## 3 Knowing the Fields: The Complete Theoretical Model
The former can be due to several factors, from changes in the central frequency of the pumps that can occur between experiments, to differences in the interaction angle as well as in the relative pulse delay, all of which can result in frequency shifts, as described previously (see, e.g., Figure 8). Figure 10: Simulated \(\theta-\lambda\) spectrum of the CFWM beams, clipped at \(10^{-6}\) of the maximum [22]. Figure 9: Spectrum of the multiple CFWM orders generated in fused silica spanning over 1.5 octaves (the two pump frequencies correspond to the two larger peaks). Dashed line: Experimental results (see [21]), Solid line: Numerical simulation, Dotted line: Numerical simulation for collinear pumps [22]. Regarding the latter, the simulated pulses exhibit some positive chirp due to the nonlinear phase imposed by the pumps. This is clearer in the right plot of Figure 12, which shows the calculated PG-XFROG traces for the synthesized field of Figure 13. Another factor that directly contributes to the chirp of the newly generated pulses is the initial chirp of the pump pulses. In the simulations the pumps are assumed to be Fourier-limited, whereas in the experiments there can be slight deviations to this condition. Figure 11: Experimental (measured and retrieved) and simulated polarization-gating FROG (PG-XFROG) traces of the second frequency-upshifted pulse (top row) and first downshifted pulse (bottom row). The retrieved pulse durations are \(30.7\) and \(40.2\) fs for the second upshift and first downshift respectively (FROG error was \(0.014\) and \(0.008\) for a 128 \(\times\) 128 grid) [22]. Figure 12: Simulated PG-XFROG, clipped at \(10^{-4}\) of the maximum, for the recombined and focused CFWM field, assuming 20 fs (**left**) and 80 fs (**right**) gate pulses [22]. Other kinds of temporal processing are also possible. For example the total electric field can be calculated by synchronizing and spatially superimposing all the CFWM orders. Since after passing the slide the different orders have acquired a slightly different group delays, synchronization can be achieved by subtracting the linear phase. This is equivalent to focusing all the orders together in a point, which would be an experimental approach for having the total field in the same region. The total electric field obtained by this numerical procedure can be seen in Figure 12a, along with the field intensity in Figure 12b and its power spectrum in Figure 12c. We see that in the time domain we obtain a train of pulses separated by the pump beat period (23 fs) with the central peak having 3.1 fs in duration. A PG-XFROG trace of this total field can also be simulated. The gate pulse can be of course arbitrarily chosen, but it looks reasonable and informative to use one of the pump pulses as gating pulse, as done in the experiments, namely the 80 fs pulse at \(\omega_{0}\). The PG-XFROG trace for the complete focused set of CFWM orders is shown in Figure 13 where the different sub-traces for the individual orders can be distinguished (on the right). For shorter gate pulses spectral resolution is lost but temporal resolution is improved revealing the fine temporal structure (pulse train) obtained in this case (on the left).