## Weak-wave advancement in nearly collinear four-wave mixing

Optics Express, Vol. 10, Issue 13, pp. 581-585 (2002)

http://dx.doi.org/10.1364/OE.10.000581

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### Abstract

We identify a new four-wave mixing process in which two nearly collinear pump beams produce phase-dependent gain into a weak bisector signal beam in a self-defocusing Kerr medium. Phase matching is achieved by weak-wave advancement caused by cross-phase modulation between the pump and signal beams. We relate this process to the inverse of spatial modulational instability and suggest a time-domain analog.

© 2002 Optical Society of America

2. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. **87**, 033902 (2001). [CrossRef] [PubMed]

3. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B **19**, 477 (2002). [CrossRef]

4. N. N. Akhmediev, “Nonlinear physics - Deja vu in optics,” Nature **413**, 267 (2001). [CrossRef] [PubMed]

5. G. P. Agrawal, “Transverse modulation instability of copropagating optical beams in nonlinear Kerr Media,” J. Opt. Soc. Am. B **7**, 1072 (1990). [CrossRef]

6. J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. **68**, 3547 (1992). [CrossRef] [PubMed]

7. R. W. Boyd and G. S. Agarwal, “Preventing laser beam filamentation through use of the squeezed vacuum,” Phys. Rev. A **59**, R2587 (1999). [CrossRef]

*P*

_{4}. For simplicity we assume that this process will be stimulated, introducing a small seed field

*E*

_{3}in the direction of the bisector line (see Fig. 1). The general expression for the polarization of the signal field is

*E*

_{j}are the (complex)fields associated with the pump (

*j*= 1, 2)and signal (

*j*= 3,4)beams. The first term in this equation represents self-phase modulation (SPM), the second cross-phase modulation (XPM)and the third four-wave mixing. For a weak signal beam, SPM is negligible. The XPM terms from the pump beams are large and play an important part in the four-wave mixing process, while the XPM term from the signal beam may be neglected. The exponential four-wave mixing term indicates that in order for this polarization to efficiently transfer energy into the signal field

*E*

_{4}, there must be energy conservation and phase matching between the different wave vectors associated with each field.

*n*

_{0}is the linear index of refraction. Consequently, there exists a shortening of the weak-wave momentum vector by Δ

*k*= - |Δ

*n*|

*ω*

_{0}/

*c*, an effect which we call

*weak-wave advancement*in analogy with “weak-wave retardation” in the case of the self-focusing sign of the Kerr nonlinearity[9

9. R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. **17**, 1158(1966). [CrossRef]

*E*

_{3}and

*E*

_{4}. In this case, we use Eq. 1 to calculate the evolution equation for

*E*

_{4}in the paraxial and plane wave limits, finding

*γ*= 6

*π*|

*k⃗*

_{1}|

*P*is the pump beam power, δ

*k*= |

*k⃗*

_{1}⃗ +

*k⃗*

_{2}- 2

*k⃗*

_{4}| is the momentum mismatch and

*ϕ*

_{1},

*ϕ*

_{2}are the phases of the pump beams. Taking the self-defocusing case (

*γP*< 0) we guess a solution to Eq. 3 given by

*A*is the signal field amplitude,

*g*is a purely real exponential gain coefficient and

*ϕ*

_{4}is the signal beam phase (for a similar treatment, see reference [10], p. 392). Since the phase

*ϕ*

_{4}is a free parameter, Eq. 4 is a solution of Eq. 3 when the momentum mismatch is in the range

*g*given by

*γP/k*. The maximum gain

*g*= 2|

*γP*| occurs when Δ

*k*= 2|

*γP*|.

11. R. Y. Chiao, M. A. Johnson, S. Krinsky, H. A. Smith, C. H. Townes, and E. Garmire, “A new class of trapped light filaments,” IEEE J. of Quant. Elec. **QE-2**467 (1966). [CrossRef]

12. A. J. Campillo, S. L Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. **23**, 628(1973). [CrossRef]

*k*as above, the signal beam gain is maximized for the phase relation

*ϕ*<

*π*) or loss (

*π*< Δ

*ϕ*< 2

*π*). However, we find from numerical solutions to Eq. 3 that as the signal beam propagates it accumulates phase so as to satisfy Eq. 7. The signal beam thus eventually experiences gain even if it was initially lossy. The distance over which this rephasing occurs increases for phases far from Δ

*ϕ*=

*π*/2, becoming infinite for Δ

*ϕ*= 3

*π*/2. It should be noted that the exact phase relation Δ

*ϕ*= 3

*π*/2 is unstable, and any phase noise will cause

*ϕ*

_{4}to evolve towards satisfying Eq. 7. Numerical solutions for the log of the signal beam amplitude are plotted against distance in Fig. 3 for several pump-signal phase relations.

*α*≈ 1°, satisfying the relation in Eq. 5 implies

^{-6}cm

^{2}/W, Eq. 8 is satisfied for

*P*≥ 3W/cm

^{2}. Nonlin-earities of this order are available in current materials including atomic vapors [13

13. H. Wang, D. Goorskey, and M. Xiao, “Dependence of enhanced Kerr nonlinearity on coupling power in a three-level atomic system,” Opt. Lett. **27**, 258(2002). [CrossRef]

7. R. W. Boyd and G. S. Agarwal, “Preventing laser beam filamentation through use of the squeezed vacuum,” Phys. Rev. A **59**, R2587 (1999). [CrossRef]

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

*ω*

_{0}+ Ω and

*ω*

_{0}-Ω through an optical fiber in the normal dispersion regime. These two optical frequency components correspond to the noncollinear pump beams in the spatial case discussed here. As a result, a stationary signal at the frequency

*ω*

_{0}should be generated, corresponding to the bisecting signal beam in the spatial case.

## References and links

1. | E. Fermi, J. Pasta, and H. C. Ulam, in |

2. | G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. |

3. | G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B |

4. | N. N. Akhmediev, “Nonlinear physics - Deja vu in optics,” Nature |

5. | G. P. Agrawal, “Transverse modulation instability of copropagating optical beams in nonlinear Kerr Media,” J. Opt. Soc. Am. B |

6. | J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. |

7. | R. W. Boyd and G. S. Agarwal, “Preventing laser beam filamentation through use of the squeezed vacuum,” Phys. Rev. A |

8. | M. W. Mitchell, C. J. Hancox, and R. Y. Chiao, “Dynamics of atom-mediated photon-photon scattering,” Phys. Rev. A |

9. | R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. |

10. | G. P. Agrawal, |

11. | R. Y. Chiao, M. A. Johnson, S. Krinsky, H. A. Smith, C. H. Townes, and E. Garmire, “A new class of trapped light filaments,” IEEE J. of Quant. Elec. |

12. | A. J. Campillo, S. L Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. |

13. | H. Wang, D. Goorskey, and M. Xiao, “Dependence of enhanced Kerr nonlinearity on coupling power in a three-level atomic system,” Opt. Lett. |

14. | K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. |

15. | R. Y. Chiao, “Bogoliubov dispersion relation for a ’photon fluid’: Is this a superfluid?,” Opt. Comm. |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 13, 2002

Revised Manuscript: June 26, 2002

Published: July 1, 2002

**Citation**

C. McCormick, R. Chiao, and Jandir Hickmann, "Weak-wave advancement in nearly collinear four-wave mixing," Opt. Express **10**, 581-585 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-13-581

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### References

- E. Fermi, J. Pasta, and H. C. Ulam, in Collected Papers of Enrico Fermi, edited by E. Segre (The University of Chicago, Chicago, 1965), vol. 2 pp. 977-988.
- G. Van Simaeys, Ph. Emplit, and M. Haelterman, �??Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,�?? Phys. Rev. Lett. 87, 033902 (2001). [CrossRef] [PubMed]
- G. Van Simaeys, Ph. Emplit, and M. Haelterman, �??Experimental study of the reversible behavior of modulational instability in optical fibers,�?? J. Opt. Soc. Am. B 19, 477 (2002). [CrossRef]
- N. N. Akhmediev, �??Nonlinear physics - Deja vu in optics,�?? Nature 413, 267 (2001). [CrossRef] [PubMed]
- G. P. Agrawal, �??Transverse modulation instability of copropagating optical beams in nonlinear Kerr Media,�?? J. Opt. Soc. Am. B 7, 1072 (1990). [CrossRef]
- J. M. Hickmann, A. S. L. Gomes, and C. B. de Araujo, �??Observation of spatial cross-phase modulation e.ects in a self-defocusing nonlinear medium,�?? Phys. Rev. Lett. 68, 3547 (1992). [CrossRef] [PubMed]
- R. W. Boyd, and G. S. Agarwal, �??Preventing laser beam .lamentation through use of the squeezed vacuum,�?? Phys. Rev. A 59, R2587 (1999). [CrossRef]
- M. W. Mitchell, C. J. Hancox, and R. Y. Chiao, �??Dynamics of atom-mediated photon-photon scattering,�?? Phys. Rev. A 62, 043819 (2000). [CrossRef]
- R. Y. Chiao, P. L. Kelley, and E. Garmire, �??Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,�?? Phys. Rev. Lett. 17, 1158(1966). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, 3rded. (Academic, San Diego, 2001).
- R. Y. Chiao, M. A. Johnson, S. Krinsky, H. A. Smith, C. H. Townes, and E. Garmire, �??A new class of trapped light .laments,�?? IEEE J. Quantum Electron. QE-2 467 (1966). [CrossRef]
- A. J. Campillo, S. L Shapiro, and B. R. Suydam, �??Periodic breakup of optical beams due to self-focusing,�?? Appl. Phys. Lett. 23, 628 (1973). [CrossRef]
- H. Wang, D. Goorskey, and M. Xiao, �??Dependence of enhanced Kerr nonlinearity on coupling power in a three-level atomic system,�?? Opt. Lett. 27, 258(2002). [CrossRef]
- K. Tai, A. Hasegawa, and A. Tomita, �??Observation of modulational instability in optical fibers,�?? Phys. Rev. Lett. 56, 135 (1986). [CrossRef] [PubMed]
- R. Y. Chiao, �??Bogoliubov dispersion relation for a �??photon fluid�??: Is this a superfluid?,�?? Opt. Commun. 179, 157 (2000). [CrossRef]

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