## 4 Towards Single-Cycle Pulse Synthesis
### Ultrashort Pulses from the UV to the VUV by CWFM  
Materials with large band gap such as alkali metal halide crystals are transparent in the VUV, so they are candidates for generating a cascade of upconverted orders via FWM up to the VUV, and we have tested this numerically [27]. The model developed in Section 3 assumes isotropic materials, and some alkali metal halides have cubic symmetry, so they are equivalent to an isotropic material and we used our model to test the feasibility of obtaining VUV pulses via CFWM in LiF using a pump beam at 400 nm and a signal beam at 800 nm. Equation (3) gives the internal crossing angle with would result in perfect phase-matching of the \(n\)th order, but for pump and signal beams as separated as a fundamental wavelength and its second harmonic, the internal angles for successive upconverted orders is too different for a cascade to be generated departing from only two beams, so the geometry of Figure 18 was devised, with a pump field at 400 nm and 3 signal beams at 800 nm, the latter satisfying the phase-matching condition for different consecutive orders. We consider a pump beam at 400 nm and three signal beams at 800 nm, all transform-limited with 25 fs duration and focused to a 40 \(\upmu\)m radius spot. All pump and signal beams have an irradiance of \(5\times 10^{12}\)\(\mathrm{Wcm^{-2}}\) (energy per pulse of \(\sim 6\)\(\upmu\)J). The pump and the three signal beams form internal angles of \(11.98\), \(15.65\) and \(20.06\) degrees in a 300 \(\upmu\)m thick LiF slide. The signal beams can be individually delayed to optimize the VUV generation. Figure 19 shows the CFWM spectrum generated for a time delay of 0 fs or 7 fs (for all signal beams) along with the spectrum generated by adding the individual spectra obtained by using the pump pulse and one of the signal beams at a time, also with a time delay of 7 fs. Figure 18: Scheme for multiple-beam CFWM with strong angular separation between pump and signal frequencies. [27]. In these conditions, we observe generation of CFWM beams up to the 6 th harmonic of 800 nm, with good efficiency up to the 5 th harmonic. Only a slight dependence on the time delay is seen and there is full evidence of the existence of CFWM processes, since the VUV spectrum obtained by addition of the individual spectra from the different pairs of pump and signal pulses is much weaker. As already seen in section 3 the generated orders are angularly separated and due to the fact that all three pump beams at 800 nm contribute to generate a specific order at a slightly different interaction angle and wavelength, because of the different phase-matching conditions, the generated orders have a complex spatial structure, although angular selection of a specific order can always be performed. Figure 20 shows the simulated \(\theta-\lambda\) spectrum for 7 fs and 0 fs delay. A cleaner structure and also a better efficiency in the 4 th and 5 th harmonic is observed in the case of 7 fs delay compared to perfect synchronization (0 fs delay). Figure 19: Simulated CFWM spectra in LiF. Solid (dashed) curve: spectrum generated by the interaction of all three signal pulses at \(800\ \mathrm{nm}\) delayed by \(7\ \mathrm{fs}\) (\(0\ \mathrm{fs}\)) with respect to the pump pulse at \(400\ \mathrm{nm}\) – (see text); dotted curve: combined spectrum for three independent simulations, each optimizing an individual consecutive harmonic generated by each pair of noncollinear pump and \(7\ \mathrm{fs}\) delayed signal pulses [27]. Figure 20: \(\theta-\lambda\) spectrum for \(7\ \mathrm{fs}\) (a) and \(0\ \mathrm{fs}\) (b) relative delays between signal and pump pulses. Insets: magnified \(\theta-\lambda\) spectra for the first three upconverted orders [27]. The temporal behavior is also as expected, with the generated orders having ultrashort durations of the order of the duration of the pump and signal pulses (31.9, 30.5, 30.1 and 20.4 fs for the third to the sixth harmonic respectively), due to the corresponding nonlinear spectral phases.