## 4 Towards Single-Cycle Pulse Synthesis
The total field in the time domain is then  
\[E(t)=\text{Re}\big{[}\sum_{n}\tilde{E}_{n}(s)e^{-i(\omega_{0}+n\omega_{m})t} \big{]} \tag{11}\]  
From Equation (11) and using the Jacobi-Anger identity, \(\sum_{n}J_{n}(z)\exp(in\theta)i^{n}=\exp(iz\cos\theta)\), we obtain  
\[E(t)=Re\big{[}e^{i\Phi_{0}}(E_{0}e^{-i\omega_{0}t}+E_{1}e^{i \delta}e^{-i\omega_{1}t})\] \[\times e^{-is\cos(\omega_{m}t-\delta)}\big{]} \tag{12}\]  
This result is similar to the frequency-modulation (FM) solution by [PERSON] [24] for coherent addition of Raman sidebands in the approximation of negligible dispersion and limited modulation bandwidth (on its hand similar to the solution previously obtained by [PERSON] _et al._[6] for nondegenerate FWM of 100 ns pulses - for which dispersion was practically negligible - propagating in 400 m long optical fibers), with the important difference that in CFWM the synthesized pulse envelope strongly depends on the initial phase difference \(\delta\). In molecular modulation, the detuning between the driving frequency and the Raman transition dictates the preparation of a phased or an anti-phased molecular state, the latter resulting in a negative effective nonlinear coefficient \(\gamma\) for which the synthesized pulses could have an approximately negative chirp and hence could be further compressed by simple propagation in a normally dispersive medium [24]. In the present case of CFWM, the sign of \(\gamma\) is independent of the pump field (and usually positive for most optically transparent \(\chi^{(3)}\) media), but the chirp of the synthesized pulses can nevertheless be controlled by adjusting \(\delta\) alone. From Equation (12) we see that \(\phi_{0}\) only affects the carrier envelope phase (CEP) for the pulse train, defined with respect to the two-color beat envelope, whereas \(\delta\) is associated with net temporal shifts and also determines the pulse chirp, as given by the oscillating nonlinear phase term. In particular, when \(\delta=\pm\pi/2\), the synthesized pulses can have a negative chirp. In practice, a given phase difference can be introduced by slightly adjusting the delay between the two (phase-locked) pump pulses without significantly affecting their temporal overlap. Also, if the pump frequencies are commensurate, this will result in the generation of identical pulses that are CEP-stabilized within each train (although their CEP may change from shot-to-shot), since even though each CFWM step is not self-CEP-stabilized, the total recombined field will be. The corresponding pulse envelopes will also be identical from shot-to-shot, provided that the relative phase difference \(\delta\) remains constant, which is a far less stringent requirement than the need of CEP stabilized pump pulses. To demonstrate the feasibility of this approach for the generation of single cycle pulses, the basic set-up of Figure 2 was improved by using fused-silica instead of BK7 to generate more orders in the UV, and complemented with additional optics for pulse synthesis and measurement. This set-up is reproduced in Figure 14-I. As in the previous case, two-horizontally polarized visible femtosecond pulses from a dye laser-amplifier at 10 Hz were used as pumps: an orange pump with \(\lambda_{0}\) = 615 nm, \(\sim\)80 fs duration, 2 mJ energy and a green pump with \(\lambda_{1}\) = 569 nm, \(\sim\)60 fs duration and 200 \(\upmu\)J energy. Special care was taken in using Glan-Taylor polarizers (GTPs, 100,000:1 extinction ratio) to have perfectly polarized beams, which is important not only for the generation, but for the subsequent temporal measurements. In this laser system, the green pump beam is directly (optically) derived from the orange pump beam: a small portion of the orange beam is first used to generate supercontinuum in a cell filled with deuterated water, and then the green portion of this supercontinuum is amplified using green laser dyes.