Weak-wave advancement in nearlycollinear four-wave mixing: comment

Eric Lantz and Thibaut SylvestreLaboratoire d’Optique P.M. Duffieux, U.M.R. CNRS/Universite de Franche-Comte 6603,

25030 BESANCON cedex, France

elantz@univ-fcomte.fr

http://lopmd.univ-fcomte.fr/onl/

Abstract: The exponential gain predicted in “Weak-wave advancement innearly collinear four-wave mixing” [Opt. Express10, 581 (2002)], disap-pears when all interacting sidebands are properly taken into account. Thedemonstration closely follows well-established literature in the formallyequivalent temporal domain.

© 2004 Optical Society of America

OCIS codes: (190.3100) Instabilities and chaos; (190.4420) Nonlinearoptics, transverse ef-fects in; (190.3270) Kerr effect.

References and links1. C. F. McCormick, R.Y. Chiao, J.M. Hickmann, “Weak-wave advancement in nearly collinear four-wave mixing,”

Opt. Express10, 581 (2002),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-13-5812. M. Kauranen, A. L Gaeta, C.J. McKinstrie, ”Transverse instabilities of two intersecting laser beams in a nonlinear

Kerr medium,” J. Opt. Soc. Am. B10, 2298 (1993).3. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett.59, 880 (1987).4. G. Millot, S. Pitois, P. Tchofo-Dinda, and M. Haelterman, “Observation of modulational instability induced by

velocity-matched cross-phase modulation in a normally dispersive bimodal fiber,” Opt. Lett.22, 686 (1997).5. J. E. Rothenberg, “Modulational instability of copropagating frequencies for normal dispersion,” Phys. Rev. Lett.

64, 813 (1990).6. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal-dispersion

regime,” Phys. Rev. E48, 2178 (1993).7. C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, and M. Haelterman,

“Symmetry-breaking instability of multimode vector solitons,” Phys. Rev. Lett.89, 083901 (2002).

Reference [1] considers the four wave-mixing process depicted on Fig. 11(a) (black arrows).Two slightly non collinear pump beams labelled 1 and 2 interact in a Kerr medium with a smallsignal beam labelled 3 whose direction is given by a bisectorline between the pump beams.The modulus of the linear phase-mismatch vector is given by

|−→∆kL| = 2|

−→k |(1−cos(α)) (1)

where|−→k | is the common wave vector modulus of all the waves at the same temporal frequency

andα is the half angle between the pump waves. In a Kerr medium, this linear phase-mismatchcan be compensated by nonlinear terms because the cross-phase modulation (CPM) of the sig-nal by the pump is twice as large as the self-phase modulationof the pump. As a result, phase-matching occurs for a negative Kerr index, that shortens more the signal wavevectors than thepump wavevectors.

See Ref. [2] for an analysis of related processes in media with positive Kerr indices.

(C) 2004 OSA 31 May 2004 / Vol. 12, No. 11 / OPTICS EXPRESS 2566#3205 - $15.00 US Received 19 November 2003; revised 18 December 2003; accepted 12 January 2004

α

3α

2α

χ(3)

Medium 2k3

k1

k2

k31

k32

k12

k21

k1

k2

0 1 2 3 4 5 6 7 8 9 10-10

0

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90

Gai

n (d

B)

propagation distance (cm)

(a) (b)

Fig. 1. (a) Wave vectors configuration for nearly collinear four-wavemixing. (b) Signalgain versus the propagation distance, forα=3.9 mrad and a wavelengthλ=1 µm. Blueline: actual results. Black line: results without the sidebands drawn in coloron Fig. 1(a).

Reference [1] claims that such a phase-matched interactionleads to exponential amplifica-tion of the signal beam until pump depletion. As suggested inRef. [1], this spatial interactionhas its formal equivalent in the temporal domain, where dispersion replaces diffraction. In sil-ica fibers, the most used Kerr medium for temporal interactions, the Kerr index is positive andphase-matching is obtained in the normal dispersion regime. However, early theoretical studieshave shown that such an interaction in a single-mode fiber does not lead to amplification of asignal wave or, equivalently, to modulation instability (MI) in the normal dispersion regime. Atfirst Agrawal [3] showed by using a linear stability analysisthat incoherent CPM between twopump beams leads to MI. This process has been experimentallydemonstrated by launching apump beam in a bimodal fiber, where light in one mode interactsincoherently with the light inthe other mode [4]. However, Rothenberg [5] showed using a nonlinear Schrodinger equation(NLS) that CPM between two pump waves at different frequencies in the same fiber mode, i.e.,the process initially considered in the Agrawal’s paper, does not lead to MI, because coher-ent coupling between all sidebands must be considered. Notethat such a NLS automaticallyencompasses four-wave mixing interactions between all sidebands. Finally, it has been shown[6] that the matrix describing the sideband evolution does not exhibit eigenvalues associated toinstability.

We have followed the same steps in the spatial domain to demonstrate that the process con-sidered in Ref. [1] does not lead to exponential amplification. The blue line in Fig. 11(b) repre-sents the gain on the signal obtained by simulating with a NLS(standard split-step algorithm)the configuration studied in Ref. [1] with the same nonlinearindex and an initial phase of thesignal corresponding to a maximum amplification. Because ofcoherent interaction between allsidebands, the exponential gain disappears. Indeed, phasematching cannot be obtained simulta-neously for all the interactions between the two pumps and the four signal and idler sidebands.The black line has been calculated with a NLS where the sidebands drawn in red and green onFig.1(a) are set to zero at each propagation step. These sidebands correspond respectively tofour-wave mixing between both pumps (green arrows) and between the signal and one pump(red arrows). With this truncation, the exponential gain isretrieved, with the exact slope cal-culated in Ref. [1]. Hence, we have demonstrated without ambiguity that exponential gain isunphysical, since it is obtained by considering only one interaction, while all interactions arenot separable in an actual experiment. To conclude, the coherent interaction proposed in Ref. [1]does not lead to exponential amplification of the seeded signal. Actual amplification would oc-cur only by incoherent coupling, that could be obtained in anisotropic non birefringent mediumwith counter-rotating circularly polarized pumps [7]. Such a scheme needs further study.

(C) 2004 OSA 31 May 2004 / Vol. 12, No. 11 / OPTICS EXPRESS 2567#3205 - $15.00 US Received 19 November 2003; revised 18 December 2003; accepted 12 January 2004