## 2 Materials and Methods
### Simulations  
Filamentation and THz generation in the conditions of our experiment (moderate focusing with a numerical aperture of \(\sim\)0.02) can be simulated using Unidirectional Pulse Propagation Equation (UPPE, [27]). The spatio-temporal spectrum \(\widehat{\mathcal{E}}(\omega,k_{r},z)=\mathcal{F}[\mathcal{E}(t,r,z)]\) represents the electric field \(\mathcal{E}(t,r,z)\) of the laser pulse together with all the secondary emission (emerging terahertz and optical harmonics). Here, \(\mathcal{F}\) denotes a Fourier-Hankel transformation from the spatio-temporal domain \((t,r,z)\) to the frequency-angular domain \((\omega,k_{r},z)\). Although the most rigorous description of a nonlinear source term requires the solution of the Schrodinger [32] or Maxwell-Bloch Equations [33], the decomposition of the medium susceptibility into the response of bound and free electrons (or intra- and inter-band responses) is successful for THz generation simulations in gases [34] and transparent condensed media [26], and can even be applied to metals [35]. To decrease the computational costs, we follow this simplified approach. The external electrostatic field \(E_{DC}\) is introduced into UPPE as the nonlinear susceptibility modification of the medium [36]. With this modification the third-order polarization is given by:  
\[P(t)=\chi^{(3)}\Big{[}(\mathcal{E}(t)+E_{DC})^{3}-E_{DC}^{3}\Big{]}, \tag{1}\]  
where \(\chi^{(3)}\) is the third-order susceptibility of air. The transient photocurrent obeys the equation  
\[\frac{\partial{I_{e}}}{\partial{t}}=\frac{e^{2}}{m_{e}}N_{e}(t)(\mathcal{E}(t )+E_{DC})-v_{c}J_{e}, \tag{2}\]  
where \(e\) and \(m_{e}\) are electron charge and mass, respectively, \(v_{c}=5\) ps\({}^{-1}\) is the collision rate. The absorption current \(J_{a}\)[37] does not require modification because the influence of the external electrostatic field on the plasma density \(N_{e}\) is negligible. The density of free electrons \(N_{e}\) was calculated from the rate equations for the major air components: molecular nitrogen (78%) and molecular oxygen (22%). The nonlinear terms are calculated in time domain and substituted into UPPE:  
\[\bigg{(}\frac{\partial{}}{\partial{z}}+ik_{z}\bigg{)}\widehat{\mathcal{E}}=- \frac{2\pi i}{c^{2}k_{z}}\bigg{(}\omega^{2}\mathcal{F}[P]-\mathcal{F}\bigg{[} \frac{\partial{I_{e}}}{\partial{t}}\bigg{]}-i\omega\mathcal{F}[J_{a}]\bigg{)}, \tag{3}\]where \(k_{z}=\sqrt{n^{2}(\omega)\omega^{2}/c^{2}-k_{r}^{2}}\), \(n(\omega)\) is the refractive index of dry air, and \(c\) is the speed of light. The derivative \(\partial f_{e}/\partial t\) is used as it vanishes at the right boundary of the temporal domain \(t\rightarrow+\infty\), whereas the photocurrent \(J_{e}\) does not. The simulations of THz generation from femtosecond filament were carried out in the experimental conditions for 800-nm pump pulse with the energy of 2 mJ, FWHM duration of 50 fs and beam diameter of 1 cm. We simulated three different THz sources implemented in three different experimental schemes designed in our lab. These schemes are mixing the pump and electrostatic field \(\omega+DC\), the pump and its second harmonic \(\omega+2\omega\), and the pump and the second harmonic in the presence of an electrostatic field \(\omega+2\omega+DC\). For the \(\omega+DC\) scheme, the THz source was a single-color filament biased by the external electrostatic field \(E_{DC}=7\) kV/cm. The \(\omega+2\omega\) case corresponds to the two-color filament formed by mixing the 800-nm pump pulse with its second harmonic (wavelength of 400 nm, energy of 200 \(\mu\)J, FWHM duration of 35 fs and the same beam shape as the pump). The \(\omega+2\omega+DC\) case was a combination of the previous two schemes, i.e., we simulated THz generation in the DC-biased two-color filament in which the second harmonic and the external field \(E_{DC}\) were the same in value and direction as in the \(\omega+2\omega\) and \(\omega+DC\) cases.