## Parametric instabilities driven by orthogonal pump waves in birefringent fibers

Optics Express, Vol. 11, Issue 20, pp. 2619-2633 (2003)

http://dx.doi.org/10.1364/OE.11.002619

Acrobat PDF (1222 KB)

### 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

## 1. Introduction

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]

*ω*

_{1-}=

*ω*

_{1}-

*ω*, where

*ω*is the frequency difference (modulation frequency). Then the modulation interaction (MI) in which 2

*ω*

_{1}=

*ω*

_{1-}+

*ω*

_{1+}produces an idler with frequency

*ω*

_{1+}=

*ω*

_{1}+

*ω*, the Bragg-scattering (BS) process in which

*ω*

_{1-}+

*ω*

_{2}=

*ω*

_{1}+

*ω*

_{2-}produces an idler with frequency

*ω*

_{2-}=

*ω*

_{2}-

*ω*and the phase-conjugation (PC) process in which

*ω*

_{1}+

*ω*

_{2}=

*ω*

_{1-}+

*ω*

_{2+}produces an idler with frequency

*ω*

_{2+}=

*ω*

_{2}+

*ω*. Each of the idlers is coupled to the others by the appropriate FWM process (BS, MI or PC).

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]

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]

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]

## 2. Four-sideband equations

*E*

_{x}(

*t*,

*z*) and

*E*

_{y}(

*t*,

*z*) be the electric-field components of a light wave in a fiber. It is often convenient to measure (angular) frequencies relative to a reference frequency

*ω*

_{0}and wavenumbers relative to the associated reference wavenumbers, and write the field components as

*A*

_{x}and

*A*

_{y}satisfy the incoherently-coupled NS equations

*β*(

*ω*)=

*ω*

^{n}/

*n*! represents the higher-order terms in the Taylor expansion of the dispersion function about

*ω*

_{0},

*γ*is the self-nonlinearity coefficient and

*∊*=2/3 is the ratio of the cross- and self-nonlinearity coefficients [13

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

*P*

_{1}and

*P*

_{2}are the pump powers,

*ω*

_{1}and

*ω*

_{2}are the pump frequencies (measured relative to

*ω*

_{0}), and the pump phases

*p*is the pump index (1 or 2) and

*l*=

*m*-

*n*. In the first line of Eq. (11) the null term

*B*

_{p}exp(-

*iω*

_{p}

*t*)] was added to simplify the algebra that follows. In the last line of Eq. (11) the term in square brackets is

*m*th derivative of the dispersion function evaluated at

*ω*

_{p}. By using this result, one finds that

*β*

_{1}(

*ω*) represents the higher-order (

*m*≥1) terms in the Taylor expansion of

*β*

_{x}about

*ω*

_{1}and

*β*

_{2}(

*ω*) represents the expansion of

*β*

_{y}about

*ω*

_{2}.

*x*-polarized signal sideband with (absolute) frequency

*ω*

_{1-}=

*ω*

_{1}-

*ω*. Because of the the way in which

*B*

_{1}was defined [Eq. (9)], this sideband is represented by a component of

*ω*. It follows from Eqs. (14) and (15) that the signal sideband produces an

*x*-polarized idler sideband with (relative) frequency

*ω*, and

*y*-polarized idler sidebands with frequencies -

*ω*and

*ω*, as shown in Fig. 1. Each

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]

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]

*∊*, which does not equal 2, and second, the wavenumber mismatches depend on the dispersion functions

*β*

_{x}and

*β*

_{y}, which are not equal.

*a priori*assumption that

*x*-polarized sidebands with (absolute) frequencies

*ω*

_{1}±=

*ω*

_{1}±

*ω*interact with

*y*-polarized sidebands with frequencies

*ω*

_{2}±=

*ω*

_{2}±

*ω*[23

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]

## 3. Results

*ik*

_{j}

*z*), where each wavenumber

*k*

_{j}is a root of the characteristic equation

*δβ*

_{1o}(

*δβ*

_{1e}) and

*δβ*

_{2o}(

*δβ*

_{2e}) denote the odd (even) terms in the Taylor expansions of

*βx*(

*ω*

_{1}+

*ω*) and

*β*

_{y}(

*ω*

_{2}+

*ω*) about the frequencies

*ω*

_{1}and

*ω*

_{2}, respectively. Physically,

*dδβ*

_{1o}/

*dω*and

*dδβ*

_{2o}/

*dω*are the average slownesses (inverse group speeds) of the sidebands of pumps 1 and 2, respectively, which are comparable to the pump slownesses

**8**, 538–547 (2002). [CrossRef]

*δβ*=(

*δβ*

_{1-}+

*δβ*

_{1+})/2 (and the second wavenumber is -

*k*). The MI of pump 1 does not depend on birefringence or the odd-order dispersion coefficients (evaluated at

*ω*

_{1}). Its (spatial) growth rate Im(

*k*) is maximal when the linear and nonlinear mismatches cancel (

*δβ*=-

*γ*

*P*

_{1}). Similar results apply to the MI of pump 2. The BS wavenumber

*δβ*=(

*δβ*

_{2-}-

*δβ*

_{1-})/2. BS depends on birefringence, all orders of dispersion and both pump powers. It is intrinsically stable. Similar results apply to the BS process that involves the 1+ and 2+ sidebands. The PC wavenumber

*δβ*=(

*δβ*

_{1-}+

*δβ*

_{2+})/2. PC depends on birefringence and dispersion. Like MI, its growth rate is maximal when the linear and nonlinear mismatches cancel [

*δβ*=-

*γ*(

*P*

_{1}+

*P*

_{2})/2]. Unlike MI, the linear mismatch depends on dispersion coefficients evaluated at both pump frequencies [even-order coefficients evaluated at the average pump frequency (

*ω*

_{1}+

*ω*

_{2})/2] and the nonlinear mismatch depends on both pump powers. Similar results apply to the PC process that involves the 1+ and 2- sidebands.

*k*

_{j}≈0. Suppose that the 1- sideband is the signal. Then a short calculation shows that the (normalized) sideband powers

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]

*s*=

*δn*/

*c*is the differential slowness and

*δn*is the differential index of refraction. [If

*s*>0 the

*x*-axis is called (colloquially) the slow axis.] The birefringent and dispersive contributions to the group speeds are independent [25

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]

^{-7}(transmission fibers) to 10

^{-5}(polarization maintaining fibers). To illustrate the use of the FS equations we chose the intermediate value

*δn*=10

^{-6}, for which

*s*=3 ps/Km. We also chose the (representative) fiber parameters

^{3}/Km,

^{-4}ps

^{4}/Km,

*γ*=10/Km-W and

*l*=0.3 Km, and the common pump power

*P*=1 W. The dispersion coefficients represent values associated with the zero-dispersion frequency

*ω*

_{0}.

*ω*and the pump frequency

*ω*

_{2}in Fig. 3, for

*ω*

_{1}=-5 THz. (In this paper all frequencies are angular frequencies: THz is an abbreviation for Trad/s.) The ridges that are parallel to the

*ω*

_{2}-axis are associated with MIs [Eq. (29)]. For most values of

*ω*

_{2}, the group speeds of the pumps are dissimilar and the MIs of the pumps are independent. Because

*ω*

_{1}is constant, the ridge associated with pump 1 has constant width. Because

*ω*

_{2}varies, the ridge associated with pump 2 has variable width and does not exist for

*ω*

_{2}>0 (normal dispersion region). The ridges that extend to large values of

*ω*are associated with PC processes [Eq. (31)]. To demonstrate this fact, the PC gain loci are plotted in Fig. 4. Negative modulation frequencies correspond to cases in which the 1- sideband is the signal, whereas positive modulation frequencies correspond to cases in which the 1+ sideband is the signal. Because Eq. (28) is an even function of the modulation frequency, both PC processes manifest themselves in Fig. 3. To connect Fig. 4 to Fig. 3 the reader should project the mirror image of the 1-PC process (-

*ω*→

*ω*) onto the right half-plane: The 1- PC process is responsible for the ridges that cover the points (15,-20) and (20, 6), whereas the 1+ PC process is responsible for the ridges that cover (20, 0) and (20, 20). When

*ω*

_{2}≈-9.2 THz group-speed matching occurs and all four sidebands interact strongly. In this case Eq. (28) is biquadratic and can be solved analytically for arbitrary

*β*

_{1e}and

*β*

_{2e}. If one ignores the difference between the pump frequencies, one finds that

*δβ*

_{o}(

*δβ*

_{e}) denote the odd (even) terms in the Taylor expansion of the (common) dispersion function. The maximal FS growth rate

*κ*≈

*γ*(1+

*∊*)

*P*is higher than the maximal growth rates associated with MI (

*κ*=

*γP*) and PC (

*κ*=

*γ∊P*). For the stated parameters, the value of the FS growth rate predicted by Eq. (38) agrees fairly well with the value obtained by solving Eq. (28) numerically (16.7 and 16.0, respectively). Group-speed matching also occurs when

*ω*

_{2}≈9.2 THz. In this case the presence of a modulationally stable pump (2) reduces slightly the growth rates associated with the unstable pump (1). The matching conditions for PC are not satisfied.

*ω*

_{1}=-5.0 THz and

*ω*

_{2}=-9.2 THz are displayed in Fig. 5(a). The solid, dot-dashed and dashed curves represent the signal gains (ratios of the output and input signal powers) associated with the FS process [Eqs. (18)–(21)], the PC processes [Eqs. (18) and (21) for the external sidebands, and Eqs. (19) and (20) for the internal sidebands] and the MIs [Eqs. (18) and (19) for pump 1, and Eqs. (20) and (21) for pump 2], respectively. Clearly, the FS gain-level is higher and the FS gain-bandwidth is broader that the levels and bandwidths of the constituent processes (PC and MI). The signal gain and idler gains (ratios of the output idler powers and the input signal power) obtained by solving the FS equations numerically are displayed in Fig. 5(b). The gains of the idlers (1+, 2- and 2+) are comparable to that of the signal (1-), which justifies our assertion that all four sidebands interact strongly.

*ω*

_{1}=-5.0 THz and

*ω*

_{2}=-9.2 THz. These spectra were obtained by solving Eqs. (3) and (4) numerically, with noise included in the spectrum of the

*x*-component at

*z*=0. An idealized model of noise was used, in which the modulus of the noise field was the same for all frequency bins, but the phase varied randomly from bin to bin. The bin width was 6.2 GHz (angular frequency) and the input noise power was 10

^{-8}W (-50 dBmW) per bin. The simulation results are shown in Fig. 6. The locations and heights of the primary peaks are consistent with the predictions of the FS model. (The peaks in Fig. 6 are 6-dB higher than those in Fig. 5 because of constructive interference between noise-driven sidebands with modulation frequencies

*ω*and -

*ω*.) The presence of the secondary peaks is a higher-order nonlinear effect driven by the pumps and primary sidebands. To check this assertion the simulation was repeated with lower pump power (

*P*=0.5 W). The heights of the primary peaks were reduced and the secondary peaks were absent.

*ω*

_{2}≈6 THz. The signal-gain curves obtained by solving the sideband equations numerically for

*ω*

_{1}=-5.0 THz and

*ω*

_{2}=5.9 THz are displayed in Fig. 7(a). (The line styles were defined in the discussion of Fig. 5.) For low modulation frequencies the FS gain is produced (mainly) by the MI of pump 1, whereas for high modulation frequencies the FS gain is produced (mainly) by PC. The signal and idler gains obtained by solving the FS equations numerically are displayed in Fig. 7(b). For low frequencies the gain of the 1+ idler is comparable to that of the 1- signal, as predicted by the MI equations. Notice that the nonlinear coupling provided by pump 2 allows the 2+ sideband to experience (sympathetic) gain.For high frequencies only the gain of the 2+ idler is comparable to the signal gain, which justifies our assertion that broad-bandwidth gain is produced by PC. The 2- idler does not experience significant gain for any frequency.

*x*-polarized noise are displayed in Fig. 8 for

*ω*

_{1}=-5.0 THz and

*ω*

_{2}=5.9 THz. There is good agreement between the predictions of the FS model (which were illustrated in Fig. 7) and the simulation results.

*ω*

_{1}=5.0 THz and

*ω*

_{2}=-2.4 THz are displayed in Fig. 11(a). (The line styles were defined in the discussion of Fig. 5.) Consistent with the predictions of Eq. (28), the signal gain is significantly higher than the MI and PC gains for a broad range of modulation frequencies. The signal and idler gains obtained by solving the FS equations numerically are displayed in Fig. 11(b). For low frequencies the gain of the 2- idler is comparable to that of the 2+ signal and the nonlinear coupling provided by pump 1 allows the 1- sideband to experience (cooperative) gain. For high frequencies only the gain of the 1- idler is comparable to the signal gain, so the broad-bandwidth gain is produced by PC.

*y*-polarized noise are displayed in Fig. 12 for

*ω*

_{1}=5.0 THz and

*ω*

_{2}=-2.4 THz. Once again there is good agreement between the predictions of the FS model (which were illustrated in Fig. 11) and the simulation results.

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]

## 4. Summary

## Acknowledgments

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]

## References and links

1. | R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. |

2. | M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E |

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. |

4. | C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. |

5. | C. J. McKinstrie and S. Radic, “Parametric amplifiers driven by two pump waves with dissimilar frequencies,” Opt. Lett. |

6. | F. S. Yang, M. C. Ho, M. E. Marhic, and L. G. Kazovsky, “Demonstration of two-pump fibre optical parametric amplification,” Electron. Lett. |

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 |

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. |

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. |

10. | R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. |

11. | K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies,” J. Lightwave Technol. |

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. |

13. | C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. |

14. | G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. |

15. | C. J. McKinstrie and R. Bingham, “Modulational instability of coupled waves,” Phys. Fluids B |

16. | G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A |

17. | C. J. McKinstrie and G. G. Luther, “Modulational instability of colinear waves,” Phys. Scripta |

18. | S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A |

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 |

20. | J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A |

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. |

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 |

24. | H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in |

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 |

26. | M. G. Forest and O. C. Wright, “An integrable model for stable: unstable wave coupling phenomena,” Physica D |

27. | M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E |

28. | M. E. Marhic, F. S. Yang, M. C. Ho, and L. G. Kazovsky, “Broadband fiber optic parametric amplifiers,” Opt. Lett. |

29. | A. H. Nayfeh and D. T. Mook, |

30. | C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B |

31. | C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A |

32. | E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase-modulationinduced modulational instabilty,” Phys. Rev. E |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(190.5040) Nonlinear optics : Phase conjugation

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 21, 2003

Revised Manuscript: September 17, 2003

Published: October 6, 2003

**Citation**

C. McKinstrie, S. Radic, and C. Xie, "Parametric instabilities driven by orthogonal pump waves in birefringent fibers," Opt. Express **11**, 2619-2633 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2619

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### References

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