## 2 Basic Experiment and Interpretation
As the interaction angle approaches \(2.9^{\circ}\), the wave vector mismatch becomes clearly asymmetric, with the appearance of a region where the material dispersion is partially compensated for by the interaction angle. Since the self-phase modulated pump pulses have very large bandwidths, the geometry that results in phase-matching of the \(n\)th-order beam can give rise to multiple emission for all orders below \(n\), provided that the maximum phase-mismatch for each order is less than approximately three times the bandwidth of the ultrafast pump pulses (\(\sim\)1350 cm\({}^{-1}\)). We see that eleven upconverted beams and two downconverted beams all fall inside this region, and so the phase-matching conditions can be met in a quite relaxed way for several orders of the cascaded processes, within the frequency range allowed by the bandwidths, and in perfect agreement which the observed asymmetry between frequency upconversion and downconversion processes. Notice that, according to this criterion, a first frequency-upshifted beam should be generated even for a collinear interaction geometry (dashed curve of Figure 6). This is indeed the case, which has been confirmed by measuring the spectra of the resulting collinear and overlapping beams (with proper attenuation of the pump and signal pulses). Experimentally, the interaction angle is adjusted for maximum overall intensity and number of orders in the projected multicolored pattern. We expect the efficiency of a particular process to be determined by the magnitude of the nonlinear susceptibilities, the strength of the input fields, the phase-mismatch of any intermediate process, and ultimately absorption for any of the fields. Moreover, we see that the optimum yield does not necessarily occur under conditions that minimize the phase-mismatch for the final step in the process. Cascading will  
Figure 5: Representation of the wavevectors of all the beams in the pattern: open squares, measured points; filled circles, calculation with the approximate cascaded processes; open circles, calculated exactly phase-matched beams, with the interaction angle given by Equation (3). The solid curve is a guide. Adapted from [11]. take place in the presence of some residual phase-mismatch in either the intermediate steps or the final step. The low degree of residual phase-mismatch also allows cascaded third-order processes to dominate over any high-order direct processes since cascaded processes will be favored if the phase-mismatch of the intermediate steps is sufficiently low. Storage of energy in the intermediate field then determines the conversion efficiency, similar to the case of an intermediate resonance. Although direct higher-order processes cannot be completely ruled out, they do not appear to take part in the observed phenomenon, as the detailed numerical simulations presented in the next section also demonstrate. The experiment and the simple description based on momentum and energy conservation laws already show that the process is of universal character and will occur in any material with a third order susceptibility. Hence, all materials can show CFWM processes. Indeed CFWM processes was shortly thereafter shown to occur in gases [13] and later found to occur in other materials such as sapphire [15] and in crystals [16] or using two color beams from a fundamental laser and the output of a hollow fiber [17], which also demonstrated the intrinsic tunability of the process. Highly nondegenerate CFWM was also shown to occur in an optical parametric oscillator via cascaded \(\chi^{(2)}\) processes giving rise to an effective third-order nonlinearity [18]. Recently, almost octave-spanning spectra were produced by CFWM of two synchronized ps lasers in optical microfibers [19]. Tunable CFWM has also been obtained by crossing two chirped pulses with the same central wavelength and variable delay in glass [20].