Temporal-Spatial Manipulation of Bi-Focal Bi-Chromatic Fields for Terahertz Radiations

Fig. 2(a) depicts the 3rd harmonic intensity $I_{\mathrm{3rd}}(d,\tau)$ versus temporal-spatial manipulation and the calibration of the zero delay of $\omega$-2$\omega$ pulses. When referring to $\tau>0\ \mathrm{fs}$, it indicates that the 2$\omega$ pulse temporally precedes the $\omega$ pulse. For $d>0\ \mathrm{mm}$, it means that the plasma induced by the 2$\omega$ pulse is spatially located ahead of the plasma induced by the $\omega$ pulse along the laser propagation direction. The $\tau$ corresponding to the maximum 3rd harmonic yield is defined as the zero delay. $I_{\mathrm{THz}}(\tau)$ and $I_{\mathrm{3rd}}(\tau)$ with sub-femtosecond time delay, as well as the determination of the zero delay for the bi-chromatic fields specifically are provided in the Supplementary Note 3. $I_{\mathrm{THz}}(\tau)$ and $I_{\mathrm{3rd}}(\tau)$ exhibit anti-correlated behavior as a function of sub-femtosecond time delay, consistent with the previous measurements [26, 20].

In Fig. 2(b), the distribution of THz intensity $I_{\mathrm{THz}}(d,\tau)$ presents a lobe landscape in the dimensions of spatial $d$ and temporal $\tau$ separations. Experimental THz intensity $I_{\mathrm{THz}}(d,\tau)$ sensitively depends on changes of $d$ and $\tau$, where a remarkable maximum distribution illuminates at certain $d$ and $\tau$ values. It reveals two counter-intuitive features that warrant attentions: (i) In the temporal dimension as shown in Fig. 2(c), the maximum THz generation appears when the bi-chromatic pulses are temporally separated. By scanning the time delay with femtosecond accuracy, we note that $I_{\mathrm{THz}}$ is insensitive versus $\tau$ at $d$ = 0 mm, where a broad distribution at about $\tau\approx$-25 fs and 100 fs is observed. However, peaked maxima of $I_{\mathrm{THz}}(\tau)$ at increasing time delays $\tau$ are more and more pronounced as $d$ increases and optimally $I_{\mathrm{THz}}$ with a remarkable 13 times high is achieved in Fig. 2(c) at $d$=2.5 mm and $\tau$=58 fs in comparison to that at $d$=0 mm and $\tau$=0 fs. (ii) In the spatial dimension as shown in Fig. 2(d), efficient optimization of THz generation occurs when the foci of bi-chromatic beams are noticeably separated. The maximum $I_{\mathrm{THz}}$ does not occur at $d=0\ \mathrm{mm}$, and $I_{\mathrm{THz}}$ is more efficient when $d>0\ \mathrm{mm}$ compared to $d<0\ \mathrm{mm}$. Analogy to the THz intensity dependence of two-pulse time delays shown in Fig. 2(c), peaked maximum distributions $I_{\mathrm{THz}}(d)$ shift to larger spatial separation $d$ as $\tau$ increases.

These findings contrast with conventional expectations, where one would anticipate the THz yield to be optimized when the foci of the $\omega$-2$\omega$ beams are spatially and temporally overlapped. This observations are counter to the common expectation that the temporal and spatial overlapping of the bi-chromatic pulses would be a prerequisite for the most efficient THz generation and the $\omega$ laser is mainly responsible for the THz generation. According to the conventional understanding, both 3rd harmonic and THz radiation can be attributed to the Brunel radiation mechanism within the framework of the single-electron approximation [27, 18, 17, 28]. For the case of present two-foci cascading plasma in time and in space, the most likely scenarios are as followings: The 2$\omega$ plasma filament plays a dominating role for THz generation (see experimental frequency spectrum later), where two color lasers are still prerequisites for highly efficient THz generation and the $\omega$ laser provides field modulation resulting in the symmetry breaking of a combined 2$\omega$-$\omega$ field. To a large extent present two-foci cascading plasma configuration, where 2$\omega$ travels ahead in time and is focused ahead in space considering laser propagation direction, enable to avoid THz absorption by the $\omega$ created plasma. Theoretical calculations [25] on the spatial modulation of bifocal field attributed to plasma absorption. In the present spatial geometry of two-foci, the $\omega$ laser field at focus spot of 2$\omega$ laser is relatively stronger only if $\tau>0$ in comparison to $\tau\leq 0$. That is to say that the optimal magnitudes of temporal and spatial separations for the highest THz emission will also be affected by laser profiles like intensity and pulse duration etc..

To gain insight into the origin of the enhancement of THz waves in the temporal-spatial manipulation of bi-chromatic fields, we measured the spectra of the bi-chromatic fields after focusing and the THz spectra emitted by the bi-chromatic fields at different $d$, which are shown in Fig. 3. The $\omega$ spectra versus $d$ are presented in Fig. 3(a), and we do not observe any shift in the $\omega$ spectra after focusing. Conversely, the red arrow in Fig. 3(b) indicates that the intensity of the 2$\omega$ spectra increases with $d$, along with a broadening on the red side of the 2$\omega$ spectra from $d$ = 0 mm to $d$ = 3 mm. It’s worth noting that non-zero frequency detuning may occur naturally during the propagation of strong-field pulses. For example, the frequency shift due to plasma is blue shift, and the Kerr effect is red shift at the leading edge of the intensity [29, 30]. The $\omega$ and 2$\omega$ spectra are depicted as a function of $d$, illustrating that the $\omega$ pulse serves solely as a perturbation during THz generation process, It is observed that laser nonlinear propagation has a greater impact on the 2$\omega$ pulse compared to the $\omega$ pulse.

Furthermore, from the results of THz spectra versus $d$ in Fig. 3(c), it can be observed that the peak of THz spectra shifts from 20 THz to 45 THz as $d$ increases from 0 mm to 2 mm, as depicted by the red arrow. This blue shift is accompanied by a broadening of the THz spectra bandwidth by a factor of 6 compared to the $\omega$ spectra bandwidth. The red shift of the 2$\omega$ spectra and blue shift of the THz spectra demonstrate the energy transfer from the 2$\omega$ beam to the THz beam in the THz generation process. Whether based on the four-wave-mixing (FWM) process ($\omega$ + $\omega$ - 2$\omega$) or the photocurrent model, it is indicated that the broadening of 2$\omega$ spectra on the red side, driven by bi-chromatic laser fields, leads to a shift of the radiated THz pulses towards the high-frequency side [31, 32]. Our experimental results are consistent with the theoretical description.

We conducted investigations on THz emission from various segments of the plasma filament, enabling us to pinpoint the spatial location of THz emission along the plasma filament, and gain valuable insights into the underlying mechanism driving THz amplification. The experimental schematic is shown in Fig. 4(a). An iris with an aperture diameter of 0.5 mm, concentric with the plasma filament, is moved along the propagation axis of the bi-chromatic beams. This manipulation results in blocking THz waves emitted from the plasma filament on the left side of the iris while enabling the detection of THz waves generated by the plasma filament on the right side of the iris. In this scenario, the plasma filament generated by the bi-chromatic fields is fixed in both spatial and temporal dimensions at $d$ = 2 mm and $\tau$ = 0 fs. The plasma fluorescence image is presented in Fig. 4(d).

The THz intensity $I_{\mathrm{THz}}(l)$ as a function of iris position $l$ is illustrated in Fig. 4(b), where $l$ = 0 mm is defined as the starting position of measurement. Notably, when the iris is positioned at the location of the plasma filament generated by the $\omega$ pulse, the detected THz intensity remains constant versus $l$. However, a sudden decrease in THz intensity is observed when blacking the THz emission from the $2\omega$ plasma filament, implying that the majority of THz pulses are generated from the $2\omega$ plasma filament rather than $\omega$ plasma filament. Here, we consider that the $\omega$ pulse plays an assisting role in the THz radiation process, and the propagation of the 2$\omega$ pulse is altered at the 2$\omega$ focus due to certain nonlinear effect, which in turn radiates THz pulses at the 2$\omega$ plasma position [33, 34]. Meanwhile, the THz spectra as a function of $l$ are measured using a Fourier transform spectrometer while translating the iris position $l$, and the results are illustrated in Fig. 4(c). Employing the differential approach, the spectral distribution radiated from different segments of the plasma filament is presented in Supplementary Note 4. Notably, the high-frequency components of THz waves predominantly originate from the 2$\omega$ plasma filament, particularly the end of the 2$\omega$ plasma filament.

Since our experimental arrangement permits us to adjust the energy of $\omega$ and 2$\omega$ beams individually, direct measurements of the emitted THz intensity $I_{\mathrm{THz}}$ versus $I_{\omega}$ at $d$ = 0 mm and $d$ = 2 mm, respectively, as depicted in Fig. 5(a). We also demonstrate the dependency of $I_{\mathrm{THz}}$ on $I_{2\omega}$ at $d$ = 0 mm and $d$ = 2 mm, respectively, as shown in Fig. 5(b). During the measurement, the energy of one beam is fixed while the energy of the other beam is changed. Typically, in the case of bifocal overlap ($d=0\ \mathrm{mm}$) was reported previously [35, 36, 37]: $\boldsymbol{E}_{\operatorname{THz}}\propto E_{\operatorname{2\omega}}{E_{\operatorname{\omega}}}^{2}$. Since the THz intensity is proportional to the square of the THz electric field, i.e. $\boldsymbol{I}_{\operatorname{THz}}\propto{\boldsymbol{E}_{\operatorname{THz}}}^{2}$, $\boldsymbol{E}_{\operatorname{THz}}\propto E_{\operatorname{2\omega}}{E_{\operatorname{\omega}}}^{2}$ can be rewritten as:

The red solid line in Fig. 5(a) and Fig. 5(b) represents the fitted results of $I_{\mathrm{THz}}$ versus $I_{\mathrm{\omega}}$ and $I_{\mathrm{2\omega}}$, respectively. These results are obtained from Eq. (1) in combined with experimental data. This analysis corresponds to the case where $d$ = 0 mm. As predicted by the theory, the emitted $I_{\mathrm{THz}}$ is proportional to the square of $I_{\omega}$ (as observed in Fig. 5(a)) and to $I_{2\omega}$ above the ionization threshold (as observed in Fig. 5(b)). The observed dependence of $I_{\mathrm{THz}}$ on $I_{\omega}$ and $I_{2\omega}$ at $d=0\ \mathrm{mm}$ is consistent with the conventional results.

Simultaneously, we measured the $I_{\mathrm{THz}}$ versus $I_{\omega}$ and $I_{2\omega}$ at $d=2\ \mathrm{mm}$, respectively. The results demonstrate that $I_{\mathrm{THz}}$ is proportional to the square of the $I_{\omega}$ at $d$ = 2 mm, which is consistent with the trend observed at $d=0\ \mathrm{mm}$. However, the relationship between $I_{\mathrm{THz}}$ and $I_{2\omega}$ shows a rapid increase followed by a slow increase, deviating from the trend at $d=0\ \mathrm{mm}$. To confirm that this phenomenon is not exclusive to high laser intensities, we decreased the $\omega$ and 2$\omega$ laser intensity to explore the relationship between THz intensity and laser intensity. The observation reveals that the phenomenon persists under lower laser intensity (for more details, see the Supplementary Note 5). Thus, another solid piece of evidence is provided here that the increase in THz intensity at $d=2\ \mathrm{mm}$ is due to the influence of the 2$\omega$ pulse.

Based on the aforementioned observations, it can be conclude that 2$\omega$ pulse plays a crucial role in both the enhancement and broadening of THz waves. An attempt was made to simulate the experiment using 3D propagation equations [38, 39]; however, a comprehensive elucidation of the experimental results cannot be achieved at present. As a result, a more thorough theoretical explanation is needed.