## 3 Results and Discussion
To clarify the origin of the THz radiation from the two-color filament placed into the electrostatic field, we subtract the \(\omega+2\omega\) waveform, obtained without the DC field, from the \(\omega+2\omega+DC\) waveform. This subtraction was performed for both measured and simulated waveforms (Figure 3c,d, black curves). As the next step, we plot the measured THz waveform, obtained without the second harmonic pulse, i.e., in \(\omega+DC\) case, on the same panel as the result of subtraction (Figure 3c, cf. red and black curves). Similar comparison was done for the simulated \(\omega+DC\) curve (Figure 3d, cf. red and black curves). In both the experiment and the simulations the difference between the result of \(\omega+2\omega\)  
Figure 2: Upper row: measured waveforms of THz radiation emitted from ((a), red, \(\omega+DC\)) DC-biased single-color filament, ((b), blue, \(\omega+2\omega\)) two-color filament and ((c), orange, \(\omega+2\omega+DC\)) DC-biased two-color filament. Middle row: THz spectra in the cases shown in upper row; (dots) THz spectra obtained from the measured waveforms; (solid lines) simulated THz spectra integrated over the overall transverse domain. Lower row: the simulated THz spectra from the middle row in the extended frequency range till \(\
u=30\) THz and normalized. waveform subtraction from the \(\omega+2\omega+DC\) waveform and the \(\omega+DC\) waveform is \(\sim\)8%. Thus, the THz field from a two-color filament placed into the electrostatic field is the superposition of THz fields from the two-color dipole source and DC-biased filament plasma dipole source. To confirm our proposition, let us perform the analysis based on the material equations for the atmospheric density gas ionized by a femtosecond pulse. The main source of THz radiation in this underdense plasma is transient photocurrent \(J_{e}\)[42]. Let the laser field \(\mathcal{E}(t)\) consist of the fast oscillating components only: the 800-nm pump and its second harmonic, i.e., \(\mathcal{E}(t)=\mathcal{E}_{\omega}(t)+\mathcal{E}_{2\omega}(t)\). The part of the transient photocurrent \(J_{\omega}\) responsible for the generation of odd harmonics satisfies Equation (2) with \(E_{DC}=0\) and \(\mathcal{E}(t)=\mathcal{E}_{\omega}(t)\). The remaining part of the transient photocurrent \(J_{e}-J_{\omega}\) provides for the generation of THz radiation (and other even optical harmonics). For \(\omega+DC\), \(\omega+2\omega\) and \(\omega+2\omega+DC\) mechanisms of THz generation in the filament, the equations for the remaining part of the photocurrent following from Equation (2) are, respectively:  
\[\frac{\partial J_{DC}}{\partial t} =\frac{e^{2}}{m_{e}}N_{e}(t)E_{DC}-v_{c}J_{DC}, \tag{4a}\] \[\frac{\partial J_{2\omega}}{\partial t} =\frac{e^{2}}{m_{e}}N_{e}(t)\mathcal{E}_{2\omega}(t)-v_{c}J_{2 \omega},\] (4b) \[\frac{\partial J_{2\omega+DC}}{\partial t} =\frac{e^{2}}{m_{e}}N_{e}(t)[\mathcal{E}_{2\omega}(t)+E_{DC}]-v_{ c}J_{2\omega+DC}. \tag{4c}\]  
Plasma density \(N_{e}\) does not depend significantly on either the external DC field \(E_{DC}\) or on the electric field of low-intensity second harmonic \(\mathcal{E}_{2\omega}\) (cf. red and blue curves in Figure 4). Therefore, Equation (4c) is the sum of Equations (4a) and (4b), so, \(J_{2\omega+DC}=J_{DC}+J_{2\omega}\), i.e., the THz source terms in DC-biased two-color (\(\omega+2\omega+DC\)) filament are additive. According to Equation (3), the additivity of THz source terms corresponds to the interference of the THz fields induced by them and the superposition of the measured and simulated THz waveforms. Figure 3: (**a**) Measured and (**b**) simulated waveforms of THz radiation generated from a two-color filament (\(\omega+2\omega\), blue curves), a DC-biased single-color filament (\(\omega+DC\), red) and a DC-biased two-color filament (\(\omega+2\omega+DC\), orange).