vertical lines locate the n=1 (pump) and n=3 (signal) FWM modes, and the arrows indicate the FWM components to which the eigenvector components refer.

Fig. 5.
Fig. 5.

(a), (c), and (e) Output spectra at z=20 from Eq. (1), for β=1 and different input detunings Ω. Panels (b), (d), and (f) show the early stage evolution of the pump components (solid green), compared with that obtained from the four-mode truncation [Eq. (7)] (dashed red).

Fig. 6.
Fig. 6.

(a) Output spectrum at z=16 obtained from dual-frequency input (Ω=1.5) with noise and additional seed along the most unstable eigenvector of band 2 (δω=3.6). (b) Evolution of pump: NLS evolution (green solid) and truncated ODE solution (red-dashed). (c) Seed–idler pair and noise floor evolution (no amplification of the latter is detected). The dashed line gives the growth rate associated with the peak gain of band 2 in Fig. 4(b).

Equations (19)

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

iuzβ22ut2+|u|2u=0,
i(u1z+δ2u1t)β22u1t2+(|u1|2+2|u2|2)u1=0,i(u2zδ2u2t)β22u2t2+(|u2|2+2|u1|2)u2=0,
uk(z,t)=[Pk+εk(z,t)]exp[iϕk(z)],
δω=Ω+Ω21.
g=2(XP1)2(βδω22+P1+δδω)2.
u(z,t)=12[Ap1(z)exp(iΩt)+Ap2(z)exp(iΩt)+As1(z)exp(i3Ωt)+As2(z)exp(i3Ωt)],
idAp1dz=(|Ap1|22+|Ap2|2+|As1|2+|As2|2)Ap1+As1As2Ap2*+As1Ap1*Ap2+Ap22As2*2+βΩ22Ap1,idAs1dz=(|Ap1|2+|Ap2|2+|As1|22+|As2|2)As1+Ap12Ap2*2+Ap1Ap2As2*+9βΩ22As1,
Ak(z,t)=[ηk(z)+εk(z,t)]exp[iϕk(z)],
dXdz=Mp(δω)X,
δω26Ωδω+8Ω2+1=0,
η˙p1=ηp22ηs22sin(ψ2)ηp1ηp2ηs1sin(ψ1)ηp2ηs1ηs2sin(ψ3),η˙p2=ηp12ηs12sin(ψ1)ηp1ηp2ηs2sin(ψ2)ηp1ηs1ηs2sin(ψ3),η˙s1=ηp12ηp22sin(ψ1)+ηp1ηp2ηs2sin(ψ3),η˙s2=ηp22ηp12sin(ψ2)+ηp1ηp2ηs1sin(ψ3),
ϕ˙p1=βΩ22+ηp122+ηp22+ηs12+ηs22+ηs1ηs2ηp2ηp1cos(ψ3)+ηs1ηp2cos(ψ1)+ηp22ηs22ηp1cos(ψ2),ϕ˙p2=βΩ22+ηp12+ηp222+ηs12+ηs22+ηs1ηs2ηp1ηp2cos(ψ3)+ηs2ηp1cos(ψ2)+ηp12ηs12ηp2cos(ψ1),ϕ˙s1=9βΩ22+ηp12+ηp22+ηs122+ηs22+ηp12ηp22ηs1cos(ψ1)+ηp1ηp2ηs2ηs1cos(ψ3),ϕ˙s2=9βΩ22+ηp12+ηp22+ηs12+ηs222+ηp22ηp12ηs2cos(ψ2)+ηp1ηp2ηs1ηs2cos(ψ3).
H(η,ψ)=12(1κ)ηη22+η(1η)cos2ψ+ηη(1η)cosψ;ψ˙=Hη;η˙=Hψ,
A˙=f(A),
a˙=Ja;JfA.
εk(z,t)=εka(z)exp(iδωt)+εks(z)exp(iδωt),
iε˙p1a=[β2(δω2+2Ωδω)+ηp2232ηsηpcos(ψ)ηs2cos(2ψ)]εp1a+[ηp22+ηpηsexp(iψ)]εp1s*+[ηp2+2ηpηscos(ψ)]εp2a+[ηp2+ηs2exp(i2ψ)]εp2s*+{ηpηs[1+exp(i2ψ)]+ηp2exp(iψ)}εs1a+ηpηsεs1s*+ηpηs[1+exp(i2ψ)]εs2a+[ηpηs+ηp22exp(iψ)]εs2s*;iε˙p1s*=[β2(δω22Ωδω)+ηp2232ηsηpcos(ψ)ηs2cos(2ψ)]εp1s*+[ηp22+ηpηsexp(iψ)]εp1a+[ηp2+2ηpηscos(ψ)]εp2s*+[ηp2+ηs2exp(i2ψ)]εp2a+{ηpηs[1+exp(i2ψ)]+ηp2exp(iψ)}εs1s*+ηsηpεs1a+ηpηs[1+exp(i2ψ)]εs2s*+[ηpηs+ηp22exp(iψ)]εs2a;
iε˙s1a=[β2(δω2+6Ωδω)+ηs22ηp2cos(2ψ)ηp32ηscos(ψ)]εs1a+ηs22εs1s*+ηs2εs2a+[ηs2+ηp2exp(i2ψ)]εs2s*+{ηpηs[1+exp(i2ψ)]+ηp2exp(iψ)}εp1a+ηpηsεp1s*+ηpηs[1+exp(i2ψ)]εp2a+[ηpηs+ηp22exp(iψ)]εp2s*;iε˙s1s*=[(δω26Ωδω)+ηs22ηp2cos(2ψ)ηp32ηscos(ψ)]εs1s*+ηs22εs1a+ηs2εs2s*+[ηs2+ηp2exp(i2ψ)]εs2a+{ηpηs[1+exp(i2ψ)]+ηp2exp(iψ)}εp1s*+ηpηsεp1a+ηpηs[1+exp(i2ψ)]εp2s*+[ηpηs+ηp22exp(iψ)]εp2a*,
X(z)=(εp1a,εp1s*,εp2a,εp2s*,εs1a,εs1s*,εs2a,εs2s*)T.

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