## 4 Towards Single-Cycle Pulse Synthesis  
The described set of cascaded FWM orders will be a coherent broadband source provided the different orders can be spatially and temporally overlapped. Coherent spectra that are sufficiently broad can be used for generating few- and single-cycle pulses if properly compressed. A well-known technique for generating high energy few-cycle pulses is the temporal compression of the broadband supercontinuum produced by self-phase modulation (SPM) in e.g., gas-filled hollow core fibers [23] (discrete spectra, composed of evenly spaced near-monochromatic waves can also lead to the synthesis of a train of single-cycle optical pulses, as demonstrated using molecular modulation in a gas driven by two independent nanosecond lasers [24]). Moreover, the central frequency of the total CFWM spectrum and the separation between sidebands can be freely adjusted by tuning the pump frequencies, so in principle it should always be possible to obtain the commensurate sidebands required to synthesize a train of  
Figure 13: Recombined and focused CFWM pulses. (**a**) Electric field as a function of the delay (in fs); (**b**) Intensity; (**c**) Power spectrum (logarithmic scale) [22]. identical pulses. Two frequencies are commensurate if \(m\omega_{0}=n\omega_{1}\), with \(m,n\) integers. This also means that each frequency is an exact multiple of the frequency spacing \(\omega_{m}=\omega_{1}-\omega_{0}\). Let us take a deeper insight into pulse synthesis by addition of a finite number of pulses with different central wavelengths. The pulses must have fixed relative phases in order to produce a stable coherent superposition of fields giving rise to an ultrashort pulse or to a train of ultrashort pulses. We also found this result in the previous section (Figure 12), where the coherent sum and focusing of the CFWM orders generated by numerically solving the equations of our model, resulted in a train of ultrashort laser pulses. For the CFWM pulses this can be understood using a simplified model based on the coupled amplitude equations between the pumps and the generated sidebands. Using the wave equation in the slowly-varying envelope approximation, for two pump fields with complex amplitudes \(\tilde{E}_{0}=E_{0}e^{i\Phi_{0}}\) and \(\tilde{E}_{1}=E_{1}e^{i\Phi_{1}}\), frequencies \(\omega_{0}\) and \(\omega_{1}\), respectively, assuming a cubic nonlinearity, neglecting dispersion and pump depletion, and assuming perfect phase-matching and constant coupling coefficients, the solution for the complex amplitude \(\tilde{E}_{n}\) of the field with frequency \(\omega_{n}\) at the exit of a medium with length \(L\) can be written as [21,25]  
\[\tilde{E}_{n}(s)=i^{n}e^{in\delta}\tilde{E}_{0}J_{n}(s)-i^{n+1}e^{i(n-1) \delta}\tilde{E}_{1}J_{n-1}(s) \tag{10}\]  
where \(s=\gamma L\) is the nonlinear phase shift acquired in the medium, \(\delta=\Phi_{1}-\Phi_{0}\) is the initial phase difference between pumps, \(J_{n}\) are the _n_th order Bessel functions of the first kind, and the nonlinear coefficient is given by \(\gamma=3\omega_{0}\chi^{(3)}E_{0}E_{1}/(2n_{0}c)\), with \(n_{0}\) the linear refractive index at frequency \(\omega_{0}\) and \(c\) the speed of light in vacuum. Equation (1) predicts well-defined phase relationships between the several CFWM orders, thus establishing mutual coherence between the beams.