Abstract

The four-sideband model of parametric instabilities driven by orthogonal pump waves in birefringent fibers is developed and validated by numerical simulations. A polynomial eigenvalue equation is derived and used to determine how the spatial growth rates and frequency bandwidths of various instabilities depend on the system parameters. The maximal growth rate is associated with a group-speed matched four-sideband process (coupled modulation instability), whereas broad-bandwidth gain is associated primarily with a two-sideband process (phase conjugation). This four-sideband model facilitates the design of parametric amplifiers driven by two pump waves with different frequencies and polarizations.

©2003 Optical Society of America

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References

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  1. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982) and references therein.
    [Crossref]
  2. 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]
  3. M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, “Broadband fiber-optical parametric amplifiers and wavelength converters with low-ripple Chebyshev gain spectra,” Opt. Lett. 21, 1354–1356 (1996).
    [Crossref] [PubMed]
  4. 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]
  5. C. J. McKinstrie and S. Radic, “Parametric amplifiers driven by two pump waves with dissimilar frequencies,” Opt. Lett. 27, 1138–1140 (2002).
    [Crossref]
  6. 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]
  7. J. M. Chavez 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]
  8. 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]
  9. S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
    [Crossref]
  10. R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
    [Crossref]
  11. K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies,” J. Lightwave Technol. 12, 1916–1920 (1994).
    [Crossref]
  12. K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2003).
    [Crossref]
  13. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
    [Crossref]
  14. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [Crossref] [PubMed]
  15. C. J. McKinstrie and R. Bingham, “Modulational instability of coupled waves,” Phys. Fluids B 1, 230–237 (1989).
    [Crossref]
  16. 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] [PubMed]
  17. C. J. McKinstrie and G. G. Luther, “Modulational instability of colinear waves,” Phys. Scripta T-30, 31–40 (1990).
    [Crossref]
  18. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38, 2018–2021 (1988).
    [Crossref] [PubMed]
  19. S. Trillo and S. Wabnitz, “Ultrashort pulse train generation through induced modulational polarization instability in a birefringent Kerr-like medium,” J. Opt. Soc. Am. B 6, 238–249 (1989).
    [Crossref]
  20. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
    [Crossref] [PubMed]
  21. 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]
  22. S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps,” Optical Fiber Communication conference, Atlanta, Georgia, 23–28 March 2003, paper ThK3.
  23. C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
    [Crossref]
  24. H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego, 2002), pp. 725–861.
  25. E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bibault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
    [Crossref] [PubMed]
  26. M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Physica D 178, 173–189 (2003).
    [Crossref]
  27. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
    [Crossref]
  28. M. E. Marhic, F. S. Yang, M. C. Ho, and L. G. Kazovsky, “Broadband fiber optic parametric amplifiers,” Opt. Lett. 21, 573–575 (1996).
    [Crossref] [PubMed]
  29. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, New York, 1979).
  30. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993).
    [Crossref]
  31. 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] [PubMed]
  32. E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase-modulationinduced modulational instabilty,” Phys. Rev. E 61, 3139–3150 (2000).
    [Crossref]

2003 (4)

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[Crossref]

M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Physica D 178, 173–189 (2003).
[Crossref]

2002 (3)

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).
[Crossref]

C. J. McKinstrie and S. Radic, “Parametric amplifiers driven by two pump waves with dissimilar frequencies,” Opt. Lett. 27, 1138–1140 (2002).
[Crossref]

2001 (1)

2000 (1)

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

1997 (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]

1996 (3)

1995 (2)

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[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] [PubMed]

1994 (1)

K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies,” J. Lightwave Technol. 12, 1916–1920 (1994).
[Crossref]

1993 (3)

R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
[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, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993).
[Crossref]

1990 (3)

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

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]

C. J. McKinstrie and G. G. Luther, “Modulational instability of colinear waves,” Phys. Scripta T-30, 31–40 (1990).
[Crossref]

1989 (3)

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

S. Trillo and S. Wabnitz, “Ultrashort pulse train generation through induced modulational polarization instability in a birefringent Kerr-like medium,” J. Opt. Soc. Am. B 6, 238–249 (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)

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

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

1982 (1)

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

Agrawal, G. P.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[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]

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] [PubMed]

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

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] [PubMed]

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] [PubMed]

Bibault, J. M.

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

Bingham, R.

C. J. McKinstrie and R. Bingham, “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. QE-18, 1062–1072 (1982) and references therein.
[Crossref]

Boggio, J. M. Chavez

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]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps,” Optical Fiber Communication conference, Atlanta, Georgia, 23–28 March 2003, paper ThK3.

Cao, X. D.

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).
[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] [PubMed]

Dinda, P. Tchofo

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

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]

Forest, M. G.

M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Physica D 178, 173–189 (2003).
[Crossref]

Fragnito, H. L.

Haelterman, M.

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

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]

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]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps,” Optical Fiber Communication conference, Atlanta, Georgia, 23–28 March 2003, paper ThK3.

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]

M. E. Marhic, F. S. Yang, M. C. Ho, and L. G. Kazovsky, “Broadband fiber optic parametric amplifiers,” Opt. Lett. 21, 573–575 (1996).
[Crossref] [PubMed]

Inoue, K.

K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies,” J. Lightwave Technol. 12, 1916–1920 (1994).
[Crossref]

Jopson, R.

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps,” Optical Fiber Communication conference, Atlanta, Georgia, 23–28 March 2003, paper ThK3.

Jopson, R. M.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
[Crossref]

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego, 2002), pp. 725–861.

Jorgensen, C.

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps,” Optical Fiber Communication conference, Atlanta, Georgia, 23–28 March 2003, paper ThK3.

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.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2003).
[Crossref]

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]

M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, “Broadband fiber-optical parametric amplifiers and wavelength converters with low-ripple Chebyshev gain spectra,” Opt. Lett. 21, 1354–1356 (1996).
[Crossref] [PubMed]

M. E. Marhic, F. S. Yang, M. C. Ho, and L. G. Kazovsky, “Broadband fiber optic parametric amplifiers,” Opt. Lett. 21, 573–575 (1996).
[Crossref] [PubMed]

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]

Kogelnik, H.

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego, 2002), pp. 725–861.

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]

Li, J. S.

Lin, Q.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

Luther, G. G.

C. J. McKinstrie and G. G. Luther, “Modulational instability of colinear waves,” Phys. Scripta T-30, 31–40 (1990).
[Crossref]

Marhic, M. E.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2003).
[Crossref]

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]

M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, “Broadband fiber-optical parametric amplifiers and wavelength converters with low-ripple Chebyshev gain spectra,” Opt. Lett. 21, 1354–1356 (1996).
[Crossref] [PubMed]

M. E. Marhic, F. S. Yang, M. C. Ho, and L. G. Kazovsky, “Broadband fiber optic parametric amplifiers,” Opt. Lett. 21, 573–575 (1996).
[Crossref] [PubMed]

McKinstrie, C. J.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[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).
[Crossref]

C. J. McKinstrie and S. Radic, “Parametric amplifiers driven by two pump waves with dissimilar frequencies,” Opt. Lett. 27, 1138–1140 (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, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[Crossref]

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993).
[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, “Modulational instability of colinear waves,” Phys. Scripta T-30, 31–40 (1990).
[Crossref]

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

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps,” Optical Fiber Communication conference, Atlanta, Georgia, 23–28 March 2003, paper ThK3.

Menyuk, C. R.

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

Millot, G.

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

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

Mook, D. T.

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, New York, 1979).

Nayfeh, A. H.

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, New York, 1979).

Nelson, L. E.

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego, 2002), pp. 725–861.

Park, Y.

Radic, S.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[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]

C. J. McKinstrie and S. Radic, “Parametric amplifiers driven by two pump waves with dissimilar frequencies,” Opt. Lett. 27, 1138–1140 (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).
[Crossref]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps,” Optical Fiber Communication conference, Atlanta, Georgia, 23–28 March 2003, paper ThK3.

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. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bibault, 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]

Santagiustina, M.

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] [PubMed]

Seve, E.

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

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

Stolen, R. H.

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

Tench, R. E.

R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
[Crossref]

Tenenbaum, S.

Trillo, S.

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

S. Trillo and S. Wabnitz, “Ultrashort pulse train generation through induced modulational polarization instability in a birefringent Kerr-like medium,” J. Opt. Soc. Am. B 6, 238–249 (1989).
[Crossref]

Uesaka, K.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2003).
[Crossref]

Wabnitz, S.

S. Trillo and S. Wabnitz, “Ultrashort pulse train generation through induced modulational polarization instability in a birefringent Kerr-like medium,” J. Opt. Soc. Am. B 6, 238–249 (1989).
[Crossref]

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

Wong, K. K. Y.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2003).
[Crossref]

Wright, O. C.

M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Physica D 178, 173–189 (2003).
[Crossref]

Xie, C.

Yang, F. S.

Yu, M.

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[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]

Electron. Lett. (3)

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]

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
[Crossref]

IEEE J. Quantum Electron. (2)

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

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

IEEE Photon. Technol. Lett. (2)

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]

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2003).
[Crossref]

J. Lightwave Technol. (1)

K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies,” J. Lightwave Technol. 12, 1916–1920 (1994).
[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. (3)

Phys. Fluids B (1)

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

Phys. Rev. A (5)

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] [PubMed]

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

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] [PubMed]

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

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bibault, 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 (3)

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

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[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. (1)

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

Phys. Scripta (1)

C. J. McKinstrie and G. G. Luther, “Modulational instability of colinear waves,” Phys. Scripta T-30, 31–40 (1990).
[Crossref]

Physica D (1)

M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Physica D 178, 173–189 (2003).
[Crossref]

Other (3)

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego, 2002), pp. 725–861.

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps,” Optical Fiber Communication conference, Atlanta, Georgia, 23–28 March 2003, paper ThK3.

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, New York, 1979).

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Figures (12)

Fig. 1.
Fig. 1.

Diagram of the pump and sideband polarizations. The long arrows represent pumps, whereas the short arrows represent sidebands.

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Fig. 2.
Fig. 2.

Illustration of the group-speed matching conditions. The solid curve represents the x-polarized pump (1), whereas the dashed curve represents the y-polarized pump (2). If ω 1=-5.0 THz (solid circle), the group speeds are matched for ω 2=±9.2 THz (hollow circles). The case in which ω 1=5.0 THz is similar.

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Fig. 3.
Fig. 3.

Spatial growth rate plotted as a function of the modulation frequency ω and the pump frequency ω 2 for ω 1=-5 THz. The highest growth rate is associated with a group-speed matched FS instability.

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Fig. 4.
Fig. 4.

Loci of PC growth for ω 1=-5 THz. Negative modulation frequencies correspond to signals at ω 2-, whereas positive modulation frequencies correspond to signals at ω 2+.

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Fig. 5.
Fig. 5.

(a) Signal gain obtained by solving the sideband equations numerically for ω 1=-5.0 THz and ω 2=-9.2 THz. The solid, dot-dashed and dashed curves represent the FS, PC and MI gains, respectively. (b) Signal and idler gains obtained by solving the FS equations numerically. The solid, long-dashed, short-dashed and dotted curves represent the 1- (signal), 1+, 2- and 2+ sidebands, respectively.

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Fig. 6.
Fig. 6.

Simulation of broad-bandwidth x-polarized noise interacting with orthogonal pumps for the case in which ω 1=-5.0 THz and ω 2=-9.2 THz. (a) Power spectrum of the x-component. (b) Power spectrum of the y-component.

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Fig. 7.
Fig. 7.

(a) Signal gain obtained by solving the sideband equations numerically for ω 1=-5.0 THz and ω 2=5.9 THz. The solid, dot-dashed and dashed curves represent the FS, PC and MI gains, respectively. (b) Signal and idler gains obtained by solving the FS equations numerically. The solid, long-dashed, short-dashed and dotted curves represent the 1- (signal), 1+, 2- and 2+ sidebands, respectively.

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Fig. 8.
Fig. 8.

Simulation of broad-bandwidth x-polarized noise interacting with orthogonal pumps for the case in which ω 1=-5.0 THz and ω 2=5.9 THz. (a) Power spectrum of the x-component. (b) Power spectrum of the y-component.

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Fig. 9.
Fig. 9.

Spatial growth rate plotted as a function of the modulation frequency ω and the pump frequency ω 2 for ω 1=5 THz. There is neither a group-speed matched FS instability nor a broad-bandwidth PC instability.

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Fig. 10.
Fig. 10.

Loci of PC growth for ω 1=5 THz. Negative modulation frequencies correspond to signals at ω 2-, whereas positive modulation frequencies correspond to signals at ω 2+.

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Fig. 11.
Fig. 11.

(a) Signal gain obtained by solving the sideband equations numerically for ω 1=5.0 THz and ω 2=-2.4 THz. The solid, dot-dashed and dashed curves represent the FS, PC and MI gains, respectively. (b) Signal and idler gains obtained by solving the FS equations numerically. The solid, long-dashed, short-dashed and dotted curves represent the 1-, 1+, 2- and 2+ (signal) sidebands, respectively.

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Fig. 12.
Fig. 12.

Simulation of broad-bandwidth y-polarized noise interacting with orthogonal pumps for the case in which ω 1=5.0 THz and ω 2=-2.4 THz. (a) Power spectrum of the x-component. (b) Power spectrum of the y-component.

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Equations (47)

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

E x ( t , z ) = A x ( t , z ) exp ⁡ [ i k x ( ω 0 ) z − i ω 0 t ] + c . c . ,
E y ( t , z ) = A y ( t , z ) exp [ i k y ( ω 0 ) z − i ω 0 t ] + c . c .
− i ∂ z A x = β x ( i ∂ t ) A x + γ ( ∣ A x ∣ 2 + ∊ ∣ A y ∣ 2 ) A x ,
− i ∂ z A y = β y ( i ∂ t ) A y + γ ( ∊ ∣ A x ∣ 2 + ∣ A y ∣ 2 ) A y ,
A x ( 0 ) ( t , z ) = P 1 1 2 exp [ i ϕ 1 ( z ) − ω 1 t ] ,
A y ( 0 ) ( t , z ) = P 2 1 2 exp [ i ϕ 2 ( z ) − ω 2 t ] ,
ϕ 1 ( z ) = β x ( ω 1 ) z + γ ( P 1 + ∊ P 2 ) z ,
ϕ 2 ( z ) = β y ( ω 2 ) z + γ ( ∊ P 1 + P 2 ) z .
A x ( t , z ) = B 1 ( t , z ) exp [ i ϕ 1 ( z ) − i ω 1 t ] ,
A y ( t , z ) = B 2 ( t , z ) exp [ i ϕ 2 ( z ) − i ω 2 t ] ,
β 0 ( i ∂ t ) [ B p exp ( − i ω p t ) ] = ∑ n = 0 ∞ β 0 ( n ) ( i ∂ t ) n n ! [ B p exp ( − i ω p t ) ]
= ∑ n = 0 ∞ ∑ m = 0 ∞ β 0 ( n ) ( i ∂ t ) m B p ω p n − m m ! ( n − m ) !
= ∑ m = 0 ∞ ∑ n = m ∞ β 0 ( n ) ω p n − m ( n − m ) ! ( i ∂ t ) m B p m !
= ∑ m = 0 ∞ [ ∑ l = 0 ∞ β 0 ( m + l ) ω p l l ! ] ( i ∂ t ) m B p m ! ,
− i ∂ z B 1 = β 1 ( i ∂ t ) B 1 + γ ( ∣ B 1 ∣ 2 − P 1 ) B 1 + γ ∊ ( ∣ B 2 ∣ 2 − P 2 ) B 2 ,
− i ∂ z B 2 = β 2 ( i ∂ t ) B 2 + γ ∊ ( ∣ B 1 ∣ 2 − P 1 ) B 2 + γ ( ∣ B 2 ∣ 2 − P 2 ) B 2 ,
− i ∂ z B 1 ( 1 ) = β 1 ( i ∂ t ) B 1 ( 1 ) + γ P 1 [ B 1 ( 1 ) + B 1 ( 1 ) * ] + γ ∊ ( P 1 P 2 ) 1 2 [ B 2 ( 1 ) + B 2 ( 1 ) * ] ,
− i ∂ z B 2 ( 1 ) = β 2 ( i ∂ t ) B 2 ( 1 ) + γ ∊ ( P 1 P 2 ) 1 2 [ B 1 ( 1 ) + B 1 ( 1 ) * ] + γ P 2 [ B 2 ( 1 ) + B 2 ( 1 ) * ] .
B 1 ( 1 ) ( t , z ) = B 1 + ( z ) exp ( − i ω t ) + B 1 − ( z ) exp ( i ω t ) ,
B 2 ( 1 ) ( t , z ) = B 2 + ( z ) exp ( − i ω t ) + B 2 − ( z ) exp ( i ω t ) .
d z B 1 − * = − i ( δ β 1 − + γ P 1 ) B 1 − * − i γ P 1 B 1 +
− i γ ∊ ( P 1 P 2 ) 1 2 B 2 − * − i γ ∊ ( P 1 P 2 ) 1 2 B 2 + ,
d z B 1 + = i γ P 1 B 1 − * + i ( δ β 1 + + γ P 1 ) B 1 +
+ i γ ∊ ( P 1 P 2 ) 1 2 B 2 − * + i γ ∊ ( P 1 P 2 ) 1 2 B 2 + ,
d z B 2 − * = − i γ ∊ ( P 1 P 2 ) 1 2 B 1 − * − i γ ∊ ( P 1 P 2 ) 1 2 B 1 +
− i ( δ β 2 − + γ P 2 ) B 2 − * − i γ P 2 B 2 + ,
d z B 2 + = i γ ∊ ( P 1 P 2 ) 1 2 B 1 − * + i γ ∊ ( P 1 P 2 ) 1 2 B 1 +
+ i γ P 2 B 2 − * + i ( δ β 2 + + γ P 2 ) B 2 + ,
δ β 1 ± = β 1 ( ± ω ) ,
δ β 2 ± = β 2 ( ± ω ) .
A 1 ± ( t , z ) = B 1 ± ( z ) exp [ i ϕ 1 ( z ) − i ω 1 ± t ] ,
A 2 ± ( t , z ) = B 2 ± ( z ) exp [ i ϕ 2 ( z ) − i ω 2 ± t ] .
δ β 1 ± = β x ( ω 1 ± ) − β x ( ω 1 ) ,
δ β 2 ± = β y ( ω 2 ± ) − β y ( ω 2 ) .
[ ( k − δ β 1 o ) 2 − δ β 1 e ( 2 γ P 1 + δ β 1 e ) ]
× [ ( k − δ β 2 o ) 2 − δ β 2 e ( 2 γ P 2 + δ β 2 e ) ]
− ( 2 γ ∊ ) 2 δ β 1 e δ β 2 e P 1 P 2 = 0 .
k = [ ( δ β + γ P 1 ) 2 − ( γ P 1 ) 2 ] 1 2 ,
k = { [ δ β + γ ( P 2 − P 1 ) 2 ] 2 + ∊ 2 γ 2 P 1 P 2 } 1 2 ,
k = { [ δ β + γ ( P 1 + P 2 ) 2 ] 2 − ∊ 2 γ 2 P 1 P 2 } 1 2 ,
P 1 − ( z ) ≈ 1 + γ 2 P 1 2 z 2 ,
P 1 + ( z ) ≈ γ 2 P 1 2 z 2 ,
P 2 − ( z ) ≈ γ 2 ∊ 2 P 1 P 2 z 2 ,
P 2 + ( z ) ≈ γ 2 ∊ 2 P 1 P 2 z 2 .
β 1 ( 1 ) = s 2 + β 0 ( 3 ) ω 1 2 2 ,
β 2 ( 1 ) = − s 2 + β 0 ( 3 ) ω 2 2 2 ,
k ≈ δ β o ± { δ β e [ 2 γ ( 1 ± ∊ ) P + δ β e ] } 1 2 ,

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