## 3 Knowing the Fields: The Complete Theoretical Model
For \(\pm\) 50 um shifts in the overlap of the pump beams (each \(100\,\upmu\)m in diameter) at the entrance plane of the medium, the overall efficiency decreases roughly by a factor of 2 with respect to the perfect overlap (0 um) case. This is again more noticeable for the higher orders in the ultraviolet and near-infrared regions. Figure 7: Interaction geometry. Hence it is possible to tune the generated spectra by varying the interaction angle and/or the delay, in very good agreement with experimental observations. This means that the terms considered in the linear and nonlinear sub-equations are sufficient to faithfully reproduce the experimental results, as seen further below. The experiment was repeated in a 150 \(\upmu\)m fused silica slide to obtain more upconverted orders in the UV and the internal interaction angle was 1.57\({}^{\circ}\). In this situation, multiple CFWM orders were generated up to the 21 st upshifted order at 209 nm (made visible with the help of a fluorescent card) and were registered with a broadband fiber-coupled spectrometer, but due to spectral limitations the measurement reached only down to 250 nm. The temporal characteristics of the beams were measured by polarization-gating FROG (PG-XFROG). More details about these experiments are given in the next section. The simulations provide the time dependent field distribution at the end of the glass slide as a function of the transverse coordinate \(x\) and frequency \(\upomega\). The corresponding total spectrum (obtained by integrating along the transverse direction) for an interaction angle of 1.57\({}^{\circ}\) is given in Figure 9 and clearly shows the asymmetric generation of up and downconverted orders. The efficiency of the generated frequency upshifted pulses follows the same pattern as the experiment, although it has a sharper cutoff after the 12 th order that depends on the interaction angle and pulse energy. This discrepancy is possibly due to experimental intensity calibration errors, which are larger at the spectral edges. A look at the fields inside the slide shows that all orders are practically generated within the first 60 \(\upmu\)m of material. The spatial resolution of the model allows determining at which angle each of the orders is generated. This can be seen in the \(\uptheta-\lambda\) plot in Figure 10.  
Figure 8: (**a**) Generated spectra for different interaction angles; (**b**) Generated spectra for \(\uptheta=1.93^{\circ}\) and different relative time delays between pumps (positive delay P\({}_{1}\) arrives before P\({}_{0}\), negative delay P\({}_{0}\) arrives before P\({}_{1}\) ); (**c**) Generated spectra for \(\uptheta=1.93^{\circ}\), 0 fs relative time delay and different entrance \(x\)-positions at the slide. The temporal characteristics of the pulses can be also calculated, since the model gives us complete temporal and spectral information of the field. A given CFWM order can be spatially selected since they are emitted at different angles and we can simulate its interaction with another beam, as done in experiments for measuring the temporal duration. For instance, one can make a given generated order interact with the beam at \(\omega_{0}\) on a second thin slide of fused silica for different relative temporal delays and hence numerically obtain polarization-gating cross-correlated frequency resolved optical gating (PG-XFROG) traces. Figure 11 shows the measured and simulated PG-XFROG traces for the second frequency upshifted beam with central frequency \(\omega_{2}\) (top) and the first downshifted beam with central frequency \(\omega_{-1}\) (bottom). I, the simulation results are in good agreement with the measurements regarding the broad bandwidth and short duration of the newly generated low-order pulses (which are shorter than the pumps). There are however differences in the central frequency and in the chirp of the pulses.