## 2 Basic Experiment and Interpretation
Thus for the upconverted beams the equation \(\mathbf{k}_{n}=n\mathbf{k}_{1}-(n-1)\mathbf{k}_{0}\) is written in this coordinate system as  
\[k_{n}=(nk_{1}-(n-1)k_{0}\cos\uptheta_{n},(n-1)k_{0}\sin\uptheta_{n}) \tag{1}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Order \(\mathbf{n}\)** & **Wave Vector \(\mathbf{k}_{n}=\mathbf{k}_{0}\)+\(\mathbf{n}\mathbf{k}_{m}\)** & **Equivalence** \\ \hline \(-n\) & \(\mathbf{k}_{-n}=\mathbf{k}_{0}-n\mathbf{k}_{m}=(n+1)\mathbf{k}_{0}-n\mathbf{k}_{1}\) & \(\mathbf{k}_{-n}=\mathbf{k}_{-n+1}-\mathbf{k}_{m}\) \. &. &. \. &. &. \. &. &. \\ \(-2\) & \(\mathbf{k}_{-2}=\  
Its square modulus is given by  
\[{k_{n}}^{2}=n^{2}{k_{1}}^{2}-2n(n-1)k_{0}k_{1}\cos\theta_{n}+(n-1)^{2}{k_{0}}^{2 }{\cos^{2}}\theta_{n}+(n-1)^{2}{k_{0}}^{2}{\sin^{2}}\theta_{n} \tag{2}\]  
and solving for \(\cos(\theta_{n})\) we obtain  
\[\cos\theta_{n}=\frac{n^{2}{k_{1}}^{2}+(n-1)^{2}{k_{0}}^{2}-{k_{n}}^{2}}{2n(n-1 )k_{0}k_{1}} \tag{3}\]  
which is valid for the upshifts (\(n\geq 2\)) and downshifts (\(n\leq-1\)). With reference to Figure 4a,b, the emission angles of the newly generated orders can be obtained by calculating the corresponding wave vectors (e.g., \(\mathbf{k}_{2}=2\mathbf{k}_{1}-\mathbf{k}_{0}\) for the first frequency-upshifted beam), with the ideal phase-matching angle given by Equation (3), and determining their angle with respect to the horizontal axis, which results in  
\[\cos\beta_{n}=\frac{n^{2}{k_{1}}^{2}-(n-1)^{2}{k_{0}}^{2}-{k_{n}}^{2}}{2n(n-1 )k_{0}k_{1}} \tag{4}\]  
Hence it is possible to calculate the interaction angle between the beams inside the medium for the different orders to appear. Thus one can chose for example the phase matching angle which optimizes the first upconverted order. This order will be generated in optimum energy conditions and the remaining orders will be generated less efficiently, since perfect phase-matching is not set for them. Also, it is not possible to exactly phase-match frequency down conversion processes in media with normal dispersion, as Equation (4) does not have a real solution, which results in the observed asymmetry between frequency upconverted and downconverted beams. The fact that the interaction angle affects the generation of the cascade of colors was experimentally observed [11] and the fact that angle phase-matching was achieved thanks to the material dispersion was settled, as detailed below. Figure 5 shows the measured and calculated wave vectors generated in the first nondegenerate CFWM experiment, where the two-color pump beams cross in the medium at an external angle of \(2.9^{\circ}\). The arrows denote the pump beam wave vectors; the ten consecutive upconverted beams appear above the horizontal axis, and the two downconverted beams are below this axis. The open squares falling outside the dotted box are the five upconverted beams for which only the wave vector direction was measured directly, and hence  
Figure 4: Wave vectors of the upconverted and downconverted beams generated by cascaded four-wave mixing (CFWM). (**a**) upconversion processes; (**b**) downconversion processes; (**c**) schematic of the wave vector distribution obtained by CFWM in real space. the amplitude was extrapolated from the phase-matching conditions. The filled circles are calculated with the phase-matching conditions, where \(\mathbf{k}_{0}\) and \(\mathbf{k}_{1}\) were obtained directly from spectral and angular measurements of the pump pulses. The open circles are the calculations for perfectly phase-matched beams and are shown for reference only, since a cascade of such ideal processes (which would required different angles for each process) cannot be obtained experimentally. Figure 6 shows the phase mismatch \(\Delta k_{n}\) for the optimized interaction angle of \(2.9^{\circ}\) as a function of beam order \(n\), where \(n>1\)\((n<0)\) correspond to upconverted (downconverted) beams. The dashed curve is a calculation assuming collinear geometry (\(0^{\circ}\)).