## 2 Basic Experiment and Interpretation
Table 1 shows, in an equivalent way, how the frequencies of the different orders are generated from the previous order plus or minus the modulation frequency. An approximate description of the cascaded FWM process is completed by considering the phase-matching condition for the wave vectors of the upconverted and downconverted beams. Let us \begin{table}
\begin{tabular}{c c c} \hline
**Order \(n\)** & **Frequency \(\omega_{n}\)=\(\omega_{0}\)+\(n\omega_{m}\)** & **Equivalence** \\ \hline \(-n\) & \(\omega_{-n}=\omega_{0}-n\omega_{m}=(n+1)\omega_{0}-n\omega_{1}\) & \(\omega_{-n}=\omega_{-n+1}-\omega_{m}\) \. &. &. \. &. &. \. &. &. \\ \(-2\) & \(\omega_{-2}=\omega_{0}-2\omega_{m}=3\omega_{0}-2\omega_{1}\) & \(\omega_{-2}=\omega_{-1}-\omega_{m}\) \\ \(-1\) & \(\omega_{-1}=\omega_{0}-\omega_{m}=2\omega_{0}-\omega_{1}\) & \(\omega_{-1}=\omega_{0}-\omega_{m}\) \\ 0 & \(\omega_{0}=\omega_{0}\) & \(\omega_{0}\) \\ 1 & \(\omega_{1}=\omega_{0}+\omega_{m}=\omega_{1}\) & \(\omega_{1}\) \\ 2 & \(\omega_{2}=\omega_{0}+2\omega_{m}=2\omega_{1}-\omega_{0}\) & \(\omega_{2}=\omega_{1}+\omega_{m}\) \\ 3 & \(\omega_{3}=\omega_{0}+3\omega_{m}=3\omega_{1}-2\omega_{0}\) & \(\omega_{3}=\omega_{2}+\omega_{m}\) \. &. &. \. &. &. \. &. &. \\ \(n\) & \(\omega_{n}=\omega_{0}+n\omega_{m}=n\omega_{1}-(n-1)\omega_{0}\) & \(\omega_{n}=\omega_{n-1}+\omega_{m}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Approximate frequencies generated by the Cascaded Four-Wave Mixing process. Figure 3: Spectra of the beams in the cascade: dashed curve, original spectra of the orange (\(P_{0}\)) and green (\(P_{1}\)) pump beams after crossing the glass slide but without temporal overlap (scaled for comparison); solid curve, spectra of the simultaneously generated beams. Adapted from [11]. call \(\mathbf{k}_{0}\) and \(\mathbf{k}_{1}\) the wave vectors of the lower frequency (orange) and higher frequency (green) pump, whereas \(\mathbf{k}_{-n}\) and \(\mathbf{k}_{n}\) denote the wave vector of a downconverted beam or an upconverted beam of order \(n\) respectively. With \(\mathbf{k}_{m}=\mathbf{k}_{1}-\mathbf{k}_{0}\) the modulation (or grating) wave vector, the newly generated wave vectors are approximately given by \(\mathbf{k}_{n}=\mathbf{k}_{0}+n\mathbf{k}_{m}\), or equivalently by \(\mathbf{k}_{n}=\mathbf{k}_{n-1}+\mathbf{k}_{m}\) or \(\mathbf{k}_{n}=n\mathbf{k}_{1}-(n-1)\mathbf{k}_{0}\). We will see that approximate phase matching can be achieved for a large number of frequency upconversion processes in a medium with normal dispersion, whereas frequency downconversion processes are intrinsically phase-mismatched, which explains the observed asymmetry between the two processes. For the same reason, the contribution of backwards processes such as \(\mathbf{k}_{n}=\mathbf{k}_{n+1}-\mathbf{k}_{m}\) is small and therefore can be neglected. Table 2 shows how the wave vectors of the different orders are generated from the previous one plus or minus the modulation wave vector. Figure 4 shows the diagrams for the vector additions in the cascaded FWM processes for both the upconverted and downconverted beams, along with their exiting angles \(\upbeta_{n}\). Moreover it is possible to calculate the interaction angle \(\uptheta_{n}\) that the main beams should have inside the medium for phase matching to occur for a particular process of order \(n\), under the assumed approximation. Let us choose a coordinate frame in which \(\mathbf{k}_{1}=k_{1}\times(1,0)\). In this frame \(\mathbf{k}_{0}\) would be given by \(\mathbf{k}_{0}=k_{0}(\cos\uptheta_{n},\sin\uptheta_{n})\) and let us have in mind that any wave vector is related with the refractive index of the medium through \(k_{i}=n_{r}(\upomega_{i})\upomega_{i}/c\).