## 1 Introduction  
Four-wave mixing (FWM) processes result from the interplay between four electromagnetic waves coupled through the optical Kerr nonlinearity of a medium, given by the third-order susceptibility \(\chi^{(3)}\)[1]. Since all media, either isotropic or anisotropic (with any kind of crystal symmetry), have a non-zero third order susceptibility, FWM processes have a universal character. The four interacting waves can all have the same frequency, in which case the process is degenerate and the corresponding susceptibility is given by \(\chi^{(3)}(\omega;\omega,\omega,-\omega)\). In this case, three fields with frequency \(\omega\) are mixed in the medium to produce another field also with frequency \(\omega\). The energy conservation law that determines the frequency of the newly generated field is given in this case by: \(\omega=\omega+\omega-\omega\equiv 2\omega-\omega\). In practice, this process can be observed by crossing two intense pulses in a nonlinear medium at a small angle. For incident wavevectors \(\mathbf{k}_{0}\) and \(\mathbf{k}_{1}\), momentum conservation dictates that the newly generated beams will be emitted in the directions given by \(\mathbf{k}_{2}=2\mathbf{k}_{1}-\mathbf{k}_{0}\) and \(\mathbf{k}_{2}^{\prime}=2\mathbf{k}_{0}-\mathbf{k}_{1}\), as shown in Figure 1. This process can be seen as the diffraction of the incident beams by the nonlinear index grating (laser induced grating) produced by the same beams [2]. In bulk dispersive media this process is intrinsically phase mismatched (see, e.g., [3]), with a mismatch given approximately by \(\delta k=2k(1-\cos\theta)\simeq k\theta^{2}\), where \(k=|\mathbf{k}_{0}|=|\mathbf{k}_{1}|\). This means that the efficiency of the process drops for increasing interaction (and hence emission) angles. The waves can also have different frequencies, so a new field with frequency \(\omega_{2}\) can be generated departing from two fields with frequencies \(\omega_{0}\) and \(\omega_{1}\) (\(\omega_{1}>\omega_{0}\)). In this case, the nonlinear polarization terms which describe diffraction from a moving laser-induced grating give rise to frequency upshifted and downshifted pulses with frequencies \(\omega_{2}=2\omega_{1}-\omega_{0}\) and \(\omega_{2}^{\prime}=2\omega_{0}-\omega_{1}\), respectively, so the output consists of four beams with different colors emitted in different directions. [PERSON] and [PERSON] [4] analized theoretically the noncollinear phase-matched amplification of ultrashort light pulses via four-wave mixing in isotropic media, having derived explicit expressions for the phase-matching angles and gain. If the fields are intense enough, the third order susceptibility \(\chi^{(3)}\) can continue mixing the fields, as long as the energy and momentum conservation laws are fulfilled for the new processes. This way, new  
Figure 1: Basic geometry of nonlinear self-diffraction. fields are generated via cascaded four-wave mixing (CFWM). For noncollinear interactions, this gives rise to additional beams that can be seen as higher orders of diffraction of the moving grating (note that the moving grating corresponds to the spatially and temporally varying nonlinear refractive index change inside the medium, since the medium itself is stationary in the laboratory frame). For the degenerate case, the larger the interaction angle, the smaller the observed number of diffracted orders (due to increasing phase-mismatch), but for the nondegenerate case we will see that the phase-mismatch in cascaded processes can be reduced by proper choice of the interaction angle between the two initial beams. Collinear fifth-harmonic generation in a crystal as a result of cascaded processes of a cubic nonlinearity was reported in an early work [5]. Extensive work has been done in this field by mixing near-degenerate nanosecond laser pulses in optical fibers and obtaining several pairs of sidebands [6, 7, 8].