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 small signal beam labelled 3 whose direction is given by a bisector line between the pump beams. The modulus of the linear phase-mismatch vector is given by

where |k⃗| is the common wave vector modulus of all the waves at the same temporal frequency and α is the half angle between the pump waves. In a Kerr medium, this linear phase-mismatch can be compensated by nonlinear terms because the cross-phase modulation (CPM) of the signal by the pump is twice as large as the self-phase modulation of the pump. As a result, phase-matching occurs for a negative Kerr index, that shortens more the signal wavevectors than the pump wavevectors.

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

 

Fig. 1. (a) Wave vectors configuration for nearly collinear four-wave mixing. (b) Signal gain versus the propagation distance, for α=3.9 mrad and a wavelength λ=1 μm. Blue line: actual results. Black line: results without the sidebands drawn in color on Fig. 1(a).

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Reference [1] claims that such a phase-matched interaction leads to exponential amplification of the signal beam until pump depletion. As suggested in Ref. [1], this spatial interaction has its formal equivalent in the temporal domain, where dispersion replaces diffraction. In silica fibers, the most used Kerr medium for temporal interactions, the Kerr index is positive and phase-matching is obtained in the normal dispersion regime. However, early theoretical studies have shown that such an interaction in a single-mode fiber does not lead to amplification of a signal wave or, equivalently, to modulation instability (MI) in the normal dispersion regime. At first Agrawal [3] showed by using a linear stability analysis that incoherent CPM between two pump beams leads to MI. This process has been experimentally demonstrated by launching a pump beam in a bimodal fiber, where light in one mode interacts incoherently with the light in the other mode [4]. However, Rothenberg [5] showed using a nonlinear Schrödinger 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 coherent coupling between all sidebands must be considered. Note that such a NLS automatically encompasses 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 to instability.

We have followed the same steps in the spatial domain to demonstrate that the process considered in Ref. [1] does not lead to exponential amplification. The blue line in Fig. 11(b) represents the gain on the signal obtained by simulating with a NLS (standard split-step algorithm) the configuration studied in Ref. [1] with the same nonlinear index and an initial phase of the signal corresponding to a maximum amplification. Because of coherent interaction between all sidebands, the exponential gain disappears. Indeed, phase matching cannot be obtained simultaneously 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 on Fig.1(a) are set to zero at each propagation step. These sidebands correspond respectively to four-wave mixing between both pumps (green arrows) and between the signal and one pump (red arrows). With this truncation, the exponential gain is retrieved, with the exact slope calculated in Ref. [1]. Hence, we have demonstrated without ambiguity that exponential gain is unphysical, since it is obtained by considering only one interaction, while all interactions are not 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 occur only by incoherent coupling, that could be obtained in an isotropic non birefringent medium with counter-rotating circularly polarized pumps [7]. Such a scheme needs further study.