## 7 Carrier-Envelope Phase  
In theory, the carrier-envelope phase (CEP) of the generated MIR pulse is passively stabilized in the present scheme [22; 49]. Following the expression of the CEP in frequency conversion processes described in [49; 50], the phase of the generated MIR pulse produced through the process, \(\omega_{1}+\omega_{1}-\omega_{2}\rightarrow\omega_{0}\), is given by \(\phi_{0}=\pi/2+\phi_{1}+\phi_{1}-\phi_{2}\), where \(\phi_{0}\), \(\phi_{1}\), and \(\phi_{2}\) are the phases of \(\omega_{0}\), \(\omega_{1}\), and \(\omega_{2}\) components, respectively. Since \(\omega_{2}\) wave was produced through SH of \(\omega_{1}\) wave, the phase of the  
Figure 6: (**a**) Experimental and (**b**) retrieved XFROG traces. The retrieved pulse in (**c**) time and (**d**) frequency domain. wave is given by \(\phi_{2}=\pi/2+2\phi_{1}\). Therefore, \(\phi_{0}\) must be 0, which fact means that the CEP of the MIR pulse is automatically stabilized even without CEP stabilization of the input pulse. This feature makes the scheme highly attractive since the CEP is a very important physical property of sub-single cycle pulses. However, it is important to check the CEP stability experimentally since the fluctuation of the delay between the fundamental and SH pulses and some noise of the input pulse can affect the CEP stability [26,51,52]. The CEP stability measurement of the MIR pulses was carried out by measuring the interference between the SH of the reference pulse (\(E_{\mathrm{shg}}(t)\propto E_{\mathrm{ref}}^{2}(t)\)) and the FWM signal (\(E_{\mathrm{fwm}}(t)\propto E_{\mathrm{ref}}^{2}(t)E_{\mathrm{test}}^{*}(t)\)), which was used for the XFROG measurement. The interference signal is explained as a cross-term of \(\left|E_{\mathrm{shg}}(t)+E_{\mathrm{fwm}}(t)\right|^{2}\), namely, \(E_{\mathrm{shg}}^{*}(t)E_{\mathrm{fwm}}(t)\). The phase of the interference signal is written as \(-\phi_{\mathrm{shg}}+\phi_{\mathrm{fwm}}=-\left(\pi/2+2\phi_{\mathrm{ref}} \right)+\left(\pi/2+2\phi_{\mathrm{ref}}-\phi_{\mathrm{test}}\right)=-\phi_{ \mathrm{test}}\), where \(\phi_{\mathrm{x}}\) denotes the CEP of \(E_{\mathrm{x}}\) for each subscript. From the equation, it is clear that the phase drift of the interference signal reflects that of the CEP of the MIR test pulse. This scheme is essentially the same as that described in [53]. Figure 7: **(a)** Interference signal between the SH of the reference pulse and the FWM signal; **(b)** Time evolution of the phase of the interference fringe; **(c)** Time evolution of the fringe pattern; Each fringe pattern was measured by single shot every 22 s; **(d)** Fringe pattern change by tilting the delay plate in Figure 1(a). The experimental setup is shown in Figure 1(d). A \(\beta\)-barium borate crystal (BBO, \(\theta=29^{\circ}\), \(t=\)50 \(\mu\)m) on a 2-mm thick fused silica substrate was inserted behind the focus where the FWM signal was generated as is shown in Figure 1(d). The crystal was placed in a way that the pulses enter from the back side, _i.e_., from the substrate side. This way, the FWM signal is delayed by \(\sim\)300 fs relative to the reference pulse due to the group delay difference in the substrate before entering the BBO crystal. A half-wave plate and a calcite polarizer were inserted behind the crystal to optimize the intensity ratio between the SH and FWM signals. In Figure 7(a) the observed interference fringe is shown. The fringe spacing was about 3 THz, which corresponds to the expected delay between the reference pulse and the FWM signal. The phase of the fringe was reasonably stable for hours without any feedback loop. The results of the phase measurements are shown in Figure 7(b) and (c). The instability of the phase was estimated as 78 mrad rms with 100 shots (in 0.1 s) and 154 mrad rms for 2.5 h. Thanks to the inline configuration, the phase stability was improved by a factor of four compared to the previous experiment [22].