Abstract

Cross-phase modulation in an optical fiber can lead to various types of modulational instability, such as polarization instability in a highly birefringent fiber and two-pump optical parametric amplification. We present unified analyses of such instabilities and clarify the general mechanisms behind them. By solving the generalized eigenvalue equation, we indicate the explicit conditions of inducing modulational instability. The eigenvector is also calculated, which allows us to explain underlying physics of instabilities using a phasor diagram. As a result, all types of cross-phase modulation-induced modulational instability are classified into three types in terms of their mechanisms.

© 2003 Optical Society of America

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  1. L. A. Ostrovskii, “Propagation of wave packets and space–time self-focusing in a nonlinear medium,” Zh. Eksp. Teor. Fiz. 51, 1189–1194 (1966) [Sov. Phys. JETP 24, 797–800 (1967)].
  2. A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
    [Crossref]
  3. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
    [Crossref]
  4. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
    [Crossref]
  5. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
    [Crossref] [PubMed]
  6. S. Trillo and S. Wabnitz, “Parametric and Raman amplification in birefringent fibers,” J. Opt. Soc. Am. B 9, 1061–1082 (1992).
    [Crossref]
  7. E. Lantz, D. Gindre, H. Maillotte, and J. Monneret, “Phase matching for parametric amplification in a single-mode birefringent fiber: influence of the non-phase-matched waves,” J. Opt. Soc. Am. B 14, 116–125 (1997).
    [Crossref]
  8. J. Hong and W. P. Huang, “Modulation instability in highly birefringent optical fibers: a coupled-mode analysis,” IEEE J. Quantum Electron. 28, 1838–1843 (1992).
    [Crossref]
  9. E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
    [Crossref] [PubMed]
  10. P. T. Dinda, G. Millot, E. Seve, and M. Haelterman, “Demonstration of a nonlinear gap in the modulational instability spectra of wave propagation in highly birefringent fibers,” Opt. Lett. 21, 1640–1642 (1996).
    [Crossref] [PubMed]
  11. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].
  12. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [Crossref] [PubMed]
  13. G. Millot, S. Pitois, P. T. Dinda, and M. Haelterman, “Observation of modulational instability induced by velocity-matched cross-phase modulation in a normally dispersive bimodal fiber,” Opt. Lett. 22, 1686–1688 (1997).
    [Crossref]
  14. G. Millot, S. Pitois, and P. T. Dinda, “Modulational instability processes in optical isotropic fibers under dual-frequency circular polarization pumping,” J. Opt. Soc. Am. B 19, 454–460 (2002).
    [Crossref]
  15. J. E. Rothenberg, “Modulational instability of copropagat-ing frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813–814 (1990).
    [Crossref]
  16. W. Huang and J. Hong, “A coupled-mode analysis of modulation instability in optical fibers,” J. Lightwave Technol. 10, 156–162 (1992).
    [Crossref]
  17. 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]
  18. F. S. Yang, M. C. Ho, M. E. Marhic, and L. G. Kazovsky, “Demonstration of two-pump fibre optical parametric amplification,” Electron. Lett. 33, 1812–1813 (1997).
    [Crossref]
  19. J. M. C. Boggio, S. Tenenbaum, and H. L. Fragnito, “Amplification of broadband noise pumped by two lasers in optical fibers,” J. Opt. Soc. Am. B 18, 1428–1435 (2001).
    [Crossref]
  20. 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) [correction: 956 (2002)].
    [Crossref]
  21. S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
    [Crossref]
  22. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
    [Crossref]
  23. B. L. van der Waerden, Algebra, Vol. I (Springer-Verlag, New York, 1991).
  24. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
    [Crossref] [PubMed]

2002 (3)

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

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) [correction: 956 (2002)].
[Crossref]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

2001 (1)

1997 (3)

1996 (2)

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[Crossref] [PubMed]

P. T. Dinda, G. Millot, E. Seve, and M. Haelterman, “Demonstration of a nonlinear gap in the modulational instability spectra of wave propagation in highly birefringent fibers,” Opt. Lett. 21, 1640–1642 (1996).
[Crossref] [PubMed]

1993 (1)

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]

1992 (3)

W. Huang and J. Hong, “A coupled-mode analysis of modulation instability in optical fibers,” J. Lightwave Technol. 10, 156–162 (1992).
[Crossref]

J. Hong and W. P. Huang, “Modulation instability in highly birefringent optical fibers: a coupled-mode analysis,” IEEE J. Quantum Electron. 28, 1838–1843 (1992).
[Crossref]

S. Trillo and S. Wabnitz, “Parametric and Raman amplification in birefringent fibers,” J. Opt. Soc. Am. B 9, 1061–1082 (1992).
[Crossref]

1990 (3)

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

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

J. E. Rothenberg, “Modulational instability of copropagat-ing frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813–814 (1990).
[Crossref]

1989 (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

1988 (1)

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

1987 (2)

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

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

1980 (1)

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

1970 (1)

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].

1966 (1)

L. A. Ostrovskii, “Propagation of wave packets and space–time self-focusing in a nonlinear medium,” Zh. Eksp. Teor. Fiz. 51, 1189–1194 (1966) [Sov. Phys. JETP 24, 797–800 (1967)].

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

Berkhoer, A. L.

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].

Bilbault, J. M.

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[Crossref] [PubMed]

Boggio, J. M. C.

Brar, K.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

Brinkman, W. F.

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

Centanni, J. C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

Chraplyvy, A. R.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

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) [correction: 956 (2002)].
[Crossref]

Dinda, P. T.

Drummond, P. D.

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

Fragnito, H. L.

Gindre, D.

Haelterman, M.

Harvey, J. D.

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

Hasegawa, A.

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

Headley, C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

Ho, M. C.

F. S. Yang, M. C. Ho, M. E. Marhic, and L. G. Kazovsky, “Demonstration of two-pump fibre optical parametric amplification,” Electron. Lett. 33, 1812–1813 (1997).
[Crossref]

Hong, J.

W. Huang and J. Hong, “A coupled-mode analysis of modulation instability in optical fibers,” J. Lightwave Technol. 10, 156–162 (1992).
[Crossref]

J. Hong and W. P. Huang, “Modulation instability in highly birefringent optical fibers: a coupled-mode analysis,” IEEE J. Quantum Electron. 28, 1838–1843 (1992).
[Crossref]

Huang, W.

W. Huang and J. Hong, “A coupled-mode analysis of modulation instability in optical fibers,” J. Lightwave Technol. 10, 156–162 (1992).
[Crossref]

Huang, W. P.

J. Hong and W. P. Huang, “Modulation instability in highly birefringent optical fibers: a coupled-mode analysis,” IEEE J. Quantum Electron. 28, 1838–1843 (1992).
[Crossref]

Jorgensen, C. G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

Kazovsky, L. G.

F. S. Yang, M. C. Ho, M. E. Marhic, and L. G. Kazovsky, “Demonstration of two-pump fibre optical parametric amplification,” Electron. Lett. 33, 1812–1813 (1997).
[Crossref]

Kennedy, T. A. B.

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

Lantz, E.

Leonhardt, R.

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

Maillotte, H.

Marhic, M. E.

F. S. Yang, M. C. Ho, M. E. Marhic, and L. G. Kazovsky, “Demonstration of two-pump fibre optical parametric amplification,” Electron. Lett. 33, 1812–1813 (1997).
[Crossref]

McKinstrie, C. J.

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) [correction: 956 (2002)].
[Crossref]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (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]

Menyuk, C. R.

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

Millot, G.

Monneret, J.

Ostrovskii, L. A.

L. A. Ostrovskii, “Propagation of wave packets and space–time self-focusing in a nonlinear medium,” Zh. Eksp. Teor. Fiz. 51, 1189–1194 (1966) [Sov. Phys. JETP 24, 797–800 (1967)].

Pitois, S.

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) [correction: 956 (2002)].
[Crossref]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

Raybon, G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

Remoissenet, M.

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[Crossref] [PubMed]

Rothenberg, J. E.

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

J. E. Rothenberg, “Modulational instability of copropagat-ing frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813–814 (1990).
[Crossref]

Seve, E.

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[Crossref] [PubMed]

P. T. Dinda, G. Millot, E. Seve, and M. Haelterman, “Demonstration of a nonlinear gap in the modulational instability spectra of wave propagation in highly birefringent fibers,” Opt. Lett. 21, 1640–1642 (1996).
[Crossref] [PubMed]

Tenenbaum, S.

Trillo, S.

van der Waerden, B. L.

B. L. van der Waerden, Algebra, Vol. I (Springer-Verlag, New York, 1991).

Wabnitz, S.

S. Trillo and S. Wabnitz, “Parametric and Raman amplification in birefringent fibers,” J. Opt. Soc. Am. B 9, 1061–1082 (1992).
[Crossref]

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

Yang, F. S.

F. S. Yang, M. C. Ho, M. E. Marhic, and L. G. Kazovsky, “Demonstration of two-pump fibre optical parametric amplification,” Electron. Lett. 33, 1812–1813 (1997).
[Crossref]

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,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].

Electron. Lett. (1)

F. S. Yang, M. C. Ho, M. E. Marhic, and L. G. Kazovsky, “Demonstration of two-pump fibre optical parametric amplification,” Electron. Lett. 33, 1812–1813 (1997).
[Crossref]

IEEE J. Quantum Electron. (4)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980).
[Crossref]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

J. Hong and W. P. Huang, “Modulation instability in highly birefringent optical fibers: a coupled-mode analysis,” IEEE J. Quantum Electron. 28, 1838–1843 (1992).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

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) [correction: 956 (2002)].
[Crossref]

IEEE Photon. Technol. Lett. (1)

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing,” IEEE Photon. Technol. Lett. 14, 1406–1408 (2002).
[Crossref]

J. Lightwave Technol. (1)

W. Huang and J. Hong, “A coupled-mode analysis of modulation instability in optical fibers,” J. Lightwave Technol. 10, 156–162 (1992).
[Crossref]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

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

Opt. Lett. (2)

Phys. Rev. A (3)

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
[Crossref] [PubMed]

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

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[Crossref] [PubMed]

Phys. Rev. E (1)

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. (2)

J. E. Rothenberg, “Modulational instability of copropagat-ing frequencies for normal dispersion,” Phys. Rev. Lett. 64, 813–814 (1990).
[Crossref]

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

Zh. Eksp. Teor. Fiz. (2)

L. A. Ostrovskii, “Propagation of wave packets and space–time self-focusing in a nonlinear medium,” Zh. Eksp. Teor. Fiz. 51, 1189–1194 (1966) [Sov. Phys. JETP 24, 797–800 (1967)].

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Zh. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)].

Other (1)

B. L. van der Waerden, Algebra, Vol. I (Springer-Verlag, New York, 1991).

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Figures (15)

Fig. 1
Fig. 1

Frequency components involved in a general incoherent XPM-MI process. Note, ω1=ω2 for a degenerate case such as PI in a Hi-Bi fiber.

Fig. 2
Fig. 2

Distribution of |Im(κ)| on a (d1, d2) plane calculated for the case of Δ=0. (a) R=0. (b) R=0.5, r=2. (c) R=1, r=2. (d) R=1, r=2/3.

Fig. 3
Fig. 3

Relative magnitudes of respective sidebands composing the MI eigenmode for the case of Δ=0,R=1, and r=2. (a) |A1+|2. (b) |A1-|2. (c) |A2+|2. (d) |A2-|2.

Fig. 4
Fig. 4

Relative magnitudes of the AM/PM components of the MI eigenmode for the case of Δ=0,R=1, and r=2. (a) |ψAM1|2. (b) |ψPM1|2. (c) |ψAM2|2. (d) |ψPM2|2.

Fig. 5
Fig. 5

Relative phases of the AM/PM components of the MI eigenmode for the case of Δ=0,R=1, and r=2. (a) arg(ψAM1). (b) arg(ψPM1). (c) arg(ψAM2). (d) arg(ψPM2).

Fig. 6
Fig. 6

Evolution of phasors corresponding to the MI eigenmode for the case of Δ=0 and d1, d2<0.

Fig. 7
Fig. 7

Evolution of phasors corresponding to the MI eigenmode for the case of Δ=0 and d1, d2>0.

Fig. 8
Fig. 8

Evolution of phasors corresponding to the MI eigenmode for the case of Δ=0 and d1d2<0.

Fig. 9
Fig. 9

Distribution of |Im(κ)| calculated for the case of (a) Δ=2 and (b) Δ=4 (R=1, r=2).

Fig. 10
Fig. 10

Relative magnitudes of respective sidebands composing the MI eigenmode for the case of R=1, r=2, and Δ=4. (a) |A1+|2. (b) |A1-|2. (c) |A2+|2. (d) |A2-|2.

Fig. 11
Fig. 11

Distribution of |Im(κ)| on a (Δ, d0) plane for the case of R=1, r=2/3 and ρ=1 (d1=d2). Solid parabolic curves represent the trajectories of point (Δ, d0) for various values of a: α=1.5, 0.3, 0.2, 0.1, -0.2, -1, and -5.

Fig. 12
Fig. 12

Gain spectra of the MI eigenmode for the case of R=1, r=2/3 and ρ=1 (d1=d2) with (a) positive and (b) negative values of a.

Fig. 13
Fig. 13

Distribution of |Im(κ)| on a (Δ, d0) plane for the case of R=1, r=2 (parallel) and ρ=-0.99 (d1-d2). Solid parabolic curves represent the trajectories of point (Δ, d0) for various values of a.

Fig. 14
Fig. 14

Gain spectra of the MI eigenmodes for the case of R=1, r=2 and ρ=-0.99 (d1-d2) with (a) positive and (b) negative values of a. The left and right sides correspond to the gain spectra of SSB-MI eigenmodes in which (A1+, A2-) and (A1-, A2+) amplify dominantly, respectively.

Fig. 15
Fig. 15

Simulated results of two parallel pumps interacting with broad-bandwidth noise for the case where β3=0.075 ps3/km, (ω1-ω2)/2π=5 THz, γ=1.5/km/W, P1=P2=0.2 W. (a) β20=0.03 ps2/km (ρ=-0.999, a=0.203). (b) β20=-0.03 ps2/km (ρ=-0.999, a=-0.203). Analytic results are also shown below the simulated spectra.

Tables (1)

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Table 1 Types of Cross-Phase Modulation-Induced Modulational Instability in Respective Domains of (d1, d2, Δ)

Equations (24)

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Ej(z, t)ejRe(Aj(z)exp[-i(ωjt-kjt)]),
Ej±(z, t)ejRe(Aj±(z)×exp{-i[(ωj±Ω)t-kj±t]}),
Aj(z)=Pjexp[iγ(Pj+rP3-j)z],
dAj±dz=iγ[(2Pj+rP3-j)Aj±+2PjAj*×exp{i[Δkj,j,j,j±+2γ(Pj+rP3-j)]z}+rPjP3-jA(3-j)*exp{i[Δkj,(3-j),(3-j),j±+γ(1+r)(Pj+P3-j)]z}+rPjP3-jA(3-j)±exp{i[Δkj,(3-j)±,(3-j),j±+γ(1-r)(Pj-P3-j)]z},
Aj±(z)Aj±0exp(iλj±z)
λj+=KNL+Kave+kj-kj++γ(Pj+rP3-j),
λj-=-(KNL*+Kave)+kj-kj-+γ(Pj+rP3-j),
M  A=KNL A,
[(κ-Δ)2-d1(d1+2)][(κ+Δ)2-R2d2(d2+2)]
=4r2R2d1d2,
κKNL/(γP1),
d1(k1++k1--2k1)/(2γP1),
d2(k2++k2--2k2)/(2γP2),
Δ(k1+-k1--k2++k2-)/(4γP1),
RP2/P1.
ΨAMjiΨPMj=111-1 Aj+0Aj-0*,
dj=β2 j2γP Ω2,
Δ=δ2γP ΩσΩ,
ρ2d1d2d12+d22=2β21β22β212+β222,
d0=αΔ2,
αγP(β21+β22)δ2,
ρ=-(β3ωd/2)2-β202(β3ωd/2)2+β202,
α=2γPβ20ωd2,
σ=β20ωd2γP=1αωd,

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