## 6 Cross-Correlation Frequency-Resolved Optical Gating  
In order to quantitatively evaluate the temporal shape of the generated MIR pulse, we measured cross-correlation frequency resolved optical gating (XFROG) [48]. In this experiment we used nitrogen as a nonlinear medium and used an FWM process (\(\omega_{1}+\omega_{1}-\omega_{0}\rightarrow\omega_{2}\)) as the nonlinear interaction between the test pulse (MIR pulse, \(E_{\mathrm{test}}(t)\)) and the reference pulse (\(E_{\mathrm{ref}}(t)\)) in the same way as our previous experiment [22]. The scheme is free from spectral filtering caused by phase mismatch in the nonlinear interaction. The system for the XFROG measurement is also shown in Figure 1(c). Small portion (\(\sim\)15 \(\mu\)J) of the fundamental 30-fs pulse was used as a reference pulse. The reference pulse and the MIR pulse (test pulse) were combined through a mirror with a hole and focused into nitrogen with an aluminum-coated parabolic mirror (\(f=50\) mm). The generated blue spectra (centered around 440 nm) were measured with a spectrometer (USB2000+, OceanOptics) by scanning the delay time between the reference pulse and the MIR test pulse. The reference pulse was independently characterized with SHG-FROG, and the result was used for retrieving the MIR pulse.  
The XFROG signal \(I_{\mathrm{FROG}}(\
u,\tau)\) is written as follows,  
\[E_{\mathrm{fwm}}(t,\tau) \propto E_{\mathrm{ref}}^{2}(t)E_{\mathrm{test}}^{*}(t-\tau) \tag{7}\] \[I_{\mathrm{FROG}}(\
u,\tau) =\left|\int dtE_{\mathrm{fwm}}(t,\tau)\exp{(i2\pi\
u t)}\right|^{2} \tag{8}\]  
where \(\tau\) is a delay time between the reference pulse and the test pulse, and \(E_{\mathrm{ref}}(t)\), \(E_{\mathrm{test}}(t)\), and \(E_{\mathrm{fwm}}(t,\tau)\), are the electric fields of the reference pulse, the test pulse, and the FWM signal, respectively.  
The measured and retrieved XFROG traces are shown in Figure 6(a) and (b). We used a retrieval program by Femtosoft (FROG 3.2.4). We chose DFG (r-t) as the nonlinear interaction and used square of \(E_{\mathrm{ref}}(t)\) as the reference pulse. The FROG error was 0.0030 with 512 \(\times\) 512 grid, which indicates the result is very reliable. The retrieved time and frequency domain pictures are shown in Figure 6(c) and (d), respectively. The pulse duration was estimated to be 6.9 fs which is 0.63 cycles for 3.3 \(\mu\)m carrier wavelength. The retrieved spectrum was nearly identical to the spectrum measured with the Fourier-transform spectrometer, as is shown in Figure 6(d). This indicates that the whole MIR spectral components were well focused and overlapped with the fundamental beam.  
It is obvious that there is a phase shift around 2200 cm\({}^{-1}\). We believe that the phase shift is due to interference between two parametric processes, \(\omega_{1}+\omega_{1}-\omega_{2}\rightarrow\omega_{0}\) and \(\omega_{2}-\omega_{1}-\omega_{1}\rightarrow\omega_{0}\) because the phase dependence of the intensity becomes significant in the frequency region lower than 2200 cm\({}^{-1}\) (see Figure 3(a)).  
The phase difference could not be compensated with any delay between the fundamental and SH. The reason is that the residual chirp of the light source is significant, then the phase cannot be compensated in the whole frequency range. We can also observe the direction of the chirp is opposite in high and low frequency components, which is consistent with the fact that the two parametric processes must have the opposite chirp.  
We did not observe such chirp and phase shift in our previous experiment [22]. The important difference is that the chirp difference between fundamental and SH can be minimized with the previous scheme since the two beams were separated and the chirp of each pulse can be controlled. However, such independent control of two color pulses is not possible with the inline configuration. If one can provide some very complicated chirped mirrors that can add some chirp to the two color pulses with certain ratio, they are helpful to achieve flat phase in the whole spectral region.