## 3 Knowing the Fields: The Complete Theoretical Model
The nonlinear part in the paraxial approximation is given by  
\[2 ik_{0}\frac{\partial A(\mathbf{r},\upomega-\upomega_{0})}{\partial z}=-2\upmu_{ 0}\upomega^{2}P_{NL}(\upomega)e^{-ik_{0}z} \tag{8}\]  
The nonlinear polarization \(P_{NL}\) can be written in terms of a slowly varying envelope \(p\) as \(P_{NL}(\mathbf{r},t)=1/2p(\mathbf{r},t)e^{i(k_{0}z-w_{0}t)}+c.c.\), with \(p(\mathbf{r},t)=3/8\epsilon_{0}\upchi^{(3)}|A(\mathbf{r},t)|^{2}A(\mathbf{r},t)\) and is better described in the time domain. By doing the inverse Fourier transform, we get  
\[\frac{\partial A(\mathbf{r},t)}{\partial z}=\frac{3\upchi^{(3)}}{8 ik_{0}c^{2}} \Big{[}-\upomega_{0}{}^{2}|A|^{2}A-2i\upomega_{0}\frac{\partial(|A|^{2}A)}{ \partial t}+\frac{\partial^{2}(|A|^{2}A)}{\partial t^{2}}\Big{]} \tag{9}\]  
where the first term on the right hand side accounts for self-phase modulation effects and the second for self-steepening. The accurate evaluation of the self-steepening term is essential, because the spectral span of the generated CFWM pulses is of the same order as the central frequencies of the pumps, \(\omega_{0}\) and \(\omega_{1}\). It should be pointed out that no additional delayed response terms were required in order to faithfully reproduce the experimental results. This equation can be solved by a second-order Runge-Kutta method, which proved sufficiently accurate. Equations (8) and (9) are very powerful in the context of the present work and can exhibit a very rich behavior, since they constitute a much better approach to the process than the simple algebraic description of the previous section. Let us consider the geometry of Figure 7 with spatial gaussian profiles \(A(\mathbf{r},t)\) or \(A(\mathbf{r},\omega-\omega_{0})\) and transform limited pump pulses, for the two beams used in the first experiment: an orange beam (\(\omega_{0}\), \(\lambda_{0}\) = 618 nm, 80 fs) and a green beam (\(\omega_{1}\), \(\lambda_{1}\) = 561 nm, 40 fs), interacting at an internal angle of 1.93\({}^{\circ}\) (corresponding to an external angle of 2.9\({}^{\circ}\)) in a BK7 glass slide with a relative delay of 0 fs between them. The resulting simulated CFWM spectrum is shown in Figure 8 (black curve). We see that several upshifted orders are generated, which are broadband and have well-defined central wavelengths. Also, the total spectral intensity has an envelope which decays at around 400 nm and then increases again, in agreement with the experimental observations shown in the next section. If the interaction occurs at a slightly different angle, which is a feasible situation in an experimental setup, the efficiency and the generated central wavelengths change, as shown in Figure 8 (collinear pumps don't generate more than 1-2 CFWM orders since there is no phase-matching). For angles smaller than 1.93\({}^{\circ}\), the generated orders don't reach so far into the UV, while for angles larger than 2\({}^{\circ}\) the overall efficiency drops rapidly (Figure 8a). For a fixed interaction angle, different time delays between the pump and the signal can be set, which also influences the central wavelength of the generated orders due to additional shifts caused by cross-phase modulation and the varying amount of temporal and spatial overlap between the two pump beams (Figure 8b). We see that for delays of \(\pm\) 20 fs (typical reading precision of a manual delay line) around the optimum zero delay point, the overall efficiency drops by only a small amount (more noticeable for higher orders) since such delays are still well within the 40 and 80 fs FWHM pump pulse durations. For delays larger than \(\pm\) 40 fs the efficiency starts to drop significantly, as illustrated in Figure 8b for the +40 fs case. The model also offers the possibility of spatially shifting the pumps at the entrance of the slide. Figure 8c shows the dependence of the generated field intensity for different spatial overlaps of the pump beams.