## 3 Spectra
Here we discuss the difference between the MIR generation and the THz generation experiments. The first point is the length of the plasma. At the THz generation, typical length of the plasma is about 5 mm, which is six times shorter than our experiment. Considering the confocal parameter of THz wave, it is understandable that the longer filament does not contribute very much to increase the efficiency of the THz generation but contribute to increase the shorter wavelength components, namely MIR components here. The second point is the adjustment of the relative delay and polarization of the input pulses. At the THz generation experiments in the references [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], only the angles of the BBO crystal for the SHG can be adjusted to optimize the efficiency of the wavelength conversion. In this case, the freedom to control the relative delay and polarizations of the fundamental and SH pulses is seriously limited. In contrast, by using the inline scheme with a delay plate and a dual wave plate, it is possible to perfectly adjust the relative delay and polarizations of the fundamental and SH pulses. The adjustment of the two input pulses affect not only the efficiency of the wavelength conversion but also that of the cross-phase modulation among the multi-color waves. As is discussed in the previous paragraphs, the cross-phase modulation is a key process to generate short wavelengths. The third point is the detection scheme for the generated pulses. In the above mentioned references, only electro-optic sampling was used to detect the generated pulses. However, the frequency detectable with electro-optic sampling is limited by the duration of the reference pulse [40]. For example, reference pulses as short as 5 fs are necessary to detect an MIR wave at 3 \(\mu\)m in reasonable accuracy. Such short pulses were never used in the above mentioned references. In such situation, even they generated MIR components, it is not possible to detect them. We also performed electro-optic sampling to detect our infrared pulses. The experimental results and the discussion about the detection range is shown in the next section. The intensity of the MIR pulse was very sensitive to the distance between the fundamental and SH pulses. The intensity changed with the period of \(\sim\)200 nm of the distance. This phenomenon was also observed at THz generation with gas plasma [30, 41]. At the THz generation, the depth of the modulation was almost 100%. However, this was not the case in our experiment. The comparison of the spectra of the in-phase and out of phase cases is shown in Figure 3(a). It is obvious that the phase dependence was stronger for low frequency components (\(<\)2500 cm\({}^{-1}\)) whereas the high frequency components (\(>\)2500 cm\({}^{-1}\)) were much less sensitive to the phase. We performed a simple simulation of the MIR pulse generation to explain this phenomenon. The MIR electric field, \(\tilde{E}_{0}(\omega)\), was calculated with a convolution of the fundamental and SH spectra (\(\tilde{E}_{1}(\omega)\) and \(\tilde{E}_{2}(\omega)\), respectively) after the filament (shown in Figure 2),  
\[\tilde{E}_{0}(\omega)\propto i\omega\left\{\iint d\omega^{\prime}d\omega^{ \prime\prime}\tilde{E}_{1}(\omega^{\prime})\exp(i\omega^{\prime}\tau)\tilde{E }_{1}(\omega^{\prime\prime})\exp(i\omega^{\prime\prime}\tau)\tilde{E}_{2}^{*} (\omega^{\prime}+\omega^{\prime\prime}-\omega)+\mathrm{c.c.}\right\} \tag{1}\]  
where \(\tau\) is a relative delay between the fundamental and SH. We assumed no chirp for the both pulses in the simulation. An important point is that two different conversion processes should be considered, in other words, \(\omega_{2}-\omega_{1}-\omega_{1}\rightarrow\omega_{0}\) components should be added to \(\omega_{1}+\omega_{1}-\omega_{2}\rightarrow\omega_{0}\) components with certain phase relation. The result is shown in Figure 3(b).