## 4 Electro-Optic Sampling  
Although the power measured with our Fourier-transform spectrometer in the frequency region lower than 500 cm\({}^{-1}\) is not so reliable since the beam splitter in the Michelson-type interferometer was designed for \(>\)500 cm\({}^{-1}\), it is very likely that the generated spectrum has significant intensity in the low frequency region. Electro-optic sampling [42] is effective to detect low frequency components (\(<\)1000 cm\({}^{-1}\)). We performed an air-biased coherent detection scheme [43] for the electro-optic sampling. The schematic of the system is shown in Figure 1(b). A 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 (\(f=150\) mm). The polarizations of the reference pulse and the MIR pulse were perpendicular to each other, and the polarization of the generated FWM signal was parallel to the MIR pulse and perpendicular to the reference pulse. To produce SH of the reference pulse, a bias field was applied to Rogowski-type electrodes with a distance of 3 mm [44] by two high voltage amplifiers (HEOPS-5B6, Matsusada) with an amplitude of \(\pm\)4 kV and a frequency of 500 Hz, which is synchronized with the 1 kHz repetition rate of the laser pulse train. The direction of the electric field was parallel to the reference pulse, and the FWM signal and the bias induced SH have polarizations perpendicular to each other. Polarization of the generated blue beam was controlled with a half-wave plate. The beam subsequently passed through a polarizer and a bandpass filter centered at 400 nm (10 nm FWHM). The beam was detected with a photomultiplier tube (H10721-210, Hamamatsu) by scanning the delay time between the reference pulse and the MIR test pulse. Interference between the SH and FWM signals was detected through a lock-in amplifier (SR830, Stanford Research). The experimental results are shown in Figure 4. We obviously observe components down to 250 cm\({}^{-1}\), which corresponds to the wavelength of 40 \(\mu\)m. On the other hand, high frequency components were filtered out due to rather long reference pulse duration. A small oscillating tail was observed due to absorption of residual water vapor. The filtering effect can be quantitatively evaluated by describing the electro-optic sampling signal with formulas. The bias induced SH (\(E_{\mathrm{shg}}\)) and the FWM (\(E_{\mathrm{fwm}}\)) signals are given as follows,  
\[E_{\mathrm{shg}}(t-\tau) \propto\chi^{(3)}_{xxxx}E_{\mathrm{ref}}^{2}(t-\tau)E_{\mathrm{ bias}} \tag{2}\] \[E_{\mathrm{fwm}}(t,\tau) \propto\chi^{(3)}_{yyxx}E_{\mathrm{ref}}^{2}(t-\tau)E_{\mathrm{ test}}^{*}(t) \tag{3}\]  
where \(\tau\) is the delay between the reference pulse and the test pulse. \(E_{\mathrm{bias}}\) is a bias field. \(\chi^{(3)}_{xxxx}\) and \(\chi^{(3)}_{yyxx}\) are the third-order tensors [45] with the four subscripts corresponding to polarizations of the FWM (or SH), MIR (or bias field), fundamental, and fundamental beams, respectively. These signals were mixed with a half wave plate and one polarization component was selected with a polarizer. The intensities of the selected component (horizontal polarization in this experiment), \(I(\tau)\), is given as follows[46]:  
\[I(\tau) =\left\langle\left|\frac{1}{\sqrt{2}}\left(E_{\mathrm{shg}}(t- \tau)+E_{\mathrm{fwm}}(t,\tau)\right)\right|^{2}\right\rangle \tag{4}\] \[=\frac{1}{2}\left\langle\left|E_{\mathrm{shg}}(t-\tau)\right|^{2 }\right\rangle+\frac{1}{2}\left\langle\left|E_{\mathrm{fwm}}(t,\tau)\right|^{2 }\right\rangle+\left\langle\Re\left\{E_{\mathrm{shg}}(t-\tau)E_{\mathrm{fwm}}^ {*}(t,\tau)\right\}\right\rangle \tag{5}\]  
Figure 4: (**a**) Signal of gas-biased coherent detection and (**b**) power spectrum obtained with Fourier analysis. The dashed line is filtering function described in the text.