## 3 Knowing the Fields: The Complete Theoretical Model  
Although the previous reasonings based on energy and momentum conservation allow us to understand the generation of multicolored light pulses departing from two-color pumps by cascaded FWM processes, enabling the estimation of the frequencies and emission angles of the generated beams, no predictions can be made for example regarding the energy carried by each color, nor the emission bandwidth and phase of the generated fields (including the time at which each beam is generated). Hence, a more complete model was required, which was developed based on the Slowly Varying Envelope Approximation (SVEA) for the propagation of the fields in 2.5 D (propagation along a plane while assuming cylindrical symmetry in each beam) that accurately reproduces the experimental results. Figure 6: Wave-vector mismatch of the beams in the pattern: open squares, measured points; filled circles, calculation for the cascaded processes; dashed curve, calculation assuming collinear geometry (\(\theta=0^{\circ}\)). Adapted from [11]. The use of two spatial dimensions is important to account for angular phase-matching and actual beam overlapping region. The wave equation for an electric field \(\mathbf{E}(\mathbf{r},t)\) propagating in a nonlinear medium with a polarization comprising a linear and a nonlinear term as given by \(\mathbf{P}(\mathbf{r},t)=\mathbf{P}_{L}(\mathbf{r},t)+\mathbf{P}_{NL}(\mathbf{r},t)\), is  
\[\
abla^{2}E(\mathbf{r},t)-\frac{1}{c^{2}}\frac{\partial^{2}E(\mathbf{r},t)}{\partial t ^{2}}=\upmu_{0}\frac{\partial^{2}P(\mathbf{r},t)}{\partial t^{2}} \tag{5}\]  
with \(c\) the velocity of light and \(\upmu_{0}\) the vacuum permeability, where we have written the electric field as a scalar quantity, which is correct for linearly polarized fields as is the case of our experiments. Considering that the electric field is well represented under the SVEA approximation, _i.e._, \(E(\mathbf{r},t)=1/2A(\mathbf{r},t)e^{i(k_{0}z-w_{0}t)}+c.c.\), changing the equation to the reference frame which moves with the group velocity of the pulse (\(k_{1}=1/v_{g}\), where \(v_{g}\) is the group velocity of the pulse at frequency \(\upomega_{0}\)), separating the polarization into its linear and nonlinear parts and Fourier transforming to the frequency domain, the wave equation becomes  
\[ \begin{split}\frac{\partial^{2}A(\mathbf{r},\upomega-\upomega_{0})}{ \partial z^{2}}+2i[k_{0}+k_{1}(\upomega-\upomega_{0})]\frac{\partial A(\mathbf{r },\upomega-\upomega_{0})}{\partial z}+\\ [k^{2}(\upomega)-{k_{0}}^{2}-2k_{0}k_{1}(\upomega-\upomega_{0}) -{k_{1}}^{2}(\upomega-\upomega_{0})^{2}+{\
abla_{\perp}}^{2}]A(\mathbf{r},\upomega -\upomega_{0})=\\ -2\upmu_{0}\upomega^{2}P_{NL}(\upomega)e^{-ik_{0}z}\end{split}  \tag{6}\]  
This equation describes both linear and nonlinear propagation. It includes dispersion and diffraction, as well as a particular form of \(P_{NL}\) which includes the instantaneous non-resonant Kerr effect and self-steepening. Note that no expansion of the wave vector \(\mathbf{k}\) was performed and therefore higher order terms of the dispersion are intrinsically included. To solve the equation, it was separated in its linear and nonlinear parts. For \(P_{NL}=0\), the linear part of Equation (6) in the paraxial approximation (\(\frac{\partial^{2}A(\mathbf{r},\upomega-\upomega_{0})}{\partial z^{2}}=0\)) is given by  
\[ \begin{split} 2i(k_{0}+k_{1}(\upomega-\upomega_{0}))\frac{ \partial A(\mathbf{r},\upomega-\upomega_{0})}{\partial z}+\\ [k^{2}(\upomega)-{k_{0}}^{2}-2k_{0}k_{1}(\upomega-\upomega_{0}) -{k_{1}}^{2}(\upomega-\upomega_{0})^{2}+{\
abla_{\perp}}^{2}]A(\mathbf{r},\upomega -\upomega_{0})=0\end{split}  \tag{7}\]  
This linear part carries information about dispersion and diffraction and can be solved in the frequency domain by using a (2,2) Pade approximant for wide angle propagation.