Theory of terahertz emission from femtosecond-laser-induced microplasmas

We present a theoretical investigation of terahertz (THz) generation in laser-induced gas plasmas. The work is strongly motivated by recent experimental results on microplasmas, but our general findings are not limited to such a configuration. The electrons and ions are created by tunnel ionization of neutral atoms, and the resulting plasma is heated by collisions. Electrons are driven by electromagnetic, convective, and diffusive sources and produce a macroscopic current which is responsible for THz emission. The model naturally includes both ionization current and transition-Cherenkov mechanisms for THz emission, which are usually investigated separately in the literature. The latter mechanism is shown to dominate for single-color multicycle laser pulses, where the observed THz radiation originates from longitudinal electron currents. However, we find that the often discussed oscillations at the plasma frequency do not contribute to the THz emission spectrum. In order to predict the scaling of the conversion efficiency with pulse energy and focusing conditions, we propose a simplified description that is in excellent agreement with rigorous particle-in-cell simulations.

Figure 1

Example of a $t_0=50\mathrm{fs},I_{\mathrm{L}}^0=4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ laser pulse at ${\lambda }_{\mathrm{L}}=800$ nm in argon gas with initial atom density $n_{\mathrm{a}}=3\times {10}^{19}{\mathrm{cm}}^{-3}$ in 1D configuration. Because we neglect laser propagation effects, the problem depends on the comoving time $\tau =t-z/c$ only. In (a) the laser intensity $I_{\mathrm{L}}$ (dashed red line) and the resulting electron density $n_0$ (solid black line) according to our model are shown. (b) presents the thermal energy $E_{\mathrm{th}}$ as captured by the model (dotted red line) for ${\lambda }_{\mathrm{ei}}=3.5$, in excellent agreement with the thermal energy $E_{\mathrm{th}}^{\mathrm{PIC}}$ obtained from a PIC simulation (solid blue line, see text for details). In (c) the collision frequency ${\nu }_{\mathrm{ei}}$ according to Eq. (9) is shown.

Figure 2

In (a) the low-frequency power spectra of the second order source term ${\iota }_{2,z}$ (topmost line) and its constituents are shown. The legend specifies the quantities according to their vertical order in the graph. The longitudinal electric field $E_{2,z}$ is presented in (b) together with the electron density $n_0$. In (c) the power spectra of the longitudinal currents corresponding to the source terms in (a) are plotted. Driving laser parameters are the same as in Fig. 1.

Figure 3

In (a) the up to $0.2{\omega }_{\mathrm{L}} \left(\nu \le 75\mathrm{THz}\right)$ integrated power spectrum of the source term ${\iota }_{1,x}$ (IC source) is shown as a function of laser pulse duration $t_0$ and intensity $I_{\mathrm{L}}^0$. The same data for ${\iota }_{2,z}$ (TC source) are shown in (b). These two quantities are compared in panel (c): In the blue region the IC source term ${\iota }_{1,x}$ dominates by at least one order of magnitude, in the red region the same is true for the TC source term ${\iota }_{2,z}$, and in the green region ${\iota }_{1,x}$ and ${\iota }_{2,z}$ are both important. The computations are performed for an argon gas with the initial atom density $n_{\mathrm{a}}=3\times {10}^{19}{\mathrm{cm}}^{-3}$.

Figure 4

Hypothetical far-field spectra integrated over all angles computed by assuming an infinitely thin $10\mu \mathrm{m}$ long plasma wire (see text) are shown for $I_{\mathrm{L}}^0=4\times {10}^{14}$ W/${\mathrm{cm}}^2,t_0=5$ fs (a), and $t_0=50$ fs (b). Power spectra are calculated from current densities associated with the IC ($x$-polarized current), TC ($z$-polarized current) mechanisms, and obtained by 1D PIC simulations and model according to the legend in (a).

Figure 5

Snapshot of the longitudinal electric field (a) $E_{2,z}$ from our model and (b) $E_z^{\mathrm{PIC}}$ from a corresponding 2D PIC simulation at the time moment when the laser pulse is at focus. The $y$-polarized Gaussian laser pulse ($t_0=50$ fs, $I_{\max }=4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$) is focused to $w_0=0.8\mu \mathrm{m}$ into a uniform preformed plasma $\left(n_0=3\times {10}^{19}{\mathrm{cm}}^{-3}\right)$.

Figure 6

The same laser pulse as in Fig. 5 is focused into argon gas at ambient pressure. A snapshot of the generated plasma (a) and the absolute value of the cosine of the angle between ${\mathbf{E}}^{\mathrm{PIC}}$ and $\mathbf{\nabla }{n}^{\mathrm{PIC}}$ (b) after the pulse has passed the focus are shown. The corresponding snapshot of the electric field component ${\mathrm{E}}_z^{\mathrm{PIC}}$ is depicted in (c). The exemplary time trace of ${\mathrm{E}}_z^{\mathrm{PIC}}$ in (d) features oscillations at the local plasma frequency, in agreement with Eq. (24). The snapshot of $B_y^{\mathrm{PIC}}$ in (e) shows the static field which is present in the interaction region after the laser pulse has passed [see corresponding time trace in (f)]. All temporal snapshots in (a), (b), (c), and (e) are taken about 100 fs after the pulse has passed the focus. Recording positions of the time traces shown in (d) and (f) are indicated by the respective arrows.

Figure 7

The same laser pulse as in Figs. 5 and 6 is focused into argon gas at ambient pressure. The angle-integrated far-field spectra obtained from 2D PIC simulation, model, and 1D wire model (see text) are presented according to the legend.

Figure 8

Snapshot of the magnetic field $B_y^{\mathrm{PIC}}$ from the same PIC simulation as shown in Fig. 6. The figure is a zoom-out of Fig. 6, so the emitted THz and SH waves are visible (denoted as “SH” and “THz”). The mean width of the focused laser is indicated by the dark red lines, and the position of the generated plasma is marked as a blue oval.

Figure 9

Far-field THz power spectra as a function of frequency ${\omega }_{\mathrm{THz}}$ and detection angle $\phi$ for the laser pulse of Figs. 6, 7, and 8. In (a) the result of the PIC simulation and in (b) those of the simplified model (see text) are presented. In (c) and (d) analogous results accounting for collisions are shown. The color scale allows for quantitative comparison of the amplitudes, which are normalized to $max\left(P_{\mathrm{far}}^{\mathrm{PIC}}\right)$.

Figure 10

Scaling of the conversion efficiency ${\eta }_{\mathrm{THz}}$ with focal spot size and pulse energy, for fixed laser wavelength ${\lambda }_{\mathrm{L}}=0.8\mu \mathrm{m}$, pulse duration $t_0=50\mathrm{fs}$, and argon gas density $n_{\mathrm{a}}=3\times {10}^{19}{\mathrm{cm}}^{-3}$. In (a), PIC results (solid dark red line) and the simplified model (dashed dark red line) for a laser pulse energy of $E_{\mathrm{p}}=0.18\mathrm{J}/\mathrm{m}$ are shown. The dashed light gray line shows model results accounting for collisions. In (b), the model is evaluated in the $(w_0,E_{\mathrm{p}})$ plane. Ratios of PIC and model conversion efficiencies are indicated in gray. In (c), ${\eta }_{\mathrm{THz}}$ as a function of the pulse energy is shown for tight focusing $\left(w_0=0.8\mu \mathrm{m}\right)$. Results from PIC simulations and the simplified model with and without collision are in good agreement. The dashed-dotted dark red line shows model results when the opaqueness of the plasma is ignored (see text).