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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    [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) and references therein.
    [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-modulation-induced modulational instabilty,�?? Phys. Rev. E 61, 3139�??3150 (2000).
    [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)

OFC (1)

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.

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-modulation-induced 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 (2)

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.

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.

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.

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.

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+.

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.

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.

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.

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.

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.

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+.

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.

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.

Equations (47)

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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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