## 2 Materials and Methods  
### Experimental Setup  
Figure 1 shows the scheme of the experimental setup. Ti:Sapphire laser system Spectra Physics Spitfire Pro XP delivered the pulses with the horizontal polarization centred at a wavelength of 800 nm with the energy up to 2.7 mJ, for a duration of 50 fs (full width at half maximum, FWHM) and a beam diameter of 12 mm (at \(e^{-2}\) level) at 1-kHz repetition rate. The pulse energy was reduced down to 2 mJ by the half-wave plate and the [PERSON] prism. The beam was focused by the lens with a focal length of 300 mm forming a \(\sim\)15-mm filament in free air (humidity of 30% at 23 \({}^{\circ}\)C). Two copper 40 \(\times\) 15 mm\({}^{2}\) plane electrodes with a gap of 10 mm between them were set parallel to the optical table so that the filament was burnt in the center between them in the transverse direction and closer to the registration system in the longitudinal direction in order to avoid the reflection of THz radiation from the electrodes surface. A voltage of \(U_{DC}=7\) kV applied to the electrodes formed a vertically directed field with a strength of \(E_{DC}=7\) kV/cm. To generate the second harmonic and form a two-color filament, the \(\beta\)-barium borate (BBO) crystal (I-type, 100-\(\upmu\)m thickness, 10% efficiency) was inserted into the pump beam after the lens. The BBO crystal was adjusted to maximize the second harmonic yield, so, the 400-nm radiation has the vertical polarization, i.e., perpendicular to the 800-nm pump and parallel to the external DC bias. For single-color experiments, the BBO crystal was removed from the optical path. Thus, our experimental setup allowed us to study the THz waveforms in three cases:  
(a) = \(\omega+DC\): DC-biased single-color filament, when BBO crystal was removed from the optical path;  
(b) = \(\omega+2\omega\): two-color filament, when the voltage on the electrodes was turned off (\(U_{DC}=0\));  
(c) = \(\omega+2\omega+DC\): DC-biased two-color filament. In all cases, the THz radiation had linear vertical polarization. Thus, in our setup the superposition of THz waveforms generated according to \(\omega+DC\) and \(\omega+2\omega\) mechanisms could be observed. For our purposes, a coherent THz detection technique is required. Among the three widely used methods, namely, photoconductive antenna detection [28; 29], THz Air Breakdown Coherent Detection (THz-ABCD) [30], and electro-optical sampling (EOS) [31], the latter is best suited for our study. Indeed, the photoconductive antennas are regularly employed with MHz repetition rate laser systems, and would not have enough sensitivity for the kHz source we have. ABCD works very well with the two-color scheme, while the DC-biased filament produces THz wave with lower frequency, which is not properly detected by THz-ABCD setups. Furthermore, the THz-ABCD method requires a high-power probe pulse and an additional DC supply, which can produce distortions for the one used to bias the filament (and vice versa). In turn, EOS has excellent sensitivity, and the frequencies below 3 THz are present in both two-color- and DC-bias-generated THz pulses. To obtain waveforms of the THz electric field, we used a standard time-domain system  
Figure 1: Scheme of the experimental setup. (TDS), which consists of the ZnTe crystal (\(10\times 10\times 0.5\) mm\({}^{3}\), \(\langle 110\rangle\) cut), the quarter-wave plate, the Glan-Taylor prism, and two photodiodes (Figure 1). The THz field induces birefringence in ZnTe crystal, so the probe pulse attains ellipticity proportional to the field strength. After passing through the quarter-wave plate, the probe pulse is divided by the Glan-Taylor prism and is registered by the photodiodes. The difference in the photodiode signals characterizes the birefringence induced in the crystal and, therefore, the THz field strength. By scanning the delay between the optical pump generating the THz wave and the probe pulse, we obtain a THz waveform. This TDS is sensitive to the radiation in the spectral range from \(\sim\)0.2 to \(\sim\)3 THz.